Saccharomyces cerevisiae is an important microorganism for both industrial processes and scientific research. Consequently, there have been extensive efforts to characterize its cellular processes. In order to fully understand the relationship between yeast's genome and its physiology, the stockpiles of diverse biological data sets that describe its cellular components and phenotypic behavior must be integrated at the genome-scale. Genome-scale metabolic networks have been reconstructed for several microorganisms, including S. cerevisiae, and the properties of these networks have been successfully analyzed using a variety of constraint-based methods. Phenotypic phase plane analysis is a constraint-based method which provides a global view of how optimal growth rates are affected by changes in two environmental variables such as a carbon and an oxygen uptake rate. Some applications of phenotypic phase plane analysis include the study of optimal growth rates and of network capacity and function.The yeast Saccharomyces cerevisiae genome-scale metabolic network was used to formulate a phenotypic phase plane that displays the maximum allowable growth rate and distinct patterns of metabolic pathway utilization for all combinations of glucose and oxygen uptake rates. In silico predictions of growth rate and secretion rates and in vivo data for three separate growth conditions  were concordant.In this study, the Taken together, this study examines the function and capacity of yeast's metabolic machinery and shows that the phenotypic phase plane can be used to accurately predict metabolic phenotypes and to interpret experimental data in the context of a genome-scale model. The development of numerous high-throughput experimental techniques such as DNA microarrays, genome sequencing, and protein chips has revolutionized the analysis of biological systems and generated a catalog of information about a cell's components -3. EfforSaccharomyces cerevisiae. Flux-balance models of yeast have appeared for small-scale network reconstructions. Most of these studies are specific for growth conditions, such as anaerobic, glucose-limited metabolism  (mM) is the substrate/product or DO concentration in the flask, q (mmole/g-dry weight/hr) is the substrate uptake rate or by-product formation rate or OURflask, and X(t) (g-dry weight/L) is the biomass concentration at time = t (hr). + is for the by-product formation and - is for the substrate consumption. The solution to this equation was fitted to the experimental data points using the nonlinear estimation routine in Statistica  or the solver in Microsoft Excel. All data, to be considered valid and included in the analysis, must have correlation coefficients of 0.95 or greater. The data that were generated in this way represented the "pseudo-steady-state" [flask.where y-state"  of the bS. cerevisiae metabolic network has been reconstructed [x, and reaction activities, v, is described by the dynamic mass balance equation [A genome-scale structed . The netequation ,44:S is an m × n matrix of stoichiometric coefficients, x is an m × 1 vector of metabolite concentrations, and v is and n × 1 vector of reaction activities. Thus, the rows of S correspond to the internal metabolites and the columns represent the reactions in the network. Under steady-state conditions, the dynamic mass balance equation simplifies to:where S • v = 0     (Eq. 6)S. cerevisiae for various growth conditions [Since the number of reactions is often greater than the number of metabolites, Eq. 6 is underdetermined and contains multiple solutions. One approach to solving Eq. 6 for microbial networks is to define a set of inputs and outputs that correspond to the growth conditions and use linear optimization to maximize the cell growth . This apnditions .S. cerevisiae PhPP displays optimal growth rates for all possible variations in two constraining environmental variables, such as the carbon substrate and oxygen uptake rates. In this study, the glucose uptake rate (x-axis) was allowed to vary from 0 to 20 mmol/gDCW/hr and the oxygen uptake rate (y-axis) ranged from 0.1 to 20 mmol/gDCW/hr. The oxygen uptake rate was not allowed to reach zero because anaerobic simulations required additional supplements to maintain cell growth (ergosterol and zymosterol). Linear programming was used to calculate the optimal growth rate for all points in the x-y plane. Growth rate values were then plotted as the z-axis to form the surface of a three-dimensional PhPP  to changes in the availability of each metabolite:i is the ith metabolite and γi is the ith shadow price. Shadow prices were calculated for each point in the x-y plane during the linear programming simulations. By definition, phases were identified as regions of the PhPP in which all of the points have the same shadow prices. Lines of optimality, which represent the optimal ratio of glucose and oxygen uptake rates for maximal biomass synthesis, were also identified using shadow price analysis [where banalysis .e.g. the value of the objective function increases if the metabolite's net production increases or its net consumption decreases. Similarly, a positive shadow price indicates that a metabolite is available in excess and a shadow price equal to zero indicates that a change in the availability of the metabolite does not affect the objective value.To obtain a physiological interpretation of the differences between the oxidative fermentative phases (phases 2–6), we analyzed how the shadow prices of key metabolites changed across the phase boundaries. Simulations were run at a fixed glucose uptake rate of 5 mmol/gDCW/hr and an oxygen uptake rate ranging from 1.5 to 15 mmol/gDCW/hr. The sign of the shadow price was used to identify whether a small change in the metabolite's availability would positively or negatively affect the objective value. According to the convention defined in , a negatin silico . Single-5 and 1.27 × 10-4 mmol/gDCW/hr, respectively, were used for the calculation at OUR = 0). The simulations were then re-run with a fixed glucose uptake rate, oxygen uptake rate, and growth rate to determine the maximum and minimum secretion rates of each metabolite with a shadow price equal to zero.The first step in generating the secretion profile was to calculate the optimal growth rate for a given glucose and oxygen uptake rate. For the simulations in Figures NCD calculated the phenotypic phase plane Fig. , carried


Mycobacterium tuberculosis continues to be a major pathogen in the third world, killing almost 2 million people a year by the most recent estimates. Even in industrialized countries, the emergence of multi-drug resistant (MDR) strains of tuberculosis hails the need to develop additional medications for treatment. Many of the drugs used for treatment of tuberculosis target metabolic enzymes. Genome-scale models can be used for analysis, discovery, and as hypothesis generating tools, which will hopefully assist the rational drug development process. These models need to be able to assimilate data from large datasets and analyze them.Mycobacterium tuberculosis H37Rv. This functional in silico bacterium, iNJ661, contains 661 genes and 939 reactions and can produce many of the complex compounds characteristic to tuberculosis, such as mycolic acids and mycocerosates. We grew this bacterium in silico on various media, analyzed the model in the context of multiple high-throughput data sets, and finally we analyzed the network in an 'unbiased' manner by calculating the Hard Coupled Reaction (HCR) sets, groups of reactions that are forced to operate in unison due to mass conservation and connectivity constraints.We completed a bottom up reconstruction of the metabolic network of in vitro and in silico or in vivo and in silico results highlight specific loci that are worth further experimental investigations. Finally, by considering the HCR sets in the context of known drug targets for tuberculosis treatment we proposed new alternative, but equivalent drug targets.Although we observed growth rates comparable to experimental observations (doubling times ranging from about 12 to 24 hours) in different media, comparisons of gene essentiality with experimental data were less encouraging . The reasons for the often conflicting results were multi-fold, including gene expression variability under different conditions and lack of complete biological knowledge. Some of the inconsistencies between  Tuberculosis continues to be a devastating pathogen throughout the world, particularly in developing nations. In 2001, the World Health Organization (WHO) estimated 8.5 million new cases of tuberculosis (based on 3.8 million new reported cases) and an estimated 1.8 million deaths from tuberculosis in 2000 , two of Mycobacterium tuberculosis can survive in a wide range of environments (many different tissues) and fairly extreme pHs [Many of the unique properties of tuberculosis are attributable to its metabolism, particularly the complex fatty acids characteristics of the organism. These mycolic acids, phenolic glycolipids, and mycoceric acids confer many of the properties such as its acid-fastness and are believed to contribute to the resilience of the organism. reme pHs . One of Mycobacterium tuberculosis H37Rv (M. tb) in 1998 [in vitro , and identifying new drug targets for treatment.DNA sequencing of the ~4.4 Mbp genome of  in 1998  enabled M. tb in silico (iNJ661), from which we developed a model to compute perform computational analyses and interpret experimental data. These bottom-up reconstructions have been described in the past as, biochemically, genetically and genomically structured (BiGG) 'databases'. We employ constraint-based reconstruction and analysis (COBRA) of this BiGG reconstruction to learn about its normal metabolic function and to infer new potential targets for drugs.In order to gain understanding about the unique characteristics of this important pathogen, we manually reconstructed the metabolic network of iNJ661, which has 661 genes and 939 intra-system reactions, are summarized in Table M. tb H37Rv and other mycobacteria strains if information was lacking. The biomass objective function was defined using the literature for chemical composition studies of M. tb [Mycobacterium bovis was used . The sioglycans . The bioiNJ661 in silico maximizing the biomass function defined in Table iNJ661 was grown in silico on three different types of media, Middlebrook 7H9 (supplemented with glucose and glycerol) [iNJ661.Flux Balance Analysis (FBA) was used to grow lycerol) , Youmanslycerol) , and thelycerol) . Uptake lycerol) ,19, and lycerol)  are withM. tb growth on Middlebrook 7H9 media requires supplementation with glucose and glycerol. Glycerol is a component of the biomass function  can be used to calculate the span of each flux while still achieving the maximum value for the objective function. Reactions with non-zero fluxes, in which the maximum flux equals the minimum flux (and is non-zero), are essential in achieving the particular objective . Flux variability [As noted by Primm et al , M. tb gon Table  accordinconstant . Furtheriability  at maximin silico. Only one of these, however, has been suggested to be essential experimentally (Rv3490) [iNJ661 in Middlebrook media (with glycerol but without glucose supplementation) severely retards the growth rate, however it is still viable in silico in contrast to M. tb growth in vitro. The discrepancy between in vitro essentiality of glucose, but non-essentiality of reactions requiring glucose derivatives and in silico non-essentiality of glucose suggest that