Accordingly, if an enzyme catalyzes a reaction with a ΔrG′ of -5.7 kJ/mol then the forward flux will be roughly ten times the reverse flux. As ΔrG′ approaches equilibrium (ΔrG′ = 0 kJ/mol), exponentially more enzyme counterproductively catalyzes the reverse reaction, reducing the net rate at which the reaction proceeds. Thus, the enzyme level required to achieve a given flux increases dramatically near equilibrium. Here, we develop a framework for quantifying the degree to which pathways suffer these thermodynamic limitations on flux. For each pathway, we calculate a single thermodynamically-derived metric , which enables objective ranking of pathways by the degree to which their flux is constrained by low thermodynamic driving force. Our framework accounts for the effect of pH, ionic strength and metabolite concentration ranges and allows us to quantify how alterations to the pathway structure affect the pathway's thermodynamics. Applying this methodology to pathways of central metabolism sheds light on some of their features, including metabolic bypasses , substrate channeling , and use of alternative cofactors . The methods presented here place another arrow in metabolic engineers' quiver, providing a simple means of evaluating the thermodynamic and kinetic quality of different pathway chemistries that produce the same molecules.In metabolism research, thermodynamics is usually used to determine the directionality of a reaction or the feasibility of a pathway. However, the relationship between thermodynamic potentials and fluxes is not limited to questions of directionality: thermodynamics also affects the kinetics of reactions through the flux-force relationship, which states that the logarithm of the ratio between the forward and reverse fluxes is directly proportional to the change in Gibbs energy due to a reaction  can pinpoint the enzymes whose expression will have the largest effect on steady-state flux through the pathway. These analyses can aid experimentalists in tuning enzyme expression levels along a metabolic pathway. In this work, we offer a framework that is complementary to MCA. Rather than focusing on the relationship between enzyme levels and pathway flux, we examine a pathway's stoichiometry and thermodynamics and ask whether it is likely to support high flux in cellular conditions. Our framework calculates a single thermodynamically-derived metric (the MDF) for each pathway, which is convenient for selecting the promising pathways from a large collection. This approach has several advantages. First, enzyme kinetic properties are laborious to measure and differ between organisms and isozymes, but no kinetic data is required to calculate the MDF. Second, as our framework accounts for pH, ionic strength and allowed concentration ranges, it is simple to model the effect of these parameters on the MDF. Finally, as it can be difficult to control the exact expression level of enzymes within cells, the MDF helps identify alternative pathways that are less sensitive to the levels of their constituent enzymes. A primary scientific goal of metabolic research is to develop an understanding of the evolutionary, chemical and physical forces that shape the structure of cellular metabolism. Specifically, to what extent are present-day metabolic pathways the result of evolutionary optimization rather than fossilized accidents? In recent years various aspects of central metabolism have been explained on the basis of specific selection pressures and constraints imposed during evolution rG′, affects the net reaction rate through the flux-force relationship rG′ = −RTln(J+/J−), R being the gas constant, T the temperature, J+ the forward flux and J− the backward flux. Consequently, an enzyme catalyzing a reaction that is far from equilibrium (ΔrG′<<0) carries almost no backwards flux (rG′∼0) “wastes” many enzyme units catalyzing substantial flux through the reverse reaction. According to the flux-force relationship, as a reaction shifts towards equilibrium we would see an exponential increase in the number of enzyme units required to catalyze a single unit of flux. For example, a ΔrG′ of −7.3 kJ/mol implies that about 5% of the enzymatic flux is in the reverse direction. Alternatively, a rather close-to-equilibrium ΔrG′ of −1 kJ/mol implies that about 40% of the enzymatic flux is in the reverse direction and the reaction rate is only ∼20% of the rate that would be achieved if all enzyme units catalyzed the forward reaction.Thermodynamic analysis is typically applied to determine whether a reaction direction or pathway is feasible in physiological conditions rds flux  while anrG′) and the amount of enzyme required to sustain a particular flux through that reaction. Therefore, the protein burden imposed by a pathway is directly related to the thermodynamic landscape of that pathway.Metabolic Control Analysis (MCA) is commonly used to describe the control that enzymes exert on metabolic fluxes We develop a quantitative framework to analyze the thermodynamic landscape of metabolic pathways. Our framework identifies those reactions within a pathway whose rates are constrained by low thermodynamic driving force. These enzymes will constrain the activity of the pathway unless they are present at high concentrations or are much faster-than-average catalysts rG′o (reactant concentrations of 1 M). This method produces a consistent set of ΔrG′o values by integrating three sources of information in decreasing priority: (1) ∼200 Gibbs energies of formation (ΔfGo) collected and published by Alberty rG′o values were transformed to a pH of 7.5 and an ionic strength of 0.2 M We used the Component Contribution method rG′oΔ) and the reactant concentrations. It is given by rG′ = ΔrG′o+RT·ln(Q)Δ, where Q is the reaction quotient . Because of its more intuitive sign we will often refer to a reaction's Driving Force, defined as rG′−ΔThe Gibbs energy dissipated by a reaction can be calculated from the standard Gibbs energy of the reaction . oG denotes a column vector of reaction energies, i.e., ojG is the standard change in Gibbs energy (rG′oΔ) of reaction j. Finally, let x be the column vector of the log-concentrations, so that ix is the natural logarithm of the concentration of compound i in molar units. Then the vector of reaction driving forces (−rG′Δ) is given by −(oG+RT·ST·x).When analyzing pathways containing multiple reactions, it is convenient to use matrix notation S are written such that the forward direction is the direction of the net flux in the pathway, and the stoichiometric coefficients represent the actual molecularities in the enzyme's reaction center i.e., the number of reactant molecular entities that are involved in the ‘microscopic chemical event’ constituting an elementary reaction). This convention obviates the situation existing in more general stoichiometric models We use a convention where all reactions in oG to be constant and allowing x to vary, a pathway is feasible if and only if there is at least one solution to the linear system defined by the constraints ln(minC)≤x≤ln(maxC) and −rG′Δ>0; i.e., there must exist a set of metabolite concentrations within the predefined range (minC to maxC) such that all reactions have a positive driving force. The concentration bounds (minC and maxC) may be the same for all compounds or can be defined individually.Considering x exists within the chosen concentration range such that −rG′Δ>0 and so the pathway reactions cannot all be made feasible simultaneously Even if a pathway is composed of thermodynamically-feasible reactions, the pathway may be thermodynamically infeasible as a whole. That is, no solution for Flux-Force Efficacy – a unitless measure between 0 and 1 – as the ratio between the net flux  as an optimization goal and maximize it – within metabolite concentration bounds – using linear programming. This can be formalized as a linear problem:B represents a tight lower bound  on the driving force of all reactions. Maximizing B yields a solution where all reactions are as far from equilibrium as possible given the defined concentration ranges. The maximal value of B is denoted as the Max-min Driving-Force (MDF) of the pathway and is measured in units of kJ/mol Because of this interdependence between thermodynamic potentials and flux, pathways operating near equilibrium will incur a kinetic penalty due to backwards flux. We therefore attempt to quantify a pathway's tendency to operate near-equilibrium. We seek a set of reactant concentrations that will maximize the driving forces of all reactions in the pathway. To achieve this, we use the minimum over all reaction driving forces (−rG′ = −7.3 kJ/mol corresponds to a J+/J− ratio of exp(7.3/RT) = 19, which in turn suggest that 95% of the overall flux is in the forward direction and 95/19 = 5% is in the backward direction. Therefore, the Flux-Force Efficacy is 95−5 = 90%  of the reaction driving forces (−oG−RT· ST · x≥B), the upper bound of metabolite concentrations (x≤ln(maxC)), and the lower bound of metabolite concentrations (ln(minC)≤x). We therefore define the Reaction Shadow Price as the shadow price associated with the constraint on the driving force of that reaction, representing how much a change in oG will affect the MDF. The Metabolite Shadow Price is the maximum of the absolute values of the two shadow prices associated with the constraints (lower and upper bound) on that metabolite's concentration.In order to quantify the extent to which a reaction or a metabolite affects the value of the MDF, we use the concept of shadow prices w, maxu, minu:The shadow prices are the solution for the variables of the dual problem w are the reaction shadow prices and max are the metabolite shadow prices. See According to the definitions above, rG′o would increase the pathway MDF by 1 kJ/mol (assuming that this reaction still limits the pathway MDF). Similarly, a metabolite shadow price of 0.5, associated with the upper bound constraint, implies that raising the upper bound concentration of this metabolite by 10 fold will result in the MDF increasing by 0.5*RTln(10)≈3 kJ/mol. Shadow prices are 0 for reactions whose ΔrG′o does not constrain the MDF, and likewise for metabolites whose concentrations do not constrain the MDF.The shadow prices represent a scaling between a change in the constraint and the resulting change in the MDF. For example, a reaction shadow price of 0.25 indicates that a 4 kJ/mol decrease in ΔminC = 1 µM and an upper bound maxC = 10 mM E. coli's cytoplasm. Cofactors participate in tens or even hundreds of reactions and so their concentrations are considerably more constrained by the endogenous metabolic network than common reaction intermediates +] = 0.1 +] = 10 reduced]/[Ferredoxinoxidized] = 1 ] = 10 µM (ambient conditions).Throughout our analyses we used metabolite concentration bounds characteristic of cellular physiology, a lower bound Enzyme abundances control the steady-state fluxes within a pathway J is the steady-state flux, iv is the rate of reaction i and iE is the abundance of the enzyme catalyzing reaction i. In the general case, control coefficients depend on the pathway structure, enzymatic parameters, and allosteric regulation. Since all control coefficients for a flux J always sum to 1 Potentially, all enzymes exert control on all fluxes within the pathway, but to different extents. Metabolic Control Analysis describes this control mathematically: if we consider small changes to a given steady-state, the effect of an enzyme's abundance on the pathway flux, can be quantified by the scaled flux control coefficient M (enzymes operate in the linear regime). We show that in both cases the control coefficients are completely determined by reaction driving forces, such that the two are always correlated: the higher the driving force of a reaction, the higher the control it exerts on the pathway flux. Specifically, reactions with low driving forces have very low control coefficients.The flux control coefficients are related to the thermodynamic driving forces, as derived in For the first case – all enzymes are substrate-saturated – we find that For the second case – when all enzymes are substrate and product sub-saturated – we find that:In both cases the scaling factor α is identical for all reactions and is determined byhttps://code.google.com/p/milo-lab/). We use IBM's ILOG CPLEX Optimization Studio 12.5 to solve the MDF primal and dual problems. The code which predicts the standard Gibbs energies http://openbabel.org) http://www.genome.jp/kegg/) The software for all the analyses presented in this paper is open source (MIT license) and stored in an online repository  equals RTln(+J/−J), as presented above. For low enough driving forces (roughly 