Clostridium thermocellum (ATCC 27405) is a thermophilic anaerobe that is both cellulolytic and ethanologenic, meaning that it can directly use the plant sugar, cellulose, and biochemically convert it to ethanol. A major challenge in using microorganisms for chemical production is the need to modify the organism to increase production efficiency. The process of properly engineering an organism is typically arduous.Microorganisms possess diverse metabolic capabilities that can potentially be leveraged for efficient production of biofuels. C. thermocellum metabolism, iSR432, for the purpose of establishing a computational tool to study the metabolic network of C. thermocellum and facilitate efforts to engineer C. thermocellum for biofuel production. The model consists of 577 reactions involving 525 intracellular metabolites, 432 genes, and a proteomic-based representation of a cellulosome. The process of constructing this metabolic model led to suggested annotation refinements for 27 genes and identification of areas of metabolism requiring further study. The accuracy of the iSR432 model was tested using experimental growth and by-product secretion data for growth on cellobiose and fructose. Analysis using this model captures the relationship between the reduction-oxidation state of the cell and ethanol secretion and allowed for prediction of gene deletions and environmental conditions that would increase ethanol production.Here we present a genome-scale model of C. thermocellum and establish a strong foundation for rational strain design. In addition, we are able to draw some important conclusions regarding the underlying metabolic mechanisms for observed behaviors of C. thermocellum and highlight remaining gaps in the existing genome annotations.By incorporating genomic sequence data, network topology, and experimental measurements of enzyme activities and metabolite fluxes, we have generated a model that is reasonably accurate at predicting the cellular phenotype of  Constraint-based modeling is a rapidly expanding approach to studying biological systems on the genome scale. Models for a range of different organisms with varying metabolic capabilities have been published over the last ten years -4. Each The constraint-based approach to metabolic modeling fundamentally represents a functional, in-context method for studying cellular metabolism. Through various analyses, such as FBA, predictions of cellular behavior can be made to test or confirm our current state of knowledge about a specific organism's metabolic network. Not only does this give us a computational means of predicting function from genomic information, but the process of building these models represents an added level of functional annotation within the context of an overall network . Both thClostridium thermocellum has generated a great deal of interest, because it both hydrolyzes cellulose and produces ethanol as a fermentation product and therefore has potential to be a model organism for consolidated bioprocessing (CBP) that eliminates the separate saccharification step [A variety of cellulolytic microorganisms exist that have the potential to utilize cellulose for biofuel production, but these organisms are typically poorly characterized. The application of constraint-based metabolic modeling to cellulolytic organisms may help increase our understanding of how these organisms function and lead to improvements in bio-based biofuel production. The utilization of cellulose for the production of ethanol or other fuels has recently been highlighted -16 as anion step . If suchC. thermocellum produces a number of industrially important fermentation products in addition to ethanol, including acetic acid, formic acid, and hydrogen (H2). Thus, C. thermocellum is a potentially important candidate for metabolic engineering, to divert energy and carbon flow toward desired fermentation products. Recent investigations of a related organism, Clostridium acetobutylicum [C. thermocellum from an application perspective.utylicum -21, suggutylicum ,12,17,22C. thermocellum (ATCC 27405) consisting of 577 reactions involving 525 distinct metabolites, 73 membrane transport reactions, and 432 genes . The reaction list was compiled based on the genome annotations available in the UniProt [In this paper, we present a genome-scale model of  UniProt , Kyoto E UniProt -26, and lum ATCC 405 consiC. thermocellum was computationally analyzed by integrating genomic, biochemical, and physiological information. The main result of this work was the development of a genome-scale metabolic model of C. thermocellum that includes a unique model representation of a cellulosome, a major functional unit in cellulose hydrolysis that accounts for a high percentage of the total protein content in C. thermocellum. The accuracy of the constructed model was determined by comparing the results of model simulations to experimental results for growth on cellobiose and fructose from two different independent studies. FBA was used to study the range of ethanol production capabilities of both wild-type and gene deletion strains of C. thermocellum in different chemical environments. Metabolic network comparisons were also conducted between C. thermocellum and a related butanol-producing species C. acetobutylicum and between C. thermocellum and the model ethanologen, Saccharomyces cerevisiae. All computational and experimental work was based upon the wild-type strain of C. thermocellum (ATCC 27405). The final reaction and metabolite list is available in Excel format  and cell envelope biosynthesis . The model includes reactions relating to the synthesis of a variety of vitamins and cofactors, including biotin, NAD, pantothenate, and riboflavin.The breakdown of no acids , and iSRC. thermocellum metabolism was assembled based on the existing annotations (see Methods) and using a database of biochemical reactions compiled from previously published genome-scale constraint-based models, resulting in an initial set of approximately 400 reactions. This initial version was incomplete in several respects, including gaps in critical metabolic subsystems and the inability to synthesize metabolites known to be crucial . This initial version was then supplemented with reactions representing processes known to be crucial, but missing in this first version. Some of these reactions were missing because they represent either processes specific to C. thermocellum, or processes that must be represented by manually constructed, lumped reactions. Examples include cellulosome production, cellulose and chitin breakdown, fatty acid synthesis, phospholipid synthesis, teichoic acid and peptidoglycan synthesis, steroid metabolism, and various transport processes known to occur in C. thermocellum. We also added missing transport reactions for important inputs  and known metabolic fermentation products . This second model version resulted in the addition of about 150 additional reactions.The initial reconstruction of C. thermocellum [C. thermocellum that might code for a missing enzyme, we used BLASTP similarity searches between the translated set of C. thermocellum genes and enzymes from public databases with the annotation of interest. For example, for pyruvate kinase (EC 2.7.1.40) all sequences from UNIPROT annotated with this EC number (2.7.1.40) were downloaded. Using BLASTP, we identified reciprocal best hits (RBH) between the set of genes in C. thermocellum and the set of genes from UNIPROT annotated to EC 2.7.1.40, using an e-value cutoff of 10-5. The complete list of reciprocal best hits can be found in Additional file C. thermocellum gene that could plausibly perform the function in question. Similar procedures were repeated for all other model subsystems, until we were able to generate a positive flux on the biomass reaction when analyzing the model by FBA. Gaps that were filled using reactions that were not associated with direct genetic evidence were noted. The results of this analysis  to identify missing components. For example, in glycolysis, there was no pyruvate kinase included in the early versions of the model, because none of the existing genome annotations included a gene with this annotated function. However, upon inspection of the glycolysis pathway, this gap became self-evident. Because essentially all of the surrounding reactions  were present and because pyruvate kinase activity has been experimentally observed in mocellum , we decie EC 2.7..40 all sC. thermocellum. Using a colorimetric assay that can detect the activities of various dehydrogenase enzymes, we measured succinate dehydrogenase and lactate dehydrogenase (LDH) activity in both E. coli (control) and C. thermocellum (LDH as a control). We were able to detect LDH activity in both C. thermocellum and E. coli as well as SDH activity in E. coli (data not shown). We were unable to detect SDH activity in C. thermocellum. Based on the absence of genetic and experimental biochemical evidence for this reaction, we elected to exclude the succinate dehydrogenase reaction from iSR432. We did, however, add a reaction for succinate:CoA ligase, based on the fact that succinate has been reported as a metabolic fermentation product of C. thermocellum [We noted during model curation that the citric acid cycle was complete except for two reactions, namely succinate dehydrogenase and succinate:CoA ligase. There was no clear genetic evidence for these genes, although similarity searches produced some probable reciprocal best hits . We then compared model predictions to experimentally observed growth rates and fermentation product secretion profiles of C. thermocellum grown in either continuous [in silico minimal medium used for simulations contained water (h2o), ammonia (nh4), sulfate (so4), phosphate (pi), calcium (ca2), ferrous iron (fe3), hydrogen sulfide (h2s), potassium (k), magnesium (mg2), pantothenate (pnto-r), and nicotinate D-ribonucleotide (nmn). In each of the three simulation conditions , we applied progressively more experimentally determined constraints [To test the predictions of ntinuous  culture.straints . Figure C. thermocellum grew in continuous culture and consumed cellobiose at 2.2472 mmol gDW-1 hr-1 [-1 [-1. Applying an additional constraint for acetate production resulted in a predicted growth rate of ~0.076 hr-1 and greatly improved the prediction for ethanol production. Fermentation products predicted by the model, but for which we had no experimental measurements, included CO2, and H2. To consider the effect of our computational representation of the cellulosome on growth predictions, we restored the cellulosome production requirement back into the objective function. This resulted in a decrease of ~4-17% in predicted growth rates, depending on which constraints were applied. This decrease in the predicted growth rate represents the metabolic impact (burden) of producing cellulosomes in C. thermocellum.Fermentation experiments in which W-1 hr-1  were thehr-1 [-1 . Based ohr-1 [-1 , we set -1[-1. Applying one further constraint for acetate production resulted in a predicted growth rate of 0.095 hr-1. Applying constraints for acetate, lactate, ethanol, and formate production resulted in a predicted growth rate of 0.090 hr-1. Removing the cellulosome production requirements resulted in an increase of ~4-8% in growth rates, depending on which constraints were applied.In the case of continuous growth on fructose, the experimentally measured growth rate was 0.125 hr-1. Experim-1, celluloC. thermocellum grew in batch culture and consumed cellobiose at 12.8 mmol gDW-1 hr-1 [iSR432 predicts a maximum growth rate of 0.518 hr-1, which correlates with high production of H2, CO2, and acetate and very low production of all other major metabolites. As other constraints are added to match observed fluxes, the maximum growth rate, as in other cases, tends to drop, and carbon flux is redistributed to other