Geobacter metallireducens was the first organism that can be grown in pure culture to completely oxidize organic compounds with Fe(III) oxide serving as electron acceptor. Geobacter species, including G. sulfurreducens and G. metallireducens, are used for bioremediation and electricity generation from waste organic matter and renewable biomass. The constraint-based modeling approach enables the development of genome-scale in silico models that can predict the behavior of complex biological systems and their responses to the environments. Such a modeling approach was applied to provide physiological and ecological insights on the metabolism of G. metallireducens.G. metallireducens was constructed to include 747 genes and 697 reactions. Compared to the G. sulfurreducens model, the G. metallireducens metabolic model contains 118 unique reactions that reflect many of G. metallireducens' specific metabolic capabilities. Detailed examination of the G. metallireducens model suggests that its central metabolism contains several energy-inefficient reactions that are not present in the G. sulfurreducens model. Experimental biomass yield of G. metallireducens growing on pyruvate was lower than the predicted optimal biomass yield. Microarray data of G. metallireducens growing with benzoate and acetate indicated that genes encoding these energy-inefficient reactions were up-regulated by benzoate. These results suggested that the energy-inefficient reactions were likely turned off during G. metallireducens growth with acetate for optimal biomass yield, but were up-regulated during growth with complex electron donors such as benzoate for rapid energy generation. Furthermore, several computational modeling approaches were applied to accelerate G. metallireducens research. For example, growth of G. metallireducens with different electron donors and electron acceptors were studied using the genome-scale metabolic model, which provided a fast and cost-effective way to understand the metabolism of G. metallireducens.The genome-scale metabolic model of G. metallireducens that features both metabolic similarities and differences to the published model for its close relative, G. sulfurreducens. Together these metabolic models provide an important resource for improving strategies on bioremediation and bioenergy generation.We have developed a genome-scale metabolic model for  Geobacter species are environmentally significant because of their capacity for dissimilatory Fe(III) reduction  acetate isotopomer labeling flux analysis. Overall, the flux distributions were similar . The computationally predicted and experimentally determined values were well matched at high fluxes, but less consistent at low fluxes. One difference between the two analysis is that the [1-13C] acetate isotopomer labeling flux analysis used a network where serine, glycine, and cysteine were derived from 3-phosphoglycerate. However, gene encoding for these functions has not been found in G. metallireducens, whereas genes for a pathway where serine, glycine, and cysteine were derived from oxaloacetate in the TCA cycle were identified and used in G. metallireducens genome-scale model and simulations. This may account for some of the differences, such as the differences of fluxes from 3-phosphoglycerate or oxaloacetate to biomass, observed between the model simulation and the 13C labeling flux analysis results., et al. . The res biomass . As show13C labeling experiments were within such feasible ranges (data not shown), and validated the consistency between the experimental and predicted results. These results suggested that in silico growth simulation optimized for biomass formation and flux variability analysis to define the feasible flux ranges together provided a fast and easy alternative method to estimate flux distribution for the metabolism of G. metallireducens.Flux variability analysis defines a feasible range of fluxes for each individual reaction . A flux In silico deletion analyses for G. metallireducens growth using electron donor/acceptor pairs of acetate/Fe(III) or acetate/fumarate were completed and the results were shown in Figure G. metallireducens growth under the conditions. These deletion analysis results are similar to the results from the G. sulfurreducens model. This suggests that the core metabolic pathways are conserved among Geobacteraceae, and that the models of G. metallireducens and G. sulfurreducens can be used to represent of the physiology of Geobacteraceae.Genome-scale metabolic model enabled the systems level gene deletion analysis for growth in defined medium. This information will provide important insight into the potential phenotypes associated with gene deletions in genetic investigations. Geobacter species are being extensively studied for their applications in bioremediation and bioelectricity production. However, similarities and differences in the metabolism and physiology of Geobacter species have not been well characterized. In this report, we have developed a genome-scale metabolic model for G. metallireducens to accelerate discovery and gain insight into its metabolism. Together with the published G. sulfurreducens model, the G. metallireducens metabolic model provides an important resource for the improving strategies for bioremediation and bioenergy generation.Environmental pollution and sustainable energy are among the most important challenges that the world is facing in the 21st century. Consequently, these areas have attracted significant research efforts. In particular, G. metallireducens was used to gain insight into the metabolism of this bacterium. The G. metallireducens model is metabolically distinct from the G. sulfurreducens model, largely due to the wider range of G. metallireducens substrate utilization. The G. metallireducens metabolic model contains many additional reactions reflecting these specific metabolic capabilities that the G. sulfurreducens model does not have. In silico modeling of these additional metabolic capabilities can be used to understand how these substrates are utilized by G. metallireducens and how these capabilities can be applied in bioremediation and bioelectricity production.The reconstructed metabolic model of G. metallireducens model suggested that its central metabolism contains several energy-inefficient reactions that are not present in the G. sulfurreducens model. Experimental biomass yield of G. metallireducens growing with pyruvate was lower than the predicted optimal in silico biomass yield, and microarray data of G. metallireducens growing with benzoate and acetate indicated that genes encoding these unique reactions were up-regulated by benzoate. These results suggested that the energy-inefficient reactions were likely turned off during G. metallireducens growth with acetate to optimize biomass yield, but were up-regulated during growth with complex electron donors to improve flux for rapid energy generation. Thus, the evolution of G. sulfurreducens and G. metallireducens might have occurred in environments with different nutrient levels: G. sulfurreducens in a predominantly acetate limiting environments, whereas G. metallireducens probably in environments in the presence of complex nutrients or nutrient abundance. These results will help understand the physiology of these Geobacter species in the subsurface environments.Detailed examination of the in silico computational modeling approaches were applied to accelerate G. metallireducens research. For example, growth of G. metallireducens with different electron donors and electron acceptors were simulated using the genome-scale metabolic model. These simulations provided an easy and cost-effective way further understanding the metabolism of G. metallireducens. Flux distribution was compared between in silico prediction and 13C labeling flux analysis results, suggesting that in silico prediction could provide a fast alternative method to estimate metabolic fluxes. Finally, the deletion analysis of the G. metallireducens metabolic model predicts phenotypes of gene knock-outs systematically and quickly. It is also important to understand that the testable hypotheses and predictions generated by in silico computational modeling with the reconstructed model should be evaluated experimentally. The experimental findings will in turn further refine and expand the reconstructed model, as well as improve our understanding of the Geobacter metabolism, in an iterative fashion.Furthermore, several ACKr: acetate kinase; ACOAH: acetyl-CoA hydrolase; ACS: acetyl-CoA synthetase; ATO: acetyl-CoA transferase; gdw: gram dry weight; ORF: open reading frame; PTAr: phosphotransacetylase; PPC: phosphoenolpyruvate carboxylase; PPCK: phosphoenolpyruvate carboxykinase; PPCKG: phosphoenolpyruvate carboxykinase (GTP); ac: acetate; accoa: acetyl-CoA; actp: acetyl phosphate; oaa: oxaloacetate; pep: phosphoenolpyruvate; succ: succinate; succoa: succinyl-CoA; ORF: open reading frames; gdw: g dry weight; GPR: gene-protein-reaction.G. metallireducens. BS and JEB carried out the growth experiments. JS and RM analyzed the experimental data and drafted the manuscript. JS, CHS, DRL, and RM conceived the study and revised the manuscript. All authors read and approved the final manuscript.JS, PP, TRF, and IF developed the genome-scale metabolic model of G. metallireducens model.Non-gene associated reactions in the  This table provides the detailed information for the non-gene associated reactions in the G. metallireducens model, including justification for the inclusion of each non-gene associated reaction in the model.Click here for fileG. metallireducens model.All the reactions, genes, and metabolites in the  The data provide comprehensive lists of genes, reactions, metabolites, and GPR associations in the G. metallireducens metabolic model.Click here for fileG. sulfurreducens model.All the reactions, genes, and metabolites in the  The data provide comprehensive lists of genes, reactions, metabolites, and GPR associations in the G. sulfurreducens metabolic model.Click here for file


In contrast to global assessments of changes in gene or protein interaction networks, we study here control aspects of the diauxic shift by performing a detailed analysis of one specific pathway–sphingolipid metabolism–which is known to have signaling functions and is associated with a wide variety of stress responses.The diauxic shift in yeast requires cells to coordinate a complicated response that involves numerous genes and metabolic processes. It is unknown whether responses of this type are mediated The approach uses two components: publicly available sets of expression data of sphingolipid genes and a recently developed Generalized Mass Action (GMA) mathematical model of the sphingolipid pathway. In one line of exploration, we analyze the sensitivity of the model with respect to enzyme activities, and thus gene expression. Complementary to this approach, we convert the gene expression data into changes in enzyme activities and then predict metabolic consequences by means of the mathematical model. It was found that most of the sensitivities in the model are low in magnitude, but that some stand out as relatively high. This information was then deployed to test whether the cell uses a few of the very sensitive pathway steps to mount a response or whether the control is distributed throughout the pathway. Pilot experiments confirm qualitatively and in part quantitatively the predictions of a group of metabolite simulations.The results indicate that yeast coordinates sphingolipid mediated changes during the diauxic shift through an array of small changes in many genes and enzymes, rather than relying on a strategy involving a few select genes with high sensitivity. This study also highlights a novel approach in coupling data mining with mathematical modeling in order to evaluate specific metabolic pathways. It is not surprising that the diauxic shift constitutes a very complicated dynamic process that requires fine tuned coordination at the genomic and biochemical levels. At the genomic level, the switch to secondary non-fermentable carbon sources necessitates sweeping changes in gene regulation, which have been assessed with microarrays measured at a series of time points [Yeast cells challenged by depletion of their preferred carbon sources in the surrounding medium begin using other available carbons for energy production. This switch, usually from glucose to ethanol and acetate, is known as the e points ,2e.g., [Specifically about