Mycobacterium tuberculosis, residing in the infected host provide novel opportunities for therapeutic intervention.Because metabolism is fundamental in sustaining microbial life, drugs that target pathogen-specific metabolic enzymes and pathways can be very effective. In particular, the metabolic challenges faced by intracellular pathogens, such as M. tuberculosis in a medium whose carbon source was restricted to fatty acids, and that of the 5'-O- adenosine inhibitor in a medium with low-iron concentration.We developed a mathematical framework to simulate the effects on the growth of a pathogen when enzymes in its metabolic pathways are inhibited. Combining detailed models of enzyme kinetics, a complete metabolic network description as modeled by flux balance analysis, and a dynamic cell population growth model, we quantitatively modeled and predicted the dose-response of the 3-nitropropionate inhibitor on the growth of in vitro growth of M. tuberculosis cells.The predicted results quantitatively reproduced the experimentally measured dose-response curves, ranging over three orders of magnitude in inhibitor concentration. Thus, by allowing for detailed specifications of the underlying enzymatic kinetics, metabolic reactions/constraints, and growth media, our model captured the essential chemical and biological factors that determine the effects of drug inhibition on  System-level networks of biological processes and functions allow us to draw inferences about the phenotype of an organism that cannot be made by considering each of its individual components -4. In paEscherichia coli  (the affected reactions are referred to as target reactions), b) how the change in the metabolite flow or flux of the target reactions decreases the growth rate of the organism [Metabolic Network], and, finally, c) how the reduced growth rate results in an effective lower bacterial cell concentration [Population Growth Model]. Figure inhibition model is defined by the enzyme inhibition kinetics governing the reactants and products of a particular metabolic reaction  and relates the manner by which the inhibitor concentration [I] modifies or adds constraint to the flux of the target reaction. The mathematical form of the inhibition model depends on the particular enzyme kinetics associated with the specific inhibitor and the metabolic reaction affected by the inhibitor. Mathematically, the inhibition model relates an inhibitor concentration [I] to the resulting metabolite flux ratio in the presence and absence of the inhibitor. Inhibitors affecting more than one reaction can also be considered.The metabolic network is used to self-consistently calculate the overall biomass growth rate μ, substrate uptake rates vC, and the fluxes of all metabolic reactions. It is coupled to the inhibition model and the population growth model. The inhibition model places constraints on the flux of the target reactions in the metabolic network that affects the total biomass growth rate, and the population growth model adds constraints to the network's substrate uptake rate from the medium. The effect of enzyme deletions (deletion mutants) on growth can be incorporated in the metabolic network model by removal of the particular reactions catalyzed by the specified enzymes  indicate the concentration of "." and  and an upper limit constraint on the limiting substrate uptake rate μ and the uptake rate vC outputs of a FBA of a metabolic network for a given set of input conditions [I] and The biomass growth rate We employed the Michaelis-Menten kinetic model  to estimVm denotes the maximum initial rate of substrate uptake and Km represents the Michaelis-Menten rate constant. Moreover, we linked the experimental readout, in this case the optical density (OD) under 600-nm-wavelength light [X] by:where th light ,53, to twhere K denotes a proportionality constant. Other experimental readouts can be similarly accounted for.OD represents the cell concentration expressed as optical density under 600-nm-wavelength light and p represents the parameter analyzed for sensitivity, the sensitivity coefficient The presence of a number of parameters in our mathematical framework warranted a sensitivity analysis as to how the assigned parameter values affected the final computational results. We used two different metrics to ascertain parameter sensitivity. In the first analysis, we gauged the variation in the results by separately setting each one of the parameter values to reasonable lower and upper bounds , in this follows ,60:OD, can be substituted for in Eq. 6. To numerically calculate the sensitivity coefficient p, we started from ∂p = +0.5p and repeated the process by reducing ∂p and calculating the sensitivity coefficient until p yielded the same p = -0.5p until convergence. In the calculation performed here, both processes converged to the same numerical value.Other observables, different from group I included parameters obtained from the literature, group II included those determined by matching experimental data, group III included those assumed to be derived from other parameters, and group IV included those that, by definition, were directly determined once the other parameters were defined. During the sensitivity analysis of the parameters in groups I, II, and III, we calculated dose-response curves while increasing and decreasing each parameter by 50%  and sensitivity coefficients spanning three orders of magnitude in inhibitor concentration. Although the parameters in group III were assumed to be derived from the parameters in groups I and II, during the sensitivity analysis for the first two groups we held the parameter values in group III fixed. Also, because the parameters in group IV were dependent on other parameters and their values changed as we performed sensitivity analysis on these independent parameters, we did not perform analysis for this group.To address the different types of parameters in our framework, we classified the modeled parameters into four groups: M. tuberculosis faces a hostile and harsh environment upon infecting a mouse or a human host. The invasion of M. tuberculosis stimulates the activation of host immunity systems initiated by the release of macrophages that ingest pathogen cells. Macrophage-ingested cells are contained in phagosomes where they confront high pH, antibacterial reactive oxygen and nitrogen, and the lack of carbohydrates ) of the ICL reaction was given by:We used the previously developed kinetic equation for the 3-NP-inhibited ICL reaction  to relatvICL and wICL1 and wICL2 represent the fractions of the overall inhibitor-free ICL reaction flux for the ICL1- and ICL2-catalyzed reaction components, respectively, SUC denotes the succinate substrate, [SUC] indicate its concentration, and KNP, ICL13-, KNP, ICL23-, KSUC, ICL1, and KSUC, ICL2 denote Michaelis constants  to 2.464 mM ) (or fMCL([3-NP])), determined from the inhibition model, and the inhibitor-free fluxes fICLfMCLM. tuberculosis deletion mutant Δicl1Δicl2 was studied, the fluxes associated with the ICL and MCL reactions were set to zero  [C'] = [C]/K and Km/K, Eqs. 1-5 can be written as:In the experimental study of the 3-NP inhibitor, cell concentrations are monitored at different time points during a 16-day growth experiment in propionate medium with and without inhibitor . We can tion [X] . By defiOD were taken directly from the experimental data [The initial values for tal data . The popC'] (t = 0), the maximum initial propionate uptake rate Vm, the Michaelis-Menten rate constant for the propionate uptake KSUC, ICL1 in Eq. 9.All parameters needed to calculate cellular growth and growth inhibition from the mathematical framework in Figure M. tuberculosis. We systematically manipulated the values for [C'] (t = 0), Vm, and C'] (t = 0), Vm, and M. tuberculosis in propionate medium containing 0.025 mM 3-NP inhibitor to estimate the value of the fourth and last unknown parameter, KSUC, ICL1. This was achieved by mathematically varying the value of KSUC, ICL1 until we obtained close agreement between experimental and predicted growth data, as shown in Figure KSUC, ICL1 to 1.5 mM  and compared the results to the experimental values  of the four measured inhibitor concentrations 0.025, 0.050, 0.100, and 0.200 mM  to deterR2) was 0.9867, indicating a strong and significant correlation  between the simulated values and experimental data.We used linear regression  to evaluM. tuberculosis in medium with 0.025, 0.050, 0.100, and 0.200 mM 3-NP was very similar to the inhibitor-free growth when only the ICL reaction was affected by the inhibitor. However, when we assumed that 3-NP only inhibited the MCL reaction, the calculated growth was virtually the same as the results in Figure M. tuberculosis growth. This observation is compatible with the metabolism of odd- and even-chain fatty acids. Both the ICL and the MCL reactions are necessary steps in transferring extracellular carbon atoms to intracellular metabolites and obtaining energy from fatty acids. However, these reactions differ in that the ICL reaction is used for even-chain fatty acids, the MCL reaction is used for odd-chain fatty acid with three carbon atoms such as propionate, and longer odd-chain fatty acids require both reactions  and wMCL1 in group I; [C'] (t = 0), Vm, and group II, and KSUC, MCL1 and KNP, MCL1 3-in group III. Figure SUC] had relatively a small effect, suggesting that it was reasonable to assign [SUC] a constant value in the model. Figure wMCL1 introduced relatively large variations in the calculated dose response. Because this parameter was originally derived from a relative ratio of experimentally determined rate constants, the ± 50% variations of wMCL1 greatly exaggerated plausible experimental errors. For this reason, the sensitivity coefficient analysis below gives a more realistic measure of the importance of this experimentally determined parameter. Changes in the initial propionate concentration [C'] (t = 0) ] for low inhibitor concentrations (≤0.003 mM). For large values of Vm, the effective biomass production per unit of propionate uptake became lower and pushed the bacterium into a growth regime where propionate could not be efficiently used. Hence, for low inhibitor concentrations, the cell concentrations at the largest Vm  Figure  and in tm Figure  had largwICL1, KNP, ICL13-, KNP, ICL23-, KSUC, ICL1, and KSUC, ICL2, used in the definition of the 3-NP inhibition model in Eq. 8, did not affect the results because 3-NP primarily inhibited growth through the MCL reaction and not the ICL reaction (see previous subsection "Growth Predictions"). Similarly, KSUC, MCL2 and KNP, MCL23-, relating to the ICL2 enzyme in the second term of Eq. 10, did not affect the calculated dose-response curves.The other parameters of the model had no effect on the calculated dose-response curves. The parameters wMCL1) the absolute changes of cell concentration around 10-1 mM 3-NP were small, the values in Table Table SUC] and the fluxes of the ICL and MCL reactions inside M. tuberculosis cells could be ascertained experimentally. This could be done