3 kJ/mol or less) the effect of ΔrG′ on the reaction flux is similar to that of kcat (or VMAX) in the sense that a fold change in either of them will change the reaction rate by the same fold change. The dependence of the reaction rate on −ΔrG′ decreases as −ΔrG′ increases above ∼3 kJ/mol. When −ΔrG′ exceeds 10 kJ/mol, the reaction rate is effectively insensitive to thermodynamic effects, reflecting the fact that there is negligible flux in the reverse direction.The kinetics of a reaction can be linked to three main factors: (1) maximal velocities and saturation levels, related to the enzyme kinetic parameters and to the concentrations of substrates and products; (2) enzyme abundances and (3) reverse flux though reactions To address the effect of driving force on reaction flux more systematically, we define the Flux-Force Efficacy as the ratio between reaction net flux and total flux, +J≈−J. In contrast, a reaction with a high Flux-Force Efficacy, characterized by +J>>−J, is thermodynamically inefficient, dissipating a considerable amount of Gibbs energy.We use the term “efficacy”, instead of “efficiency”, to distinguish between Flux-Force Efficacy and thermodynamic efficiency, as the two are antagonistic. For example, a reaction operating close to equilibrium is often considered to be thermodynamically efficient since it dissipates almost no Gibbs energy. Yet, such a reaction is characterized by a low Flux-Force Efficacy as For a fixed enzyme level, a high Flux-Force Efficacy implies a high net reaction rate, as backward flux is negligible. On the other hand, a low Flux-Force Efficacy indicates considerable backward flux, leading to a decreased reaction rate. In a complementary manner, we can use this relationship to estimate the amount of enzyme required to sustain a particular flux through the reaction Max-min Driving-Force (MDF).When analyzing an entire pathway, it is essential to consider the interplay between the driving forces of all participating reactions: varying the concentration of a given metabolite can modulate the driving force of multiple reactions. We developed a method for finding metabolite concentrations – within an allowed range, see rG′o values of the different pathway reactions, while the magenta line represents the ΔrG′ values for each of the reactions after optimizing reactant concentrations to maximize the MDF. After this optimization, all reactions in the pathway have a positive driving force  and so it is clear that the EMP pathway is thermodynamically feasible.rG′o values used are accurate and that our concentration bounds reflect cellular concentrations, pathways are thermodynamically feasible if and only if they have a positive MDF. Moreover, the value of the MDF indicates the degree to which a pathway is expected to be kinetically constrained by backward flux. A pathway with a high MDF can achieve a steady-state with very low backward flux as all of its constituent reactions can achieve high driving forces simultaneously. On the other hand, a pathway characterized by a low MDF contains reactions that are expected to have low driving force in physiological conditions. Due to the flux-force relationship, these reactions must either sustain low flux or be catalyzed by an abundant enzyme. For example, the green pathway shown in Presuming the ΔrG′o value of a reaction with a positive shadow price would lead to an increase in the MDF. Similarly, if the concentration of a metabolite with a positive shadow price is permitted to violate the allowed concentration range (becomes too high or too low), the MDF increases (Methods).The shadow prices determine whether a specific reaction or metabolite constrains the pathway MDF, as described in detail in the cat values. While it is tempting to test this hypothesis systematically, it is, unfortunately, very challenging using available experimental data. Specifically, the MDF analysis requires that the magnitudes of all fluxes are precisely determined (taking into account that the flux through enzymes participating in the same pathway might differ due to an overlap with other metabolic routes). However, even for the relatively simple case of E. coli's central metabolism, flux and metabolite concentration measurements from different groups vary significantly, even when performed under similar conditions . How, then, are cells able to sustain high flux through the TCA cycle? The MDF framework enables us to suggest solutions for this apparent paradox.In most organisms, the TCA cycle is the pathway responsible for the catabolic oxidation of organic compounds to CO2 . Figure cat of 100 s−1 and catalyzing a reaction with a driving force of only 0.3 kJ/mol (Flux-Force Efficacy ∼6%) is equivalent to an enzyme having a kcat of 10 s−1  will have a higher propensity to be channeled between enzymes, therefore potentially guiding experimental efforts to such locations in search of evidence for substrate channeling. When high throughput methods for identifying channeling are developed, it will be possible to test this hypothesis systematically.A further possibility, also supported by experimental studies, is that oxaloacetate is channeled between malate dehydrogenase and citrate synthase Another solution to this thermodynamic puzzle might be the use of electron acceptors with a higher reduction potential than that of NAD. For example, various organisms operate a malate:quinone oxidoreductase enzyme e.g., i.e., oxaloacetate reduction), eliminating the thermodynamic constraints due to malate oxidation. Remarkably, it was recently suggested that E. coli uses a forked TCA cycle even during aerobic growth, despite the low ATP yield associated with this mode Finally, it is important to note that the TCA cycle is not actually a cycle in many organisms and under various conditions e.g., . Instead2E. coliSeveral natural alternatives to the TCA cycle are also known to support the complete oxidation of organic compounds to COde novo ATP synthesis in the pathway (the downstream pyruvate kinase only recoups the ATP invested at the beginning of the pathway) EMP glycolysis  is the mrG′o value then the MDF of the entire pathway would increase.Notably, the trend shown in e.g., Fructose bisphosphate is one of the two non-cofactor metabolites with a positive shadow prices , and theSeveral glycolytic variants are known to exist in nature and their structures and shown in i.e., no other pathway has both a higher MDF and a higher ATP yield. Notably, despite their prevalence in nature, neither the EMP nor the ED pathways are on the Pareto front, which suggests that thermodynamic properties alone are insufficient to explain the structure of central metabolism pathways, as we previously analyzed in detail . While near-equilibrium reactions can significantly increase the protein burden of a pathway, they may have certain advantages. For example, if the direction of a reaction must change quickly in response to some stimulus, operating near equilibrium (and at high enzyme level) is a good strategy: a small change in reactant concentrations can reverse the reaction direction while maintaining a similar absolute flux. This may be particularly important for glycolysis, where some carbon sources require glycolytic flux  and others require flux in the direction of gluconeogenesis . Therefore, fast environmental fluctuations in the availability of carbon sources may require speedy reversal of most glycolytic reactions, which is consistent with recent measurements indicating that reactions in glycolysis mostly operate with low driving-force in E. coliWe introduce a quantitative framework for analyzing the thermodynamic profile of metabolic pathways and identifying reactions that limit metabolic flux within feasible pathways  and especially low (<1 µM) metabolite concentrations have been measured e.g., . Furtherin-vivo metabolite concentrations are constrained by many other factors, including their stability, permeability and their participation in other metabolic routes, and so cellular concentrations are unlikely to match these optima precisely. Hence, many of the pathways we analyzed might be more thermodynamically constrained than suggested by the MDF analysisIt is important to remember that the MDF methodology assumes that metabolite concentrations are optimized to achieve the most favorable thermodynamics. These optima are calculated using thermodynamic and stoichiometric data with respect to a single pathway and ignoring the rest of the endogenous metabolic network. Yet, i.e., what counts as independent metabolic steps. Merging reactions into a single metabolic step or splitting them into several steps can considerably affect the MDF of a pathway and the Flux-Force Efficacies of its reactions. For example, consider a reaction dissipating 2 kJ/mol and hence operating at a Flux-Force Efficacy of ≈40%. If this reaction is split into two steps, each of these will optimally dissipate 1 kJ/mol and its Flux-Force Efficacy will be only ≈20%. Hence, dividing a pathway into more steps results in lower MDF and Flux-Force Efficacies. Yet, the definition of metabolic steps is not arbitrary. A reaction should be treated as an independent metabolic step if all of its substrates and products are soluble. On the other hand, if two reactions involve a common reactant which remains bound to the enzyme(s), they can be treated as a single metabolic step Our analysis is sensitive to the definition of reactions, i.e., kcat, KM) cat/KM should be compensated by a high driving force  and still sustain a high net flux  the binding and release of the reactants from the enzyme's active site; (ii) the different reaction intermediates formed during catalysis; and (iii) the activation energies of converting one reaction intermediate to another ng force , becauseng force . On the net flux . Finallynet flux .Interestingly, the thermodynamic driving forces of reactions can be directly connected to their control coefficients On the other hand, a reaction with a high driving force will exert high control on the pathway flux. For such a reaction the net flux roughly equals the forward flux, which is much larger than the reverse flux. In this case, increasing the enzyme abundance will mostly increase the forward flux (in absolute terms). Even if the driving force decreases somewhat, the flux-force efficacy will remain high see . Hence, Our ultimate goal is to establish a single framework that integrates pathway thermodynamics and enzyme kinetics. We believe it should be possible to reformulate measured kinetic data as thermodynamic potentials and analyze pathways in purely energetic terms. By considering the chemical potential of reaction intermediates and integrating these data with the concentrations of soluble pathway intermediates, one can arrive at a more complete analysis of pathway activity. It remains for future research to develop such an integrated framework.Figure S1E. coli under glucose starvation. (C) A pathway used by Pseudomonas fluorescens under conditions of aluminum toxicity. (D) The oxidative pentose-phosphate cycle, which can be used to fully oxidize sugars into CO2, thus providing NADPH for cellular activity. The reactions marked in red are those with positive shadow prices (at pH 7.5 and ionic strength of 0.2 M). Non-cofactor metabolites shaded in green show positive shadow prices.Structure of oxidative pathways. (A) The ubiquitous TCA cycle. (B) The PEP-glyoxylate pathway which was found to operate in (EPS)Click here for additional data file.Figure S2Structure of fermentation pathways. (A) Entner-Meyerhof-Parnas (EMP) glycolysis. Blue arrows correspond to the pyruvate formate lyase shunt. Magenta arrows correspond to a bypass of substrate-level phosphorylation. (B) Entner-Doudoroff (ED) glycolysis. (C) Semi-phosphorylative ED glycolysis, as known to operate in some hyperthermophilic archaea. (D) Non-phosphorylative ED glycolysis, as known to operate in some hyperthermophilic archaea. (E) The methylglyoxal pathway, in which dihydroxyacetone phosphate is converted into the highly reactive compound methylglyoxal when the concentration of inorganic phosphate becomes limiting. (F) The suggested phosphoketolase pathway, which uses the pentose phosphate pathway in conjunction with the enzyme phosphoketolase that cleaves xylulose-phosphate to glyceraldehyde-phosphate and acetyl-phosphate. The reactions marked in red are those with positive shadow prices (at pH 7.5 and ionic strength of 0.2 M). Non-cofactor metabolites shaded in green show positive shadow prices.(EPS)Click here for additional data file.Text S1A mathematical derivation and characteristics of the Max-min Driving Force and its relation to Metabolic Control Analysis.(PDF)Click here for additional data file.