there may be non-metabolic functions related to the need for glucose, in addition to alternative metabolic pathways for the production and/or use of glucose-6-phosphate.The non-zero fluxes with no flexibility directly involving glucose were the glucose transporter  and glucokinase . Glucose-6-phosphate is subsequently essential for a number of other reactions involving various aspects of membrane metabolism, including the synthesis of trehalose phosophate  and inositol-phosphate , in addition to phosphoglucomutase . These three reactions are required not only for optimal growth, but they are in fact essential for any growth at all in silico. Although the uptake fluxes are small compared to the uptake of the carbon sources and oxygen, both are required for growth of M. tb. Consequently, the sulfate and phosphate transporters may serve as bacteriostatic and possibly bacteriocidal drug targets. A robustness analysis (see Methods) was carried for biomass production by iNJ661 for phosphate and sulfate [see Additional file in silico E. coli strain iJR904 [iNJ661 and iJR904 are both approximately 1. However, the slope for sulfate uptake in iNJ661 is approximately four times that of iJR904 (about 16 for iNJ661 compared to about 4 for iJR904). This difference reflects the relatively larger amount of sulfur as a percent of total biomass in M. tb compared to E. coli, which is consistent with knowledge of M. tb's composition [M. tb may be much more sensitive to sulfate depletion than E. coli.Figure n iJR904  on aerobn iJR904 . The senposition . CollectM. tb is its ability to grow in culture with carbon monoxide (CO) serving as the sole carbon source [M. tb lacks ribulose-bisphosphate carboxylase (RuBisCO), an enzyme found in plants enabling fixation of carbon dioxide. Although it was argued that a reductive citric acid cycle may enable the fixation of carbon dioxide into by M. tb [M. tb growth on CO [iNJ661 to grow on CO, since the only experimentally characterized CO associated metabolism in M. tb was CO dehydrogenase (CODH). The uptake of glycerol versus the uptake of CO while optimizing for growth can be seen in Figure 2). Looking further into this issue, we tested the hypothesis that RuBisCO would enable growth on CO (even though M. tb is not known to have it). We added RuBisCO in addition ribose 1-phosphokinase and ribose 1,5-bisphosphate isomerase (to complete the pathway) and optimized for growth on a modified Middlebrook 7H9 media (glycerol and glucose were removed and CO was added). iNJ661 was unable to grow under these conditions, despite conferring the ability to fix CO2 with ribose sugars. This observation in conjunction with Figure M. tb is able to grow using CO as the carbon source is that it somehow is able to fix CO into a glycolytic intermediate that does not appear to occur through a RuBisCO pathway or a reductive citrate cycle.A remarkable property of n source ,27. Unliby M. tb , experimM. tb in different environmental conditions, Sassetti et al [M. tb in vitro (this will be referred to as the OptGro dataset throughout the rest of the text), Gao et al [M. tb survival in vivo during infection in mice [in vivo and in vitro patterns of gene expression may be expected, the OptGro and ConstExp datasets were both done in vitro. The heterogeneity among the datasets  suggests that defining an objective function based on in vitro data, may not necessarily translate to useful results in vivo (it may constrain the solution space from the wrong directions).This BiGG reconstruction can be used as a 'context for content' in analyzing large biological datasets. Multiple investigators have employed high-throughput data analysis methods to characterize ti et al  used traao et al  identifi in mice   results between iNJ661 and the OptGro dataset .We evaluated these three datasets for iNJ661 were reactions associated with the biosynthesis of vitamins and cofactors, such as dihydrofolate and tetrahydrofolate. 16 such metabolites were added [see Additional file iNJ661 gene essentiality, but in vitro gene non-essentiality) results .Since quantitative data was not available for vitamins and cofactors, they were not included in the original definition of the objective function Table . HoweverM. tb. The AL cases are enumerated in Table in silico growth of iNJ661, the lack of a biochemically characterized gene protein relationship can result in false predictions.Genes categorized as AL or AR suggest that there was an error in the sequence based annotation, there are regulatory control loops that we are not aware of, or there was a false positive result from the experimental data (TraSH is often used as a screening tool). Sassetti et al  dicussedM. tb that are not in iNJ661. Almost half of these cases (46) involve fatty acid, membrane, or peptidoglycan metabolism, since the fatty acid composition in M. tb can change significantly over time in vitro [in silico simulations have fixed biomass compositions). These cases highlight areas in which experimental studies documenting the changes in composition and growth rates may yield interesting results. Five of the cases involve loci associated with the transport of phosphate, ammonium, and a sugar transporter. These cases likely involve as of yet unidentified alternative transporters in M. tb. Another case involves glycerol kinase  which was one of the fluxes with no flexibility (see M. tb growth in silico), since M. tb in vitro requires glycerol supplementation for growth [M. tb. The ability to grow using CO as the sole carbon source and maintain optimal growth even without glycerol kinase activity imply unique and unidentified aspects to central metabolism in the M. tb network. Further experimental investigations may provide clues towards controlling the growth of M. tb (increasing the growth rate in culture and inhibiting growth in vivo).The 103 false positive cases likely reflect incomplete knowledge about the network  or an alternative objective function [see Additional file in vitro , many ofr growth . HoweverOut of the 194 genes identified in the Infect dataset, only 17 were shared with the OptGro dataset and 37 were shared with the ConstExp dataset Figure . From thM. tb in culture over the span of weeks [M. tb causes one to adopt the view that single enzyme drug targets actually knock out complete pathways. As a result, terminating the activity of any other enzyme in that pathway should have the same effect. Similar to the correlated reaction sets calculated by sampling or flux coupling, HCR sets are defined by mass balance constraints. The definition of HCRs is stricter in the sense that it is based on metabolites with one-to-one connectivity. However in contrast to the other approaches, it is independent of the exchange reactions with the environment and any requirements for demand functions. Consequently, the sets can be calculated directly from the stoichiometric matrix. There is a significant degree of overlap between HCRs and the Enzyme Subsets (ES) described by Pfeiffer et al [Biomass functions can help improve predictions under well-defined conditions by constraining the steady state solution space. However, different growth conditions can appreciably alter the composition and the 'objectives' of bacteria, as reflect by Figure of weeks . These cof weeks  and idenof weeks . Correlaof weeks  and fluxof weeks  and implof weeks . Applyiner et al . Since Eer et al ). Additiin vitro. These HCR sets may reflect underlying core sets of reactions vital to growth and survival in vitro.147 HCR sets  were calculated for the network, the average size of each set was 2.93, with median and modes both equal to 2. The largest set, HCR 40, consisted of 13 reactions involved in Cofactor Metabolism and Porphyrin Metabolism. The summary of how the HCRs are split among the 35 subsystems in the network are outlined in Table M. tb reasonably comprehensively [iNJ661  or for optimizing growth (in the lab).• Analysis of the capabilities under maximal growth conditions can help identify critical nutrient targets for killing M. tb using CO as the carbon source is still unclear. The computational studies suggest that there must be an alternative pathway (to reductive citrate cycle and RuBisCO) which enables entry of CO2 into central metabolism.• The pathway enabling growth of in vitro growth and the almost diametrically opposed results for in silico growth, suggest that there may be additional pathways involving glucose metabolism and potentially non-metabolic functions related to the glucose requirement.• The seemingly paradoxical need for glucose supplementation in Middlebrook media but non-essentiality of directly related reactions involving the metabolism of glucose-6-phosphate for • The definition of the biomass function will largely dictate the results of optimal growth/gene deletion experiments, consequently, positive results (in agreement with the experiments) are not as interesting as the false negative . The false negative results may be due to an error in the annotation of the gene or an underlying physiological mechanism that we are not aware of. In either case, investigating the particular genes experimentally may help clarify the issue.• The model can be used as a Context for Biological Content and provides a consistent, coherent framework to analyze various datasets focusing on metabolism.Unbiased analysis methods, methods that calculate properties of the network independent of objective functions, can provide interesting and valuable insights into network capabilities . The defTaken together, this study adds another high resolution reconstruction of microbial metabolism. There now exist a growing number of such reconstructions that have been particularly useful for basic studies of network properties and for bioprocessing applications -44. As tReconstruction and model development of the metabolic network has been described previously ,8. The rM. tuberculosis, many of these gaps could not be filled. Legacy data in the form of primary articles, review articles, and textbooks were employed in addition to the database resources [see Additional file iNJ661's metabolic network was visualized by creating a map of the network organized by lumped subsystems of metabolism [see Additional files M. tb was found, no additional reactions or transporters were added.Figure The stoichiometric matrix, S, was constructed based on the reactions described in the reconstruction. The biomass function was defined using available legacy data [see Additional file m rows by n columns, where there are m metabolites and n reactions in the network. Mass conservation dictatesThe metabolic network is represented mathematically by the stoichiometric matrix, which simplifies toat the steady state. Further constraints can be placed on the system based on thermodynamics  and limits on substrate uptake ratesthe Mycobacteria Bovis BCG strain of 25 μL/hr/mg dry weight (0.98 mmol O2/hr/g dry weight) [in silico culture media are listed in Table The oxygen uptake rate measured from  weight)  was usedConstraining the biomass objective function, c, to the optimal value calculated in FBA,every flux in the network is then minimized and maximized. The different between the maximum and minimum for each flux defines the flux variability for that reaction.Robustness plots are performed by varying a particular flux through a pre-defined range and recalculating the objective function. The slope of the curve describes the sensitivity of the objective function on that particular flux .