metabolites. When constraints for acetate, formate, ethanol, H2, and CO2 are added, flux balance analysis predicts a maximum growth rate of 0.387 hr-1 and very low or absent production of lactate and succinate Furthermore, when succinate and lactate production are also constrained, excess carbon is relieved by an increase in the production of aspartate to 8.48 mmol gDW-1 hr-1 , the maximum growth rate gradually drops to below the observed value until in silico growth is impossible with no additional amino acid production.Fermentation experiments in which W-1 hr-1  were thein silico single gene deletions with iSR432, using cellobiose as a carbon source and the other minimal media components (as described above), and constraining cellobiose uptake to its experimentally observed value for batch growth . Among the most vulnerable subsystems are the cell envelope biosynthesis, nucleobase metabolism, and fatty acid metabolism.We conducted comprehensive For each gene deletion predicted to be non-lethal, we conducted flux variability analysis to determine the effect of the deletion on the lower and upper bounds for ethanol secretion. Results are shown in Figure 2 production and ethanol production. A number of studies [C. thermocellum. To investigate this phenomenon, we tested the relationship between H2 escape and the production of other fermentation products during batch growth on cellobiose. Over the entire ranges of hydrogen and ethanol production that would allow in silico growth, we constrained H2 and ethanol escape and used FBA to determine the maximum growth rate. The results are displayed in Figure 2 production yield isIn examining the trends shown in Figure  studies -39 have in silico organism is forced to utilize less efficient pathways to varying degrees. For example, when ethanol and hydrogen escape fluxes are both forced to zero, the carbon flux to ethanol is rerouted at acetyl-CoA to acetate production, and since H2 can no longer act as an electron acceptor for reduced ferredoxin produced by pyruvate:ferredoxin oxidoreductase , acetyl-CoA is instead produced by pyruvate formate lyase , resulting in a drastic increase in formate export  and a different ethanologenic microorganism (S. cerevisiae). We conducted our analysis by comparing the content of the metabolic models for these three organisms, shown in Figure C. thermocellum are commonly associated to pathways such as Starch and Sucrose Metabolism  and Porphyrin Metabolism . EC numbers unique to C. acetobutylicum are commonly associated to pathways such as Butanoate Metabolism (CAC is the only one of the three organisms that natively produces butanol) and Pentose and Glucuronate Interconversions. EC numbers unique to S. cerevisiae are commonly associated to pathways such as Tryptophan Metabolism and Purine Metabolism. Finally, the EC numbers shared by all three models are commonly associated to pathways such as Purine and Pyrimidine Metabolism and aromatic amino acid biosynthesis. No doubt, especially in the latter set, one reason that these particular pathways are well represented is because they are associated with many EC numbers, i.e., they are "big" subsystems. As expected, core pathways of metabolism, such as glycolysis and the pentose phosphate pathway, are also among the most commonly represented in the set of EC numbers common to all three organisms.One consequence of constructing genome-scale models is the ability to obtain a comprehensive overview of an organism's metabolic network. Using our reconstructed network for iSR432 to make context-dependent predictions of growth and fermentation product secretion profiles. Specifically, the additions of 35 metabolites were systematically simulated, both individually and in combinations of two, and flux variability analysis was performed to determine the resulting maximum growth rate and concomitant ethanol production range. This analysis was repeated for each of the reaction deletion strains that were shown to result in an increase in ethanol production  for growth on cellobiose and fructose in continuous and batch culture. Specific results of significance were: 1) the generation of a weighted amino acid representation of a cellulosome based upon proteomic data, 2) suggestions for additional genome annotations and identification of unanswered questions related to metabolism, and 3) identification of a general design principle related to intracellular reduction-oxidation balance that strongly influences the selection of gene deletions and chemical environments to increase ethanol production in C. thermocellum.Current and future demands for renewable energy sources have spurred research in developing biofuels. One promising route for biofuel production is to use an organism-based bioprocess where cellulose could be converted to biofuel. One of the main challenges to this approach is that there are relatively few cellulolytic organisms capable of biofuel production, and none of these are especially well-characterized at present. Here we have implemented a computational modeling approach to study C. thermocellum metabolism. The model accounts for 432 genes and includes 577 reactions involving 525 intracellular metabolites. Some thermodynamic constraints are placed on the metabolic solution space in the form of irreversible reactions. This reconstruction was tested by comparing computational predictions to experimentally measured growth rates and fermentation product secretion fluxes. By including as few as two experimentally determined values (substrate uptake rate and one fermentation-product secretion rate) as model constraints, we find that the model's predicted growth rate closely matches the experimentally observed value for continuous culture growth on cellobiose and fructose [C. acetobutylicum and S. cerevisiae reflects the expected relationships between the organisms; i.e., the two Clostridia models show much more overlap with each other than with the S. cerevisiae model. Shared functionality tends to be concentrated in central metabolism and nucleic acid metabolism, while primary differences tend to occur in how each organism can process starch and sucrose. Significantly, despite a large overlap between the clostridial models, significant functional differences are manifest between the models based on a relatively small number of reaction discrepancies. This points to the utility of constraint based models for making detailed functional predictions based on genome content, as well as the importance of correct genome annotation.We have constructed a genome-scale constraint-based model of C. thermocellum, we developed a weighted amino acid representation of a cellulosome that can be included as a component of the cellular objective required for growth. Experimental evidence suggests that the cellulosome, the extracellular structure responsible for cellulose degradation, comprises as much of 20% of the dry weight of C. thermocellum in certain conditions [In vivo cellulosome production is of course dynamic, however, and the in silico representation is therefore designed to be tunable to available experimental data for different conditions by varying the relative contribution of the mass of the cellulosome to biomass or the ATP input required for cellulosome production. To our knowledge, our inclusion of an amino acid-weighted representation of a cell substructure, i.e. the cellulosome, as part of the biomass equation is the first such formulation used in constraint-based models.In the process of constructing the model for nditions , and thunditions , thus weiSR432, two examples of poorly characterized areas of metabolism in C. thermocellum that require further investigation were identified during the gap filling phase of model construction. These two areas are: the fate of succinate and the possibility of anaerobic respiration in this organism. Genomic evidence suggests an incomplete citric acid cycle in C. thermocellum. In our simulations, we found the citric acid cycle operates in the "forward" direction up through succinate, resulting in production and secretion of succinate. This behavior is not necessarily preferred by our model, unless constraints are placed on the production of other, more energetically favorable fermentation products. Secretion of succinate by C. thermocellum has been previously reported [C. thermocellum, however no activity was found. This leads us to believe that the citric acid cycle in C. thermocellum is not complete and there remains an open question regarding the cellular fate of intracellular succinate. If succinate is not secreted by C. thermocellum, or is secreted in very small amounts, this indicates that there must be intracellular fates of succinate that are not captured by current genome annotations or our current model. Possibilities include further processing via propanoate, glyoxalate, or tyrosine metabolism; though none of these seems likely on the basis of the available genomic evidence.One of the useful features of genome-scale constraint-based models is that they can focus attention on areas of metabolism that are relatively unexplored, or illuminate high-priority areas for future research . In lighreported , althougreported . Thus, iiSR432. We found genomic evidence for NADH-quinone oxidoreductase. While there is no clear genomic evidence for a nitrate reductase, there is a C. thermocellum gene, Cthe_0199, that is a reciprocal best hit for a gene in the prokaryotic molybdopterin-containing oxidoreductase family [C. thermocellum does not reduce nitrate under the conditions studied [C. thermocellum does not possess nitrate reductase or some other similar reaction, it is not clear how reduced quinone generated by NADH-quinone oxidoreductases might be reoxidized and it may be possible that the NADH-quinone oxidoreductase may be incorrectly annotated.Anaerobic respiration is another area of interest highlighted in our studies using e family . There i studied . If C. tC. thermocellum was to create a useful tool for strain design [in silico prior to experimental laboratory work. A simple illustration of this is provided by the results in Figure C. thermocellum and the production of ethanol. This finding was demonstrated more specifically in Figure 2 and ethanol production is shown. As H2 production increases, ethanol production falls, eventually to zero. These results indicate that the capacity of C. thermocellum for ethanol production is strongly influenced by intracellular reduction-oxidation balance. Future strain design work in C. thermocellum likely needs to consider this aspect of C. thermocellum's cellular physiology. This finding also points to the utility of using genome-scale metabolic models to facilitate the strain design process by reducing the workload associated with manually accounting for redox considerations.One of our goals in constructing a model of n design ,22,45, sC. thermocellum. It was found that the addition of lactate or malate to the growth medium of C. thermocellum should induce a marked increase in ethanol secretion for a number of gene deletion strains. These results allude to the utility of specifying environmental conditions as a design parameter for engineering strains. In addition, the change in fermentation production secretion profile occurs naturally in C. thermocellum over the course of the its growth cycle as fermentation products secreted early in fermentation can influence what fermentation products are secreted in the late stages of fermentation.Additional computational analyses were conducted to study the effects of modifying the chemical environment of C. thermocellum in an effort to expedite research on this organism that has a high biofuel production potential. By incorporating genomic sequence data, network topology, and experimental measurements of enzyme activities and metabolite fluxes, we have generated a model that is reasonably accurate at predicting the cellular