The model facultative aerobe Escherichia coli, the microbial cell factory par excellence, has elaborate sensing and signal transduction mechanisms that respond to the availability of electron acceptors and alternative carbon sources in the surrounding environment. In particular, the ArcBA and CreBC two-component signal transduction systems are largely responsible for the metabolic regulation of redox control in response to O2 availability and carbon source utilization, respectively. Significant advances in the understanding of the biochemical, genetic, and physiological duties of these regulatory systems have been achieved in recent years. This situation allowed to rationally-design novel engineering approaches that ensure optimal carbon and energy flows within central metabolism, as well as to manipulate redox homeostasis, in order to optimize the production of industrially-relevant metabolites. In particular, metabolic flux analysis provided new clues to understand the metabolic regulation mediated by the ArcBA and CreBC systems. Genetic manipulation of these regulators proved useful for designing microbial cells factories tailored for the synthesis of reduced biochemicals with added value, such as poly(3-hydroxybutyrate), under conditions with restricted O2 supply. This network-wide strategy is in contrast with traditional metabolic engineering approaches, that entail direct modification of the pathway(s) at stake, and opens new avenues for the targeted modulation of central catabolic pathways at the transcriptional level.Bioprocesses conducted under conditions with restricted O E. coli remains as the microbial cell factory (MCF) par excellence, mainly because this well- characterized enterobacterium can easily and rapidly grow on cheap substrates and can be modified as desired through a broad variety of molecular tools. As fermentation technologies designed for the production of bulk bioproducts need further innovation to became economically and environmentally sound , the environmental performance of PHAs equals that of petrochemical polymers. This situation clearly calls for alternative bioprocesses and biocatalysts to meet the technical and environmental requirements of a sound production process.The design and development of efficient MCFs for PHA production received much attention in the last few decades. Although different microorganisms have been exploited for polymer production, catalyst  as it docatalyst . The maiE. coli arc mutants are unregulated for respiration under micro-oxic conditions. Some enzymes of the TCA cycle are not repressed and the pool of reducing equivalents is elevated, and thus available to be funneled into reduced biochemicals. The PHB biosynthetic genes from Azotobacter sp. strain FA8  circumventing overflow metabolism by re-directing central metabolic fluxes and preserving sufficient precursors for amino acids synthesis, the actual building blocks of proteins.Acetate formation in organism , 97, 98.organism . This, iorganism . These aE. coli. Vemuri et al.  enhancement in the rate of substrate uptake, [ii] reduction of flux(es) to undesirable by-products, and enhancement of precursors and cofactor availability, [iii] introduction of relevant heterologous pathway(s) and optimization of the activity of its constituent enzymes, and [iv] export of the product to the extracellular medium in order to shift the concentration equilibrium towards product formation.In the first place, genome-scale and specific metabolic models are starting to play an important role in deciphering the cellular and metabolic characteristics of microbial cells n silico , 110. ThThis exciting and extremely dynamic scenario will certainly lead to the development of better strategies to manipulate central and peripheral pathways to enhance the production of reduced biochemicals and other molecules of industrial interest.Click here for additional data file.


Geobacter sulfurreducens PCA with a fixed acetate uptake rate. Simulation results indicated the fluxes of reactions directing acetate towards dissimilation to generate electrons increased under the suboptimal growth condition, resulting in an increase in the respiration rate and a decrease in the growth rate. The results revealed the competitive relationship between oxidative respiration and cell growth during the metabolism of microbe current generation. The results helped us quantitatively understand why microbes growing slowly have the potential to make good use of fuel in MFCs. At the same time, slow growth does not necessarily result in speedy respiration. Alternative respirations may exist under the same growth state due to redundant pathways in the metabolic network. The big difference between the maximum and minimum respiration mainly results from the total formate secretion. With iterative flux variability analysis, a relatively ideal model of variant of G. sulfurreducens PCA was reconstructed by deleting several enzymes in the wild model, which could reach simultaneous suboptimal growth and maximum respiration. Under this ideal condition, flux towards extracellular electron transfer rather than for biosynthesis is beneficial for the conversion of organic matter to electricity without large accumulations of biomass and electricigens may maximize utilization of limited fuel. Our simulations will provide an insight into the enhanced current-generating mechanism and identify theoretical range of respiration rates for guiding strain improvement in MFCs.Microbial fuel cells (MFCs) are a class of ideal technologies that function via anaerobic respiration of electricigens, which bring current generation and environmental restoration together. An in-depth understanding of microbial metabolism is of great importance in engineering microbes to further improve their respiration. We employed flux balance analysis and selected Fe(iii) as a substitute for the electrode to simulate current-generating metabolism of  Geobacter sulfurreducens have been utilized to enhance the capacity for current production [Microbial Fuel Cells ,2 are deoduction .2、Mn4+. Unsurprisingly, direct correlation exists among current production, extracellular electron transfer and oxidative dissimilation of electricigens to generate electrons. The stronger oxidative dissimilation is, the faster extracellular electron transfer is, and consequently more power is produced.The activity of electricigens ,5 that pUnder natural conditions where there are short of evolutionary pressure on electricigens to oxidize electron donors rapidly, electricigens tend to maximize growth companied by slow electron generation. Anodes of MFCs are likely to favor slow-growing, also called suboptimal-growing microbes, rather than microbes capable of oxidizing a great deal of fuel ,5,6. Exein silico models of biological systems, and subsequently simulating the cellular processes employing these computational models, providing biologists with thorough insights into cellular behaviors [Cellular metabolism state is governed by metabolic flux distribution. Investigating the metabolic process quantitatively at a system level enables strain improvement ,12. Althehaviors ,14.Geobacter species have the ability to oxidize completely a variety of organic compounds to carbon dioxide under anaerobic conditions coupled with electron transfer outside the cell over long distances [in silico algorithms to simulate different current-generating metabolic states with a fixed fuel uptake rate, and reconstructed a relatively ideal model of variant of G. sulfurreducens PCA. The results enabled us to quantitatively understand the enhanced current-generating mechanism and develop electricigens that utilize electron donors efficiently.istances ,16. The in silico model of G. sulfurreducens PCA, a model reconstructed in 2005 and updated in 2008 using the SimPheny  platform was applied in the following simulations [G. sulfurreducens secretory products, 64 reactions corresponding to transport between the cytoplasm and extracellular space were added. The rest were exchange reactions, which allowed extracellular metabolites to enter into or end products to be secreted from the cell.The ulations ,18. The 3+ as the electron acceptor. Biomass reaction agg_GS13m_2, which represents growth-associated biosynthetic demands, contains biomass components and essential growth metabolites, and the flux through