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1.  Reconstruction and flux-balance analysis of the Plasmodium falciparum metabolic network 
In the paper we present a metabolic reconstruction and flux-balance analysis (FBA) of Plasmodium falciparum, the primary agent of malaria. The compartmentalized metabolic network of the parasite accounts for 1001 reactions and 616 metabolites. Enzyme–gene associations were established for 366 genes and 75% of all enzymatic reactions.The model was able to reproduce phenotypes of experimental gene knockout and drug inhibition assays with up to 90% accuracy. The model also can be used to efficiently integrate mRNA-expression data to improve the accuracy of metabolic predictions.Using FBA of the reconstructed metabolic network, we identified 40 enzymatic drug targets (i.e. in silico essential genes) with no or very low sequence identity to human proteins.We experimentally tested one of the identified drug targets, nicotinate mononucleotide adenylyltransferase, using a recently discovered small-molecule inhibitor.
Malaria remains one of the most severe public health challenges worldwide (WHO, 2008). Although several available drugs have been successful in controlling malaria in the past, most of them are rapidly losing efficacy due to acquired drug resistance in the most lethal causative agent, Plasmodium falciparum (Mackinnon and Marsh, 2010). This creates an urgent need for new drugs and treatments, as well as better understanding of the parasite physiology. With this in mind, we built a genome-scale flux-balance model of the P. falciparum metabolism. Given the complex life cycle of Plasmodium, the flux-balanced model is of direct relevance to the ongoing search to identify new therapeutic drug targets. The model can be used to explore diverse metabolic states of the parasite and identify essential metabolic genes in the context of known alternative pathways (Oberhardt et al, 2009).
The reconstructed model, which is based on Plasmodium-specific databases, genomic annotations, and literature reports, includes 366 genes, 1001 reactions, 616 metabolic species, and 4 cellular compartments. We applied flux-balance analysis (FBA) (Orth et al, 2010) to identify the genes and reactions that are required to produce a set of necessary biomass components. Interestingly, compared with the yeast metabolic network (Duarte et al, 2004), a model eukaryote with a similar genome size, the Plasmodium network has a significantly higher proportion of essential genes; we confirmed this result using a comparative analysis of known gene knockouts in the two microbes. This low level of genetic robustness, which is likely due to the parasitic lifestyle, suggests that many metabolic genes of the parasite can be used as effective drug targets. Indeed, based on the in silico analysis we identified 40 essential P. falciparum genes with no or very little sequence identity to their human homologs.
We used a recently described small-molecule inhibitor (compound 1_03; Sorci et al, 2009) to experimentally verify one of the enzymes identified as essential: nicotinate mononucleotide adenylyltransferase (NMNAT; Figure 2A). This enzyme, and the corresponding NAD synthesis and recycling pathway, have been recently used for anti-microbial development (Magni et al, 2009). However, to the best of our knowledge, they have not been used against P. falciparum. The compound 1_03 was able to completely block host cell escape and reinvasion by arresting parasites in the trophozoite growth stage (Figure 2B). These results demonstrate that the inhibitory compound may be a good starting lead for new anti-malarials.
Importantly, the metabolic model of the parasite can be also used to integrate various genomic data, such as gene expression (Oberhardt et al, 2009). To illustrate these possibilities, we applied gene-expression data as constraints for the flux-balance model (Colijn et al, 2009) in order to predict changes in metabolic exchange fluxes. We found that the model was able to correctly predict the changes in external metabolite concentrations (Olszewski et al, 2009) with about 70% accuracy (Figure 3). The availability of a human metabolic network reconstruction (Duarte et al, 2007) would allow, in the future, to analyze the combined parasite–host network, which would deepen understanding of the P. falciparum metabolic vulnerabilities.
Future improvements of the presented P. falciparum metabolic model, for example incorporation of missing activities and yet undiscovered pathways, will lead to a better understanding of parasite physiology. Ultimately, the improved understanding should significantly accelerate the identification and development of desperately needed new drugs against this devastating disease.
Genome-scale metabolic reconstructions can serve as important tools for hypothesis generation and high-throughput data integration. Here, we present a metabolic network reconstruction and flux-balance analysis (FBA) of Plasmodium falciparum, the primary agent of malaria. The compartmentalized metabolic network accounts for 1001 reactions and 616 metabolites. Enzyme–gene associations were established for 366 genes and 75% of all enzymatic reactions. Compared with other microbes, the P. falciparum metabolic network contains a relatively high number of essential genes, suggesting little redundancy of the parasite metabolism. The model was able to reproduce phenotypes of experimental gene knockout and drug inhibition assays with up to 90% accuracy. Moreover, using constraints based on gene-expression data, the model was able to predict the direction of concentration changes for external metabolites with 70% accuracy. Using FBA of the reconstructed network, we identified 40 enzymatic drug targets (i.e. in silico essential genes), with no or very low sequence identity to human proteins. To demonstrate that the model can be used to make clinically relevant predictions, we experimentally tested one of the identified drug targets, nicotinate mononucleotide adenylyltransferase, using a recently discovered small-molecule inhibitor.
doi:10.1038/msb.2010.60
PMCID: PMC2964117  PMID: 20823846
flux-balance analysis; Plasmodium falciparum metabolism; systems biology
2.  Dynamic Analysis of Integrated Signaling, Metabolic, and Regulatory Networks 
PLoS Computational Biology  2008;4(5):e1000086.
Extracellular cues affect signaling, metabolic, and regulatory processes to elicit cellular responses. Although intracellular signaling, metabolic, and regulatory networks are highly integrated, previous analyses have largely focused on independent processes (e.g., metabolism) without considering the interplay that exists among them. However, there is evidence that many diseases arise from multifunctional components with roles throughout signaling, metabolic, and regulatory networks. Therefore, in this study, we propose a flux balance analysis (FBA)–based strategy, referred to as integrated dynamic FBA (idFBA), that dynamically simulates cellular phenotypes arising from integrated networks. The idFBA framework requires an integrated stoichiometric reconstruction of signaling, metabolic, and regulatory processes. It assumes quasi-steady-state conditions for “fast” reactions and incorporates “slow” reactions into the stoichiometric formalism in a time-delayed manner. To assess the efficacy of idFBA, we developed a prototypic integrated system comprising signaling, metabolic, and regulatory processes with network features characteristic of actual systems and incorporating kinetic parameters based on typical time scales observed in literature. idFBA was applied to the prototypic system, which was evaluated for different environments and gene regulatory rules. In addition, we applied the idFBA framework in a similar manner to a representative module of the single-cell eukaryotic organism Saccharomyces cerevisiae. Ultimately, idFBA facilitated quantitative, dynamic analysis of systemic effects of extracellular cues on cellular phenotypes and generated comparable time-course predictions when contrasted with an equivalent kinetic model. Since idFBA solves a linear programming problem and does not require an exhaustive list of detailed kinetic parameters, it may be efficiently scaled to integrated intracellular systems that incorporate signaling, metabolic, and regulatory processes at the genome scale, such as the S. cerevisiae system presented here.
Author Summary
Cellular systems comprise many diverse components and component interactions spanning signal transduction, transcriptional regulation, and metabolism. Although signaling, metabolic, and regulatory activities are often investigated independently of one another, there is growing evidence that considerable interplay occurs among them, and that the malfunctioning of this interplay is associated with disease. The computational analysis of integrated networks has been challenging because of the varying time scales involved as well as the sheer magnitude of such systems (e.g., the numbers of rate constants involved). To this end, we developed a novel computational framework called integrated dynamic flux balance analysis (idFBA) that generates quantitative, dynamic predictions of species concentrations spanning signaling, regulatory, and metabolic processes. idFBA extends an existing approach called flux balance analysis (FBA) in that it couples “fast” and “slow” reactions, thereby facilitating the study of whole-cell phenotypes and not just sub-cellular network properties. We applied this framework to a prototypic integrated system derived from literature as well as a representative integrated yeast module (the high-osmolarity glycerol [HOG] pathway) and generated time-course predictions that matched with available experimental data. By extending this framework to larger-scale systems, phenotypic profiles of whole-cell systems could be attained expeditiously.
doi:10.1371/journal.pcbi.1000086
PMCID: PMC2377155  PMID: 18483615
3.  Omic data from evolved E. coli are consistent with computed optimal growth from genome-scale models 
Proteomic and transcriptomic data from wild-type and laboratory-evolved strains of Escherichia coli are consistent with predicted pathway usage from optimal growth rate solutions.In laboratory-evolved strains, there is an upregulation of the pathways in the computed optimal growth states, and downregulation of non-functional pathways.Known regulatory mechanisms are only partially responsible for altered metabolic pathway activity.