by jointly measuring intracellular metabolite concentrations and metabolic fluxes, as recently done in a study of E. coli metabolism  as one parameter  to the flux ratio of the mycobactin synthesis reaction arameter :vMS and in vitro experimental assay and set [E] = 20 nM [E] is unknown but can be inferred, as described below .where s 0.7 nM . To stud = 20 nM . For intfMS and To duplicate the experimental conditions of the sAMS inhibitor study, we constrained the substrate uptake based on the medium used in the experiment . The medX] in Eq. 1 was directly integrated to give:The experimental study of the sAMS inhibitor reports the relative cell concentrations, which represent the ratios of cell concentrations in the presence to the absence of the inhibitor after eight days of growth . The expXt = 0] denotes the initial cell concentration, which is the same whether sAMS was present or not. The relative cell concentration RC after eight days was obtained as:where [X0] denotes the inhibitor-free cell concentration, t is set to eight days, and the inhibitor-present biomass growth rate μ and the inhibitor-free biomass growth rate μ0 were inferred from the metabolic network using FBA.where  in Eq. 15 was not readily available from the experimental data. To obtain this parameter value, we selected a relative cell concentration RC of 0.49 at a sAMS concentration [sAMS] of 1.7 μM from the growth-inhibition experiment in iron-deficient medium [E] in Eq. 15 until our framework reproduced this RC value. This point was selected because 0.49 is close to the mid-range of RC values, 0 ≤ RC ≤ 1. After trying different values for [E], we found that [E] = 40 μM yielded good agreement between the calculated (0.47) and selected (0.49) relative cell concentrations , the measured relative cell concentration becomes close to zero , t could not be considered to be associated with any experimental variation. The sensitivity analysis identified the most critical parameters of the model to be [E] and E] and Table activity  and nutractivity .in vitro bacterial growth under different nutritional conditions.We developed a mathematical framework connecting kinetic models of enzyme inhibition with metabolic network analysis and a population growth model. The three components correspond to the three major steps through which a metabolic inhibitor affects bacterial growth. First, the inhibition model describes how the particular inhibitor affects the enzyme kinetics and the flux(es) of one or more metabolic reactions. Second, the metabolic network analysis connects the changes in the affected metabolite flux(es) to the growth rate of the organism. Finally, the population growth model takes the altered growth rate and converts it to an effective bacterial cell concentration. This framework allowed us to quantitatively simulate the effect of two distinct metabolic inhibitors on M. tuberculosis cells on propionate medium and on iron-deficient GAST medium, respectively. Both reactions affected by these two inhibitors are required for the survival of the pathogen in the host environment and could potentially become important therapeutic targets. 3-NP inhibits key reactions in the glyoxylate shunt and the methylcitrate cycle, effectively blocking the utilization of fatty acids, the major carbon source of M. tuberculosis in the host environment [M. tuberculosis within an iron-deficient host environment [We applied this framework to model the effect of two separate metabolic inhibitors, 3-NP and sAMS, on the growth of ironment ,46. sAMSironment . Our modin vitro environments designed to duplicate aspects of the nutritional conditions encountered in the host. However, intracellular pathogens have complex interactions with their hosts [in vitro environment may not be operative in the in vivo host environment [In this work, the mathematical framework was used to model an inhibitor's effect on cellular growth of a pathogen in different ir hosts  and the ironment . The curironment . The impFor the two inhibitors studied, the essentiality of the protein targets was a necessary condition. Essentiality of a gene can be imparted by the network itself or any other condition that alters of restricts the flux of metabolites in the network. Thus, some genes become essential only under specific nutritional conditions, while others may become essential when one or more nonessential genes are knocked out. It is also possible to envision certain scenarios where drugs affecting parts of the metabolic network induce essentiality to uninhibited enzymes in the network. Quantitative models, such as the one developed here, could be used to rapidly investigate such conditions and assist future experimental studies. For example, using our framework, we suggested that 3-NP was effective in fatty acid medium but not in glucose medium (data not shown), which was supported by experimental observations . In addiM. tuberculosis cells. Our results suggest that this type of inclusive modeling approach would be valuable in proposing new experimental studies by extending, combining, and exploring novel chemical and biological inhibition concepts.The current work introduces a systems biology approach using enzyme kinetics, metabolic networks, and population growth models that is capable of capturing the essential chemical and biological variability of the system under study. This enabled us to simulate and understand the underlying chemical and biological factors that give rise to the experimental observables, in this case growth inhibition of M. tuberculosis cells in an in vitro environment, which was modeled to represent the nutritional challenges encountered in a host cell. We calculated dose-response curves corresponding to the cellular growth versus drug concentration for the growth in a medium whose carbon source was restricted to fatty acids and was infused with varying concentrations of the 3-NP inhibitor. Similarly, we obtained dose-response curves for cells grown in medium with low-iron concentration and exposed to different amounts of the sAMS inhibitor. These results quantitatively reproduced experimentally measured dose-response curves, ranging over three orders of magnitude in inhibitor concentration. The ability of the proposed models to capture in vitro drug inhibition confirms that relevant features of intracellular metabolism of M. tuberculosis can be modeled by a metabolic network-based framework.We implemented a systems biology framework, which combines detailed models of enzyme kinetics, a complete metabolic network analysis, and a cell population growth model, to represent and understand cellular growth inhibition in response to drugs. We used this mathematical framework to simulate two separate inhibition mechanisms for the growth of All authors contributed to the design and coordination of the study. XF performed the computational implementations, and XF and AW prepared the original draft, which was revised by JR. All authors read and approved the final manuscript.Supplementary information. Supplementary information provides the details for metabolic network modification and intermediate results during computation.Click here for fileMetabolic network used to model the inhibitory effect of 3-nitropropionate. The network file is in format of Systems Biology Makeup Language.Click here for fileO- adenosineMetabolic network used to model the inhibitory effect of 5'-. The network file is in format of Systems Biology Makeup Language.Click here for file


Metabolic reconstructions contain detailed information about metabolic enzymes and their reactants and products. These networks can be used to infer functional associations between metabolic enzymes. Many methods are based on the number of metabolites shared by two enzymes, or the shortest path between two enzymes. Metabolite sharing can miss associations between non-consecutive enzymes in a serial pathway, and shortest-path algorithms are sensitive to high-degree metabolites such as water and ATP that create connections between enzymes with little functional similarity.We present new, fast methods to infer functional associations in metabolic networks. A local method, the degree-corrected Poisson score, is based only on the metabolites shared by two enzymes, but uses the known metabolite degree distribution. A global method, based on graph diffusion kernels, predicts associations between enzymes that do not share metabolites. Both methods are robust to high-degree metabolites. They out-perform previous methods in predicting shared Gene Ontology (GO) annotations and in predicting experimentally observed synthetic lethal genetic interactions. Including cellular compartment information improves GO annotation predictions but degrades synthetic lethal interaction prediction. These new methods perform nearly as well as computationally demanding methods based on flux balance analysis.We present fast, accurate methods to predict functional associations from metabolic networks. Biological significance is demonstrated by identifying enzymes whose strong metabolic correlations are missed by conventional annotations in GO, most often enzymes involved in transport vs. synthesis of the same metabolite or other enzyme pairs that share a metabolite but are separated by conventional pathway boundaries. More generally, the methods described here may be valuable for analyzing other types of networks with long-tailed degree distributions and high-degree hubs. High quality metabolic reconstructions are available for many organisms and provide a rich scaffold for interpreting data from high-throughput biological experiments. The topology of a metabolic network, defined by connections between enzymes and metabolites, can be used to predict genetic interactions, transcriptional correlations and disease co-morbidity -3.Previous studies have used the topology of the metabolic network to predict co-expression of transcripts for yeast metabolic enzymes . This stMethods that involve calculation of optimal fluxes subject to constraints, such as flux coupling , have peOur goal is to provide improved topological measures for enzyme functional associations from metabolic networks without the need for expensive calculations of optimal fluxes or sampling over feasible flux space. The motivation of our approach is that methods that count shared metabolites, or methods that generate a p-value for shared metabolites based on a hypergeometric distribution, essentially assume a flat degree distribution for metabolites. High-degree metabolites violate the assumption of a flat degree distribution, and the hypergeometric distribution is inappropriate for calculating p-values for metabolite sharing. Randomization methods based on rewiring, which maintain the observed degree distribution, are robust to high-degree metabolites but unfortunately are computationally expensive.In this work, we provide a series of scores that are the Bayesian equivalent of the hypergeometric distribution, but adjusted for the known metabolite degree distributions. These scores are fast to calculate, essentially no more expensive