Mycoplasma genitalium is one of the smallest known self-replicating organisms and, additionally, has extremely fastidious nutrient requirements. The reduced genomic content of M. genitalium has led researchers to suggest that the molecular assembly contained in this organism may be a close approximation to the minimal set of genes required for bacterial growth. Here, we introduce a systematic approach for the construction and curation of a genome-scale in silico metabolic model for M. genitalium. Key challenges included estimation of biomass composition, handling of enzymes with broad specificities, and the lack of a defined medium. Computational tools were subsequently employed to identify and resolve connectivity gaps in the model as well as growth prediction inconsistencies with gene essentiality experimental data. The curated model, M. genitalium iPS189 , is 87% accurate in recapitulating in vivo gene essentiality results for M. genitalium. Approaches and tools described herein provide a roadmap for the automated construction of in silico metabolic models of other organisms.With a genome size of ∼580 kb and approximately 480 protein coding regions,  Mycoplasma genitalium is believed to be a close approximation to the minimal set of genes required for bacterial growth. In this paper, we constructed a genome-scale metabolic model of M. genitalium that mathematically describes a unified characterization of its biochemical capabilities. The model accounts for 189 of the 482 genes listed in the latest genome annotation. We used computational tools during the process to bridge network gaps in the model and restore consistency with experimental data that determined which gene deletions led to cell death . We achieved 87% correct model predictions for essential genes and 89% for non-essential genes. We subsequently used the metabolic model to determine components that must be part of the growth medium. The approaches and tools described here provide a roadmap for the automated metabolic reconstruction of other organisms. This task is becoming increasingly critical as genome sequencing for new organisms is proceeding at an ever-accelerating pace.There is growing interest in elucidating the minimal number of genes needed for life. This challenge is important not just for fundamental but also practical considerations arising from the need to design microorganisms exquisitely tuned for particular applications. The genome of the pathogen  Genome-scale metabolic reconstructions are already in place or under development for a growing number of organisms including eukaryotic, prokaryotic and archaeal species in vivo behavior E. coli to correct some of the in silico growth predictions. In our group, we have taken the next step for gene deletion data by attempting to correct all such growth inconsistencies by allowing not just additions but also eliminations of functionalities in the model  . As outlined in Mycoplasma genitalium model construction process (as opposed to an a posteriori mode of deployment).In response to this flood of present and future genomic information, automated tools such as Pathway Tools M. genitalium has received considerable attention as it is the smallest organism that can be grown in pure culture, having a genome size of ∼580 kb and approximately 480 protein coding regions M. genitalium to quantify this minimal set. For example, Mushegian and Koonin have carried out a detailed comparison of M. genitalium and H. influenzae proteins to derive a set of 256 genes that they suggested are necessary for viability Mycoplasma genitalium is not only the closest known approximation of a minimal cell but also an important sexually transmitted human pathogen. It is a cause of nongonococcal urethritis in men and is associated with genital tract inflammatory diseases in women, including endometritis, cervicitis, pelvic inflammatory disease, and tubal factor infertility . We then extend the list to include non-gene associated reactions . Subsequently, we append spontaneous reactions for chemical transformations that are not enzyme catalyzed and for the transport of molecules that can freely diffuse across the cell membrane and are likely to be present in the natural or laboratory-adopted environment of M. genitalium. A distribution of the ORFs, metabolites, and reactions included in the final model iPS189 are provided  metabolites of the system.SConversion of model into computations-ready form. The stoichiometric matrix for iPS189 after step 1 involves 179 rows  and 292 columns . We also established gene-protein-reaction (GPR) associations to link reactions to genes. Only 48 reactions contained more than a single GPR association. Of these, none contained single-protein isozymes, and 47 were catalyzed by just one of the various enzyme complexes (among which was the ribosome and its associated proteins). Only one contained both isozymes and protein complexes.Biomass equation derivation. The derived biomass equation drains all metabolites present in cellular biomass in their appropriate molar biological ratios  and growth prediction inconsistencies (step 4).The initial model was constructed almost exclusively based on homology searches within model libraries. This procedure led to the presence of many network gaps M. genitalium  were excluded as gap filling candidates. We first applied GapFill to unblock constituents of biomass guided by the known components in the growth medium. We unblocked biomass production by adding 65 reactions, for which most  were involved in metabolite transport, such as for the uptake of amino acids (14), folate, riboflavin, metal ions (8), and cofactors such as CoA. Among the remaining reactions were those responsible for the hydrolysis of dipeptides (15) and eight reactions involving other biotransformations. We performed an additional round of BLASTp comparisons of genes annotated with these reactions against the M. genitalium genome to determine if we could associate any of these reactions with specific genes in M. genitalium. We found five proteins catalyzing these reactions that had BLASTp scores smaller than 10−5 (see Through the use of GapFill 10−5 see . For exai.e. MG066) for 1-deoxy-D-xylulose 5-phosphate synthase was already included in the model but with a different functionality . The secondary synthase functionality, revealed by GapFill/BLASTp, was subsequently associated with gene MG066 in the model. A similar situation occurred with MG053, which was already associated with phosphomannomutase in the model. In addition, gene MG259  was added to the model to carry out the glutamine-N5 methyltransferase activity elucidated by GapFill/BLASTp. The model statistics after correcting for network gaps are summarized in Even after unblocking biomass formation, 43 metabolites remained blocked and were subsequently analyzed by GapFill. The results from GapFill are summarized in in vivo gene essentiality data in vivo gene essentiality experiments were performed using non-defined medium containing serum and yeast hydrolysate among other rich components. During the in silico model predictions/comparisons, we allowed the uptake of all extracellular metabolites with transport reactions, except for sugars other than glucose, in order to computationally approximate this medium.Based on in vivo) instead of under-predictions . We subsequently deployed the GrowMatch method  to rectify as many as possible of the erroneous essentiality predictions by the model. GrowMatch functions by identifying the minimal number of model modifications required to restore consistency between growth predictions and gene essentiality experiments , which had been associated with macromolecule formation during the automodel construction. We observed that deletion mutants of the homolog in E. coli (lepA) are viable Model under-predictions include mutants (MG410 and MG411), which encode the subunits for the phosphate transporter, preventing in both cases the uptake of phosphate. This implies that in vivo, at least under the experimental conditions, and perhaps is not an activity encoded by M. genitalium. An additional six over-predictions involved two metal ion ABC transporters. GrowMatch identified each transporter to be essential when the other one was suppressed. We rejected co-regulation of the two transporters as a model restoration mechanism. Instead, we restored consistency for three of the six genes by assigning the cobalt uptake to the complex with a better homology to characterized cobalt transporters . The remaining three genes were removed from the model. An alternative interpretation of the GrowMatch results is that some other ion uptake reaction(s) are uniquely associated with these transporters and are thus responsible for the in vivo phenotype.Interestingly, three of the 37 erroneous over-predictions were corrected by adding three membrane components to the biomass equation see . These cin vivo data, suggesting possible regulation events. iPS189) from 79 to 87%. By recalculating the diagnostics of the percentage of correctly- identified essential genes Overall, the application of GrowMatch to the metabolic model led to the generation of a number of testable hypotheses regarding the presence or absence of specific functionalities and emphasized the importance of determining the substrate specificity of the transporters. We also identified reactions that had to be inactivated only for certain knock-outs suggesting their dependence on the genetic background in addition to the specific environmental conditions. For example, MG112 (ribulose-phosphate 3-epimerase) had to be suppressed in conjunction with two single gene deletions  to restore consistency with the iPS189 model predicts that M. genitalium uptakes fructose via a PTS system. The fructose is converted to fructose 1,6-bisphosphate (fdp) and finally enters the glycolytic pathways to produce lactate via lactate dehydrogenase. Neither fructose nor glycerol uptake was found to be essential as glucose could be efficiently taken up and converted. Specifically, glucose is transformed to glucose 6-phosphate and finally to fdp via phosphofructokinase. As expected, the model also indicates that co-enzyme A (CoA) is taken up, since M. genitalium has no coA biosynthesis genes. Additionally, accetal-CoA (accCoA) is not formed via pyruvate formate lyase but rather by pyruvate dehydrogenase. Interestingly, we find that should acetate be taken up, it is converted to acetyl phosphate (actp) and finally to accoa by phosphotransacetylase. Sources for acyl-CoA (aCoA) and CDP-glucose are also required for lipid production. The metabolites riboflavin and nicotinic acid (niacin) are taken up for synthesis of the cofactors FAD and NAD, respectively.The d-ribose 1-diphosphate, which enables the conversion of adenine to amp. We also deduced that only adenine and cytidine are precursors to nucleotides and nucleosides . Interestingly, the model required the direct uptake of GTP and could not be produced through the uptake of guanine. Model modifications that eliminate this requirement using GrowMatch resulted in a number of incorrect gene essentiality predictions. The need for the direct uptake of GTP is consistent with the fact that in M. mycoides the guanine nucleotide pathways depend on transport of preformed guanine derivatives M. genitalium is an arginine nonfermenting species, and not surprisingly arginine deiminase activity was not present in the model. Furthermore, in iPS189, the only participation of the amino acid arginine is its direct incorporation into biomass. Finally, flux predictions revealed that lactate is the main product of M. genitalium fermentation.In addition, both spermidine and putrescene are directly imported as biomass components. Similarly, we also found that D-ribose (rib-D) is needed to fuel the truncated pentose phosphate pathway. Examination of fluxes indicated that the uptake of rib-D results in production of 5-phospho-α-iPS189 metabolic model is to drive the development of a defined growth medium. As noted above, gene essentiality experiments were performed using a non-defined medium, SP-4, which contains beef heart infusion, peptone supplemented with yeast extract and fetal bovine serum. The use of an undefined medium can confound the characterization of gene essentiality, as the exact environmental conditions are not fully specified. Furthermore, the lack of a defined growth medium complicates the understanding of nutritional control and regulation of pathologies, evaluation of drug susceptibility, characterization of auxotrophic mutants and performing genetic analysis.A key targeted application of the Mycoplasma capricolumAcholeplasma laidlawiiSpiroplasmasMycoplasma mycoides subspecies Helicobacter pyloriHaemophilus influenzaeUsing trial-and-error researchers have already attempted to formulate defined media by systematically deleting components from an undefined or complex media iPS189 metabolic model as a roadmap of the available transporters, metabolites and internal interconversions to seek out the minimum number of growth medium components necessary for biomass production. We used as a starting point the components of the C5 medium , which is used as a component of SP-4, with the addition of folate and biotin and the replacement of thiamine by the four individual deoxynucleosides  to introduce and bring to bear automated procedures that streamline the construction of metabolic models. The procedure does not require a fully annotated genome and can serve to complement existing annotations In this paper, our focus was two-fold: (a) to construct the metabolic model for the minimal organism and pathogen M. genitaliumLactococcus