it is equal to the strain growth rate vgr. G. sulfurreducens could completely oxidize acetate to carbon dioxide under anaerobic conditions, together with electron transfer outside the cell to reduce Fe3+. The flux through exchange reaction EX_fe3(e) is equal to the rate of Fe3+ reduction, also named as respiration rate vres, and its value indicates the speed of extracellular electron transfer. Both the anode of MFC systems and Fe3+ are insoluble electron acceptors that can be reduced by G. sulfurreducens with a similar mechanism, hence, Fe3+ was taken as the electron acceptor to simulate cellular metabolism in MFCs [The metabolic reconstruction used electron donor acetate as carbon source and energy source simultaneously, with soluble fumarate or insoluble Fe in MFCs .m metabolites and n reactions is given below:Flux Balance Analysis (FBA) is a linear programming approach, an approach assuming the metabolic network at steady state to ensure the rate at which every metabolite is consumed is equal to that of being produced. FBA allows predictive computation of the cellular flux distribution that maximizes objective function with a particular meaning under certain cultivation conditions . The forSi,j is the stoichiometric coefficient of metabolite i in reactionj, vj  is the flux through reactionj, lbj is the minimum allowable flux through reactionj, ubj is the maximum allowable flux through reactionj, and Parameters description: vgr asm−1, and the following two formulations was applied to examine alternate respiration states vres：Flux Variability Analysis (FVA)  has the G. sulfurreducens can use acetate as carbon source and energy source. When simulating optimal growth using Flux Balance Analysis (FBA) [vuptake was set as a fixed value, and the rate of biomass reaction agg_GS13m_2, vgr, was defined as objective function with the need to be maximized.is (FBA) , acetatevgr =k﹒vgrmax,k∈   could be employed to describe the suboptimal growth state, where vgrmax is the optimal value for growth. Flux Variability Analysis (FVA) [If microbes grow suboptimally, the growth rate is less than that of optimal growth. The equation is (FVA)  was carris (FVA) .G. sulfurreducens [In order to examine the impact of growth state on the current production capacity, we simulated optimal and suboptimal growth by constraining acetate uptake rate at 13.630mmol/g.dw/h. We confined our attention to central metabolism see 1, a coureducens ,22. Simuvgr and vres of vgr, vres would decreased. The result indicated that there may be a competitive relationship between oxidative respiration and fuel assimilation into biomass. So microbes growing suboptimally could make good use of fuel for current generation in MFC systems.From the conlums of After being transported into the cell, acetate was activated to acetyl-CoA via acetate kinase followed by acetyl-CoA transferase . Over 90G. sulfurreducens was completely dependent on electrogenic electron transport. Electrons carried in reducing equivalents were transferred into the inner membrane via NADH or NADPH dehydrogenase, and protons were pumped out of the cytoplasm for ATP synthesis via ATP synthase. Although it was well known that c-type cytochromes were important for extracellular electron transfer, a specific electron transfer chain for electron transfer out of the inner membrane to Fe3+ or electrodes had not been determined for G. sulfurreducens [G. sulfurreducens, a simplified extracellular electron transfer process was developed by NADH or NADPH dehydrogenase, cytochrome-c reductase and then Fe3+ reductase.ATP production of reducens . In the Simulation results presented here indicated that when microbes grew suboptimally, more fuel was directed to the TCA cycle to generate electron and energy, resulting in less being used for biomass synthesis compared with that optimal growth. Therefore microbes growing suboptimally may make good use of fuel for current generation in MFC systems.Computational results presented in the above section indicated that suboptimal growth was good for the improvement of respiration rate. However, alternative respiratory states may exist under the same growth state as a result of the inherent redundancies built into metabolic networks . FVA was-1, a value that was 10% of that of optimal growth, and we thought this may be the best circumstance for electron extraction from the substrate. G. sulfurreducens, while The genome-scale flux distributions were examined to illustrate the respiratory rate variance between the maximum and minimum respiration corresponding to all simulated growth states (from 0% to 90%). Excluding the reactions whose flux variances were less than 1% of their original flux values, those reactions whose flux distributions existed remarkable variance  included five metabolic subsystems: central metabolism, transport, energy metabolism, amino acid metabolism, and exchange. Because of the same reason leading to the respiratory rate variance, we only paid attention to the suboptimal growth rate of 0.006hG. sulfurreducens, had three pathways . With regard to maximum respiration, only two pathways were activated and no formate was produced. In contrast, regarding minimum respiration, three pathways were all activated and formate was produced at a rate of 22.907mmol/g.dw/h. Formate was chosen to be secreted as an end product outside the cell totally via proton symport instead of oxidation via formate dehydrogenase, resulting in the loss of electrons. At the same time, less acetate (13.544 vs 14.309 mmol/g.dw/h) was directed toward the TCA cycle, from which the majority of electrons were derived for G. sulfurreducens [Interconversion from acetyl-CoA to pyruvate, the first step of assimilation in reducens . The con3+ reductase) and ATP synthetic reaction (6.333 vs 0.990 mmol/g.dw/h) also increased. Unexpectedly the flux through NADPH dehydrogenase reaction was predicted to decrease (9.619 vs 13.438 mmol/g.dw/h). Conversion of pyruvate to acetyl-CoA, one of the redundancies in central metabolism of G. sulfurreducens [Cell need for energy was crucial to metabolism regulation. Increased cellular energy consumption inspired reducing equivalent-producing reactions to increase their rates to meet more energy demand . We extrreducens , contribreducens .-1, a value that was 90% of that of optimal growth. The flux difference through pyruvate dehydrogenase reaction was 2.389 mmol/g.dw/h. (999.406 vs 997.017 mmol/g.dw/h) between maximum and minimum respiration. The value increased to 22.142 mmol/g.dw/h. (999.180 vs 977.038 mmol/g.dw/h) at a growth rate of 0.006 h-1. It was clear that the discrepancy between maximum and minimum respiration depended on the flux difference through reducing equivalent-producing reactions.The discrepancy between maximum and minimum respiration increased along with the decrease of growth rate 1. We haIn the above section, an ideal current-generating metabolism state was given. The ideal state consisted of simultaneous suboptimal growth and maximum respiration. Under this ideal condition, flux towards extracellular electron transfer rather than for biosynthesis is beneficial for the conversion of organic matter to electricity without large accumulations of biomass that might otherwise plug specific parts of an aquifer. And at the same time, electricigens may maximize utilization of limited fuel.in silico metabolic model predicted that when G. sulfurreducens grew suboptimally at a rate of 0.006h-1, minimum respiratory rate was 62.177mmol/g.dw/h and caused by formate secretion. We deleted formate enzyme GSU0234 by constraining the flux through the reaction catalyzed by it at 0 mmol/g.dw/h, and after FVA, minimum respiratory rate increased to 75.267 mmol/g.dw/h, which was caused by pyruvate secretion. Then enzymes for pyruvate, L-valine, L-leucine, citrate, L-isoleucine, L-proline, L-lysine, L-cysteine, h2 and n2 were determined through iterative FVA and were deleted one by one. A relatively ideal model of variant for G. sulfurreducens strain was reconstructed finally by modifying the wild metabolic model in this way Click here for additional data file.Table S2This table provides information on all reducing equivalent-producing reactions.(XLS)Click here for additional data file.Table S3This table provides information on the full flux range through every reaction.(XLS)Click here for additional data file.