than a hypergeometric p-value, and much faster than any methods that require rewiring permutations, flux sampling, or flux optimization. Results from applying these methods to metabolic networks in yeast demonstrate performance better than previous methods based on local connectedness. The results also reveal functional associations that are not captured by conventional metabolic pathway definitions, but which are inherent in the network structure.Metabolic networks can be represented as bipartite graphs with edges between enzymes and metabolites. An enzyme can use a metabolite in multiple unique reactions involving distinct subsets of other metabolites, and the number of unique reactions defines an integer-valued edge weight.A graphical overview shows how metabolic network information is incorporated to yield increasingly sophisticated models Figure . The modIn this work, we introduce a more sophisticated local method that also corrects for metabolite degree, discounting the contribution of highly connected metabolites like water, protons, and ATP Figure . Our newMethods termed "Global" are capable of generating rankings for enzymes that are not directly connected by metabolites by using the full network topology. Examples are shortest paths and the more robust graph diffusion kernel (GDK) Figure . GDKs arYet more elaborate flux balance analysis methods sample flux states that are feasible under steady-state constraints. Flux correlations can then be used to rank enzyme pairs for functional associations Figure . Other fPerformance is assessed primarily by the ability to predict synthetic lethal genetic interactions between metabolic enzymes, and secondarily by the ability to identify classes of enzymes with similar Gene Ontology (GO) annotations. The synthetic lethal interactions provide a direct link to testable experiments. The database annotations are not necessarily testable, but instead show whether inference from computational models is consistent with known biology.We generated rank ordered lists of enzyme pairs based on the local methods. Performance was assessed from the receiver operating characteristic (ROC) curve using the area under the curve (AUC), and from the precision-recall (PR) curve using the maximum F-score, the harmonic mean of precision and recall. Known positives were taken from experimentally reported synthetic lethal/growth defect interactions recorded in the BioGRID database. There were 170 growth defect/lethal interactions in which both genes involved were part of the metabolic network model. Known negatives were defined as gene pairs where each gene has at least one synthetic lethal interaction, and one of the two has at least 5 synthetic lethal interactions as a query in a high-throughput screen, to exclude pairs that might not have been tested experimentally.Performance metrics for the local methods in predicting synthetic lethal genetic interactions are the AUC and F-score Figure . Raw couWe then compared the performance of the best local method (Poisson score using metabolite degree distribution) with methods that use global topology and incorporate local edge weights Figure . Global Overall, incorporating global information through the graph diffusion kernel improves the performance in identifying synthetic lethal pairs. Adding edge weights to the graph diffusion kernel does not appear to improve performance. Further comparisons with the graph diffusion kernel use the unweighted model only, as it is simpler.We next considered possible improvements that use the knowledge that the network edges represent a flux balance model for metabolism. While others have investigated models that investigate the robustness of metabolism to pairwise gene deletions , correlaComparison of the flux sampling method to the best local and global topology methods considers only the enzymes present in the reduced model Figure . The fluThe epistatic estimates, obtained from previous work , do not -6), and removing compartment information neglects this information.Reactions in the metabolic network are localized to specific compartments . Of the total 1266 metabolic reactions in the model, 47% are entirely contained in the cytosol, roughly 10% are either mitochondrial or extracellular, and 24% couple metabolites from different compartments  and the global method (graph diffusion kernel with an unweighted network) is that removing compartments improves the precision at high recall at the expense of worse precision at low recall Figure .We used published methods to assess the ability of different ranking methods to identify enzyme pairs with similar database annotations . This asOf all the methods, the graph diffusion kernel clearly performs best overall in producing consistent GO annotations Table . The perWe investigated whether the models with or without compartments perform better using a binomial test under the null hypothesis that each of the two models has equal probability of performing better for a particular method. Tests were performed separately for AUC/ROC, PR/F-score, and the three GO categories, and separately for the full and reduced network. In all cases except GO/Cellular Compartment, two-tailed tests showed no significant deviations from the null hypothesis at p = 0.05. The test for Cellular Compartment is significant (p = 0.0396), with improved consistency for models that include compartment information.Flux correlation performs best in the Biological Process category. A surprising result here is that overall flux correlation performs worse than GDK and Poisson  with only 70% consistent GO terms. It is possible the flux correlation method could be improved with longer runs that reduce the statistical noise in the correlations, although the flux calculations already require over 100× more computer time than any of the other methods .The different methods generate a similar set of consistent GO term, and terms missed by the graph diffusion kernel are almost always missed by the other methods as well Figure . For celWe have investigated the performance of three classes of methods for predicting functional associations in metabolic networks: (i) local methods, based primarily on the metabolites shared by two metabolic enzymes; (ii) global methods, based on the probability that a random walk started at one enzyme will visit a second enzyme; (iii) flux-based methods that use flux balance to identify enzymes with correlated fluxes. The local and global methods are fast and generally applicable to other types of networks, whereas the flux-based methods are computationally expensive and dedicated to metabolism (or other networks that have similar conservation-of-mass constraints). In terms of performance in predicting functional associations, however, the dedicated flux-based methods have typically been superior. Developing fast methods with performance similar to expensive flux-based methods has been a challenge.Previous local and global methods have had difficulties with high-degree metabolites. For local methods, metabolites such as water and ATP are often shared by enzymes with very different functions. For global methods, these metabolites introduce many short paths through the network. Often, high-degree metabolites are removed from a network prior to analysis. This approach is undesirable because it introduces an ad hoc tuning parameter, which can lead to over-fitting, and it excludes potentially interesting metabolites from the analysis.The hypothesis motivating this work is that the difficulties from high-degree metabolites arise from an implicit assumption of a narrow metabolite degree distribution, as opposed to the known long-tailed degree distribution. The hypergeometric distribution for shared metabolites corrects for enzyme degree, but not for metabolite degree. Consequently a high-degree metabolite such as water is given the same weight as a low-degree metabolite when counting shared metabolites. Intuitively, high-degree metabolites should be down-weighted. Our improved local method uses the known enzyme and metabolite degrees to generate a degree-corrected score with excellent performance.The global method we examine, a graph diffusion kernel on the bipartite enzyme-metabolite network, also includes a degree normalization that down-weights the contribution of high-degree metabolites (and high-degree enzymes). This method is somewhat more expensive than the local methods, requiring a full matrix inverse rather than sparse matrix multiplication.The graph diffusion kernel explores the topology of the metabolic network using random walks that visit metabolites and enzymes. Enzyme-metabolite edges are treated as undirected, permitting random walkers to traverse both directions even for a unidirectional reaction. There are no constraints on the flux of random walkers through any enzyme, and the stoichiometry of a metabolite as a reactant or product is ignored.Flux-balance methods go beyond graph diffusion by adding constraints specific to metabolic networks. Enzyme fluxes are coupled by mass balance and reaction stoichiometry, and correlations between enzyme fluxes can propagate through the network. These additional constraints capture more of the biological reality of metabolism than either shared metabolites or graph diffusion. Predictive performance is also better, presumably because of the biological constraints. A curious point is that flux sampling, with a uniform sample over the feasible space, performs better than calculations of epistatic effects based on reductions to an optimized fitness objective function. This may indicate errors in the assumed objective function for cellular fitness. The main drawback of flux-balance methods is the high computational cost.In summary, graph diffusion methods have become a method of choice for analyzing many types of networks. While the degree-corrected local methods provide a substantial improvement over previous local methods, the graph diffusion kernel using the entire network topology performs somewhat better. Dedicated flux-sampling methods are slightly better for predicting genetic interactions, but take over 100× longer to calculate and are much more difficult to implement.As the graph diffusion kernel (GDK) method provides a good trade-off between performance (assessed by genetic interaction prediction) and computational efficiency, it is likely to be a method of choice. The average semantic similarity for gene pairs highly ranked by the GDK provides an additional assessment of performance Figure . GDK scoThese average results, however, do not always hold for individual gene pairs. A cross-tabulation of semantic similarity vs. GDK score demonstrates that many gene pairs with high GDK scores nevertheless have essentially no semantic similarity Figure . These GWe therefore selected examples with high GDK scores and low semantic similarity. A GDK threshold of score ≥50 focused attention on the top-ranked GDK pairs, where on average the semantic similarity corresponds to only 23 genes annotated to the parent category of a pair. We then set a threshold of semantic similarity ≤2. This