When prokaryotes are maintained at early- to mid-log phase growth through serial passaging for hundreds of generations, the strains improve fitness and evolve a higher growth rate (Lenski and Travisano, 1994; Ibarra et al, 2002). This increased growth rate is the result of the appearance of a few causal mutations (Herring et al, 2006; Conrad et al, 2009). In Escherichia coli, these altered growth phenotypes are consistent with predictions from genome-scale models of metabolism (GEMs) (Ibarra et al, 2002; Fong and Palsson, 2004). However, it is still not known (1) whether absolute gene and protein expression levels and expression changes are consistent with optimal growth predictions from in silico GEMs or (2) whether measured expression changes can be linked to physiological changes that are based on known mechanisms or pathways. In this study, we begin to address these questions using constraint-based modeling of E. coli K-12 metabolism (Feist and Palsson, 2008) to analyze omic data that document the expression changes in E. coli under adaptive evolution in three different growth conditions.
Mapping high-throughput data to a network can be useful for interpretation. However, it does not account for upstream and downstream effects of gene and protein expression changes. The analysis of data in the context of GEMs can suggest if predicted activity is consistent with the data. For this work, we used a variant of flux balance analysis (FBA), called Parsimonious enzyme usage FBA (pFBA) (Figure 1), to classify all genes according to whether they are used in the optimal growth solutions. Results from these models were compared with the data to assess whether the data were consistent with genes and proteins within the predicted optimal solutions, and whether the expression changes were consistent with measured physiology. Through this analysis, we find that the data provide a high coverage of genes that contribute to the optimal growth solutions (Figure 1B). In fact, the union of the proteomic and transcriptomic data for non-essential genes provides support for 97.7% of all non-essential gene-associated reactions within the optimal growth predictions. Thus, the spectrum of expressed genes and proteins is consistent with the pathway utilization that is predicted for these optimal growth phenotypes.
Laboratory-evolved strains attain a higher growth rate. This higher growth rate is usually associated with an increased substrate uptake rate (Ibarra et al, 2002; Fong et al, 2005) and in some cases more efficient metabolism (Ibarra et al, 2002). Both of these properties are also witnessed in the strains studied here. It has been reported that in most cases, evolved strain growth phenotype is consistent with GEM predictions (Ibarra et al, 2002; Teusink et al, 2009). Here, we evaluate whether the laboratory-evolved strains adjust the gene and protein expression levels in accordance with pathway usage in the optimal growth predictions. Essential and non-essential genes and proteins within the optimal growth solutions are significantly upregulated (Figure 1B). This suggests that these proteins may be acting as bottlenecks that are relieved through the adaptive process, thereby allowing for a higher substrate uptake rate and growth rate. However, genes and proteins associated with reactions that cannot carry a flux in the given growth conditions are downregulated in the evolved strains (Figure 1B). Furthermore, there is downregulation of genes associated with less efficient pathways (Figure 5C). Thus, the omic data support the emergence of the predicted optimal growth states, consistent with the increased substrate uptake upstream and the increased biomass production downstream of these internal pathways.
Regulatory mechanisms, both known and unknown, are responsible for the changes seen here. Across all data sets, several metabolic regulons are significantly downregulated. However, no known regulons were enriched among upregulated genes or proteins for all but one data set. Aside from just regulating the metabolic pathways directly, these mechanisms lead to additional physiological changes. For example, in the minimal media growth conditions used here, the stringent response normally represses growth while upregulating amino-acid biosynthetic processes. However, evolved strain gene expression shows a suppression of the stringent response, as evolved strain gene expression shows either no expression change or changes opposite to the normal stringent response.
The implications of this work are as follows: (1) genome-scale gene and protein expression data are consistent with FBA computed optimal growth states, and evolved strains reinforce these optimal states; (2) genome-scale models will have an important function bridging the gap between genotype and phenotype; and (3) the development of additional genome-scale models of other growth-related processes such as transcription and translation (Thiele et al, 2009) will have an important function in elucidating the mechanisms that contribute the most to altered phenotypes (Lewis et al, 2009a). In addition, reconstruction of the transcriptional regulation network will aid in identifying the control of expression changes seen in the other systems.
After hundreds of generations of adaptive evolution at exponential growth, Escherichia coli grows as predicted using flux balance analysis (FBA) on genome-scale metabolic models (GEMs). However, it is not known whether the predicted pathway usage in FBA solutions is consistent with gene and protein expression in the wild-type and evolved strains. Here, we report that >98% of active reactions from FBA optimal growth solutions are supported by transcriptomic and proteomic data. Moreover, when E. coli adapts to growth rate selective pressure, the evolved strains upregulate genes within the optimal growth predictions, and downregulate genes outside of the optimal growth solutions. In addition, bottlenecks from dosage limitations of computationally predicted essential genes are overcome in the evolved strains. We also identify regulatory processes that may contribute to the development of the optimal growth phenotype in the evolved strains, such as the downregulation of known regulons and stringent response suppression. Thus, differential gene and protein expression from wild-type and adaptively evolved strains supports observed growth phenotype changes, and is consistent with GEM-computed optimal growth states.
doi:10.1038/msb.2010.47
PMCID: PMC2925526  PMID: 20664636
Escherichia coli; genome-scale models; microarray; optimality; proteomics
4.  Economics of membrane occupancy and respiro-fermentation 
The authors propose that prokaryotic metabolism is fundamentally constrained by the cytoplasmic membrane surface area available for protein expression, and show that this constraint can explain previously puzzling physiological phenomena, including respiro-fermentation.
We propose that prokaryotic cellular metabolism is fundamentally constrained by the finite cytoplasmic membrane surface area available for protein expression.A metabolic model of Escherichia coli updated to include a cytoplasmic membrane constraint is capable of predicting a variety of puzzling phenomena in this organism, including the respiro-fermentation phenomenon.Because the surface area to volume ratio is directly related to the morphology of the cell, this constraint provides a direct link between prokaryotic morphology and physiology.The potential relevance of this constraint to eukaryotes is discussed.
Many heterotrophs can produce ATP through both respiratory and fermentative pathways, allowing them to survive with or without oxygen. Since the molar ATP yield (molar ATP yield: mole of ATP produced/mole of substrate consumed) from respiration is about 15-fold higher than that from fermentation, ATP production via respiration is more efficient. Surprisingly, at high catabolic rate, many facultative aerobic organisms employ fermentative pathways simultaneously with respiration, even in the presence of abundant oxygen to produce ATP (Pfeiffer et al, 2001; Vemuri et al, 2006; Molenaar et al, 2009). This leads to an observable tradeoff between the ATP yield and the catabolic rate (Pfeiffer et al, 2001; Vemuri et al, 2006). This respiro-fermentation physiology is commonly observed in microorganisms, including Escherichia coli, Bacillus subtilis, Saccharomyces cerevisiae (Molenaar et al, 2009), as well as cancer cells (Vander Heiden et al, 2009). Despite extensive research, existing theories (Majewski and Domach, 1990; Varma and Palsson, 1994; Pfeiffer et al, 2001; Vazquez et al, 2008; Molenaar et al, 2009) cannot fully explain the respiro-fermentation phenomenon.