S cerevisiae and E coli in order to assess if there are relationships between systemic variables that are present in both organisms. Strong correlations between the total flux per unit of substrate and the ATP turnover rate per unit of substrate and between the growth rate divided by the total flux and the total flux per unit of substrate were observed for both organisms. We also observed that the common assumption of biomass yield optimization is not consistent with the experiments.The search for optimization principles in microbial metabolism, such as biomass or ATP yields or growth rate optimization, has attracted substantial research efforts in the recent years. Here we use the results of C13 labeling experiments together with genome scale metabolic networks of  The recent development of sequencing technologies and bioinformatics has made possible the reconstruction of genome-scale metabolic networks for different microorganismsGenome-scale metabolic networks contain only stoichiometric information. In order to use these networks as tools to predict in-vivo metabolic flux distributions and how these flux distributions change after genetic manipulations such as gene deletions, other assumptions about the operation principles of metabolism are necessary.S cerevisiae and E coli, with the aim of identifying possible additional criteria (objective functions or extra constrains), that could be used in order to improve the predictive power of genome-scale metabolic models.In this paper we use experimental flux distributions measured in two different microorganisms, E. coli seem to be more consistent with the maximization of ATP production rate per flux unitE. coli's central carbon metabolism, it has been argued that the experimental flux distributions of E. coli and other bacterial species are close to the optimality Pareto surface that defines the trade-offs between maximal biomass yield, maximal ATP production yield and minimal sum of metabolic fluxes9Linear objective functions  have been extensively used to compute metabolic flux distributions65Another limitation is the fact that the experimental techniques available to measure metabolic flux distributions, which are based on using C13 labeled substrates121314It is a common practice to infer the flux distributions in a larger network from a reduced number of measured fluxes by first computing one solution compatible with the constraints in the larger network in which the set of measured fluxes is as close as possible to the measured valuesS cerevisiae and E coli), we aim to identify possible global operation principles that govern the metabolic flux distributions and that are general enough to be conserved between two microorganisms that are very far apart from an evolutionary perspective. The genome-scale metabolic models that we used are iTO980S cerevisiae and iJR904E coli. We have the experimental flux distributions of a reference S. cerevisiae strain and 36 deletion mutantsE coli strain, 3 deletion mutants and 6 strains evolved from the deletion mutants (2 evolved strains per mutant)16Using available C13 labeling experimental data for two of the best known eukaryotic and prokaryotic microorganisms , and what we computed are optimal estimators of these variables (See methods).S cerevisiae and E coli were observed, these relations corresponded to: a positive correlation with ordinate at the origin equal to zero between the ATP turnover rate and the NADH turnover rate (correlation coefficients of 0.953 for E coli and 0.938 for S cerevisiae), a positive correlation with negative ordinate a the origin between the total flux per unit of substrate and the ATP turnover rate per unit of substrate (correlation coefficients of 0.879 for E coli and 0.980 for S cerevisiae) and a negative correlation with positive ordinate at the origin between the growth rate divided by the total flux and the total flux per unit of substrate (correlation coefficients of (0.919 for E coli and 0.909 for S cerevisiae).Here we computed the correlation coefficients between the estimators for the systemic variables in order to identify possible relations of the types: y = a + bx or z = a + bx + cy. Only 3 relations with correlation coefficients higher than 0.8 and conserved both for The slopes and the ordinates at the origin are different for each organism  but the We also aimed to test how consistent with the experimental evidence is the assumption of the existence of an objective function maximized by the metabolic network. As we have mentioned previously the main difficulty for assessing the suitability of an objective function by comparing predictions and experimental data is the fact that besides the objective function one or several constraints must be also imposed in order to predict metabolic flux distributions. The choice of constraints determines the output of the predictions as much as the objective functions. For example it is common to constrain the carbon source uptake rate, which in reality (given the fact that it changes a lot between different conditions and as a result of gene deletions) seems to be far from being constrained. In order to avoid any ad hoc assumption of constraints, we are not comparing predictions and experiments, but the experimental values for the wild type strain with the values of deletion strains. Deletion strains have a metabolic network which is a sub-network of the wild type . Therefore if an objective function exists its value should be smaller or equal in the strains with a deletion than in the wild type strain. A significance test was performed (see methods) in order to identify the number of mutant strains in which each of the considered systemic variables was significantly higher or lower than in the wild type strain. The systemic variables that are not significantly higher in any of the mutant strains are possible objective functions. We see in We also tested how far the biomass yields are from the assumption of anabolic optimality. By anabolic optimality we mean that the produced ATP is optimally allocated to biomass production. To test this we constrained for each strain the glucose and the oxygen consumption rates to their experimental values and compared the predicted biomass production rate with its experimental value. The real strains seemed to be well below the anabolic optimality .In summary, using two genome scale metabolic models of high quality, for a prokaryotic and a eukaryotic organism, and high quality C13 labeling experimental data16Based on our results, we suggest the utilization of the identified relationships as extra constrains in the genome-scale metabolic models. This is likely to lead to more realistic predictions of the metabolic flux distribution, at least in the two organisms we have analyzed. Using objective functions such as biomass yield does not seem to be a good option, and other objective functions , such asS, which contains the stoichiometric coefficients of each metabolite (columns) in each reaction (rows) of the network. If each internal metabolite is in steady state, the following relationship must be satisfied. A metabolic network can be described by its stoichiometric matrix C is defined by the following condition: if x and y are vectors satisfying the previous constrains, any vector defined as αx + (1 − α)y will also satisfy the mentioned constraints.If the irreversibility constraints (the flux in the irreversible reactions has to be zero or positive) are added to z of the convex set C can only be expressed as z = αx + (1 − α)y  if z = x or z = y then z belongs to the convex basis of C.If an element x of C can be expressed as a linear combination of the elements of the convex basis of C {iz}, so that i being positive and their sum is equal to 1.Any element i, n elements, the most likely flux distribution will be: It is possible to define the Shannon's information entropy of the set of values αIn a genome scale metabolic network with thousands of reactions, the number of elements of the convex basis follows a combinatorial explosion that makes unfeasible its complete enumeration. It was recently shown by Kelk and co-workers, than the high number of elements in a convex basis can be explained by a small number of alternative flux values in several metabolic sub-networks, which by a mechanism of combinatorial explosion; give rise to millions of corners. Based on this property we can conclude that sampling a limited number of corners in the solution space (3000 in our case) is enough to explore all the rank of values that each reaction can take. The sampling was carried out as described in a previous paperWe are also interested in knowing in how many mutants a systemic variable was higher or lower than in the wild type strain. To do that we compared the first sample of the wild type with all the samples of the mutant and computed how many times the mutant had a higher (or lower) value for that systemic variable. Then we repeated the same for the second sample of the wild type and continued doing the same till the last sample of the wild type. If in 95% or more of the 3000 × 3000 comparisons the mutant had a higher or lower value for the systemic variable, we considered that systemic variable as significantly higher or lower in the mutant than in the wild type.S.B. did all the work.S1S2