value is substantially below the semantic similarity of gene pairs selected at random, and was chosen to yield a number of example that was feasible for case-by-case analysis, a data set of 101 pairs denoted putative false positives [Additional file Manual inspection suggests that these cases can be classified into 4 main categories: unannotated, transport-synthesis, pathway-boundary, and secondary activity  of the same metabolite. Thus, both enzymes are fulfilling the same role of increasing the intracellular concentration of a metabolite. The structure of the GO annotations does not reflect this close functional association, however. Examples include transport of choline/ethanolime (HNM1-CKI1), allantoate (DAL5-DAL1), sterols (AUS1-ERG27), and uridine (FUI1-URK1).The final category, pathway-boundary, arises when the boundary between two well-accepted pathways cuts through a metabolite. Enzymes that connect to this metabolite are then annotated to very different pathways, despite a close network-level association. These cases are responsible for 20% of the total. Examples include associations between enzymes in the TCA cycle and those using TCA metabolites for amino acid synthesis, enzymes with different roles in glycoprotein synthesis, and enzymes responsible for quinone metabolism [Additional file In analyzing large networks, it has become common to delete high-degree vertices. This practice is questionable. It depends on an arbitrary high-degree cutoff, usually without any clear break in a vertex degree distribution. It can remove vertices that are of interest, and it can introduce unknown biases into the analysis.Here we have introduced methods that are readily applied to networks with high-degree hubs. Local methods use known degree distributions to correct for high-degree enzymes and metabolites, and global methods use graph diffusion kernels to rank the association between pairs of enzymes. We show that these methods outperform previous methods that eliminate high-degree vertices from the networks. The context is cellular metabolism, where high-degree metabolites like water and ATP are shared by many enzymes. Our methods are able to infer functional associations between enzymes, without being misled by sharing of these high-degree metabolites.In several cases, enzymes predicted by network analysis to have high functional association have very little similarity in database annotations. Some of these cases are due to a discrepancy between the metabolic reconstruction, which records a reaction for an enzyme, and the annotation database, which lacks information or omits a secondary activity. Two additional patterns were observed, however, which relate to the structure of Gene Ontology hierarchies. First, enzymes that are responsible for synthesis and transport of the same metabolite often have little annotation similarity. Second, conventional pathway definitions may place two enzymes with strong network-level associations on opposite sides of a pathway boundary.The methods developed here should be applicable in general to other bipartite networks, particularly those with high-degree hubs.-iND750-GlcMM.xml" corresponding to the minimal media condition was obtained from http://gcrg.ucsd.edu/Downloads/Cobra_Toolbox. This network has 1061 metabolites, 1266 reactions and 750 genes. The stoichiometry matrix S provides the number of metabolites m consumed or produced in reaction r. The reaction-gene association matrix E in the metabolic network indicates whether reaction r can be catalyzed by enzyme e.The Yeast metabolic network used in our study was obtained from the database maintained by systems biology group, University of California, San Diego ,13. The e and the metabolites m as disjoint vertices in a bipartite graph. We use various metabolic coupling measures between two enzymes in this graph to predict synthetic lethal genetic interactions. Towards this goal, we use both methods from literature  and oi in a reaction j with a value 1 and 0 otherwise  . In our n, n1, n2 and n12 represent the total number of metabolites in the network, the number connected to enzyme 1, the number connected to enzyme 2, and the number connected to both. The score is thenWe used our previous method based on 2 × 2 contingency table for calculating the metabolites shared between two enzymes [C is n!/k!(n - k)!. This score increases when n12 is either larger or smaller than the value n1n2/n expected under the null hypothesis . For n12 smaller than the null expectation, we used score(n12) - |score(n12) - score(n1n2 / n)| to restrict attention to enrichment.The combinatorial factor The score based on the hypergeometric p-value for characterizing the metabolic coupling between the enzymes 1 and 2 isλ. For the alternative, λalt = n12, the observed count; for the null, λnull = n1n2/ntot. The total number of metabolite-enzyme edges is ntot; n1 and n2 are the numbers of metabolites connected to each enzyme; and n12 is the intersection of the metabolites in n1 and n2. The Poisson score isThis score is also obtained as the log-likelihood ratio of an alternative to null model for the observed number of shared partners. Both the alternative and the null employ a Poisson distribution with a single parameter The absolute value for the second term in Eq. 3 ensures that large scores come from enrichment rather than depletion of shared metabolites.λnull using the degree distribution of the metabolites connected to the enzymes. Let ki1, ki2,..., mi1, mi2,..., min connected to enzyme i in the enzyme pair . The probability of the metabolite m11 to be connected to enzyme 2 isThe null model in the Poisson score was further improved by considering a different value for The average number of metabolites connected to enzyme 1 that are also connected to enzyme 2 is thenλnull used in the improved null model is obtained by arithmetic mean of The values of λnull, accounting for the full metabolite degree distribution. The probability of k connections given the status of the metabolites m1,..., mi connected to enzyme E, PE isA further improved Poisson score can be generated using a more elaborate i mare estimated asThe probability of enzyme connected and not connected to the metabolite EP = 0, EP(k = 0|-) = 1 in the calculations and obtain EP for the enzymes A and B in the pair (and respectively connected to n1 and n2 metabolites) to obtain the λnull,We take λnull (Eq. 9) to obtain the Poisson score that takes metabolite associations and degree into account,We use Results from this more complicated model are included in Table Graph diffusion kernels are solutions to the steady-state density distribution for continuous-time random walk or diffusive process on a graph with sources and sinks . The adjM defined as MT = E and captures the direct links from metabolites to enzymes. The degrees of the nodes are summarized by the diagonal matrix D . The graph diffusion kernel score with normalization for the node degrees isThe matrix γ controls the extent of diffusion, or equivalently the length of the random walks. These lengths are distributed exponentially, with the probability of a d-step walk proportional to e-γd. The results shown in this work are for a value of γ = 1. Results were not sensitive to the value of γ, with similar results over a range from 0.5 to 120. The entries in the kernel corresponding to the enzyme-enzyme relationships were then extracted to predict genetic interactions. For readability, GDK scores displayed in the figures are multiplied by 104.The parameter M(MT = E) in the adjacency matrix A rather than a binary version as used in the non-weighted case (Eq. 11). The element T )ij . Then we calculated all the metabolic coupling measures (except enzyme flux correlation measure) with the compartment-free metabolic network. There were 646 metabolites in the compartment-free network.in silico medium similar to YPD [The performance of various scores considered in this work were compared with enzyme flux correlation score used in our previous study . Brieflyr to YPD . Then rer to YPD . This enr to YPD . The abshttp://kishony.med.harvard.edu/prism/index.html. For calculating the AUC and F-score from scaled epistasis score, only enzyme pairs with epistasis score were considered. We did not calculate GO consistency scores for this method because it performed poorly for predicting genetic interactions.The scaled epistasis values corresponding to the minimal media were obtained from a previous study . The filSynthetic lethality data for this study was obtained from the BioGRID database (version 2.0.46) . There wThe ROC curves, AUC, F-score and PR curves were generated in R using the ROCR package . The ROCWe used a procedure proposed in a previous study for validating the functional gene similarity measures . We used-association.sgd was obtained from the Saccharomyces genome database, http://downloads.yeastgenome.org/literature_curation/.The percent of consistent GO terms were calculated for each bipartite coupling measure. We considered only GO terms associated with 5 though 100 genes. For each GO term, the pairwise coupling score between metabolic genes associated with it are averaged. The statistical significance of this averaged score is assessed by random shuffling of gene GO annotation associations, maintaining both the annotation and gene distribution. We calculated an empirical p-value based on 10,000 iterations for each GO term. These empirical p-values were corrected for multiple testing of many GO terms to control for false discovery rates . The conln(NT/NP ), where the total number of genes NT = 6310 for yeast, and the number of genes annotated to the closest parent category of two genes is NP [Semantic similarity was calculated as es is NP ,20. SemaThe methods proposed in this work are computationally less expensive as compared to the flux correlation based approaches. The computation of flux correlation takes about 15 hours (each sampling run taking about 5 hours). Computing the other scores was performed as a single calculation that required only 9 minutes. The graph diffusion kernel, part of this single calculation, was computed directly as the inverse of the graph Laplacian rather than as a repeated matrix multiplication.BV and JSB conceived the study and wrote the manuscript. BV performed the work. All authors read and approved the final manuscriptPerformance of raw neighbor count. The data from Figure Click here for fileConsistent GO terms and category size. GO consistency analysis results are provided for category sizes of 2-100 and 2-200, compared with 5-100 in the main text.Click here for filePutative false positives. The 101 gene pairs with high network association and low semantic similarity are tabulated and the reasons for disagreement are annotated.Click here for file