Concurrent with the efforts currently underway in mapping microbial genomes using high-throughput sequencing methods, systems biologists are building metabolic models to characterize and predict cell metabolisms. One of the key steps in building a metabolic model is using multiple databases to collect and assemble essential information about genome-annotations and the architecture of the metabolic network for a specific organism. To speed up metabolic model development for a large number of microorganisms, we need a user-friendly platform to construct metabolic networks and to perform constraint-based flux balance analysis based on genome databases and experimental results.Thermoanaerobacter sp. strain X514, to predict its biomass growth and ethanol production.We have developed a semi-automatic, web-based platform (MicrobesFlux) for generating and reconstructing metabolic models for annotated microorganisms. MicrobesFlux is able to automatically download the metabolic network (including enzymatic reactions and metabolites) of ~1,200 species from the KEGG database (Kyoto Encyclopedia of Genes and Genomes) and then convert it to a metabolic model draft. The platform also provides diverse customized tools, such as gene knockouts and the introduction of heterologous pathways, for users to reconstruct the model network. The reconstructed metabolic network can be formulated to a constraint-based flux model to predict and analyze the carbon fluxes in microbial metabolisms. The simulation results can be exported in the SBML format (The Systems Biology Markup Language). Furthermore, we also demonstrated the platform functionalities by developing an FBA model (including 229 reactions) for a recent annotated bioethanol producer, 13 C- metabolic flux analysis results. MicrobesFlux is available at http://tanglab.engineering.wustl.edu/static/MicrobesFlux.html and will be continuously improved based on feedback from users.MicrobesFlux is an installation-free and open-source platform that enables biologists without prior programming knowledge to develop metabolic models for annotated microorganisms in the KEGG database. Our system facilitates users to reconstruct metabolic networks of organisms based on experimental information. Through human-computer interaction, MicrobesFlux provides users with reasonable predictions of microbial metabolism via flux balance analysis. This prototype platform can be a springboard for advanced and broad-scope modeling of complex biological systems by integrating other “omics” data or  In the past decade, over 100 genome-scale metabolic models have been constructed for E.coli29], we aain X514 . Since tThermoanaerobacter sp. strain X514 is not yet available. In the drafted model, we used the reported biomass composition of a related species, Clostridium acetobutylicum[-1 reported in [Clostridium thermocellum) [C. thermocellum during cell growth [-1 to 0.027 h-1). By inhibiting the acetate production from 2.0 mmol/g DCW/h to 0.6 mmol/g DCW/h, the ethanol production under the optimal growth conditions could be improved by 33%. Therefore, we have shown the platform is able to make reasonable predictions for the biomass growth in response to metabolites synthesis.The biomass composition of butylicum. Besidesbutylicum-32. In oorted in ) to be 2ocellum) . Such a l growth . This rel growth . The strThermoanaerobacter sp. strain X514 (used in the MicrobesFlux case study) is capable of growing under high temperatures and converting sugars to ethanol, which can be simulated by a medium-scale metabolic model. Such a convenient FBA model has decent predictive power for in silico studies of a non-model environmental microorganism for bioethanol production at different growth rates  in MicrobesFlux to improve the accuracy and predictive power of the drafted metabolic models.In summary, we have developed MicrobesFlux to draft the metabolic models of environmental organisms from the KEGG database and to serve as a high-throughput tool for systems biology. The user manual is also provided along with this software (Additional file Project name: MicrobesFlux· Project homepage:http://tanglab.engineering.wustl.edu/static/MicrobesFlux.html· Operating systems: Platform independent· Programming language: Java and Python· License: MicrobesFlux is freely available for noncommercial purposes.· Any restrictions to use by non-academics: none· Xueyang Feng and You Xu are the authors contributed equally to this work.The authors declare that they have no competing interests.XF and YJT initiated this study. YX, XF and YC developed the MicrobesFlux software. XF and YX performed the flux analysis. XF and YJT wrote the manuscript. All authors revised and approved the final manuscript.SBML of TOY model used in MicrobesFlux.Click here for fileSimulated results of TOY model used in MicrobesFlux.Click here for fileMATLAB Code for FBA of TOY model.Click here for fileSBML of drafted model of Thermoanaerobacter sp. strain X514.Click here for fileSimulated results of drafted model of Thermoanaerobacter sp. strain X514.Click here for fileUser manual of MicrobesFlux.Click here for file


In prokaryotic genomes the number of transcriptional regulators is known to beproportional to the square of the total number of protein-coding genes. Atoolbox model of evolution was recently proposed to explain this empiricalscaling for metabolic enzymes and their regulators. According to its rules, themetabolic network of an organism evolves by horizontal transfer of pathways fromother species. These pathways are part of a larger “universal”network formed by the union of all species-specific networks. It remained to beunderstood, however, how the topological properties of this universal networkinfluence the scaling law of functional content of genomes in the toolbox model.Here we answer this question by first analyzing the scaling properties of thetoolbox model on arbitrary tree-like universal networks. We prove that criticalbranching topology, in which the average number of upstream neighbors of a nodeis equal to one, is both necessary and sufficient for quadratic scaling. Wefurther generalize the rules of the model to incorporate reactions with multiplesubstrates/products as well as branched and cyclic metabolic pathways. Toachieve its metabolic tasks, the new model employs evolutionary optimizedpathways with minimal number of reactions. Numerical simulations of thisrealistic model on the universal network of all reactions in the KEGG databaseproduced approximately quadratic scaling between the number of regulatedpathways and the size of the metabolic network. To quantify the geometricalstructure of individual pathways, we investigated the relationship between theirnumber of reactions, byproducts, intermediate, and feedback metabolites. Ourresults validate and explain the ubiquitous appearance of the quadratic scalingfor a broad spectrum of topologies of underlying universal metabolic networks.They also demonstrate why, in spite of “small-world” topology,real-life metabolic networks are characterized by a broad distribution ofpathway lengths and sizes of metabolic regulons in regulatory networks. It has been previously reported that in prokaryotic genomes the number oftranscriptional regulators is proportional to the square of the total number ofgenes. We recently offered a general explanation of this empirical powerlawscaling in terms of the “toolbox” model in which metabolic andregulatory networks co-evolve together. This evolution is driven by horizontalgene transfer of co-regulated metabolic pathways from other species. Thesepathways are part of a larger “universal” network formed by theunion of all species-specific networks. In the present work we address thequestion of how topological properties of this universal network influence thepowerlaw scaling of regulators in the toolbox model. We also generalize itsrules to include reactions with multiple substrates and products, branched andcyclic metabolic pathways, and to account for optimality of metabolic pathways.The main conclusion of our analytical and numerical modeling efforts is that thequadratic scaling is the robust feature of the toolbox model in a broad range ofuniversal network topologies. They also demonstrate why, in spite of“small-world” topology, real-life metabolic networks arecharacterized by a broad distribution of pathway lengths and sizes of regulonsin regulatory networks. In prokaryotic genomes the number of transcriptional regulators is known toquadratically scale with the total number of protein-coding genes The question we address in this study is how the topology of the universal networkdetermines this scaling exponent. To answer this question we first consider andsolve a more realistic  case in which theuniversal metabolic network is a directed tree of arbitrary topology. While beingcloser to reality than previously solved To make our approach even more realistic we propose and numerically study acompletely new version of the toolbox model incorporating metabolic reactions withmultiple substrates and products as well as branched and cyclic metabolic pathways.Furthermore, unlike random linear pathways on a universal network We will first consider the case where the universal metabolic network is adirected tree. For simplicity in this section we will consider the case ofcatabolic pathways, while identical arguments  apply to anabolic pathways. The root of the tree corresponds tothe central metabolic core of the organism responsible for biomass production.Peripheral catabolic pathways (branches of the tree) convert external nutrients(leaves) to this core, while the internal nodes of the tree representintermediate metabolites. Each of metabolites is characterized by its distanceevel see. Anorgaevel see at distaThe toolbox model specifies rules by which organism acquires new pathways in thecourse of its evolution. It consists of the following steps: 1) randomly pick anew nutrient metabolite  2) use the universalnetwork to identify the unique linear pathway which connects the new nutrient tothe root of the tree (the metabolic core) and finally 3) add the reactions andintermediate metabolites in the new pathway to the metabolic network of theorganism . To derive the equation for We further assume that due to random selection d starting from This equation allows one to iteratively calculate p2, p1 andp0 correspondingly. If the average number ofparents The Galton-Watson branching process The principal geometric difference between supercritical and critical trees isthat in the former case the number of nodes in a layerA random critical network where each node has at most has two parents in theprevious layer is defined by dsatisfies A critical branching process that has not terminated by level Here This quadratic relation is exact in a critical branching tree where each node canbranch out into at most two nodes at the next layer, and it is still correct toa leading order in As was explained in the previous section the ratioThe geometrical properties of the universal network such as its total number ofnodes/edges cal tree  verifiedFor a supercritical branching process Here Note that this equation connecting In conclusion, while the toolbox model on a critical universal network ischaracterized by a quadratic scaling between To test our mathematical results for a more realistic version of the universaltree we linearized pathways and reactions in the network formed by the union ofall reactions in the KEGG database see Ref. .To better understand the origins of this scaling we investigated the topology ofthe underlying universal tree. The criticality of a tree is defined by theasymptotic value of the ratio In addition to using a random spanning tree to linearize the KEGG network we alsotried a version using minimal paths. In this version the universal network isgenerated by randomly picking a metabolite and connecting it to the root of thetree (pyruvate) by the shortest path. At a first glance such “minimalpath” selection appears to be reasonable from evolutionary standpoint.Indeed, evolution would favor simpler and shorter pathways in order to minimizethe expenditure of resources to achieve a given metabolic goal 2 →cyanate. As a consequence of such artificial shortcuts branches of