Zymomonas mobilis. Combined use of Flux Balance, Elementary Flux Mode, and thermodynamic analysis of its central metabolism, together with dynamic modeling of the core catabolic pathways, can help to design novel substrate and product pathways by systematically analyzing the solution space for metabolic engineering, and yields insights into the function of metabolic network, hardly achievable without applying modeling tools.Mathematical modeling of metabolism is essential for rational metabolic engineering. The present work focuses on several types of modeling approach to quantitative understanding of central metabolic network and energetics in the bioethanol-producing bacterium  Zymomonas mobilis, a member of the family of Sphingomonadaceae, is an unusual facultatively anaerobic Gram-negative bacterium, which has a very efficient homoethanol fermentation pathway. High ethanol yields, outstanding ethanol productivity  glycolytic pathway is not operating in this bacterium. Absence of the EMP pathway has been confirmed by [1-13C]glucose experiments  could not be identified in the sequenced genomes.The TCA cycle is truncated, and consists of two branches, leading to 2-oxoglutarate and fumarate as the end products . The gen mobilis . Also, t mobilis . The act mobilis . SubsequZ. mobilis metabolism that differs from that typically found in other bacteria. Z. mobilis is one of the few known bacteria in which both NADH and NADPH can serve as electron donors for the respiratory type II NADH dehydrogenase . This was done by Flux Balance Analysis approach, using the COBRA Toolbox H, generated by the glycerolphosphate dehydrogenase reaction could be oxidized by the respiratory chain. If succinate is the desired product H withdrawal by the respiratory chain . Apparently, while the estimated overall stoichiometry of aerobic glycerol conversions is correct, thermodynamic analysis suggests the need to search for alternative reaction sequences to avoid excessive intracellular accumulation of metabolites.The pathway from glycerol to glyceraldehyde-3-phosphate via phosphorylation and following oxidation and isomerization steps is presented in biochemistry textbooks as the pathway of glycerol catabolism after breakdown of triacylglycerols in higher animals and humans . Though feasible from the stoichiometric point of view, it reveals problems when subjected to thermodynamic analysis. The equilibrium of the glycerolphosphate dehydrogenase reaction appears to be shifted very much toward formation of glycerol phosphate. E. coli, EMF-based mutation analysis helped to eliminate catabolite repression and to increase carbon flux toward the target product ethanol (A metabolic network can function according to many different pathway options. Elementary flux mode (EFM) analysis has emerged as a systems biological tool that dissects a metabolic network into its basic building blocks, the EFMs , 2002. A ethanol .Z. mobilis, which expresses heterologous gnd and enzymes for pentose conversion, using the metabolic modeling package ScrumPy  that have considerable interest for study of aerobic energy-coupling in this bacterium. With both monosaccharides, knocking out edd (encoding 6-phosphogluconate dehydratase), and overexpressing heterologous gnd (encoding 6-phosphogluconate dehydrogenase), would lead to generation of additional NAD(P)H and CO2 in the pentose phosphate pathway, while lowering the ethanol yield. Yet, most importantly, decrease of the ethanol yield would not be accompanied by accumulation of acetaldehyde and acetoin.By decomposing a network of highly interconnected reactions, the EFM analysis may reveal unexpected flux options. Recently we applied EFM analysis to the interaction between the ED, pentose phosphate pathway and respiratory chain in an engineered  ScrumPy . We wereZ. mobilis aerobic metabolism so that its electron transport chain would receive more reducing equivalents without accumulation of inhibitory byproducts. Strains with such metabolic modifications might be very useful for study of the mechanisms underlying the uncoupled mode of oxidative phosphorylation in this bacterium.Thus, a simple EFM analysis suggests how to modify Z. mobilis physiology and genetics, little has been done so far to combine the accumulated knowledge in a form of kinetic model of central metabolism that would be comparable to the existing models for E. coli and yeast, and could be used to develop efficient metabolic engineering strategies . A kinetic model reported by Z. mobilis ED glycolysis. Providing predictions for optimization of expression levels of the heterologous genes, this study contributed to strategies for maximizing xylose conversion to ethanol. However, the authors assumed constant intracellular concentrations of all adenylate cofactors. Since the ED pathway itself is a major player in ANP and NAD(P)(H) turnover, this might lead to erroneous conclusions on the pathway kinetics and restrict the range of model application. The recent kinetic model by Despite the diverse studies of Z. mobilis growth rate (Z. mobilis must be controlled at some point(s) outside the ED pathway itself. The negative effects of overexpression apparently did not result from intrinsically negative flux control coefficients of the ED enzymes, but were attributable to the protein burden effect  and both alcohol dehydrogenase isoenzyme(adhA and adhB) overexpression, in accordance with previous experimental observations, predicted little or no increase of glycolytic flux (pdc) reaction suggested that overexpression of this enzyme by more than 3-fold, might lead to an increase of glycolytic flux of almost 23% , has the potential to increase the glycolytic flux by up to 25%  might disturb the intracellular ATP homeostasis, with successive suspension of glycolysis . Co-response analysis indicates (Obviously, another option would be to raise ATP dissipation. That could be done by overexpression of the Hndicates  that, atZ. mobilis metabolism has been subject to extensive research, and genome sequence data for several strains are now also available, it is only quite recently that modeling of its central metabolic network has started to gain momentum. These latest results of modeling Z. mobilis illustrate the relevance of combined stoichiometric, thermodynamic and kinetic analysis of central metabolism at different scales for microorganisms producing biorenewables. Concerted application of structural and dynamic modeling will help to identify targets for future metabolic engineering in a systematic manner, and provide novel insights into the biotechnological potential of this bacterium.Although All authors have equally contributed to the manuscript and have accepted the final version to be published.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


E. coli’s energy metabolism. We first constructed a metabolic weighted network using the rates of free energy consumption within metabolic reactions as the weights. We then analyzed some structural characters of the metabolic weighted network that we constructed. We found that the distribution of the weight values was uneven, that most of the weight values were zero while reactions with abstract large weight values were rare and that the relationship between w  and v  was not of linear correlation. At last, we have done some research on the equilibrium of free energy for the energy metabolism system of E. coli. We found that E. coli.Genome-scale models of metabolism have only been analyzed with the constraint-based modelling philosophy and there have been several genome-scale gene-protein-reaction models. But research on the modelling for energy metabolism of organisms just began in recent years and research on metabolic weighted complex network are rare in literature. We have made three research based on the complete model of  E. coliE. coli_iAF1260, and apart from 304 EX_ & DM_ reactions (The text ‘EX_’ denotes an exchange reaction for a metabolite that can enter or leave the extra-cellular compartment. ‘DM_’ reactions are similar and signify compounds that the degradation pathway is unknown), the reconstructed reaction number is 2077 E. coli_iAF1260 Since various ‘Omics’ datasets are becoming available, biology has transited from a data-poor to a data-rich environment. Systems biology has become a rapidly growing field as well E. coli’s energy metabolism. First, we construct a metabolic weighted network using the rates of free energy consumption in metabolic reactions as the network weights. Then we did some research on some structural characters of the metabolic weighted network we constructed. At last, we did some research