Elementary Flux Modes Research Articles

Logic programming-based Minimal Cut Sets reveal consortium-level therapeutic targets for chronic wound infections

Minimal Cut Sets (MCSs) identify sets of reactions which, when removed from a metabolic network, disable certain cellular functions. The traditional search for MCSs within genome-scale metabolic models (GSMMs) targets cellular growth, identifies reaction sets resulting in a lethal phenotype if disrupted, and retrieves a list of corresponding gene, mRNA, or enzyme targets. Using the dual link between MCSs and Elementary Flux Modes (EFMs), our logic programming-based tool aspefm was able to compute MCSs of any size from GSMMs in acceptable run times. The tool demonstrated better performance when computing large-sized MCSs than the mixed-integer linear programming methods. We applied the new MCSs methodology to a medically-relevant consortium model of two cross-feeding bacteria, Staphylococcus aureus and Pseudomonas aeruginosa. aspefm constraints were used to bias the computation of MCSs toward exchanged metabolites that could complement lethal phenotypes in individual species. We found that interspecies metabolite exchanges could play an essential role in rescuing single-species growth, for instance inosine could complement lethal reaction knock-outs in the purine synthesis, glycolysis, and pentose phosphate pathways of both bacteria. Finally, MCSs were used to derive a list of promising enzyme targets for consortium-level therapeutic applications that cannot be circumvented via interspecies metabolite exchange.

Adapting tree algorithms for partial enumeration of Extreme Pathways sets

Analysis of metabolic networks through characterising flux vectors such as Elementary Flux Modes and Extreme Pathways has shown to be valuable in many application within systems biology. However, their application is still limited to medium-scale metabolic networks due to the combinatorial explosion of the set size. Enumeration of the full set in genome-scale metabolic networks remains impossible, but few techniques exist for partial enumeration. Currently, an adapted version of the Canonical Basis Approach for generation of Extreme Pathways has allowed partial enumeration with good sampling quality through a filter. However, computational efficiency currently limits its application, especially when moving towards larger networks. One approach to increase this efficiency significantly is through candidate narrowing based on bit pattern trees. Here, many of the candidate modes are removed every iteration based on their ability to pass a rank test. However, the depth of the trees should be adapted based on the filter setting and the current amount of candidates to process. In this work, a novel algorithm is presented using candidate narrowing in a stochastic partial enumeration of Extreme Pathways. A case study for the central carbon metabolism of Escherichia coli showcases the adaptive tree depth and the overall performance increase for different settings.

From metagenomes to metabolism: Systematically assessing the metabolic flux feasibilities for “Candidatus Accumulibacter” species during anaerobic substrate uptake

With the rapid growing availability of metagenome assembled genomes (MAGs) and associated metabolic models, the identification of metabolic potential in individual community members has become possible. However, the field still lacks an unbiassed systematic evaluation of the generated metagenomic information to uncover not only metabolic potential, but also feasibilities of these models under specific environmental conditions. In this study, we present a systematic analysis of the metabolic potential in species of "Candidatus Accumulibacter", a group of polyphosphate-accumulating organisms (PAOs). We constructed a metabolic model of the central carbon metabolism and compared the metabolic potential among available MAGs for “Ca. Accumulibacter” species. By combining Elementary Flux Modes Analysis (EFMA) with max-min driving force (MDF) optimization, we obtained all possible flux distributions of the metabolic network and calculated their individual thermodynamic feasibility. Our findings reveal significant variations in the metabolic potential among “Ca. Accumulibacter” MAGs, particularly in the presence of anaplerotic reactions. EFMA revealed 700 unique flux distributions in the complete metabolic model that enable the anaerobic uptake of acetate and its conversion into polyhydroxyalkanoates (PHAs), a well-known phenotype of “Ca. Accumulibacter”. However, thermodynamic constraints narrowed down this solution space to 146 models that were stoichiometrically and thermodynamically feasible (MDF > 0 kJ/mol), of which only 8 were strongly feasible (MDF > 7 kJ/mol). Notably, several novel flux distributions for the metabolic model were identified, suggesting putative, yet unreported, functions within the PAO communities. Overall, this work provides valuable insights into the metabolic variability among "Ca. Accumulibacter" species and redefines the anaerobic metabolic potential in the context of phosphate removal. More generally, the integrated workflow presented in this paper can be applied to any metabolic model obtained from a MAG generated from microbial communities to objectively narrow the expected phenotypes from community members.