the universalnetwork linearized by minimal paths are much shorter than they are in reality..This problem is at least partially alleviated by 1) removing unusuallyhigh-degree nodes corresponding to common co-factors such as H2O,ATP, NAD in the metabolic network so that some unrealistic paths are eliminated,and also 2) using random spanning tree instead of the shortest paths. In Ref.How do we reconcile the evolutionary pressure apparently selecting for minimalpathways with dramatically wrong scaling properties of this model? We believethat most of the ultra-short “small world” pathways generated byminimal paths on the KEGG network are unrealistic from biochemical standpoint.Indeed, highly connected co-factors often position metabolites with verydifferent chemical formulas in close proximity to each other. For example, theKEGG reaction R00134: Real metabolic reactions routinely include multiple inputs (substrates) andmultiple outputs (products) . The new pathways areAt the beginning of the simulation, the model organism starts with a“seed” metabolic network consisting of 40 metabolitesclassified by the KEGG as parts of central carbohydrate metabolism, plusa number of “currency” metabolites such as water, ATP andNAD . In case of multi-layer expansion processsome substrates of this reaction are not among the core metabolites(otherwise this reaction would be in the first layer). One then goesdown one layer and adds the reactions fabricating these missingsubstrates. This is repeated all the way down to the first level of theoriginal expansion. The resulting pathway includes the minimal  set of reactions needed to generate the targetmetabolite from the current metabolic core of the organism. Startingfrom the next step of the model the target and all intermediatemetabolites become part of the metabolic core. Genes for enzymescatalyzing these new reactions are assumed to be horizontallytransferred to the genome of the organism. The newly added metabolicpathway is assumed to have a dedicated transcriptional regulator so thatthe number of transcription factors in our model is always equal to thenumber of pathways or their target metabolites.Steps 1–5 are repeated until metabolic network of the organismreaches its maximal size. At this stage it includes the entire scopeNumerical simulation of this model shows that the number of transcriptionalregulators scales with the number of metabolites with power . This isThe mathematical formalism derived in the previous sections is limited totree-like universal networks and thus does not directly apply to the new model.Nevertheless, one generally expects the quadratic scaling to be limited only tocritical, “large world” networks in which organisms with smallgenomes initially tend to acquire sufficiently long pathways. As noted before,from purely topological standpoint the KEGG network has a “smallworld” property making long pathways unlikely. It is important to check ifthe realistic treatment of multi-substrate reactions did in fact restore the“large world” property and criticality to the KEGG universal networkby increasing the minimal number of steps required for connecting targetmetabolites to the metabolic core. To quantify the criticality of the expansionprocess as before we use the ratio nborder rxn–thenumber of added reactions that are connected (as a substrate or a product) withat least one metabolite in the core, 2)nbase–the number of metabolites in the corethat serve as substrates to reactions in the added pathway, 3)nfeedback–the number of core metabolitesthat are products of reactions in the new pathway, 4)nbyproduct–the number of final metabolicproducts of the added pathway that are neither core metabolites nor the target,5) length-the number of steps (layers of the scope expansion process) it takesto transform core metabolites into the target product. 4 illustrates thedefinition of these parameters while Unlike linearized pathways in the original version of the toolbox model nborder rxnvs. nbase) or as products(nfeedback). Further analysis indicates that“currency metabolites” (common co-factors that serve as substratesor products of many reactions) constitute a significant fraction(∼25%) of all core metabolites involved in border reactions  metabolicuniverse of all reactions in the KEGG database. The only simplifying approximationsremaining in the new most realistic version of the toolbox model is random selectionof metabolic targets in the course of evolution and easy availability of any subsetof KEGG reactions for horizontal transfer. Both these approximations can be relaxedin later versions of the model. Another promising direction is to extend the toolboxmodel to artificial chemistry universal networks of Refs. ftp.genome.jp/pub/kegg/ligand/reaction/reaction?mapformula.lst. Theuniversal network with linearized pathways used to construct The universal network used in our study consists of the union of all reactions listedin the KEGG database. The directionality of reactions and connected pairs ofmetabolites were inferred from the map version of the reaction formula: Text S1Supplementary information.(0.37 MB DOC)Click here for additional data file.


Motivation: Metabolic pathways are complex systems of chemical reactions taking place in every living cell to degrade substrates and synthesize molecules needed for life. Modeling the robustness of these networks with respect to the dysfunction of one or several reactions is important to understand the basic principles of biological network organization, and to identify new drug targets. While several approaches have been proposed for that purpose, they are computationally too intensive to analyze large networks, and do not properly handle reversible reactions.Results: We propose a new model—the flux balance impact degree—to model the robustness of large metabolic networks with respect to gene knock-out. We formulate the computation of the impact of one or several reaction blocking as linear programs, and propose efficient strategies to solve them. We show that the proposed method better predicts the phenotypic impact of single gene deletions on Escherichia coli than existing methods.Availability:https://sunflower.kuicr.kyoto-u.ac.jp/∼tyoyo/fbid/index.htmlContact:takutsu@kuicr.kyoto-u.ac.jp or Jean-Philippe.Vert@mines.orgSupplementary information:Supplementary data are available at Bioinformatics online. Metabolic pathways are complex systems of biochemical reactions taking place in every living cell to degrade substrates and synthesize molecules needed for life. Any metabolic dysfunction may lead to the impossibility to degrade or produce crucial molecules for the organism, potentially inducing disease or death. Yet cells seem to be able to maintain their normal functions despite many perturbations, such as the gene knock-out or DNA mutations perturbing the functions of proteins, while being sensitive to some specific attacks -catalyzed reactions that bridge these metabolites to transformation processes. A gene perturbation, such as knock-out or DNA mutation, can inhibit one or several reactions in a metabolic system. The impact of this perturbation on the cell phenotype can vary widely, ranging from no effect to cell death, depending on how many other reactions and crucial metabolites are impacted in cascade.Several approaches have been proposed to model and predict the phenotypic impact of inhibiting one or several genes through metabolic network perturbation. Flux balance analysis (FBA) is a constraint-based mathematical model, which uses the stoichiometry of a given metabolic network along with a biologically relevant objective function to identify optimal reaction flux distributions . It can P-complete, and although Other approaches model the effect of gene knock-out using the concept of elementary modes (EMs), which are minimal sets of reactions that can operate at the steady state, such that all irreversible reactions involved are used in the appropriate direction . Acuna eAlternatively, An alternative route to model the impact of a perturbation on a metabolic network is to start from a dynamic model of metabolism and assess how the model is impacted when a reaction is inhibited. Boolean models, in particular, are popular to describe and analyze large-scale metabolic networks , builds on the concept of steady-state fluxes and variability of FBA and FVA. The FBID of a perturbation is defined as the number of reactions that become inactive in all steady states after perturbation. We show that the FBID can be computed either by enumerating all EMs of the metabolic network, or by solving a series of linear programs, the later scaling much better to large networks. In contrast to techniques like FBA, FVA and MOMA, the new FBID does not require the definition of a specific objective function to model growth. Experiments on the Escherichia coli metabolic network show that FBID is competitive with existing approaches. It is computationally efficient even for global metabolic networks, where it outperforms existing approaches in terms of prediction accuracy.In this study, we propose a new model to assess the impact of gene perturbations on a metabolic network, together with efficient algorithms to compute it of large-scale networks. The model, which we call S, where m is the number of metabolites and n is the number of reactions in the network. The activity of the network is represented by a flux vector We represent a metabolic network by its b are set to 1. This ensures that fluxes are bounded by  for reversible reactions, and by  for irreversible ones.In addition, flux vectors must satisfy additional constraints of the form x in The metabolic networks we consider are usually under-determined because there are usually more reactions than metabolites  to solve the LP instances of the LP-based method, and CellNetAnalyzer which is a free software running under MATLAB to compute the EMs of a network , collected by The second dataset , while for iJO1366, we use the core growth biomass proposed by Because these experimental datasets are under different conditions, we used different input sources and output biomass in the networks for the different datasets, as listed in E.coli metabolic networks listed in For each of the three We proposed two algorithms to compute the FBID of a perturbation: one approach based on enumerating EM (Section 2.2), and one approach based on an LP formulation (Section 2.3). Supplementary Information).We see that the EM-based method is fast for a small network but not efficient for large ones; in fact, CellNetAnalyzer did not manage to compute the EMs of both large networks within a week. This is coherent with the exponential complexity of the method. On the other hand, although many LP instances need to be solved for the LP-based method, we see that its polynomial complexity allows it to better scale to large networks. fastFVA  manages E.coli global metabolic networks (iJE660 and iJO1366) are summarized in the Supplementary Materials.Based on these observations, in what follows we only run the LP-based implementation with fastFVA to compute the FBIDs corresponding to the different genes and networks investigated for phenotypic prediction. The total computation times for both msbA (b0914) gene, a bacterial lipid flippase whose knock-out blocks ATP synthesis by oxidative phosphorylation, or 448 for acpP (b1094), a acyl carrier protein that catalyzes polyketide biosynthesis of holo-ACPS; unsurprisingly, both are essential genes.The FBIDs computed on each network vary significantly between different genes. For example, E.coli metabolic network (iJE660 and iJO1366). In each experimental data, the genes are separated in two classes corresponding to genes with a large or small phenotypic impact. By thresholding the FBID to some level, we can predict that genes with an FBID above the threshold should have large phenotypic impact, while those below the threshold should not. To assess more quantitatively how predictive the FBID is for gene essentiality, we systematically compared the FBID corresponding to each gene deletion with the corresponding experimental phenotypic data, for both versions of the large E.coli networks are overall similar, with an advantage for the former on all phenotype datasets. The