The membrane economics theory
We propose the hypothesis that the prokaryotic metabolism is fundamentally constrained by the finite cytoplasmic surface area available for protein expression—in order to maximize fitness, prokaryotic organisms such as E. coli must economically manage the expression of membrane proteins based on the membrane cost and the fitness benefit of the proteins. This hypothesis is proposed based on theoretical considerations (in this work), numerical analysis (Phillips and Milo, 2009), and experimental observation that the overexpression of non-respiratory membrane protein significantly reduces the oxygen consumption rate and induces aerobic fermentation (Wagner et al, 2007). Such a constraint on transmembrane protein expression may have significant physiological consequences in prokaryotes, such as E. coli, at higher catabolic rates. First, since both substrate transporters and respiratory enzymes are localized on the cytoplasmic membrane in prokaryotes, increased substrate uptake rates necessitates a decrease in the respiratory rate. This decrease in the respiratory rate, forces prokaryotes to process the additional substrate through the fermentative pathways, which are not catalyzed by transmembrane proteins, for continued ATP production. Furthermore, since the membrane requirement of an enzyme is inversely related to its turnover rate (see Materials and methods section in the manuscript), the faster and inefficient respiratory enzymes (such as Cyd-I and Cyd-II in E. coli) might be preferred over the slower and efficient enzymes (such as Cyo in E. coli), leading to an altered respiratory stoichiometry at higher catabolic rates. Finally, the absence of the respiratory enzymes under anaerobic conditions explains why the maximum glucose uptake rate (GUR) of E. coli is much higher.
Applying membrane economics theory to E. coli
To illustrate that the ‘membrane economics' theory could satisfactorily explain the physiological changes associated with the respiro-fermentation phenomenon in E. coli, we modified the genome-scale metabolic model of E. coli (Feist et al, 2007) to include a cytoplasmic membrane occupancy constraint. Using ‘relative membrane costs' calculated from experimental data, the new modeling framework—FBA with membrane economics (FBAME)—predicted that wild-type E. coli has a GUR of 10.7 mmol/gdw/h, an oxygen uptake rate (OUR) of 15.8 mmol/gdw/h, and a specific growth rate of 0.69 per hour during aerobic growth with excess glucose. FBAME also predicted that under the same growth condition, an E. coli knockout strain with no cytochromes has a GUR of 18 mmol/gdw/h and growth rate of 0.42. These values agree very well with the reported experimental values for E. coli grown in batch cultures (Vemuri et al, 2006; Portnoy et al, 2008), which supports our hypothesis that the higher GUR of E. coli during glucose-excess anaerobiosis than under aerobic conditions is due to the absence of the respiratory enzymes. We also simulated the aerobic growth of E. coli in glucose-limited chemostat using both conventional FBA and FBAME. FBAME successfully predicted the growth rate and yield changes with respect to increasing GUR (Figure 2A and B), as well as the aerobic production of acetate (Figure 2C) and concomitant repression of oxygen uptake (Figure 2D). On the other hand, traditional FBA significantly overestimated the growth rate and yield at higher GURs (this overestimation cannot be explained by varying the growth-associated maintenance (GAM) energy parameter; Figure 2A), and failed to predict the decrease in yield independent of acetate overflow and reduction in oxygen uptake at higher GURs (Figure 2). In addition, FBAME was able to predict the reduction of the TCA cycle activities at higher uptake rates (Figure 3C and D) as well as the selective expression of Cyo and Cyd-II at lower uptake rates (Figure 3A and B), whereas conventional FBA cannot predict the expression of inefficient Cyd-II. These predictions agree with the gene expression data from glucose-limited chemostat (Figure 3). Given the simplicity of the constraint we imposed, our model predictions agree surprisingly well with experimental observations, lending strong credibility to the membrane economics hypothesis.
Concluding remarks
Although it has been long suggested that cellular evolution are governed by non-adjustable mechanistic constraints (Palsson, 2000; Papin et al, 2005; Novak et al, 2006), to date, most metabolic models rely on empirically derived parameters such as glucose and OUR. In this article, we showed that complex phenomena, such as the respiro-fermentation in E. coli, could be satisfactorily explained and accurately predicted by using constraint-based optimization by introducing a simple mechanistic constraint on membrane enzyme occupancy. Given that the cytoplasmic membrane occupancy constraint is directly related to the surface area to volume (S/V) ratio of the cell, it is possible that this constraint resulted in the evolution of mitochondria in eukaryotes as mitochondria allows for a significantly increased S/V ratio. Further efforts to elucidate such fundamental cellular constraints as well as the underlying design principles could significantly improve our understanding of the regulation and evolution of metabolism.
The simultaneous utilization of efficient respiration and inefficient fermentation even in the presence of abundant oxygen is a puzzling phenomenon commonly observed in bacteria, yeasts, and cancer cells. Despite extensive research, the biochemical basis for this phenomenon remains obscure. We hypothesize that the outcome of a competition for membrane space between glucose transporters and respiratory chain (which we refer to as economics of membrane occupancy) proteins influences respiration and fermentation. By incorporating a sole constraint based on this concept in the genome-scale metabolic model of Escherichia coli, we were able to simulate respiro-fermentation. Further analysis of the impact of this constraint revealed differential utilization of the cytochromes and faster glucose uptake under anaerobic conditions than under aerobic conditions. Based on these simulations, we propose that bacterial cells manage the composition of their cytoplasmic membrane to maintain optimal ATP production by switching between oxidative and substrate-level phosphorylation. These results suggest that the membrane occupancy constraint may be a fundamental governing constraint of cellular metabolism and physiology, and establishes a direct link between cell morphology and physiology.
doi:10.1038/msb.2011.34
PMCID: PMC3159977  PMID: 21694717
constraint-based modeling; flux balance analysis; membrane occupancy; overflow metabolism; respiro-fermentation
5.  Including metabolite concentrations into flux balance analysis: thermodynamic realizability as a constraint on flux distributions in metabolic networks 
BMC Systems Biology  2007;1:23.
Background
In recent years, constrained optimization – usually referred to as flux balance analysis (FBA) – has become a widely applied method for the computation of stationary fluxes in large-scale metabolic networks. The striking advantage of FBA as compared to kinetic modeling is that it basically requires only knowledge of the stoichiometry of the network. On the other hand, results of FBA are to a large degree hypothetical because the method relies on plausible but hardly provable optimality principles that are thought to govern metabolic flux distributions.
Results
To augment the reliability of FBA-based flux calculations we propose an additional side constraint which assures thermodynamic realizability, i.e. that the flux directions are consistent with the corresponding changes of Gibb's free energies. The latter depend on metabolite levels for which plausible ranges can be inferred from experimental data. Computationally, our method results in the solution of a mixed integer linear optimization problem with quadratic scoring function. An optimal flux distribution together with a metabolite profile is determined which assures thermodynamic realizability with minimal deviations of metabolite levels from their expected values. We applied our novel approach to two exemplary metabolic networks of different complexity, the metabolic core network of erythrocytes (30 reactions) and the metabolic network iJR904 of Escherichia coli (931 reactions). Our calculations show that increasing network complexity entails increasing sensitivity of predicted flux distributions to variations of standard Gibb's free energy changes and metabolite concentration ranges. We demonstrate the usefulness of our method for assessing critical concentrations of external metabolites preventing attainment of a metabolic steady state.
Conclusion
Our method incorporates the thermodynamic link between flux directions and metabolite concentrations into a practical computational algorithm. The weakness of conventional FBA to rely on intuitive assumptions about the reversibility of biochemical reactions is overcome. This enables the computation of reliable flux distributions even under extreme conditions of the network (e.g. enzyme inhibition, depletion of substrates or accumulation of end products) where metabolite concentrations may be drastically altered.
doi:10.1186/1752-0509-1-23
PMCID: PMC1903363  PMID: 17543097
6.  Systematic evaluation of objective functions for predicting intracellular fluxes in Escherichia coli 
The in vivo distribution of metabolic fluxes in Escherichia coli can be predicted from optimality principles At least two different sets of optimality principles govern the operation of the metabolic network under different environmental conditionsMetabolism during unlimited growth on glucose in batch culture is best described by the nonlinear maximization of ATP yield per unit of flux
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, bEdwards 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.
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).
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 13C-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.
doi:10.1038/msb4100162
PMCID: PMC1949037  PMID: 17625511
13C-flux; evolution; flux balance analysis; metabolic network; network optimality
7.  Construction and completion of flux balance models from pathway databases 
Bioinformatics  2012;28(3):388-396.