Whole-cell metabolic control analysis

Since its conception some fifty years ago, metabolic control analysis (MCA) aims to understand how cells control their metabolism by adjusting the activity of their enzymes. Here we extend its scope to a whole-cell context. We consider metabolism in the evolutionary context of growth-rate maximisation by optimisation of protein concentrations. This framework allows for the prediction of flux control coefficients from proteomics data or stoichiometric modelling. Since genes compete for finite biosynthetic resources, we treat all protein concentrations as interdependent. We show that elementary flux modes (EFMs) emerge naturally as the optimal metabolic networks in the whole-cell context and we derive their control properties. In the evolutionary optimum, the number of expressed EFMs is determined by the number of protein-concentration constraints that limit growth rate. We use published glucose-limited chemostat data of S. cerevisiae to illustrate that it uses only two EFMs prior to the onset of fermentation and that it uses four EFMs during fermentation. We discuss published enzyme-titration data to show that S. cerevisiae and E. coli indeed can express proteins at growth-rate maximising concentrations. Accordingly, we extend MCA to elementary flux modes operating at an optimal state. We find that the expression of growth-unassociated proteins changes results from classical metabolic control analysis. Finally, we show how flux control coefficients can be estimated from proteomics and ribosome-profiling data. We analyse published proteomics data of E. coli to provide a whole-cell perspective of the control of metabolic enzymes on growth rate. We hope that this paper stimulates a renewed interest in metabolic control analysis, so that it can serve again the purpose it once had: to identify general principles that emerge from the biochemistry of the cell and are conserved across biological species.

A Markov constraint to uniquely identify elementary flux mode weights in unimolecular metabolic networks

Elementary flux modes (EFMs) are minimal, steady state pathways characterizing a flux network. Fundamentally, all steady state fluxes in a network are decomposable into a linear combination of EFMs. While there is typically no unique set of EFM weights that reconstructs these fluxes, several optimization-based methods have been proposed to constrain the solution space by enforcing some notion of parsimony. However, it has long been recognized that optimization-based approaches may fail to uniquely identify EFM weights and return different feasible solutions across objective functions and solvers. Here we show that, for flux networks only involving single molecule transformations, these problems can be avoided by imposing a Markovian constraint on EFM weights. Our Markovian constraint guarantees a unique solution to the flux decomposition problem, and that solution is arguably more biophysically plausible than other solutions. We describe an algorithm for computing Markovian EFM weights via steady state analysis of a certain discrete-time Markov chain, based on the flux network, which we call the cycle-history Markov chain. We demonstrate our method with a differential analysis of EFM activity in a lipid metabolic network comparing healthy and Alzheimer’s disease patients. Our method is the first to uniquely decompose steady state fluxes into EFM weights for any unimolecular metabolic network.

On the geometry of elementary flux modes

Elementary flux modes (EFMs) play a prominent role in the constraint-based analysis of metabolic networks. They correspond to minimal functional units of the metabolic network at steady-state and as such have been studied for almost 30 years. The set of all EFMs in a metabolic network tends to be very large and may have exponential size in the number of reactions. Hence, there is a need to elucidate the structure of this set. Here we focus on geometric properties of EFMs. We analyze the distribution of EFMs in the face lattice of the steady-state flux cone of the metabolic network and show that EFMs in the relative interior of the cone occur only in very special cases. We introduce the concept of degree of an EFM as a measure how elementary it is and study the decomposition of flux vectors and EFMs depending on their degree. Geometric analysis can help to better understand the structure of the set of EFMs, which is important from both the mathematical and the biological viewpoint.