Pichia pastoris has been recognized as an effective host for recombinant protein production. A number of studies have been reported for improving this expression system. However, its physiology and cellular metabolism still remained largely uncharacterized. Thus, it is highly desirable to establish a systems biotechnological framework, in which a comprehensive in silico model of P. pastoris can be employed together with high throughput experimental data analysis, for better understanding of the methylotrophic yeast's metabolism.P. pastoris (iPP668), composed of 1,361 reactions and 1,177 metabolites, was reconstructed based on its genome annotation and biochemical information. The constraints-based flux analysis was then used to predict achievable growth rate which is consistent with the cellular phenotype of P. pastoris observed during chemostat experiments. Subsequent in silico analysis further explored the effect of various carbon sources on cell growth, revealing sorbitol as a promising candidate for culturing recombinant P. pastoris strains producing heterologous proteins. Interestingly, methanol consumption yields a high regeneration rate of reducing equivalents which is substantial for the synthesis of valuable pharmaceutical precursors. Hence, as a case study, we examined the applicability of P. pastoris system to whole-cell biotransformation and also identified relevant metabolic engineering targets that have been experimentally verified.A fully compartmentalized metabolic model of P. pastoris, thus allowing us to gain valuable insights into the metabolism of methylotrophic yeast and devise possible strategies for strain improvement through in silico simulations. This computational approach, combined with synthetic biology techniques, potentially forms a basis for rational analysis and design of P. pastoris metabolic network to enhance humanized glycoprotein production.The genome-scale metabolic model characterizes the cellular physiology of  Pichia pastoris, enabled the synthesis of humanized glycoproteins with substantially improved bioactivity [P. pastoris is expected to be one of the promising hosts for industrial production of recombinant protein in the near future. Indeed, companies are already beginning to use P. pastoris for the production of several therapeutic proteins [In the biopharmaceutical industry, over 70% of the therapeutic proteins under preclinical and clinical development are glycosylated and there has been an increasing need for highly efficient glycoprotein expression systems. Mammalian systems such as Chinese hamster ovary (CHO) cells have been most widely used since they have been extensively characterized and are capable of human-like glycosylation. However, they typically exhibit low survivability and low recombinant protein productivity unless sophisticated experimental techniques were employed . On the activity ,3. Furthactivity ,3. Thus,proteins ,5.P. pastoris. These studies have elucidated various favorable characteristics of the organism, which makes it more attractive for large-scale production of recombinant humanized glycoprotein [P. pastoris as a more superior expression host than the well-characterized Saccharomyces cerevisiae [P. pastoris strain in order to achieve economic feasibility and efficiency [A number of studies have been carried out for understanding and enhancing physiological properties of oprotein -10. In previsiae ,11. Hencficiency -14. In tficiency .in silico modeling and analysis to understand and design cellular system to achieve desirable properties [S. cerevisiae, which was metabolically engineered for improved production of chemicals and biopharmaceuticals [P. pastoris by resorting to the systems biotechnological approach, which requires a comprehensive in silico metabolic model [Systems biotechnology integrates high-throughput omics data and operties . This apeuticals ,18. For euticals ,20 and heuticals . Therefoic model .Escherichia coli [Zymomonas mobilis  to complex mammalian systems such as Mus musculus [Homo sapiens [E. coli [E. coli [P. pastoris metabolism which allows us to elucidate interesting features of the methylotrophic yeast and identify engineering targets for achieving enhanced physiological properties of the strain.To date, more than 50 genome-scale metabolic models have been reconstructed for over 30 species from three main domains of life, i.e. Archaea, Bacteria and Eukarya ,24. Varihia coli  and Zymomusculus  and even sapiens ; phenoty[E. coli ; and ide[E. coli ,30. TowaP. pastoris metabolism was reconstructed (see Methods). During the reconstruction process, significant efforts were required in the gap-filling process. Essential metabolic reactions are added sequentially to the draft model, through repeated cell growth simulations using constraints-based flux analysis, to ensure in silico viability. For example, a metabolic gap exists in the initial draft of the methonine biosynthetic pathway as the gene coding for the enzyme catalyzing transamination of 2-keto-4-methylbutyrate to form methionine [Following the procedure illustrated in Figure thionine  was not P. pastoris, iPP668, accounts for 668 metabolic genes, 1,361 reactions and 1,177 metabolites segregated into 8 compartments: cytosol, endoplasmic reticulum, extracellular matrix, Golgi apparatus, mitochondria, nucleus, peroxisome and vacuole as summarized in Table The reconstructed metabolic model of iPP668, we have included a biomass equation which is composed of biosynthetic precursors and energy requirement. Appropriate coefficients of amino acids and carbohydrates were obtained from a recent experimental composition analysis of P. pastoris [S. cerevisiae [In pastoris  while lipastoris . The reppastoris  forms threvisiae  accordinrevisiae . We alsoP. pastoris metabolism were further elucidated by comparing iPP668 with two model organisms: S. cerevisiae (iMM904) [E. coli (iAF1260) [P. pastoris and S. cerevisiae, are generally classified under the lipid and carbohydrate biosynthetic pathways while the 79 reactions and 46 metabolites unique to P. pastoris are mainly from the methanol utilization pathway   and E. ciAF1260) . It shouy Figure  and certiPP668 metabolic model by checking the difference between predicted cellular phenotypes and experimental observations obtained from two sets of chemostat culture conditions, glucose minimal medium (see Methods) and glycerol/methanol mixed medium [In this study, we validated the d medium . In bothIn the case of glucose minimal medium, we observed that the predicted cell growth, oxygen uptake and carbon dioxide evolution rates are highly consistent with the data obtained from our chemostat experiments Figure . DiscrepAOX by glycerol [Another set of chemostat culture data  also allglycerol , that arAOX promoter which is commonly used for heterologous protein production in P. pastoris. However, the yeast typically grows very slowly on this carbon source [P. pastoris using the reconstructed model and as such, identify the best candidate for recombinant protein production. To do so, we simulated cell growth by constraining the supply of each carbon source to one C-mmol/gDCW-hr .Among the various carbon sources available, methanol is the inducer for n source . Thus, an source . Based oP. pastors.Based on the first criterion of high growth yield, glycerol is the most promising candidate for recombinant protein production, followed by sorbitol Figure . GeneralAOX promoter is used for recombinant protein expression, sorbitol will be more superior as glycerol is known to repress the promoter [Not surprisingly, flux-sum distribution profiles on various carbon sources confirmed that sorbitol and glycerol are capable of generating higher amino acid flux-sum compared to others. Thus, this large pool of amino acids can potentially be diverted to synthesize recombinant protein during the induction phase, indicating sorbitol and glycerol are desirable carbon sources for recombinant protein production in terms of both growth yield and amino acid biosynthetic capability Figure . Howeverpromoter . TherefoP. pastoris in methanol. From the in silico analysis, it was observed that growing P. pastoris in methanol minimal media can exhibit high turnover of NADH, implying the potential capability of P. pastoris for whole-cell biotransformation. For example, value-added chemical precursors, e.g. 2,3-butanediol, can be synthesized via enzymatic reduction of ketones, e.g. acetoin, which requires NADH or NADPH regeneration [P. pastoris is suitable for biocatalysis of butanediol dehydrogenase reaction to convert acetoin into 2,3-butanediol [P. pastoris by identifying gene targets to be overexpressed through the modified flux analysis implementation (see Methods).Apart from recombinant protein production through methanol induction, we managed to identify another potential application of culturing neration -49. In ttanediol . To achiP. pastoris is supplied with 1 mmol/gDCW-hr of acetoin, attenuating the flux activity of some reactions can give rise to a proportional decrease in the maximum achievable butanediol dehydrogenase reaction flux , glutathione-dependent formaldehyde dehydrogenase (FLD1), peroxisomal matrix protein (PEX8) and secretion GTPase (YPT1) promoters, have been proposed since methanol, a petroleum compound, may not be perceived as an appropriate raw material for production of proteins for human consumption [iPP668 can provide us with the corresponding best carbon source selection.The genome-scale metabolic modeling and analysis procedure presented in this study can potentially be used to design culture media for enhancing the performance of e Figure . Furtherpromoter . The genpromoter -55. Althsumption . For theiPP668 is the rational identification of metabolic engineering targets. From our analysis of carbon source utilization, we found that growing P. pastoris in methanol minimal medium is suitable for biocatalysis of ketone reduction such as the conversion of acetoin into 2,3-butanediol, which requires NADH regeneration [in silico analysis have been demonstrated to be appropriate metabolic engineering targets for enhancing 2,3-butanediol production [iPP668 can be a useful tool for tackling other metabolic engineering problems associated with P. pastoris.Another application of neration . In thisoduction  and the iPP668 to account for glycosylation processes can potentially increase the utility of the model. The design of glycosylation pathway through glycoengineering is critical for recombinant protein production in yeasts [Pichia pastoris.Protein glycosylation is an important post-translational process which can affect the protein's secretion, folding and bioactivity ,58. Thusn yeasts . When con yeasts . Hence, n yeasts -63. Althn yeasts . We can n yeasts . We can n yeasts . By addiet al. [P. pastoris metabolism, PpaMBEL1254. A model comparison between iPP668 and PpaMBEL1254 shows that both models are fully compartmentalized into 8 subcellular compartments, and capable of describing P. pastoris cellular metabolism under various carbon source uptake conditions. Nonetheless, several differences between the current study and Sohn et al. can be found with respect to the model size, validation and subsequent analysis. The size of iPP668  is larger, providing more information than PpaMBEL1254 . In addition, iPP668 was validated by two sets of chemostat experimental data while Sohn et