the time of diauxic shift, the cells begin up-regulating hundreds of genes, which are associated with respiration, fatty acid metabolism and the launch of an environmental stress response, while down-regulating other genes whose products are no longer needed in prior amounts . In turAs such, published microarray data contain a hidden wealth of information, and often specific aspects of cell regulation are of interest to particular investigators. Therefore, there are increasing needs to develop approaches that allow extraction of relevant data and then applying specific analytical methods on these data in order to predict functional consequences. In this study, we focus on sphingolipid metabolism and changes that occur during the diauxic shift. The choice of this pathway system was based on the fact that sphingolipids have been recognized in yeast and other eukaryotes as important signaling molecules that respond to a variety of stresses and are crucially involved in the coordination of stress responses . The oveSaccharomyces cerevisiae following the initiation of heat stress [In a pilot study using a similar strategy, we previously translated global mRNA microarray results into a mathematical pathway model, which was then employed to study the coordination of the glycolytic pathway in t stress . Using st stress . Analyzit stress  used mict stress ,2 into aThe two published microarray data on the diauxic shift consist of global mRNA measurements at seven time points, spanning a period of about 12 hours [Specifically, we are interested in changes within the (sphingo)lipidomic profile between a baseline fermentative state during exponential growth (at 11 hours of batch culturing) and a later time point at 21 hours, which corresponds to respiration after the diauxic shift [Complementing the microarray data ,2, our aKM values in Michaelis-Menten models or rate constants and kinetic orders in power-law models, while the latter typically refer to enzyme activities and input variables, such as substrates and other precursors or modulators. Relative changes in model output that are caused by small perturbations in independent variables are called logarithmic gains . These Sphingolipid metabolism constitutes an interesting system, as it is biochemical in nature and should therefore be robust, exhibiting small gains. At the same time, some of the sphingolipids and their relative amounts serve as signaling molecules, which therefore have to respond forcefully to the sensing of specific, and often adverse, environmental conditions such as heat shock or oxidative stress. For these reasons of contrasting demands, it is interesting to study log gain profiles of the sphingolipid pathway in detail. We execute this analysis here, focusing on functional clusters of variables and fluxes of primary significance, and compare our findings to results characterizing diauxic shift conditions. Given the complexity of the pathway one should expect that there are multiple ways of genomic and metabolic switching from the pre-diauxic metabolic profile to one that is suited for post-diauxic conditions. To gain insight into this switch, we will study the specific question of whether yeast employs a few independent variables (enzymes) with high log gains that are able to effect appropriate changes in metabolic profile during diauxic shift, or whether larger numbers of enzymes are adjusted only slightly. We will also explore whether there is a preference for exerting control through changes in precursors or in enzyme activities.Finally, we discuss the utility of this approach as a prototype that can be employed towards 'mining' pathway-specific data from the ever-increasing numbers of published microarrays, and then using these data to predict functional metabolic consequences.The analysis is overall divided in three parts, which are all executed with a recent mathematical model of sphingolipid metabolism . SinOne of the most widely used quantitative criteria of model quality and robustness is parameter sensitivity.e.g., [e.g., [In a comprehensive sensitivity analysis, each parameter is modulated by a small amount, and the effects of this modulation on steady-state concentrations and fluxes e.g., ,15), or , or e.g., [e.g., ,17) are logarithmic gains (LG), which have been successfully applied to moderately large biological systems , are, arelogaAn LG with magnitude greater than 1 implies amplification of the perturbation; thus, a 1% change in the independent variable evokes more than 1% in the steady-state output quantity. A magnitude less than 1 indicates attenuation. A positive sign for the LG indicates that the changes are in the same direction, so that both increase in value or both decrease. A negative sign indicates that the changes are in opposite directions.e.g., further metabolism) of this variable but are not included in the model. These additional processes tend to buffer the variable against perturbations. Third, a variable associated with a high LG is not modeled with sufficient accuracy. It could be that a very inaccurate value is assigned to a parameter or that some production or degradation processes are missing. True high gains are interesting because they allow the cell to effect a desired change or adaptation to a new situation with relatively modest effort. At the same time, high gains are obviously difficult to control. We will analyze in a later section to what degree yeast may employ high-gain variables to organize the diauxic shift from fermentation to respiration.In typical, robust models of metabolic pathways, the majority of LGs are in a range between -1 and 1, which indicates that perturbations in most independent variables are attenuated by the system. LGs with a magnitude between 1 and 5 characterize the effect of moderate amplification. LGs of much higher magnitude typically have one of three causes. The particular independent variable may truly have a high gain, which is, for instance, the case in signaling systems whose role it is to amplify weak incoming signals. Second, the independent or the dependent variable associated with the LG is at the fringes of the model, and the high gain is an artifact due to processes that in reality contribute to the dynamics , dihydrosphingosine , dihydroceramide , dihydrosphingosine-1P , phytosphingosine , phytosphingosine-1P , phytoceramide , inositol phosphorylceramide , palmitoyl-CoA , and serine . They are represented in the diagram of Fig. For the current analysis, we focused on LGs for metabolites and fluxes of the sphingolipid core, e KDHS, X, dihydroe KDHS, X, dihydroThe LG analysis described above characterizes the robustness of the model with respect to a given, small perturbation. In contrast to such small alterations in values, a coordinated cellular response such as the diauxic shift from fermentation to respiration is associated with multiple changes in gene expression and enzyme activities, which are not necessarily small. To analyze this response, we used two sets of published time series of yeast microarray data ,2 one foet al. [X36), phosphatidate phosphatase (X39), G3P acyltransferase (X49), and ELO1p (X59), or by different subunits such as FAS (X52) and SPT (X57). The contributions of these isozymes and subunits were weighted against their corresponding mRNA isoenzymes or subunits.DeRisi et al.  quantifiX63) isoenzymes were not weighted because their product (Ac-CoA) is considered an independent variable in the model.The ACSp (X39), represented in Table For example, at 9 hrs, the two reported isoenzymes for phosphatidate phosphatase X9, represAs a second example, the weighted phosphatidate phosphatase mRNA fold change at 19 hrs is computed asIn a more refined analyses, one could represent each isozyme separately, which however would require more input data for model design.600 ≅ 0.1 and incubated at 30°C and 220 rpm. Samples were taken after 6 hours (OD600 = 0.34) and 24 hours (OD600 = 2.2), spun down at 3000 rpm for 5 minutes, the supernatant removed, and the remaining cell pellet frozen at -80°C until lipid analysis.Background strain BY4742 (MATα his3Δ1 leu2Δ0 lys2Δ0 ura3Δ0) from the yeast deletion library was first grown in an overnight culture of YPD from a freshly streaked plate of the frozen stock. Flasks of SC medium were then inoculated to a starting ODet al. [Samples were fortified with internal standards, extracted with a solvent system modified from Mandala et al.  and thenet al. . Peaks cQuantitative analysis was based on the calibration curves generated by spiking an artificial matrix with the known amounts of the target analyte synthetic standards and an equal amount of the internal standards (ISs). The target analyte/IS peak areas ratios were plotted against analyte concentration. The target analyte/IS peak area ratios from the samples were similarly normalized to their respective ISs and compared to the calibration curves, using a linear regression model.The lipid concentrations from the mass spectrometry analysis were normalized by total lipid phosphate as determined with a standard curve analysis and colorimetric assay of ashed phosphate: aliquots of extracted samples were re-extracted via Bligh and Dyer  to separInitially, the most relevant metabolites were grouped in three functional blocks and analyzed with respect to the flux and metabolite LG within each block. The blocks were chosen as: a) precursor block, including fatty acid metabolism and serine metabolism; b) sphingolipid block, including complex and backbone sphingolipids, which are crucial in cell regulation ,28, and In Fig. X4) to perturbations in independent variables, as shown in Fig. X4. The dark and light blue lines indicate that alterations in PS synthase and PI kinase activities have essentially no effect on the steady-state value of DHS-P. Following the spoke inward, one can see that a 1% increase in G3P acyltansferase leads to a steady-state DHS-P value that is decreased by about 4%. Thus, the LG is about -4. Looking at the corresponding spoke in Fig. In Fig. The widest metabolite LG range was obtained inside the precursor block  followed by the sphingolipid block , and lastly by the glycerolipid block . The fluxes in all three blocks have smaller LG values than the metabolites, which means that the metabolic profile is more sensitive than the overall flux pattern.X12) and Ac-CoA (X25), followed by serine (X13). Also high are LG for ACSp (X63) and FAS (X52), which is consistent with the crucial biological importance of these two enzymes for yeast viability: indeed, ACS1/2 double null mutant yeast strains and FAS3 knockout have been reported as non-viable [The LG pattern in this block is extreme Figs. . Some ken-viable ,30.e.g., [The LG for Ac-CoA precursors have mostly an inverse effect on the sphingoid phosphates, which suggests that even small increases in Ac-CoA could lead to significant decreases in these metabolites. While the importance of Ac-CoA for sphingolipid dynamics is clear from this LG profile, the specific numerical values of the LG associated with Ac-CoA should at this point be considered merely as a measure of tendency. First, a 1% increase in Ac-CoA corresponds to an available Ac-CoA concentration of 8.7 μM of material into the system. This amount is very large in comparison to the normal sphingolipid concentrations. Second, it is known that Ac-CoA is involved in many processes that are not modeled here , with te.g., , Fig 2A–The high positive LG associated with external serine transport is in accordance with experiments from our laboratory where this process was identified as the determinant for the control of sphingolipid flux and even more important than external palmitate input . Again, It might be interesting to note that the concentration of DHS-P is more strongly affected than that of PHS-P, while the opposite is true with respect to fluxes. Given the high LG, one could expect sphingoid base kinase and lyase to be more influential, but that does not seem to be the case.X57) has the strongest effect  complex exhibits negative LG for the sphingoid phosphates and backbones, indicating that increases tend to short-circuit production of sphingoid phosphates (or compete with it) and instead channels fatty acid precursors directly into ceramide, which is immediately  used for IPC-g and the production of complex sphingolipids.Interestingly, the Elo1p (X49) tends to have negative LG values because increases in this enzyme divert its substrate, palmitoyl-CoA (X12), away from sphingolipid metabolism and toward the glycerolipid