on the equilibrium of free energy for the energy metabolism system of E. coli.In this paper, we have done three research on the complete model of E. coli _iAF1260. Then we construct the model using the rates of free energy consumption in metabolic reactions as the network weights. At first, we draw the metabolic unweighted network of E. coli; we then calculate the flux distribution of E. coli_iAF1260; the third, we calculate the weights of the metabolic weighted network of E. coli that we will construct; at last, we calculated the input and output of free energy about E. coli.Before constructing the metabolic weighted network, we complement the remained unknown S, is the center-piece of a mathematical representation of genome-scale metabolic networks. It represents each reaction as a column and each metabolite as a row, where each numerical element is the corresponding stoichiometric coefficient. In the calculation of The stoichiometric matrix, S. For a certain compound even in different compartments, the free energy change of formation is the same. In these 1039 metabolites, Gibbs free energy of formation of 872 compounds (84%) can be estimated by group contribution method E. coli_iAF1260. There are 2381 reactions in the iAF1260 model, so there are 2381 columns in the stoichiometric matrix S. For those 2077 reconstructed reactions, Gibbs free energy change of 1996 reactions (96%) can be estimated by group contribution method E. coli_iAF1260.There are 1039 metabolites in the iAF1260 model, and if distinguishing the different compartments in the cell, i.e. [c] (cytoplasm), [e] (extracellular space), and [p] (periplasm), the total number of metabolites in the model is 1668. So there are 1668 rows in the stoichiometric matrix From the equation (1) and (2) of the paper i in the reaction, and m is the number of species involved in the reaction; j; nj is the number of instances of group j in the molecular structure; Ngr is the number of groups for which i-th species. Where From those 167 compounds with unknown X (with dimension 167×1) to indicate the values of those unknown P (with dimension 1668×1) to indicate the values of E. coli_iAF1260; Let the value of P(i), i-th sub-variable of P, be 0, if the i-th compound is unknown; Use vector F (with dimension 2381×1) to indicate the values of E. coli_iAF1260. From Now we infer these unknown T (with dimension 1668×167) is the transfer matrix from vector X to the vector indicating the values of E. coli_iAF1260; S (with dimension 1668×2381) is the stochiomatrix of E. coli_iAF1260, ST is its transpose. Further we obtainWhere Write it asWhere X includes the sub-variables of X and while the remained 1752 rows of X. So we can use these 244 rows from The dimension of vector Where the dimensions of matrix X0) of F0 of Conversely, we now use the obtained F0 are known, so we can now obtain the Now all the sub-variables of The 1M reference state for the metabolite concentrations on which R is the universal gas constant; T is the temperature assumed to be 298 K; ni is the stoichiometric coefficient of compound i in the reaction (ni is negative for reactants and positive for products); PR is the set of products and reactants in this reaction. Note also that for H2, we should substitute 0.000034 for 0.001; For O2, we should substitute 0.000055 for 0.001; For H2O and H+, we should not include these compounds in the concentration portion of the calculation at all Where E. coli_iAF1260 are given in Ref. E. coli_iAF1260 can be downloaded from the supplementary information of Ref. in silico model that we used is E. coli_iAF1260_flux1.xml. SBML file properties are also given in Ref. rxns, mets, and genes are respectively 2382, 1668, 1261. The minimal media of in silico model is also an important aspect. The computational minimal media of E. coli_iAF1260 is also included in the supplementary information of Ref. E. colirxnGeneMat matrix, which is a matrix with as many rows as there are reactions in the model and as many columns as there are genes in the model. The ith row and jth column contains a one if the jth gene in genes is associated with the ith reaction in rxns and zero otherwise. The simulation condition (the nutrients and the uptake rates of the nutrients) of this paper is the same as in the file.The general features of E. coli_iAF1260. The computational method we use is flux balance analysis (FBA) −1 and defined in the biomass objective function), the maximum possible growth rate of the cells can be predicted in silico.We now calculate the flux distribution of Subject toS is the stoichiometric matrix, and iα and iβ define the bounds through each reaction iv. The flux range was set arbitrarily high for all internal reactions so that no internal reaction restricted the network, with the exception of irreversible reactions, which have a minimum flux of zero. The inputs to the system were restricted to a minimal media. We use the COBRA toolbox E. coli_iAF1260 is illustrated in Where E. coli can be obtained. .By the newest group contribution method, Now we can construct a metabolic weighted network. There is not a standard manner of determining reaction edge weights and Almaas has used flux value as the weight of metabolic network i-th edge (i.e. reaction) of metabolic network, i-th reaction and i-th reaction.Where For an open system at nonequilibrium steady state, from the theory of system science, its free energy rate dissipated by the system, We also distinguish the free energy rate E. coli, we have known all of its reactions, the values of corresponding For i-th reaction in i-th reaction inWhere j-the reaction in j-th reaction in Where Table S1) and Gibbs free energy of formation (Table S2) which are previously unknown for E. coli_iAF1260. We add our computed E. coli_iAF1260. The entire E. coli_iAF1260 are consistent with the known Table S2 and ). So we conclude that our computed With our method, we obtain Gibbs free energy change . We can easily find that the distribution of the weight values is uneven and that most of the weight values are near zero while reactions with abstract large weight values are rare, illustrating in E. coli. We can calculate the weight values of the metabolic network of w scopes, corresponding reaction number within these scopes and these reaction names, and we call these reactions the highly-dissipative reactions in the energy metabolism of E. coli. We examined into these large weights and found that they were the result of joint action from flux and free energy dissipation, while their E. coli, the deletion or loss of one gene will not result in death, and this may be due to the result of evolution.w, v), 2077 data pairs in total. Many data pairs are superposition and locate at the same place. From the diagram, we can easily find that the relationship between w and v is not of linear correlation. So we can’t say that a reaction with high flux has a corresponding high weight and vice verse, and in other words, we can’t say that a reaction with high flux will dissipate more free energy. In fact, many different flux values correspond to the same weight value.By comparing the distribution of the weight values Fig. 2 wAlthough there is no consistency between flux values and energetic weights, energetic weights are very useful and important in determine the distribution of reaction fluxes. We have defined energetic weights as the rates of free energy dissipations, i.e. the multiplying product of the reaction fluxes and the free energy changes of reactions. Free energy dissipation can be regarded as the counterpart of entropy increase. Based on maximum entropy production principle (MEPP), the authors of the paper have developed a method to improve the prediction accuracy of flux balance analysis E. coli_iAF1260. The values of There are 2077 reactions in E. coli, we found that it is chemical process other than physical process that plays a great role in the dissipation of free energy in cells. By these research and to a certain extend, we can understand more about the energy metabolism of E. coli.In this paper, we constructed a metabolic weighted network by using the rates of free energy consumption within metabolic reactions as the network weights. We found several important and interesting results: 1) the distribution of the weight values was uneven; 2) the relationship between w  and v  was not of linear correlation; 3) by analyzing of the free-energy equilibrium for the energy metabolism system of In our next step, we will conduct a similar type of analysis for different organisms using some of the other readily available constraint-based models and run the baseline simulations for growth in different carbon substrate environments. We will also do FBA analysis including energetic weighting as an additional constraint to bias flux distributions.Table S1Standard Gibbs free energy formation (XLS)Click here for additional data file.Table S2Gibbs free energy change ((XLS)Click here for additional data file.