-, and the output/product part of the stoichiometric matrix, S+, then it is transparent that,Denoting the binary form of the stoichiometric matrix, - and +, respectively. The input:output ratio for each metabolite can be determined and all metabolites with 1:1 connectivity will be hard-coupled by the network. Metabolites with 0:2 or 2:0 connectivity reflect metabolites that are hard-coupled in the event that the corresponding reaction is reversible. Metabolites with 0:1 or 1:0 connectivity are either source/sink transporters or blocked reactions.The input and output connectivities of each metabolite can be calculated for A general algorithm for calculating hard coupled reaction sets.1. Identify 1:1, 0:2, and 2:0 metabolites2. Define HCRs0a. Identify the sets of reactions corresponding to the metabolites, denote the entire set as HCR0 subset that share 1 or more reactions, denote the entire set as HCR1b. Join any HCRc. i ← 1i-1 ≠ HCRi)3. while  and since they had many irreversible reactions throughout, the 1:1 set and the 0:2/2:0 set were not combined .Since 0:2 or 2:0 reactions may be irreversible reactions, an intermediate processing step may be needed to verify that there is biological evidence supporting the catalysis of the reaction in the reverse direction. If one does not wish to consider these potential effects and assume that all of the reactions are reversible then step 1 can be simplified by finding all of the 2 metabolites for © v4.2 .The reconstruction process, the metabolic maps, and the majority of the calculations were performed using Simpheny™ v1.11 software . HCR Sets were calculated with MathematicaNJ carried out the reconstruction, performed the analyses, and drafted the manuscript. BØP conceived the study and helped revise the manuscript. Both authors approved the content of the final manuscript.Molecular composition of biomass functions. A detailed list of the biomass constituents and their respective coefficients in the biomass function in addition to the references from which this information was found.Click here for fileiNJ661 and iJR904. Plotting the biomass objective function versus phosphate or sulfate uptake rates for in silico strains of M. tb and E. coli.Robustness plots for phosphate and sulfate for Click here for fileiNJ661 lethal gene deletions. A catalog of the false negative results produced by iNJ661 when compared to the OptGro data set. Each of false negative results is categorized into at least one of four different categories. Specific loci are listed in cases with suggested alternative loci or alternative paths.Comprehensive list of false negatives between OptGro data set and Click here for fileHCRs mapped to experimental data sets. The list of HCRs that mapped to the three experimental data, based on gene loci.Click here for fileExplicit list of references used in the reconstruction process. The list of references used to define particular reactions and GPR relationships.Click here for fileiNJ661.Amino acid metabolism. A map of the amino acid pathways in Click here for fileiNJ661.Central metabolism. A map of the central metabolic pathways in Click here for fileiNJ661.Fatty acid metabolism. A map of the fatty acid pathways in Click here for fileiNJ661.Membrane metabolism. A map of the membrane metabolism pathways in Click here for fileiNJ661.Nucleotide metabolism. A map of the nucleotide synthesis and degradation pathways in Click here for fileiNJ661.Peptidoglycan metabolism. A map of the peptidoglycan pathways in Click here for fileiNJ661.Transporters. A map of the transport reactions found in Click here for fileiNJ661.Vitamin and cofactor metabolism. A map of the vitamin and cofactor pathways in Click here for fileM. tb.HCRs mapped to reaction and metabolic subsystems. The complete list of HCRs, highlighting those known to be drug targets in Click here for fileiNJ661. A complete list of the reactions (abbreviations and full names), the associated mass and charge balanced reactions, confidence scores, reaction subsystems, and the list of the Gene-Protein-Reaction relationships, including logical AND/OR relationships, for all of the reactions in the model. A complete list of metabolites appearing in iNJ661, including abbreviation, full name, formula, and charge at pH 7.2 can be found in the second tab in the file.Model content for Click here for file


Despite recent algorithmic and conceptual progress, the stoichiometric network analysis of large metabolic models remains a computationally challenging problem..SNA is a interactive, high performance toolbox for analysing the possible steady state behaviour of metabolic networks by computing the generating and elementary vectors of their flux and conversions cones. It also supports analysing the steady states by linear programming. The toolbox is implemented mainly in Mathematica and returns numerically exact results. It is available under an open source license from: Thanks to its performance and modular design, SNA is demonstrably useful in analysing genome scale metabolic networks. Further, the integration into Mathematica provides a very flexible environment for the subsequent analysis and interpretation of the results. Advances in genomics have enabled the large scale reconstruction of biochemical reaction networks where the main information provided concerns the stoichiometry and reversibility of the involved reactions. While such models disregard the kinetics of the reactions, biologically meaningful predictions can nevertheless be obtained by analysing the stoichiometrically viable steady states of the network . The aimIn mathematical terms the stoichiometrically viable steady states form a convex polyhedral cone which is called the flux cone. Recently  it has bLinear Programming, as in Flux Balance Analysis , can be By computing the set of elementary flux vectors a complete and very explicit description of the possible steady state behaviour is obtained and the answers to many questions about the metabolism, such as the maximal yield of some compound and the reactions essential for its synthesis are immediately found by inspecting this set. However, for large networks, enumerating all elementary fluxes may not be desirable, or even feasible, since their number can be very large. It is then often useful to first consider the conversion cone which is much simpler than the flux cone when many elementary fluxes are equivalent with regard to the consumed and produced compounds, differing only in the details of the internal synthesis. While this provides a less complete description of the metabolism than the flux cone, many interesting questions can nevertheless be answered by computing the elementary conversions. Cases in point are (a) the determination of minimal media  and (b) I shall only consider the architecture of the toolbox here since the main mathematical and computational concepts implemented in SNA have been described elsewhere. In particular the Nullspace algorithm used by SNA for calculating elementary fluxes, which runs significantly faster than previous procedures , is descThe user interface for SNA is Mathematica and the functionality described in the preceeding section is provided by a Mathematica package called SNAsym. The matrix level computational geometry routines underpinning the operation of SNAsym are delegated to the lower level package SNAmat. SNAmat may be of independent interest to people wishing to avoid using SNA proper and only interested in a computational engine for calculating elementary modes from a stoichiometry matrix.5 elementary fluxes for the model of the central carbon metabolism of E. coli [In contrast to SNAsym, which is pure Mathematica code, the lower level package is a hybrid consisting of Mathematica code communicating with a binary program compiled from Matlab and C sources. The division of labour is that the Mathematica part of SNAmat does the arithmetic, whereas, for speed, the purely logical operation are done by the binary. An important motivation for this arises from the fact that in mathematical terms only rational operations are required in the network analysis tasks implemented by SNA, i.e. no irrational functions such a root taking are needed. Hence, if the stoichiometric factors in the networks are represented by fractions the entire analysis can only lead to numbers which are rational as well and thus have an exact numerical representation. SNA exploits this, by doing all of the arithmetic in Mathematica and using the built-in functionality of this software, to always provides numerically exact results. Due to the division of labour between Mathematica and binary code, SNA is nevertheless fast enough to handle large scale problems. For instance, it takes only 7 minutes to calculate the 5 × 10 E. coli  on a staBiochemical reactions are represented in SNA by Mathematica expressions such asXt denotes an external metabolite which can be used both as an input and an output to the network, whereas metabolites with subscript Xtin or Xtout are restricted to being inputs or, respectively, outputs. Note that rôle and compartment are treated as quite distinct concepts. The reason is that in a study focusing on just an organelle, such as mitochondria, one might wish to treat all cytosolic metabolites as external.Each metabolite has a body which is a string denoting the chemical species (e.g. adp) and a superscript designating the compartment. Above the strings 'm', 'c', and 'e' are used for mitochondrial, cytosolic and, respectively, extra-cellular metabolites. In addition each metabolite has a rôle, the default rôle being internal while other rôles are specified by a subscript. The subscript mnet (metabolic net). The toolbox provides a large set of functions for the basic manipulation of mnet's: e.g. joining them, extracting subnetworks and changing the rôle of the metabolites in the network. Less basic functions include, determining which reactions are feasible, i.e. admit a non-zero steady state flow, or simplifying the network in a way that preserves e.g. the conversion cone.Based on a list of such reactions and an appropriate naming scheme for the reactions, SNA is used to construct the abstract data type But, of course, the main functionality of SNA is related to the tasks mentioned in the first section. Besides supporting flux balance analysis, the toolbox provides functions analysing the flux or the conversion cone by calculating a minimal generating set or enumerating all the elementary vectors. A session, demonstrating some of the functions of SNA, is shown in Fig. E. coli [S. cerevisiae [mnet given a table of numbers representing a stoichiometry matrix. The second shows how to parse textual representations of metabolic networks such as the ones used by the Palsson group. e.g. [The main documentation for SNA consists of 9 tutorial Mathematica notebooks. After a basic introduction to the toolbox, the tutorials quickly move on to show how SNA can be used to analyse quite large metabolic networks, considering as examples models of the human cardiac mitochondria , the cenrevisiae . In addiup. e.g. ,11. Furtup. e.g. .S. cerevisiae. Further, compared to Metatool, SNA has as much expanded functionality and, in particular, provides functions for calculating elementary conversions. Finally, the design of SNA is very modular. While this may make the learning curve steeper, it provides the flexibility required to tackle the computational challenge still posed by the analysis of large metabolic networks.Currently, the most widely used program for calculating elementary flux vectors probably is Metatool . The lat. The toolbox runs under PC-Linux and additionally requires Mathematica (5.0 or higher). Further, if one wants to recompile the binary program mentioned in the implementation section, Matlab 6.5 is needed.SNA is distributed under an open source license and can be downloaded from