lactisM. genitalium specific model with growth requirement as a constraint revealed novel uptake and non-gene associated reactions that previous studies were unable to identify.Many genome annotation errors are caused M. genitalium and its fastidious growth-requirements. Notably, no other metabolic model reconstruction efforts to date were complicated by the lack of both a well-defined biomass composition and a defined growth medium. Here, we allowed the uptake of all metabolites known to be present in the current medium. We also included metabolites either with identified transporters or those necessary for reconnecting blocked metabolites/reactions. Even though missing metabolites and pathways still exist in iPS189, we were able to achieve a high degree of agreement between the model predictions and in vivo gene essentiality data (87%). We note that the most recent iteration of the metabolic model for E. coli, an organism which has both a well-defined biomass composition as well as chemically defined growth media, has an overall agreement with in vivo gene essentiality data of 91% under aerobic glucose conditions in silico models correctly predict less than 45% of essential genes  because of, in part, the much higher percentage of essential genes in M. genitalium but also its careful construction. Additional in vivo gene essentiality studies using a fully defined medium could usher a more accurate elucidation of the true metabolic capabilities of M. genitalium, as well as suggest improvements to the reconstruction.The identification of which metabolites are produced internally or transported directly from the extracellular environment was complicated by the lack of a defined medium for iPS189 is smaller than the 256 genes suggested as a minimal gene set based on comparison of M. genitalium and H. influenzae proteins in vivo have unknown function, and thus cannot (yet) be incorporated into the model. It is possible that they could, in part, carry out some of the non-gene associated reactions that were proposed during the GapFill procedure. Determining their metabolic function through biochemical and molecular biology techniques, as well as determining the substrate specificity for non-characterized transporters, would improve subsequent metabolic models.The metabolic model de novo synthesis and assembly of the complete M. genitalium genome ab initio design of microbes from scratch that are exquisitely tuned for specific biotechnological applications. The constructed metabolic model iPS189 could serve as a core of metabolic functions to add upon so as to bring about the desired biological functionalities and/or production capabilities.Looking to the future, we note that it was recently shown that it is feasible to transplant the genome from one mycoplasma species to another of M. genitalium are detailed.The general principles of the metabolic reconstruction process have been previously outlined M. genitalium involved analyzing the annotated M. genitalium G-37 genome sequence with the SimPheny automated model generation platform developed by Genomatica . This automated procedure leveraged the content contained in Genomatica's manually curated models to expedite the initial reconstruction of M. genitalium metabolism. For the M. genitalium model, reconstructions for E. coli, Haemophilus influenzae, Geobacter sulfurreducens, B. subtilis, and Saccharomyces cerevisiae were used in the comparisons. Subsequently genes-protein-reaction (GPR) associations were established based on the evidence provided by homology identity provided by sequence analysis and annotation information. Specifically, forward and reverse protein-protein BLAST (BLASTp) comparisons were carried out to identify genes in the library of manually curated models that most closely resemble those in M. genitalium. In the forward BLASTp step, each M. genitalium ORF was compared against each gene in the manually curated models. Then, in the reverse BLASTp step, the best hit from manually curated models was compared against the M. genitalium genome ORFs. The top hits for each of the searches were stored, and a third list was compiled consisting of the cases in which the top hits are identical in both the forward and reverse BLASTp searches. For only these pairs, the GPR association(s) from the manually curated model was used as a template to assign those in the M. genitalium model. This two-way BLASTp search procedure improves the likelihood that M. genitalium orthologs, not paralogs, to genes in the manually curated models are identified and assigned by the automodel process.The first step in our reconstruction of the genome-scale model of metabolism of M. genitalium model and performed BLASTp analysis to identify additional proteins with biochemically characterized functionalities in other organisms. The target database was the nr (non-redundant) database at NCBI (http://www.ncbi.nlm.nih.gov/blast). Next, all target proteins that the BLASTp search reported Expect (E) values less than 10−5 were further investigated for biochemical characterization. Next, we carried out a reverse BLASTp analysis of each of these characterized proteins against sequences in the M. genitalium genome. If the initial M. genitalium gene was ranked with the best score and had an acceptable E value (less than 10−5) at the end of this step, that gene and its associated reactions were included in the model as with the library of curated models.Although the automated genome-model comparisons enabled a fast generation of a draft model, the auto-generated reconstruction was extended by searches for relevant reactions not included in the library of curated models. First, all high-quality annotated “non-metabolic” genes, such as those encoding transcription factors and proteins involved in replication and repair, were excluded from subsequent searches. We then extracted the remaining sequences of all the ORFs not yet included in the M. genitalium) were updated to occur in the cytosol, three reactions for ATP synthase were merged, and two macromolecule synthesis reactions were combined and their GPR associations were manually updated. We also added GPR associations for missed characterized members of complexes  when the automodel associated the other members, as well as corrected those GPR associations for proteins that were misidentified as isozymes instead of complexes. Furthermore, we investigated the cluster of orthologous groups (COGs) ontology M. genitalium model. Those having classification within non-metabolic classifications, such as transcription (K); replication, recombination and repair (L); general function prediction only (R); function unknown (S); or none assigned (N/A) were inspected further. Although most of the ORFs within (R) and (N/A) were retained, we removed all those within (K), (L), and (S) from the model  as compared to E. coli (49.8%) and B. subtilis (40.9%) and its smaller genome size. The amino acid relative percentages were likewise adjusted based on utilization patterns and codon biases. Amino acid utilization was incorporated as charged and uncharged tRNA molecules in the biomass equation as reactants and products, respectively. Note that the charged gln-tRNA becomes an uncharged glu-tRNA and the charged fmet-tRNA becomes an uncharged met-tRNA through the biomass equation. Metal ions known to be present in the active sites of catalytic enzymes were also included in the biomass equation. Various membrane and lipid components were subsequently added in accordance with the GrowMatch predictions.The biomass equation was generated by accounting for as many as possible of the constituents that form the cellular biomass of Testing the metabolic model using optimization-based approaches requires the definition of a number of sets and parameters.Sets:Parameters:Variables:jUB and jLB, were chosen as not to exclude any physiologically relevant metabolic flux values. The upper bound for all reactions was set to 1,000. The lower bound was set equal to zero for irreversible reactions and to −1,000 for reversible reactions. The non-growth associated ATP maintenance limit was set to jLB = 8.4 gDW−1 h−1. The maximum transport rate into the cell was 5 mmol gDW−1 h−1 for any external carbon containing metabolite . The lower bound for the remaining source exchange reactions was −20 mmol gDW−1 h−1Upper and lower bounds, Using the principle of stoichiometric analysis along with the application of a pseudo-steady-state hypothesis to the intracellular metabolites When constructing the model, we also generated the gene-protein-reaction (GPR) associations that link the ORFs to the reactions that are catalyzed by their gene products using standard conventions i.e., cannot carry any net influx), given the availability of all substrates supported by the model. We next applied GapFill to modify the existing genome-scale model in order to reconnect these metabolites to the model. The reaction source databases used by GapFill in this work were the KEGG M. genitalium .Once a mathematical representation of the metabolic model was generated, we first determined using GapFind The introduction of the GPR associations introduces additional complexity in that an additional layer of detail is needed to fully characterize the network when a single gene is deleted. We made the following definitions to this end:Sets:Parameters:Variables:s0001, as in 2 across the cell boundaries), but this fictitious gene was not included when enumerating the total genes included in the model, nor could it be knocked out. Note that a gene k may have more than one GPR association when it is involved with more than one reaction. Likewise, a reaction j may be involved in more than one GPR association, as is the case with isozymes or multi-protein complexes. We restricted reaction fluxes, jv, using the binary variable jw as follows:j can take a non-zero value only if the reaction is active .Set k on reaction j. For instance, if only a single gene 1k is associated with reaction j, then we assign 1k and 2k encode isozymes that catalyze reaction j, then we assign 1k and 2k are required for the formation of a multi-protein complex that catalyzes reaction j, we set both Parameter in silico growth predictions of the M. genitalium metabolism network by examining the flux of the biomass equation. Given the deletion of a single gene k, we solved the following formulation:j is zero when the gene k that is necessary for its activity is deleted. Note that this equation takes advantage of the fact that in this work we only examined single gene deletions and thus needed not write more complicated GPR-related constraints. This formulation was solved for each gene j in the model using CPLEX version 11 accessed within the GAMS modeling environment. The predictions were compared against in vivo gene essentiality data average growth exhibited by all the single gene deletions under consideration. However, we found that the in silico growth predictions for iPS189 were insensitive to this value.We tested the in silico and in vivo growth predictions across single gene deletion mutants . To this end, we first classified growth prediction inconsistencies into two categories: (a) a mutant is termed a “Grow/NoGrow” (GNG) mutant if the in silico model predicts growth whereas there is no observed growth in vivo and (b) a mutant is termed a “No Grow/Grow” (NGG) mutant if the in silico model predicts no growth in contrast with observed in vivo growth. In GNG mutants, the model overpredicts the metabolic capabilities of the organism. GrowMatch automatically restores consistency in these mutants by suppressing reaction activities to prevent in silico growth . Conversely, in NGG mutants, the model underpredicts the metabolic capabilities of the organism. GrowMatch restores consistency in these mutants by adding functionalities that ensure in silico growth consistent with in vivo data. As when GapFill was applied, the additional reactions were carefully monitored. In all cases, GrowMatch operates so as not to perturb any correct growth predictions.We next applied the GrowMatch method to reconcile inconsistencies between We pose the problem of identifying the minimum number of added components to the growth medium so as to allow for the formation of all biomass constituents as an optimization problem. To this end, we introduce the following additional sets and variables:Sets:Variables:j is active (ju = 1), Equation (7) permits its flux to assume non-zero negative values, thus allowing the uptake of the corresponding substrate. Conversely, when an exchange reaction j is inactive (ju = 0), Equation (7) ensures that the exchange reaction can only remove the corresponding product from the extracellular environment . Constraint (8) ensures a minimum amount of biomass formation, with the cutoff set to be the same as that used for the gene essentiality predictions above . This MILP problem was also solved using CPLEX version 11 accessed within the GAMS modeling environment.We minimized the total number of activated substrate exchange reactions that enable the uptake of growth medium components through the use of the following optimization formulation:Table S1Reconstruction metabolite and reaction content(0.15 MB XLS)Click here for additional data file.Table S2Composition of the biomass equation(0.03 MB XLS)Click here for additional data file.Table S3Gene content of the model during each step of the reconstruction(0.05 MB XLS)Click here for additional data file.Table S4Components included in modified C5 medium(0.02 MB XLS)Click here for additional data file.Text S1iPS189 in SBML format(0.02 MB ZIP)Click here for additional data file.