phenotype of C. thermocellum (at least for the environments shown in this study). The iSR432 model establishes a foundation for the integration and interpretation of large-scale systems biology data sets because the model's predictions can be further refined through the incorporation of thermodynamic constraints, gene regulatory data, and enzyme kinetics and because resulting models can be used for future strain design work involving combinations of gene modifications and chemical environments. Even at the level of studying single gene deletions in combination with chemical additions, analysis using the iSR432 model resulted in an interesting observation that ethanol production is influenced by the reduction-oxidation state of the cell. These observations illustrate the utility of this model as a predictive tool for rational manipulation of C. thermocellum metabolism.In this study, we applied constraint-based modeling to a genome-scale metabolic reconstruction of the cellulolytic, ethanologenic bacterium C. thermocellum, compiled based on evidence from genome annotations and experimental observations. An initial list of biochemical reactions was assembled based on predicted enzymatic functions in the genomic annotations available from IMG, UniProt, and KEGG [C. thermocellum genes were used to select reactions from a set of database reactions, assembled using previously published constraint-based metabolic models [C. thermocellum genome and the Transport Classification Database (TCDB) [C. thermocellum genes and genes in the Transport Classification Database (TCDB) using BLASTP with an e-value cutoff of 10-5, and then added reactions according to mechanisms proposed for the genes from TCDB, where available. This initial assembly contained many broken, incomplete, and/or isolated pathways and subsystems. Furthermore, many reactions involving the synthesis of large, complex molecules, e.g., cell wall teichoic acids and extracellular proteins, were missing from this initial reaction list.The core of the stoichiometric metabolic model is a list of metabolic reactions occurring in and KEGG -27. Specc models ,3,21,46.e (TCDB) . For theC. thermocellum as determined by Chan et al. [To represent the synthesis of large molecules such as phospholipids and cell wall teichoic acid, we created lumped reactions that produced an "average" of that molecular species from a representative fraction of small molecules. For example, to account for the synthesis of 1-acyl-glycerol 3-phosphate, we first created lumped reactions synthesizing necessary fatty acyl-CoA molecules from malonyl-CoA, acetyl-CoA, and other cofactors. Next, we used percent w/w fatty acid content for n et al.  and Herrn et al.  to constn et al. .C. thermocellum's metabolic objective. As in previous work, we represented biomass production by a lumped reaction, with metabolites crucial for biomass formation drained in proportion to their respective requirements [C. thermocellum using experimental data whenever possible. In the absence of experimental data specific for C. thermocellum, we used similar data from the genome scale metabolic model of the gram positive bacterium Bacillus subtilis [B. subtilis no doubt differ from those of C. thermocellum, the results of FBA are relatively insensitive to minor misspecification of biomass requirements [In order to analyze a constraint-based model by FBA, one must specify a metabolic objective. We specified maximization of biomass as irements ,50. The subtilis . Althougirements . The celGPR relationships specify the putative relationship between genes and enzymatic activities in an organism. Following previous work , we reprC. thermocellum, we used the annotations described above and information on protein complexes from UniProt. For each EC number in the C. thermocellum annotations, we searched UniProt [C. thermocellum, conforming to known enzyme complex architecture whenever possible. As a specific example of this process, consider the reaction corresponding to carbamoyl-phosphate synthase ('R_CBPS'), EC 6.3.5.5. Several UniProt entries  with this EC number are annotated as being members of a complex composed of two chains, a small glutamine-hydrolyzing chain and a large chain that synthesizes carbamoyl phosphate. We found two C. thermocellum genes annotated as "carbamoyl-phosphate synthase, small subunit"  and two genes annotated as "carbamoyl-phosphate synthase, large subunit" . Thus, we expressed the GPR relationship for this reaction as: (Cthe_1867 or Cthe_0950) and (Cthe_1868 or Cthe_0949) -> R_CBPS.To develop the GPR relationships for  UniProt  to deterC. thermocellum is one of a number of cellulolytic microorganisms that breaks down cellulose via a large , multi-functional polypeptide assembly of extracellular enzymes and scaffold proteins called the cellulosome [C. thermocellum varies considerably across specific conditions. For example, the amount was found to be ninefold greater in Avicel-grown batch cultures compared to cellobiose-grown batch cultures [iSR432 includes a means of including an additional "cellulosome" requirement in the biomass objective. A lumped reaction was created to represent the formation/export of the cellulosome from the constituitive amino acids and using ATP. The pseudometabolite "cellulosome," representing the outcome of this reaction, can be optionally added to the "cellmass" fraction of biomass discussed above to form total biomass. The molar coefficients of each amino acid to form a single "mole" of cellulosome were determined based on a proteomics study of the C. thermocellum cellulosome, which produced the relative abundances of individual peptide constituents of the cellulosome [lulosome . The amocultures . Cellulocultures , and thilulosome . These dlulosome .i" to denote "in silico," followed by the initials of the first author ("SR"), followed by the number of genes included in the model ("432").We followed the previously established convention  for namiC. thermocellum (ATCC 27405) was cultured anaerobically at 55°C in serum bottles sealed with butyl rubber stoppers and purged using ultra high purity nitrogen gas. Standard growth medium for C. thermocellum contained in final concentrations 5 g/L cellobiose, 3 g/L sodium citrate tribasic dihydrate, 1.3 g/L ammonium sulfate, 1.43 g/L potassium phosphate monobasic, 1.8 g/L potassium phosphate dibasic, 0.13 g/L calcium chloride dihydrate, 6 g/L glycerol-2-phosphate disodium, 4.5 g/L yeast extract, 2.6 g/L magnesium chloride hexahydrate, 1.1 mg/L ferrous sulfate heptahydrate, 0.5 g/L cystein-HCl, 50 mM MOPS buffer, and 0.1% resazurin. Escherichia coli was cultured aerobically at 37°C in M9 minimal growth medium containing 2 g/L glucose.C. thermocellum was tested using a modification of the method used by Kun and Abood [E. coli) or 50°C (C. thermocellum) water bath for 20 minutes. Tubes were removed, and the reaction was stopped by adding 7 mL acetone and shaking well. A blue color indicated reduction of the tetrazolium dye, and thus, the presence of the query enzyme activity.The enzymatic activity of succinate dehydrogenase in nd Abood  in whichC. thermocellum genome-scale constraint-based model using FBA [Once constructed, we analyzed the sing FBA . In esseMaximize: ZS • v = 0,Subject to: ai <vi <bi for all reactions i,v is the vector representing the flux values of each reaction in the model (the flux distribution) [i = bi = 0) or other experimentally-determined flux values. If no information was available, the upper and lower flux bounds were set to 1000 and -1000 mmol gDW-1 hr-1, respectively. The stoichiometric matrix S is the mathematical representation of the reaction list. It has one row for each metabolite and one column for each reaction. Each element of S, sij, represents the stoichiometric coefficient of the ith metabolite in the jth reaction (the coefficients are positive when the metabolite is a product of the given reaction and negative when the metabolite is a reactant). The vector v contains the flux values for all reactions in the model, i.e., vi is the flux on the ith reaction.where Z is the flux on the biomass reaction, S is the stoichiometric matrix (discussed below), and ibution) . The firin silico gene essentiality predictions [in silico gene knockout on reaction activities is assessed. If the gene is crucial to the activity of a reaction in the model, then that reaction is constrained to have zero flux to simulate the effect of the gene deletion. Finally, the model with this new constraint is analyzed by FBA, maximizing the growth objective for batch growth on cellobiose. If the maximum flux on the biomass reaction is zero, then the deleted gene is predicted to be essential. If the maximum flux on the biomass reaction is greater than zero, the deleted gene is predicted to be nonessential. All constraints are reset to their default values, and the process is repeated for the next gene in the model.Constraint-based models can be analyzed by FBA to make comprehensive dictions . For eacFBA is guaranteed to produce an optimal solution to the linear programming problem stated above. However, there generally are many optimal solutions, i.e., different flux distributions that give the same optimal objective value ,57,58. IC. acetobutylicum (represented by two models: Cac1 [C. thermocellum (Cth), and S. cerevisiae (Sce) [C. thermocellum and S. cerevisiae models, this reaction is written as,We compared genome-scale constraint based models for three key organisms in ethanol production, ls: Cac1 ,20 and Cls: Cac1 ) C. therae (Sce) . Direct atp + glc-D --> adp + g6p + hwhile in the Cac1 model, this reaction is written as,ATP + beta-D-Glucose < = > ADP + beta-D-Glucose 6-phosphateand in the Cac2 model, the reaction is written,bDG6P + ADP < -> ATP + bDGLC.C. thermocellum and S. cerevisiae models, while the opposite is true in the Cac1 and Cac2 models. In addition, the Cac1 and Cac2 models distinguish between different metabolite anomers  while the others do not. For these reasons and others, we decided to use EC numbers to compare the contents of each model. EC numbers have the advantage that they are relatively stable over time, represent international consensus on biochemical reaction categorization, and can overcome ambiguities regarding how to represent a given metabolic reaction in a constraint-based model.Aside from the different metabolite identifiers, reactions in the Cac1 and Cac2 models are written using the "biochemical" convention, as opposed to the charge-balanced "chemical reactions" used in the others. Usually, this difference is manifested in whether hydrogen is represented as a distinct metabolite in a reaction or not (see above). However, it also results in other differences, e.g., the metabolite NH4 is used instead of NH3 in the Conventions in assigning EC numbers to reactions can vary. In the Cac1 and Cth models, a single reaction could be associated with multiple EC numbers while only one EC number was assigned to each reaction in the Cac2 and Sce models. To ensure that the comparison was uniform, the lists of EC numbers used to represent Cac1 and Cth were compared with the other models, and EC numbers unique to these models (Cac1 and Cth) were only kept if no other EC number for that same reaction occurred in the other models.There are also no universally adhered to guidelines for assigning reactions (and thus their associated EC numbers) to subsystems. To minimize the impact of this on our EC based model comparisons, we attempted to use KEGG as an "external standard." Specifically, for each EC number encountered in any of the models, we assigned all pathways associated to that EC by KEGG.This EC-based method of comparison necessarily excludes reactions not linked to an EC number. This means that significant numbers of reactions from subsystems such as extracellular transport and cell envelope biosynthesis are not a part of the comparison. However, the EC-based