pathway. The strongest effects are again seen in the sphingoid phosphates.The LG of this block Figs.  are geneX46) are relatively high. The reasons are not intuitively evident, and we will explore in the laboratory whether this enzyme might be a regulator of the sphingolipids pathway. The fluxes through the IPC-g require phosphatidylinositol (PI), so that increases in enzymes that generate PI might significantly affect the production of complex sphingolipids, while enzymes that use PI in other processes, like PI kinase, have the opposite effect.Interestingly, the LG associated with inositol-1-phosphate synthase (X40) and PS synthase (X38) that are consistent with the mRNA fold changes in the microarrays and therefore did not require adjustments.Table r = 0.88) is very similar to the value reported for replicate experiments in DeRisi's complete dataset (r = 0.87) [p = 0.93). All of the above confirms that the two datasets are consistent and that the model implementation is reproducible.Likewise, Table  = 0.87) . To confi.e., within a factor of 2, which used to be considered the signal-to-noise threshold at the time when DeRisi's data were obtained. During Phase 2, ACSp and serine transport spike to 10- and 5-fold levels, respectively. The situation is similar for the glycerolipid and sphingolipid blocks . By and large, all other metabolites are at or below the 11-hr baseline level. Between 17 and 19 hrs, some dramatic changes take place. Most prominent are several-fold increases in the sphingoid phosphates DHS-P and PHS-P. DHS, PHS, PA, and DAG are also increased. Interestingly, C26-CoA becomes significantly depleted, while IPC increases four-fold.Microarray results on the sphingolipid pathway, taken by themselves, are difficult to interpret. However, application of the current dynamic model facilitates an additional type of exploration. Under the assumption that most regulation is transcriptional, it is easy to adjust enzyme activities in accordance with changes in mRNA levels and to compute the corresponding metabolic profiles by letting the model reach steady state. The results of this analysis are shown in Fig. 26-CoA is produced faster than it is utilized. This is not accomplished with one or two high-gain enzymes that are strongly up- or down-regulated, but through the coordinated regulation of many enzyme activities that are slightly increased or decreased. Initially Elo1p is up-regulated, but quickly returns to its original level, so that it alone cannot be responsible for the increase in C26-CoA levels. Notably, all backbone sphingolipids and sphingoid phosphates are reduced in concentration, which through the lyase reaction may lead to recycled palmitoyl-CoA, which is subsequently converted into C26-CoA. One could surmise that some important factor might be missing from the model, but even if that was the case, it would yield no explanation, because the mathematical model generates these higher levels of C26-CoA without additional input or manipulation. Further analysis of Figs. e.g. [Considering Figs. gs. e.g. ).de novo palmitoyl-CoA production 2C synthase are also up-regulated, suggesting utilization of remaining sphingolipids in membrane material.During Phase 2, drastic changes occur. Both ACSp and serine transport are strongly up-regulated, along with enzymes that shift production to sphingoid phosphates Fig. . These mOverall, the cell seems to use a mixture of high-gain variables, including serine transport (with positive LG) and SHMT (with negative LG), and low-gain variables, such as ceramidases, to mount the sphingolipid responses to glucose depletion. ACSp, which has a very strong negative gain on sphingoid phosphates, is strongly up-regulated in Phase 2. One can only speculate that the effects of this up-regulation may be compensated by a lack of substrates for this reaction.de novo experiment, subsequent to the model predictions, wild type sphingolipid levels were determined after 6 and 24 hours of growth, representing two clearly defined pre- and post- diauxic time points (Table In a ts Table . The folts Table  show genX2) and especially DHS-P (X4) in the experimental data vs. the major increases shown for those metabolites in the simulations. This disparity may result from the difference in strains and growth media in our experiments and those of DeRisi et al. [et al. [cf. [There is a large quantitative difference for DHS .The results from this study advance the development, analysis, and utility of a previous mathematical model for sphingolipid metabolism in yeast. The model was further refined with minor modifications encompassing the inclusion of enzymes for PI degradation, serine import, and GPI remodeling, and adjustments to ceramide synthase fluxes based on experimental data (see Appendix). LG analysis was conducted, and this provided further insight into the model and into the structure of the sphingolipid pathway in yeast. Finally, the model was applied to data extracted from the literature on changes in enzymes of sphingolipid metabolism, and this allowed for specific and novel insights into sphingolipid metabolism in during the diauxic shift. This use of a mathematical model, coupled with the integration of data from several sources, promises to have applicability beyond the system discussed here.de novo fatty acid synthesis , ATP, and the serine hydroxymethyl transferase have the strongest effect on the sphingolipid backbone metabolites  for further study.Several key features emerged from the calculated changes in sphingolipids during the diauxic shift.2C in the post diauxic phase are consistent with this requirement  by a 32% and 64% at 19 and 21 hrs respectively. This decrease exerted great influence on the metabolomic profile because Elo1p shows one of the highest LGs reported DPL1 pre and post diauxic shift, the breakdown of sphingoid phosphates through lyase (X50) increased 45% at 19 hr and returned to the basal values at 21 hr , increased 100% at 19 hr. On the other hand, mRNA of LCB3, which codes for one of the sphingoid base phosphate phosphatases (X41) decreased 40% from pre to post diauxic shift.RSB1, a suggested ATP dependent flippase or transporter for LCBs which may catalyze LCB movement from the inner to the outer leaflet of the plasma membrane [RSB1 increased four fold with respect to its value at the early exponential phase, suggesting a requirement for increased plasma membrane LCBs translocation.Other notable changes are those in membrane . In the SVF1 and YPK1 to about 50% [Svf1p was recently shown to be involved in the regulation of sphingoid bases and their phosphates . Ypk1p iet al. [+2 influx, such as BNI1, FAR1, MID1, and CRZ1 [PMC1 (a gene that codes for vacuolar Ca2+-ATPase) suggest the tendency to maintain a low cytoplasmatic Ca2+ level at this cellular moment.Third, there are many possible relationships between the changes in enzymes of sphingolipid metabolism and calcium homoestasis. According to Birchwood et al. , the accand CRZ1 ,49, we nFourth, the model predicts some notable changes in glycerolipids, which increase at later times. It is important to remark here that the glycerolipids represented in the model are restricted to the metabolites directly related with the sphingolipid pathway and that not all routes involved in glycerolipid metabolism are included. For example, the cardiolipin pathway is not included in the model. This route becomes important when the cell needs to increase the mitochondrial biomass after the diauxic shift. In fact the deletion of the phosphatidylglycerophosphate synthase, the first and rate limiting enzyme of the cardiolipin pathway, is not essential to cell viability but causes growth dependence on fermentable carbon sources because of mitochondrial dysfunction . Pilot se.g., the 19 hr time point).Table Table In conclusion, the computational study presented here suggests that the coordination of sphingolipid involvement in the diauxic shift in yeast is achieved through multiple small modifications of functional clusters of mRNAs rather than through large alterations in just a few "key" genes or metabolic steps. In most cases, the variations in mRNA expression are less than 2-fold, which implies that expression studies have to achieve a level of relatively high accuracy, if they are to detect subtle, coordinated control mechanisms.The combined approaches of this study may also represent an important new paradigm in metabolomic studies whereby mathematical modeling of metabolic pathways is employed to mine and analyze gene expression studies. Given the rapidly increasing accumulation of microarray-based gene expression datasets, such an approach may result in very important insights into the changes, behavior, and possibly function of specific metabolic pathways.FAV developed the model, conducted the simulations and drafted the initial version of the manuscript. KJS performed the experiments. YAH and EOV conceived and supervised the collaboration and overall strategy of the project, and edited the manuscript.et al. [e.g., [The model of Alvarez-Vasquez et al. ,9 is bas. [e.g., -54). In et al. [Enzyme Plb3p, which catalyzes the degradation of phosphatidylinositol (PI), was added to the collection of processes metabolizing PI. Merkel et al.  reportedX43) was not associated with a specific enzyme. Recently, Bosson et al. [Gup1p was included explicitly as a representative enzyme for GPI remodeling. In the original model, the independent variable responsible for driving the phytoceramide flux toward GPI remodeling . This result was implemented as a numerical alteration of the former ceramide synthase fluxes in this model.The quantification of ceramide synthase fluxes was fine-tuned. e.g., [et al. [To simplify the assessment of LG terms, the model was implemented as an S-system by aggregating all influxes and effluxes for each time-dependent variable into a single power-law term each e.g., -54). Thi. Thie.g. [et al.  that altmetabolite LG is defined as the ratio of the percent change in a dependent variable  Xi to the percent change in an independent variable  Xk, while all other independent concentrations and parameters are held constant. It is thus defined asLogarithmic gains come in two varieties. Each where the subscript 0 refers to the chosen operating point, which usually coincides with the steady state.flux LG is defined as:In an analogous fashion, a Vi represents a given (aggregated) flux, and Xk and the subscript 0 are defined as before.where cf. [The conversion of gene expression profiles into changes in enzyme activities cf. -7) relie reliecf.The direct proportionality between a change in mRNA level and the corresponding enzymatic activity presupposes strictly transcriptional regulation, while posttranslational modifications are ignored. There is circumstantial evidence suggesting that this assumption is quite realistic in the case of heat stress responses in yeast.It is assumed that the mathematical model ,9 accuraX13), inositol (X16), and acetyl-CoA (X25) do not change significantly between the time points of analysis. The rationale for this assumption is that these metabolites are involved in very many cellular processes that are not adequately represented in the model. Thus, while they are buffered in reality by a slew of metabolic processes, the model does not allow for such buffering, because peripheral reactions and pathways outside sphingolipid metabolism are not modeled. In fact, experimental results from our lab  suggest that internal serine is maintained close to its baseline concentration of 2.6 mM throughout the diauxic shift. The production of inositol depends directly on G-6-P through catalysis mediated by I-1-P synthase. However, G-6-P is not comprehensively modeled because, for instance, utilization for trehalose or pentose production is not included in the model. Because the branch point at G-6-P is highly regulated and we do not have specific information on changes in inositol concentrations in vivo, we consider inositol as constant by default.It is assumed that the values of internal serine (X52) and ACCp (X60) mRNA's are down-regulated at the diauxic shift, thereby probably affecting the acetyl-CoA influx (Fig. Acetyl-CoA is increasingly notably at the diauxic shift  because lux Fig. .The steady states obtained for each new mRNA profile were obtained with a model where all fluxes are balanced, while the previous model contained some unbalanced fluxes among the complex sphingolipids, reflecting the need for membrane in the exponentially growing culture .