Minimizing enzyme mass to decompose flux distribution for identifying biologically relevant elementary flux modes

The flux distribution in metabolic network can be decomposed as non-negative linear combinations of elementary flux modes (EFMs). Identifying biologically relevant EFM combination by decomposing flux distribution in metabolic network is a useful method to study metabolisms in systems biology. However, the occurrence of biologically irrelevant EFMs hinders the application of such methods. In this paper, we introduce a novel method for identifying EFM combination by minimizing enzyme mass. Our proposed method, called EMMD (Enzyme Mass Minimization Decomposition), takes into consideration both thermodynamic and enzymatic constraints in stoichiometry metabolic models. By implementing EMMD, we can decompose the flux distributions in metabolic network to detect biologically relevant EFM combinations. We demonstrate the effectiveness of our method by applying it to the core Escherichia coli metabolic network and show that the optimal EFM combinations identified by EMMD are unique. Moreover, the optimal EFM combination identified by EMMD not only aligns more closely with experimental values in terms of estimated growth rate, but it also demonstrates more favorable thermodynamics. Finally, we investigated the growth of the core Escherichia coli metabolic network in Luria-Bertani medium containing different carbon sources, revealing the impact of various carbon sources on the growth rate of flux distribution. EMMD thus could be a promising complement to the existing flux decomposition tools.

Dynamic genome-based metabolic modeling of the predominant cellulolytic rumen bacterium Fibrobacter succinogenes S85.

Fibrobacter succinogenes is a cellulolytic bacterium that plays an essential role in the degradation of plant fibers in the rumen ecosystem. It converts cellulose polymers into intracellular glycogen and the fermentation metabolites succinate, acetate, and formate. We developed dynamic models of F. succinogenes S85 metabolism on glucose, cellobiose, and cellulose on the basis of a network reconstruction done with the automatic reconstruction of metabolic model workspace. The reconstruction was based on genome annotation, five template-based orthology methods, gap filling, and manual curation. The metabolic network of F. succinogenes S85 comprises 1,565 reactions with 77% linked to 1,317 genes, 1,586 unique metabolites, and 931 pathways. The network was reduced using the NetRed algorithm and analyzed for the computation of elementary flux modes. A yield analysis was further performed to select a minimal set of macroscopic reactions for each substrate. The accuracy of the models was acceptable in simulating F. succinogenes carbohydrate metabolism with an average coefficient of variation of the root mean squared error of 19%. The resulting models are useful resources for investigating the metabolic capabilities of F. succinogenes S85, including the dynamics of metabolite production. Such an approach is a key step toward the integration of omics microbial information into predictive models of rumen metabolism. IMPORTANCE F. succinogenes S85 is a cellulose-degrading and succinate-producing bacterium. Such functions are central for the rumen ecosystem and are of special interest for several industrial applications. This work illustrates how information of the genome of F. succinogenes can be translated to develop predictive dynamic models of rumen fermentation processes. We expect this approach can be applied to other rumen microbes for producing a model of rumen microbiome that can be used for studying microbial manipulation strategies aimed at enhancing feed utilization and mitigating enteric emissions.

On the representativeness and stability of a set of EFMs.

Elementary flux modes are a well-known tool for analyzing metabolic networks. The whole set of elementary flux modes (EFMs) cannot be computed in most genome-scale networks due to their large cardinality. Therefore, different methods have been proposed to compute a smaller subset of EFMs that can be used for studying the structure of the network. These latter methods pose the problem of studying the representativeness of the calculated subset. In this article, we present a methodology to tackle this problem. We have introduced the concept of stability for a particular network parameter and its relation to the representativeness of the EFM extraction method studied. We have also defined several metrics to study and compare the EFM biases. We have applied these techniques to compare the relative behavior of previously proposed methods in two case studies. Furthermore, we have presented a new method for the EFM computation (PiEFM), which is more stable (less biased) than previous ones, has suitable representativeness measures, and exhibits better variability in the extracted EFMs. Software and additional material are freely available at https://github.com/biogacop/PiEFM.