al. used batch fermentation data for qualitative model validation. Interestingly, Sohn et al. explored the capability of P. pastoris to produce recombinant proteins such as human serum albumin and human superoxide dismutase under various oxygenation rates using PpaMBEL1254, while we have discussed the general procedure to represent the protein production in the model based on amino acid biosynthetic rates. However, the PpaMBEL1254 model did not clearly describe the methanol utilization pathway of P. pastoris which is an important metabolic characteristic of great interest for applications in recombinant protein production and biotransformation, as discussed in the current study. Therefore, we suggest the combination of iPP668 and PpaMBEL1254 for future model expansion and systems metabolic engineering studies to harness the useful information provided by two genome-scale metabolic models.During the peer review process of the current work, we identified another independent reconstruction effort that has been made by Sohn et al. , presentP. pastoris, which has been developed for recombinant protein production as well as whole-cell biotransformation. The metabolic model was manually curated with information from literature and various databases to provide a good description of cell growth under various culture conditions through the use of constraints-based flux analysis. Validation of the curated model has shown that the model was able to reproduce key characteristics of P. pastoris metabolism reported in various experimental studies. The analysis of P. pastoris growth on different substrates has also allowed us to understand some useful application of the methylotrophic yeast and to generate testable hypotheses which can help to improve its biochemical production capability. Future works in the experimental analysis of P. pastoris can potentially enhance the predictive capability of the current model which can be integrated with high-throughput omics analysis for yeast systems biotechnology.In this study, we have reconstructed a genome-scale metabolic model of the methylotrophic yeast P. pastoris was reconstructed based on a recently published genome annotation of the organism [The metabolic network of organism . An initorganism  based onorganism , BRENDA organism , ExPASy organism , MetaCycorganism  and KEGGorganism  . The inP. pastoris under various culture conditions. Mathematically, the rate of biomass synthesis forms the objective function to be maximized, subjected to stoichiometric, thermodynamic and reaction capacity constraints, resulting in the following linear programming problem:Cellular metabolism can be simulated by constraints-based flux analysis as described elsewhere -74. We uSij refers to the stoichiometric coefficient of metabolite i involved in reaction j and vj is the flux of reaction j. Reaction capacity and thermodynamic constraints, based on reaction reversibility information, are specified using the parameters αj and βj indicating the lower and upper bounds of reaction j, respectively. Unless experimental measurements are available, we typically specify βj = inf and αj = -inf (for reversible reactions) or αj = 0 (for irreversible reactions). In addition, gene deletion can be simulated by setting αj = βj = 0 for the associated metabolic reactions. In this study, the linear programming problem was solved using General Algebraic Modeling System (GAMS) Integrated Development Environment (IDE) version 22.7 [where ion 22.7 .vATPS3m/vCYOOm) where ATPS3m and CYOOm refer to ATP synthase and cytochrome c oxidase reactions in mitochondria compartment, respectively. It can be changed by adjusting stoichiometric coefficients of hydrogen ions in the cytochrome c oxidase reaction.  and P/O ratio, vary with culture media and other environmental conditions. Thus, these values have to be determined for simulation of different experimental conditions. NGAM requirement is specified by fixing the flux value of ATP maintenance reaction. The experimental value is determined by plotting the uptake rates of carbon source against dilution rates from a chemostat run . In our computational framework, vtarget is maximized under different levels of reaction flux perturbation. Taking into consideration that the enhancement of biochemical production typically results in attenuated growth rate, we set the lower limit of cell growth to 50% of the maximum value , we can evaluate the effect of individual reaction flux activity on the flux of any target reaction. The MIP is implemented and solved in GAMS IDE version 22.7 [We define "flux activity" as the absolute value of a reaction flux. Thus Pichia pastoris X-33  was used for the chemostat culture. A single colony of the strain was inoculated into 20 mL of yeast protein dextrose (YPD) medium containing (per liter): 20 g of glucose, 10 g of yeast extract, and 20 g of Bacto-peptone and incubated overnight at 30°C. The culture was transferred to a 500 mL Erlenmeyer flask containing 100 mL of YPD broth and incubated overnight at 30°C. This culture was used as a seed for the chemostat culture using 2.5 L jar fermentors  with a constant volume of 800 mL. Cells were first batch-cultured in a defined glucose minimal medium containing (per liter): 50 g glucose, 1.5 g MgSO4·7H2O, 0.2 mg Biotin, 3 g KH2PO4, 15 g (NH4)2SO4, 2.0 mL of trace salts stock solution and 1.0 mL of a solution containing 2 g/L biotin in 1M NaOH. The trace element solution contained (per liter) 5 mL H2SO4, 4.0 g CuSO4·5H2O, 14.0 g MnSO4·H2O, 2.6 g Na2MoO4·2H2O, 4.0 g H3BO3, 4.0 g CoCl2·6H2O, 22.0 g ZnSO4·7H2O, 55.0 g CaCl2·2H2O and 37.5 g FeCl3·6H2O. Then, the chemostat culture was initiated after the depletion of glucose at the end of batch culture and continued at different dilution rates under the same medium condition. The cultures were carried-out at 30°C and pH 6.0 controlled by 24% NH4OH. Agitation and aeration were fixed at 800 rpm and 1 vvm, respectively, and the dissolved oxygen in the culture medium level was maintained above 10% air saturation during the entire culture period. Cultures were assumed to be in steady-state after at least 5 volumes changes.A wild type 2 and CO2 analyzer . The carbon dioxide evolution rate (CER) and oxygen uptake rate (OUR) were deduced from the inlet and outlet carbon dioxide and oxygen compositions.The concentration of biomass was determined as dry cell weight (DCW). Culture samples were collected after achieving steady-state for different dilution rates. The collected samples were centrifuged and washed twice with ultrapure water and then the pellet was dried at 100°C. The culture supernatants were used for measuring the concentration of glucose, organic acids, glycerol and ethanol by high-pressure liquid chromatography  equipped with an HPX 87H column , an ERC-7515A RI detector  and a UV detector . Analysis of carbon, nitrogen and hydrogen content was performed with a CHN analyzer  of lyophilized cells as suggested by . The ashThe authors declare that they have no competing interests.BKSC performed the model reconstruction and drafted the manuscript. BKSC and SS performed the model analysis and simulations. JR, HL(1), JA and HL(2) performed the chemostat experiments. DYL coordinated and directed the project, and prepared the final manuscript. All authors have read and approved the final manuscript.Excel file contains metabolic reaction, metabolite and gene lists.Click here for fileWord document contains details on the calculation of biomass composition, carbon balance and ATP maintenance requirement.Click here for file


Metabolism and its regulation constitute a large fraction of the molecular activity within cells. The control of cellular metabolic state is mediated by numerous molecular mechanisms, which in effect position the metabolic network flux state at specific locations within a mathematically-definable steady-state flux space. Post-translational regulation constitutes a large class of these mechanisms, and decades of research indicate that achieving a network flux state through post-translational metabolic regulation is both a complex and complicated regulatory problem. No analysis method for the objective, top-down assessment of such regulation problems in large biochemical networks has been presented and demonstrated.We show that the use of Monte Carlo sampling of the steady-state flux space of a cell-scale metabolic system in conjunction with Principal Component Analysis and eigenvector rotation results in a low-dimensional and biochemically interpretable decomposition of the steady flux states of the system. This decomposition comes in the form of a low number of small reaction sets whose flux variability accounts for nearly all of the flux variability in the entire system. This result indicates an underlying simplicity and implies that the regulation of a relatively low number of reaction sets can essentially determine the flux state of the entire network in the given growth environment.We demonstrate how our top-down analysis of networks can be used to determine key regulatory requirements independent of specific parameters and mechanisms. Our approach complements the reductionist approach to elucidation of regulatory mechanisms and facilitates the development of our understanding of global regulatory strategies in biological networks. Metabolic network reconstructions have beeE. coli [Our procedure is comprised of five main steps Figure . The recE. coli  was usedE. coli , and in 6) of flux vectors. Because the sampling procedure is a linear one and because we sought a basis for the sampled space, we performed Principal Components Analysis using Singular Value Decomposition. The cumulative fractional eigenvalue distribution , which catalyzes the first reaction step of acetate secretion in eigenflux 1, is allosterically activated by pyruvate and inhibited by NADH and NADPH. High concentrations of NADH are indicative of a redox imbalance, which eigenfluxes 6 and 7 serve to correct by oxidizing NADH.3oxidase (CYTBO3) is utilized under high oxygen concentrations and has a higher bioenergetic efficiency (protons translocated per electron) than the other cytochrome bd oxidase (CYTBD). Even though cytochrome bd is energetically less efficient it is operational at low oxygen levels.2. Eigenflux 2 is associated with establishing the proton motive force. By flexibly tuning the electro-chemical gradient of protons across the cellular membrane through translocation in the electron transport chain, a cell can economically provide the energy for essentially all cellular activity–such as ATP synthesis, solute transport, and flagellar motility. The cytochrome oxidases that catalyze the two opposing reactions in bimodal eigenflux 2 have slightly different functional behaviors. The cytochrome bo3. Eigenflux 3 is the flux variation through glycolysis, and functions with eigenflux 5 to determine the absolute magnitudes of the fluxes in glycolysis and the pentose phosphate pathway (PPP).4. Eigenflux 5 is associated with the dominant tradeoff in central metabolism for providing the metabolic precursors, ATP, and