Several approaches, including metabolic control analysis (MCA), flux balance analysis (FBA), correlation metric construction (CMC), and biochemical circuit theory (BCT), have been developed for the quantitative analysis of complex biochemical networks. Here, we present a comprehensive theory of linear analysis for nonequilibrium steady-state (NESS) biochemical reaction networks that unites these disparate approaches in a common mathematical framework and thermodynamic basis.A obtained in the standard, linear-dynamic-stability analysis of the steady-state can be decomposed as A = SRT where R and S are directly related to the elasticity-coefficient matrix for the fluxes and chemical potentials in MCA, respectively; the control-coefficients for the fluxes and chemical potentials can be written in terms of RTBS and STBS respectively where matrix B is the inverse of A; the matrix S is precisely the stoichiometric matrix in FBA; and the matrix eAt plays a central role in CMC.In this theory a number of relationships between key matrices are introduced: the matrix One key finding that emerges from this analysis is that the well-known summation theorems in MCA take different forms depending on whether metabolic steady-state is maintained by flux injection or concentration clamping. We demonstrate that if rate-limiting steps exist in a biochemical pathway, they are the steps with smallest biochemical conductances and largest flux control-coefficients. We hypothesize that biochemical networks for cellular signaling have a different strategy for minimizing energy waste and being efficient than do biochemical networks for biosynthesis. We also discuss the intimate relationship between MCA and biochemical systems analysis (BSA). Developing methodologies for quantitative analysis, both experimental and mathematical, of complex biochemical networks has become one of the central themes of post-genomic biochemistry and mathematical biology. Several disparate approaches, including metabolic control analysis (MCA) ,2, flux The physical concept of a NESS  applies to that of a driven system, such as a living organism, with a constant flow of matter (transforming nutrients to waste) through the system and a corresponding dissipation of free energy . Even thA obtained in linear stability analysis, the flux control-coefficient matrix C in MCA, the stoichiometric matrix S of FBA, and the correlation matrix R from CMC are intimately related. We develop the relationships in the context of complex biochemical networks and their NESS thermodynamics. We show that all the key matrices in MCA and CMC can be computed if one knows the fluxes, chemical potential differences, and the stoichiometric matrix.We distinguish between two types of linear analysis: (i) linear stability analysis , and , fluxes (currents), and resistances. It becomes immediately clear that the resistances are related to reaction effectors, such as the expression levels of enzymes, that affect both the forward and backward fluxes of a given reaction, but do not change the chemical potential difference. It is this thermodynamic perspective of a NESS that can be used to develop a comprehensive theory that unites these approaches.J of each metabolic reaction in a NESS metabolic network is decomposed into J = φ+ - φ- and the chemical potential difference of the reaction in the NESS is Δμ = kBT ln(φ-/φ+), where kB is Boltzmann's constant and T is absolute temperature. Therefore, knowing the chemical potential difference and the flux of a reaction givesIn the context of the newly developed biochemical circuit theory (BCT) ,16, the μo is known from equilibrium thermodynamics, then Furthermore, if the standard-state free energy of reaction Δβ-cells and ketogenesis in hepatocytes, where cellular organelles, such as mitochondria, play a key role in the handshake between metabolism and signaling. Ultimately, our goal is to understand how these cellular systems behave in response to stress and disease [In many theoretical approaches, metabolic networks and signaling pathways are often separated into disparate mechanisms with only peripheral interactions. This is far from reality, however, as these mechanisms are intimately intertwined and often indistinguishable. For example, in whole-body metabolism, glucose and fatty acids serve as the fuel for cellular metabolism, but also as signals for insulin secretion in pancreatic  disease . It is iS, and measures the correlation matrix, R, to obtain the control coefficients, and vice versa . The jth biochemical reaction  is characterized by a set of stoichiometric coefficients Let us consider a network of ν's and κ's can be zero. The N × M matrix S = κ - ν is known as the stoichiometric matrix. Eq. (2) assumes that the forward and backward reaction kinetics are characterized by the constants Some of the integer s. Segel , for exaFor the reaction system (2), kinetic equations of the time-dependent concentration changes for each biochemical species can be written according to the law of mass action as ,20xℓto denote the concentration of respective Xℓ, and i = 1, 2, ..., N) are external injection fluxes to species i . We use the convention that subscripts index species and superscripts index reactions. To further simplify the notation, we introduce forward and backward fluxes where we use j=φ+j−φ−j:dx/dt = SJ + Je, in which x and Je are N-dimensional column vectors, and J is an M-dimensional column vector.Then Eq. (3) becomes closed reaction system, Je = 0. In this case, it can be shown that thermodynamic equilibrium is the only positive stationary solution to Eq. (3): the internal fluxes J = φ+ - φ- = 0 for each and every reaction. For a closed biochemical reaction system, since all the fluxes are necessarily zero in its unique equilibrium , all the control coefficients are necessarily zero. Therefore, in a NESS, at least one injection flux and one effux are nonzero , i.e., there is no unique stationary solution for different initial conditions.same NESS. We use the notations For an open system that approaches a NESS, the stationary solution is not usually degenerate. If we assume {In the linear regime near the NESS ,δxj = xj - in which A = {Aij} is called the linear stability matrix. We introduce the N × M matrix R = A = SRT. We also denote A-1 = B.S does not have full rank and A is singular [A into a nonsingular matrix by replacing its linearly dependent rows with vectors in its left null space (see Ref. 18 of [A is understood to be the transformed nonsingular matrix.If the nonlinear dynamics scheme of Eq. (3) conserves certain quantities, such as total element, motif, or enzyme concentrations, singular ,19,20. IEq. (3) is a general scheme for biochemical reaction networks. For metabolic reactions involving enzymes, the enzyme and the enzyme-substrate complexes are treated as additional "species." In this section, however, we study MCA, and assume that every reaction involves an enzyme which is not explicitly expressed as a species. Rather, we assume enzyme activity is absorbed into the rate constants for the forward and backward reactions. This is clearly not an accurate approximation for many enzymatic reaction mechanisms. Nevertheless, it provides a first-order approximation for treating a metabolic network. More complex rate laws for enzymatic reactions have been discussed for biochemical systems analysis ,22.th reaction or the Nth  species.MCA focuses on how the NESS {N is changed: XN. The immediate, local change is characterized byFirst, we consider the case where the concentration of external species ε = (dg J)-1 RT. Here, we use the same notation used by Heinrich and Schuster [v) denotes the diagonal matrix containing the components of the vector v along its diagonal. The unscaled, local, elasticity-coefficient matrix is given by ε' = (dg J) ε (dg x)-1 = RT (dg x)-1. The prime symbol is used to distinguish unscaled coefficients from scaled coefficients, both here and in what is to follow.which is called the scaled, local elasticity-coefficient . These cSchuster , where  satisfyThe coefficient response . When a BiN is the Nth column vector of the matrix A-1. The new NESS established near {Jm + δJm, are given bySolving Eq. (6), we have Hence, the scaled, steady-state elasticity-coefficient is ε = (dg J)-1 RTB (dg B)-1. Here, (dg B) denotes the diagonal matrix containing only the diagonal components of the matrix B along its diagonal. The unscaled, steady-state, elasticity-coefficient matrix is given by ε' = (dg J) ε (dg x)-1 = RTB (dg B)-1 (dg x)-1.which may also be known as a response coefficient ,25. Thesε and ε are different. In principle, the ε can be experimentally measured as a system response, but obtaining ε can be more difficult since it is a transient response that must be measured individually.As we shall show, some connectivity theorems for the two quantities th reaction EM is changed: EM → EM + δEM. Since we have assumed that the rate constants for reaction M, M, EM, we haveNext, we consider the case where the enzyme for the MEM → EM + δEM, the new NESS satisfies [When atisfies δxj/j = 1, 2, ..., N), we haveSolving for From Eq. (10), we get the scaled, concentration control-coefficient ,BS(dg J). The unscaled, concentration, control-coefficient matrix is x) J)-1 = -(dg x)BS.which is an element of the scaled, concentration, control-coefficient matrix given by Combining Eqs. (7) and 10), we get the scaled, flux control-coefficient [0, we getδmM is the Kronecker delta function. These coefficients are elements of the scaled, flux, control-coefficient matrix given by C = I - (dg J)-1 RTBS (dg J). The unscaled, flux, control-coefficient matrix is C' = (dg J) C (dg J)-1 = I - RTBS.where ε, ε, C, and Among between the scaled coefficients, and similarly,between the unscaled coefficients.SJ = -Je = 0, we haveSecond, from Eqs. (13)–(16), it is straightforward to show that if there is no injection flux, i.e., in NESS 1i × j and 0i × j denote the i × j matrices of ones and zeros, respectively. Eqs. (17) and (18) are known as the summation theorems for the scaled control-coefficients. For the unscaled control-coefficients, we have the generalized summation theorems, C'K = K and K is the