Speeding Up the Structural Analysis of Metabolic Network Models Using the Fredman-Khachiyan Algorithm B.

The problem of computing the Elementary Flux Modes (EFMs) and Minimal Cut Sets (MCSs) of metabolic network is a fundamental one in metabolic networks. A key insight is that they can be understood as a dual pair of monotone Boolean functions (MBFs). Using this insight, this computation reduces to the question of generating from an oracle a dual pair of MBFs. If one of the two sets (functions) is known, then the other can be computed through a process known as dualization. Fredman and Khachiyan provided two algorithms, which they called simply A and B that can serve as an engine for oracle-based generation or dualization of MBFs. We look at efficiencies available in implementing their algorithm B, which we will refer to as FK-B. Like their algorithm A, FK-B certifies whether two given MBFs in the form of Conjunctive Normal Form and Disjunctive Normal Form are dual or not, and in case of not being dual it returns a conflicting assignment (CA), that is, an assignment that makes one of the given Boolean functions True and the other one False. The FK-B algorithm is a recursive algorithm that searches through the tree of assignments to find a CA. If it does not find any CA, it means that the given Boolean functions are dual. In this article, we propose six techniques applicable to the FK-B and hence to the dualization process. Although these techniques do not reduce the time complexity, they considerably reduce the running time in practice. We evaluate the proposed improvements by applying them to compute the MCSs from the EFMs in the 19 small- and medium-sized models from the BioModels database along with 4 models of biomass synthesis in Escherichia coli that were used in an earlier computational survey Haus et al. (2008).

Modeling the metabolic dynamics at the genome-scale by optimized yield analysis

The hybrid cybernetic model (HCM) approach is a dynamic modeling framework that integrates enzyme synthesis and activity regulation. It has been widely applied in bioreaction engineering, particularly in the simulation of microbial growth in different mixtures of carbon sources. In a HCM, the metabolic network is decomposed into elementary flux modes (EFMs), whereby the network can be reduced into a few pathways by yield analysis. However, applying the HCM approach on conventional genome-scale metabolic models (GEMs) is still a challenge due to the high computational demands. Here, we present a HCM strategy that introduced an optimized yield analysis algorithm (opt-yield-FBA) to simulate metabolic dynamics at the genome-scale without the need for EFMs calculation. The opt-yield-FBA is a flux-balance analysis (FBA) based method that can calculate optimal yield solutions and yield space for GEM. With the opt-yield-FBA algorithm, the HCM strategy can be applied to get the yield spaces and avoid the computational burden of EFMs, and it can therefore be applied for developing dynamic models for genome-scale metabolic networks. Here, we illustrate the strategy by applying the concept to simulate the dynamics of microbial communities.

Hybrid Dynamic Models of Bioprocesses Based on Elementary Flux Modes and Multilayer Perceptrons

The derivation of minimal bioreaction models is of primary importance to develop monitoring and control strategies of cell/microorganism culture production. These minimal bioreaction models can be obtained based on the selection of a basis of elementary flux modes (EFMs) using an algorithm starting from a relatively large set of EFMs and progressively reducing their numbers based on geometric and least-squares residual criteria. The reaction rates associated with the selected EFMs usually have complex features resulting from the combination of different activation, inhibition and saturation effects from several culture species. Multilayer perceptrons (MLPs) are used in order to undertake the representation of these rates, resulting in a hybrid dynamic model combining the mass-balance equations provided by the EFMs to the rate equations described by the MLPs. To further reduce the number of kinetic parameters of the model, pruning algorithms for the MLPs are also considered. The whole procedure ends up with reduced-order macroscopic models that show promising prediction results, as illustrated with data of perfusion cultures of hybridoma cell line HB-58.