reducing power (in the form of NADPH) to generate macromolecular building blocks against the need for reducing power (in the form of NADH) for sustaining the proton gradient. Until recently it was assumed that nearly all of the NADPH needed for the biosynthesis reactions was produced in the PPP and the TCA cycle, but recent experiments have demν can be calculated from the equationThe flux carried through eigenfluxes in a given environment will be a result of molecular regulatory mechanisms, including mass action kinetics. The contribution of an eigenflux to any given steady-state flux vector U is the matrix of eigenfluxes and <ν > is the vector of mean flux values  are presented as colored β vectors in Figure β can be similarly interpreted, and altogether allow one to assess the flux state of the entire metabolic network at essentially a glance.To illustrate, we identified the two flux distributions that most oppositely utilized eigenflux 1 and computed their respective The faithfulness of the computational results of the presented procedure to biological reality depends critically on the completeness and integration of the molecular system reconstructions, assumptions made in the transformation of the integrated reconstructions into a model, and the manner in which the range of the system's functional capabilities are sampled. Cells are more than a combination of transcriptional regulatory and metabolic systems, so the exclusion of other cellular systems limits the scope of the decomposition that can be performed–as does the completeness of the included systems. For instance, with other systems included, the decomposition procedure could potentially identify osmotic or movement  mechanisms in addition to purely metabolic mechanisms. Similarly, more complete regulatory information would allow a more accurate setting of reactions that can potentially carry flux. The fixed biomass composition in the utilized model does not accurately describe a cell in all growth environments for all growth rates. Such a fixed composition limits the exploration of how the relative amounts of the biomass components can vary or how the cell can utilize different operating regimes. The ranMolecular network reconstructions enable the objective, top-down assessment of regulatory challenges and functional capabilities associated with particular phenotypic states. In the context of metabolism, the method presented herein is also applicable to kinetic and free energy parameter spaces and to concentration space -26. In gOur results also shed light on the network topology-function relationship, which is the result of a lengthy evolutionary process in varying environments. Robustness,28, defiSimilarly, the demonstrated procedure can be used to elucidate potential evolutionary mechanisms. By modifying the sampling procedure it is possible to simulate the evolution  of an organism in a particular growth environment. The decomposition procedure would identify the flux adjustment routes (i.e. eigenfluxes) by which evolution would be achieved.With a top-down view comes an understanding of what must be controlled to attain a network flux state, and with a reductionist view comes an understanding of the mechanisms that achieve such control. The top-down view will provide a context for the seemingly overlapping and redundant regulatory strategies that must make sense in not one environment, but in a large and varied range of environments. The analysis method presented here will help bridge the systems and molecular biology approaches for understanding cellular regulation in the context of large biochemical networks.v1 [E. coli reconstruction because there are still unresolved issues with infeasible reaction cycles when performing Monte Carlo sampling. The iMC1010v1 reconstruction is composed of 906 genes supporting 931 reactions  involviThe model was utilized in this work to comprehensibly compute the range of flux states that the network can support in aerobic glucose minimal media conditions. The regulation included in the model plus the EcoCyc  and ReguThe COBRA Toolbox  for MatlMetabolic models contain reaction cycles that are responsible for some thermodynamically-impossible fluxes in computed model-wide flux distributions. These reaction cycles are dealt with in Energy Balance Analysis (EBA)  and can For sampling we used a modified version of an existing sampling algorithm ,36 that The covariance matrix for reaction fluxes was calculated in the standard way  by firstThe Singular Value Decomposition  was usedThe oblique and orthogonal rotation methods tested were oblimin, promax, varimax, quartimax, parsimax, equimax, and orthomax .CLB conceptualized and developed the method and MJH assisted in the development process. CLB wrote the manuscript, and all authors participated in manuscript editing.Correlation between eigenfluxes. A histogram of the correlation between eigenfluxes derived by SVD of the flux correlation matrix.Click here for fileModel components. Two tables listing the full and abbreviated names of the metabolites and the reactions comprising the model used in this work.Click here for fileResults from application of presented procedure for glucose anaerobic conditions. A figure showing the cumulative fractional eigenvalue spectrum of the eigenfluxes and a table describing the reactions in each eigenflux.Click here for fileMean and variance of reaction rates computed from sampling data. A table showing the mean and variance fluxes resulting from Monte Carlo sampling in glucose aerobic conditions.Click here for file


A modular approach is presented that allows the observation of the transcriptional activity of metabolic functions at the genome scale. High-throughput techniques have multiplied the amount and the types of available biological data, and for the first time achieving a global comprehension of the physiology of biological cells has become an achievable goal. This aim requires the integration of large amounts of heterogeneous data at different scales. It is notably necessary to extend the traditional focus on genomic data towards a truly functional focus, where the activity of cells is described in terms of actual metabolic processes performing the functions necessary for cells to live.In this work, we present a new approach for metabolic analysis that allows us to observe the transcriptional activity of metabolic functions at the genome scale. These functions are described in terms of elementary modes, which can be computed in a genome-scale model thanks to a modular approach. We exemplify this new perspective by presenting a detailed analysis of the transcriptional metabolic response of yeast cells to stress. The integration of elementary mode analysis with gene expression data allows us to identify a number of functionally induced or repressed metabolic processes in different stress conditions. The assembly of these elementary modes leads to the identification of specific metabolic backbones.This study opens a new framework for the cell-scale analysis of metabolism, where transcriptional activity can be analyzed in terms of whole processes instead of individual genes. We furthermore show that the set of active elementary modes exhibits a highly uneven organization, where most of them conduct specialized tasks while a smaller proportion performs multi-task functions and dominates the general stress response. The increasing availability of high-throughput data has allowed more and more analyses to be performed at the cell scale. After completion of genome sequencing for many species, the focus is shifting towards getting a global understanding of cell physiology. This task requires the integration of heterogeneous data at different scales, including genomic, transcriptomic, proteomic, and metabolomic data.At the level of metabolism, good knowledge of the structure of metabolic networks has now been achieved for several species. A number of genome-wide models of metabolism have been reconstructed -4, but tet al. [Metabolic networks are also highly complex, formed by several hundreds of densely interconnected chemical reactions. To characterize such complex systems at the genome scale, it is necessary to identify smaller building blocks. Cellular networks have been shown to have a high degree of modularity, and are composed of groups of interacting elements and molecules that carry out specific biological functions . In receet al. , who shoA complementary approach is provided by the concept of an 'elementary mode'. Elementary modes, and the very similar concept of 'extreme pathways', are minimal sets of reactions that can operate in steady state in a metabolic network -20. TheySaccharomyces cerevisiae by relying on elementary modes. Our study is based on the integration of dynamic gene expression data in various stress conditions into a genome-scale model of metabolism, modularly structured in elementary modes. We used a bioinformatics tool called BlastSets [Here, we address the problem of identifying metabolic units in a genome-scale model of the yeast lastSets  to combiThe computation of elementary modes in genome-wide models of metabolism is seriously hampered by the problem of combinatorial explosion. Even though the number of elementary modes is usually smaller in a real system than its theoretical limit and can be further reduced by taking into account various environmental or regulatory constraints, it is of no practical use to handle systems of thousands of elementary modes because such systems become impossible to interpret ,29. One We have developed an alternative approach for computing elementary modes at the genome scale. In the Kyoto Encyclopedia of Genes and Genomes (KEGG) database, metabolic pathways are represented as a series of maps, where each map covers a precise biological function . These mBecause of their combinatorial nature, a number of different elementary modes usually share common reactions along their path. It often occurs that several elementary modes are almost identical except for a few branches at their extremities. Similarly, a given reaction can belong to a large number of different elementary modes. Figure The use of KEGG maps for defining subnetworks aims at having entities that are as much as possible biologically coherent. The start and end points of elementary modes are compounds located at the boundaries between subnetworks. One drawback of this approach is that active metabolic routes that are spread over different KEGG maps may not be easily identified. To overcome this problem, we constructed two different collections of elementary modes, EM1 and EM2. EM1 contains the full set of single elementary modes computed with each KEGG pathway map being used as a subnetwork; each elementary mode from EM1 is entirely included in a single pathway map. EM2 was formed by combining all pairs of elementary modes from EM1 that are connected through a common boundary compound; elementary modes from EM2 thus spread over two different pathway maps Figure . The useP values than whole pathways , which should already guarantee that identified elementary