comparisons do cover most reactions in all four models. Excluding transport reactions, the percentages of reactions not associated with an EC number are 8%, 17%, 12%, ~5%, for the Cth, Sce, Cac1, Cac2 models, respectively.iSR432 was used to simulate the addition of an arbitrary amount of various common metabolites in order to determine the effect on ethanol productivity and yield during batch growth on cellobiose. To simulate an added metabolite, an exchange flux for that metabolite was allowed with the constraints -20 mmol gDW-1 hr-1 ≤ v ≤ 0 mmol gDW-1 hr-1. This reflects the thermodynamic prevention of any production of that metabolite and allows the use of it as a source within a reasonable range. For each alternative media formulation, FVA was performed for the ethanol exchange flux and the maximum ethanol flux was used along with the predicted maximum growth rate to calculate the ethanol:biomass yield.SBR constructed the metabolic model, conducted computational analyses, and drafted the manuscript. CMG participated in model construction and computational analyses. JPB participated in model construction and developed computational algorithms for model construction and analysis. SSF conceived of the study and participated in the design and coordination. All authors read and approved the final manuscript.Model details and results. This Excel workbook contains the final reaction list, metabolite definitions, biomass function, cellulosome reaction formulation, and gene-protein-reaction (GPR) relationships. In addition, it contains the complete list of results for the reciprocal best hits (RBH) of EC numbers missing from the current C. thermocellum annotations, and the complete results from the single gene- and reaction-deletion simulations and the model comparisons to growth on cellobiose and fructose.Click here for fileThis model file includes tab delimited GPR relationships, reaction list, and sources and escapes for growth on minimal media.Click here for fileThis file contains the model in SBML format.Click here for fileRemoved futile cycles. This results file shows reactions which were deleted from the original model build to remove futile cycles.Click here for fileCalculations and additional methods. This file contains detailed methods and sample calculations for determining the biomass objective reaction and the fatty acid content reactions as well as details about simulations of alternative media formulations.Click here for file


Recent research shows that a faulty or sub-optimally operating metabolic network can often be rescued by the targeted removal of enzyme-coding genes – the exact opposite of what traditional gene therapy would suggest. Predictions go as far as to assert that certain gene knockouts can restore the growth of otherwise nonviable gene-deficient cells. Many questions follow from this discovery: What are the underlying mechanisms? How generalizable is this effect? What are the potential applications? Here, I approach these questions from the perspective of compensatory perturbations on networks. Relations are drawn between such synthetic rescues and naturally occurring cascades of reaction inactivation, as well as their analogs in physical and other biological networks. I specially discuss how rescue interactions can lead to the rational design of antagonistic drug combinations that select against resistance and how they can illuminate medical research on cancer, antibiotics, and metabolic diseases. When the rats were deprived of a specific fat in their diet, their body cells compensated by overproducing it. (…) So, maybe, we could stop Lorenzo's body from producing saturated C24 and C26 by loading his diet with another kind of fat… you know, one that is less… is less harmful,” said the character of Susan Sarandon in the true-story movie Lorenzo's Oil.“decentralized and enjoy a certain level of redundancy.Potentially many more than previously anticipated. Contrary to what common wisdom may suggest, most cellular functions are carried out by the coordinated activity of multiple interacting elements, including genes, proteins, and biochemical reactions.in silico studies of the metabolism of Escherichia coli. For E. coli fully adapted to arabinose minimal medium, for example, the knockout of gene fbaA is observed to be lethal, owing to the inability of the mutant strain to metabolize building blocks of the biomass, such as phenylalanine, tyrosine, and l-lysine  implies that it will generally respond to perturbations in a non-optimal manner with respect to any  objective function. Cells will, in particular, not grow as fast as they could under the constraints imposed by a perturbation even if the pre-perturbation state is optimal. The fact that the network is redundant  implies that the activity of certain pathways, such as glyoxylate in the example above, can be constrained without compromising cellular function. More important, by doing so, fluxes can be forced to be routed through pathways that upon perturbation, such as the knockout of gene e.g., “optimal growth”); second, regard reactions with fluxes much larger than the desired ones as candidates for  knockouts. The knockout of one or few such reactions, implemented through the knockout of the corresponding enzyme-coding genes, often brings the entire system closer to the desired state. This is precisely the criterion used to design the synthetic rescue of the fbaA-deficient mutant described above. Although this is admittedly a simplified description, it is not necessarily model dependent as it allows for implementations based on experimental flux measurements using, for example, increasingly accurate 13C-tracer techniques.Synthetic rescues can be identified through a fairly general two-step procedure: first, identify the actual reaction fluxes and the reaction fluxes that correspond to the desired metabolic states  of the reaction fluxes. Complications due to unknown regulatory mechanisms and kinetic parameters are avoided by focusing on steady-state dynamics and describing the response to perturbations as transitions between steady states. The steady-state approximation is generally appropriate to describe the short-term behavior of individual cells as well as the average long-term behavior of large populations of cells. For a network with n fluxes and m species, the steady-state solutions are determined byS = (Sij) is the stoichiometric matrix accounting for the network structure. This matrix typically consists of hundreds or thousands of reactions and a large but smaller number of biochemical species, rendering Eq. νj− ≤ νj ≤ νj+ to model, for example, constraints imposed by thermodynamics, availability of nutrients, and maximum reaction rates.νj± = 0 is used for uptake reactions of nutrients not available in the medium and νj− = 0 for reactions that are irreversible. The resulting system is still underdetermined, as expected since cell regulation is not explicitly incorporated into the model. Phenomenological methods such as flux balance analysis (FBA),νj± = 0 for the corresponding reactions. MOMA provides a solution compatible with the constraints imposed by the perturbation that is closest to the original metabolic state in the space of fluxes, while ROOM minimizes the number of significant flux changes. Thus, a perturbed metabolic network will generally depart from its optimal states.Metabolic networks can be modeled using The observation that optimal states activate fewer reactions than non-optimal ones bears a strong relation to the rescuing effect of targeted reaction knockouts. The top candidates for rescue knockouts are precisely the reactions that are active in non-optimal states but predicted to be inactive in the desired state see Fig. . It is iE. coli or growth rate functions, but is in fact a general behavior also found in other organisms and for many other linear functions of fluxes. Even more compelling, for optimal states under growth conditions, the number of active reactions was found to be ∼300 for all typical linear functions and all four organisms – Helicobacter pylori, Staphylococcus aureus, Escherichia coli, and Saccharomyces cerevisiae – studied in detail by Nishikawa et al.in silico model and with the caveat that the model ignores unknown side effects.What is the theoretical maximum number of reactions that can be knocked out? This question has a surprisingly simple and general answer for each fixed nutrient condition. The key observation that a reduced number of reactions are recruited by optimal states is not unique to φ, as illustrated schematically in Fig. φ1 of this function that is smaller than its optimum φ. By doing so, the new best state available may change to a value φ2 larger than φ1. Such an improvement is expected to be possible in general by constraining (rather than augmenting) the dynamics on the network and, in particular, by removing nodes and edges as in the case of reaction knockouts. This intervention only exploits resources and states originally available in the system. Accordingly, I refer to this intervention as a restraint-based approach to control the network response.In all likelihood, these are instances of a general and ubiquitous behavior, which we are now only starting to understand. This behavior can be conceptualized in terms of a fitness landscape for a given objective function φ have to be eliminated for this to be achieved. In fact, states in which most fluxes are zero or close to zero always exist simply because they are always solutions of Eq. In the specific case of cellular metabolism, if the reaction rates are constrained to (but not less than) their optimal values,Eq.  Box .partially knocking out  the genes, and hence reactions, that are found to be over-expressed in the observed non-optimal states . Therefore, the systematic study of synthetic rescues may add a new dimension to microbial optimization efforts, a field in which gene knockouts have been previously used to optimize production of specific metabolic compounds as byproducts of adaptation-induced growth optimization.46Synthetic rescue interactions improve fitness at the expense of robustness. The rescued organisms will grow faster in the environmental conditions under consideration but will (presumably) lose flexibility to adapt to other conditions. For many applications, however, such as in the microbial production of compounds of industrial interest, this is a secondary problem. The primary problem is to improve tolerance to the specific industrial environment  function of interest. But the study of synthetic rescuesAs argued here, the response of a decentralized complex network can be largely controlled and optimized by constraining its structure and/or dynamics or, more generally, the resources available in the system. Moreover, biological networks appear to have evolved to operate with a mechanism of selection These compensatory perturbations are by no means evident from the structure of the network, highlighting the importance of a modeling and experimental framework that can account for both the nonlinear and the system-level nature of the network response to perturbations. Moreover, because synthetic rescues involve the inactivation of two (and often more) genes, it is imperative to develop methods to systematically study the response of cellular networks to multiple perturbations. The expected results of this research are very promising, as they will address outstanding questions about collective gene interactions and potentially lead to new approaches for drug development that, like in the story of Lorenzo's oil, involve interactions that challenge common sense. The latter can have direct implications for medical research related to metabolic diseases and, combined with current studies on antagonistic drug interactions,Therefore, whether you call it systems biology, integrative biology, or network biology, it is clear that we need more of it.