Models have been reconstructed for almost 20 bacterial species, so far mainly through expert curation efforts integrating information from the literature with genome annotation. A wide variety of computational methods exploiting metabolic models have been developed and applied to bacteria, yielding valuable insights into bacterial metabolism and evolution, and providing a sound basis for computer-assisted design in metabolic engineering. Recent advances in computational systems biology and high-throughput experimental technologies pave the way for the systematic reconstruction of metabolic models from genomes of new species, and a corresponding expansion of the scope of their applications. In this review, we provide an introduction to the key ideas of metabolic modeling, survey the methods, and resources that enable model reconstruction and refinement, and chart applications to the investigation of global properties of metabolic systems, the interpretation of experimental results, and the re-engineering of their biochemical capabilities.Genome-scale metabolic models bridge the gap between genome-derived biochemical information and metabolic phenotypes in a principled manner, providing a solid interpretative framework for experimental data related to metabolic states, and enabling simple  Combined with increasingly efficient annotation methods, these set the stage for the systematic identification of most enzymes encoded in the genomes of the corresponding bacterial species. A variety of so-called ‘-omics’ technologies now routinely provide large-scale functional clues on molecular interactions and cellular states, offering snapshots of the dynamic operation of metabolism under specified conditions, and adding to the store of accumulated knowledge on microbial biochemistry and physiology.The flow of genome sequencing, metagenome sequencing and other high-throughput experimental efforts aimed at exploring the space of microbial biochemical capabilities has been steadily growing in recent years. At the time of writing, more than 1800 bacterial genome-sequencing projects have been initiated and nearly 650 have been completed  and associated molecular components  that are most directly related to the metabolic capabilities mentioned above.Investigations of a bacterium's metabolism are typically fed by knowledge (ultimately from observations) at two different scales of description of the chemistry at work within cells. The larger scale focuses on the physiology of the whole bacterial cell. For instance, which media is it able to grow on? What are the relative quantities of chemical nutrients it requires for growth? How efficient is the cell at converting chemicals from the environment into its own components? Such metabolic capabilities result from the coordinated action of the enzymes expressed in the respective species, the knowledge of which belongs to the finer, molecular scale. Each of the corresponding biochemical conversions can be identified either directly by performing enzymatic assays, or indirectly, from the genome sequence, through a homology relationship with proteins whose function has been previously elucidated. Together, the reactions that have been demonstrated to potentially occur in the cell form the metabolic pathways, typically grouping together reactions that allow the conversion of one or more ‘input’ metabolites into ‘output’ metabolites. Pathways boundaries are somewhat arbitrary, even though inputs and outputs tend to be metabolites involved in several reactions. Pathway-based analyses are thus focused on the possible fates of a restricted number of compounds, and are amenable to manual expertise thanks to the simplification brought by the modularized view  ultimately unused byproducts. The latter transformations typically involve several pathways, and are dependent on the stoichiometry and rates of the reactions. Manual approaches are insufficient to assess their feasibility by a given network for at least two reasons: metabolic networks are too large, and the question requires a quantitative analysis.Bridging that gap between knowledge of the metabolic network structure and observed metabolic phenotypes is precisely where metabolic models come into play. Generally speaking, a model of a natural system is one of many possible mathematical representation of that system, explicitly describing some of its features and supporting predictions on some other features, the latter being typically time- or environment dependent. In this particular case, knowledge of the metabolic network alone is not quite sufficient to predict the metabolic capabilities of a cell. Also needed are a structured  representation of that network, together with a set of rules and possibly quantitative parameters enabling simulations or predictions on the joint operation of all network reactions in a given environment, and in particular predictions on the values of metabolite fluxes and/or concentrations  constraint . Because the goal is to describe the joint operation of many metabolic reactions, it is convenient to define a flux distribution as a collection of reaction fluxes covering the entire system. Under the steady-state approximation, the concentrations of balanced metabolites being constant, a flux distribution carries sufficient information to completely describe a state of the system. Using the stoichiometric matrix, a simple matrix equation – summarizing all mass balance equations shown above – can then be used to enforce the mass balance constraints on all reactions fluxes: S.ν=0, where S is the stoichiometric matrix and ν the flux distribution represented as a vector.Mass balance, irreversibility and upper-bound constraints result from the application of simple laws of physics to individual reactions or metabolites from the network. These constraints propagate from reaction to reaction throughout the metabolic network; the constraint-based modeling framework is designed to automatically compute the resulting balance. To that end, it makes use of a succinct mathematical representation of all reaction stoichiometries: the trix see . In thistrix see . ReactioA precise definition of the boundary of the system to be modeled is also needed to formulate an explicit mathematical representation. The system typically includes the whole cell and its vicinity, in order to encompass all the exchanges of matter between the cell and its environment. Transport reactions that allow for exchange of specific metabolites with the extracellular space through the membrane are also included in the model. Environmental conditions are then modeled by acting on the balance of the external metabolites: metabolites that are available from the environment can be taken up by transporters while the others can only be excreted.A flux distribution that is compatible with all the constraints in a given environment is considered achievable (or feasible) by the cell, whereas a distribution that violates at least one of these constraints is not. The simplicity of the system of linear equations that represent constraints is one of the main strengths of the framework, because it permits fast assessments of the feasibility of a flux distribution using a computer and standard algorithms.The simplicity of constraint-based models comes at the expense of a number of limitations in their predictive capabilities. Such models focus solely on reaction fluxes, and completely ignore the influence of metabolites and enzymes. In reality, however, enzyme kinetics, and transcriptional or metabolic regulation may significantly influence reaction fluxes. Regulation can for instance limit the use of a pathway by downregulating some of its enzymes when particular environmental conditions are met. These mechanisms, if they could somehow be taken into account, would eliminate flux distributions otherwise allowed by constraint-based models. In other words, models may allow ‘false-positive’ metabolic states, which respect the enforced metabolic constraints but are inconsistent with other biological processes. Several attempts have been made to extend the constraint-based modeling framework, in order to account for regulatory interactions , or metaSome predictions of constraint-based models may be wrong in cases where modeling assumptions do not hold. For instance, some metabolites do accumulate in the cell, and the mass balance assumption clearly does not hold for these. In general, the concentration of specific metabolites may be high enough relatively to the fluxes they are involved in for the mass balance approximation to become clearly false.In practice, many of the analytical methods that have been developed for constraint-based models focus on defining and characterizing sets of feasible flux distributions. Others focus on a single distribution. The diversity of flux distributions compatible with constraints in a given environment can be viewed as reflecting the diversity of the metabolic states the cell may find itself in. Nevertheless, the space of feasible flux distributions features biologically informative properties whose determination requires adequate techniques; these will be introduced in the next sections of this review.metabolic network reconstruction. We will then review the techniques commonly used to assess the consistency of reconstructed models, and show how missing biochemical activities can be identified to complete the model.The level of detail necessary to build a constraint-based model of a bacterium's metabolism is relatively low; the only information required is the precise reaction stoichiometries and directions, in order to account for mass balance and irreversibility constraints. To reflect the global biochemical capabilities of the organism, the model also needs to encompass the complete set of metabolic activities that can occur within it – or a reasonable approximation thereof. This comprehensiveness requirement and the high number of metabolic reactions make the actual construction of such models a challenging task in itself. In this section, we will review the main methods and resources helping in this task. We will first show how information from genome annotation can be used to infer biochemical reactions at large scale, a task commonly called Escherichia coli metabolism – includes references to more than 320 articles  numbers classification offers an unambiguous way to identify enzyme-catalyzed reactions. When provided by the enzyme annotations, these numbers directly specify which reactions they catalyze. Several enzyme and metabolic databases provide the correspondence between EC numbers and reactions see  and 2. Tpathologic, the metabolic network reconstruction software tied to the BioCyc metabolic databases, includes an algorithm performing the identification of gene-reaction links from textual annotations  or ontology based [as in Gene Ontology  may catalyze the degradation of several distinct alcohols. Similarly, enzymes acting on lipids are often not specific to the length of their carbon chain. In such cases, functional annotations often report the activity using generic metabolites  representing the entire set of possible substrates. Instantiating reactions with specific metabolites is required when building a metabolic model, however, as accounting for the mass balance constraint requires that all metabolites should be well defined. It is thus necessary to identify for each generic compound the corresponding set of specific compounds, as much for primary substrates as for cofactors. This task is complicated by the combinatorial effect, because the number of substrate combinations may significantly increase the number of specific reactions. To address this issue, databases of chemical species can be used to identify all metabolites of a given chemical category see  and 2. Igem) system  may show that the reconstructed model is incomplete and lacks some metabolic reactions. Resolving these metabolic gaps entails expanding the model by identifying and including missing biochemical activities. This process basically consists of two steps: (1) identifying plausible candidate reactions that could complete the model and (2) finding genes that could catalyze the hypothesized activities.Reactions contained in metabolic databases are the primary source of information for completing the metabolic model see . The seaThe set of reactions referenced in metabolic databases is far from being comprehensive: the right candidates for completing the model may not yet be known. Computational and experimental approaches have been proposed to extend this ‘universe of possible reactions.’ On the computational side, several methods originating from the field of chemo-informatics have been designed to infer chemical transformations . Some ofab initio pathway inference methods by selecting candidate pathways that are compatible with observed isotopic patterns. Finally, untargeted enzyme activity screenings have recently been performed to identify the substrates of enzymes of unknown function and discover novel activities  techniques are able to identify and quantify large sets of metabolites at high throughput . They doThe search for candidate genes for orphan metabolic activities is in some ways the reverse of the classical genome annotation problem (i.e. searching the function of identified genes). Yet, many of the tools developed to determine gene functions can be adapted for this purpose. Sequence homology to already characterized genes is central to most methods for candidate gene detection, but combining it with additional types of evidence can significantly improve performance. For instance, several approaches exploit functional links, such as gene neighborhood, gene co-expression, protein interaction, or phylogenetic co-occurrence, to relate candidate genes with genes involved in the same metabolic pathways or close in the metabolic network  in Geobacter sulfurreducens) consumption rate vs. carbon source consumption rate and growth rate . With regard to the metabolic network itself, metabolic gap filling approaches have been adapted to propose network corrections that resolve wrongly predicted growth phenotypes  found in products of metabolism given known isotope patterns in nutrient metabolites  simulates time courses of gene expression states: at each time step, the new transcriptional state is computed from the metabolic state predicted at the previous time step, and is used to constrain FBA prediction of the current metabolic state  . Its unified dictionary of metabolite and reaction names enables direct comparisons between its metabolic models.Currently, the only freely accessible (to academic users) repository dedicated to constraint-based models is the BiGG database (matlab (http://www.mathworks.com/) and mathematica (http://www.wolfram.com/), or optimization modeling packages, for example gams (http://www.gams.com/), are well suited. Specialized optimization packages can be added for greater efficiency. In addition, modules dedicated to constraint-based modeling have been developed for matlab: fluxanalyzer (cobra toolbox (metatool (systems biology research toolbox (scrumpy (metafluxnet (fluxexplorer (sympheny platform (http://www.genomatica.com/) associates a metabolic database with several analytical methods, while yanasquare  and output compounds (products) – can bring a useful decomposition of metabolism into basic biochemical functional units, in the spirit pioneered by SEED , working


Reduced glycolytic and mitochondrial respiration rates are common features of apoptosis that may reflect key events contributing to cell death. However, it is unclear to what extent the rate changes can be explained by direct alterations in the kinetics of the participating reactions, as changes in the concentrations of intermediates also affect reaction rates. Direct kinetic changes can be identified, ranked, and compared to the indirect effects mediated by the intermediates using top-down control analysis. Flux changes that are explained primarily by direct effects are likely to be prime targets of the pathways that signal death, and thus important contributors to apoptosis. Control analysis concepts relevant to identifying such effects are reviewed. Metabolic flux measurements are essential for this approach, but can be technically difficult, particularly when using adherent cells such as neurons. A simple method is described that renders such measurements feasible. This Reduced glycolytic and mitochondrial fluxes are common features of physiological apoptosis induced by limited trophic factor support, yet the basis for the change in flux is uncertain. Growth factor restriction is associated with reduced glycolytic flux in non-neuronal cells, possibly due to inhibition of one or more glycolytic steps –28. Glut2.in vitro when cultured with fetal bovine serum as a source of growth factors, and non-physiologically high extracellular K+  to mimic tonic excitatory synaptic input [+ depolarizes the plasma membrane potential (ΔΨp ≈ −35 mV) [2+ channels. This approximately doubles the cytoplasmic Ca2+ concentration ([Ca2+]c) from the low K+  state, where the majority of L-type Ca2+ channels are closed (ΔΨp ≈−75 mV) c from high to low K+ significantly affects CGN metabolism [2+]c, as the effect can be reproduced by omitting Ca2+ from high K+ buffer [2+]c to a level similar to that in low K+. Lower [Ca2+]c could affect metabolism in two ways. First, the kinetics of the enzymes may be affected, either by Ca2+ directly , or by the phosphorylation state of the enzymes or regulatory proteins that affect them, because of altered Ca2+-dependent kinase  activity. Second, ATP demand may decline as a result of decreased plasma membrane Ca2+ cycling which, as described below, will alter enzyme activity through changes in the concentrations of intermediates. It is unclear to what extent each might contribute to the changes in glycolytic and oxidative phosphorylation fluxes.The change in [Catabolism . Specifitabolism . The cha+ buffer , a condi2+-dependent changes are those mediated by growth factor deprivation, which acts synergistically with low K+ to facilitate CGN apoptosis [2+-dependent mechanisms. More importantly, RTKs stimulate a number of survival kinase pathways, the most important of which in many cells is phosphatidylinositol-3-kinase (PI-3K) and its downstream target protein kinase B/Akt. The PI-3K/Akt pathway may target steps in glycolysis, potentially explaining the flux changes observed during apoptosis [Superimposed on the Capoptosis ,46,47. Ppoptosis ,4,48–50.poptosis , therebypoptosis ,52. Intepoptosis ,27. Insupoptosis . Inhibitpoptosis ,52–55, ppoptosis  and glucpoptosis  has beenpoptosis  in a groThe emergence of glycolytic suppression as a potential contributor to apoptosis complements studies showing abnormally high flux rates in cancer cells, many of which are resistant to apoptosis. However, it is crucial to know the extent to which the decreased flux observed during apoptosis can be attributed to changes in (1) the kinetics of the glycolytic reactions and (2) the concentrations of intermediates to which the glycolytic enzymes respond. If the flux decreases primarily because of altered reaction kinetics, then this could have important consequences on mitochondrial function and ATP levels, whereas if it decreases due to changes in the concentrations of key intermediates, then the apoptotic signaling must be acting elsewhere in the system. In this respect, it is important to view glycolysis as a reaction embedded within a larger pathway (whose principle function is to produce ATP) so that potential signaling routes that do not target glycolysis but nonetheless affect its flux can be considered. By broadening the scope of metabolism to include those reactions connected to glycolysis through key intermediates, a more comprehensive picture of the actions of apoptosis will be achieved.2+ cycling and subsequent decreased ATP utilization may explain much of the decrease in glycolysis. Increased ATP could also occur by activation of the mitochondrial phosphorylation reactions. Thus, the processes altering glycolytic flux could be through direct actions on the glycolytic enzymes, or signaling that acts elsewhere to influence key metabolites, or a combination of these.Glycolytic flux is strongly affected by ATP levels, which simultaneously affect the rates of multiple reactions. The balance of rates of ATP utilization and production establishes the ATP level. Most studies have not determined the extent to which ATP utilization changes with apoptosis, and, if so, how much the change can explain the suppression of glycolysis through effects on ATP levels. Granule neurons are a good example, where decreased CaAssume that at least some of the decrease in glycolysis can be explained by direct inhibition of one or more glycolytic reactions. It would be useful to know how much this affects mitochondrial substrate oxidation and ATP synthesis. Glycolytic impairment has been implicated in restricting cellular ATP supply, both through reduced substrate level phosphorylation and by restricting mitochondrial substrate supply, thereby promoting mitochondrial dysfunction. However, in most cells glycolysis accounts for considerably less than half of total ATP synthesis, so decreased glycolysis may have little effect on ATP levels. It has been suggested that mitochondrial ATP synthesis decreases because of substrate limitation, but this has not been rigorously tested. Much of the glucose that enters glycolysis is not oxidized by mitochondria, but ends up exported from the cell as lactate. It can be argued that impaired glycolysis results in a greater proportion of pyruvate being oxidized by the mitochondria rather than being converted to lactate, resulting in no diminution in mitochondrial substrate supply under physiological conditions. For these reasons, when apoptotic stimuli directly affect glycolysis, it is important to assess the extent to which this functionally alters mitochondrial substrate oxidation and/or phosphorylation. A systems biology approach to metabolism, through the use to top-down control analysis, can help resolve some of these uncertainties.+/serum deprived CGNs, direct kinetic changes and indirect effects mediated through the intermediates were not distinguished [+/serum deprivation is decreased Ca2+ cycling and hence decreased ATP utilization. This primary disturbance is propagated throughout other reactions within the metabolic network through changes in the concentrations of other key intermediates in the system. Reduced ATP utilization produces a transient imbalance in supply and demand: ATP synthesis rate by glycolysis and the mitochondrial phosphorylating system exceeds utilization, resulting in increased ATP/ADP ratio. Consistent with this scheme, higher ATP levels have been measured after K+/serum deprivation [Direct changes in the kinetics of those reactions which together exert significant control over a pathway flux indicate a concerted action that is likely important to apoptosis. Despite extensive analysis of the early bioenergetic changes occurring in Knguished . As mentrivation . This wiLooking one step further in this scheme, reduced glycolytic flux will increase glucose-6-phosphate levels as its rate of production by glucose uptake and phosphorylation reactions exceeds the now lower consumption through glycolysis. The increased concentration of glucose-6-phosphate negatively affects hexokinase and thus the rate of glucose phosphorylation. At the same time, decreased mitochondrial phosphorylation system flux causes transient inequity between production and consumption of the mitochondrial protonmotive force. Proton extrusion by the respiratory chain exceeds influx through the now slightly less active ATP synthase. The protonmotive force increases, resulting in inhibition of the oxidation reactions that produce it.2+) begins a series of transient inequalities in reaction fluxes that produce and consume intermediates within the energy metabolism pathway. This produces changes in the concentrations of the intermediates connecting them, which then alter the reaction rates of the enzymes. Thus, the initial disturbance propagates throughout the metabolic network until a new steady-state is achieved. Flux changes arising from these indirect changes in the concentrations of key intermediates are probably not important in signaling apoptosis, as these would be changes that occur in the normal course of metabolism of any healthy cell. This description, however, is a qualitative one, where it has been assumed that once a primary disturbance occurs, any flux change elsewhere arises entirely as a result of the changes in the concentrations of intermediates. It is possible that one or more reactions are affected to a greater (or lesser) extent than would be expected simply by the changes in intermediate concentrations. A number of kinases, phosphatases, and/or pro-apoptotic proteins affected by K+ /serum deprivation could mediate these effects. Such direct changes would be candidate signaling steps contributing to apoptosis, but they would be obscured because of