This definition overcomes limitations of purely graph based analyses of metabolism by exploiting the functional links between reactions. We call a metabolic network viable in a given chemical environment if it can synthesize all of an organism's biomass compounds from nutrients in this environment. An organism's metabolism is highly versatile if it can sustain life in many different chemical environments. We here ask whether versatility affects the modularity of metabolic networks.The ubiquity of modules in biological networks may result from an evolutionary benefit of a modular organization. For instance, modularity may increase the rate of adaptive evolution, because modules can be easily combined into new arrangements that may benefit their carrier. Conversely, modularity may emerge as a by-product of some trait. We here ask whether this last scenario may play a role in genome-scale metabolic networks that need to sustain life in one or more chemical environments. For such networks, we define a network module as a maximal set of reactions that are fully coupled, in silico metabolic networks that differ in their versatility. We find that highly versatile networks are also highly modular. They contain more modules and more reactions that are organized into modules. Most or all reactions in a module are associated with the same biochemical pathways. Modules that arise in highly versatile networks generally involve reactions that process nutrients or closely related chemicals. We also observe that the metabolism of E. coli is significantly more modular than even our most versatile networks.Using recently developed techniques to randomly sample large numbers of viable metabolic networks from a vast space of metabolic networks, we use flux balance analysis to study Our work shows that modularity in metabolic networks can be a by-product of functional constraints, e.g., the need to sustain life in multiple environments. This organizational principle is insensitive to the environments we consider and to the number of reactions in a metabolic network. Because we observe this principle not just in one or few biological networks, but in large random samples of networks, we propose that it may be a generic principle of metabolic network organization. The architectures of most multi-cellular organisms are strikingly modular. On the one hand, such modularity can be spatial. Organisms are partitioned into organs and tissues whose cells have specialized functions ,2, and wOn the other hand, modularity can be topological, as research of the last ten years has shown. Such modularity is evident in biological networks such as protein-protein interaction networks ,4, transrate of adaptation [The prevalence of modularity  in living systems might have several ultimate evolutionary origins  This approach  allows uFor our study, we use genome-scale metabolic network modeling. The set of chemical reactions that can take place in an organism and their associated metabolites define the organism's metabolic network. Each reaction is typically catalyzed by an enzyme that allows the transformation of substrate molecules into product molecules. With the advent of genome-scale metabolic network modeling -34, it hN possible reactions, any genotype can be represented by a string of N bits, G =  as illustrated in Additional File i is encoded in the organism's genome, then i b= 1, while i b= 0 otherwise. In this framework, the space of all metabolic genotypes contains 2N elements. Following previous work [An organism's set of enzyme coding genes, identified here with a list of reactions, can be viewed as a discretized binary metabolic genotype; for brevity we refer to it from here on as the organism's genotype or metabolic genotype. Specifically, given a total universe of ous work ,40, we hous work ,42. Thisviable. This leads us to define an organism's phenotype via its ability to synthesize biomass in a number of given chemical environments. Note that the mapping from genotype to phenotype in our approach is completely determined by the FBA framework. Previous research has shown that this map is highly degenerate, meaning that a huge number of genotypes will produce the same phenotype; indeed, many reactions in a metabolic network are typically non-essential and can be replaced by other reactions. Furthermore, genotypes of identical phenotype are such that small genotypic changes  connect these genotypes into a vast graph; we refer to this graph as a genotype network. A consequence of this connectivity property is that gradual evolution of genotypes is possible, while leaving the phenotype unchanged [If an organism can grow in a specific chemical environment (defined through the nutrients it contains), its metabolic network is able to produce all of its biomass precursors (see Methods); we then call the organism (and by extension its metabolic network) nchanged ,40. For nchanged . Such prnchanged -46, as wnchanged .To characterize metabolic networks of a given phenotype, we cannot examine all genotypes because of their astronomical number. Instead, we use a Markov Chain Monte Carlo (MCMC)  approachflux of matter through each reaction in a network. Doing so permits an analysis of modularity that is based on network function, not just topology. Here we take advantage of the notion of coupling between reaction fluxes to identify sets of reactions that form a metabolic module. Such sets have been referred to as reaction/enzyme subsets or correlated reaction sets or Fully Coupled Sets [The analysis of modularity in large graphs or networks is a mature field. Not surprisingly, multiple different measures of modularity have been developed ,49-54. Iled Sets -29. Hereled Sets ,29,57,58led Sets . By defiE. coli metabolic network; it includes reactions that are involved in cell envelope biosynthesis.The simplest possible FCS involves reactions in a linear biochemical pathway, arguably the most intuitive form of a functional module in biochemistry. However pathways with branches and cycles can also form FCSs . For illE. coli, we find that reactions in the same FCS typically belong to a common pathway. To quantify whether this property was statistically significant, we implemented the following test.We first asked how modules, as defined by FCSs, relate to conventional biochemical pathways, the classical functional modules of metabolism. To this end, we mapped reactions in many different FCSs onto biochemical pathways, as defined by standardized annotations ,59. We rE. coli. We observed that in most FCSs  all reactions belonged to the same pathway, and nearly 75 percent of FCSs had more than 75 percent of their reactions belonging in the same pathway. This strong association of reactions in an FCS with one pathway is not expected by chance alone, as a randomization test shows (P < 0.001). Thus, most of the FCSs in E. coli can be viewed as biochemical pathways or parts thereof.For each FCS, we identified the pathway annotation for all of its reactions. Because each reaction can be annotated as belonging to multiple pathways, we identified for each FCS the pathway annotation that is shared by most of its reactions. We defined the quantity Q as the fraction of reactions that are annotated as belonging to that pathway, and computed Q for each FCS in the metabolic network of in silico metabolic genotypes viable in all of the 89 carbon source environments we consider (see Methods). In this analysis, we observed that in most FCSs  all reactions belonged to the same pathway, and nearly 80 percent of FCSs had more than 80 percent of their reactions in the same pathway . Thus, both for E. coli and for our random samples, most FCSs can be viewed as biochemical pathways or as parts thereof. To illustrate such FCSs, Additional File The same analysis can be applied to random samples of metabolic networks with specific properties, as generated by our MCMC sampling procedure (see Methods). Specifically, we first identified FCSs from 1000 M and s. The first index is the number M of reactions in a network that belong to FCSs. Then we calculate the average <M>for a sample of networks generated by our MCMC procedure, where each network in the sample needs to be viable in a given set env Vof chemical environments (see Methods). We consider env Vas an index of environmental versatility for these metabolic networks. In our analysis, we study up to 89 minimal chemical environments that differ only in the sole carbon source they contain .We next asked quantitatively how network modularity is affected by environmental versatility. To answer this question, we defined two indices of network modularity, which we call M> depends on versatility envV, both on average (yellow dots), and for multiple different nested sets of carbon sources (symbols with different colors and shapes). The analysis is based on metabolic networks with the same number of reactions as E. coli [In Figure  E. coli . The datM - the number of reactions in FCSs - occur through an increase in the size of the FCSs, or through an increase in the number of FCSs, while their size remains constant? To address this question, we next studied the number of FCSs, which we denote by s, our second index of modularity. We applied the procedures we described earlier to the same genotypes as before, averaging now the number of modules (FCSs) rather than the number of reactions in these modules.As a network's versatility rises, does an increase in s>, for the same 1000 metabolic genotypes, the same choices of envV, and the same nested sets of environments as above. The figure shows that greater versatility leads to higher values of this modularity index. This holds for the averages over different nested sets (yellow dots), and also without averaging, i.e., for different nested sets of carbon sources (symbols of different shapes and colors). Both <M> and <s> show a monotonic increase with env Vbut with possible deviations from linearity.Figure n of reactions equaled that of the E. coli metabolic network, i.e., n = 831 [<M>, increases with versatility env Valso for networks with n = 500  and n = 700  reactions. The sole difference to the data of Figure <M>is beginning to level off as env Vreaches the largest values investigated here, in particular for