Motivation: Flux balance analysis (FBA) is a well-known technique for genome-scale modeling of metabolic flux. Typically, an FBA formulation requires the accurate specification of four sets: biochemical reactions, biomass metabolites, nutrients and secreted metabolites. The development of FBA models can be time consuming and tedious because of the difficulty in assembling completely accurate descriptions of these sets, and in identifying errors in the composition of these sets. For example, the presence of a single non-producible metabolite in the biomass will make the entire model infeasible. Other difficulties in FBA modeling are that model distributions, and predicted fluxes, can be cryptic and difficult to understand.
Results: We present a multiple gap-filling method to accelerate the development of FBA models using a new tool, called MetaFlux, based on mixed integer linear programming (MILP). The method suggests corrections to the sets of reactions, biomass metabolites, nutrients and secretions. The method generates FBA models directly from Pathway/Genome Databases. Thus, FBA models developed in this framework are easily queried and visualized using the Pathway Tools software. Predicted fluxes are more easily comprehended by visualizing them on diagrams of individual metabolic pathways or of metabolic maps. MetaFlux can also remove redundant high-flux loops, solve FBA models once they are generated and model the effects of gene knockouts. MetaFlux has been validated through construction of FBA models for Escherichia coli and Homo sapiens.
Availability: Pathway Tools with MetaFlux is freely available to academic users, and for a fee to commercial users. Download from: biocyc.org/download.shtml.
Contact: mario.latendresse@sri.com
Supplementary information: Supplementary data are available at Bioinformatics online.
doi:10.1093/bioinformatics/btr681
PMCID: PMC3268246  PMID: 22262672
8.  Genome-level transcription data of Yersinia pestis analyzed with a New metabolic constraint-based approach 
BMC Systems Biology  2012;6:150.
Background
Constraint-based computational approaches, such as flux balance analysis (FBA), have proven successful in modeling genome-level metabolic behavior for conditions where a set of simple cellular objectives can be clearly articulated. Recently, the necessity to expand the current range of constraint-based methods to incorporate high-throughput experimental data has been acknowledged by the proposal of several methods. However, these methods have rarely been used to address cellular metabolic responses to some relevant perturbations such as antimicrobial or temperature-induced stress. Here, we present a new method for combining gene-expression data with FBA (GX-FBA) that allows modeling of genome-level metabolic response to a broad range of environmental perturbations within a constraint-based framework. The method uses mRNA expression data to guide hierarchical regulation of cellular metabolism subject to the interconnectivity of the metabolic network.
Results
We applied GX-FBA to a genome-scale model of metabolism in the gram negative bacterium Yersinia pestis and analyzed its metabolic response to (i) variations in temperature known to induce virulence, and (ii) antibiotic stress. Without imposition of any a priori behavioral constraints, our results show strong agreement with reported phenotypes. Our analyses also lead to novel insights into how Y. pestis uses metabolic adjustments to counter different forms of stress.
Conclusions
Comparisons of GX-FBA predicted metabolic states with fluxomic measurements and different reported post-stress phenotypes suggest that mass conservation constraints and network connectivity can be an effective representative of metabolic flux regulation in constraint-based models. We believe that our approach will be of aid in the in silico evaluation of cellular goals under different conditions and can be used for a variety of analyses such as identification of potential drug targets and their action.
doi:10.1186/1752-0509-6-150
PMCID: PMC3572438  PMID: 23216785
Flux balance analysis; Gene-expression; Yersinia pestis; Stress response; Metabolism
9.  Predicting selective drug targets in cancer through metabolic networks 
The authors develop a genome-scale model of cancer metabolism and use it to predict genes that are essential for cancer cell growth. An array of target combinations are then identified that could potentially provide novel selective treatments for specific cancers.
The first genome-scale network model of cancer metabolism is developed and validated by successfully identifying genes essential for cellular proliferation in cancer cell lines.The model predicts 52 cytostatic drug targets, of which 40% are targeted by known, approved or experimental anticancer drugs, and the rest are new.Combinations of synthetic lethal drug targets are predicted, whose synergy is validated using available drug efficacy and gene expression measurements across the NCI-60 cancer cell line collection.Potential selective treatments for specific cancers that depend on cancer type-specific downregulation of gene expression and somatic mutations are compiled.
During tumor development, cancer cells modify their metabolism to meet the requirements of cellular proliferation, thus facilitating the uptake and conversion of nutrients into biomass. Many key metabolic alterations are similar across tumor cells, including changes in glucose metabolism that give rise to the Warburg effect, and an increase in biosynthetic activities (such as nucleotide, lipids and amino-acid synthesis) (DeBerardinis et al, 2008; Tennant et al, 2009; Vander Heiden et al, 2009). The observation that many types of cancer cells adapt their metabolism toward increased proliferation makes flux balance analysis (FBA), a constraint-based modeling (CBM) approach, suitable for modeling cancer metabolism as it assumes that cells are under selective pressure to increase their growth rate (Price et al, 2003).
Building on previous reconstructions of a generic (non-tissue specific) human metabolic network (Duarte et al, 2007; Ma et al, 2007), we develop here the first large-scale FBA model of cancer metabolism that aims to capture the main metabolic alterations that are common across many cancer types. The model reconstruction is based on our recent computational method for the automatic reconstruction of human tissue metabolic models (Jerby et al, 2010), integrating the human metabolic model with cancer gene expression data. The construction of the cancer model focuses on activating a core set of metabolic enzyme-coding genes that are highly expressed across cancer cell lines in the NCI-60 collection, with additional reactions enabling their activation and the biosynthesis of a set of biomass compounds required for cellular proliferation. This generic cancer model enables the successful prediction of the metabolic state of cancer cells across different gene knockdowns and modeling the effects of drug applications on a large scale. As a first demonstration of the predictive performance of the cancer model, we applied it to predict 199 growth-supporting genes whose knockdown is expected to inhibit cellular proliferation, showing that the model predictions indeed match results of shRNA gene silencing experiments (Luo et al, 2008).
To identify viable anticancer drug targets, we predicted whether the knockdown of the growth-supporting genes is likely to be toxic to normal cells. Out of the 199 genes that are predicted to be growth supporting in the cancer model, 52 are predicted to have negligible effects on energy production in normal cells. However, the knockdown of the majority of the latter is yet predicted to potentially cause damage to proliferation of normal cells, suggesting that the targeting of these genes would cause similar side effects to those observed with current cytostatic drugs (Partridge et al, 2001). Next, we predicted 342 synthetic lethal drug targets, whose predicted synergy was validated based on (i) comparison with genetic interactions between the corresponding yeast orthologs (Costanzo et al, 2010) and (ii) by analyzing the efficacy of metabolic drugs targeting these genes, finding that drugs that target a single gene (participating in a predicted synthetic lethal pair) indeed have higher efficacy in cell lines in which the synergistic gene is lowly expressed. In contrast to the single targets described above, the knockdown of a third of these synergistic pairs is predicted to leave the proliferation of normal cells intact. Most importantly, the specific targeting of a gene participating in a synergistic pair is especially appealing in tumors in which its interacting gene is specifically inactivated—the targeting of such a gene solely is likely to selectively damage the tumor, without affecting the function of healthy tissues in which the interacting gene in the pair is active. We utilized genomic and transcriptomic data to infer gene inactivation across an array of cancers, which has led to the identification of cancer type-specific targets based on the intersection of this data with our predicted synergistic gene pairs.
In summary, the model presented here lays down a fundamental computational approach for interpreting the rapidly accumulating proteomics (Bichsel et al, 2001) and metabolomics (Fan et al, 2009) data characterizing cancer metabolic alterations. We hope that the publication of this first step will spur further studies aimed at obtaining a systems level understanding of cancer metabolism and at designing new therapeutic means that selectively target them.