M × (M - N) loop (or nullspace) matrix, i.e., its columns form a basis of the nullspace of S so that SK = 0N × (M - N) [where  (M - N) ,16.Euler's theorem on homogeneous functions can be used to understand the significance of the injection fluxes ,26. If tBSRT = I ⇒ (RTBS)(RTB) = RTB, which yields the connectivity theorems for the scaled coefficients:Third, and the generalized connectivity theorems for the unscaled coefficients :Finally, by combining the generalized summation and connectivity theorems, we getK ε') has been proved to be invertible [which is the central equation in MCA . The matvertible ,27. TherUsing the steady-state elasticity-coefficients, we haveK ε') is also invertible (see Methods). Therefore, if the steady-state elasticity-coefficients are known, the control-coefficients can be calculated. This is an important result because ε can be experimentally measured as a system response, but obtaining ε can be more difficult since it is a transient response and must be measured individually.The matrix , and vice versa. In general, the flux J and the chemical potential difference Δμ are nonlinearly related [Both  related :J and Δμ are linearly related with the biochemical conductance being φ+/kBT (= φ-/kBT) [φ- is relatively unperturbed. Then we have δJ/δΔμ = -φ+/kBT. Similarly, if the reaction is perturbed from its downstream, then δJ/δΔμ = -φ-/kBT.In the linear regime with small flux and chemical potential difference,  φ-/kBT) ,16. In tscaled, local elasticity-coefficients for chemical potentials [Combining BCT and MCA, we can define the tentials εμ Δ= kBT (dg Δμ)-1 ST, the scaled, steady-state elasticity-coefficients for chemical potentialsas elements of the matrix εμ Δ= kBT (dg Δμ)-1 STB (dg B)-1, and the scaled, potential control-coefficients [as elements of the matrix ficients Cμ Δ= -kBT (dg Δμ)-1 STBS (dg J), where reaction potentials and substrate concentrations in the reference state are used for normalization. The corresponding unscaled coefficients are given in matrix form asas elements of the matrix ST, STB, and STBS in the local and steady-state elasticity-coefficients and the control-coefficient for the potentials, as compared to the matricies RT, RTB, and RTBS in the coefficients for the fluxes. Furthermore, note the relationshipsNote the appearance of the matrices Eqs. (19) and (29) in combination yield the summation  and connEq. (31) can also be derived using the homogeneous function theory (discussed above with flux and concentration control-coefficients). For the unscaled coefficients, we get the generalized summation and connectivity theorems by combining Eqs. (21) and (30) so thatEq. (33) is a manifestation of Kirchhoff's loop law ,16.By combining the generalized summation and connectivity theorems, we getK ε') has been proved to be invertible [The matrix  has an inverse (see Methods). Therefore, if the steady-state elasticity-coefficients are known, the control-coefficients can be calculated.The matrix (xN ∈ Correlation metric construction (CMC) focuses on how the NESS {If the random perturbation has a sufficiently long correlation time compared to the relaxations of all the reactions in the system, then we are dealing with a shift in the NESS where, according to Eq. (6),δxi} with an uncorrelated perturbation on δxN, we haveFrom the time series of {i, j = 1, 2, ..., N.for δxN is a Markov process characterized byCMC extracts information from reaction systems by using perturbations with insufficiently long correlation times. kd controls the relaxation time of the perturbation, a is its random amplitude with Gaussian distribution, and ξ(t) is a uncorrelated noise. δxN in Eq. (38) is called an Ornstein-Uhlenbeck (OU) process with auto-covariance functionwhere With this type of random perturbation, the temporal correlations between the concentration fluctuations give information on the connectivity of a network following the kinetic equation (4)i = 1, 2, ...,(N - 1). More precisely , instead of Eq. (40) CMC follows an autoregressive (AR) process, the discrete version of an OU process, with discrete time T, 2T, ...,for in which A matrix into a (N - 1) × (N - 1) matrix Let's rewrite the which leads tog(τ) in Eq. (39) is ≫ all the relaxation times of Two limits of Eq. (42) are particularly interesting. If the correlation time of Rij(t) is the response curve, as a function of t, for the ith species to an impulse of the jth species at time 0. The relation between the correlation matrix and the fundamental solution is expected for a linear system. For directly connected species i and j, the i and j are not directly connected, The matrix analyzed .Both flux control-coefficients and biochemical conductances can be used as indicators for rate-limiting steps in a biochemical network . TraditiLet's consider the linear pathwayx0 and xn for X0 and Xn, respectively. Analogous to a continuous energy landscape, we have the discrete energy functionunder NESS with clamped concentrations j = 1, 2,..., n. If there is a rate-limiting step, say for example that reaction j is nearly irreversible, then Ej is maximal and that is where the highest energy barrier is located. We now show that it is also where the largest flux control-coefficient and the smallest conductance are.for The NESS flux of reaction system (44) can be solved for analytically to yield ,32Therefore, the flux control-coefficient isEj. The biochemical conductance, on the other hand, iswhich reaches its maximum at maximal Ej where concentration xℓis the lowest.which is at its minimum when J given, the smallest conductance also corresponds to the largest |Δμ|. In general, the rate-limiting step is not necessarily where the k's are smallest. The most efficient situation for reaction system (44) is when all the conductances are equal. This result is analogous to the constant torque principle suggested for molecular motors [Note that with r motors ,13,33.The analysis provided in the present paper poses the following biological question. When is a NESS of a biochemical network controlled by a constant flux injection and when is it controlled by a clamped concentration ? Clearly in real biological systems, the appropriate answer is a combination of both. Just as a real battery, having both finite internal resistance and conductance, in an electrical circuit is a combination of constant-current  and constant-voltage  ideal batteries, so too would a real biological system be a combination of constant flux injections and clamped concentrations . NeverthJ to a simple unimolecular reaction in a NESS, we have the heat dissipation rate (hdr)First, let's see what are the respective consequences of these two different types of "forcing" to a biochemical system. For a constant flux injection J a reaction with larger φ-  dissipates a smaller amount of heat. For a constant clamped chemical potential Δμ, we haveThis indicates that for the same injection flux hdr increases with φ- for the constant potential difference (concentration clamping) case. These results are analogous to an electric circuit in which the heat dissipation equals I2R and U2/R, respectively.We note that This observation leads us to the following hypothesis. In a biochemical network for biosynthesis, the fluxes are essential for supply and demand. Hence, such networks will in general have large conductances in order to minimize the energy waste and be efficient. On the other hand, in a biochemical network for cellular signaling, the concentrations are the essential signal. Thus, in this case the conductances are generally small in order to minimize the energy waste.J in the enzymatic reaction as a rational function of the concentration of a substrate x [MCA shares an intimate relationship with the biochemical systems analysis (BSA) originally proposed by M.A. Savageau . A critistrate x , such thK ≤ L due to substrate saturation or effector inhibition. The coefficients ai, bi, α0, and β0 contain the concentrations of all the other substrates. BSA provides insight to the properties of ai and bj. The network of metabolic reactions is then modeled [where  modeled  by a sysJ can be written asIn terms of Eq. (49) the local elasticity-coefficient for the flux which also has the standard form of a Pade approximation . Furtherwhich is essentially the integral form of the definition of local elasticity-coefficients.In contrast, MCA assumes no specific rate laws for an enzymatic reaction except that the rate is linearly proportional to the enzyme concentration. This assumption is certainly valid for most biochemical reactions except when a substrate concentration is significantly less than that of an enzyme.K ε'), that appears in the central, control-analysis equations, i.e., Eqs. (24) and (36), is invertible, we note that since we are dealing with systems without conserved quantities, rank(S) = N. Furthermore, we note that the matrix KTK is invertible because K contains linearly independent columns, and because all columns of K are in the right nullspace of S, the matrixTo demonstrate that the matrix,  loop (or nullspace) matrix (SK = 0); R: N × M correlation matrix defined such that A = SRT; S: N × M stoichiometric matrix; x:N-dimensional vector of species' concentrations; J: M-dimensional vector of reaction fluxes; Δμ: M-dimensional vector of reaction potentials; ε:M × N local, elasticity-coefficient matrix; ε: M × N steady-state, elasticity-coefficient matrix; N × M concentration, control-coefficient matrix; C: M × M flux, control-coefficient matrix; CμΔ: M × M biochemical potential, control-coefficient matrix; (dg v): If v is a vector, (dg v) denotes the diagonal matrix with the components of v along its diagonal. If v is a matrix, (dg v) denotes the diagonal matrix with only the diagonal components of v along its diagonal; *: Denotes steady-state; ': Distinguishes unscaled coefficients from scaled coefficients.BCT: Biochemical Circuit Theory; BSA: Biochemical Systems Analysis; CMC: Correlation Metric Construction; FBA: Flux Balance Analysis; MCA: Metabolic Control Analysis; NESS: Nonequilibrium Steady-State; HQ initiated this work. All authors contributed to and wrote the manuscript together. All authors read and approved the final manuscript.