Saccharomyces cerevisiae, and its entire space of optimal and near-optimal flux distributions. We show that this space reveals underlying evolutionary constraints on expression regulation, as well as on the conservation of the underlying gene sequences. Genes that have a high range of optimal flux levels tend to display divergent expression levels among different yeast strains and species. This suggests that gene regulation has diverged in those parts of the metabolic network that are less constrained. In addition, we show that genes that are active in a large fraction of the space of optimal solutions tend to have conserved sequences. This supports the possibility that there is less selective pressure to maintain genes that are relevant for only a small number of metabolic states.Variation in gene expression levels on a genomic scale has been detected among different strains, among closely related species, and within populations of genetically identical cells. What are the driving forces that lead to expression divergence in some genes and conserved expression in others? Here we employ flux balance analysis to address this question for metabolic genes. We consider the genome-scale metabolic model of  The regulation of gene product activity allows cells to efficiently cope with various tasks under varying conditions. Given that, one may have expected that striving for efficiency would cause genetically similar cells to have similar regulation. However, in reality, high variations in gene expression levels are detected between different strains and even between genetically identical cells taken from the same culture. What are the driving forces that lead to expression divergence in some genes and conserved expression in others?To address this question, the authors study the conservation of regulation in yeast metabolism, using a computational model. They find that genes coding for reactions whose flux rates are narrowly constrained by the cellular need to maximize growth rate tend to have strictly conserved regulation and expression. However, when a wide range of flux rates is compatible with high cellular growth rates, the corresponding regulation and expression patterns are free to diverge. Furthermore, enzymes that participate in a large number of alternative metabolic behaviors tend to be encoded by genes with a highly conserved sequence. Taken together, these findings support the hypothesis that maintaining large variability in the overall expression and metabolic repertoire of the cell is under marked evolutionary selection. Recent comparative studies of genomic-scale gene expression levels have revealed substantial variation among different strains ,2, amongA key tool in studying metabolic networks is constraint-based modeling, which permits analysis of large-scale networks. Accurate prediction of dynamic metabolic activity requires kinetic models, but these rely on detailed information of the rates of enzyme activity, which is mostly unavailable, and are thus limited to small-scale networks. In contrast, constraint-based models use genome-scale networks to predict steady-state metabolic activity, regardless of specific enzyme kinetics. Stoichiometric, thermodynamic, flux capacity, and possibly other constraints are used to limit the space of possible flux distributions attainable by the metabolic network. Flux balance analysis (FBA) ,9 is a soptimal solution space, the space of all flux distributions leading to an optimal growth rate. On one hand, there are missing constraints in the model, and one line of research aims at reducing the solution space by adding biologically plausible constraints, e.g., by explicitly incorporating regulatory constraints in the model 29], whicProtein abundance, as measured through GFP fluorescence, was taken from ,38,40, aExpression divergence measures were taken from the supplementary results of , 581 of Expression variance values are based on the dataset compiled by , and werEvolutionary rates were taken from , and werThe space of all flux distributions that obey the constraints imposed in FBA is a polyhedral set defined by the half-spaces corresponding to the constraints. Finding a flux distribution that maximizes the biomass reaction  can thus be found using a simplex algorithm. Roughly, the algorithm travels from one vertex of the polyhedral set to a neighboring one, while improving the value of the objective function, until the function can no longer by improved . See, e.g.,  for morepivot rule. In a randomized simplex algorithm , the pivertices of the polyhedral set will be chosen and not internal points. But this is an appropriate sampling scheme when one is interested in sampling optimal flux distributions with extreme flux values since they correspond to vertices.We note that this is not a uniform sample of the solution space, since only optimal solution is a non-negative linear combination of them.Indeed, the vertices of the polytope of optimal solutions are analogous to the much-studied extreme pathways  elementap, the maximal uptake rate was set to 0—i.e., only excretion was allowed. Otherwise, uptake rate was limited to a value chosen uniformly at random between 0 and 1.Random growth media were generated by setting limiting values to the uptake reactions independently at random. With probability p = 0.5 and p = 0.95 were tested, simulating rich and poor media, respectively. Eight of the uptake rates were taken positive in all media, to ensure viability .The values A similar sampling method was used in .Comparing flux level with mRNA level and protein level requires inferring a “flux level” for a gene, based on the reactions it is associated with. For each gene, we defined this to be the maximal flux level predicted for its associated reactions, as this is the level that most constraints it.Analysis of isozymes is deferred to In The flexibility score of a reaction in a given medium is the difference between the maximal flux that can flow through it in an optimal flux distribution, and the minimal one. This is computed using flux variability analysis .The flexibility score of a gene in a given medium is the maximum of the flexibility scores for the reactions it is associated with. The rationale for using the maximum value is that this value is the one that most constrains the required enzyme quantity for obtaining optimal flux. That is, if a protein is associated with several reactions, for its expression level to comply with all optimal flux values, it most complies with the highest one.R can attain values between x and y, and hence its flexibility score is y minus x. If a single gene is associated with this reaction, then, as defined above, the reaction's contribution towards the gene's flexibility score is simply y minus x, since we think of this range of fluxes as defining the flexibility of the gene's expression. However, if there are several isozymes associated with the reaction, then their flexibility can, potentially, be larger.For isozymes, the definition is slightly more complicated. Suppose a reaction x > 0, then the values an associated isozyme I can attain are from 0 to y, since if the expression level of I is smaller than x, the other isozymes can compensate for it, putting the total reaction rate with the optimal range of . Hence, in this case the contribution of the reaction towards the gene's flexibility is y, rather than y minus x. Similarly, if y < 0, an isozyme could attain any value in the range , and hence the contribution of the reaction towards the gene's flexibility would be |x|. Taken together, we define contribution of a reaction attaining flux values in the range  towards the flexibility score of its associated isozymes as the maximum among y minus x, y, and |x|.If Transcription factors' binding sites were considered as regulating a gene if they appear in the map of  within 5S. paradoxus, S. mikatae, and S. bayanus; a score of 0.5 if it conserved in two of them; and a score of 0 otherwise. The promoter conservation score for a gene is the mean score for the binding sites associated with it.Each binding site received a score of 1, if it is conserved in all three species from among Data was available for 382 genes.The activity score of a gene in a given medium is obtained by sampling the solution space, and by counting the number of solutions that have reactions associated with it that are active.When doing so by sampling the space of optimal solutions for YPD, we got an activity score of either 0 or 1 for 90% of the genes. To obtain a range of activity scores that can be better differentiated between activity scores we sampled the space of near-optimal solutions, as was done in ,45. SpecGenes that encode enzymes which have isozymes were excluded from the analysis, since their knockout has no effect on the network. In p = 1.3 × 10−12, 270 genes.Genes that are active in all solutions are also excluded, since we expect many of them to be essential genes, and it is known that essential genes tend to have conserved sequence. If they are included, the Spearman rank correlation between the YPD activity score and sequence divergence is indeed more distinct: r = −0.41, The intercondition activity score of a gene was computed from 1,000 random growth media , and computed similarly to the medium-specific activity score: For each gene, its interactivity score is the number of random media in which it is active, where a gene is considered active if one the reactions for which the gene is required is active.As for the medium-specific activity score, we omitted from the analysis genes that are active in all conditions. If these genes are included, the Spearman rank correlation between the intercondition activity score and sequence divergence is more distinct: r = −0.31 for rich growth conditions, and r = −0.37 for poor growth conditions.Dataset S1(5 KB TXT)Click here for additional data file.Protocol S1(58 KB DOC)Click here for additional data file.Protocol S2(73 KB DOC)Click here for additional data file.Protocol S3(70 KB DOC)Click here for additional data file.Table S1(136 KB DOC)Click here for additional data file.Saccharomyces Genome Database (http://www.yeastgenome.org), see For the accession numbers of the yeast genes analyzed in this paper, from the