The interest in studying metabolic alterations in cancer and their potential role as novel targets for therapy has been rejuvenated in recent years. Here, we report the development of the first genome-scale network model of cancer metabolism, validated by correctly identifying genes essential for cellular proliferation in cancer cell lines. The model predicts 52 cytostatic drug targets, of which 40% are targeted by known, approved or experimental anticancer drugs, and the rest are new. It further predicts combinations of synthetic lethal drug targets, whose synergy is validated using available drug efficacy and gene expression measurements across the NCI-60 cancer cell line collection. Finally, potential selective treatments for specific cancers that depend on cancer type-specific downregulation of gene expression and somatic mutations are compiled.
doi:10.1038/msb.2011.35
PMCID: PMC3159974  PMID: 21694718
cancer; metabolic; metabolism; modeling; selectivity
10.  Understanding the Adaptive Growth Strategy of Lactobacillus plantarum by In Silico Optimisation 
PLoS Computational Biology  2009;5(6):e1000410.
In the study of metabolic networks, optimization techniques are often used to predict flux distributions, and hence, metabolic phenotype. Flux balance analysis in particular has been successful in predicting metabolic phenotypes. However, an inherent limitation of a stoichiometric approach such as flux balance analysis is that it can predict only flux distributions that result in maximal yields. Hence, previous attempts to use FBA to predict metabolic fluxes in Lactobacillus plantarum failed, as this lactic acid bacterium produces lactate, even under glucose-limited chemostat conditions, where FBA predicted mixed acid fermentation as an alternative pathway leading to a higher yield. In this study we tested, however, whether long-term adaptation on an unusual and poor carbon source (for this bacterium) would select for mutants with optimal biomass yields. We have therefore adapted Lactobacillus plantarum to grow well on glycerol as its main growth substrate. After prolonged serial dilutions, the growth yield and corresponding fluxes were compared to in silico predictions. Surprisingly, the organism still produced mainly lactate, which was corroborated by FBA to indeed be optimal. To understand these results, constraint-based elementary flux mode analysis was developed that predicted 3 out of 2669 possible flux modes to be optimal under the experimental conditions. These optimal pathways corresponded very closely to the experimentally observed fluxes and explained lactate formation as the result of competition for oxygen by the other flux modes. Hence, these results provide thorough understanding of adaptive evolution, allowing in silico predictions of the resulting flux states, provided that the selective growth conditions favor yield optimization as the winning strategy.
Author Summary
Being able to predict the metabolic fluxes and growth rate of a microorganism is an important topic in microbial systems biology. One approach, constraint-based modeling, uses a reconstructed metabolic network and optimization techniques to make such predictions. Although widely used, the success of this approach depends on a number of important assumptions. First, it assumes that evolutionary forces have shaped the metabolism towards optimality of, in most cases, growth rate. Second, through the nature of the modeling approach, it assumes that microorganisms maximize the growth rate through optimizing the yield on the growth substrate. Despite successes of the approach in model organisms such as Saccharomyces cerevisiae and Escherichia coli, we have previously observed that the approach fails in Lactobacillus plantarum, a lactic acid bacterium that clearly does not optimize its yield on glucose but “wastes” glucose by producing lactic acid. In the current study we provide evidence that L. plantarum does optimize its yield when grown under a poor carbon condition, i.e., when grown on glycerol as its main carbon source. The study provides new insight in when the application of in silico optimization techniques can be expected to be predictive.
doi:10.1371/journal.pcbi.1000410
PMCID: PMC2690837  PMID: 19521528
11.  Efficiently gap-filling reaction networks 
BMC Bioinformatics  2014;15:225.
Background
Flux Balance Analysis (FBA) is a genome-scale computational technique for modeling the steady-state fluxes of an organism’s reaction network. When the organism’s reaction network needs to be completed to obtain growth using FBA, without relying on the genome, the completion process is called reaction gap-filling. Currently, computational techniques used to gap-fill a reaction network compute the minimum set of reactions using Mixed-Integer Linear Programming (MILP). Depending on the number of candidate reactions used to complete the model, MILP can be computationally demanding.
Results
We present a computational technique, called FastGapFilling, that efficiently completes a reaction network by using only Linear Programming, not MILP. FastGapFilling creates a linear program with all candidate reactions, an objective function based on their weighted fluxes, and a variable weight on the biomass reaction: no integer variable is used. A binary search is performed by modifying the weight applied to the flux of the biomass reaction, and solving each corresponding linear program, to try reducing the number of candidate reactions to add to the network to generate a working model. We show that this method has proved effective on a series of incomplete E. coli and yeast models with, in some cases, a three orders of magnitude execution speedup compared with MILP. We have implemented FastGapFilling in MetaFlux as part of Pathway Tools (version 17.5), which is freely available to academic users, and for a fee to commercial users. Download from: biocyc.org/download.shtml.
Conclusions
The computational technique presented is very efficient allowing interactive completion of reaction networks of FBA models. Computational techniques based on MILP cannot offer such fast and interactive completion.
doi:10.1186/1471-2105-15-225
PMCID: PMC4094995  PMID: 24972703
Flux Balance Analysis (FBA); Gap-filling; Systems biology; Reaction network; Linear Programming (LP); Mixed-Integer Linear Programming (MILP)
12.  Community Flux Balance Analysis for Microbial Consortia at Balanced Growth 
PLoS ONE  2013;8(5):e64567.
A central focus in studies of microbial communities is the elucidation of the relationships between genotype, phenotype, and dynamic community structure. Here, we present a new computational method called community flux balance analysis (cFBA) to study the metabolic behavior of microbial communities. cFBA integrates the comprehensive metabolic capacities of individual microorganisms in terms of (genome-scale) stoichiometric models of metabolism, and the metabolic interactions between species in the community and abiotic processes. In addition, cFBA considers constraints deriving from reaction stoichiometry, reaction thermodynamics, and the ecosystem. cFBA predicts for communities at balanced growth the maximal community growth rate, the required rates of metabolic reactions within and between microbes and the relative species abundances. In order to predict species abundances and metabolic activities at the optimal community growth rate, a nonlinear optimization problem needs to be solved. We outline the methodology of cFBA and illustrate the approach with two examples of microbial communities. These examples illustrate two useful applications of cFBA. Firstly, cFBA can be used to study how specific biochemical limitations in reaction capacities cause different types of metabolic limitations that microbial consortia can encounter. In silico variations of those maximal capacities allow for a global view of the consortium responses to various metabolic and environmental constraints. Secondly, cFBA is very useful for comparing the performance of different metabolic cross-feeding strategies to either find one that agrees with experimental data or one that is most efficient for the community of microorganisms.
doi:10.1371/journal.pone.0064567
PMCID: PMC3669319  PMID: 23741341
13.  Development and experimental verification of a genome-scale metabolic model for Corynebacterium glutamicum 
Background
In silico genome-scale metabolic models enable the analysis of the characteristics of metabolic systems of organisms. In this study, we reconstructed a genome-scale metabolic model of Corynebacterium glutamicum on the basis of genome sequence annotation and physiological data. The metabolic characteristics were analyzed using flux balance analysis (FBA), and the results of FBA were validated using data from culture experiments performed at different oxygen uptake rates.
Results
The reconstructed genome-scale metabolic model of C. glutamicum contains 502 reactions and 423 metabolites. We collected the reactions and biomass components from the database and literatures, and made the model available for the flux balance analysis by filling gaps in the reaction networks and removing inadequate loop reactions. Using the framework of FBA and our genome-scale metabolic model, we first simulated the changes in the metabolic flux profiles that occur on changing the oxygen uptake rate. The predicted production yields of carbon dioxide and organic acids agreed well with the experimental data. The metabolic profiles of amino acid production phases were also investigated. A comprehensive gene deletion study was performed in which the effects of gene deletions on metabolic fluxes were simulated; this helped in the identification of several genes whose deletion resulted in an improvement in organic acid production.
Conclusion
The genome-scale metabolic model provides useful information for the evaluation of the metabolic capabilities and prediction of the metabolic characteristics of C. glutamicum. This can form a basis for the in silico design of C. glutamicum metabolic networks for improved bioproduction of desirable metabolites.
doi:10.1186/1475-2859-8-43
PMCID: PMC2728707  PMID: 19646286
14.  MicrobesFlux: a web platform for drafting metabolic models from the KEGG database 
BMC Systems Biology  2012;6:94.