Dinoroseobacter shibae and Phaeobacter gallaeciensis was studied at the level of metabolic fluxes. These two strains belong to the marine Roseobacter clade, a dominant bacterial group in various marine habitats, and represent surface-associated, biofilm-forming growth  and symbiotic growth with eukaryotic algae (D. shibae). Based on information from recently sequenced genomes, a rich repertoire of pathways has been identified in the carbon core metabolism of these organisms, but little is known about the actual contribution of the various reactions in vivo.In the present work the central carbon metabolism of 13C labelling techniques in specifically designed experiments, it could be shown that glucose-grown cells of D. shibae catabolise the carbon source exclusively via the Entner-Doudoroff pathway, whereas alternative routes of glycolysis and the pentose phosphate pathway are obviously utilised for anabolic purposes only. Enzyme assays confirmed this flux pattern and link the lack of glycolytic flux to the absence of phosphofructokinase activity. The previously suggested formation of phosphoenolpyruvate from pyruvate during mixotrophic CO2 assimilation was found to be inactive under the conditions studied. Moreover, it could be shown that pyruvate carboxylase is involved in CO2 assimilation and that the cyclic respiratory mode of the TCA cycle is utilised. Interestingly, the use of intracellular pathways was highly similar for P. gallaeciensis.Using Roseobacter group. Fluxes through major intracellular pathways of the central carbon metabolism, which are closely linked to the various important traits found for the Roseobacter clade, could be determined. The close similarity of fluxes between the two physiologically rather different species might provide the first indication of more general key properties among members of the Roseobacter clade which may explain their enormous success in the marine realm.The present study reveals the first insight into pathway utilisation within the  Roseobacter lineage, representing a group of Alphaproteobacteria  glucose by each individual pathway leads to a different labelling pattern in specific fragments of alanine and serine, which can be taken as a clear differentiation of flux  revealed that the use of the ED pathway amounts to >99%, whereas glycolysis and PPP contribute only <1%   glucose  was used as substrate. The cultivations were carried out on orbital shakers at 200 rpm in 500 ml shaken flasks with a culture volume of 50 ml at 37°C (D. shibae) and 28°C . To ensure comparable conditions between the two microorganisms and avoid any potential influencing effects of phototrophy in D. shibae, both organisms were cultivated in the light. Under these conditions, no bacteriochlorophyll is synthesised D. shibae  - always contains the C1 and C2 carbon of the corresponding amino acid.Metabolic flux calculations were performed as described previously . BrieflyIn the case of alternative pathways yielding a specific product, the fractional contribution of each pathway can be determined concerning the mass distributions of the reactants and the product according to Eq. (1) .(1)X indicates the product molecule whereas the consecutive numbers 1 through n represent reactant molecules of alternative pathways contributing to the mass distribution of the product pool. The corresponding fractional amount of each pathway f can then be calculated by considering two additional constraints: (i) all fractions must have a positive value and (ii) their sum has to equal 1. A more detailed description will be given in the following respective sections.In Eq. (1) index D. shibae and P. gallaeciensis, a metabolic network was constructed based on genome data . As we focused on the central carbon metabolism, the major catabolic routes for glucose as well as the reactions linking the C3 and C4 pools were considered. In terms of glucose catabolism, the annotated genome revealed the presence of the genes encoding for glycolytic enzymes, enzymes of reactions in both the PPP and the ED pathway and TCA cycle. For D. shibae, pyruvate carboxylase (PYRCx) and phosphoenolpyruvate carboxykinase (PEPCk) were found to be the interconnecting reactions between the C3 and the C4 pool. Furthermore, a gene encoding for pyruvate orthophosphate dikinase (PPDK) is annotated, indicating a potential exchange flux between the PYR and PEP pool. A summary of all reactions considered is presented in Figure To carry out metabolic flux calculations for 3 pool depending on the activity of the glycolysis, PPP and ED pathways are presented. It can be taken from the illustration that the combined analysis of two fragments derived from PYR (Ala [M-57] and Ala [M-85]) enables the contributions of each pathway to be resolved. The scheme for the estimation of the major catabolic pathways is shown in Figure In Figure 1 and C2) of the amino acids directly connected to the PEP pool according to Equations 4 and 5.The metabolic reaction network around the PEP node is presented in Figure 13C] glucose: Glucose labelled at C1-position; 3PG: 3-phosphoglycerate; 6PG: 6-phosphogluconate; AcCoA: Acetyl-Coenzyme A; Ala: Alanine; CLE: labelling experiment; EDA: 2-keto-3-deoxy-6-phosphogluconate aldolase; EDD: 6-phosphogluconate dehydrogenase; EDP: Entner-Doudoroff pathway; EMP: Embden-Meyerhof-Parnas; F6P: Fructose-6-phosphate; FUM: Fumarate; G6P: Glucose 6-phosphate; GAP: Glyceraldehyde 3-phosphate; GC/MS: Gas chromatography/mass spectrometry; KDPG: 2-keto-3-deoxy-6-phosphogluconate; MAL: Malate; MDV: Mass distribution vector; OAA: Oxaloacetate; OD: Optical density; OGA: 2-oxoglutarate; PEP: Phosphoenolpyruvate; PEPCk, pckA: Phosphoenolpyruvate carboxykinase; PFK: 6-phosphofructokinase; Phe: Phenylalanine; PPP: Pentose phosphate pathway; PPDK: ppdK Pyruvate orthophosphate dikinase; PYR: Pyruvate; PYRCx, pycA: Pyruvate carboxylase; R5P: Ribose 5-phosphate; Ser: Serine; SUC: Succinate; TCA: Tricarboxylic acid; Tyr: Tyrosine[1-D. shibae. HZ performed the cultivation experiments for P. gallaeciensis. JT assisted in method set-up for cultivation and analytics. IWD helped to draft the manuscript. RR helped to draft the manuscript. CW conceived, designed and coordinated the study and drafted the manuscript together with TF. All authors read and approved the final manuscript.TF carried out the labelling analytics and data processing, performed the flux calculations and drafted the manuscript together with CW. MP performed the cultivation experiments for


Escherichia coli, allowing for the estimation of energy generation and cell synthesis, based on intracellular fluxes and that may be used to characterize phenotypic growth.It is quite important to simulate the metabolic changes of a cell in response to the change in culture environment and/or specific gene knockouts particularly for the purpose of application in industry. If this could be done, the cell design can be made without conducting exhaustive experiments, and one can screen out the promising candidates, proceeded by experimental verification of a select few of particular interest. Although several models have so far been proposed, most of them focus on the specific metabolic pathways. It is preferred to model the whole of the main metabolic pathways in E. coli. Once intracellular fluxes were computed by this model, the specific ATP production rate, the specific CO2 production rate, and the specific NADPH production rate could be estimated. The specific ATP production rate thus computed was used for the estimation of the specific growth rate. The CO2 production rate could be used to estimate cell yield, and the specific NADPH production rate could be used to determine the flux of the oxidative PP pathway. The batch and continuous cultivations were simulated where the changing patterns of extracellular and intra-cellular metabolite concentrations were compared with experimental data. Moreover, the effects of the knockout of such pathways as Ppc, Pck and Pyk on the metabolism were simulated. It was shown to be difficult for the cell to grow in Ppc mutant due to low concentration of OAA, while Pck mutant does not necessarily show this phenomenon. The slower growth rate of the Ppc mutant was properly estimated by taking into account the lower specific ATP production rate. In the case of Pyk mutant, the enzyme level regulation was made clear such that Pyk knockout caused PEP concentration to be up-regulated and activated Ppc, which caused the increase in MAL concentration and backed up reduced PYR through Mez, resulting in the phenotypic growth characteristics similar to the wild type.In the present study, we considered the simulation of the main metabolic pathways such as glycolysis, TCA cycle, pentose phosphate (PP) pathway, and the anapleorotic pathways using enzymatic reaction models of E. coli for understanding metabolic changes inside the cell in response to specific pathway gene knockouts, considering the whole main metabolic pathways. The comparison of the simulation result with the experimental data indicates that the present model could simulate the effect of the specific gene knockouts to the changes in the metabolisms to some extent.It was shown to be useful to simulate the main metabolism of  One of the most challenging goals of metabolic engineering is to design the cell metabolism based on the analysis of metabolic regulation. For this, it is strongly desired to develop a mathematical model which can describe the dynamic behaviour of the cell in response to the changes in the culture environment and/or specific genetic modifications. Although an attempt has been made to develop a platform for the whole cell model , the totSaccharomyces cerevisiae  glucose, 0.4 g of [1-13C] glucose, and 3.2 g of naturally labeled glucose per liter. Biomass samples for GC-MS analysis were taken after one residence time. Sample preparation, analytical procedures for GC-MS analysis, and flux computation are given elsewhere  denotes the concentration, and the abbreviations for the metabolites are explained in Nomenclature. ll grows . As showThe estimation for the cell growth rate is by far important. In particular, it is critical for the prediction of the specific gene knockout mutant. The most typical equation for the cell growth rate is the Monod equation:S is the substrate concentration and S ≡ [ex Glc] in the present case. The drawback of this equation is that the cell will keep growing at μm as far as S ≫ Ks where Ks is usually quite small in the case of the utilization of glucose as the substrate. This does not reflect the fermentation characteristics in practice. Therefore, we introduced another term such aswhere X is the cell concentration and Xm is the final value of X in the batch culture, and n is the model parameter. Xm may be set as S(0)YS/X where S (0) is the initial substrate concentration, and YS/X is the yield coefficient. Although this equation can be used to simulate the dynamic behavior in batch culture, Eq. (2b) may not be able express the cell growth rate for the mutants. We, therefore, introduce another idea by taking into account the effect of ATP generation on the cell growth rate based on experimental observation.where 2 can contribute in generating ATP via oxidative phosphorylation. The pathways involved in electron transfer and oxidative phosphorylation have variable stoichiometry due to the use of different dehydrogenases and cytochromes. Namely, the NADH dehydrogenase NDH-I encoded by nuo gene transports 2 H+/e-, while NDH-II encoded by ndh gene transports 0 H+/e- /[NAD] ratio increases as given by Eq.10) of Table 0 of Tablm = 8 mM) and Icl  compete for ICIT. This branch point is under both gene level regulation by aceBAK operon and the enzyme level regulation by the phosphorylation/dephosphorylation of ICDH. Here, we considered the rate equation for ICDH as given by Eq.(15) of Table For citrate synthase reaction, Eq.14) in Table 4 in TablAs for the glyoxylate pathway, Eq. 16) of Table 6 of TablFor G6PDH and 6PGDH reactions in the PP pathway, Eqs.