Metabolic networks perform some of the most fundamental functions in living cells, including energy transduction and building block biosynthesis. While these are the best characterized networks in living systems, understanding their evolutionary history and complex wiring constitutes one of the most fascinating open questions in biology, intimately related to the enigma of life's origin itself. Is the evolution of metabolism subject to general principles, beyond the unpredictable accumulation of multiple historical accidents? Here we search for such principles by applying to an artificial chemical universe some of the methodologies developed for the study of genome scale models of cellular metabolism. In particular, we use metabolic flux constraint-based models to exhaustively search for artificial chemistry pathways that can optimally perform an array of elementary metabolic functions. Despite the simplicity of the model employed, we find that the ensuing pathways display a surprisingly rich set of properties, including the existence of autocatalytic cycles and hierarchical modules, the appearance of universally preferable metabolites and reactions, and a logarithmic trend of pathway length as a function of input/output molecule size. Some of these properties can be derived analytically, borrowing methods previously used in cryptography. In addition, by mapping biochemical networks onto a simplified carbon atom reaction backbone, we find that properties similar to those predicted for the artificial chemistry hold also for real metabolic networks. These findings suggest that optimality principles and arithmetic simplicity might lie beneath some aspects of biochemical complexity. Metabolism is the network of biochemical reactions that transforms available resources (“inputs”) into energy currency and building blocks (“outputs”). Different organisms have different assortments of metabolic pathways and input/output requirements, reflecting their adaptation to specific environments, and to specific strategies for reproduction and survival. Here we ask whether, beneath the intricate wiring of these networks, it is possible to discern signatures of optimal  pathway architectures. A systematic search for such optimal pathways between all possible pairs of input and output molecules in real organic chemistry is computationally intractable. However, we can implement such a search in a simple artificial chemistry, which roughly resembles a single atom  version of real biochemistry. We find that optimal pathways in our idealized chemistry display a logarithmic dependence of pathway length on input/output molecule size. They also display recurring topologies, including autocatalytic cycles reminiscent of ancient and highly conserved cores of real biochemistry. Finally, across all optimal pathways, we identify universally important metabolites and reactions, as well as a characteristic distribution of reaction utilization. Similar features can be observed in real metabolic networks, suggesting that arithmetic simplicity may lie beneath some aspects of biochemical complexity. The prominent role of metabolism in any biological process and the fact that a large portion of the environmental factors shaping living systems are ultimately metabolic in nature, suggest that strong selective forces have been acting on metabolic networks throughout the history of life. In laboratory evolution experiments In a 1961 review, Baldwin and Krebs suggested that biochemical network topologies may reflect the adaptation toward optimally efficient metabolic strategies, and that manifold use of certain molecules may be a crucial element of this adaptation, as “it is indeed a general principle of evolution that multiple use is made of given resources.”In the present work, we combine the study of an extremely simple artificial chemistry N possible molecules {a1, a2, a3, …, Na}, that can participate in reversible ligation/cleavage reactions of the form i + aja ↔ ka, with i+j = k. This model could be viewed as the simplest possible string-based artificial chemistry NR that includes all metabolites up to length N and all possible reactions  from a single available nutrient . We define an optimal pathway as one that satisfies the following conditions: (i) it allows a steady state solution, i.e., a mass-conserving flow from input to output; (ii) it has maximal yield, and no waste out = vin·j/iv; and (iii) it has the fewest reaction steps possible. A pathway satisfying these conditions is termed a minimal balanced pathway (MBP) between ia and ja, and will be denoted ia ⇒ ja. MBPs  can be thought of as the pathways that are most efficient for a specific metabolic task, in the sense that they require the smallest possible number of different enzymes for producing the maximal possible yield Our artificial chemistry consists of a set of NR is a challenge for large N. We implement three algorithms to approach this problem: a mixed integer linear programming (MILP) akin to flux balance analysis (FBA) Despite the simplicity of our artificial chemistry, identifying the MBPs between all possible input-output pairs in a given artificial chemistry Behind the apparent complexity of the topologies encountered in each of the different pathways, it is possible to observe the recurrence of three fundamental categories: each MBP functions either as a pure “addition chain” a9 ⇒ a6 can be performed in 2 steps, but the neighbor task a9 ⇒ a7 requires a minimum of 6 steps. Moreover, while most MBPs have only one or a few optimal realizations, selected instances display a peak in possible redundant solutions (xa ⇒ xa+1), or to the inherent complexity of a specific molecule . These regular patterns suggest that it may be possible to reproduce the MBP length curves without having to actually compute the MBPs.This modular architecture of recurring graph types provides a topological signature of optimally efficient pathways in our idealized chemistry. Since these pathways are chosen based on their minimal length, one may expect that a systematic analysis of all MBP lengths will display additional distinctive properties. Indeed, pathway lengths increase roughly logarithmically with the size of the input (or output) molecule  and S3, olutions , usuallyi-th entry is either the sum or difference of any two previous entries in the sequence. These chains are often used in calculating large exponents of numbers n128 can either be performed in 127 multiplications  or in a chain of 7 exponent multiplications . The latter can be further simplified by tracking the sums of the exponents in each calculation, which form an addition chain . Shortest addition-subtraction chains are commonly used to calculate very large numbers in the fewest number of steps, thus speeding up computation time. These are often applied to methods in cryptography where the calculated exponents can have on the order of thousands to tens of thousands of bits A similar search for patterns associated with minimal steps had been previously encountered in the mathematics of addition-subtraction chains, of high importance in cryptography a128 from a1 is formally equivalent to the addition chain example described above. However while typical addition-subtraction chains start with the number 1, in our MBPs we explore minimal paths starting from any molecule ia (i≥1). As described in detail in the L is the number of reactions in the MBP with input ia and output ja, and gcd is the greatest common divisor of i and j. As seen in The pathways explored in our model resemble optimal addition-subtraction chains. For example, the problem of obtaining 6H14O12P2) into dihydroxyacetone phosphate (C3H7O6P) and glyceraldehyde-3-phosphate (C3H7O6P), can be mapped onto a carbon atom backbone, becoming simply C6 ↔ C3 + C3 , the specific peaks and valleys of the predicted function are closely followed by the E. coli network. While this does not prove that MBPs are indeed used in real metabolic networks, it suggests that the logarithmic strategy of MBPs is embedded in their architecture. However, because this second method focuses on shortest paths across a metabolite-to-metabolite network rather than on flow-conserving MBPs, the predicted values are likely an underestimate of the number of reactions necessary to construct one metabolite from another in a mass-conserved manner.The second method is aimed at identifying all shortest pathways between any two carbon compounds in the whole genome-scale metabolic network of  results  and S3 sE. coli metabolism. However, some fundamental aspects of the architecture of metabolism may be visible only at the ecosystem-level, namely by collectively analyzing the metabolic network obtained as the union of the metabolic maps of known individual species . For example, previous work using an algorithm of network expansion applied to this meta-metabolome has identified potential signatures of major evolutionary events So far, we have analyzed the properties of individual MBPs in our idealized chemistry, as well as analogous minimal length pathways in ai ⇒ aj corresponds to a different organism, which has filled a specific metabolic niche (availability of ia), and found an optimal solution (the MBP) for its main metabolic task (produce ja). The question we ask next is whether, in this ecosystem of MBPs, all metabolites and reactions are used in roughly the same number of pathways, or if specific metabolites or reactions seem to be essential for many optimal tasks, hence representing “universal tools”.Here, we build a meta-metabolome for our idealized chemistry by considering the collection of MBPs. One could imagine that each task R19 network using the MILP method. One first result of this analysis is that every metabolite of an even length is used in many more MBP reactions than their odd length neighbors, compared to the underlying chemistry . In addition, this same pathway, with one additional reaction, can also be used to optimally produce a9 and a10 . This value is close to our theoretically predicted value of −1  that perform elementary metabolic tasks with maximal yield and minimal length in an idealized chemistry. Using the results from the model chemistry, we asked whether similar signatures of optimally efficient organization could be found in real metabolic networks.R100 network, though in this case the ensuing pathways are not of minimal length , or the dependence of degeneracy on the numerical properties of input/output pairs. One of our algorithms (the elementary modes one) can find a large number of degenerate solutions, including autocatalytic cycles. This algorithm is currently not scalable, because of the difficulty of computing elementary flux modes, especially in the highly connected artificial chemistry network we have used, though very recent improvements in elementary flux mode calculations ngth see . In any ngth see .7a to 8a) are simultaneously optimal for multiple metabolic tasks. In this specific case, the MBP 7a ⇒ 8a is also an MBP for the production of each intermediate in the cycle , to the complex interdependence of multiple elements, and to the properties of stability of intermediates.To determine whether predictions of minimal MBP length in our idealized chemistry could have implications in real biochemistry, we searched for pathways of minimal length between compounds with different counts of carbon atoms in the 6 compound to C5, with a yield of 5/6 ∼83% and an additional C1 byproduct; yet, obtaining maximal yield in this transformation, would cost at least 3 more reactions. One could argue that the combination of all of these criteria, including the maximization of ATP production and optimization of enzymatic catalysis may have played a key role in the evolution of modern metabolism, leading to compromise solutions. Exploring pathways that produce multiple compounds from multiple inputs with the addition of thermodynamic constraints might constitute an interesting model extension for further investigation.Previous work had addressed the question of optimality in specific metabolic pathways. For example, Meléndez-Hevia and Torres We found that the statistical properties of the usage of reactions across MBPs recapitulate the statistics of reaction usage in the union of all known metabolic pathways (represented by the KEGG metabolic database). Both across the set of all MBPs for the idealized chemistry, and in the KEGG metabolic map, we observed that a few reactions are used far more often than many of the others in the set. Another way of determining the importance of individual reactions in the context of the global functionalities of a meta-metabolome would be to perform perturbation experiments. We implemented such an experiment in our idealized chemistry, by progressively removing reactions and checking how many metabolites can still be produced. Depending on whether reactions are removed in random order or in the order determined by their usage across MBPs, the outcome is quite different . This anDetermining to what extent real metabolic networks obey optimality principles like the ones described here will take additional effort. Even if an underlying arithmetic simplicity governs idealized optimal pathways, deviations from ideal behavior should be expected. For example, parallel selection pressures for energy production and biochemical stability would likely sacrifice pathway minimality. However, guiding principles as the ones we are proposing could serve as reference points for future research, including circumstances in which metabolism can be different from what we are used to. Using synthetic biology techniques, for example, it might be possible to redesign metabolic pathways so as to approach predicted ideal efficiencies and minimal enzyme cost N) of a monomeric unit a. Since no specific assumption is made in the model about the nature of these molecules, they could