Saccharomyces cerevisiae is its ability to perform simultaneous respiration and fermentation at high growth rate even under fully aerobic conditions. In the present work, this Crabtree effect called phenomenon was investigated in detail by comparative 13C metabolic flux analysis of S. cerevisiae growing under purely oxidative, respiro-fermentative and predominantly fermentative conditions.One of the most fascinating properties of the biotechnologically important organism S. cerevisiae exhibited a by-pass of pyruvate dehydrogenase in all physiological regimes. During oxidative growth this by-pass was mainly provided via pyruvate decarboxylase, acetaldehyde dehydrogenase, acetyl-CoA synthase and transport of acetyl-CoA into the mitochondrion. During fermentative growth this route, however, was saturated due to limited enzyme capacity. Under these conditions the cells exhibited high carbon flux through a chain of reactions involving pyruvate carboxylase, the oxaloacetate transporter and malic enzyme. During purely oxidative growth the PPP alone was sufficient to completely supply NADPH for anabolism. During fermentation, it provided only 60 % of the required NADPH.The metabolic shift from oxidative to fermentative growth was accompanied by complex changes of carbon flux throughout the whole central metabolism. This involved a flux redirection from the pentose phosphate pathway (PPP) towards glycolysis, an increased flux through pyruvate carboxylase, the fermentative pathways and malic enzyme, a flux decrease through the TCA cycle, and a partial relocation of alanine biosynthesis from the mitochondrion to the cytosol. S. cerevisiae possesses different metabolic by-passes to channel carbon into the mitochondrion. This involves the conversion of cytosolic pyruvate either into acetyl CoA or oxaloacetate followed by intercompartmental transport of these metabolites. During oxidative growth mainly the NAD specific isoforms of acetaldehyde dehydrogenase and isocitrate dehydrogenase catalyze the corresponding reactions in S. cerevisiae, whereas NADPH supply under fermentative conditions involves significant contribution of sources other than the PPP such as e. g. NADPH specific acetaldehyde dehydrogenase or isocitrate dehydrogenase.We conclude that, in order to overcome the limited capacity of pyruvate dehydrogenase,  S. cerevisiae is an important biotechnological organism e.g. for production of ethanol [S. cerevisiae is able to perform simultaneous respiration and fermentation at high growth rates even under fully aerobic conditions [S. cerevisiae by a shift from biomass formation towards fermentative products at increased dilution rates [13C metabolic flux analysis which allows the quantification of in vivo activity of pathways and reactions [S. cerevisiae and related yeast strains [S. cerevisiae by quantifying metabolic fluxes through the underlying reactions in the central metabolism at these different physiological states. For this purpose chemostat cultures with [1-13C] glucose as substrate were grown at different growth rates under aerobic glucose-limited conditions to metabolic and isotopic steady-state. The use of a continuous culture hereby allowed the precise adjustment of the physiological state of the cells, i.e. the relative activity of the fermentative pathways.The budding yeast  ethanol , recombi ethanol , antibio ethanol  or bioma ethanol . Additio ethanol ,6. This nditions ,8. This on rates . Despiteon rates -12, a clon rates ,14. The on rates ,16. A poeactions -19. In r strains -27. In t-1 and 0.45 h-1 , mixed respiro-fermentative (0.30 h-1), and mainly fermentative metabolism (0.40 h-1) were selected for the flux studies. The resulting ethanol yield under these conditions was 0.0, 0.5, and 1.1 mol ethanol (mol glucose)-1, respectively. The stoichiometric data obtained from the 13C cultures agreed well with the corresponding values from the initially performed cultivations  The metabolic shift had a strong effect on the flux partitioning between glycolysis and PPP. Whereas, at purely oxidative metabolism, the major carbon flux was channeled into the PPP, the opposite was observed for fermentative metabolism. The data of the present work together with previous flux studies of t Figure . A simil 0.4 h-1 . The red 0.4 h-1 . Due to S. cerevisiae  glucose . Cultivations were performed in a continuously operated bioreactor  with a culture volume of 100 ml at 30°C and 500 rpm. The aeration rate was kept at 100 mL/min by a mass flow controller . Composition of inlet and exhaust gas was determined on-line by mass spectrometry as described previously  was obtained. Enzymatic test-kits were used for quantification of glucose, glycerol, acetate  and ethanol  in the supernatant. Concentrations of 2-oxoglutarate, pyruvate, succinate, and fumarate in the supernatant were analyzed by HPLC [S. cerevisiae was quantified by HPLC [i.e. the quantification of the yield coefficients, was 3.7 % (biomass yield), 4.0 % (ethanol yield), 9.3 % (acetate yield) and 11.3 % (glycerol yield).Optical density (OD by HPLC  after hy0) to the fully 13C labeled (xn) variant could be measured with good signal intensity and accuracy  is shown in Figure S. cerevisiae [The metabolic network of revisiae . For glyrevisiae ,46 was t1 - ν2 - ν3 + ν4 - ν5 = 0     (1)Glucose 6-phosphate: ν3 - ν4 - ν6 + ν8 - ν9 + ν10 - ν11 = 0     (2)Fructose 6-phosphate: ν2 - ν7 - ν8 + ν9 - 2ν12 + 2ν13 = 0     (3)Pentose phosphate: ν8 + ν9 + ν10 - ν11 - ν14 = 0     (4)Erythrose 4-phosphate: - ν12 - ν13 - ν10 + ν11 - ν5 = 0     (5)Sedoheptulose 7-phosphate: ν6 - ν10 + ν11 + ν8 - ν9 + ν12 - ν13 + ν16 - ν17 - ν18 = 0     (6)Glyceraldehyde 3-phosphate: ν6 - ν15 - ν16 = 0     (7)Dihydroxyacetone-phosphate: ν17 - ν19 - ν20 - ν24 = 0     (8)3-Phosphoglycerate: ν20 - ν21 - ν22 = 0     (9)Serine: ν22 - ν23 + ν29 = 0     (10)Glycine: ν28 - ν29 - ν30 = 0     (11)Threonine: ν24 - ν25 - ν26 = 0     (12)Phosphoenolpyruvate: νcyt: ν26 - ν27 - ν32 - ν33 - ν34 = 0     (13)Pyruvate32 - ν37 - ν38 = 0     (14)Acetaldehyde: ν38 - ν39 - ν40 = 0     (15)Acetate: νcyt: ν40 - ν41 - ν47 + ν48 = 0     (16)Acetyl CoAcyt: ν33 - ν42 = 0     (17)Alaninecyt: ν27 - ν28 - ν31 - ν35 + ν36 = 0     (18)Oxaloacetatemit: ν34 - ν43 - ν44 + ν50 - ν51 = 0     (19)Pyruvatemit: ν43 - ν45 = 0     (20)Alaninemit: ν44 - ν46 + ν47 - ν48 - ν49 = 0     (21)Acetyl CoA49 - ν52 - ν53 = 0     (22)2-Oxoglutarate: ν53 - ν54 + ν55 = 0     (23)Succinate: νmit: ν35 - ν36 - ν49 - ν50 + ν54 - ν55 = 0     (24)Oxaloacetate/MalateS. cerevisiae was considered  combining metabolite balancing and isotopomer modeling. The performed approach utilized metabolite balancing  during each optimization step considering stoichiometric data on anabolic demand for biomass precursors Table  and prod13C tracer studies and all analytics involved.OF carried out the experimental work including the chemostat cultivations, the S. cerevisiae, the software implementation, the calculation of carbon fluxes, and the statistical analysis. He drafted and composed the manuscript.CW designed the study and carried out all computational work involving the construction of the metabolic network model of Both authors read and approved the final manuscript.


Lactococcus lactis and developed a genome-scale flux model able to simulate and analyze network capabilities and whole-cell function under aerobic and anaerobic continuous cultures. Flux balance analysis (FBA) and minimization of metabolic adjustment (MOMA) were used as modeling frameworks.Genome-scale flux models are useful tools to represent and analyze microbial metabolism. In this work we reconstructed the metabolic network of the lactic acid bacteria L. lactis ssp. lactis IL1403 together with physiological and biochemical information. The established network comprised a total of 621 reactions and 509 metabolites, representing the overall metabolism of L. lactis. Experimental data reported in the literature was used to fit the model to phenotypic observations. Regulatory constraints had to be included to simulate certain metabolic features, such as the shift from homo to heterolactic fermentation. A minimal medium for in silico growth was identified, indicating the requirement of four amino acids in addition to a sugar. Remarkably, de novo biosynthesis of four other amino acids was observed even when all amino acids were supplied, which is in good agreement with experimental observations. Additionally, enhanced metabolic engineering strategies for improved diacetyl producing strains were designed.The metabolic network was reconstructed using the annotated genome sequence from L. lactis metabolic network can now be used for a better understanding of lactococcal metabolic capabilities and potential, for the design of enhanced metabolic engineering strategies and for integration with other types of 'omic' data, to assist in finding new information on cellular organization and function.The  They are applied worldwide in the industrial manufacture of fermented food products, mainly in the dairy industry. During fermentation LAB primarily produce lactic acid from the available carbon source, resulting in the rapid acidification of the food raw material, which is an important parameter in the preservation and extension of shelf life of food products. LAB metabolism also contributes for the development of desired product properties such as flavor and texture ,2. Becauof foods .Lactococcus lactis is, by far, the most extensively studied with respect to its physiology, metabolic pathways and regulatory mechanisms, and its genome was the first LAB genome to be completely sequenced and annotated [de novo from a carbon source. The major part of the carbon from carbohydrates is converted into fermentation end-products. For example, during growth on glucose, only about 5% of the metabolized sugar is converted into biomass [L. lactis implies that, for optimal growth, they require exogenous supply of a fermentable sugar, numerous vitamins and amino acids, phosphate, potassium and magnesium [Among LAB, nnotated . Lactoco biomass . The verL. lactis is a facultative anaerobic bacterium. Some strains are capable of growing in the presence of oxygen and adjust their metabolism accordingly, while others are strongly inhibited under aerobic conditions. As this microorganism lacks a functional respiratory chain, the ability to grow aerobically has been linked to the presence of NADH-oxidases. Since L. lactis is not able to perform respiration, ATP is only formed through substrate level phosphorylation. Thus, in order for Lactococci to grow at a high specific growth rate, a high carbohydrate degradation rate  is called for. The main function of the sugar metabolism in L. lactis is to generate the energy necessary for rapid growth and for maintenance of intracellular pH during acidification of the environment [ironment .L. lactis has been an organism of choice for many metabolic engineering purposes [Due to its major importance as a laboratorial and industrial microorganism, and because of its relatively simple metabolism and limited biosynthetic capabilities, purposes ,6,7. HenThe reconstruction of the entire metabolic reaction network of a cell and subsequent application of genome-scale flux models has been conducted for many organisms, including bacteria, yeast, fungi and animal cells -12. ThesL. lactis is presented. Network reactions were collected using the annotated genome sequence, biochemical and metabolic pathways databases, biochemistry books and journal publications. Once the network was established, FBA and MOMA were applied to analyze the network capabilities and to model phenotypic behavior under anaerobic and aerobic conditions. Simulation results were compared with experimental data available in the literature. Furthermore, the model was used to identify possible metabolic engineering targets to design an efficient diacetyl producing strain.In this work the reconstruction process of the metabolic network of . A total of 358 ORFs out of the detected 2310 ORFs in the sequenced genome were considered, corresponding to 476 associated reactions. The remaining 145 reactions were included based on biochemical/physiological considerations or inferred by the demands imposed on the metabolic network. Namely, reactions were included based on experimental information reported in literature to account for pathway gaps, transport steps and biomass assembly. From the entire set of reactions, 492 correspond to intracellular reactions while 129 are transport fluxes over the cytoplasmic membrane. From the 358 ORFs considered, 63 ORFs account for genes that have been isolated and identified from L. lactis, 286 ORFs have high homology with identified genes from other organisms, 2 ORFs are annotated sequences derived from low level