the indirect changes that may also occur. Regulation analysis is a branch of metabolic control analysis that can be used to distinguish between primary and secondary changes, and additionally provide quantitative information on how strongly different reactions are directly affected [This series of events illustrates how a direct change in flux through one group of reactions (those that utilize ATP) due to a primary disturbance . Control analysis was developed to quantitatively describe the contribution of each reaction to bringing about small variations in the steady-state flux through any given point in a pathway –61. It i1 and E2 are constant, any change in the reaction rates can be attributed to changes in the concentration of intermediate x. Prior to reaching a steady-state, as E1 produces x, its concentration will increase such that (1) E2 consumes x at an increasing rate as its concentration approaches the E2 Michaelis constant, and (2) E1 produces x more slowly because of product inhibition. The rates ultimately converge to establish a steady-state flux, with the kinetics of both enzymes to x determining when this is achieved. Since the kinetics of E1 to x partly determines the steady-state concentration of x, which in turn influences the rate of E2, then E1 will exert control over the reaction catalyzed by E2. The flux control coefficient is a measure of this control and is defined as:dJE2 is the flux change through E2 as a result of an infinitesimal change dvE1 in the rate through E1. The E1 rate change is denoted v to indicate that its rate has been directly manipulated ; the E2 rate is denoted JE2 to indicate that it is a measured system flux dependent on the concentrations of the intermediates in the system. If E1 exerts only partial control over E2 flux , then E2 exerts the remaining control over its own flux, such that 1 and E2 are part of a larger pathway as illustrated below, then the control E1 and E2 exert over E2 flux will also depend on how the remaining enzyme(s) affect the concentrations of pathway intermediates to which E2 responds.For simplicity, assume an infinite source of substrate S and infinite sink of product P such that their concentrations do not change with time; since their effects on E2- intermediate y- is the substrate for a third reaction catalyzed by enzyme E3. Assuming y exerts inhibition on E2, the kinetics of E3 to y will partly determine the concentration of y, and thus the flux through E2. Enzyme E3 will therefore exert some degree of control over E2 through its effects on establishing the concentration of y, and in so doing lessens some of the control by E1. The concentration of x is no longer the only influence on E2; the product y is now explicitly considered as also affecting E2. As the pool of intermediates that influence a reaction flux increases, so too will the number of reactions involved in controlling the flux because of their effects on determining the concentrations of the intermediates. If, in one particular steady-state, a reaction has more influence on dictating the concentrations of intermediates than in another steady-state, then it will exert more control in the former state, while the remaining enzymes collectively exert less control. Flux control coefficients are therefore properties of the system operating at a given steady-state and are unique to that state.In this example, the product of the reaction catalyzed by EdvE2 is the rate change through enzyme E2 as a result of an infinitesimal change dx in the concentration of intermediate x. Here the rate of the reaction catalyzed by E2 is designated v rather than J to emphasize that elasticities denote how the reaction rate of an isolated enzyme is affected by a small concentration change in one intermediate while all other intermediates are held constant. For elasticities to be applicable to the conditions existing within cells, each must be determined in the presence of the physiological concentrations of all other relevant intermediates and effectors. As with control coefficients, the elasticities are only valid for the steady-state condition in which they are determined.From this example, it is apparent that multiple reactions in a pathway have the potential to control the flux through a reaction of interest in the same pathway because of their effects in determining the concentrations of the intermediates. Quantifying the kinetics of the reactions to the intermediates is central to control analysis, as the balance of these responses determines the control pattern that emerges in a steady-state. In control analysis, the responsiveness of an enzyme to an intermediate is given by the elasticity coefficient:Elasticities and flux control coefficients have been determined for limited segments of metabolic pathways but not complete ones due to the enormous experimental effort required. However, Ainscow and Brand  have shoEnergy metabolism can broadly be defined as those reactions involved in the production and consumption of ATP. Evidence to date suggests that the glycolytic and/or mitochondrial reactions are targeted by apoptosis signaling, so a relatively detailed division of energy metabolism around these blocks is required. m), product of the PYR and NAO blocks, and substrate for the mitochondrial H+ leak (PRL) and phosphorylating (PHO) blocks; (5) ATP/ADP ratio, product of the GLY and PHO blocks, and substrate for the ATP consumer (ATC) block; and (6) NADPH/NADP ratio, product of the PPP block and substrate for the NPO block.The blocks are based on those conceived by Ainscow and Brand to investigate the control of hepatocyte energy metabolism , with a 4.Within cells, reaction block elasticities cannot be directly determined since it is impossible to vary the amount of one intermediate while holding all others constant. They can, however, be calculated using a series of modulations designed to perturb the original steady-state. To see this, first consider a hypothetical isolated reaction block that responds to an explicit intermediate x. The elasticity of the block to x can be estimated by making a finite, small change in x (Δx) and measuring the resulting finite fractional change in velocity.If the block also responds to a second intermediate y, then the process can be repeated by holding x constant while making a small finite change in y to obtain in situ, embedded within the complex energy metabolism pathway and responsive to at least some of the 6 intermediates. If a finite change in the steady-state has occurred due to a change in some physiological parameter that does not directly affect the kinetics of the glycolytic enzymes, then the finite fractional change in flux is approximated by:Now consider the GLY block If glycolytic flux and the concentrations of all intermediates are experimentally measured, then the elasticities can be solved once five additional equations of the same type are generated. This can be done by manipulating different parameters to perturb the original steady-state of healthy neurons in different ways. This is known as the multiple modulation approach to determining elasticities ,67. Each5.IE) coefficient [Jblock/Jblock, which is known as the integrated response IThe indirect effects imply that the parameter change exerts direct effects on at least one block in the system to cause the intermediates to change. Thus, at least one block must have a non-zero integrated elasticity to the parameter. The flux change brought about indirectly by each intermediate is referred to as the partial integrated response of a block acting through an intermediate, and is given by the individual summation terms in In summary, a measured fractional flux change 2+ had very small effects on the control pattern of isolated skeletal muscle mitochondria despite significant flux changes [This is known as the partial integrated response of a block flux to a parameter change due to the direct effect of the parameter on another block. It gives a different perspective from the partial integrated response acting through an intermediate  by indic changes . If gros+ and serum, maintain a constant respiration rate for approximately 3–4 hr [For this analysis to be valid, the system studied must maintain a steady-state after the parameter change. When the parameter change is a stimulus which induces cell death, is a steady-state possible? Granule neurons deprived of Ky 3–4 hr . This is6.m. Most oxygen electrodes require continual stirring to assure uniform oxygen mixing between the electrode and the buffer into which it is inserted. With adherent cells, this is problematic for two reasons: (1) magnetic stirrers commonly used with oxygen electrodes mix from the chamber bottom where attached cells reside, and (2) as a consequence of (1), most commercially available chambers are designed for cell suspensions. The problem can be circumvented by scraping or trypsinizing cells from their attached substrate, but this is not ideal because of potential damage and loss of attachment-mediated signaling. We recently developed a perfusion method that can be used to measure the respiration of attached cells [Respiration rate is central to determining fluxes through the blocks which directly produce and consume ΔΨed cells , but it Granule neurons isolated from rat cerebellum are plated in 2-well Lab-Tek chambers (Nunc), with the intent of turning each chamber into a closed system for measuring respiration rate. An acrylic lid was custom fabricated to fit tightly into the top of the chamber. It was made with two ports - one to hold a conventional micro-oxygen electrode  and one to make additions. Chamber mixing required for proper operation of the oxygen electrode is accomplished with a minineomagnetic stirrer  mounted above the lid to a custom-made removable post that sits in an acrylic base. The neo-magnetic is sufficiently strong to control a micro stir bar through the 1cm thick lid. The lid with oxygen electrode is assembled into the chamber while holding the micro stir bar up against it with the magnet. In this way, the stir bar does not contact the cell layer  and mixeThe setup is kept in a temperature-controlled chamber to record the rates at 37°C. Because the chamber walls are made of oxygen permeable plastic, the rate of oxygen back-diffusion was assessed by oxygen-depleting buffer and measuring the increase in oxygen tension over time . To date+/serum buffer exhibit respiration rates similar to that measured by the perfusion method [Back-diffusion rates were detectable only when >20% oxygen was depleted. In this configuration, neurons in high Kn method ,38, and + leak flux at different ΔΨm values so that PRL, PHO, and ATC fluxes can be calculated  and lactate  . Using t.g., see ).7.Glycolytic and mitochondrial flux changes have been measured in response to trophic factor withdrawal from a variety of cell types. Measured changes in the activity of one or more glycolytic/mitochondrial proteins have been suggested to account for the flux changes. However, metabolic flux is controlled by a number of interactions, all of which should be considered to obtain a comprehensive picture of the routes through which an effector induces a flux change. These interactions include the responsiveness of the enzymes to the intermediates in the pathway (reflected in the elasticity coefficients), and the influence that each enzyme has on the concentrations of the intermediates, and thus the flux catalyzed by the other enzymes in the pathway (reflected in the flux control coefficients). There are few reports indicating the extent to which a direct kinetic change in a particular protein accounts for a pathway flux change. Moreover, the contribution of indirect effects- changes in the concentrations of the intermediates within the pathways of interest- have not been determined during apoptosis. Identifying kinetic changes brought about directly by an apoptotic stimulus, and the importance of this in causing a flux change, would be useful to understanding the possible role of metabolism in signaling apoptosis. Such a comprehensive picture can be attained using the top-down approach to control analysis. The utility of the approach to understanding how hormones alter cellular metabolism has been described, and its direct applicability to understanding the routes responsible for an observed flux change during apoptosis is apparent. Measurement of respiration rate is central to the determination of metabolic fluxes. This can be difficult when using adherent cells such as neurons, but through the use of a new technique such measurements are feasible.