Strain engineering and metabolic flux analysis of a probiotic yeast Saccharomyces boulardii for metabolizing l-fucose, a mammalian mucin component

BackgroundSaccharomyces boulardii is a probiotic yeast that exhibits antimicrobial and anti-toxin activities. Although S. boulardii has been clinically used for decades to treat gastrointestinal disorders, several studies have reported weak or no beneficial effects of S. boulardii administration in some cases. These conflicting results of S. boulardii efficacity may be due to nutrient deficiencies in the intestine that make it difficult for S. boulardii to maintain its metabolic activity.ResultsTo enable S. boulardii to overcome any nutritional deficiencies in the intestine, we constructed a S. boulardii strain that could metabolize l-fucose, a major component of mucin in the gut epithelium. The fucU, fucI, fucK, and fucA from Escherichia coli and HXT4 from S. cerevisiae were overexpressed in S. boulardii. The engineered S. boulardii metabolized l-fucose and produced 1,2-propanediol under aerobic and anaerobic conditions. It also produced large amounts of 1,2-propanediol under strict anaerobic conditions. An in silico genome-scale metabolic model analysis was performed to simulate the growth of S. boulardii on l-fucose, and elementary flux modes were calculated to identify critical metabolic reactions for assimilating l-fucose. As a result, we found that the engineered S. boulardii consumes l-fucose via (S)-lactaldehyde-(S)-lactate-pyruvate pathway, which is highly oxygen dependent.ConclusionTo the best of our knowledge, this is the first study in which S. cerevisiae and S. boulardii strains capable of metabolizing l-fucose have been constructed. This strategy could be used to enhance the metabolic activity of S. boulardii and other probiotic microorganisms in the gut.

A systematic elementary flux mode selection procedure for deriving macroscopic bioreaction models from metabolic networks

Elementary flux modes (EFMs) are an important concept in metabolic pathway analysis and for the derivation of macroscopic dynamic models. However, the computation of elementary flux vectors is facing combinatorial explosion with the size of the metabolic network, which hinders widespread application. This study proposes a systematic elementary flux mode reduction procedure to derive reduced-order dynamic models starting from an initial set of EFMs either generated by complete enumeration or subset selection. The procedure proceeds in several steps, including geometric and optimization-based criteria. The methodology ends up with a macroscopic bioreaction scheme with a reaction number smaller than that of the measured species, and shows very satisfactory prediction results, as illustrated with data of batch cultures of CHO-320 cells.

Thermodynamic limitations of PHB production from formate and fructose in Cupriavidus necator

The chemolithotroph Cupriavidus necator H16 is known as a natural producer of the bioplastic-polymer PHB, as well as for its metabolic versatility to utilize different substrates, including formate as the sole carbon and energy source. Depending on the entry point of the substrate, this versatility requires adjustment of the thermodynamic landscape to maintain sufficiently high driving forces for biological processes. Here we employed a model of the core metabolism of C. necator H16 to analyze the thermodynamic driving forces and PHB yields from formate for different metabolic engineering strategies. For this, we enumerated elementary flux modes (EFMs) of the network and evaluated their PHB yields as well as thermodynamics via Max-min driving force (MDF) analysis and random sampling of driving forces. A heterologous ATP:citrate lyase reaction was predicted to increase driving force for producing acetyl-CoA. A heterologous phosphoketolase reaction was predicted to increase maximal PHB yields as well as driving forces. These enzymes were then verified experimentally to enhance PHB titers between 60 and 300% in select conditions. The EFM analysis also revealed that PHB production from formate may be limited by low driving forces through citrate lyase and aconitase, as well as cofactor balancing, and identified additional reactions associated with low and high PHB yield. Proteomics analysis of the engineered strains confirmed an increased abundance of aconitase and cofactor balancing. The findings of this study aid in understanding metabolic adaptation. Furthermore, the outlined approach will be useful in designing metabolic engineering strategies in other non-model bacteria.