homologues and 7 ORFs are described as probable homologues of unknown function (ORF reliability classification based on [L. lactis.The reconstruction process resulted in a network comprising 621 reactions and 509 metabolites. The entire reaction database can be consulted in the based on ). The re-1. One reaction was included to account for non-growth dependent ATP maintenance. For growth under glucose limitation this value has been previously experimentally determined and equals 1 mmol ATP gDW-1h-1 [From the 358 ORFs considered in the reconstruction process, ORFs that are assigned to energy metabolism, amino acid biosynthesis and nucleotides metabolism account for more than half of the total number. These ORFs are involved in 476 reactions. The relatively high number of associated reactions is mainly due to the existence of low substrate-specific enzyme activities catalyzing different reactions. For instances, the model includes a low-specific aminotransferase (araT) involved in the metabolism of several amino acids and defined to catalyze 18 reactions in the model. An equation on biomass formation was developed to account for the drain of precursors and building blocks into biomass. A detailed description of its assembly can be found in the gDW-1h-1 .L. lactis contains 509 metabolites, 422 of which are unique metabolites. It is through metabolites that reactions are connected, as the product of one reaction becomes the substrate of another. At a zoom-out level, different biochemical pathways within a cell are interconnected by virtue of metabolites that participate in more than one pathway [The metabolic network of  pathway . In partL. lactis is able to metabolize a variety of different sugars and other carbon sources [L. lactis include fructose, galactose, glucosamine, glucose, lactose, maltose, mannitol, mannose, ribose, sucrose and trehalose. sources ,21 to obin silico model to utilize different carbon sources to grow were evaluated using FBA, using comparable uptake rates and carbon-limited conditions. All the above mentioned carbon sources were supplied, one at a time, together with all amino acids and nucleotides. A sugar consumption rate of 13.6 mmol gDW-1 h-1 was considered (6.8 mmol gDW-1 h-1 in the case of disaccharides), both under aerobic and anaerobic conditions.The capabilities of the -1 h-1 [-1, while the aerobic specific growth rate was 0.62 h-1. In case oxygen uptake is not constrained  the specific growth rate was 0.82 h-1. Constrained aerobic growth is slightly lower than anaerobic growth due to the different flux constraints applied to the enzymes that metabolize pyruvate into acetyl-CoA (see Methods). Pyruvate-formate lyase is only active under anaerobic conditions, while the pyruvate dehydrogenase (PDH) complex is only active during aerobic growth [The specific growth rate under anaerobic conditions was always higher than at aerobic conditions when the specific oxygen consumption was set to 3.61 mmol gDW-1 h-1 , which i-1 h-1 . The prec growth . This foet al. (2002) [Growth on mannose, galactose, sucrose, lactose and glucosamine was found to be the same as for growth on glucose. The capacity to grow on trehalose and maltose was slightly higher, due to a decreased ATP requirement for the synthesis of the corresponding phosphorylated sugars. Anaerobically, growth on fructose and mannitol led to a decrease of 3% and 16% in biomass formation, respectively, compared with growth on glucose. This decrease has been qualitatively observed experimentally ,25,26. Iin silico growth, even when all other amino acids are supplied and despite the presence of the biosynthetic pathways for these amino acids in the reconstructed network. In the case of valine this result is trivial, since simulations were run under the constraint that the reaction catalyzed by bcaT_1 can only occur in the catabolic direction, and there is no alternative pathway for valine synthesis. In the case of methionine, the reason appears to be the lack of available FAD (cofactor) to sustain growth. Finally, if no arginine is supplied, the linear problem becomes infeasible. Nevertheless, arginine synthesis is observed when maximizing for an arginine drain.In addition to a carbohydrate source, the minimal amino acid requirements were determined by omission of each amino acid at a time. The single omission of arginine, methionine and valine was found to prevent L. lactis strains are usually auxotrophic for 7 to 9 amino acids, including these four [L. lactis IL1403 [Growth was not observed when only these three amino acids were supplied. If glutamate  is allowed to be taken up in the presence of these three amino acids, growth occurs. Therefore, glucose, arginine, methionine, glutamate (or glutamine) and valine were found to be the minimal required medium for growth. ese four ,27. GrowL. lactis [Bacillus subtilis [Single gene deletion analyses were computed to predict the lethal effect of deleting each individual gene. Prediction capability was assessed by comparing simulation results with experimental data on gene lethality reported for . lactis  and Bacisubtilis , a relatThe effect of the deletion of each reaction was also assessed. These computational studies were performed both in rich media and minimal media, and both under aerobic and anaerobic conditions. FBA and MOMA were used to compute these simulations, with MOMA predicting around 10 more lethal genes/reactions than FBA Table . We focu2 production and to the pyruvate dehydrogenase (PDH) complex. Under aerobic conditions, the PDH complex is the only pathway leading to the formation of acetyl-CoA from pyruvate, and as acetyl-CoA is an essential metabolite in many processes (such as lipid formation), deletion of the PDH complex results in a lethal phenotype.Aerobic and anaerobic simulations led to similar results, with a few more genes/reactions being lethal under aerobic conditions, namely those associated to oxygen utilization, COWhen comparing the growth capabilities in both rich and minimal media, it can be observed that approximately 40 additional genes are lethal during growth in minimal media. These are mainly genes associated with biosynthesis of amino acids. From the 83 lethal genes in minimal media, 19 encode for gene products that catalyze more than one reaction. Only one of these 83 genes encoded for a protein that has an isoenzyme.L. lactis growing anaerobically. This process consisted in comparing simulation results with experimental observations reported by Thomas, T.D. et al. (1979) [To further evaluate the modeling capabilities of FBA we used the genome-scale flux model to simulate the shift from homolactic to heterolactic metabolism in . (1979)  for glucL. lactis, product formation at high dilution rates during continuous cultivations is similar to product formation during batch growth at high glucose concentrations, resulting in lactic acid as the sole metabolic product [Experimental observations reported in literature suggest that, for  product ,31. On t product . While iin silico strain was auxotrophic for some amino acids, which could act as a carbon source, and no information was found available in the literature for amino acid uptake rates under chemostat conditions. To solve this problem, consumption rates were approximated to be equal to the amino acids requirement for biomass formation. In the absence of experimental information for amino acid uptake rates another approach was followed, in which phenotypic behavior was simulated by constraining the growth rate and minimizing for substrate uptake. Both approaches led to different and complementary qualitative results regarding the network capabilities, as discussed below.For simulations of phenotypic behavior, the first obvious objective function was thought to be "maximization of growth" while constraining for substrate uptake. However, it became clear that constraining values for amino acid uptake rates would pose a problem, as the -1 h-1 revealed the occurrence of multiple solutions for products of metabolism. From these simulations it was observed that growth rate does not change with different glucose uptake rates, indicating that growth is nitrogen-limited and reaches a maximum at 0.82 h-1 (in good agreement with the fact that amino acid uptake rates were established based on the amino acid cell content at 0.8 h-1), while for glucose uptake rates lower than 14.1 mmol gDW-1 h-1 growth is glucose-limited. Since lactate production is reported under high glycolytic fluxes, and this is known to be due to regulation exerted by the NADH/NAD+ ratio on the pyruvate-formate lyase [et al [-1 h-1 to simulate homolactic growth when glucose uptake rates equal 24.6 and 18 mmol gDW-1 h-1, respectively.When maximizing for biomass formation, the metabolic phenotype cannot be correctly predicted from the reconstructed network by simply stating anaerobic constraints (see Methods). The process of tuning and refining the model is discussed below and detailed results are summarized in Figures 5 and 6 see . Simulatte lyase , additioe [et al  describeIn all simulations it can be observed that the metabolic model predicts trace production of amino acid catabolism products , which is in good agreement with experimental results -37. IntePlotting experimental against model results for biomass, lactate, formate, ethanol and biomass formation Figure  shows th-1 h-1 (see Table 6) by fitting simulation results against experimental dataTo overcome this difficulty another approach was used: the minimization of substrates uptake while constraining the growth rate. From simulation results, the formation of end-products of the pyruvate metabolism is observed. No by-products from the amino acid catabolism were predicted using this approach, as it minimizes amino acids uptake, and therefore no excess of amino acids is available for catabolism of amino acids. Without further constraints, mixed-acid fermentation is observed, as this is energetically more favorable for the cell (more ATP is formed when acetate is synthesized). However, as mentioned above, high fluxes through glycolysis lead to regulation effects towards homolactic fermentation. Consequently, the reaction catalyzed by pyruvate-formate lyase was constrained to values between 0 and 9 mmol gDWFrom Figure in silico growth interestingly shows that, for most of the cases, amino acid uptake rates linearly increase with the growth rate . Remarkably, this observation is in very good agreement with experimental data reported by Jensen, N.B. et al. (2002) [L. lactis subsp cremoris. In their work, they analyzed the capacity for the de novo biosynthesis of amino acids when all amino acids except glutamate were supplied, having observed that alanine, aspartate, phenylalanine and threonine, were synthesized de novo by the cell. Analyzing the flux distribution of the reactions involved in amino acid biosynthesis, it can be found that those preferences are associated with an increased production of 2-oxoglutarate, leading to an increased formation of L-glutamate.In all simulations under the objective of minimizing the substrate uptake, it could be noted that some amino acids are not taken up from the medium even if they are present in the medium. The cell seems to prefer to synthesize some of them. Amino acids completely synthesized by the . (2002)  for L. lL. lactis fermentative metabolism and it is produced chemically by oxidative decarboxylation of the metabolic intermediate 2-acetolactate (which is derived from the condensation of two molecules of pyruvate). Hence, knockout strategies leading to an increased yield of 2-acetolactate