In recent years it has become clear that carcinogenesis is a complex process, both at the molecular and cellular levels. Understanding the origins, growth and spread of cancer, therefore requires an integrated or system-wide approach. Computational systems biology is an emerging sub-discipline in systems biology that utilizes the wealth of data from genomic, proteomic and metabolomic studies to build computer simulations of intra and intercellular processes. Several useful descriptive and predictive models of the origin, growth and spread of cancers have been developed in an effort to better understand the disease and potential therapeutic approaches. In this review we describe and assess the practical and theoretical underpinnings of commonly-used modeling approaches, including ordinary and partial differential equations, petri nets, cellular automata, agent based models and hybrid systems. A number of computer-based formalisms have been implemented to improve the accessibility of the various approaches to researchers whose primary interest lies outside of model development. We discuss several of these and describe how they have led to novel insights into tumor genesis, growth, apoptosis, vascularization and therapy. Living organisms are complex systems. Nowhere is this complexity more evident than in the genesis and development of cancer. While cancer may originate from genetic and molecular changes that occur in a single cell, the subsequent proliferation, migration and interaction with other cells is crucial to its further development. In their landmark paper, Hanahan and Weinberg described six hallmarks they thought necessary for the transition from normal cells to invasive cancers . These iThe molecules that govern the cell growth and division cycle in response to external and internal signals are numerous and interact through complex, multiply-connected pathways over a wide range of temporal and spatial scales. Tumors reflect this complexity in that they are composed of several different cell types that interact to create malignant growth . DespiteComputational systems biology, which is a sub-discipline of systems biology, has developed both as a tool supporting the processing of these massive amounts of data and as a modeling discipline, building upon this “omic” data in order to predict biological behavior . Not surThis review provides an overview of how computational systems biology can be, and is being used to model cancer at multiple levels and scales, ranging from molecules to cells to tissues. Specifically, we begin with a general description of the different computational modeling methods that can be used along with a discussion of their relative strengths and weaknesses. Following that, we describe some of the emerging formalisms, software and standards for representing biological systems. Finally, we provide a number of examples illustrating how computational systems biology has enriched our understanding of a variety of cancer-related processes including genetic instability, tumor growth, apoptosis, angiogenesis and anti-cancer therapy. Overall it is our hope that this review will provide an improved understanding of modeling issues and thereby assist the reader in selecting an appropriate method for their own research.in silico to save time or cost; 4) yield non-intuitive insights into how a system or process works; 5) identify missing components, processes or functions in a system; 6) allow complex processes to be better understood or visualized and 7) facilitate the consolidation of quantitative data about a given system or process.To be truly useful to a biologist or physician, computational modeling should: 1) produce useful predictions or extrapolations that match experimental results; 2) permit data to be generated that is beyond present-day experimental capabilities; 3) allow experiments to be performed Simulations encompass many different spatial and temporal scales, ranging from nanometers to meters and nanoseconds to days . ProcessBuilding models of complex biological processes is an iterative process that requires considerable attention to detail. The network topology or structure of a model may arise through literature surveys or directly by computational analysis of high-throughput data . QuantitBiological systems are essentially multicomponent chemical reactors and thus can be represented as systems of chemical reactions. This view permits mathematical analysis using powerful techniques developed from chemistry. Many standard biochemistry texts provide thorough derivations of ordinary differential equations (ODEs) for both simple and complex reactions. In fact, ODE based modeling is the most common simulation approach in computational systems biology, reflecting both its rigor and adaptability .http://www.llnl.gov/CASC/odepack/).Simple ODEs may have exact solutions. However, most complex ODEs do not have exact solutions and must be solved numerically. Based on methods first derived by Newton and Gauss, numerical integrators utilize linear approximations of smooth curves over small time intervals to compute subsequent values of reactant concentrations. Improving the accuracy of these linear estimates may require using smaller time intervals, leading to computationally intense processes that use considerable machine time. Various methods have been derived to improve the speed and computational accuracy of these approximate methods, including Runge-Kutta algorithms and implicit methods . These algorithms are encapsulated in such publicly-available packages as the LSODA (Liver-more Solver for ODEs with Automatic method switching for stiff and non-stiff problems) or CVODE (C Variable-coefficient ODE solver) integrators . DDEs can reduce the computational effort required to model signal cascades, such as phosphorylation-dephosphorylation cycle networks . This alSystems of ODEs have been used to mathematically model a wide variety of processes including metabolic pathways , mitosisDictyostelium discoideum , a process whereby initial estimates of coefficients and exponents are stochastically perturbed while the system is slowly “cooled” from an initial high pseudo-temperature . SA methBoth conventional systems of ODEs and their Power Law approximations do not automatically reflect compartmentalization, transport and diffusion of molecular species unless explicitly specified. This is because reaction kinetics equations, upon which these mathematical models are formulated, assume steady-states in well-mixed solutions with abundant reactants and few enzymes. The simplifications in standard representations of enzymatic reactions (such as Michaelis-Menton equations) incorporate assumptions about the relative rates of intermediate steps which are independent of transient localized differences in concentrations. In reality, the cellular space even inside bacteria is crowded with macromolecules, having 300 to 400 g/l of macromolecules (e.g. protein and RNA) occupying 20 to 30 percent of cytoplasmic space, compared to the 1 to 10 g/l under which biochemists normally study reaction kinetics . MacromoMacromolecular crowding also affects reaction kinetics. Equilibrium rate constants for macromolecular association reactions under crowded conditions can increase by two to three orders of magnitude compared to dilute concentrations . Thus thWithin the cell, macromolecular crowding also leads to the formation of numerous non membrane-bound three-dimensional compartments and one-dimensional channels where reactions may occur. In addition, membranes can act as two-dimensional surfaces, leading to localized concentrations of proteins and complexes . While cP, the reaction probability density function; where Pdt = the probability that, given the state  at time t, the next reaction, will occur in the infinitesimal time interval . For any reactant in any time interval in a chemical reaction system P can be calculated in a straightforward fashion from a uniform pseudorandom generator . Classical ODE solvers are highly unstable and may fail to accurately reflect the granularity and discrete system behaviour typically found at such low concentrations. Because of this, some modelers have added “noise terms” to their systems of equations by using so-called stochastic differential equations (SDEs) . An alteator see  for a coHox gene expression in the developing vertebrate hindbrain  must be used if one wishes to incorporate explicit spatial distribution of components into a model. PDEs have been used to model several processes in cancers, including chemotactically-directed tumor growth , growth http://www.cellml.org/examples/repository/) as well as in the JWS online system (http://jjj.biochem.sun.ac.za) which is part of the silicon cell project (http://www.siliconcell.net) (http://www.ebi.ac.uk/biomodels/). Historically, the effort and mathematical skill required to set up a useful ODE or PDE model put this approach beyond the reach of most experimental biologists. However, recent advances in graphical interface (GUI) design, improved standards in displaying and generating reaction models along with the development of standardized mark-up languages such as SBML (Systems Biology Mark-up Language) and Cell-ML (Cell Mark-up Language) are making the generation and exchange of interesting biological models relatively simple . A largeell.net) . AdditioPetri nets are a discrete alternative for representing time-dependent processes such as those occuring within biological systems . Petri nPetri nets were originally designed to model discrete processes but later enhancements have added the ability to deal with continuous quantities . In addiE. coli lac operon . So-called “lattice-free” systems use the lattice to represent real, physical space and individual entities may span more than one cell . Time is a discrete entity in CA models. At time tn the state and neighbors of each cell are tallied and rules applied to determine the state transition for that cell in the next interval tn+1.An alternative approach to modeling the complex systems of discrete molecules that are found in living organisms is to use cellular automata (CA) to represent individual molecules and the rules that govern their interactions. Cellular automata (CA) are simple computer simulation tools that can be used to model both temporal and spatio-temporal processes using discrete time and/or spatial steps. Cellular automata were invented in the late 1940’s by von Neumann and Ulam  who concRules of varying complexity govern the interactions between adjacent or nearby molecules. Rules may be quite simple, for example, specifying binding of adjacent molecules with a certain probability . AlternaIn order to realistically simulate complex systems of reacting molecules, stochasticity needs to be added to the deterministic rules of the original cellular automata formalism. Dynamic Cellular Automata (DCA) permit “Brownian-like” motion of individual molecules through the incorporation of a random number generator which selects a direction of motion in the next time step . DependiChemical reaction rates are emergent properties of DCA models although molecular reaction probabilities may be derived from conventional reaction rates. However, given the problems in deriving biologically relevant reaction rates that take into account macromolecular crowding, extremely low concentrations and