equally represent aggregates or branched polymers of different sizes, as well as molecules with different counts of a specific atom. A network NR  = {NM, NC} is defined by the set of N molecules NM  =  {ia | ∀ i  = 1,…,N} and the set of all possible uni-bi ligation/lysis reactions between them, NC  =  {ia + ja ↔ ka | ∀ i, j, and k, such that i≤j, i+j = k, and k = 2,…,N}. The number of reactions in NC can be shown to grow quadratically with N. Since each reaction describes how to reversibly combine two molecules to make a larger one, there is a fixed number of ways to produce any given metabolite ia, equal to ⌊i/2⌋. The number of reactions in NC is then given by N2/4.We define an artificial chemistry inspired by previous string-based artificial chemistries  that participate in reaction j  . Each reaction can be associated with a flux, jv. Under the assumption of a steady state the following set of mass conservation constraints on the fluxes is generated:Flux Balance Analysis (FBA) is a steady state constraint-based approach to study the flow of mass through metabolic networks jα is the minimal allowed rate of a reaction and jβ is its maximal rate. Taken together, the above constraints define a convex polyhedron (the “feasible space”) in the n-dimensional space of fluxes. Linear programming (LP) can be used to identify, within the feasible space, flux vectors that maximize or minimize a given linear objective function. In microbial systems it has been often hypothesized that a biologically meaningful objective is the maximization of the flux through the reaction that represents cellular growth, or biomass production Saccharomyces cerevisiae SC288 E. coli K-12 Additional constraints  can be imposed on the fluxes as inequalities of the formNR network that can optimally perform a given metabolic task. A task is defined as the production of a specific end product  from a single available nutrient . A pathway between two molecules is a MBP if (i) it satisfies a steady state solution, analogous to Eq. (1); (ii) it produces the final product with maximal yield, i.e., out = vin·j/iv; and (iii) it contains the smallest possible number of reaction steps. The MBP between ai and aj will be indicated as ai ⇒ aj.Minimal Balanced Pathways (MBPs) are defined as sets of reactions in the We have developed three different algorithms for computing MBPs, as described below:out = vin·y/xv for the MBP xa ⇒ ya be satisfied. We then search for a solution that minimizes the number of active (nonzero) fluxes. Towards this goal, we use a modification of the LP problem described above to introduce binary variables (jb) that represent flux activity: jb = 0 if jv = 0, and jb = 1 otherwise. To identify a minimal path, we can then search for the set of fluxes that minimize Σibi. Because of the nature of the variables involved – the fluxes are continuous, and the number of active fluxes is an integer – this problem must be solved using a mixed integer linear programming (MILP) algorithm. Our MILP problem for the optimal MBP satisfying xa ⇒ ya is formulated as follows:We use a modified FBA approach to formulate the MBP problem in a constrained optimization framework. Specifically, we impose the same constraints used in an FBA problem, and further require that the maximal yield condition v that uses the fewest nonzero values to maximize the objective. In our MBP computations, the only flux constraints used were those that limit the uptake of the single nutrient to an arbitrary value of 10 mmol·gDW−1·h−1, and the production of the target metabolite to the known maximal yield out/vin = j/iv.The optimal solution for this problem will give the flux distribution S (as described in the above FBA section), a vector of fluxes v is said to correspond to an elementary flux mode (EFM) if it satisfies the following three conditions Given a metabolic network defined by a stoichiometric matrix Sv = 0).It satisfies the steady state condition , and a1 ⇒ a2 . To compute a1 ⇒ a3, we identify all the ways in which we can decompose 3 into two smaller addends . Next we combine together the previously computed MBPs that progress from a1 to each of these two addends, giving a new putative MBP for the desired new task . This procedure can be then iterated to give a prediction of MBP ia ⇒ ja .We designed and implemented an algorithm to produce most MBPs R100 network. However, it has two main drawbacks. First, it will miss pathways that “overshoot” the target value then subtract down to it. Second, it may miss MBPs that are not built modularly from smaller ones. From a comparison of the MBPs predicted by the different algorithms, one can see that the approximations introduced in this algorithm cause 18 out 361 MBPs (5%) in R19 to overestimate pathway length by one reaction. Also, this algorithm correctly identifies 204 of the 384 degenerate MBPs that the EFM algorithm finds in R10. The reaction usage using this method is highly correlated with that of the MILP method applied to R19 , and the EFM method applied to R10 .This algorithm is fast and efficient compared to the previous methods, allowing us to apply it to even the NR metabolic network and real metabolism was gathered from the KEGG LIGAND database 3 ↔ C3). These reactions were ignored as well, without consequences on the results (data not shown).Data used for the comparison between the a9 to a10 in a9 participates in only one reaction, but a10 participates in two.We counted how often each metabolite and reaction was used in the artificial chemistry pathways as well as in the KEGG-derived single-atom networks. In the model pathways, reaction usage was calculated by counting how many times each reaction was used across all pathways. Metabolite usage was similarly calculated by counting the occurrence of reactions in which each metabolite participates. For example, in the pathways that convert In the KEGG-derived networks, a similar counting scheme was used. The reaction usage was calculated by counting how many times each reduced reaction appears, and the metabolite usage was calculated by counting how many times each metabolite appears across all reactions.Escherichia coli4). Also, we effectively ignored reactions involving transport and exchange. We then used the EFM method described above to find the number of reactions in each of the MBPs for this reduced network.The first method used EFMs to find all shortest pathways in the central carbon metabolism of j of carbons. For each j, we select among these paths the shortest one, giving an estimate of the shortest path between any individual compound and the closest j-carbon compound. Finally, for each value of i, we averaged these path lengths over all input molecules with i carbons. The end result is a matrix that provides the average of the shortest paths from any i-carbon compound to its nearest j-carbon compound.For each input compound, we listed the lengths of the MBPs for all output compounds containing a number E. coliThe larger, genome-scale metabolic network of As described above, we are only interested in the connections between carbon compounds, so we removed any non-carbonaceous metabolites . Also, we removed the following cofactors that are used in many reactions, but do not participate in the transformation of carbons: ATP, ADP, AMP, NAD+, NADH, NADP+, NADPH, coenzyme A, acetyl-CoA, and the acyl carrier protein.E. coli's metabolic network. This set of pathways was collated as in the previous section, to produce a matrix of the average of the shortest paths from any i-carbon compound to its nearest j-carbon compound.Next, we used Johnson's all-pairs shortest paths algorithm (available as a Matlab function) to find shortest pathways between any two carbon compounds in ia ⇒ ja. We begin with a simplified version of the artificial chemistry model in which only irreversible addition reactions of the formja from 1a. We shall denote by l(j) the smallest possible number of such reactions (we count the use of each reaction (5) once). This problem is equivalent to the problem of addition chains l(j) grows logarithmically with j:We developed an analytical approximation for the expected numbers of reactions to be found in any MBP i can be used to produce an output metabolite of any length j. If we still assume that only addition reactions are possible (i.e. molecules cannot be broken down), a chain from ia to ja will exist only when i is a divisor of j. The problem can then be reduced to the case with a1 input and jai/ output. Therefore, in the irreversible case, we can assume that inputs consist of monomers without loss of generality. Let L(j) be the length of the shortest reaction chain in this case. Because not every reactant exists when dividing by the input length i, we have the obvious inequalityOur artificial chemistry represents a generalization, in which a metabolite of any length 0, 21, …, 2k} is obviously the shortest chain from 1 to j = 2k whose length is k+1. This suggests the general lower bound on the shortest length L(j) of the addition chain:x⌉ represents the ceiling of x, or the smallest integer not less than x. Likewise, as seen below, ⌊x⌋ represents the floor of x, or the largest integer not greater than x . The longest minimal addition chain arises when the output length is j = 2m−1. From this fact, we have the upper bound υ(j) is the number of 1s in the binary representation of j. Since Sometimes the shortest chain can be found easily. For instance, {2L(16) = 5, and L(17) = L(18) = 6 for both the lower and upper bounds, while L(31) = 8 is between the lower bound and the upper bound .The above bounds give precise values in some cases and act as bounds in others. For instance, L(j); one of the most famous L(j) is NP-hard. Nonetheless, the computation of L(j) has been pushed up to n≤225. Two other conjectures There are various conjectures regarding a1.While algorithms for generating the shortest addition chains are discussed by Thurber l of the shortest reaction chains (MBPs) with ia input and ja output. Addition-subtraction chains have also been studied previously as an expansion of addition chains, although these correspond to MBPs with only a1 as an input. Sometimes, in these cases, l(j) is readily computable, e.g.k. Both lengths can also be equal, i.e. l(j)  =  L(j). For example,We are interested in the general case involving both addition and subtraction, and specifically the lengths Note also an inequality:All of these features explain the growth law in equation (6).l has a rich behavior, e.g., there is only a trivial lower bound since l = 1. To ignore this non-interesting effect, let us divide i and j by their greatest common divisor as it never affects the length of the MBP:The quantity ia to produce ja:Recalling (6) we finally arrive at an approximation for the number of reactions in an MBP that uses ja from ia. For each  pair, take an MBP and mark all reactions. Let the reaction in (5) occur pqE times: that is, there are pqE MBPs that use (5). We now divide pqE by the total number of MBPs and call p, q) but to their ranking j, so that the reaction of rank j = 1 is the most frequent, that of rank j = 2 is the second in frequency, etc. This gives je. How does je decrease with rank? To infer the answer we note thatThe approximation in (19) can also be used to estimate the rank distribution of reaction usage. Consider all possible MBPs producing l〉 of the shortest reaction chain scales as log N. This is consistent with (20) if and only if we have jr ∼ j−1. Thus we predict the power-law decay in (21).From (19) it is clear that the average length 〈Figure S1A version of (2.81 MB TIF)Click here for additional data file.Figure S2Heatmaps describing MBP length and degeneracy(0.02 MB PDF)Click here for additional data file.Figure S3Logarithmic trend of metabolic pathway length(0.04 MB PDF)Click here for additional data file.Figure S4Metabolite and reaction usage frequencies for KEGG hydrogen, nitrogen, and oxygen sets(0.09 MB PDF)Click here for additional data file.Figure S5Fast Fourier transforms showing periodicity of metabolite usage(0.02 MB PDF)Click here for additional data file.Figure S6KEGG carbon reaction and metabolite usage with cofactors removed PDF(0.04 MB PDF)Click here for additional data file.Figure S7Pathway length comparison between iterative algorithm and algorithm minimizing absolute values of fluxes(0.06 MB PDF)Click here for additional data file.Figure S8Reaction usage is similar between R10 EFM and MILP results(0.02 MB PDF)Click here for additional data file.Figure S9Removing reactions from the R19 network affects the system's ability to yield balanced pathways(0.04 MB PDF)Click here for additional data file.Table S1Comparison of the three different MBP search algorithms(0.04 MB DOC)Click here for additional data file.Table S2List of MBPs found in the R19 network(0.07 MB XLS)Click here for additional data file.Table S3List of all degenerate MBPs found in the R10 network(0.07 MB XLS)Click here for additional data file.Table S4Metabolite and reaction usage from MBPs in the R19 network(0.03 MB XLS)Click here for additional data file.Table S5Metabolite and reaction usage from MBPs in the R10 network(0.02 MB XLS)Click here for additional data file.Table S6Metabolite and reaction usage from the KEGG carbon dataset(0.08 MB XLS)Click here for additional data file.Table S7Metabolite and reaction usage from the KEGG hydrogen dataset(0.17 MB XLS)Click here for additional data file.Table S8Metabolite and reaction usage from the KEGG nitrogen dataset(0.02 MB XLS)Click here for additional data file.Table S9Metabolite and reaction usage from the KEGG oxygen dataset(0.09 MB XLS)Click here for additional data file.Table S10Metabolite and reaction usage from the KEGG phosphorous dataset(0.02 MB XLS)Click here for additional data file.Table S11Metabolite and reaction usage from the KEGG sulfur dataset(0.01 MB XLS)Click here for additional data file.