performance on the first dataset (collected from literature) is rather disappointing. This can be explained, to some extent, by the fact that we had to modify the network by using the minimal inputs together with distinct carbon sources, which resulted in many metabolites and reactions being always inactive at steady state. The performance on the insertional mutants dataset is also not good and may also be due in part to the particular context of using minimal inputs. For the Gerdes dataset and KEIO collection, FBID performs pretty well on both networks, reaching an AUC of 0.66 and 0.78 for iJE660 and 0.6 and 0.72 for iJO1366, respectively.We can see that phenotype prediction for the iJE660 and iJO1366 E.coli network that was used by To compare the performance of FBID with existing approaches, we first focus on the iJE660 On the mutants collected from literature, our approach based on FBID is clearly worse than SA, FBA and EMA, which reach high accuracy (90% for EMA). This can be explained, to some extent, because this collection includes genes that only catalyze the central metabolism  where alTo further investigate the performance of FBID on large networks, we compare it with FBA on the largest iJO1366 network for the prediction of gene essentiality as defined in the KEIO collection. Supplementary Information.As shown on We have proposed FBID, a new definition of impact degree, which can not only efficiently deal with the reversible reactions in metabolic networks but also have the state conditions being taken into account. To compute the FBID against perturbations, we proposed two algorithms, an LP-based method and an EM-based algorithm. The advantage of the LP-based method is that it can solve all LP instances individually, can strongly benefit from warm restart techniques and is amenable to parallelization. Contrary to other LP-based formalisms like FBA, FVA or MOMA, it does not depend on a subjective definition of a relevant objective function. Although computational cost of the LP-based method grows with the network size and the number of perturbations to be tested, the overall time complexity is still bounded polynomially. If we are interested in only a few candidate perturbations, then only the corresponding LP need to be solved. The EM-based method, on the other hand, can compute the FBID of specific perturbations fast for middle-scale networks. The main computational advantage of this approach is that the computation of EMs needs to be performed only once, no matter how many perturbation we want to test—including perturbations involving several reactions. This advantage vanishes for large-scale metabolic networks, however, because of the exponential complexity of computing EMs and the lack of efficient algorithms for that purpose.E.coli metabolic networks. The results on computational time for calculating the FBID of different sized networks show that the LP-based method implemented with fastFVA is efficient, while the EM-based method did not return any result for large networks owing to the difficulty of computing the EMs. In terms of phenotype prediction, we obtained poor results when we tested metabolic networks with a minimal source input because many metabolic paths are always closed in this case. Comparison of the performance of phenotype prediction with some existing methods indicates that the FBID performs as well as other models or even better, particularly on large networks.We carried out computational experiments by using number of EMs remaining, we focus instead on the number of reactions that can still be activated in the remaining EMs. Although the number of EMs in a network has been used as a measure of flexibility and as an estimate of fault-tolerance (number of reactions inactivated is itself a crude measure, and investigating variants such as weighting reactions by their ‘importance’ before counting them may be interesting future work.The interpretation we give of the FBID in terms of EMs makes an interesting link with existing EM-based methods that measure how many EMs disappear when we inhibit a reaction (


E. coli and reference networks downloaded from the Kyoto Encyclopedia of Genes and Genomes (KEGG) database, we have found that the developed method can appropriately solve MRI in the Boolean model and is applicable to large scale-networks for which an exhaustive search does not work. We have also compared the developed method with the existing connectivity-based methods and FBA-based methods, and show the difference between the solutions of our method and the existing methods. A theoretical analysis of MRI is also conducted, and the NP-completeness of MRI is proved in the Boolean model. Our developed software is available at “http://sunflower.kuicr.kyoto-u.ac.jp/~rogi/minRect/minRect.html.”In this paper, we consider the Minimum Reaction Insertion (MRI) problem for finding the minimum number of additional reactions from a reference metabolic network to a host metabolic network so that a target compound becomes producible in the revised host metabolic network in a Boolean model. Although a similar problem for larger networks is solvable in a flux balance analysis (FBA)-based model, the solution of the FBA-based model tends to include more reactions than that of the Boolean model. However, solving MRI using the Boolean model is computationally more expensive than using the FBA-based model since the Boolean model needs more integer variables. Therefore, in this study, to solve MRI for larger networks in the Boolean model, we have developed an efficient Integer Programming formalization method in which the number of integer variables is reduced by the notion of feedback vertex set and minimal valid assignment. As a result of computer experiments conducted using the data of metabolic networks of  In (a), partial pathways can be recruited from independent organisms and co-localized in a single host. For example, 1,3-propanediol is synthesized by Nakamura et al. in which pathways from Saccharomyces cerevisiae and Klebsiella pneumonia were assembled in E. coliMetabolism is one of the most important biological processes in organisms. Relations between reactions and chemicals in the metabolism are often represented by metabolic networks The “pathway prediction system” (PPS) of the University of Minnesota Biocatalysis and Biodegradation Database (UM-BBD) is designed to predict routes for the biodegradation of xenobiotic compounds connectivity model, flow model, and Boolean model. For each of them, Minimum Reaction Insertion (MRI) problem can be defined for finding the minimum number of additional reactions from a reference metabolic network to a host metabolic network so that a target compound becomes producible in the revised host metabolic network. In the connectivity model such as seeds and the producible compounds are called the scope of the seed. In this model, a Boolean function of “AND” is attached to each reaction node and “OR” is attached to each compound node in the metabolic networks.In the type (a) problem, it seems that there are three major models for judging the producibility of target compounds, that is, substrates whereas C and D are called products. In the connectivity model, either A or B is necessary to produce C and D, whereas both A and B are necessary for the Boolean model. In the flow model including FBA, in addition to the condition that both A and B must exist, both C and D are necessary to be consumed by other reactions. Thus, each model outputs a different solution for producing desired compounds.For example, suppose that there is a chemical reaction “A+B→C+D”, where A and B are called From the view point of computational complexity, although the connectivity model is very simple and then applicable even to very large networks, its logical analysis ability is not strong since it cannot detect the lack of necessary substrates. The good point of the flow model is its computational efficiency since problems in the flow model can often be formalized by linear programming, for which there exist polynomial time algorithms et al.et al.et al.et al.et al.et al.Although the computational time of the FBA-based method for MRI is very small and scalable for genome-scale metabolic reconstruction As for the shortcoming of the FBA-based method for MRI, it tends to be considerably affected by the redundancy of the given metabolic network since each node is affected not only by the incoming flows but also by the outgoing flows. For example, suppose that a metabolic network of host network, is given; it cannot produce the target compound in its initial form. However, an integrated metabolic network of many other organisms are given as the reference network from which we should find the minimum number of additional reactions so that the target compound becomes producible. We prove that this problem is NP-complete.Therefore, in this research, we study the problem of designing a pathway for producing target compounds in metabolic networks of the Boolean model since its logical analysis ability is more stable than that of the FBA, particularly when the flexible parts of the metabolic networks are large. Our approach is based on (a), that is, the combination of existing pathways. In our problem setting, a base metabolic network of a host organism, which we call the E. coli is used as the host network and the reference pathway of the KEGG database http://sunflower.kuicr.kyoto-u.ac.jp/~rogi/minRect/minRect.html”.Although both the FBA-based model and the Boolean model for MRI are considered to be NP-complete, the former is likely to have a faster exponential time algorithm than the latter since FBA has fewer integer variables. Although the computational complexity of the Boolean model is large, we develop an efficient method based on integer programming (IP) Minimum Reaction Insertion (MRI) in a Boolean model is first explained with an example and then mathematical formalization is described.In this section, the main problem Suppose that a metabolic network shown in metabolic network can be represented by a directed graph compound nodes and reaction nodes. source nodes and target node. Source nodes have no incoming edges and correspond to the seed compounds of MRI is mathematically defined as follows: A host networkreference networkadditional compound nodes .Suppose that a valid assignment if the following conditions are satisfied: (i) for each Let source connected if there is a directed path from minimal valid assignment, which is similar to the notion of maximal valid assignment defined in minimal if If Minimum Reaction Insertion as follows:Now we define the Input: A host metabolic network Output: A minimum cardinality set of Minimum Reaction Cut (MRC) As mentioned in the section of Theoretical Results, a minimal valid assignment is uniquely determined if MRC is a problem to find a minimum set of reactions that interfere with the production of target compounds Here, we show IP formalization methods for MRI in the Boolean model. To apply IP, problems must be formalized to maximize or minimize a given objective function which is a linear function of integer variables and constraints must also be given as linear equations or inequations of integer variables.IP-MRI-A for solving Minimum Reaction Insertion is as follows where the time step increases by 1 when the Boolean calculation is synchronously conducted for every node:Suppose that the host network and the reference network are given as shown in IP-MRI-AMinimizeSubject tofor all(3) represents the Boolean relation For a compound node with indegree 1, the value of the predecessor node is just copied. For example, since For a compound node with indegree more than 1, it is necessary to convert the “As for the reaction nodes not included in the host network, TERSince we assume minimal valid assignment, at IP-MRI-B is obtained.The above formalization can clearly solve MRI and obtain the correct solution IP-MRI-BMinimizeSubject toWhen compared to IP-MRI-A, (18),(19),(20)–(24), (25)–(29), (30), and (31) correspond to (1),(2),(3)–(7), (8)–(12), (14), and (17) respectively although the notion of time is not used in IP-MRI-B. (13) in IP-MRI-A means that the value of In IP-MRI-B, the solution of IP is Thus, IP-MRI-A can solve MRI, but In the