compartmentalization, it may be more accurate to compute parameters that match the biochemical profile of a well-known test model.DCA models have been used to model a wide variety of processes including diffusion , micelleModels of complete cells, even simple bacteria, could potentially incorporate billions of individual molecules. Interactions between molecules must be specified in a computationally efficient manner if such models are to compete with solution times for the mere hundreds of equations potentially found in comparable ODE systems. Nonetheless interactions must be firmly based on real forces or accepted approximations. For example, a recent DCA model of lipid bilayer dynamics in a 60 nm diameter proto-cell with over 10,000 individual components used the Lennard-Jones potential as the basis for attractive and repulsive forces to realistically model the membrane .Agent Based Modeling (ABM) is similar in concept and design to Dynamic Cellular Automata. In ABMs genes, proteins, metabolites or cells can all be “agents”. Agents are allowed to interact with each other over space and time according to a pre-defined set of rules. The motions may be directed or random (Brownian) and the rules may be simple or highly complex. Unlike CA models, agent based systems do not formally require spatial grids or synchronized time steps, although practical coding considerations usually force these constraints on ABMs. Space is usually represented in a lattice-free grid. ABMs share many of the same advantages and disadvantages as DCA or CA models. Agent-based models have been used to simulate bacterial chemotaxis , to modeODE, Petri Net, DCA and ABM approaches all have their unique advantages and disadvantages. While DCA and ABM systems have molecular-grain accuracy and implicit simulation of compartmentalization, diffusion and stochasticity, they can also be computationally more intensive as the number of components in a model can increase rapidly . ODE sysA number of researchers have recommended hybrid or hierarchical hybrid systems to combine the strengths of both discrete and continuous approaches . For exasonic hedgehog signaling in the development of medulloblastomas  which incorporate Content MathML (http://www.w3.org/TR/MathML2/) and Resource Description Framework (RDF – http://www.w3.org/RDF/) for metadata description, only made this situation worse. Fortunately a large number of support programs have enabled biologists to define components of an SBML system using simple graphical descriptions of reactions and compartments that are much more easily interpretable. Links to web-sites supporting SBML models can be found at http://www.sbml.org.Representing a simple reaction such as KineticLaw reaction definition which incorporates compartment volumes into standard reaction rate equations  Human Physiome Project . CellML http://www.cellml.org/specifications and over 350 models are currently contained in a repository at http://www.cellml.org/models/.The modeling power of CellML is based on the ability to explicitly define the mathematical relations between all components. Compartments within a cell are considered components into which other components may be grouped. Thus, movement of components between compartments must be defined explicitly. Stochastic modeling is not yet supported by CellML nor is the modeling of discrete objects. The CellML and SBML project teams are cooperating to incorporate each other’s strengths, where possible, and to provide for models in one formalism to be translated into the other . CompletNo single formalism represents the Cellular Automata approach to modeling Usually the definition of each CA formalism is encapsulated in its specific implementation. Recently, however, some effort has been made to formalize a more consistent approach to generating and sharing hybrid CA models of cellular processes . This hyFrequently, translation into a specific formalism is the most difficult step in converting a drawing of some process into a formal model for that process. The notation systems for all formalisms look nothing like the diagrams that biologists and biochemists are used to . Fortunately computer-based graphical tools have been developed permitting the entry of a model with symbols familiar to biologists (or easily learned) and subsequent machine conversion into a particular formalism. A number of these are listed in http://sbml.org/index.psp) should make computational systems biology far more accessible to a much larger community of life sciences researchers.In addition to defining a model, simulation requires some computer program to run the model over a period of time or iterations producing output for interpretation. Output usually consists of graphs of the concentration of some component over time  but may also include dynamic representations of the system or other analysis. The programs listed in As was noted earlier biological simulations should not be considered as simple academic exercises. Rather they should aim to inform researchers about some unobvious characteristic of the simulated process. It is appropriate to ask, then, how models of cancer have led to novel insights into the disease. In the following section we review how modeling and computational systems biology has contributed to our understanding of the underlying molecular, cellular and tissue-level mechanisms of cancer.4 at the lower limit to a maximum of about 1012 , insensitivity to inhibitory signals, evasion of apoptosis, limitless replication, self-sufficiency in growth and sustained angiogenic signaling. Growth of the vascular system under angiogenic signaling was overlaid on the grid.A more recent stochastic cellular automata model based on Hanahan and Weinberg’s “hallmarks of cancer” was proposed to resolve the issue of the importance of genetic instability . A 100 ×Simulated early onset tumors in this model were dominated by genetic instability while late onset tumors were driven largely by acquisition of limitless growth. The model further emphasized the importance to tumor growth of acquiring multiple different mutations and demonstrated large fluctuations in tumor heterogeneity and size as they developed. However, though it wasn’t required for initiation of tumor formation and development, angiogenesis was found to be the primary factor driving tumor growth beyond early stages. It is interesting that cells supporting angiogenesis did not necessarily dominate larger tumors. Instead other cell types with growth or survival advantages would “piggyback” their development off the increased blood supply stimulated by a nearby angiogenic sub-population .Pre-vascular tumor tissue growth can be modeled mathematically from macroscopic growth curve data. Tumor size often assumes a sigmoidal growth pattern which many earlier researchers interpreted as demonstrating the dominant role played by surrounding growth inhibitors . HoweverEpidermal growth factor receptor  overexpression has been strongly implicated in highly malignant brain tumors. A multi-scale agent-based model has been developed by Deisboeck and colleagues over the past several years to investigate the relationship between EGFR dynamics, tumor cell proliferation and cell migration . The modThe model confirmed previous experimental findings that increasing EGFR density on the cell surface correlates with an increase in the rate of tumor expansion . The mod4 cells in a full 3-D simulation, though most data was collected using a 3-D monolayer. Growth inhibitory cell-cell interactions were modeled as well as cell-substrate division inhibition and anoikis, a form of programmed cell death initiated when cells lose contact with their underlying matrix. The strength of the cell-substrate adhesion was found to be critical in inhibiting formation of spheroids atop the epithelial layers  or low (125/μm2) densities. Diffusion probabilities were calculated using random walk theory from diffusivity constants while reaction probabilities were similarly calculated from kinetic constants. To improve computational efficiency only lattice-sites containing receptors were randomly selected for movement or reaction at each time step. Three possible pathways to EGFR activation were simulated: 1) dimerization followed by ligand-binding, 2) ligand-binding followed by heterodimer formation followed by more ligand-binding, and 3) ligand-binding followed by homodimer formation of ligand-receptor complexes. The model demonstrated a dependence on both ligand and receptor concentration as to the specific mechanism that was most favored.At the molecular scale, the dynamics of EGFR ligand binding and receptor dimerization is still largely unknown. A dynamic cellular automata (DCA) model incorporating Monte Carlo methods has been used to simulate EGF receptor activation and to compare the predictions to single-particle tracking studies . ReceptoTumors do not always develop in the relatively uniform mesenchymal environment. Lung adenocarcinomas originate and develop within the epithelia of the alveoli and the resulting tumor has a distinct non-spherical shape . The alvWhether in a mesenchymal or epithelial environment, tumor growth is actively inhibited by interactions with NK cells and cytotoxic T cells of the immune system. A hybrid CA-PDE model has recently been used to simulate the complex response of tumors to growth signals while under attack by the immune system . Cells wEscaping apoptotic signaling is an important mechanism cancer cells use to avoid being cleared from the body early in tumor develpoment . In normApplication of the model led to the discovery of a threshold mechanism in apoptotic signaling. The c-FLIP protein inhibits DISC activity by strongly binding to the caspase-8 activating site of DISCs. Below threshold levels of CD95 signaling, c-FLIP binding was found to limit the number of active DISCs. As a result no cells died. Above the threshold level, the number of activated DISCs exceeded the cytoplasmic c-FLIP pool and apoptosis was predicted to occur in all cells. Following chemotherapeutic treatment, many cancers develop apoptotic resistance, which may be due to the abnormally high c-FLIP expression found in certain cancer cells . SubsequNF-κB (Nuclear factor—kappa B) is an important downstream transcription factor also implicated in the regulation of apoptosis as well as in cell signaling, growth and response to stress. It has been identified as a therapeutic target in chronic inflammatory diseases as well as in cancer . An ODE Simulations using the adjusted parameters led to the surprising discovery that the different inhibitory proteins  act together to produce different components of NF-κB expression levels. By itself, IκBα results in strongly oscillatory levels of NF-κB expression, while increasing levels of IκBβ and IκBɛ led to first a dampening of the oscillations and then a levelling out of NF-κB expression at a plateau. This mechanism predicted that transient stimulation by TNF-α should lead to prolonged expression of NF-κB resulting in sustained production of downstream genes and protection from apoptosis.In general, the growth of solid tumors is limited to about 0.5 mm diameter without access to an oxygenated blood supply. To develop beyond this size, tumors may stimulate the growth of new blood vessels. This process is called angiogenesis. It is dependent upon the secreted growth factor Angiopoietin 