Unfortunately, our ability to identify cycles in stoichiometric networks, such as signal transduction and genome-scale metabolic networks, has been hampered by the computational complexity of the methods currently used.Identifying cyclic pathways in chemical reaction networks is important, because such cycles may indicate H. pylori, M. barkeri, E. coli, and S. cerevisiae. Our algorithm can substantially decrease both the execution time and maximum memory usage in comparison to the two previous algorithms.We describe a new algorithm for the identification of cycles in stoichiometric networks, and we compare its performance to two others by exhaustively identifying the cycles contained in the genome-scale metabolic networks of The algorithm we describe improves our ability to study large, real-world, biochemical reaction networks, although additional methodological improvements are desirable. Flux balance analysis is becoming a well developed and frequently used theoretical tool to study the capabilities of large stoichiometric networks . Flux baThe incorporation of energetic constraints in flux balance analysis is not the only motivation to identify cycles. Cycles may also point towards important aspects of network function. For instance, it has been proposed that cycles in metabolic networks can affect the sensitivity and robustness of network function and allow for regulation of biochemical pathways . In signinternal and exchange reactions. Internal reactions are those that are actually being studied. For metabolic networks, for example, the internal reactions are those that occur in or around the cell, such as reactions involved in glycolysis, respiration, transport of metabolites across cellular membranes, etc. Exchange reactions are pseudo-reactions that are used to supply (remove) chemical species to (from) the reaction system. The flux, or rate, of each internal reaction is constrained to be positive, while the fluxes of exchange reactions may be left unconstrained. If reversible reactions are present in the network, they are broken apart into a pair of unidirectional forward-reverse reactions, each with its own flux. These reactions are used to construct a stoichiometry matrix S which is used to formulate the equation Sv = 0, where v is a vector of fluxes of all reactions in the network. The solutions to this equation represent the allowable steady states of the network, where steady refers to the fact that the concentrations of internal chemical species remain constant.Two algorithms have been explicitly described for the identification of steady state cycles in stoichiometric networks. Schilling, Letscher, and Palsson (SLP) described the first of these in 2000 . The SLPextreme pathways of a network, which are a unique set of flux vectors whose superposition can generate all steady-state fluxes in a network that do not violate the principle of mass conservation. The extreme pathways are then categorized based on the types of active reactions they contain. We use the word cycles from this point forward to refer exclusively to internal cycles, that is type III extreme pathways, which are the extreme pathways that do not contain active exchange reactions  very briefly described the second algorithm for cycle identification in 2003 . In thisIn this paper, we describe a new algorithm, which we refer to here as the WW algorithm. The WW algorithm is an extension of the MS algorithm, and it can reduce the size of the sub-network supplied to the SLP algorithm far beyond that of the MS algorithm. We also measure and compare, for the first time, the performances of all three algorithms using five genome-scale metabolic networks.Step 1. Ensure that the network does not contain exchange reactions, and constrain the flux of each internal reaction to the interval [0, ∞).U denote an empty set, and for each reaction R in the network, do the following:Step 2. Let R.Step i. By pair-wise comparison, identify all reactions in the network that are the reverse of Step ii. For each of the reverse reactions identified in Step i, constrain its flux to be zero.R can assume a value other than zero, while still satisfying all currently defined constraints. If so, add reaction R to the set U.Step iii. Determine if the flux of reaction Step iv. For each of the reverse reactions identified in Step i, restore the constraint on its flux to [0, ∞).U.Step 3. Identify the extreme pathways in the network defined by the reactions contained in the set RF and RR, in the stoichiometric network. A particular RF-RR pair forms a single type III extreme pathway (see Appendix).Step 4. By pair-wise comparison, identify each pair of internal forward-reverse reactions, Step 5. Combine the results of Steps 3 and 4 to obtain the set of all cycles present in the network (see Appendix).not participate in such cycles must be zero. Any technique capable of making this distinction, such as linear programming, can be used during this step. The reason for this property is that if a reaction R only participates in cycles composed of two reactions, it must be inactive when its reverse reactions are constrained to be inactive. In Step 3, a sub-network is created that is composed of the reactions identified in Step 2, and an algorithm is applied to that network to identify the type III extreme pathways it contains. That algorithm could be the SLP algorithm, or another that achieves the same result. Step 4 is used to identify all of the extreme pathways composed of only two reactions. This could also be accomplished during Step 2.i, but it is listed separately here for the sake of clarity. The appendix contains a more explicit justification of the principle behind the WW algorithm.Step 1 ensures that the only extreme pathways present in the network are of type III, since all other types of extreme pathways must contain active exchange reactions. Step 2 is used to identify all reactions that participate in cycles composed of three or more reactions. Under these circumstances, the fluxes of reactions that participate in such cycles can assume any value on the interval [0, ∞), and the fluxes of reactions that do There are two key differences between the WW and MS algorithms. Firstly, the MS algorithm, as it is defined, individually maximizes the flux of each reaction in a network. Flux maximization can also be performed to accomplish Step 2.iii of the WW algorithm, but other  techniques could be used as well. Secondly, the MS algorithm creates sub-networks composed of all of the reactions in a network that participate in cycles. The WW algorithm, however, creates sub-networks composed only of those reactions that participate in cycles composed of three or more reactions. Consequently, the WW algorithm can sometimes produce sub-networks drastically smaller than those of the MS algorithm, which is demonstrated below.Homo sapiens (human), comprising between 486 and 2786 chemical species, and between 642 and 4482 reactions. Our results show that both execution time efficiency  as the operating system. A bash script was used to execute 10 programs sequentially for each algorithm and network. The time was recorded before each program began and after each algorithm finished execution to determine the total running time. A perl script (memmon) was used to frequently sample the contents of/proc/meminfo to monitor the memory usage during each program execution. Memory monitoring began before each algorithm was executed to establish a baseline, and the maximum memory consumption during algorithm execution was measured relative to this baseline.Both JW and AW contributed to the algorithm's conception and participated in drafting the manuscript. JW carried out all performance comparisons.The following is a list of properties of extreme pathways that have been published previously ,16, and Property 1. Extreme pathways are composed of the minimum number of reactions needed to function  at steady state .Property 2. Extreme pathways are systematically independent, that is, extreme pathways cannot be composed of other extreme pathways ,16.Property 3. Type III extreme pathways contain only internal reactions, that is, they never contain exchange reactions. All other types of extreme pathways contain at least one exchange reaction .Deduction 1. Given Property 1, if any reaction is removed from an extreme pathway, or inactivated by constraining its flux to zero, the entire pathway is rendered inactive, that is, the flux of each reaction in the pathway must be zero.Deduction 2. Given Property 1, an extreme pathway can be viewed as a single, independent functional unit. For example, if all reactions in a network are inactivated, except those participating in a given extreme pathway, that particular pathway can still have non-zero flux.Deduction 3. Given Property 1, a single unidirectional internal reaction can never be an extreme pathway, because it cannot function at steady state by itself .Deduction 4. A pair of unidirectional forward-reverse reactions can form a type III extreme pathway. The pair can function at steady state without the need for other reactions and only by functioning together, satisfying Property 1. The pair cannot be composed of other extreme pathways, since a single reaction can never be an extreme pathway (Deduction 3), satisfying Property 2.F, RR). The only way a reaction pair can maintain steady state concentrations is if RR consumes the products of RF at the same rate that RF consumes the products of RR.Deduction 5. If an extreme pathway is composed of only two unidirectional reactions, they must be a forward-reverse reaction pair Deduction 8. If a unidirectional internal reaction R is active (i.e. its flux is greater than zero), all reverse reactions of R are inactivated (i.e. their fluxes have been constrained to zero), and the network contains no exchange reactions, R necessarily participates in a type III extreme pathway composed of more than two reactions. This statement is a direct consequence of Deductions 1, 4, 6 and 7.Supplementary Figures. Algorithmic performances represented on a log scale.Click here for file