Background
Concurrent with the efforts currently underway in mapping microbial genomes using high-throughput sequencing methods, systems biologists are building metabolic models to characterize and predict cell metabolisms. One of the key steps in building a metabolic model is using multiple databases to collect and assemble essential information about genome-annotations and the architecture of the metabolic network for a specific organism. To speed up metabolic model development for a large number of microorganisms, we need a user-friendly platform to construct metabolic networks and to perform constraint-based flux balance analysis based on genome databases and experimental results.
Results
We have developed a semi-automatic, web-based platform (MicrobesFlux) for generating and reconstructing metabolic models for annotated microorganisms. MicrobesFlux is able to automatically download the metabolic network (including enzymatic reactions and metabolites) of ~1,200 species from the KEGG database (Kyoto Encyclopedia of Genes and Genomes) and then convert it to a metabolic model draft. The platform also provides diverse customized tools, such as gene knockouts and the introduction of heterologous pathways, for users to reconstruct the model network. The reconstructed metabolic network can be formulated to a constraint-based flux model to predict and analyze the carbon fluxes in microbial metabolisms. The simulation results can be exported in the SBML format (The Systems Biology Markup Language). Furthermore, we also demonstrated the platform functionalities by developing an FBA model (including 229 reactions) for a recent annotated bioethanol producer, Thermoanaerobacter sp. strain X514, to predict its biomass growth and ethanol production.
Conclusion
MicrobesFlux is an installation-free and open-source platform that enables biologists without prior programming knowledge to develop metabolic models for annotated microorganisms in the KEGG database. Our system facilitates users to reconstruct metabolic networks of organisms based on experimental information. Through human-computer interaction, MicrobesFlux provides users with reasonable predictions of microbial metabolism via flux balance analysis. This prototype platform can be a springboard for advanced and broad-scope modeling of complex biological systems by integrating other “omics” data or 13 C- metabolic flux analysis results. MicrobesFlux is available at http://tanglab.engineering.wustl.edu/static/MicrobesFlux.html and will be continuously improved based on feedback from users.
doi:10.1186/1752-0509-6-94
PMCID: PMC3447728  PMID: 22857267
15.  COMPARATIVE DETERMINATION OF BIOMASS COMPOSITION IN DIFFERENTIALLY ACTIVE METABOLIC STATES 
Flux Balance Analysis (FBA) has been successfully applied to facilitate the understanding of cellular metabolism in model organisms. Standard formulations of FBA can be applied to large systems, but the accuracy of predictions may vary significantly depending on environmental conditions, genetic perturbations, or complex unknown regulatory constraints. Here we present an FBA-based approach to infer the biomass compositions that best describe multiple physiological states of a cell. Specifically, we seek to use experimental data (such as flux measurements, or mRNA expression levels) to infer best matching stoichiometrically balanced fluxes and metabolite sinks. Our algorithm is designed to provide predictions based on the comparative analysis of two metabolic states (e.g. wild-type and knockout, or two different time points), so as to be independent from possible arbitrary scaling factors. We test our algorithm using experimental data for metabolic fluxes in wild type and gene deletion strains of E. coli. In addition to demonstrating the capacity of our approach to correctly identify known exchange fluxes and biomass compositions, we analyze E. coli central carbon metabolism to show the changes of metabolic objectives and potential compensation for reducing power due to single enzyme gene deletion in pentose phosphate pathway.
PMCID: PMC3245841  PMID: 19425132
flux balance analysis; systems biology; data integration; metabolic objectives
16.  Flux Imbalance Analysis and the Sensitivity of Cellular Growth to Changes in Metabolite Pools 
PLoS Computational Biology  2013;9(8):e1003195.
Stoichiometric models of metabolism, such as flux balance analysis (FBA), are classically applied to predicting steady state rates - or fluxes - of metabolic reactions in genome-scale metabolic networks. Here we revisit the central assumption of FBA, i.e. that intracellular metabolites are at steady state, and show that deviations from flux balance (i.e. flux imbalances) are informative of some features of in vivo metabolite concentrations. Mathematically, the sensitivity of FBA to these flux imbalances is captured by a native feature of linear optimization, the dual problem, and its corresponding variables, known as shadow prices. First, using recently published data on chemostat growth of Saccharomyces cerevisae under different nutrient limitations, we show that shadow prices anticorrelate with experimentally measured degrees of growth limitation of intracellular metabolites. We next hypothesize that metabolites which are limiting for growth (and thus have very negative shadow price) cannot vary dramatically in an uncontrolled way, and must respond rapidly to perturbations. Using a collection of published datasets monitoring the time-dependent metabolomic response of Escherichia coli to carbon and nitrogen perturbations, we test this hypothesis and find that metabolites with negative shadow price indeed show lower temporal variation following a perturbation than metabolites with zero shadow price. Finally, we illustrate the broader applicability of flux imbalance analysis to other constraint-based methods. In particular, we explore the biological significance of shadow prices in a constraint-based method for integrating gene expression data with a stoichiometric model. In this case, shadow prices point to metabolites that should rise or drop in concentration in order to increase consistency between flux predictions and gene expression data. In general, these results suggest that the sensitivity of metabolic optima to violations of the steady state constraints carries biologically significant information on the processes that control intracellular metabolites in the cell.
Author Summary
Cellular metabolism is composed of a complex network of biochemical reactions that convert environmental nutrients into biosynthetic building blocks and energetic currency. Genome-scale mathematical models of metabolic networks focus largely on trying to predict the rates – or fluxes - of these reactions. By assuming that the concentrations of intracellular metabolites are at steady-state (flux balance), and invoking optimality, these constraint-based methods for modeling metabolism have offered abundant insight into how metabolic flux is routed through the cell. Here we ask how cellular growth would respond to deviations from steady state (flux imbalance) of every possible intracellular metabolite. This question can be addressed through a sensitivity analysis inherent to linear optimization theory, known as duality. We show how some features of metabolite concentrations, such as their growth-limitation and their transient response, are captured by this sensitivity analysis. Our results suggest that, in addition to predicting fluxes, stoichiometric models offer a valuable route towards probing the metabolites themselves and their relevance to growth dynamics.
doi:10.1371/journal.pcbi.1003195
PMCID: PMC3757068  PMID: 24009492
17.  Reconstruction and flux analysis of coupling between metabolic pathways of astrocytes and neurons: application to cerebral hypoxia 
Background
It is a daunting task to identify all the metabolic pathways of brain energy metabolism and develop a dynamic simulation environment that will cover a time scale ranging from seconds to hours. To simplify this task and make it more practicable, we undertook stoichiometric modeling of brain energy metabolism with the major aim of including the main interacting pathways in and between astrocytes and neurons.
Model
The constructed model includes central metabolism (glycolysis, pentose phosphate pathway, TCA cycle), lipid metabolism, reactive oxygen species (ROS) detoxification, amino acid metabolism (synthesis and catabolism), the well-known glutamate-glutamine cycle, other coupling reactions between astrocytes and neurons, and neurotransmitter metabolism. This is, to our knowledge, the most comprehensive attempt at stoichiometric modeling of brain metabolism to date in terms of its coverage of a wide range of metabolic pathways. We then attempted to model the basal physiological behaviour and hypoxic behaviour of the brain cells where astrocytes and neurons are tightly coupled.
Results
The reconstructed stoichiometric reaction model included 217 reactions (184 internal, 33 exchange) and 216 metabolites (183 internal, 33 external) distributed in and between astrocytes and neurons. Flux balance analysis (FBA) techniques were applied to the reconstructed model to elucidate the underlying cellular principles of neuron-astrocyte coupling. Simulation of resting conditions under the constraints of maximization of glutamate/glutamine/GABA cycle fluxes between the two cell types with subsequent minimization of Euclidean norm of fluxes resulted in a flux distribution in accordance with literature-based findings. As a further validation of our model, the effect of oxygen deprivation (hypoxia) on fluxes was simulated using an FBA-derivative approach, known as minimization of metabolic adjustment (MOMA). The results show the power of the constructed model to simulate disease behaviour on the flux level, and its potential to analyze cellular metabolic behaviour in silico.