2 (Ang2) binding to the transmembrane Tie2 receptor tyrosine kinase . In the An ODE simulation of the effect of this vascular remodeling on tumor growth has recently been developed . The modAngiogenic factors are secreted by hypoxic cells on the periphery of the central necrotic region of the tumor . The celThe hypoxic response pathway was modeled by a system of ODEs reflecting the molecular kinetics of 17 compounds and validated to data from several independent experiments . The modTherapeutic approaches to cancers include surgical removal (in whole or part), partial or whole body irradiation and a vast array of chemotherapies. Chemotherapeutic agents may be broadly classified as cell-cycle phase-specific, cell-cycle non-specific, or cytostatic/anti-angiogenic . Phase-sChemotherapeutics are frequently administered in combination, because tumors are comprised of heterogeneous cell populations with different metabolic profiles and susceptibilities to attack by drugs. Gardner developed a kinetically tailored treatment (KITT) model which utilized a pair of ODEs (plus other supporting equations) to incorporate rates of tumor genesis, growth, apoptosis, necrosis, drug-induced death, development of drug resistance, cytotoxic side effects and drug pharmacokinetics . Tumor gThe effectiveness of this two-component chemotherapeutic approach has been verified by a simple delayed differential equations model, relating tumor growth, immune system attack and drug-induced cell cycle inhibition . In thisSimilar findings were reported by Crispini and colleagues in their model of the interaction between tumor morphology and invasiveness and the micro-vasculature that develops under angiogenic conditions . They usHowever, anti-angiogenic therapies, rather than suppressing all vascular extension into the tumor, have actually been shown to normalize tumor vasculature by pruning immature vessels . Recent In addition to optimizing therapeutic components and dosages, another challenge facing clinicians relates to selecting the most efficacious administration schedule. For example, in order to effectively prevent mammary carcinogenesis in HER-2/neu transgenic mice the Triplex vaccine must be administered beginning at six weeks after birth and repeated every two weeks for the entire lifespan of the mouse . An ABM Novel cancer therapeutics can have even more complex pharmacodynamics. For example, radio-virotherapy combines radiation therapy with the increased sensitivity of cancer cells to some viral infections. A viral genome (e.g. measles) is altered so that infection of a cell also introduces the gene encoding a transmembrane ion import pump. Expression of the ion pump in infected (but not dead) cells causes them to concentrate intravenously admitted radioactive ions leading to tumor regression . The intAlthough a number of important insights have already resulted from the relatively simple and non-comprehensive models presented above, it is clear that a number of challenges still exist. One of the most obvious is the lack of precise, quantitative data for gene and protein networks so that realistic parameters can be entered into these models. While a considerable amount of transcriptional and translational data is available, good quality metabolic data lags far behind. The situation is complicated in metazoans because, even in a disease state, not all cells in a tissue will express the same genes, proteins or metabolites. Single-cell isolation and analysis, including lab-on-a-chip and protein-nanoarrays, may improve this situation somewhat.Ultimately, models of living systems attempt to simulate extremely complex mixtures of proteins, genes and metabolites. Approximations to the actual reactions occuring in such a complex mixture are made of necessity and the resulting simplifications may lose important, relevant information. For example, when modeling new candidate drugs, it may be impossible to predict all potential undesirable interactions with non-target molecules. While network models may be able to help identify some of these interactions based on resulting physiological changes, only accurate prediction of all possible aberrant reactions at the molecular level could obviate this issue entirely.The proliferation of approaches and implementations makes it difficult to select the “best” approach for any particular modeling problem see . NeverthMany researchers support the move towards a clear, consistent formalism to depict biological processes. Kitano has defined a System Biology Graphical Notation  which haIn recognition of the deleterious effect this proliferation of approaches has on the advancement of modeling, particularly in relation to cancer research, the National Cancer Institute has recently established the Integrative Cancer Biology Program. The Center for the Development of a Virtual Tumor (CViT) at the Massachusetts General Hospital in Boston is one member of this program . The CVi


To solve the problem more efficiently, improved numerical optimization techniques are necessary.The importance and power of isotope-based metabolic flux analysis and its contribution to understanding the metabolic network is increasingly recognized. Its application is, however, still limited partly due to computational inefficiency. Bacillus subtilis with only succinate and glutamate as carbon sources. This creates difficulties caused by symmetry of succinate leading to limited introduction of 13C labeling information into the system. The algorithm was found to be superior to its parent algorithms and to global optimization methods both in accuracy and speed. The hybrid optimization with tolerance adjustment quickly converged to the minimum with close to zero deviation and exactly re-estimated flux variables. In the metabolic network studied, some fluxes were found to be either non-identifiable or nonlinearly correlated. The non-identifiable fluxes could correctly be predicted a priori using the model identification method applied, whereas the nonlinear flux correlation was revealed only by identification runs using different starting values a posteriori.For flux computation, we developed a gradient-based hybrid optimization algorithm. Here, independent flux variables were compactified into [0, 1)-ranged variables using a single transformation rule. The compactified parameters could be discriminated between non-identifiable and identifiable variables after model linearization. The developed hybrid algorithm was applied to the central metabolism of a posteriori identification of possible parameter correlations, and also for Monte Carlo simulations to obtain statistical qualities for flux estimates. In this way, it contributes to future quantitative studies of central metabolic networks in the framework of systems biology.This fast, robust and accurate optimization method is useful for high-throughput metabolic flux analysis,  This design was selected to study flux identifiability and to examine the developed hybrid optimization.The choice of input tracer substrates was made by means of the riterion . Since psurement . Therefo-4.For simulation studies, we assumed that the mass isotopomers of proteinogenic amino acids such as alanine, valine, serine, threonine, glycine, aspartate, isoleucine, leucine, phenylalanine, tyrosine, arginine, histidine, and glutamate are measurable. In addition, the mass isotopomers of ribose 5-phosphate from RNA hydrolysate and hexose from carbohydrate hydrolysate are also assumed to be measurable. This totally gives a mass isotopomer data size of 188 for the parallel experiment. The measurements of mass isotopomer distributions were assumed to have a unit standard deviation of 1 × 10ϕi) were bounded such that the relative values of the corresponding dependent fluxes (νdepend) lie between 0 and 20.All simulations were carried out on a PC equipped with a CPU of 2.4 GHz. The hybrid optimization algorithm depicted in Figure An example of the hybrid optimization for the problem described in the current work was coded using MATLAB (Hybridoptimizer.m) and supplemented , the model function kth iterate Θk to get the values corresponding to η = T. This is done by generating a certain flux state by νk = Fflux(Θk) and, subsequently, solving Fcarbon = 0 for xk at the kth flux state νk. Since Θk, the gradient ∇f can be computed analytically by:At each H that specifies the curvature of the search surface can be obtained by linearization [Further to this, the Hessian matrix rization ,15, i.e.In practice, the constrained problem (1) is formulated as the Lagrangian function, a linear combination of the objective function and the constraints .kth trial is acceptable , the (k + 1)th trial restarts with a 10-fold decreased tolerance, where the kth local minimizer Θ*k is used for the new starting point. The previous trial provides a more adequate starting point for the next, and it can be advantageous for local search methods. This is because the efficiency of local methods depends heavily on the quality of starting points [Θ° recorded for the smallest function value up to the current trial in the feasible region th trial with αk-n, the [0, 1)-fluxes have to be rescaled in accordance with the new scaling constant αk. Since ν = αk-nϕk-n/(1 - ϕk-n) from (2), substituting ν in ϕk = ν/(αk + ν) results in:At the beginning and at every failed trial, When the random generation of starting points  evaluated at the current iterate Θk, and n the number of the balances. By the first term in the objective and the first constraint, the minimization is directed such that all fluxes become non-negative. The second term is to prevent flux values calculated that may not obey stoichiometry due to the limited floating-point accuracy (round-off error). This approach is useful for preparing initial values while preserving random nature for Monte Carlo simulations. In practice, the upper bound of ϕi can be set to a value smaller than 1 that gives sufficiently large flux values so that ϕi has a closed interval.Here, Stationary State Stoichiometric Balances are set up around 15 metabolites specified in Figure XS equals the strain specific precursor demand for growth (mmol precursor per gram biomass).ν2 - (ν1 + νr 2+ 0.4217YXS) = 0glucose 6-P: νr 2+ ν4 + ν5 + ν6) - (ν2 + νr 4+ νr 5+ νr6) = 0fructose 6-P: (ν3 + νr 4+ ν5 + 2νr 6+ ν7) - (νr 3+ ν4 + νr 5+ 2ν6 + νr 7+0.4659YXS) = 0glyceraldehyde 3-P: (νr 7+ ν8) - (ν7 + νr 8+ 0.2209YXS) = 03-phosphoglycerate: (ν16 + νr8) - (ν8 + ν9 + 1.7854YXS) = 0phosphoenolpyruvate: (ν9 + ν15) - (ν10 + νr 15+ 0.9964YXS) = 0pyruvate: (ν1 + 2νr 3+ νr5) - (2ν3 + ν5 + 1.1622YXS) = 0pentose 5-P: (ν4 + νr5) - (νr 4+ ν5 + 0.4659YXS) = 0erythrose 4-P: (ν3 + νr4) - (νr 3+ ν4) = 0sedoheptulose 7-P: (ν10 - (ν11 + acetateex + 3.1585YXS) = 0acetyl-CoA: ν11 - ν12 = 0isocitrate: α-ketoglutarate: (ν12 + ν17) - (ν13 + νr 17+ α-ketoglutarateex) = 0νr 17+ glutamateex) - (ν17 + 1.4769YXS) = 0glutamate: (ν13 + νr 14+ succinateex) - (ν14 + fumarateex) = 0succinate: (ν14 + νr15) - (ν11 + νr 14+ ν15 + ν16 + 1.3131YXS) = 0oxaloacetate: (13C-glutamate and 13C-succniate for single and two parallel experiments.Optimal Experimental Design. the relative information index expected from the tracer designs consisting of different Click here for fileHybridoptimizer. an example of the hybrid optimization for the problem described in the current manuscript coded using MATLAB.Click here for file