n = 500. Additional File s>, also increases with versatility at n = 500  and n = 700  reactions. However, in contrast to the trend for <M> in Additional File s> does not slow down for the largest values of env Vwe have examined.The results of Figure  n = 831 . AdditioM and s based on samples of random networks of a given versatility. In such a sample, modularity has a distribution, where some networks are more modular, and others less so. We can use this distribution to ask whether the modularity observed in the metabolic network of an organism such as E. coli is atypically high or low. In other words, the distribution of modularity arising in our samples of in silico metabolic networks can provide a null hypothesis to evaluate whether a biological network shows unusual modularity.So far we have shown averages of our modularity measures M of reactions in modules, and Figure s of modules (FCSs) in a sample of 1000 random networks with n = 831 reactions , where s E.coli . This phE. coli metabolic network has higher modularity than anticipated, but the large value of M may come from either a greater number of FCSs or from an increased size of the FCSs. Figure E. coli is just slightly above the position of the distribution's peak in our ensemble, well within one standard deviation. From this observation one can conclude that the atypically high modularity of the E. coli network stems from the fact that E. coli has larger modules (FCSs) but not much more modules than typical networks allowing growth on 89 carbon sources.The architecture of the env V= 89) to FCSs of genotypes with env V= 1. Specifically, we first extracted the reactions that belonged to FCSs and that occurred in more than 50 percent of the genotypes in each of the two samples. Call these sets of reactions R89 and R1, for the ensembles with env V= 89 and env V= 1, respectively. At a qualitative level, we find that about 90% of reactions in R1 also belong to R89. We then examined the reactions that belong to R89 but that are not part of R1, and called this set of reactions R89\R1. Are the reactions in R89\R1 immediately downstream of the nutrients? The notion of downstream can be made quantitative through the Scope algorithm [89\R1 to the scope distances of all reactions in our universe of reactions. Figure 89\R1 generally have smaller scope distance than other reactions. A statistical test (see Methods) shows that this difference is significant with a p-value of 10-5. In sum, reactions of modules involved in increased versatility tend to be more closely downstream of nutrients, suggesting that they typically belong to pathways metabolizing such nutrients. To illustrate this property with concrete examples, we determined which FCSs in R89\R1 involved any of the 24 reactions occurring at scope distance 1 in Figure Thus far, we saw that metabolic networks sustaining growth on more nutrients have higher modularity, that is, more reactions contained in modules and more modules (FCSs)  ,35 is a Z. The LP formulation of the FBA problem can be written as:so as to satisfy mass conservation. Eq. 1 represents stoichiometric and mass balance constraints on the metabolic network. For genome-scale metabolic networks, Eq. 1 leads to an under-determined system of linear equations in the metabolic fluxes, leading to a large solution space of allowable fluxes. The space of allowable solutions can be reduced by incorporating thermodynamic constraints associated with irreversible reactions, as well as flux capacity constraints which limit the maximum flux through certain reactions. To obtain a particular solution, linear programming (LP) is used to find a set of flux values - a point in the solution space - that maximizes a biologically relevant linear objective function c corresponds to the coefficients of the objective function Z, and vectors a and b contain the lower and upper limits of different metabolic fluxes in v. The objective function Z is often chosen to be the so-called growth flux. This is the flux through an artifactual reaction that reflects the synthesis of biomass, which requires biosynthesis of biomass precursor molecules, such as amino acids and nucleotides, in specific proportions. The stoichiometry of this reaction is based on the experimentally measured biomass composition of an organism. The predictions from the FBA framework and related approaches are often in good agreement with experimental results [where the vector  results -38,68.E. coli metabolic model iJR904 [E. coli metabolic model iJR904.In this work, we have used a hybrid database compiled by Rodrigues and Wagner  containil iJR904 , followeE. coli iJR904 model. These 143 external metabolites were assumed to be the set of possible imported and secreted metabolites. Further, the hybrid database includes an objective function Z in the form of a biomass reaction that reflects synthesis of the E. coli biomass components, as defined in the iJR904 model.In addition to the 5870 metabolic reactions, the hybrid database has transport reactions for 143 external metabolites contained in the blocked reactions [N = 2902 reactions as the global reaction set.Genome-scale metabolic networks typically contain eactions ,69 whichE. coli metabolic model iJR904 has 931 reactions (which of course are contained in the hybrid database of 5870 reactions). Our global reaction set (having 2902 reactions) was derived from the hybrid database by excluding blocked reactions; after this exclusion, the E. coli metabolic model iJR904 is left with 831 reactions. Here, we consider this set of 831 reactions to be the E. coli metabolic genotype.The n reactions taken from the global reaction set is considered to specify a discretized binary metabolic genotype. For simplicity, we shall refer to this as a metabolic genotype or as a genotype. Specifically, a metabolic genotype G can be represented by a bit string of length N, i.e., G = , where N is the number of reactions in the global reaction set  or absent (i b= 0) in the genotype. We denote the set or space of metabolic genotypes with a given number n of reactions as Ω(n).Any subset of viable in a given environment if and only if the maximum biomass flux predicted by FBA for the genotype is nonzero; otherwise we consider the genotype to be non-viable . We consider a genotype to be n) that have a specific phenotype - growth on a specific list of environments - as V(n). In this work, we use only aerobic minimal environments containing one carbon source. Each environment also contains unlimited amounts of the following inorganic metabolites: ammonia, iron, potassium, protons, pyrophosphate, sodium, sulfate, water and oxygen. Based on FBA applied to the metabolic model iJR904, it was found earlier that E. coli can support nonzero biomass growth on 89 different aerobic minimal environments [For our purpose, the metabolic phenotype of a metabolic network (genotype) is determined by the network's viability in a list of well-defined chemical environments (media). We shall denote the subset of genotypes within Ω sampling algorithm  can be used to explore the set of genotypes having a given phenotype. In our case, this phenotype is viability on a given set of minimal environments; if this set consists of i.e., ensembles whose networks have env V= 1, 2, 5, 10, 20, 30, 40, 50, 70 and 89. The genotypes with env V= 89 are the most versatile among them as they are viable in all 89 minimal environments. There are many ways of choosing 1, 2, or more specific environments out of 89 environments to sample genotypes having a phenotype with env V= 1, 2, through 89. The properties of sampled genotypes in an ensemble with a given env Vwill depend on the choice of those env Venvironments. The computations we carry out are computationally very expensive, and they become more expensive with every additional environment in which viability is determined. To limit this expense, we pursued two strategies. First, we used nested sets of environments to sample genotypes in ensembles with different envV, e.g., the set of 70 environments chosen for env V= 70 is a subset of that for env V= 89, and the set of 50 environments chosen for env V= 50 is a subset of that used for env V= 70, and so forth. Second, we used only one subset of 70 environments within env V= 89 to sample an ensemble with env V= 70, and only one subset of 50 environments within the choice for env V= 70 for sampling an ensemble with env V= 50. For env Vbelow that, we did tackle the variability coming from different environmental choices; specifically, for env V= 40, we used 10 different subsets of 40 environments within the choice for env V= 50; thus we generated 10 different genotype ensembles, where each genotype in each ensemble had env V= 40. Each of these 10 different choices of 40 environments was then used to create a single nested sequence for env V= 30, 20, 10, 5, 2, and 1. This allowed us to have 10 different ensembles to sample at each of these env Vand to follow for each sequence of nested sets the consequences of modifying envV.  relative to Ω(n) is of the order of 10-22 for genotypes with n = 2000 reactions. This size decreases even further if one requires viability in multiple environments. Such tiny probabilities of finding a desired phenotype in Ω(n) make it infeasible to sample genotypes in V(n) by simply drawing random genotypes in Ω(n) with the correct number n of reactions, followed by determining the phenotype of each genotype. Thus, we relied on the Markov Chain Monte Carlo (MCMC) method described in Ref. [V(n).It was shown in previous work  for a si in Ref.  to unifoV(n) and produces a sequence of genotypes, wherein the (k+1)th genotype in the sequence is generated from the kth genotype using a probabilistic transition rule. At each transition step, one proposes a small modification to the current genotype in the sequence; if the modified genotype has the correct phenotype, one accepts the modified genotype as the next genotype of the sequence; otherwise the next genotype becomes identical to the current genotype. The modification introduced at each transition step is a reaction swap. It consists of removal of one reaction from the current genotype, followed by addition of new reaction from the global reaction set to generate a modified