modes are statistically significant. Nevertheless, in order to further assess the reliability of our results, we created random gene expression values by random permutation of gene expression values in several stress responses. These random sets of induced/repressed genes were compared to elementary modes in BlastSets, in the same way as for stress-induced/repressed genes. No active elementary mode was identified using these random sets. The procedure was repeated for several conditions, always with the same result. This finding confirms that elementary modes found to be active in specific environmental stress conditions have a high statistical significance.BlastSets applies a stringent threshold on To identify complete metabolic routes that are spread over several KEGG pathway maps, we constructed the EM2 collection containing elementary modes grouped in pairs. Two elementary modes are grouped as a set in EM2 if they share a common boundary compound. These compounds act as bridges between individual pathway maps, enabling more extended induced/repressed routes to be identified by this approach.P values and connecting them to each other, thanks to results from the EM2 collection . These backbones can be viewed as the main modules characterizing metabolic activity in terms of expression data in a given condition. They are provided for each individual condition in Additional data file 2.In each stress situation, we could then infer a 'backbone' of induced/repressed metabolic routes. Backbones were constructed by selecting the pairs of elementary modes with the lowest P(k) to find a given induced/repressed elementary mode in k stress conditions  The que [et al. . DiffereChallenges also remain to obtain a more accurate description of the transcriptional activity of elementary modes in a cell. Our approach can be seen as 'discrete', since an elementary mode can only be assigned to three possible categories, that is, induced, repressed, or inactive. This division into three categories relies on a threshold on expression fold-change values, but enhanced statistical approaches could be researched to obtain a more subtle classification and avoid the need to set a threshold. Furthermore, with BlastSets, the localization of induced/repressed genes 'inside' an elementary mode is not taken into account, although this information could be relevant. For example, a repressed gene belonging to a group of genes coding for the same enzyme may have no influence on the activity of the elementary mode as a whole, while repression of a gene that is the only one to encode a particular enzyme would be important. Finally, the computation of elementary modes is based on a steady-state assumption and it remains to be seen to what extent these concepts can be extended to dynamic activity.Combinatorial explosion prevents the computation of elementary modes in large networks. For example, in a network of about 110 reactions the number of elementary modes was shown to be higher than two million . FurtherS. cerevisiae as a source for the metabolic model [We used the KEGG XML files for ic model . These f2 appears on the map of the citrate cycle and must be considered external for the cycle to be able to operate, otherwise this route would contain a dead end and become inactive for the reason stated above. The complete list of metabolites covered by rule three comprises H2O, O2, P, CoA, CO2, NH3, UDP, H2, and reduced and oxidized thioredoxin.A stoichiometric matrix was constructed for each pathway based on its XML description. A point of major importance to the computation of elementary modes is the definition of 'external metabolites'. They act as start and end points of elementary modes, and in our hierarchical approach they additionally enable elementary modes from different pathway maps to be connected to each other. We adopted the following rules for defining external metabolites: one, a metabolite located at the interface between two or more pathway maps is considered external to all of them; two, a metabolite that can only be either produced or consumed is considered external; and three, unbalanced ubiquitous metabolites are considered external. Rule one creates the vast majority of entry/exit points to elementary modes and allows connections between pathway maps. Rule two prevents the existence of 'inactive' metabolic branches, that is, branches of a metabolic network that cannot participate in any elementary mode. This happens, for example, when a branch ends up in a dead end: as steady state conditions are assumed when elementary modes are computed, no flux can be present in a dead-end branch since this would lead to accumulation of the compound at its extremity. Rule three was introduced to prevent particular branches of the metabolic network from collapsing due to inappropriate balancing. For example, COS. cerevisiae is provided as Additional data file 1, and was used to create the EM1 and EM2 BlastSets collections.Once stoichiometric matrices had been constructed, elementary modes were computed using a classical algorithm . The comS. cerevisiae to various environmental stresses. Three sets of microarray experiments have been selected for our study. Causton et al. [et al. [et al. [We have chosen experiments analyzing the gene expression responses of the yeast n et al.  describen et al. . Gasch e [et al.  analyzed [et al. . Iwahash [et al.  studied  [et al. .These three datasets enabled us to study a total of 31 stress conditions. Some of these stresses involved environmental changes or nutrient depletion, while others involved exposure to toxic compounds such as pesticides or fungicides. The latter include: ash, which refers to exposure to burned ash from an industrial incinerator; maneb, which is a fungicide used in the control of several diseases of fruit, vegetable, field crops and ornamentals; pentachlorophenol (PCP), which is an effective fungicide, herbicide and algicide used as a wood preservative; tetrachloro-isophthalonitrile (TPN), which is a fungicide used to prevent biofouling on ships and in agriculture; thiuram, which is a compound used as fungicide to prevent crop damage and to protect harvested crops; and zineb, which was rated as a pesticide of low toxicity and may be a weak mutagen.We plotted the distributions of the natural logarithm of fold-change values for the Causton, Gasch and Iwahashi datasets. For each of the three sets of data, the standard deviation was determined. A threshold was defined by multiplying the standard deviation by a constant, and this threshold was used to determine which genes were considered as significantly induced or repressed in each condition. Genes whose fold change was higher than the threshold were considered induced; genes whose fold change was lower than 1 divided by the threshold were considered repressed. For each condition, a set of induced genes and a set of repressed genes were constructed. Table For three particular conditions (one from each dataset), we re-assigned gene expression values randomly to all genes of the experiment. We then processed these random expression data using the same procedures as described above. The resulting sets of random induced/repressed genes were compared to elementary modes using BlastSets in the same way as real expression data.P value). As an example, one can check if the genes found co-expressed in an experiment correspond to a set containing proteins involved in the same pathway.BlastSets is a bioinformatics tool that enables the integration of various biological data. This tool uses a standard representation for all types of data: data are structured in collections of sets of genes or proteins. Each collection corresponds to a biological source of information, and sets are composed of genes that share a similar property or value . The sets stored in the BlastSets database can be compared to each other or to submitted custom sets. To evaluate the similarity between two sets, their composition in terms of genes/proteins is compared, and the hypergeometric distribution is used to decide if the number of genes in common between the two compared sets is statistically significant  and KEGG_EM_2 (EM2) in BlastSets. EM1 is a collection of single elementary modes, that is, enzymes involved in a given elementary mode are gathered in a set labeled by the name of the mode. EM2 is a collection of pairs of elementary modes: all enzymes involved in two elementary modes that are connected through a common external link form one set in EM2. These two collections of sets of elementary modes are stored in the BlastSets database, and can be queried against user-submitted data via the BlastSets website .Sets of induced and repressed genes in various stress conditions were compared to elementary modes using BlastSets, and lists of elementary modes found to be similar to the submitted sets of induced/repressed genes were obtained. A Perl script was developed to analyze these results and, thus, make it possible to reconstruct the chain of elementary modes that have been activated or repressed in response to each stress condition.We retrieved the elementary modes (single or pair) that had the highest similarity with the set of induced/repressed genes, called the 'best hit'. First, if the best hit was a single elementary mode, we browsed subsequent hits until we found a pair of elementary modes containing this best hit. Second, once this 'best elementary mode pair' had been found, the rest of the list was browsed in order to find further pairs of elementary modes that were connected to the best pair, that is, pairs of elementary modes having one mode in common with the best elementary mode pair. Third, we could display a chain of pairs of elementary modes that defines the backbone of the metabolic response. If the best hit was a pair of elementary modes, only the second and third steps were performed. Among the elementary modes that could be added to the backbone, we removed all those that were composed of less than three enzymes to ensure that they were significant enough and to avoid the inclusion of short modes that are not specific to a single pathway.Using the files containing BlastSets results with EM2, we constructed a matrix representing the usage of elementary modes in response to the different stresses, each row corresponding to an elementary mode and each column corresponding to a stress condition. In each element of the matrix, 1 was entered if the elementary mode was identified in response to the stress, 0 if it was not. A program was developed to compute, for each pair of pathways, the number of conditions where at least one pair of elementary modes spanning both pathways was found to be induced.In Figure Saccharomyces cerevisiae. Additional data file The following additional data are available with the online version of this paper. Additional data file is a list of elementary modes for Saccharomyces cerevisiae.Elementary modes for Click here for fileInduced and repressed metabolic backbones for all stress conditions.Click here for fileClustering tree of stress conditions.Click here for file