Conclusion
The predictive power of the constructed model for the key flux distributions, especially central carbon metabolism and glutamate-glutamine cycle fluxes, and its application to hypoxia is promising. The resultant acceptable predictions strengthen the power of such stoichiometric models in the analysis of mammalian cell metabolism.
doi:10.1186/1742-4682-4-48
PMCID: PMC2246127  PMID: 18070347
18.  Semi-automated Curation of Metabolic Models via Flux Balance Analysis: A Case Study with Mycoplasma gallisepticum 
PLoS Computational Biology  2013;9(9):e1003208.
Primarily used for metabolic engineering and synthetic biology, genome-scale metabolic modeling shows tremendous potential as a tool for fundamental research and curation of metabolism. Through a novel integration of flux balance analysis and genetic algorithms, a strategy to curate metabolic networks and facilitate identification of metabolic pathways that may not be directly inferable solely from genome annotation was developed. Specifically, metabolites involved in unknown reactions can be determined, and potentially erroneous pathways can be identified. The procedure developed allows for new fundamental insight into metabolism, as well as acting as a semi-automated curation methodology for genome-scale metabolic modeling. To validate the methodology, a genome-scale metabolic model for the bacterium Mycoplasma gallisepticum was created. Several reactions not predicted by the genome annotation were postulated and validated via the literature. The model predicted an average growth rate of 0.358±0.12, closely matching the experimentally determined growth rate of M. gallisepticum of 0.244±0.03. This work presents a powerful algorithm for facilitating the identification and curation of previously known and new metabolic pathways, as well as presenting the first genome-scale reconstruction of M. gallisepticum.
Author Summary
Flux balance analysis (FBA) is a powerful approach for genome-scale metabolic modeling. It provides metabolic engineers with a tool for manipulating, predicting, and optimizing metabolism for biotechnological and biomedical purposes. However, we posit that it can also be used as tool for fundamental research in understanding and curating metabolic networks. Specifically, by using a genetic algorithm integrated with FBA, we developed a curation approach to identify missing reactions, incomplete reactions, and erroneous reactions. Additionally, it was possible to take advantage of the ensemble information from the genetic algorithm to identify the most critical reactions for curation. We tested our strategy using Mycoplasma gallisepticum as our model organism. Using the genome annotation as the basis, the preliminary genome-scale metabolic model consisted of 446 metabolites involved in 380 reactions. Carrying out our analysis, we found over 80 incorrect reactions and 16 missing reactions. Based upon the guidance of the algorithm, we were able to curate and resolve all discrepancies. The model predicted an average bacterial growth rate of 0.358±0.12 h−1 compared to the experimentally observed 0.244±0.03 h−1. Thus, our approach facilitated the curation of a genome-scale metabolic network and generated a high quality metabolic model.
doi:10.1371/journal.pcbi.1003208
PMCID: PMC3764002  PMID: 24039564
19.  Flux balance analysis of primary metabolism in Chlamydomonas reinhardtii 
Background
Photosynthetic organisms convert atmospheric carbon dioxide into numerous metabolites along the pathways to make new biomass. Aquatic photosynthetic organisms, which fix almost half of global inorganic carbon, have great potential: as a carbon dioxide fixation method, for the economical production of chemicals, or as a source for lipids and starch which can then be converted to biofuels. To harness this potential through metabolic engineering and to maximize production, a more thorough understanding of photosynthetic metabolism must first be achieved. A model algal species, C. reinhardtii, was chosen and the metabolic network reconstructed. Intracellular fluxes were then calculated using flux balance analysis (FBA).
Results
The metabolic network of primary metabolism for a green alga, C. reinhardtii, was reconstructed using genomic and biochemical information. The reconstructed network accounts for the intracellular localization of enzymes to three compartments and includes 484 metabolic reactions and 458 intracellular metabolites. Based on BLAST searches, one newly annotated enzyme (fructose-1,6-bisphosphatase) was added to the Chlamydomonas reinhardtii database. FBA was used to predict metabolic fluxes under three growth conditions, autotrophic, heterotrophic and mixotrophic growth. Biomass yields ranged from 28.9 g per mole C for autotrophic growth to 15 g per mole C for heterotrophic growth.
Conclusion
The flux balance analysis model of central and intermediary metabolism in C. reinhardtii is the first such model for algae and the first model to include three metabolically active compartments. In addition to providing estimates of intracellular fluxes, metabolic reconstruction and modelling efforts also provide a comprehensive method for annotation of genome databases. As a result of our reconstruction, one new enzyme was annotated in the database and several others were found to be missing; implying new pathways or non-conserved enzymes. The use of FBA to estimate intracellular fluxes also provides flux values that can be used as a starting point for rational engineering of C. reinhardtii. From these initial estimates, it is clear that aerobic heterotrophic growth on acetate has a low yield on carbon, while mixotrophically and autotrophically grown cells are significantly more carbon efficient.
doi:10.1186/1752-0509-3-4
PMCID: PMC2628641  PMID: 19128495
20.  Estimating the size of the solution space of metabolic networks 
BMC Bioinformatics  2008;9:240.
Background
Cellular metabolism is one of the most investigated system of biological interactions. While the topological nature of individual reactions and pathways in the network is quite well understood there is still a lack of comprehension regarding the global functional behavior of the system. In the last few years flux-balance analysis (FBA) has been the most successful and widely used technique for studying metabolism at system level. This method strongly relies on the hypothesis that the organism maximizes an objective function. However only under very specific biological conditions (e.g. maximization of biomass for E. coli in reach nutrient medium) the cell seems to obey such optimization law. A more refined analysis not assuming extremization remains an elusive task for large metabolic systems due to algorithmic limitations.
Results
In this work we propose a novel algorithmic strategy that provides an efficient characterization of the whole set of stable fluxes compatible with the metabolic constraints. Using a technique derived from the fields of statistical physics and information theory we designed a message-passing algorithm to estimate the size of the affine space containing all possible steady-state flux distributions of metabolic networks. The algorithm, based on the well known Bethe approximation, can be used to approximately compute the volume of a non full-dimensional convex polytope in high dimensions. We first compare the accuracy of the predictions with an exact algorithm on small random metabolic networks. We also verify that the predictions of the algorithm match closely those of Monte Carlo based methods in the case of the Red Blood Cell metabolic network. Then we test the effect of gene knock-outs on the size of the solution space in the case of E. coli central metabolism. Finally we analyze the statistical properties of the average fluxes of the reactions in the E. coli metabolic network.
Conclusion
We propose a novel efficient distributed algorithmic strategy to estimate the size and shape of the affine space of a non full-dimensional convex polytope in high dimensions. The method is shown to obtain, quantitatively and qualitatively compatible results with the ones of standard algorithms (where this comparison is possible) being still efficient on the analysis of large biological systems, where exact deterministic methods experience an explosion in algorithmic time. The algorithm we propose can be considered as an alternative to Monte Carlo sampling methods.
doi:10.1186/1471-2105-9-240
PMCID: PMC2483728  PMID: 18489757
21.  Metabolic stasis in an ancient symbiosis: genome-scale metabolic networks from two Blattabacterium cuenoti strains, primary endosymbionts of cockroaches 
BMC Microbiology  2012;12(Suppl 1):S5.
Background
Cockroaches are terrestrial insects that strikingly eliminate waste nitrogen as ammonia instead of uric acid. Blattabacterium cuenoti (Mercier 1906) strains Bge and Pam are the obligate primary endosymbionts of the cockroaches Blattella germanica and Periplaneta americana, respectively. The genomes of both bacterial endosymbionts have recently been sequenced, making possible a genome-scale constraint-based reconstruction of their metabolic networks. The mathematical expression of a metabolic network and the subsequent quantitative studies of phenotypic features by Flux Balance Analysis (FBA) represent an efficient functional approach to these uncultivable bacteria.
Results
We report the metabolic models of Blattabacterium strains Bge (iCG238) and Pam (iCG230), comprising 296 and 289 biochemical reactions, associated with 238 and 230 genes, and 364 and 358 metabolites, respectively. Both models reflect both the striking similarities and the singularities of these microorganisms. FBA was used to analyze the properties, potential and limits of the models, assuming some environmental constraints such as aerobic conditions and the net production of ammonia from these bacterial systems, as has been experimentally observed. In addition, in silico simulations with the iCG238 model have enabled a set of carbon and nitrogen sources to be defined, which would also support a viable phenotype in terms of biomass production in the strain Pam, which lacks the first three steps of the tricarboxylic acid cycle. FBA reveals a metabolic condition that renders these enzymatic steps dispensable, thus offering a possible evolutionary explanation for their elimination. We also confirm, by computational simulations, the fragility of the metabolic networks and their host dependence.