genotype. Note that the reaction swap preserves the number n of reactions in the genotype , as illustrated in Additional File This MCMC method starts with a genotype in in the k+th genotyin the k+th genotyV(n), we have first carried out 105 attempted swaps to erase the memory of the starting genotype. After this initial phase, we continued the MCMC procedure to sample genotypes in V(n). During this later phase, it is not useful to keep all of the genotypes produced, as many of them may be highly similar to one another. We thus saved only every 1000th genotype generated in a sequence of 106 steps. This procedure produces a random ensemble of 1000 genotypes in V(n) [V(n) for three different values of the number of reactions n = 500, 700 and 831.In our simulations, starting from an initial genotype in  in V(n) . We sampE. coli metabolic network that have nonzero flux in an optimal flux distribution with maximum biomass flux for each of the 89 minimal environments considered here. . The number of nonzero fluxes is ~300 in a typical optimal flux distribution for each of the 89 environments. We generated a genotype with n reactions and phenotype env V= 1  by starting with the set of nonzero flux reactions for E. coli in that environment, and then adding randomly other reactions until we reached a metabolic genotype with exactly n reactions. We generated a genotype with n reactions and env V= 2  by starting with the union of the two sets of reactions that had nonzero flux when the E. coli metabolic network synthesized biomass in the two different environments; then we added randomly other reactions until we reached a metabolic genotype with exactly n reactions. We generated starting genotypes with env V= 5, 10, 20, 30, 40, 50, 70 and 89 analogously.To start the MCMC sampling, a first genotype having the correct phenotype is required. To this end, we first determined those reactions in the 1 vand 2 vare said to be fully coupled to each other if a nonzero flux for 1 vimplies a proportionate (nonzero) flux for 2 vin any steady state and vice versa [fully coupled set (FCS) in a metabolic network is a maximal set of reactions that are mutually fully coupled to each other . A simple argument shows that FCSs of a network are non-overlapping entities. Indeed, if a reaction were to belong to two FCSs, then all reactions in those two sets would be fully coupled pairwise, resulting in one larger FCS.A reaction pair ce versa . A fullys. This is one index of modularity. We also define the modularity index M for a genotype as the number of reactions contained in the FCSs of that genotype . Burgard et al [We denote the number of FCSs in a metabolic network genotype by rd et al  have proRmax = Rmin then v1 and v2 are fully coupled. In the above equations, S is the stoichiometric matrix, and vectors a and b contain the lower and upper limits of different metabolic fluxes in v.If et al. to determine all FCSs in our metabolic network genotypes. We have computed the coupled reaction pairs under conditions where all external metabolites were allowed to be imported or secreted. Further, coupled reaction pairs were computed without assuming a constant biomass composition to avoid coupling a large set of fluxes to the biomass reaction. Hence, all biomass components were allowed to be synthesized independently of one another, without constraining their stoichiometry in the biomass.We have used the algorithm of Burgard scope based on a network expansion algorithm for the structural analysis of genome-scale metabolic networks. Their approach calculates for a given metabolic network/reaction database and predefined external metabolites (referred to as seed metabolites) the set of metabolites - the scope - which the reaction network is in principle able to produce. In other words, the scope describes the synthesizing capacity of a given set of seed metabolites given a list of metabolic reactions.Ebenhöh and colleagues ,61 have A of metabolites that a reaction system can synthesize. In the algorithm this set A is initialized to the set of nutrient metabolites. At each iteration i of the algorithm, one takes the current set A(i) of producible metabolites and expands it to set A(i+1) as follows. First one initializes, A(i+1) to contain all metabolites in A(i). Then one considers successively each reaction in the database and adds that reaction's products to A(i+1) if and only if all of its substrates are in A(i). This procedure ends when A(i) = A(i+1), that is when in a given iteration no new molecules can be synthesized. We have used the Scope algorithm to define a distance of a reaction in the global reaction set from nutrient metabolites. Specifically, the distance of a reaction from the nutrient metabolites in the seed set is defined as the iteration number i of the Scope algorithm when that particular reaction contributes its products to A(i+1).The Scope algorithm iteratively updates a set A limitation of the Scope algorithm in comparison to constraint-based frameworks like FBA is its inability to deal properly with the self-generating  nature of certain cofactor metabolites  in the network ,71. The scope distance of that reaction from the nutrient metabolites.We have determined the distance from nutrient metabolites for each reaction in the global reaction set for 89 different seed sets corresponding to the 89 aerobic minimal environments. For each reaction, we have designated the minimum of the 89 distances obtained for the 89 different environments as the M and the number s of FCSs is larger for sampled genotypes with env V= 89 than with env V= 1  to create a merged list of FCSs that occur in at least one such sampled genotype. Since the FCSs are non-overlapping entities, multiple copies of a FCS in the merged list signify the FCS's presence in multiple sampled genotypes. We then determined the set of reactions that occurred in at least 500 FCSs in the merged list of FCSs for env V= 89. We refer to it as the consensus set R89 of FCS reactions for env V= 89 and n = 831. In a similar way, we obtained the consensus set R1 of FCS reactions for our sampled genotypes with env V= 1.Since the modularity index 89 for env V= 89 is larger than the set R1 for env V= 1, and the complement set R89\R1 consisting of the reactions belonging to R89 but not R1 gives the set of reactions that mostly account for the additional FCSs in env V= 89. We then considered the scope distances of reactions from nutrient metabolites for two choices of reaction sets. The first set is this complement set R89\R1, the second is the set of all reactions in the global reaction set.  test allowed us to reject the null hypothesis that the two distributions are the same (P < 10-5). Further, a two sample Welch t-test allowed us to reject the hypothesis that the means of the two distributions are the same (P < 8.10-8).The consensus set RE. coli metabolic model iJR904 [We have classified reactions in our global reaction set into different biochemical pathways using the pathway information  for reacl iJR904 . We haveQ which is the fraction of reactions sharing the dominant annotation for that FCS. We computed Q for each FCS in the merged list of FCSs from our 1000 sampled genotypes with phenotype env V= 89 and n = 831 (see the previous section for merged lists of FCSs). We then considered the cumulative distribution of Q for FCSs in the merged list, namely, the probability that Q is at least as large as a given value X. The cumulative distribution of Q for FCSs in the merged list with env V= 89 and n = 831 is shown in Additional File h of FCSs in the merged list with Q ≥ h, a quantity that is analogous to the h-index commonly used to measure scientific productivity [h-index has a value of h = 0.79, as can be seen from the point of intersection of the cumulative distribution of Q with the bisecting line in Additional File For a given FCS, we define the quantity uctivity . This h-h-index obtained from the merged list of FCSs for sampled genotypes with env V= 89 and n = 831, we performed the following randomization test. Starting from the merged list of FCSs and the pathway annotations of their reactions, we generated 1000 equivalent random lists by swapping the annotations among reactions in different FCSs, while preserving the frequency of each annotation in the merged list. We swapped annotations as follows. We first recorded the multiplicity of each distinct FCS within the merged list. We then randomly picked two FCSs in the merged list with the same multiplicity, and one random reaction in each of the two FCSs, and then swapped the annotations of these two reactions in the FCSs. We performed at least 107 swaps starting from the merged list before saving a random list, that is, a list of FCSs whose reaction annotations had been randomized in this way. None of the 1000 random lists we generated had an h-index greater than 0.79 obtained for the merged list with env V= 89. On this basis, we can reject the hypothesis that reaction annotations are not similar within FCSs at a p-value of less than 0.001.To test the significance of the The project was defined by all three authors. OCM and AS conceived the algorithmic procedures. AS wrote the code and performed the numerical simulations. All authors contributed in designing research, analyzing the data, and writing the paper. All authors have read and approved the manuscript.Figures S1 to S7. This file contains the following additional figures: Figure S1 - MCMC sampling of genotypes with a given phenotype; Figure S2 - a FCS predominantly consists of a reactions belonging to one biochemical pathway; Figure S3 -Example of a frequently arising FCS in sampled genotypes; Figure S4 - Environmental versatility enhances the modularity index M regardless of the value of n; Figure S5 -Environmental versatility enhances the number of modules s regardless of the value of n; Figure S6 - Example of FCSs that account for the increase in modularity with versatility; Figure S7 - Schematic diagram describing nested sets of environments used to sample genotypes with decreasing envV.Click here for fileTables S1 and S2. This file contains the following additional tables: Table S1 - The list of 89 minimal environments; Table S2 - The list of autocatalytic metabolites.Click here for file