Elucidating Plant-Microbe-Environment Interactions Through Omics-Enabled Metabolic Modelling Using Synthetic Communities.

With a growing world population and increasing frequency of climate disturbance events, we are in dire need of methods to improve plant productivity, resilience, and resistance to both abiotic and biotic stressors, both for agriculture and conservation efforts. Microorganisms play an essential role in supporting plant growth, environmental response, and susceptibility to disease. However, understanding the specific mechanisms by which microbes interact with each other and with plants to influence plant phenotypes is a major challenge due to the complexity of natural communities, simultaneous competition and cooperation effects, signalling interactions, and environmental impacts. Synthetic communities are a major asset in reducing the complexity of these systems by simplifying to dominant components and isolating specific variables for controlled experiments, yet there still remains a large gap in our understanding of plant microbiome interactions. This perspectives article presents a brief review discussing ways in which metabolic modelling can be used in combination with synthetic communities to continue progress toward understanding the complexity of plant-microbe-environment interactions. We highlight the utility of metabolic models as applied to a community setting, identify different applications for both flux balance and elementary flux mode simulation approaches, emphasize the importance of ecological theory in guiding data interpretation, and provide ideas for how the integration of metabolic modelling techniques with big data may bridge the gap between simplified synthetic communities and the complexity of natural plant-microbe systems.

Elementary vectors and autocatalytic sets for resource allocation in next-generation models of cellular growth.

Traditional (genome-scale) metabolic models of cellular growth involve an approximate biomass “reaction”, which specifies biomass composition in terms of precursor metabolites (such as amino acids and nucleotides). On the one hand, biomass composition is often not known exactly and may vary drastically between conditions and strains. On the other hand, the predictions of computational models crucially depend on biomass. Also elementary flux modes (EFMs), which generate the flux cone, depend on the biomass reaction. To better understand cellular phenotypes across growth conditions, we introduce and analyze new classes of elementary vectors for comprehensive (next-generation) metabolic models, involving explicit synthesis reactions for all macromolecules. Elementary growth modes (EGMs) are given by stoichiometry and generate the growth cone. Unlike EFMs, they are not support-minimal, in general, but cannot be decomposed “without cancellations”. In models with additional (capacity) constraints, elementary growth vectors (EGVs) generate a growth polyhedron and depend also on growth rate. However, EGMs/EGVs do not depend on the biomass composition. In fact, they cover all possible biomass compositions and can be seen as unbiased versions of elementary flux modes/vectors (EFMs/EFVs) used in traditional models. To relate the new concepts to other branches of theory, we consider autocatalytic sets of reactions. Further, we illustrate our results in a small model of a self-fabricating cell, involving glucose and ammonium uptake, amino acid and lipid synthesis, and the expression of all enzymes and the ribosome itself. In particular, we study the variation of biomass composition as a function of growth rate. In agreement with experimental data, low nitrogen uptake correlates with high carbon (lipid) storage.

Linking process and metabolic modelling for the estimation of carbon flux distribution in Corynebacterium glutamicum growth in spent sulfite liquor

Process monitoring in microbial cultures became feasible thanks to the development of accurate measurement devices, including in-situ probes to monitor biomass growth, oxygen, carbon dioxide and sugar consumption. In comparison, estimating the metabolic fluxes of the cell factories still rely on analysis based on an under-determined set of equations, and requires an expensive and time consuming methods of verification. This problem intensifies in the presence of complex substrates, in which different sugars are utilized in parallel by the cell factories. In the present study, a growth experiment of Corynebacterium glutamicum in spent sulfite liquor was studied. The bioprocess was monitored during batch and fed-batch phases, and a parameter estimation routine was conducted to define a process model and the corresponding uptake rates. A tracking optimization algorithm minimized the error between the measured process fluxes and the equivalent fluxes of the elementary flux modes. The results indicate that the optimization technique obtained a set of elementary modes that are closer to reality than the computed from the metabolic analysis. Taken together, we show that an online estimation of metabolic flux distribution of C glutamicum based on a set of process measurement signals was possible with an optimization function that links the process and metabolic model. The procedure can be complementary to the sophisticated and expensive C NMR experimental analytical technique.