(23) and (24) of Table 2 production rate Once the fluxes were determined, the specific CO2 production rate was computed by the product of net production of CO2 and biomass concentration, i.e. and the total COvNADPH may also be estimated byMoreover, the specific NADPH production rate μ and vNADPH computed based on the flux estimates obtained from 13C-labeling experiments [It should be noted that the flux determination in the PP pathway is not easy for the enzyme-based kinetic equations. In particular, it is not easy to accurately predict the flux of G6PDH at the branch point of G6P. Figure eriments ,15, indikNADPH value from Figure kNADPH after μ was computed. Then, the flux of G6PDH may be estimated.Once we identified As stated in introduction, it is not easy, even if it is not impossible, to express gene-level regulation by mathematical equations. Here, we alternatively introduced several rules for the simulation based on the experimental observations and the roles of global regulators such as given in Additional file Metabolites2KG: 2-Keto-D-gluconate; 6PG: 6-Phosphogluconolactone; ACE: Acetate; AcP: Acetyl phosphate; AcCoA: Acetyl-CoA; ASP: Aspartate; ADP: Adenosine diphosphate; ATP: Adenosine-5'-triphosphate; AMP: Adenosine monophosphate; DHAP: Dihydroxyacetone phosphate; E4P: Erythrose 4-phosphate; F6P: Fructose 6-phosphate; FDP: Fructose 1,6-bisphosphate; FUM: Fumarate; G6P: Glucose-6-phosphate; GAP: Glyceraldehyde 3-phosphate; ICIT: Isocitrate; NAD/NADH: Nicotinamide adenine dinucleotide; NADP/NADPH: Nicotinamide adenine dinucleotide phosphate; MAL: Malate; OAA: Oxaloacetate; P: Phosphate; PEP: Phosphoenolpyruvate; PYR: Pyruvate; R5P: Ribulose 5-phosphate; Ru5P: Ribose 5-phosphate; S7P: Sedoheptulose 7-phosphate; SUC: Succinate; X5P: Xylulose 5-phosphateProtien (enzyme)Glc,/EIICBGlc: Enzyme for glucose; Eno: Enolase; Fum: Fumarase; G6PDH: Glucose-6-phosphate dehydrogenase; GAPDH: Glyceraldehyde 3-phosphate dehydrogenase; ICDH: Isocitrate dehydrogenase; HPr: Histidine containing protien; Icl: Isocitrate lyase; MDH: Malate dehydrogenase; Mez: Malic enzyme; Ms: Malate synthase; Pck: Phosphoenolpyruvate carboxykinase; PDH: Pyruvate dehydrogenase; Pfk/Pfk-1: Phosphofructokinase-1; Pgi: Phosphoglucose isomerase/Glucosephosphate isomerase; Pgk: Phosphoglycerate kinase; Pgm: Phosphoglycerate mutase; Ppc: PEP carboxylase; Pps: Phosphoenolpyruvate synthase; Pta: Phosphotransacetylase; PTS: Phosphotransferase system; Pyk: Pyruvate kinase; Rpe: Ribulose phosphate 3-epimerase; Rpi: Ribulose 5-phosphate 3-isomerase; SDH: Succinate dehydrogenase; Tal: Transaldolase; TktA: TransketolaseI; TktB: TransketolaseII2KGDH: 2-Keto-D-gluconate Dehydrogenase; Ack: Acetate kinase; Acs: Acetyl coenzyme A synthetase; AcP: Acetyl phosphate; Aldo: Aldolase; CS: Citrate synthase; EI: Cytoplasmatic protein (enzyme1); EII, EIIB, EIIC,: Carbohydrate specific (enzyme II); EIIAGeneaceBAK: operon which encodes the metabolic and regulatory enzymes of the glyoxylate bypass; cra: Catabolite repressor/activator; crr: Catabolite repressor; fadR: Fatty acid metabolism regulator; icdA: Isocitrate dehydrogenase A; iclR: Transcriptional repressor IclR; pckA: Phosphoenolpyruvate carboxykinase gene; pfkA, pfkB: Phosphofructokinase gene; ppsA: Phosphoenolpyruvate synthase gene; ptsG, ptsHI: Pts gene; pykA, pykF: Pyruvate kinase geneThe authors declare that they have no competing interests.TAAK made modeling, experiment and computer simulation, AAM made some TCA cycle modeling and simulation, AMK supervised computer simulation, JM made formulation, KS made modeling and manuscripts preparation. All authors read and approved the final manuscripts.Effects of initial metabolite concentrations on the fermentation characteristics.Click here for fileSimulation result of wild type and Ppc mutant in batch culture.Click here for fileTable S1 - Ruled used for the simulation.Click here for fileSimulation result of wild type and Ppc mutant in continuous culture.Click here for fileSimulation result of wild type and Pck mutant in continuous culture.Click here for fileSimulation result of wild type and Pyk mutant in batch culture.Click here for fileSimulation result of wild type and Pyk mutant in continuous culture.Click here for fileComputational procedure for the simulation of fitting experimental data.Click here for file


Saccharomyces cerevisiae. The enzymes with transcriptional regulation showed enrichment in certain transcription factors. This has not been previously reported. The information provided by the presented method could guide the discovery of new metabolic engineering strategies or the identification of drug targets for treatment of metabolic diseases.Genome-scale metabolic models are available for an increasing number of organisms and can be used to define the region of feasible metabolic flux distributions. In this work we use as constraints a small set of experimental metabolic fluxes, which reduces the region of feasible metabolic states. Once the region of feasible flux distributions has been defined, a set of possible flux distributions is obtained by random sampling and the averages and standard deviations for each of the metabolic fluxes in the genome-scale model are calculated. These values allow estimation of the significance of change for each reaction rate between different conditions and comparison of it with the significance of change in gene transcription for the corresponding enzymes. The comparison of flux change and gene expression allows identification of enzymes showing a significant correlation between flux change and expression change  as well as reactions whose flux change is likely to be driven only by changes in the metabolite concentrations (metabolic regulation). The changes due to growth on four different carbon sources and as a consequence of five gene deletions were analyzed for  The sequencing of full genomes and the development of high-throughput analysis technologies have made available both genome-scale metabolic networks and simultaneous transcription data for all the genes of an organism. Genome-scale metabolic models, with the assumption of steady state for the internal metabolites, allow the definition of a region of feasible metabolic flux distributions. This space of solutions can be further constrained using experimental flux measurements . Here a random sampling method was used to obtain average values and standard deviations for all the reaction rates in a genome-scale model. These values were used to quantify the significance of changes in metabolic fluxes between different conditions. The significance in flux changes can be compared to the changes in gene transcription of the corresponding enzymes. Our method allowed for identification of specific reactions that are transcriptionally regulated, and we further identified that these reactions can be ascribed to a few key transcription factors. This suggests that the regulation of metabolism has evolved to contain a few flux-regulating transcription factors that could be the target for genetic manipulations in order to redirect fluxes. Systems Biology aims to use mathematical models to integrate different kinds of data in order to achieve a global understanding of cellular functions. The data to be integrated differ both in their nature and measurability. The availability of DNA microarrays allows for the comparative analysis or mRNA levels between different strains and conditions. These data provide genome-wide information, and changes in expression at different conditions are expressed in statistical terms such as p-values or Z-scores that quantify the level of significance in transcriptional changes. The availability of annotated genome-scale metabolic networks allowed mapping of the transcriptional changes in metabolic genes on to their corresponding metabolic pathways and defining significantly up or down regulated sub-networks 13C labeling experiments Metabolic fluxes are the result of a complex interplay between enzyme kinetics, metabolite concentrations, gene expression and translational regulation. Metabolic fluxes can be directly measured using In this paper we propose a method to integrate gene expression data with flux data by transforming a limited amount of quantitative flux data into a genome-scale set of statistical scores similar to the one obtained from DNA microarrays. In order to do that, a set of experimental exchange fluxes are fixed for each of the studied conditions or for each of the strains investigated, and a sampling algorithm is then used to obtain a set of flux distributions satisfying the experimental values. This approach allows for obtaining means and standard deviations for each flux in the genome-scale network. From the mean and standard deviation it is possible to derive statistical scores for the significance of flux change between conditions Saccharomyces cerevisiae grown at different growth conditions and for the analysis of different deletion mutants.By dividing the average difference among two conditions (e.g. carbon sources or mutant strains) by its standard deviation, it is possible to obtain Z scores for each metabolic flux. These scores can be transformed into p-values that measure the significance of change of each flux see . By compThe combined use of random sampling of genome-scale metabolic networks and expression data allows for global mapping of reactions that are either transcriptionally or metabolically regulated. This information can be used to guide the engineering of microbial strains or as a diagnosis tool for studying metabolic diseases in humans. In particular we should highlight that reactions in which there is no relation between gene transcription level and metabolic flux are not suitable targets for flux increase via gene over-expression. Through analysis of different data sets the method revealed that many changes in gene expression are not correlated with a corresponding change in metabolic fluxes. The use of gene-expression data alone can therefore be misleading. However, our method allowed for identification of many specific reactions that are indeed transcriptionally regulated, and we further identified that the expression of these enzymes is regulated a few key transcription factors. This fact suggests that the regulation of metabolism has evolved to contain a few flux-regulating transcription factors that could be the target for genetic manipulations in order to redirect fluxes.Here we propose a sampling method that finds extreme solutions among the feasible flux distributions of the metabolic network. These solutions correspond to the corners in the region of allowed flux distributions, and in mathematical terms they are elements of the convex basis of the region of feasible solutions (which is a convex set). The COBRA Toolbox 13C labeling In order to assess the accuracy of our sampling method to estimate the average fluxes and their standard deviations, we compared a set of internal fluxes measured with The means obtained by the two sampling methods are very similar for most of the reactions; however the standard deviations found using the HR algorithm are significantly smaller. With the HR method the real values for the fluxes in many cases deviate several standard deviations from the mean, A high value of Z indicates that the real value has a very low chance of being obtained using the considered sampling method . The number of samples with the HR algorithm was increased up to 5000 to check possible effects of the sample size on the standard deviation. Only small increases were observed for the standard deviations of the studied fluxes.Using the CB algorithm we obtain higher standard deviations and the real flux is for most reactions less than one standard deviation away from the mean flux. We can therefore conclude that the CB sampling method gives more realistic standard deviations for the fluxes. This is important if we want to compare the significance of flux changes between conditions. An underestimated standard deviation would make some flux changes appear as being significant even though they may not be in reality, and our method therefore gives a more conservative list of significantly changed reaction fluxes than the HR algorithm.S. cerevisiae. Data from growth on four different carbon sources  in chemostat cultures and five deletion mutants  grown in batch cultures were used. The exchange fluxes and gene expression data for the mentioned conditions have been published earlier To evaluate our method we used data for the yeast Our method obtains probability scores for each enzyme in the metabolic network see  and thisThe method to identify the significance of flux changes relies on a set of measured external fluxes, and in some cases strains that don't show significant changes in external fluxes have changes in internal fluxes max changes), e.g. through over-expression, or changes in enzyme affinity (Km changes), e.g. through expression of heterologous enzymes, in order to alter the fluxes.The reactions showing transcriptional regulation form a set of putative targets where enzyme over-expression or down regulation will influence the flux through these reactions. The reactions showing metabolic regulation points to parts of the metabolism where the pools of metabolites are possibly increasing or decreasing in connection with transcriptional changes and hereby counteracting possible changes in enzyme concentration as a result of transcriptional changes. This knowledge can be used to identify whether one should target changes in enzyme concentration .The glucose to ethanol and the glucose to acetate transitions showed widespread flux and expression changes, and they are therefore more interesting study cases. In the glucose-ethanol transition 19 enzymes showed transcriptional regulation and 22 other enzymes changed in expression but not in flux. For the glucose-acetate transition the same numbers were 33 and 23 respectively. We can see that about one half of the genes that changed in transcription level also showed significant flux changes, this proportion is higher than in the case of deletion mutants (as it will be discussed later). Among the enzymes showing transcriptional regulation, 14 were shared between the glucose-ethanol and glucose-acetate transitions. Interestingly, no overlap was found between the sets of enzymes that don't change in flux. Metabolic regulation was observed in 21 reactions for each case, among which 8 overlap.