in glucose were investigated using FBA and MOMA.A relevant application of genome-scale metabolic models is the simulation of cellular behavior in response to genetic perturbations. Namely, genome-scale metabolic models can be used as tools in the design of metabolic engineering strategies, aiming at finding genetic targets leading to enhanced desired properties . We exemet al (2001) [et al (2000) [Lactococci metabolism around pyruvate is depicted in Figure l (2001)  have sucl (2000)  construc2 / mmol glucose. The simulated growth rate and yield of 2-acetolactate on glucose are presented in Table The first step in our modeling strategy was to find a set of appropriate constraints that lead to results comparable with experimental observation. By simply allowing unconstrained uptake of oxygen it was possible to obtain both 2-acetolactate and acetate as the main by-products of metabolism, which is in good agreement with experimental reports . Oxygen PTA leads to a slight yield increase. The enzyme Pta catalyzes the conversion of acetyl-CoA to acetyl-P, and its deletion eliminates the production of acetate. Furthermore, another single gene deletion was run on the "mutant" ΔldhΔaldΔadhAΔpta and three deletions leading to a higher production of 2-acetolactate were found, ΔFHS, ΔSERA and ΔZWF. The enzyme Fhs is a formyltetrahydrofolate synthetase (EC 6.3.4.3), involved in the folate metabolism. Its contribution towards a higher yield of 2-acetolactate is due to an increase in carbon availability for by-product formation, since this reaction is used by the cell to generate ATP and formate (in rich media). The enzyme SerA, a 3-phosphoglycerate dehydrogenase, is involved in serine metabolism. This part of the metabolism connects with Fhs. Hence, this deletion increases 2-acetolactate yield similarly to the deletion of SERA. Finally, ZWF deletion is only predicted by MOMA as being advantageous. ZWF encodes for glucose-6-phosphate 1-dehydrogenase, the first step of the pentose-phosphate pathway. This deletion is accompanied by a decrease in biomass formation, because the availability of NADPH decreases. Therefore, more carbon is redirected towards other products of metabolism (C4 and C5 compounds), resulting in an increase of 2-acetolactate yield. Observed yields and biomass formation rates are summarized in Table Next, the impact of single gene deletions on product formation was simulated by maximizing for growth. It was found that the deletion of e EC 6.3..3, involLactococcus lactis was reconstructed based on genomic, physiological and biochemical information, comprising a total of 621 reactions and 509 metabolites. Lactococcal network characteristics are comparable with other bacterial genome-scale reconstructed networks.The metabolic network of L. lactis was shown to be robust and able to predict many experimental observations, when considering additional constraints derived from available experimental data. The model proved to be a useful tool to analyze the metabolic capabilities of L. lactis and to understand how the individual components in the system interact and influence the overall cell function. For example, the model could predict that, if all the amino acids were supplied, the cell will prefer to synthesize de novo alanine, aspartate, glycine and phenylalanine. The model can now be used as a useful tool to test or develop novel metabolic engineering strategies to redirect fluxes towards the production of important products such as diacetyl, alanine and exopolysaccharides [Metabolic network analysis was carried out using FBA and MOMA. The genome-scale metabolic model for charides .Reconstructed metabolic networks are finding many other applications than the ones described in this work. Optimization methods have also been used to assess maximum capabilities of the network and to analyze gene dispensability . More reL. lactis involved a comprehensive search of the current knowledge of its metabolism. The process started based on the ORFs information from the annotated genome of L. lactis IL 1403 [in vivo reaction reversibility.The reconstruction process of the metabolic network of An equation for biomass formation was developed to account for the drain of precursors and building blocks into biomass. Biomass synthesis was set as a linear combination of seven macromolecular components – proteins, DNA, RNA, lipids, lipoteichoid acids, peptidoglycan and polysaccharides – which were considered to account for the cell overall biomass composition. The individual composition of every component was maintained at a fixed stoichiometry, independent of the specific growth rate. Cellular energy requirements were also considered, by taking into account information on the ATP cost of polymerisation and growth and non-growth associated ATP maintenance. The detailed calculation of the biomass composition can de found in the in silico strain, CH1.95O0.63N0.22P0.02S0.01, corresponding to a molecular weight of 27.8 g/C-mol. This is in good agreement with values found in the literature. Novak, L. et al. (2000) have measured a value of 27.7 g/C-mol for L. lactis.From the developed equations, an elemental biomass composition was determined for the reconstructed Flux balance analysis (FBA) is a linear programming posed problem where equations are defined by mass balance-derived stoichiometric reactions, constrains are imposed by fluxes limitations and the objective function is established based on a biological meaningful objective.N metabolites and M metabolic reactions, the stoichiometric matrix can be written as:Given a stoichiometric matrix derived from mass balance around all metabolites of a cell and assuming steady state, a system of linear equations is produced which simply states that the incoming fluxes are balanced by the outgoing fluxes. For a metabolic network comprising Sji is the stoichiometric coefficient of metabolite j in reaction i, vi represents the flux of reaction i, and bj quantifies the network's uptake (or secretion) of metabolite j. Reversible reactions are defined simply as two irreversible reactions in opposite directions, constraining all fluxes to nonnegative values, ie:where vi ≥ 0, ∀ i     (2)Some other constraints based on physiological or physicochemical aspects may be applied, such as thermodynamics considerations, regulation effects and maximal enzymatic rates:α ≥ vi ≥ βi, ∀ i     (3)Z, written as a linear combination of existing variables, the optimal solution can then be found at one corner of the set of feasible solutions. For metabolic applications, typical objective functions are maximization of biomass formation or minimization of substrate consumption.Establishing a particular objective function, Flux balance analysis can then be simply summarized as a linear programming problem posed as:Z = eg, biomass formation     (4)maximize subject tovi ≥ 0, ∀ iα ≥ vi ≥ βi, ∀ iAnother mathematical framework that can be used to find flux distribution solutions is the so-called minimization of metabolic adjustment (MOMA) . MOMA re and Object-Oriented software for Quadratic Programming .Both the FBA and the MOMA problems were solved with an in-house developed software using the following solvers: GNU Linear Programming Kit -1 h-1, which is the value experimentally estimated for carbon-limitated chemostat cultures [All simulations were run allowing free uptake of nucleotides , phosphate, biphosphate and sulfate. Additionally, non-growth associated ATP requirement was always set to 1 mmol ATP gDWcultures .To simulate growth under anaerobic conditions, the oxygen uptake rate was set to zero. Experimental insight on pyruvate metabolism led to two additional constraints. Since it is known that the pyruvate dehydrogenase (PDH) complex is not active in the absence of oxygen , pdhA_1 -1 h-1 [Aerobic growth was simulated by allowing oxygen to be taken up by the model. Unless otherwise stated, the oxygen consumption was set to 3.61 mmol gDW-1 h-1 . Under a-1 h-1 . pdhA_1 -1 h-1 was allowed, both under aerobic and anaerobic conditions . When the specific growth rate was higher than 0.01 h-1, the sugar was considered to be used for growth. This value was established based on the observation that when supplying only amino acids but no sugar, a specific growth rate of 0.01 h-1 is predicted by the model, which is not observed experimentally [Network capabilities to utilize different sugar sources were evaluated using FBA. A number of different sugars were supplied, one at a time, together with all amino acids and nucleotides. A sugar consumption rate of 13.6 mmol gDWmentally .in silico cell growth were analyzed under anaerobic conditions. The dilution rate was fixed to 0.18 and 0.76 h-1 and the objective function used in this investigation was the minimization of substrates uptake rate. Beginning with all the amino acids available, single-omissions were simulated by setting the corresponding uptake rate to zero, one at a time. If no biomass is formed, the omitted amino acid was defined as essential for growth. Simulations were run with the reactions catalyzed by bcaT_1, bcaT_2 and bcaT_3 constrained to the catabolic direction.Amino acid requirements for -1 h-1 and uptake of the previously determined essential amino acids at a rate of 0.5 mmol gDW-1 h-1 per amino acid. Remaining amino acids were individually supplied until growth was observed.A minimal medium was established by allowing uptake of glucose at a rate of 13.6 mmol gDW-1. For simulation of growth on rich media, glucose uptake was set at 13.6 mmol gDW-1 h-1, and all amino acids were allowed to be taken up (at a rate corresponding to the amino acid cell content assuming a specific growth rate of 0.8 h-1). Simulating growth on minimal media, only glucose at 13.6 mmol gDW-1 h-1, arginine, methionine, valine and glutamante (0.5 mmol gDW-1 h-1) were supplied.Single gene deletion (SGD) and single reaction deletion (SRD) analyses were performed using both FBA and MOMA. Reaction deletions were simulated by setting the corresponding flux to zero. Gene deletions were simulated by setting to zero all fluxes catalyzed by the corresponding gene product. Lethality was evaluated based on the deletions that led to infeasible problems or to biomass formation lower than 0.01 hThe shift from homolactic to heterolactic metabolism was simulated under anaerobic conditions, using FBA. Appropriated flux constraints were determined by fitting the model to experimental results, which is described in the following.-1 h-1. Amino acids uptake rates were calculated from the amino acid cell content assuming a specific growth rate of 0.8 h-1 [Two different approaches were selected to predict cell growth and product formation. First, glucose and amino acid uptake rates were set to fixed values and biomass production was maximized. Growth rate and products formation were determined as a function of different glucose uptake rates. Values for sugar uptake rates were taken from the literature ,30, rang 0.8 h-1 . A secon 0.8 h-1 .-1 h-1). Pyruvate secretion was not allowed and oxygen uptake was unconstrained. In silico gene deletions were simulated by constraining the respective fluxes to zero and solving the FBA problem while optimizing for biomass. Single gene deletions leading to high 2-acetolactate formation rates while allowing for growth (specific growth rate higher than 0.01 h-1) were selected as targets for metabolic engineering, as described in the Results and discussion section.Simulations were performed under aerobic conditions and on rich media (as described above), with glucose as the carbon source . Sheet "alphabetic_order" lists the reactions alphabetically.List of reactions, ordered by metabolic pathway and alphabetically. List of reactions included in the metabolic network of Click here for fileBiomass composition. Derivation of the equation for biomass composition, based on literature information.Click here for fileNon-connected Metabolites. List of the non-connected metabolites and reactions.Click here for fileSingle gene and reaction deletions. List of genes and reactions from the metabolic network found to be lethal, ie, no growth was observed when setting the corresponding flux to zero and applying FBA or MOMA. Results are described both under anaerobic and aerobic conditions and both during growth on rich and minimal media.Click here for fileModeling the shift from homolactic ot heterolactic metabolism. These tables summarize the modeling procedure for inclusion of appropriate constrains when applying FBA to simulate anaerobic growth.Click here for file