improved method, IP-MRI-C, we firstly find an FVS Although finding the minimum FVS is known to be NP-complete, it is not necessary to use the minimum FVS in our problem setting. We use a simple greedy algorithm to choose FVS as follows:Procedure fordo   fordo    while there exists    if there exists then        for alldo          else if there exists do        else if there exists do     returnSince the reaction nodes for FVS are chosen by a greedy algorithm, the size of FVS is not always optimal. However, it is important to note that even if the size of FVS is not optimal, the solution of MRI calculated by IP-MRI-C is always optimal. If there are multiple optimal solutions in MRI, there is a possibility that the solutions are different since IP outputs only one solution. However, it may be possible to enumerate all optimal solutions of MRI by iteratively solving IP with a constraint to avoid the already chosen solutions.For example, IP-MRI-C for IP-MRI-CMinimizeSubject tofor allWhen compared to IP-MRI-A, (32),(33),(34)–(38), (39)–(43), (44), (46), and (49) correspond to (1),(2),(3)–(7), (8)–(12), (13), (14), and (17), respectively. Note that the number of time steps is 2 = Additionally, if we use the FVS-based method and no cycles are included in We conducted computer experiments for solving MRI with data downloaded from the KEGG database. The experiment was conducted on a PC with an Intel(R) Xeon(R) 3.33 GHz CPU and 10 GB RAM having the SUSE Linux (version 12.2) operating system, where CPLEX (version 12.4.0.0) was used as the solver of integer programming.E. coli are used for the host networks. The major difference between the pathway alignment methods by KEGG and our developed method is that our method is based on a Boolean model, whereas the pathway alignment methods consider only the topology of networks.In this study, a reference network consists of the central metabolism and the related modules necessary for producing the target compound. A map of the KEGG PATHWAY is a minimum unit, and three or four maps of the KEGG PATHWAY are chosen and integrated as the reference network in each of our experiments. As for species, a reference network includes the chemical reactions of all species, whereas the metabolic networks of E. coli. Because the purpose of this study is not focused on the reconstruction of genome-scale metabolic network model, but the design of a minimal genome in addition to the existing pathways to produce a desired compound, each reference network consists of the maps of the KEGG pathway located between the central metabolism and each target compound.In synthetic biology, it is of great interest to construct a minimal genome that realizes the desired functions E. coli (eco00010.xml), and the reference network covers glycolysis, gluconeogenesis and glycerolipid metabolism of other species (ko00010.xml and ko00561.xml). The numbers of compound and reaction nodes are 58 and 85, respectively, where 30 reactions are reversible. The source nodes are D-glucose (C00031), oxaloacetate (C00036), salicin (C01451), arbutin (C06186), UDP-glucose (C00029), acyl-CoA (C00040), and diglucosyl-diacylglycerol (C06040), which are represented by black circles in E. coli and works as a connection between glycolysis and glycerolipid metabolism, the obtained In the first computer experiment, the target compound is propanol (C00479 in KEGG ID), the host network is glycolysis and gluconeogenesis of http://www.genome.jp/tools/pathcomp/”) was used to calculate the solution of the shortest path-based model. In the experiment, the host network consists of glycolysis, gluconeogenesis and citrate cycle of E. coli (eco00010.xml and eco00020.xml), and the reference network consists of glycolysis, gluconeogenesis, citrate cycle and pentose phosphate pathway of other species . The numbers of compound and reaction nodes are 64 and 108, respectively, where 59 reactions are reversible. There are four source nodes, D-glucose(C00031), arbutin(C06186), salicin(C01451), and acetate (C00033), and the number of candidates for the additional reactions is 66. When the target compound is sedoheptulose 7-phosphate (C05382), as shown in To show the difference between the developed model and the shortest path-based models, we conducted the second experiment where PathComp of KEGG  and sedoheptulose 7-phosphate (C05382). However, it is clear that R01827 does not occur if D-erythrose 4-phosphate (C00279) does not exist. Thus the difference between the shortest path-based method and the developed method is that the developed method considers Boolean constraints for each reaction and compound whereas the shortest path-based method only considers the connectivity of nodes.E. coli (eco00010.xml), that is, D-glucose(C00031), arbutin(C06186), salicin(C01451), oxaloacetate(C00036) and acetate (C00033). The reference network consists of glycolysis, gluconeogenesis, citrate cycle, pentose phosphate pathway and butanol metabolism of other species , where R01172 is treated as a reversible reaction. The target compound is butanol (C06142). The numbers of compound and reaction nodes are 93 and 150, respectively, where 87 reactions are reversible. It took 919.79 s (15m19s) for the developed method to solve MRI and the solution was Next, we conducted the third experiment to show the scalability of our method. The host network consists of the source nodes of glycolysis and gluconeogenesis of Finally, we conducted an experiment to show the difference between the developed model and the FBA-based model. We assume that the reference network consists of glycolysis, gluconeogenesis, citrate cycle, pentose phosphate pathway and butanol metabolism of other species , and the host network includes only one reaction R04394 between salicin (C01451) and salicin 6-phosphate (C06188) as shown in Then, the solution of MRI in our Boolean model is {R05134, R02736, R02035, R02036, R05605, R00344, R00342, R01082, R01087}, whereas the solution of FBA-based model is {R05134, R02736, R02035, R02036, R05605, R01058, R01518, R00658, R00200, R00344, R00342, R01082, R01087}. It is to be noted that {R01058, R01518, R00658, R00200} is not necessary for the Boolean model, but necessary for the FBA-based model. In the Boolean model, R01058 is not necessary to produce C01384 since the lack of reactions in downstream does not affect. However, in the FBA model, R01058 is necessary. Otherwise, C00118 is not consumed and then R05605 , which is a problem for a given graph to find the minimum number of nodes so that each edge is incident to at least one of the selected nodes. For example, for the graph shown in Let The reference network For example, MVC for the graph shown in In the following paragraphs, we show that MVC for Theorem 2: Given a host network, a reference network and a set of additional reactions, a minimal valid assignment is uniquely determined.Proof: For any valid assignment In this paper, we formalized an optimization problem MRI in a Boolean model with a notion of minimal valid assignment. We proved that MRI in the Boolean model is NP-complete and the minimal valid assignment is uniquely determined when E. coli and the reference network of KEGG was used as the reference network. The results of the computer experiments confirmed the correctness and the scalability of the developed method, and the appropriateness of the problem setting of MRI.We also conducted four computer experiments in which data were downloaded from the KEGG database, CPLEX was used as the IP solver, and propanol, butanol, sedoheptulose 7-phosphate, and maleic acid were used as the target compound for each experiment. The host network was a metabolic network of An important advantage of our Boolean model is its capability of detecting the lack of substrates, whereas the connectivity-based methods cannot appropriately handle this point. An extended type of connectivity-based method is BNICE, which enumerates all possible pathways from the source nodes to the target compound, and uses thermodynamical feasibility and pathway length to evaluate each candidate pathway. In contrast, the developed method evaluates each candidate pathway based on the number of additional reactions. Another advantage of the developed model is its capability of handling branches and/or cycles in a pathway from the source compounds to the target compound, whereas BNICE considers only the non-branching paths. However, since BNICE nicely evaluates each pathway by the thermodynamic free energy of the included compounds and length, considering the thermodynamic free energy in a Boolean model represents an important direction of our future work.It is to be noted that the solution of MRI in the FBA-based model is different from that in the Boolean model. In particular, if the reference network includes a large redundant part, the FBA-based model tends to output a larger solution than the Boolean model, although the FBA-based model is very fast when compared to the Boolean model. Therefore, one of our future works is to develop a hybrid method combining the FBA-based method and the Boolean-based method. Petri-net-based methods


The formation of intracellular aggregates is a common etiology of several neurodegenerative diseases. Mitochondrial defects and oxidative stress has been pointed as the major mechanistic links between the accumulation of intracellular aggregates and cell death. In this work we propose a “metabolic cell death by overcrowding” as an alternative hypothesis. Using a model of neuron metabolism, we predict that as the concentration of protein aggregates increases the neurons transit through three different metabolic phases. The first phase (0–6 mM) corresponds with the normal neuron state, where the neuronal activity is sustained by the oxidative phosphorylation of lactate. The second phase (6–8.6 mM) is characterized by a mixed utilization of lactate and glucose as energy substrates and a switch from ammonia uptake to ammonia release by neurons. In the third phase (8.6–9.3 mM) neurons are predicted to support their energy demands from glycolysis and an alternative pathway for energy generation, involving reactions from serine synthesis, one carbon metabolism and the glycine cleavage system. The model also predicts a decrease in the maximum neuronal capacity for energy generation with increasing the concentration of protein aggregates. Ultimately this maximum capacity becomes zero when the protein aggregates reach a concentration of about 9.3 mM, predicting the cessation of neuronal activity. The cell has a high density of macromolecules occupying about 30–40% of the intracellular space Brain metabolism has been studied extensively using different experimental techniques. Time resolved experiments have shown that the early neuronal activity (up to about 10 sec) is supported by oxidative phosphorylation of lactate from an extracellular pool In this work we provide evidence that these altered metabolic phenotypes could be the consequence of an increase in molecular crowding due to accumulation of protein aggregates.in silico model of human cell metabolism To investigate the impact of protein aggregates accumulation on cell metabolism we use an We will utilize energy demand (moles of ATP consumed per unit time and cell volume) as a surrogate for neuronal activity. The scope of the model is to predict which metabolic pathways satisfy this energy demand with the minimal use of nutrients. We note that resting human cells require energy to maintain their basal functions, which has been estimated to be about 5 mM ATP/min In this work we focus on stationary metabolic states of neuronal activity. This is a reasonable approximation for resting neurons with a steady energy demand for cell maintenance. A stationary state approximation also applies to modeling sustained neuronal activity with a steady energy demand. Under the steady state approximation the production and consumption of every metabolite balances and the metabolite concentrations remain constant . There are of course other physiological regimes that are dynamic in nature. However, we will show that even within the context of the steady state approximation we obtain very interesting predictions regarding the differential utilization of metabolic pathways.For any given