Corynebacterium glutamicum, this enzyme was regarded as promising target for improved production of lysine, one of the major amino acids in animal nutrition. In pyruvate kinase deficient strains the required equimolar ratio of the two lysine precursors oxaloacetate and pyruvate can be achieved through concerted action of the phosphotransferase system (PTS) and phosphoenolpyruvate carboxylase (PEPC), whereby a reduced amount of carbon may be lost as CO2 due to reduced flux into the tricarboxylic acid (TCA) cycle. In previous studies, deletion of pyruvate kinase in lysine-producing C. glutamicum, however, did not yield a clear picture and the exact metabolic consequences are not fully understood.Pyruvate kinase is an important element in flux control of the intermediate metabolism. It catalyzes the irreversible conversion of phosphoenolpyruvate into pyruvate and is under allosteric control. In pyk gene, encoding pyruvate kinase, was carried out in the lysine-producing strain C. glutamicum lysCfbr, expressing a feedback resistant aspartokinase, to investigate the cellular response to deletion of this central glycolytic enzyme. Pyk deletion was achieved by allelic replacement, verified by PCR analysis and the lack of in vitro enzyme activity. The deletion mutant showed an overall growth behavior  which was very similar to that of the parent strain, but differed in slightly reduced lysine formation, increased formation of the overflow metabolites dihydroxyacetone and glycerol and in metabolic fluxes around the pyruvate node. The latter involved a flux shift from pyruvate carboxylase (PC) to PEPC, by which the cell maintained anaplerotic supply of the TCA cycle. This created a metabolic by-pass from PEP to pyruvate via malic enzyme demonstrating its contribution to metabolic flexibility of C. glutamicum on glucose.In this work, deletion of the C. glutamicum to compensate for external perturbation. The organism could almost maintain its growth and production performance through a local redirection of the metabolic flux, thereby fulfilling all anabolic and catabolic needs. The formation of the undesired overflow metabolites dihydroxyacetone and glycerol, in the deletion mutant, however, indicates a limiting capacity of the metabolism down-stream of their common precursor glyceraldehyde 3-phosphate and opens possibilities for further strain engineering.The metabolic flux analysis performed illustrates the high flexibility of the metabolic network of  Corynebacterium glutamicum requires a continuous improvement of the lysine production process with a special focus on optimization of the production strains  and 50 % [13C6] glucose and a significantly extended labeling data set with consideration of additional GC-MS fragments. In addition to the previous studies, the extended approach allowed to completely resolve the fine structure of the network around the pyruvate node, which was of special interest in the present work.The exact metabolic consequences of deletion of pyruvate kinase in lysine producing pathways ,14, we c mutants -19, duri mutants  or on di mutants ,22. As cpyk gene (1428 bp) was obtained by allelic replacement with a shortened DNA fragment containing only the two flanking regions, but not the coding sequence of the gene. The resulting excision of the whole nucleotide sequence of pyk was verified by PCR analysis. Using a site specific primer set of forward and reverse primer , whereas the parent strain C. glutamicum lysCfbr showed a strong specific activity of 1099 mU mg-1 for this enzyme. PCR analysis and in vitro measurement of enzyme activity in crude cell extracts ensured the deletion of the pyk gene and further demonstrated that there is no remaining pyruvate kinase like activity in C. glutamicum lysCfbr Δpyk. In the deletion strain, direct conversion of PEP into pyruvate thus is restricted to glucose uptake by the PTS.C. glutamicum lysCfbr and its pyruvate kinase-deficient derivative C. glutamicum lysCfbr Δpyk were grown in batch culture  a mixture of [13C6] glucose and unlabeled glucose to combine the information content available from the GC-MS labelling analysis of metabolites from the different tracer substrates for the flux calculation. Since it is known, that more detailed information for flux calculation can be obtained with GC-MS via additional analysis of fragment ions which contain only specific parts of the carbon skeleton of the analyte, we additionally considered a number of fragment ions from the proteinogenic amino acids which have previously proven useful for flux analysis in complex metabolic networks of prokaryotes [13C6] glucose and unlabeled glucose as tracer substrate and variation of the free flux parameters ΦPEPC (flux partitioning between PEPC and PC), ζPEPC/PEPCK (exchange flux by PEPC and PEPCK) and ζPC/MAE  which strongly influences the labelling pattern of analyzed metabolites  and malic enzyme, was required, in order to study the exact influence of the pyruvate kinase deletion to these closely linked reactions. First, the metabolic network of ion flux . Concernate node ,24. Due karyotes  and eukakaryotes . Overall13C labeling patterns of the metabolites remained constant over time  and the membrane bound malate dehydrogenase (MDH)  glucose or by an equimolar mixture of naturally labeled and 99 % [13C6] glucose.Complex medium containing 5 g Lsolution . In trac13C labeled glucose (100 mL flasks with 10 mL medium) for each strain. All cultures were inoculated with exponentially growing cells from the second pre-culture and incubated at 30°C and 230 rpm on a rotary shaker . During cultivation, dissolved oxygen concentration was monitored via immobilized sensor spots on the flask bottom containing a fluorophor with oxygen dependent luminescent decay time  glucose and 99 % [13C6] glucose were purchased from Campro Scientific . All other chemicals were purchased from Sigma, Merck , or Fluka  and were of analytical grade.Yeast extract and tryptone were obtained from Difco Laboratories . Ninety-nine % [1-C. glutamicum lysCfbr and 0.393 × OD (gram per litre) for C. glutamicum lysCfbr Δpyk. For the analysis of extracellular substrates and products, cultivation supernatant was obtained by centrifugation of 1 mL broth . Concentration of glucose, trehalose, glycerol, dihydroxyacetone (DHA) and organic acids was quantified in 1:10-diluted cultivation supernatant by HPLC . Separation was carried out on an Aminex HPX-87H column  at 45°C with 5 mM H2SO4 as mobile phase and a flow rate of 0.5 ml min-1. Refraction index  and UV absorption at 210 nm  was used for detection. Amino acid quantification in cultivation supernatant was performed as described previously [Samples taken during the cultivation were analyzed for concentrations of biomass, substrates and products. Cell concentration was determined by measurement of the optical density at 660 nm . If necessary, samples were diluted on an analytical balance  to obtain absorbance values below 0.3. The correlation factor between cell dry mass (CDM) and optical density was determined as CDM = 0.382 × OD (gram per litre) for eviously .2, 1 mM ADP, 0.25 mM NADH, 5.5 U of lactate dehydrogenase, 10 mM PEP and 10 μl of the crude cell extract. Negative controls were carried out without PEP or without cell extract, respectively.Crude cell extract was prepared as described previously . Cell deMass isotopomer fractions of amino acids from hydrolyzed and lyophilized cell protein and of trehalose from lyophilized culture supernatant, harvested during the exponential phase from the tracer studies, were determined by GC-MS ,39. SampC. glutamicum comprised all central metabolic pathways, i. e. glycolysis, PPP and TCA cycle [C. glutamicum, pyruvate carboxylase (PC), PEP carboxylase (PEPC), malic enzyme (MAE) and phosphoenolpyruvate carboxykinase (PEPCK) link glycolysis and TCA cycle through inter conversion of C3 and C4 metabolites [C. glutamicum [The metabolic network of CA cycle . In C. gabolites . In contabolites . Enzymatutamicum . For theC. glutamicum was determined by minimizing the deviation between the experimental and simulated GC-MS mass isotopomer fractions using a metabolic model implemented in Matlab 7.0  on a personal computer with isotopomer and metabolite balancing and the solver fmincon implemented in Matlab as previously described [13C label distribution in order to consider the labelling data from both parallel tracer experiments in the same parameter optimization. The network was over determined. A least square approach was therefore possible. As error criterion a weighted sum of least squares (SLS) was used. For both strains, identical flux distributions were obtained with multiple randomized initialization values for the free flux parameters, suggesting that global minima were identified. Statistical analysis of the obtained fluxes was carried out by a Monte-Carlo approach [Metabolic flux distribution of escribed . Hereby,escribed ,43. The approach . From thFlux partitioning ratio (Φ) and flux reversibility (ζ) were defined as relative flux into one of the two branches, and as ratio of backward or exchange flux to the net flux in the forward direction, respectively . For theThe author(s) declare that they have no competing interests.pyk deletion mutant was carried out by CK. JB performed the experimental studies of the present work, i.e. the physiological studies for determination of growth and production characteristics, the enzymatic analysis and the 13C tracer studies including GC/MS labeling analysis. CW carried out the modeling work including design and validation of the extended approach, metabolic modeling, parameter estimation and statistical analysis. The manuscript was drafted by CW and JB.Construction of the