In order to improve understanding of metabolic systems there have been attempts to construct S-system models from time courses. Conventionally, non-linear curve-fitting algorithms have been used for modelling, because of the non-linear properties of parameter estimation from time series. However, the huge iterative calculations required have hindered the development of large-scale metabolic pathway models. To solve this problem we propose a novel method involving power-law modelling of metabolic pathways from the Jacobian of the targeted system and the steady-state flux profiles by linearization of S-systems.The results of two case studies modelling a straight and a branched pathway, respectively, showed that our method reduced the number of unknown parameters needing to be estimated. The time-courses simulated by conventional kinetic models and those described by our method behaved similarly under a wide range of perturbations of metabolite concentrations.The proposed method reduces calculation complexity and facilitates the construction of large-scale S-system models of metabolic pathways, realizing a practical application of reverse engineering of dynamic simulation models from the Jacobian of the targeted system and steady-state flux profiles. Systematic modelling has emerged as a powerful tool for understanding the mathematical properties of metabolic systems. The rapid development of metabolic measurement techniques has driven advances in modelling, especially using data on the effects of perturbations of metabolite concentrations, which contain valuable information about metabolic pathway structure and regulation . A powerS-system modelling from time-course data is often difficult due to its non-linear properties. Non-linear-fitting algorithms, such as genetic algorithms or artificial neural networks, have been used to resolve this problem -14. AlthWe present an approach to power-law modelling of metabolic pathways from the Jacobian of the targeted system and steady-state flux profiles with linearization of the S-system. This reduces the number of underspecified parameters. Two numerical experiments show that the S-system model generated by this method describes similar dynamic behaviour to that indicated by conventional kinetic models.et al [As a first step, the Jacobian must be obtained from metabolic time-course data. In this section, we summarize the method of Sorribas et al , which wIn biochemical systems, the Jacobian can be defined as:J is the Jacobian matrix, and δX represents a small perturbation and contains the concentration Xi as its elements. The elements of the Jacobian can be obtained from perturbed time-courses using linear least-squares fitting [where  fitting ,18. This fitting . The matThe S-system is a power-law representation constructed of two terms: the production rate and the degradation rate:i and βi are rate constants, gij and hij are kinetic orders, and xi represents the concentration of a compound.where αIn steady state conditions, it can be expressed simply as:Flux =  V+ =  V -     (3)i, and V+ and V- are production and consumption terms, respectively.where Flux is the sum of the steady-state fluxes into xIn steady state conditions, the production term and the consumption term in Eq.(2) can therefore be represented as:j,0 represents the steady-state concentration of xj.where xi and βi yield:Consequently αi as:Defining X = yi           (7)the Jacobian of Eq. (2) can be represented as:i and Eq. (6) with βi, giving:In steady state conditions, Eq. (8) can be simplified by substitution of Eq. (5) with αand thus:Jij, xj,0, and Flux are given, Eq. (10) is a constraint for the determination of the kinetic orders gij and hij. Savageau has described the linearization of S-system as an "F-factor" for stability analysis [Once Jacobian analysis . We use ij and/or hij are available from the structure of the metabolic pathway; however, in the absence of known kinetic orders, parameter estimates are needed to determine them. In such cases Eq. (10) is adopted as limiting the parameter search range.In most cases gIn the first case study, we applied this approach to the published biochemical model of yeast galactose metabolism shown in Figure In this case, all parameters of the S-system model were determined, without the need for estimation. The biochemical model could be converted into the following S-system form.1 is an independent variable, it was omitted from the listed rate equations.Since X22 = g32, h23 = g33, h33 = h43 = g53, h34 = h44 = g54, h54 = g44, h55 = g45, β2 = α3, β3 = β4 = α5, and α4 = β5. The steady-state concentrations were: X1,0 = 0.50 mM , X2,0 = 0.146 mM, X3,0 = 0.007 mM, X4,0 = 0.817 mM, and X5,0 = 0.243 mM. The steady-state fluxes were: V1 = V2 = V3 = V4 = 0.081 mM/min. The Jacobian was:The following constraints were provided from the pathway structure: h21 and g32 were determined by:Our method was able to generate the S-system model of the linear pathway from the steady-state metabolite concentrations, the steady-state fluxes, and its Jacobian. For example, g22 is equal to g32,Because hIn this modelling process, all the parameters were determined by simple calculations of this kind. The resulting S-system model was:3 is the inhibitor of both V3 and V4.The branched biochemical pathway shown in Figure Most of the parameters were obtained by our method. However one parameter could not be determined by calculation and a parameter estimation method was used. The model could be converted into the following S-system form:11 = g21. Steady-state concentrations were: X1,0 = 0.067, X2,0 = 0.049, X3,0 = 0.081, X4,0 = 0.041, and steady-state fluxes were: V1 = V2 = 0.1, V3 = V5 = 0.043, and V4 = V6 = 0.057. The Jacobian was:The following constraint was provided by the pathway structure: h3, g33, β3, and h33. For example, h11 is a parameter which could be determined by:More than half of the parameters were obtained by simple calculations using the steady-state concentrations, the steady-state fluxes, and the Jacobian. The following four parameters could not be determined: α33 is a parameter that could not be determined by a calculation, an estimate provided by some constraint was needed. As α3, h33, and β3 could be treated as dependent parameters, once g33 was obtained all four parameters were determined:Because g33 could be limited by the definition that h33 has a positive value since X3 is the substrate of the reaction V3. The search range of g33 was set from 0 to -0.95. When the number of underspecified parameters was one whose search range was limited, a linear-fitting algorithm could be used to fit the time-course data in the original kinetic model. In this modelling, g33 was determined by a linear optimization method. Consequently, the following S-system model was generated:Moreover, the search range of gThe list of parameters used in the above calculation is shown in Table To evaluate the versatility of this method, the transient dynamics of the S-system model in response to various perturbations were compared with those of the original Michaelis-Menten model, by calculating the mean relative errors (MRE):i(t) represents the time course calculated from the created S-system model, and xi(t), the time course calculated from the original Michaelis-Menten model. In this experiment there were 10 sampling points, the interval between sampling points was 0.5, and X2 in case study 1 and X1 in case study 2 were the targets of perturbations ranging from 0% to 200%. The time-courses of metabolites in response to perturbations of 100% are shown in Figure where n is the number of metabolites in the model, m, the number of sampling points in the time course, x'The changes of MREs in response to the perturbation range are shown in Figure Power-law modelling from time-course data of metabolite concentrations often requires parameter estimates that depend on the size of the target metabolic network. Especially when developing a large-scale metabolic model, this requires problem-specific simplifications. Our method can reduce the number of underspecified parameters by using steady-state flux profiles and the Jacobian of the targeted system derived from time courses of metabolites, and is thus suitable for large-scale power-law modelling. To validate our methods we used two existing biochemical pathway models, the straight and branched pathway models, described by dynamic equations. In the S-system, modelling of a linear metabolic pathway with 12 parameters (case study 1), our method determines all parameters accurately, whereas the method developed by Diaz-Sierra and Fairén  leaves aRobustness against experimental noise is an important requirement for the practical application of modelling methods. Since the Jacobian is rather sensitive to the noise in time series , a robusTo assess the robustness of our method against experimental noise, we measured the calculation error when numerical errors were manually inserted into the Jacobian or steady state fluxes. Table Sorribas and Cascante proposed a method for identifying regulatory patterns using a given set of logarithmic gain measurements . In theiComprehensive metabolome data will undoubtedly accumulate as a consequence of advances in metabolic measurement techniques -26. In oOur method provides stable and high-throughput S-system modelling of metabolic pathways because it drastically reduces underspecified parameters by employing the steady-state flux profile and Jacobian retrieved from time-course data. S-system models generated by this method can provide accurate simulations within a wide range around the steady-state point. In combination with the metabolome measurement techniques it should permit high-throughput modelling of large-scale metabolic pathways.et al. [The model of the yeast galactose utilization pathway was constructed by Atauri et al. , whose rThe rate equations of the model are:where the rate expressions are:1; 0.5 mM (fixed), X2; 0.146 mM, X3; 0.00703 mM, X4; 0.817 mM, X5; 0.243 mM, and the flux through the pathway; 0.0081 mM·s-1.The parameters used are available in reference . The calThe rate equations of the branched model the reaction scheme of which is shown in Figure  = V1 - V2 = V2 - V3 - V4 = V3 - V5 = V4 - V6where the rate expressions are:V = 0.1The author(s) declare that they have no competing interests.Kitayama contributed to the development of the modelling method. Kinoshita supported the development of the mathematical theory of this method and wrote this manuscript. Sugimoto designed two experiments for method verification. Nakayama provided the basic ideas and directed the project, and Tomita was the project leader.