stationary distribution of metabolic fluxes there is a corresponding distribution of metabolic enzymes, ribosomes and mitochodria that is required to catalyze those reactions. The total volume occupied by these macromolecules/organelles should not exceed 40%, which is the typical macromolecular volume fraction in human cells (molecular crowding constraint). In addition, we take into account the volume fraction occupied by non-metabolic macromolecules, including structural proteins and protein aggregates. In a normal state, most of the cellular proteome is composed of structural proteins  and macromolecules/organelles necessary to maintain their basal metabolism. The concentration of non-metabolic proteins is about 3 mM Summarizing, the metabolic model studied here predicts the metabolic fluxes that will result in a net generation of a predefined ATP demand  with the minimal use of nutrients and satisfying the flux balance and molecular crowding constraints. Since glucose and lactate are the major energy sources for neurons, we focus on their differential utilization depending on the energy demand and the concentration of protein aggregates. In addition, we assume that the cell media contains all essential nutrients  and oxygen.First, we focus on the impact of protein aggregates accumulation on the metabolism of resting neurons. We assume that the metabolic requirements of resting neurons are a basal ATP demand for cell maintenance and basal rates of RNA and protein synthesis to compensate for the basal turnover rates of RNA and protein, respectively.We observe significant changes in the model predicted exchange fluxes when increasing the concentration of protein aggregates . SpecifiThese changes in exchange fluxes are accompanied by changes in the relative abundance of mitochondria and cytosolic enzymes. For protein aggregate concentrations below 6 mM mitochondria occupy about 12% if the intracellular volume . HoweverIn Phase 1 lactate is the preferred energy substrate . After uIn this phase there is a mixed utilization of lactate and glucose as energy substrates . There aIn this phase there is a complete switch to glucose as the energy substrate . GlucoseThese results can be generalized to situations where the neuronal activity demands more energy than what is required for cell maintenance. More precisely, we can depict the different metabolic phases in the plane of protein aggregate concentration and energy demand . To thisThe model predicts that the transition from Phase 1 to Phase 2 and from Phase 2 to Phase 3 is shifted to lower protein aggregate concentrations as the energy demand is increased. We also note that the maximum ATP demand that can be satisfied by the cell metabolism decreases as the protein aggregate concentration increases  and (Phase 3/Unfeasible region) overlap. This observation simple means that for certain choices of kinetic parameters the Phase 3 is not observed.in silico study leads to very interesting predictions. It indicates that neurons can be in three different metabolic phases depending on the magnitude of the protein aggregate concentrations and the energy demand. Phase 1 represents the normal state of neurons, where the energy requirements are satisfied through oxidative phosphorylation of lactate. The formation of protein aggregates and/or an increase in the energy demand to satisfy neuronal activity can shift the neuron to the Phase 2, characterized by a mixed oxidative phosphorylation of lactate, the production of 3-phosphoglycerate from lactate and glucose, and the metabolism of 3-phsophoglycerate via the SOG pathway, with concomitant histidine uptake and ammonia excretion. A further increase in the protein aggregate concentration and/or a further increase of the energy demand can shift the neuron to Phase 3, where the energy demand is sustained by glycolysis, with concomitant lactate excretion, and the SOG pathway, with concomitant histidine uptake and ammonia excretion.This The contribution of the SOG pathway to energy production is at the current stage theoretical. This pathway has been predicted to be upregulated in highly proliferating cells E. coli as well The model also predicts that neurons have a maximum capacity for energy generation and that this maximum capacity decreases with increasing the concentration of protein aggregates. The maximum capacity becomes zero at protein aggregate concentrations of about 9.3 mM. Beyond the latter value the cell cannot even satisfy is basal energy demand for cell maintenance and therefore the neuron is predicted to die. We call this predicted cell death “metabolic cell death by overcrowding”. This prediction is supported by empirical data showing a negative correlation between the formation of intracellular aggregates and cell survival in human embryonic kidney cells in silico predictions comes from measurements of mitochondrial content and of metabolic byproduct accumulation in the brain of neurodegenerative disease patients. The model prediction of a decrease in mitochondria content as the concentration of protein aggregates increases agrees with the experimental observation of a lower content of mitochondria in the neurons of Alzheimer's Further support for the in silico predictions are also a motivation for further experimental work. According to our analysis the switch from ammonia uptake to ammonia excretion in Phase 2 precedes the switch from lactate uptake to lactate excretion in Phase 3. This prediction points to measurements of ammonia levels as an earlier indicator of neurodegeneration than lactate levels. It also remains to be verified whether the neurons follow the metabolic switches Phase 1 → Phase 2 → Phase 3 as the neuronal activity is increased.The miS to denote the stoichiometric coefficient of metabolite m in reaction i. We use if to denote the steady state reaction rate (flux) of the ith reaction in the metabolic network, where all reversible reactions are represented by a forward and backward rate, respectively. Reactions are divided into nutrient import reactions (RI), reactions taking place outside the mitochondria (RnM) and reactions taking place in the mitochondria (RM). We use cφ to denote the relative cell volume fraction occupied by the cth cellular compartment, where a compartment represents the overall contribution of macromolecules of certain type  or of certain cell organelle . Specifically, here we consider proteins that do not form part of enzyme complexes or ribosomes (P0), all metabolic enzymes catalyzing reactions outside the mitochondria (EnM), all metabolic enzymes catalyzing reactions in the mitochondria (EM), ribosomes (R), and mitochondria (M). We assume the energy demand ATPf and the total relative volume fraction occupied by macromolecules and organelles (maxφ) are known and are given as input parameters of the model. Finally, we estimate the metabolic fluxes and compartment densities as the solution of the following optimization problem:As starting point, we utilize a genome-scale metabolic reconstruction of a generic human cell if and cφ that minimize the sum of nutrient import costs ic is the nutrient import cost associated with the uptake reaction i, ia = iv/eff,ik are the crowding coefficients of metabolic enzymes (enzyme molar volume/enzyme effective turnover)Ra = Rv/Rk is the ribosome crowding coefficient (ribosome molar volume/protein synthesis rate per ribosome), and M,ATPa = s,Mv/Mr the crowding coefficient of mitochondria ATP generation (ATP synthesis rate per mitochondria mass/mitochondria specific volume) Find the ic is equal to the molecular weight of the molecule imported in the auxiliary uptake reaction i , where k is the enzyme turnover number, K the half-saturation concentration and S the substrate concentration. The turnover numbers of some human enzymes are reported in the BRENDA database −1 and a significant variation from 1 to 100 sec−1 .The effective turnover numbers 00 sec−1 . HoweverMa values between 0.0026 to 0.032 min/mM. Except when specified, we use the median 0.017 min/mM. Dividing the ribosome molar volume  by the rate of protein synthesis per ribosome  by the enzymes specific volume  divided by the median molecular weight of a human protein . The median molecular weight of a human protein was estimated from the median protein length  is formulated as follows. We account for three major categories, proteins not associated with metabolism, proteins that are components of enzyme complexes, and ribosomal proteins, with their concentrations (moles/cell volume) denoted by P0, EP, and RP, respectively. In non-proliferating cells, these concentrations will decrease at a rate Dk(P0+EP+RP). The total concentration of proteins in enzyme complexes can be estimated as EP = PEEn = PEnΣifi/eff,ik, where PEn is the average number of proteins in an enzyme complex (about 2.4) and E is the total concentration of metabolic enzymes. Similarly, RP = PRφRn/Rv, where PRn is the number of proteins in a ribosome (82 for the 80S ribosomes) and Rφ/Rv is the concentration of ribosomes. Putting all these elements together, the balance between protein turnover and synthesis implies Protein_sysnthesisf-Dk[P0+ PEnΣi + (PRn/Rv)Rφ] = 0.The flux balance equation for proteins  with m =eff,ik. Flux bounds were set to i,minv = 0 and i,maxv = ∞ unless specified. The lower and upper bounds of the ATP maintenance, basal RNA degradation, and basal protein degradation were set to the maintenance parameters reported above. The lower and upper bounds of the biomass producing reaction were set to zero.The optimization problem in k are mostly concentrated in the range from 1 to 100 sec−1 (10(effk) values from a uniform distribution in the range between log10(1) to log10(100). For each specified condition we run 100 simulations. On each simulation, for each reaction, a value of eff,ik is extracted from the distribution described above. With this set of eff,ik parameters we then solve the optimization problem (1)–(5) and obtain estimates for the reaction rates. Because the eff,ik are real value random numbers, the solution of the linear optimization problem is unique for each set of kinetic parameters. Based on the 100 simulations we finally estimate the median and 90% confident intervals for the rate of each reaction. 100 simulations were proven to be sufficient to capture the overall range of behavior, since running 1,000 simulations did not result in any significant change of the flux distributions. Specifically, for a given ATP demand and a give reaction, we have performed a Kolmogorov-Smirnov test to determine whether the flux distribution as obtained from 100 kinetic parameter sets is statistical equivalent to that obtained using 1,000 kinetic parameter sets. We performed this test for a protein aggregate density of P = 3 mM and different ATP demands. Given that we performed this test for thousands of reactions and to correct for multiple hypothesis tests, we set 0.005 = 0.05/1,000 as the threshold to call statistical significance. In all cases the hypothesis that the distributions are equivalent (p>0.005) was satisfied, indicating that the flux distributions obtained for 100 kinetic parameter sets are equivalent o that for 1,000 parameter sets. Therefore, 100 kinetic parameter sets is sufficient to have a reasonable exploration of the flux distribution. The results of this statistical test are reported in the The turnover numbers of human enzymes 00 sec−1 . Based oTable S1Turnover numbers of selected human enzymes.(XLS)Click here for additional data file.Table S2Sensitivity analysis. The metabolic flux predictions are dependent on the choice of kinetic parameters. For each reaction, we determined its flux distribution in 100 kinetic parameter sets or 1,000 parameters sets. This table reports the mean flux of each reaction over 100 kinetic parameter sets and the probability that the flux distribution as obtained using 100 kinetic parameters is different from that using 1,000 kinetic parameters. Values above 0.05 indicate that the null hypothesis is correct, i.e. the distributions are similar and, therefore, 100 kinetic parameter sets is sufficient.(XLS)Click here for additional data file.File S1Matlab code to run the metabolic model reported here.(ZIP)Click here for additional data file.