Valorisation of xylose to renewable fuels and chemicals, an essential step in augmenting the commercial viability of lignocellulosic biorefineries.

Biologists and engineers are making tremendous efforts in contributing to a sustainable and green society. To that end, there is growing interest in waste management and valorisation. Lignocellulosic biomass (LCB) is the most abundant material on the earth and an inevitable waste predominantly originating from agricultural residues, forest biomass and municipal solid waste streams. LCB serves as the renewable feedstock for clean and sustainable processes and products with low carbon emission. Cellulose and hemicellulose constitute the polymeric structure of LCB, which on depolymerisation liberates oligomeric or monomeric glucose and xylose, respectively. The preferential utilization of glucose and/or absence of the xylose metabolic pathway in microbial systems cause xylose valorization to be alienated and abandoned, a major bottleneck in the commercial viability of LCB-based biorefineries. Xylose is the second most abundant sugar in LCB, but a non-conventional industrial substrate unlike glucose. The current review seeks to summarize the recent developments in the biological conversion of xylose into a myriad of sustainable products and associated challenges. The review discusses the microbiology, genetics, and biochemistry of xylose metabolism with hurdles requiring debottlenecking for efficient xylose assimilation. It further describes the product formation by microbial cell factories which can assimilate xylose naturally and rewiring of metabolic networks to ameliorate xylose-based bioproduction in native as well as non-native strains. The review also includes a case study that provides an argument on a suitable pathway for optimal cell growth and succinic acid (SA) production from xylose through elementary flux mode analysis. Finally, a product portfolio from xylose bioconversion has been evaluated along with significant developments made through enzyme, metabolic and process engineering approaches, to maximize the product titers and yield, eventually empowering LCB-based biorefineries. Towards the end, the review is wrapped up with current challenges, concluding remarks, and prospects with an argument for intense future research into xylose-based biorefineries.

Metabolic Pathway Analysis in the Presence of Biological Constraints

Metabolic pathway analysis is a key method to study a metabolism in its steady state, and the concept of elementary fluxes (EFs) plays a major role in the analysis of a network in terms of non-decomposable pathways. The supports of the EFs contain in particular those of the elementary flux modes (EFMs), which are the support-minimal pathways, and EFs coincide with EFMs when the only flux constraints are given by the irreversibility of certain reactions. Practical use of both EFMs and EFs has been hampered by the combinatorial explosion of their number in large, genome-scale systems. The EFs give the possible pathways in a steady state but the real pathways are limited by biological constraints, such as thermodynamic or, more generally, kinetic constraints and regulatory constraints from the genetic network. We provide results on the mathematical structure and geometrical characterization of the solution space in the presence of such biological constraints (which is no longer a convex polyhedral cone or a convex polyhedron) and revisit the concept of EFMs and EFs in this framework. We show that most of the results depend only on very general properties of compatibility of constraints with vector signs: either sign-invariance, satisfied by regulatory constraints, or sign-monotonicity (a stronger property), satisfied by thermodynamic and kinetic constraints. We show in particular that the solution space for sign-monotone constraints is a union of particular faces of the original polyhedral cone or polyhedron and that EFs still coincide with EFMs and are just those of the original EFs that satisfy the constraint, and we show how to integrate their computation efficiently in the double description method, the most widely used method in the tools dedicated to EFs computation. We show that, for sign-invariant constraints, the situation is more complex: the solution space is a disjoint union of particular semi-open faces (i.e., without some of their own faces of lesser dimension) of the original polyhedral cone or polyhedron and, if EFs are still those of the original EFs that satisfy the constraint, their computation cannot be incrementally integrated into the double description method, and the result is not true for EFMs, that are in general strictly more numerous than those of the original EFMs that satisfy the constraint.