Conclusions
The minimized Blattabacterium metabolic networks are surprisingly similar in strains Bge and Pam, after 140 million years of evolution of these endosymbionts in separate cockroach lineages. FBA performed on the reconstructed networks from the two bacteria helps to refine the functional analysis of the genomes enabling us to postulate how slightly different host metabolic contexts drove their parallel evolution.
doi:10.1186/1471-2180-12-S1-S5
PMCID: PMC3287516  PMID: 22376077
22.  Predicting biological system objectives de novo from internal state measurements 
BMC Bioinformatics  2008;9:43.
Background
Optimization theory has been applied to complex biological systems to interrogate network properties and develop and refine metabolic engineering strategies. For example, methods are emerging to engineer cells to optimally produce byproducts of commercial value, such as bioethanol, as well as molecular compounds for disease therapy. Flux balance analysis (FBA) is an optimization framework that aids in this interrogation by generating predictions of optimal flux distributions in cellular networks. Critical features of FBA are the definition of a biologically relevant objective function (e.g., maximizing the rate of synthesis of biomass, a unit of measurement of cellular growth) and the subsequent application of linear programming (LP) to identify fluxes through a reaction network. Despite the success of FBA, a central remaining challenge is the definition of a network objective with biological meaning.
Results
We present a novel method called Biological Objective Solution Search (BOSS) for the inference of an objective function of a biological system from its underlying network stoichiometry as well as experimentally-measured state variables. Specifically, BOSS identifies a system objective by defining a putative stoichiometric "objective reaction," adding this reaction to the existing set of stoichiometric constraints arising from known interactions within a network, and maximizing the putative objective reaction via LP, all the while minimizing the difference between the resultant in silico flux distribution and available experimental (e.g., isotopomer) flux data. This new approach allows for discovery of objectives with previously unknown stoichiometry, thus extending the biological relevance from earlier methods. We verify our approach on the well-characterized central metabolic network of Saccharomyces cerevisiae.
Conclusion
We illustrate how BOSS offers insight into the functional organization of biochemical networks, facilitating the interrogation of cellular design principles and development of cellular engineering applications. Furthermore, we describe how growth is the best-fit objective function for the yeast metabolic network given experimentally-measured fluxes.
doi:10.1186/1471-2105-9-43
PMCID: PMC2258290  PMID: 18218092
23.  Adaptive bi-level programming for optimal gene knockouts for targeted overproduction under phenotypic constraints 
BMC Bioinformatics  2013;14(Suppl 2):S17.
Background
Optimization procedures to identify gene knockouts for targeted biochemical overproduction have been widely in use in modern metabolic engineering. Flux balance analysis (FBA) framework has provided conceptual simplifications for genome-scale dynamic analysis at steady states. Based on FBA, many current optimization methods for targeted bio-productions have been developed under the maximum cell growth assumption. The optimization problem to derive gene knockout strategies recently has been formulated as a bi-level programming problem in OptKnock for maximum targeted bio-productions with maximum growth rates. However, it has been shown that knockout mutants in fact reach the steady states with the minimization of metabolic adjustment (MOMA) from the corresponding wild-type strains instead of having maximal growth rates after genetic or metabolic intervention. In this work, we propose a new bi-level computational framework--MOMAKnock--which can derive robust knockout strategies under the MOMA flux distribution approximation.
Methods
In this new bi-level optimization framework, we aim to maximize the production of targeted chemicals by identifying candidate knockout genes or reactions under phenotypic constraints approximated by the MOMA assumption. Hence, the targeted chemical production is the primary objective of MOMAKnock while the MOMA assumption is formulated as the inner problem of constraining the knockout metabolic flux to be as close as possible to the steady-state phenotypes of wide-type strains. As this new inner problem becomes a quadratic programming problem, a novel adaptive piecewise linearization algorithm is developed in this paper to obtain the exact optimal solution to this new bi-level integer quadratic programming problem for MOMAKnock.
Results
Our new MOMAKnock model and the adaptive piecewise linearization solution algorithm are tested with a small E. coli core metabolic network and a large-scale iAF1260 E. coli metabolic network. The derived knockout strategies are compared with those from OptKnock. Our preliminary experimental results show that MOMAKnock can provide improved targeted productions with more robust knockout strategies.
doi:10.1186/1471-2105-14-S2-S17
PMCID: PMC3549844  PMID: 23368729
24.  Conservation of high-flux backbone in alternate optimal and near-optimal flux distributions of metabolic networks 
Systems and Synthetic Biology  2009;2(3-4):83-93.
Constraint-based flux balance analysis (FBA) has proven successful in predicting the flux distribution of metabolic networks in diverse environmental conditions. FBA finds one of the alternate optimal solutions that maximizes the biomass production rate. Almaas et al. have shown that the flux distribution follows a power law, and it is possible to associate with most metabolites two reactions which maximally produce and consume a given metabolite, respectively. This observation led to the concept of high-flux backbone (HFB) in metabolic networks. In previous work, the HFB has been computed using a particular optima obtained using FBA. In this paper, we investigate the conservation of HFB of a particular solution for a given medium across different alternate optima and near-optima in metabolic networks of E. coli and S. cerevisiae. Using flux variability analysis (FVA), we propose a method to determine reactions that are guaranteed to be in HFB regardless of alternate solutions. We find that the HFB of a particular optima is largely conserved across alternate optima in E. coli, while it is only moderately conserved in S. cerevisiae. However, the HFB of a particular near-optima shows a large variation across alternate near-optima in both organisms. We show that the conserved set of reactions in HFB across alternate near-optima has a large overlap with essential reactions and reactions which are both uniquely consuming (UC) and uniquely producing (UP). Our findings suggest that the structure of the metabolic network admits a high degree of redundancy and plasticity in near-optimal flow patterns enhancing system robustness for a given environmental condition.
doi:10.1007/s11693-009-9025-8
PMCID: PMC2735644  PMID: 19484377
Complex network; Flux balance analysis (FBA); Alternate optima and near-optima; Flux variability analysis (FVA); Flux plasticity
25.  Conservation of high-flux backbone in alternate optimal and near-optimal flux distributions of metabolic networks 
Systems and Synthetic Biology  2009;2(3-4):83-93.
Constraint-based flux balance analysis (FBA) has proven successful in predicting the flux distribution of metabolic networks in diverse environmental conditions. FBA finds one of the alternate optimal solutions that maximizes the biomass production rate. Almaas et al. have shown that the flux distribution follows a power law, and it is possible to associate with most metabolites two reactions which maximally produce and consume a given metabolite, respectively. This observation led to the concept of high-flux backbone (HFB) in metabolic networks. In previous work, the HFB has been computed using a particular optima obtained using FBA. In this paper, we investigate the conservation of HFB of a particular solution for a given medium across different alternate optima and near-optima in metabolic networks of E. coli and S. cerevisiae. Using flux variability analysis (FVA), we propose a method to determine reactions that are guaranteed to be in HFB regardless of alternate solutions. We find that the HFB of a particular optima is largely conserved across alternate optima in E. coli, while it is only moderately conserved in S. cerevisiae. However, the HFB of a particular near-optima shows a large variation across alternate near-optima in both organisms. We show that the conserved set of reactions in HFB across alternate near-optima has a large overlap with essential reactions and reactions which are both uniquely consuming (UC) and uniquely producing (UP). Our findings suggest that the structure of the metabolic network admits a high degree of redundancy and plasticity in near-optimal flow patterns enhancing system robustness for a given environmental condition.
doi:10.1007/s11693-009-9025-8
PMCID: PMC2735644  PMID: 19484377
Complex network; Flux balance analysis (FBA); Alternate optima and near-optima; Flux variability analysis (FVA); Flux plasticity

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