Biological processes such as metabolism, signaling, and macromolecular synthesis can be modeled as large networks of biochemical reactions. Large and comprehensive networks, like integrated networks that represent metabolism and macromolecular synthesis, are inherently multiscale because reaction rates can vary over many orders of magnitude. They require special methods for accurate analysis because naive use of standard optimization systems can produce inaccurate or erroneously infeasible results.E. coli.We describe techniques enabling off-the-shelf optimization software to compute accurate solutions to the poorly scaled optimization problems arising from flux balance analysis of multiscale biochemical reaction networks. We implement lifting techniques for flux balance analysis within the openCOBRA toolbox and demonstrate our techniques using the first integrated reconstruction of metabolism and macromolecular synthesis for Our techniques enable accurate flux balance analysis of multiscale networks using off-the-shelf optimization software. Although we describe lifting techniques in the context of flux balance analysis, our methods can be used to handle a variety of optimization problems arising from analysis of multiscale network reconstructions. S ∈ Rm×n be a stoichiometric matrix that represents a biochemical network consisting of m species interacting via n reactions. Flux balance analysis (FBA) predicts steady state reaction rates (fluxes) of such a biochemical network by solving the linear program Let vl,vu ∈ Rn are lower and upper bounds on the fluxes and c represents a biologically motivated objective function. We refer to [where refer to  for detaE. coli metabolism and macromolecular synthesis that represents the function of almost 2000 genes. This Metabolic-Expression model explicitly accounts for the demands of macromolecular synthesis at single nucleotide resolution. To enforce consistency between the state of metabolism and macromolecular synthesis, Thiele et al. introduce coupling constraints on certain pairs of fluxes : cmin,cmax > 0. Each coupling constraint can be formulated as a pair of linear inequality constraints, as described later. We predict the steady state reaction rates of such integrated networks by solving the linear program where Cv ≤ d includes constraints equivalent to (2) for many pairs of fluxes.where poorly scaled. Conducting FBA for such networks has been unsatisfactory because even state-of-the-art linear programming solvers can produce inaccurate (or erroneously infeasible) results. In particular, for the E. coli Metabolic-Expression model, applying CPLEX [Given the inherent multiscale nature of integrated reconstructed networks, the constraint matrices of the FBA linear programs (1) and (3) often contain entries that vary over many orders of magnitude. We say that the problems are ng CPLEX  and Gurong CPLEX  to (3) wn-space. The condition of a basis matrix associated with a vertex of the polytope provides a quantitative measure of either the “sharpness” or the “flatness” of the vertex. Poorly scaled constraints tend to create a polytope with very sharp and/or very flat vertices. To alleviate numerical difficulties for problem (1), linear programming systems typically compute row and column scaling matrices Dr ∈ Rm×m and Dc ∈ Rn×n such that the nonzero entries of the scaled constraint matrix DrSDc are of order 1. Scaling can improve the condition of many bases, but it may be at the expense of making other bases more ill-conditioned . For some problems, such as (3), the scaled constraints Sv = 0 significantly. We refer to [In the context of the simplex method for linear programming, the constraints (including bounds) form a polytope in refer to  for a coLifting techniques are commonly used in optimization to create an efficient representation of a feasible set. By using auxiliary variables to “lift” the feasible set into a higher-dimensional space, they can dramatically reduce the computational expense  Our explicit aim is to reduce the magnitude of the largest matrix entries so that the reformulated constraints do not need scaling.Here we apply lifting techniques to poorly scaled constraints to make the vertices of the “lifted” polytope more regular. Note that small entries in Sv = 0 often contain poorly scaled reactions such as In problem (1), the mass balance constraints which may represent the synthesis of a macromolecule in a reconstruction. We can decompose such reactions into sequences of reactions involving dummy metabolites with reasonably scaled coefficients. For example, (4) is equivalent to two reactions involving a dummy metabolite Cx ≤ d include equivalents of the coupling constraints (2). These enforce consistency between the states of the metabolic and macromolecular synthesis reactions and are often poorly scaled because reaction rates can vary over many orders of magnitude. For example, two fluxes could be related by In problem (3), the constraints v1 ≤ 10000v2, we would transform it to two constraints involving an auxiliary variable s1: As before, we can decompose these constraints into sequences of constraints involving auxiliary variables with reasonable coefficients. If the second inequality in (6) were presented to our implementation as v1 ≥ 0.0001v2, we would leave it alone, but the equivalent inequality 10000v1 ≥ v2 would be transformed to If the first inequality in (6) were presented as τ, set to 1024 in our experiments. Constraints containing entries larger than τ are reformulated.Our implementation of lifting techniques uses a parameter Very large entries might require more than one auxiliary variable and constraint. In these cases, we choose the reformulated constraint coefficients to be equally spaced in logarithmic scale. For example, the poorly scaled reaction 9| > τ) would be reformulated as (with |10τ)., but the added constraints are extremely sparse and should have little impact on the performance of a typical large-scale solver ) could be improved by applying a single step of classical iterative refinement [After a simplex solver has returned an allegedly optimal basic solution, the accuracy of satisfying the general linear constraints  accounts for 76664 reactions, and the second (ME76589) accounts for 76589 reactions. Because of the dependencies between pairs of metabolic reactions and macromolecular synthesis reactions, the resulting flux balanced steady state v has reaction rates that vary by four orders of magnitude [τ = 1024), with 1825 entries exceeding 105 and biggest entry 8×105.We use our implementation of lifting techniques to conduct FBA on two Metabolic-Expression models of E. coli. The modConducting FBA on ME76664 using the CPLEX and Gurobi simplex and barrier solvers with default settings  resulted in erroneous reports of infeasibility or “optimal” solutions that were significantly infeasible. Our own simplex solver SQOPT  with scacrossover (the procedure for finding a basic solution from the barrier solution). These do not alter the optimal objective value and may not be essential in practice.With the CPLEX solvers, our lifting techniques eliminate infeasible reports and significantly reduce the infeasibility of the computed steady states; see Table E. coli metabolic model (iAF1260) [We also used lifting to conduct flux variability analysis (FVA)  for the iAF1260) . The chiTables E. coli, previously impossible because of numerical difficulties encountered by solvers.We described techniques that enable off-the-shelf optimization software to be applied to multiscale network reconstructions, such as integrated networks that represent both metabolism and macromolecular synthesis. The techniques enable accurate FBA and FVA of an integrated model of metabolism and macromolecular synthesis in in silico biologists create increasingly complex models that capture more of the multiscale nature of biological systems [As  systems , the opt systems  that rel systems .MATLAB package for constraint-based reconstruction and analysis of biochemical networks.Lifting techniques for poorly scaled reactions and coupling constraints have been implemented in the openCOBRA toolbox 2.05 , a MATLAProject name: openCOBRA toolboxProject home page:http://opencobra.sourceforge.net/Operating system: platform independentProgramming language:MATLABOther requirements:MATLAB 2008a or higherLicense: GNU GPLv3Any restrictions to use by non-academics: A separate license must be acquired.The authors declare that they have no competing interests.YS developed and implemented the lifting techniques. YS and MAS wrote the manuscript. RMTF and IT provided examples, interpreted results, and edited the manuscript. All authors read and approved the final manuscript.