+ to oxidize ethanol or acetate in the cytoplasm and maintain the redox balance in the cell. Isocitrate lyase, a key component of the glyoxylate cycle, is also transcriptionally up regulated and this allows for net formation of malic acid that can further be converted to phopshoenolpyruvate  that fuels the gluconeogenesis. All these changes in fluxes are consistent with general knowledge about the changes in metabolism from glucose to C2 carbon sources like ethanol and acetate, but what is interesting to see is that not all the reactions associated with these flux changes are transcriptionally regulated, but the cell have selected a few key reactions to regulate at the transcriptional level and these are identified using our method.The enzymes showing transcriptional regulation clearly show a down regulation of enzymes involved in glucose uptake and utilization (e.g. Glucose transporter HXT4 or Hexokinase 2) and the up-regulation of enzymes involved in the gluconeogenesis  and the TCA cycle (e.g. Succinate dehydrogenase or Citrate synthase). The AcCoA synthetase 2, responsible for supplying AcCoA to the TCA cycle is also transcriptionally up-regulated as well as the ATP synthetase (involved in the respiratory chain) and the external NADH-ubiquinone oxidoreductase 2, which supplies the necessary NADIn order to make a deeper analysis, we performed an enrichment test to compare the transcription factors involved in the expression of the enzymes showing transcriptional regulation and the enzymes showing changes in expression but not in flux. We found three transcription factors that were strongly overrepresented in the metabolic genes showing transcriptional regulation. In the glucose-ethanol transition, the transcription factors Gcr1 and Gcr2 both appeared in 11 transcriptionally regulated genes and in none of the other genes, whereas the transcription factor Hap4 appeared in 11 transcriptionally regulated genes and 5 of the other regulated genes. For the glucose-acetate transition these numbers were 15-0, 11-0 and 15-0 for the same transcription factors. This means that certain transcription factors are especially involved in the transcriptional regulation of metabolic fluxes , and to our knowledge this has not been previously reported. It basically implies that there is global regulation of major flux alterations, which is similar to what has experimentally been shown to be the case for galactose metabolism The top scoring metabolically regulated reactions, both for glucose-ethanol and glucose-acetate, are the Fructose biphosphate aldolase and the Triosephosphate isomerase. These reactions are known to operate close to the equilibrium and are therefore very sensitive to changes in the metabolic pools, which is consistent with metabolic regulation of the fluxes. In the considered cases the direction of these reactions is inverted. This can only be explained by a decrease in the fructose-1,6-diphosphate pool and an increase in the glyceraldehyde-3-phosphate and dihydroxyacetone pools. This hypothesis is supported by the fact that in chemostat cultures, there was not found any correlation between the glycolytic flux and the expression of the genes encoding these two enzymes The results for the glucose-ethanol change are summarized in As mentioned above several different deletion strains were evaluated and the number of enzymatic reactions showing transcriptional regulation was 26 for the grr1Δ strain, 25 for the hxk2Δ strain, 11 for the mig1Δ strain, 8 for the mig1Δmig2Δ strain and 0 for the gdh1Δ strain. The reactions showing post-transcriptional regulation were 73, 70, 46, 36 and 89 for the same strains, respectively. These numbers clearly show, that in contrast to growth on different carbon sources, most of the transcriptional changes do not result in correlated changes in metabolic fluxes. This indicates that many transcriptional changes are indeed happening in order to minimize the metabolic adjustment resulting from a gene deletion, and the most extreme case is the gdh1Δ strain, where no transcriptional changes seem to be correlated to flux changes. This behaviour is consistent with the MOMA (Minimization of Metabolic Adjustment) hypothesis There is a substantial overlap between the strains grr1Δ and hxk2Δ, 15 transcriptionally regulated reactions were shared between both strains and 28 post-transcriptionally regulated reactions were also shared. An enrichment test was performed in order to find transcription factors regulating the enzymes showing transcriptional regulation. For the grr1Δ, the most significantly enriched transcription factors were Pho2  and Bas1 (which regulates 10 out of 26 and 7 out of 73 enzymes in the mentioned groups). For the hxk2Δ strain the most significant transcription factor enrichment was found for the factors Pho2 (with 7 and 5 enzymes in each of the groups) and Bas1 (with 7 and 6 enzymes in each of the groups).Pho2 and Bas1 are partner proteins that regulate the transcription of genes involved in purine and histidine biosynthesis −1 respectively −1 and 0.2 g-DW g−1. The specific glucose uptake rate for the hxk2Δ strain is significantly lower than for the grr1Δ strain as well as for the reference strain. This is associated with the observation that the glycolytic flux in the hxk2Δ strain shows transcriptional down-regulation of five enzymes. In the upper glycolysis, the Hexokinase 2 has been deleted and the Phosphofructokinase 1 is down-regulated. The Phosphofructokinase 1 was also strongly down-regulated in the grr1Δ strain but the decrease in flux was not as large as in the hxk2Δ strain. In the lower glycolysis of the hxk2Δ, all the three iso-enzymes of Phosphoglycerate mutase were down-regulated as well as the phosphoglycerate kinase. No down-regulation for these enzymes was seen in the grr1Δ strain. The glycerol-3-phosphate dehydrogenase has two iso-enzymes. The first of those isoenzymes was up-regulated in the grr1Δ and the hxk2Δ strains; however its expected flux decreased in both cases (more significantly in hxk2Δ). The second iso-enzyme did not show important changes in grr1Δ but was down-regulated in hxk2Δ. This is not the only case in which different iso-enzymes show different regulatory patterns, and in these cases our method for having flux estimations independent from transcriptome analysis is particularly useful.Strains grr1Δ and hxk2Δ show specific growth rates of 0.23 and 0.22 hThe strain hxk2Δ showed a strong decrease in ethanol production compared to grr1Δ. All the alcohol dehydrogenase iso-enzymes were down-regulated in a similar way in both strains. The explanation of the differences in flux towards ethanol should be found in the pyruvate decarboxylase. Pyruvate decarboxylase 3 was strongly down-regulated in both strains, however, pyruvate decarboxylase 2 was up-regulated. This up-regulation was much more significant in grr1Δ, which could explain a higher flux from pyruvate to AcCoA in this strain. The results for the hxk2Δ mutant are summarized in MIG1 activates a response against starvation that results in an increased specific growth rate. Among the transcriptionally regulated reactions, a slight down-regulation of the PP pathway is observed together with an up-regulation of several amino-acid production pathways.The mig1Δ mutant shows a higher specific growth rate than the wild type. In general it is transcriptionally very similar to the wild type MIG1 is cancelled by an opposite effect due to MIG2.In the mig1Δmig2Δ mutant there is a slight decrease in the specific growth rate. All the 8 transcriptionally regulated reactions were down-regulated and belonged to amino-acid biosynthesis pathways. The enrichment test found the factors Cbf1 and Gcn4 (represented in 4 and 1 out of 8 reactions and 5 and 6 out of 36 reactions). Gcn4 is known to regulate amino-acid biosynthetic genes In all the mutants discussed above, the AcCoA carboxylase and the fatty acid synthesis showed metabolic regulation. This could indicate that in the studied cases the AcCoA pool was the main parameter responsible for adjusting the rate of lipid biosynthesis to match the changes in specific growth rates.The experiments for the gdh1Δ mutant were performed in chemostat cultures In S. cerevisiaeThe steady state condition and the irreversibility of some reactions impose limitations on the flux distributions attainable by the cell expv) are different between conditions. This fact changes the shape of the region of feasible solutions between different conditions. S is the stoichiometric matrix of the network.In order to obtain corners in the space of allowed solutions we used the simplex method with a random set of objective functions to be maximized. The maximization of each of these objective functions will give a corner in the space of solutions. The constraints imposed upon each optimization are:−1g-DW−1 depending on each condition. Also no weights (eq. 4) were assigned to the reactions involved in loops in order to avoid objective functions that maximize the activity of loops.In order to reduce the effects of internal loops we first identified all the reactions that can get involved in loops using the FVA  option in the COBRA Toolbox. The reactions that can be involved in loops are unbounded and show the default maximal or minimal value set in the COBRA Toolbox (1000 or −1000). If these bounds were kept, the means and standard deviations for these reactions would be unrealistic iw) assigned to each reaction were generated by dividing a random number between 0 and 1 by the maximal flux for this reaction obtained using FVA. This normalization was made to account for the different size orders of the different reactions. The objective functions take the form:Random objective functions were generated by selecting random pairs of reactions and assigning them random weights (the reactions involved in loops were excluded from these choices). The weights .The Z score measures the significance of change in terms of standard deviations. If the error in the Z score is lower than 0.15, no information would be lost in terms of classifying a reaction as significantly changed or not. The order of size of a genome-scale model is about 1000 reactions. A reasonable accuracy for the Z-scores would be to expect errors higher than 0.15 on the Z score only for 1 reaction in the whole model. This means a p-value of 0.001. If we want to keep the error on the Z score under 0.15 with a probability of 0.999 we need 500 samples, and this was therefore selected as the sampling number.The Z-scores can be transformed into probabilities of change by using the cumulative Gaussian distribution. Once we have Z-scores for the significance of flux changes and Z-scores for the significance of gene-expression changes we can obtain probabilities of having correlated expression and flux changes for each enzyme.tri of zero is assigned by default.An increase in enzyme expression can result in an increase of flux . In order to evaluate the probability for a reaction of being transcriptionally regulated we multiply the probability of its enzyme level changing by the probability of its flux changing in the same direction .In the same way as in eq. (6) we can define probabilities for the expression level changing and for the flux not changing .The probability of a change in flux but not in transcription (metabolic regulation) can be obtained for each reaction as follows: