Combinatorial complexity of pathway analysis in metabolic networks

A hierarchy of metabolite exchanges in metabolic models of microbial species and communities

The metabolic network of an organism can be analyzed as a constraint-based model. This analysis can be biased, optimizing an objective such as growth rate, or unbiased, aiming to describe the full feasible space of metabolic fluxes through pathway analysis or random flux sampling. In particular, pathway analysis can decompose the flux space into fundamental and formally defined metabolic pathways. Unbiased methods scale poorly with network size due to combinatorial explosion, but a promising approach to improve scalability is to focus on metabolic subnetworks, e.g., cells' metabolite exchanges with each other and the environment, rather than the full metabolic networks. Here, we applied pathway enumeration and flux sampling to metabolite exchanges in microbial species and a microbial community, using models ranging from central carbon metabolism to genome-scale and focusing on pathway definitions that allow direct targeting of subnetworks such as metabolite exchanges (elementary conversion modes, elementary flux patterns, and minimal pathways). Enumerating growth-supporting metabolite exchanges, we found that metabolite exchanges from different pathway definitions were related through a hierarchy, and we show that this hierarchical relationship between pathways holds for metabolic networks and subnetworks more generally. Metabolite exchange frequencies, defined as the fraction of pathways in which each metabolite was exchanged, were similar across pathway definitions, with a few specific exchanges explaining large differences in pathway counts. This indicates that biological interpretation of predicted metabolite exchanges is robust to the choice of pathway definition, and it suggests strategies for more scalable pathway analysis. Our results also signal wider biological implications, facilitating detailed and interpretable analysis of metabolite exchanges and other subnetworks in fields such as metabolic engineering and synthetic biology.

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.

Microbial Pathway Thermodynamics: Stoichiometric Models Unveil Anabolic and Catabolic Processes

The biotechnological exploitation of microorganisms enables the use of metabolism for the production of economically valuable substances, such as drugs or food. It is, thus, unsurprising that the investigation of microbial metabolism and its regulation has been an active research field for many decades. As a result, several theories and techniques were developed that allow for the prediction of metabolic fluxes and yields as biotechnologically relevant output parameters. One important approach is to derive macrochemical equations that describe the overall metabolic conversion of an organism and basically treat microbial metabolism as a black box. The opposite approach is to include all known metabolic reactions of an organism to assemble a genome-scale metabolic model. Interestingly, both approaches are rather successful at characterizing and predicting the expected product yield. Over the years, macrochemical equations especially have been extensively characterized in terms of their thermodynamic properties. However, a common challenge when characterizing microbial metabolism by a single equation is to split this equation into two, describing the two modes of metabolism, anabolism and catabolism. Here, we present strategies to systematically identify separate equations for anabolism and catabolism. Based on metabolic models, we systematically identify all theoretically possible catabolic routes and determine their thermodynamic efficiency. We then show how anabolic routes can be derived, and we use these to approximate biomass yield. Finally, we challenge the view of metabolism as a linear energy converter, in which the free energy gradient of catabolism drives the anabolic reactions.

Basic pathway decomposition of biochemical reaction networks within growing cells

This contribution treats linear, steady-state dynamics for a metabolic network within a growing cell. Admissible steady-state reaction fluxes are assumed to form a pointed, convex, polyhedral, conical subset of the stoichiometric null-space. A solution of the problem is defined to consist of a linear basis for the stoichiometric null-space consisting of admissible fluxes called basic pathways. The algorithm used to construct the set of basic pathways scales as a polynomial of the system size in contrast to the NP-hard algorithms employed in the traditional notions of solution named extreme pathways, elementary flux modes, MEMos, and MinSpan, and that therefore suffer from the curse of dimensionality. The basic pathways approach is applied to a metabolic network consisting of a simplified version of the TCA cycle coupled to glycolysis highlighting that each basic pathway has a readily understood chemical interpretation. Generic admissible pathways are simply expressed in terms of basic pathways.

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.

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.

On the representativeness and stability of a set of EFMs

MOTIVATION

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.

RESULTS

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.

AVAILABILITY AND IMPLEMENTATION

Software and additional material are freely available at https://github.com/biogacop/PiEFM.

Comparison of metabolic states using genome-scale metabolic models

Genome-scale metabolic models (GEMs) are comprehensive knowledge bases of cellular metabolism and serve as mathematical tools for studying biological phenotypes and metabolic states or conditions in various organisms and cell types. Given the sheer size and complexity of human metabolism, selecting parameters for existing analysis methods such as metabolic objective functions and model constraints is not straightforward in human GEMs. In particular, comparing several conditions in large GEMs to identify condition- or disease-specific metabolic features is challenging. In this study, we showcase a scalable, model-driven approach for an in-depth investigation and comparison of metabolic states in large GEMs which enables identifying the underlying functional differences. Using a combination of flux space sampling and network analysis, our approach enables extraction and visualisation of metabolically distinct network modules. Importantly, it does not rely on known or assumed objective functions. We apply this novel approach to extract the biochemical differences in adipocytes arising due to unlimited vs blocked uptake of branched-chain amino acids (BCAAs, considered as biomarkers in obesity) using a human adipocyte GEM (iAdipocytes1809). The biological significance of our approach is corroborated by literature reports confirming our identified metabolic processes (TCA cycle and Fatty acid metabolism) to be functionally related to BCAA metabolism. Additionally, our analysis predicts a specific altered uptake and secretion profile indicating a compensation for the unavailability of BCAAs. Taken together, our approach facilitates determining functional differences between any metabolic conditions of interest by offering a versatile platform for analysing and comparing flux spaces of large metabolic networks.

Cellular metabolism is a highly complex and interconnected system. As many lifestyle diseases in humans have a strong metabolic component, it is important to understand metabolic differences between healthy and diseased states. In systems biology, metabolic behaviours are investigated using genome-scale metabolic models. In addition to the sheer size and complexity of the genome-scale metabolic models of human systems, using existing analysis methods is challenging and the parameter selection is not straightforward. Therefore, novel methodological frameworks are necessary for analysing metabolic conditions despite the challenges posed by human models. Particularly, an ongoing challenge has been that of comparing several phenotypes for identifying condition- or disease-specific metabolic signatures. We address this significant challenge by developing a scalable and model-driven approach, ComMet (Comparison of Metabolic states). ComMet enables an in-depth investigation and comparison of metabolic phenotypes in large models while also identifying the underlying functional differences. Novel hypotheses can be generated using ComMet for not only understanding known metabolic phenotypes better but also for guiding the design of new experiments to validate the processes predicted by ComMet.

Metabolic control of nitrogen fixation in rhizobium-legume symbioses

Rhizobia induce nodule formation on legume roots and differentiate into bacteroids, which catabolize plant-derived dicarboxylates to reduce atmospheric N₂ into ammonia. Despite the agricultural importance of this symbiosis, the mechanisms that govern carbon and nitrogen allocation in bacteroids and promote ammonia secretion to the plant are largely unknown. Using a metabolic model derived from genome-scale datasets, we show that carbon polymer synthesis and alanine secretion by bacteroids facilitate redox balance in microaerobic nodules. Catabolism of dicarboxylates induces not only a higher oxygen demand but also a higher NADH/NAD⁺ ratio than sugars. Modeling and ¹³C metabolic flux analysis indicate that oxygen limitation restricts the decarboxylating arm of the tricarboxylic acid cycle, which limits ammonia assimilation into glutamate. By tightly controlling oxygen supply and providing dicarboxylates as the energy and electron source donors for N₂ fixation, legumes promote ammonia secretion by bacteroids. This is a defining feature of rhizobium-legume symbioses.

Understanding FBA Solutions under Multiple Nutrient Limitations

Genome-scale stoichiometric modeling methods, in particular Flux Balance Analysis (FBA) and variations thereof, are widely used to investigate cell metabolism and to optimize biotechnological processes. Given (1) a metabolic network, which can be reconstructed from an organism’s genome sequence, and (2) constraints on reaction rates, which may be based on measured nutrient uptake rates, FBA predicts which reactions maximize an objective flux, usually the production of cell components. Although FBA solutions may accurately predict the metabolic behavior of a cell, the actual flux predictions are often hard to interpret. This is especially the case for conditions with many constraints, such as for organisms growing in rich nutrient environments: it remains unclear why a certain solution was optimal. Here, we rationalize FBA solutions by explaining for which properties the optimal combination of metabolic strategies is selected. We provide a graphical formalism in which the selection of solutions can be visualized; we illustrate how this perspective provides a glimpse of the logic that underlies genome-scale modeling by applying our formalism to models of various sizes.

Deep learning approaches for natural product discovery from plant endophytic microbiomes

Plant microbiomes are not only diverse, but also appear to host a vast pool of secondary metabolites holding great promise for bioactive natural products and drug discovery. Yet, most microbes within plants appear to be uncultivable, and for those that can be cultivated, their metabolic potential lies largely hidden through regulatory silencing of biosynthetic genes. The recent explosion of powerful interdisciplinary approaches, including multi-omics methods to address multi-trophic interactions and artificial intelligence-based computational approaches to infer distribution of function, together present a paradigm shift in high-throughput approaches to natural product discovery from plant-associated microbes. Arguably, the key to characterizing and harnessing this biochemical capacity depends on a novel, systematic approach to characterize the triggers that turn on secondary metabolite biosynthesis through molecular or genetic signals from the host plant, members of the rich 'in planta' community, or from the environment. This review explores breakthrough approaches for natural product discovery from plant microbiomes, emphasizing the promise of deep learning as a tool for endophyte bioprospecting, endophyte biochemical novelty prediction, and endophyte regulatory control. It concludes with a proposed pipeline to harness global databases (genomic, metabolomic, regulomic, and chemical) to uncover and unsilence desirable natural products.

SUPPLEMENTARY INFORMATION

The online version contains supplementary material available at 10.1186/s40793-021-00375-0.

Modelling Cell Metabolism: A Review on Constraint-Based Steady-State and Kinetic Approaches

Studying cell metabolism serves a plethora of objectives such as the enhancement of bioprocess performance, and advancement in the understanding of cell biology, of drug target discovery, and in metabolic therapy. Remarkable successes in these fields emerged from heuristics approaches, for instance, with the introduction of effective strategies for genetic modifications, drug developments and optimization of bioprocess management. However, heuristics approaches have showed significant shortcomings, such as to describe regulation of metabolic pathways and to extrapolate experimental conditions. In the specific case of bioprocess management, such shortcomings limit their capacity to increase product quality, while maintaining desirable productivity and reproducibility levels. For instance, since heuristics approaches are not capable of prediction of the cellular functions under varying experimental conditions, they may lead to sub-optimal processes. Also, such approaches used for bioprocess control often fail in regulating a process under unexpected variations of external conditions. Therefore, methodologies inspired by the systematic mathematical formulation of cell metabolism have been used to address such drawbacks and achieve robust reproducible results. Mathematical modelling approaches are effective for both the characterization of the cell physiology, and the estimation of metabolic pathways utilization, thus allowing to characterize a cell population metabolic behavior. In this article, we present a review on methodology used and promising mathematical modelling approaches, focusing primarily to investigate metabolic events and regulation. Proceeding from a topological representation of the metabolic networks, we first present the metabolic modelling approaches that investigate cell metabolism at steady state, complying to the constraints imposed by mass conservation law and thermodynamics of reactions reversibility. Constraint-based models (CBMs) are reviewed highlighting the set of assumed optimality functions for reaction pathways. We explore models simulating cell growth dynamics, by expanding flux balance models developed at steady state. Then, discussing a change of metabolic modelling paradigm, we describe dynamic kinetic models that are based on the mathematical representation of the mechanistic description of nonlinear enzyme activities. In such approaches metabolic pathway regulations are considered explicitly as a function of the activity of other components of metabolic networks and possibly far from the metabolic steady state. We have also assessed the significance of metabolic model parameterization in kinetic models, summarizing a standard parameter estimation procedure frequently employed in kinetic metabolic modelling literature. Finally, some optimization practices used for the parameter estimation are reviewed.

Cooperative metabolic resource allocation in spatially-structured systems

Natural selection has shaped the evolution of cells and multi-cellular organisms such that social cooperation can often be preferred over an individualistic approach to metabolic regulation. This paper extends a framework for dynamic metabolic resource allocation based on the maximum entropy principle to spatiotemporal models of metabolism with cooperation. Much like the maximum entropy principle encapsulates 'bet-hedging' behaviour displayed by organisms dealing with future uncertainty in a fluctuating environment, its cooperative extension describes how individuals adapt their metabolic resource allocation strategy to further accommodate limited knowledge about the welfare of others within a community. The resulting theory explains why local regulation of metabolic cross-feeding can fulfil a community-wide metabolic objective if individuals take into consideration an ensemble measure of total population performance as the only form of global information. The latter is likely supplied by quorum sensing in microbial systems or signalling molecules such as hormones in multi-cellular eukaryotic organisms.

Unlocking Elementary Conversion Modes: ecmtool Unveils All Capabilities of Metabolic Networks

The metabolic capabilities of cells determine their biotechnological potential, fitness in ecosystems, pathogenic threat levels, and function in multicellular organisms. Their comprehensive experimental characterization is generally not feasible, particularly for unculturable organisms. In principle, the full range of metabolic capabilities can be computed from an organism's annotated genome using metabolic network reconstruction. However, current computational methods cannot deal with genome-scale metabolic networks. Part of the problem is that these methods aim to enumerate all metabolic pathways, while computation of all (elementally balanced) conversions between nutrients and products would suffice. Indeed, the elementary conversion modes (ECMs, defined by Urbanczik and Wagner) capture the full metabolic capabilities of a network, but the use of ECMs has not been accessible until now. We explain and extend the theory of ECMs, implement their enumeration in ecmtool, and illustrate their applicability. This work contributes to the elucidation of the full metabolic footprint of any cell.

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Elementary conversion modes (ECMs) specify all metabolic capabilities of any organism

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Ecmtool computes all ECMs from a reconstructed metabolic network

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ECM enumeration enables metabolic characterization of larger networks than ever

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Focusing on ECMs between relevant metabolites even enables genome-scale enumeration

Understanding the metabolic capabilities of cells is of profound importance. Microbial metabolism shapes global cycles of elements and cleans polluted soils. Human and pathogen metabolism affects our health. Recent advances allow for automatic reconstruction of reaction networks for any organism, which is already used in synthetic biology, (food) microbiology, and agriculture to compute optimal yields from resources to products. However, computational tools are limited to optimal states or subnetworks, leaving many capabilities of organisms hidden. Our program, ecmtool, creates a blueprint of any organism's metabolic functionalities, drastically improving insights obtained from genome sequences. Ecmtool may become essential in exploratory research, especially for studying cells that are not culturable in laboratory conditions. Ideally, elementary conversion mode enumeration will someday be a standard step after metabolic network reconstruction, achieving the metabolic characterization of all known organisms.

Flux Coupling and the Objective Functions' Length in EFMs

Structural analysis of constraint-based metabolic network models attempts to find the network's properties by searching for subsets of suitable modes or Elementary Flux Modes (EFMs). One useful approach is based on Linear Program (LP) techniques, which introduce an objective function to convert the stoichiometric and thermodynamic constraints into a linear program (LP), using additional constraints to generate different nontrivial modes. This work introduces FLFS-FC (Fixed Length Function Sampling with Flux Coupling), a new approach to increase the efficiency of generation of large sets of different EFMs for the network. FLFS-FC is based on the importance of the length of the objective functions used in the associated LP problem and the imposition of additional negative constraints. Our proposal overrides some of the known drawbacks associated with the EFM extraction, such as the appearance of unfeasible problems or multiple repeated solutions arising from different LP problems.

Speeding up the core algorithm for the dual calculation of minimal cut sets in large metabolic networks

Background

The concept of minimal cut sets (MCS) has become an important mathematical framework for analyzing and (re)designing metabolic networks. However, the calculation of MCS in genome-scale metabolic models is a complex computational problem. The development of duality-based algorithms in the last years allowed the enumeration of thousands of MCS in genome-scale networks by solving mixed-integer linear problems (MILP). A recent advancement in this field was the introduction of the MCS² approach. In contrast to the Farkas-lemma-based dual system used in earlier studies, the MCS² approach employs a more condensed representation of the dual system based on the nullspace of the stoichiometric matrix, which, due to its reduced dimension, holds promise to further enhance MCS computations.

Results

In this work, we introduce several new variants and modifications of duality-based MCS algorithms and benchmark their effects on the overall performance. As one major result, we generalize the original MCS² approach (which was limited to blocking the operation of certain target reactions) to the most general case of MCS computations with arbitrary target and desired regions. Building upon these developments, we introduce a new MILP variant which allows maximal flexibility in the formulation of MCS problems and fully leverages the reduced size of the nullspace-based dual system. With a comprehensive set of benchmarks, we show that the MILP with the nullspace-based dual system outperforms the MILP with the Farkas-lemma-based dual system speeding up MCS computation with an averaged factor of approximately 2.5. We furthermore present several simplifications in the formulation of constraints, mainly related to binary variables, which further enhance the performance of MCS-related MILP. However, the benchmarks also reveal that some highly condensed formulations of constraints, especially on reversible reactions, may lead to worse behavior when compared to variants with a larger number of (more explicit) constraints and involved variables.

Conclusions

Our results further enhance the algorithmic toolbox for MCS calculations and are of general importance for theoretical developments as well as for practical applications of the MCS framework.

An extended and generalized framework for the calculation of metabolic intervention strategies based on minimal cut sets

The concept of minimal cut sets (MCS) provides a flexible framework for analyzing properties of metabolic networks and for computing metabolic intervention strategies. In particular, it has been used to support the targeted design of microbial strains for bio-based production processes. Herein we present a number of major extensions that generalize the existing MCS approach and broaden its scope for applications in metabolic engineering. We first introduce a modified approach to integrate gene-protein-reaction associations (GPR) in the metabolic network structure for the computation of gene-based intervention strategies. In particular, we present a set of novel compression rules for GPR associations, which effectively speedup the computation of gene-based MCS by a factor of up to one order of magnitude. These rules are not specific for MCS and as well applicable to other computational strain design methods. Second, we enhance the MCS framework by allowing the definition of multiple target (undesired) and multiple protected (desired) regions. This enables precise tailoring of the metabolic solution space of the designed strain with unlimited flexibility. Together with further generalizations such as individual cost factors for each intervention, direct combinations of reaction/gene deletions and additions as well as the possibility to search for substrate co-feeding strategies, the scope of the MCS framework could be broadly extended. We demonstrate the applicability and performance benefits of the described developments by computing (gene-based) Escherichia coli strain designs for the bio-based production of 2,3-butanediol, a chemical, that has recently received much attention in the field of metabolic engineering. With our extended framework, we could identify promising strain designs that were formerly unpredictable, including those based on substrate co-feeding.

Dynamic metabolic resource allocation based on the maximum entropy principle

Organisms have evolved a variety of mechanisms to cope with the unpredictability of environmental conditions, and yet mainstream models of metabolic regulation are typically based on strict optimality principles that do not account for uncertainty. This paper introduces a dynamic metabolic modelling framework that is a synthesis of recent ideas on resource allocation and the powerful optimal control formulation of Ramkrishna and colleagues. In particular, their work is extended based on the hypothesis that cellular resources are allocated among elementary flux modes according to the principle of maximum entropy. These concepts both generalise and unify prior approaches to dynamic metabolic modelling by establishing a smooth interpolation between dynamic flux balance analysis and dynamic metabolic models without regulation. The resulting theory is successful in describing 'bet-hedging' strategies employed by cell populations dealing with uncertainty in a fluctuating environment, including heterogenous resource investment, accumulation of reserves in growth-limiting conditions, and the observed behaviour of yeast growing in batch and continuous cultures. The maximum entropy principle is also shown to yield an optimal control law consistent with partitioning resources between elementary flux mode families, which has important practical implications for model reduction, selection, and simulation.

EFMviz: A COBRA Toolbox extension to visualize Elementary Flux Modes in Genome-Scale Metabolic Models

Elementary Flux Modes (EFMs) are a tool for constraint-based modeling and metabolic network analysis. However, systematic and automated visualization of EFMs, capable of integrating various data types is still a challenge. In this study, we developed an extension for the widely adopted COBRA Toolbox, EFMviz, for analysis and graphical visualization of EFMs as networks of reactions, metabolites and genes. The analysis workflow offers a platform for EFM visualization to improve EFM interpretability by connecting COBRA toolbox with the network analysis and visualization software Cytoscape. The biological applicability of EFMviz is demonstrated in two use cases on medium (Escherichia coli, iAF1260) and large (human, Recon 2.2) genome-scale metabolic models. EFMviz is open-source and integrated into COBRA Toolbox. The analysis workflows used for the two use cases are detailed in the two tutorials provided with EFMviz along with the data used in this study.

Engineering microbial chemical factories using metabolic models

Living organisms in analogy with chemical factories use simple molecules such as sugars to produce a variety of compounds which are necessary for sustaining life and some of which are also commercially valuable. The metabolisms of simple (such as bacteria) and higher organisms (such as plants) alike can be exploited to convert low value inputs into high value outputs. Unlike conventional chemical factories, microbial production chassis are not necessarily tuned for a single product overproduction. Despite the same end goal, metabolic and industrial engineers rely on different techniques for achieving productivity goals. Metabolic engineers cannot affect reaction rates by manipulating pressure and temperature, instead they have at their disposal a range of enzymes and transcriptional and translational processes to optimize accordingly. In this review, we first highlight how various analytical approaches used in metabolic engineering and synthetic biology are related to concepts developed in systems and control engineering. Specifically, how algorithmic concepts derived in operations research can help explain the structure and organization of metabolic networks. Finally, we consider the future directions and challenges faced by the field of metabolic network modeling and the possible contributions of concepts drawn from the classical fields of chemical and control engineering. The aim of the review is to offer a current perspective of metabolic engineering and all that it entails without requiring specialized knowledge of bioinformatics or systems biology.

Chemical Transformation Motifs-Modelling Pathways as Integer Hyperflows

We present an elaborate framework for formally modelling pathways in chemical reaction networks on a mechanistic level. Networks are modelled mathematically as directed multi-hypergraphs, with vertices corresponding to molecules and hyperedges to reactions. Pathways are modelled as integer hyperflows and we expand the network model by detailed routing constraints. In contrast to the more traditional approaches like Flux Balance Analysis or Elementary Mode analysis we insist on integer-valued flows. While this choice makes it necessary to solve possibly hard integer linear programs, it has the advantage that more detailed mechanistic questions can be formulated. It is thus possible to query networks for general transformation motifs, and to automatically enumerate optimal and near-optimal pathways. Similarities and differences between our work and traditional approaches in metabolic network analysis are discussed in detail. To demonstrate the applicability of the mathematical framework to real-life problems we first explore the design space of possible non-oxidative glycolysis pathways and show that recent manually designed pathways can be further optimized. We then use a model of sugar chemistry to investigate pathways in the autocatalytic formose process. A graph transformation-based approach is used to automatically generate the reaction networks of interest.

Novel column generation-based optimization approach for poly-pathway kinetic model applied to CHO cell culture

Mathematical modelling can provide precious tools for bioprocess simulation, prediction, control and optimization of mammalian cell-based cultures. In this paper we present a novel method to generate kinetic models of such cultures, rendering complex metabolic networks in a poly-pathway kinetic model. The model is based on subsets of elementary flux modes (EFMs) to generate macro-reactions. Thanks to our column generation-based optimization algorithm, the experimental data are used to identify the EFMs, which are relevant to the data. Here the systematic enumeration of all the EFMs is eliminated and a network including a large number of reactions can be considered. In particular, the poly-pathway model can simulate multiple metabolic behaviors in response to changes in the culture conditions.

We apply the method to a network of 126 metabolic reactions describing cultures of antibody-producing Chinese hamster ovary cells, and generate a poly-pathway model that simulates multiple experimental conditions obtained in response to variations in amino acid availability. A good fit between simulated and experimental data is obtained, rendering the variations in the growth, product, and metabolite uptake/secretion rates. The intracellular reaction fluxes simulated by the model are explored, linking variations in metabolic behavior to adaptations of the intracellular metabolism.

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Novel method to model multiple states by a poly-pathway kinetic model.

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EFMs relevant to data identified by column generation (CG)-based optimization.

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CG optimization enables use of networks much larger than systematic enumeration.

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A kinetic model simulates changes in metabolic rates linked to available amino acids.

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The flux distribution of each metabolic state is visualized in the original network.

Thermodynamic constraints for identifying elementary flux modes

Metabolic pathway analysis is a key method to study metabolism and the elementary flux modes (EFMs) is one major concept allowing one to analyze the network in terms of minimal pathways. Their practical use has been hampered by the combinatorial explosion of their number in large systems. The EFMs give the possible pathways at steady state, but the real pathways are limited by biological constraints. In this review, we display three different methods that integrate thermodynamic constraints in terms of Gibbs free energy in the EFMs computation.

The Vast Complexity of the Epigenetic Landscape during Neurodevelopment: An Open Frame to Understanding Brain Function

Development is a well-defined stage-to-stage process that allows the coordination and maintenance of the structure and function of cells and their progenitors, in a complete organism embedded in an environment that, in turn, will shape cellular responses to external stimuli. Epigenetic mechanisms comprise a group of process that regulate genetic expression without changing the DNA sequence, and they contribute to the necessary plasticity of individuals to face a constantly changing medium. These mechanisms act in conjunction with genetic pools and their correct interactions will be crucial to zygote formation, embryo development, and brain tissue organization. In this work, we will summarize the main findings related to DNA methylation and histone modifications in embryonic stem cells and throughout early development phases. Furthermore, we will critically outline some key observations on how epigenetic mechanisms influence the rest of the developmental process and how long its footprint is extended from fecundation to adulthood.

Sequential computation of elementary modes and minimal cut sets in genome-scale metabolic networks using alternate integer linear programming

Motivation

Elementary (flux) modes (EMs) have served as a valuable tool for investigating structural and functional properties of metabolic networks. Identification of the full set of EMs in genome-scale networks remains challenging due to combinatorial explosion of EMs in complex networks. It is often, however, that only a small subset of relevant EMs needs to be known, for which optimization-based sequential computation is a useful alternative. Most of the currently available methods along this line are based on the iterative use of mixed integer linear programming (MILP), the effectiveness of which significantly deteriorates as the number of iterations builds up. To alleviate the computational burden associated with the MILP implementation, we here present a novel optimization algorithm termed alternate integer linear programming (AILP).

Results

Our algorithm was designed to iteratively solve a pair of integer programming (IP) and linear programming (LP) to compute EMs in a sequential manner. In each step, the IP identifies a minimal subset of reactions, the deletion of which disables all previously identified EMs. Thus, a subsequent LP solution subject to this reaction deletion constraint becomes a distinct EM. In cases where no feasible LP solution is available, IP-derived reaction deletion sets represent minimal cut sets (MCSs). Despite the additional computation of MCSs, AILP achieved significant time reduction in computing EMs by orders of magnitude. The proposed AILP algorithm not only offers a computational advantage in the EM analysis of genome-scale networks, but also improves the understanding of the linkage between EMs and MCSs.

Availability and Implementation

The software is implemented in Matlab, and is provided as supplementary information .

Contact

hyunseob.song@pnnl.gov.

Supplementary information

Supplementary data are available at Bioinformatics online.

A graph-based approach to analyze flux-balanced pathways in metabolic networks

An elementary flux mode (EFM) is a pathway with minimum set of reactions that are functional in steady-state constrained space. Due to the high computational complexity of calculating EFMs, different approaches have been proposed to find these flux-balanced pathways. In this paper, an approach to find a subset of EFMs is proposed based on a graph data model. The given metabolic network is mapped to the graph model and decisions for reaction inclusion can be made based on metabolites and their associated reactions. This notion makes the approach more convenient to categorize the output pathways. Implications of the proposed method on metabolic networks are discussed.

Designing Optimized Production Hosts by Metabolic Modeling

Many of the complex and expensive production steps in the chemical industry are readily available in living cells. In order to overcome the metabolic limits of these cells, the optimal genetic intervention strategies can be computed by the use of metabolic modeling. Elementary flux mode analysis (EFMA) is an ideal tool for this task, as it does not require defining a cellular objective function. We present two EFMA-based methods to optimize production hosts: (1) the standard approach that can only be used for small and medium scale metabolic networks and (2) the advanced dual system approach that can be utilized to directly compute intervention strategies in a genome-scale metabolic model.

Identification of metabolic pathways using pathfinding approaches: a systematic review

Metabolic pathways have become increasingly available for various microorganisms. Such pathways have spurred the development of a wide array of computational tools, in particular, mathematical pathfinding approaches. This article can facilitate the understanding of computational analysis of metabolic pathways in genomics. Moreover, stoichiometric and pathfinding approaches in metabolic pathway analysis are discussed. Three major types of studies are elaborated: stoichiometric identification models, pathway-based graph analysis and pathfinding approaches in cellular metabolism. Furthermore, evaluation of the outcomes of the pathways with mathematical benchmarking metrics is provided. This review would lead to better comprehension of metabolism behaviors in living cells, in terms of computed pathfinding approaches.

Resource allocation in living organisms

Quantitative prediction of resource allocation for living systems has been an intensive area of research in the field of biology. Resource allocation was initially investigated in higher organisms by using empirical mathematical models based on mass distribution. A challenge is now to go a step further by reconciling the cellular scale to the individual scale. In the present paper, we review the foundations of modelling of resource allocation, particularly at the cellular scale: from small macro-molecular models to genome-scale cellular models. We enlighten how the combination of omic measurements and computational advances together with systems biology has contributed to dramatic progresses in the current understanding and prediction of cellular resource allocation. Accurate genome-wide predictive methods of resource allocation based on the resource balance analysis (RBA) framework have been developed and ensure a good trade-off between the complexity/tractability and the prediction capability of the model. The RBA framework shows promise for a wide range of applications in metabolic engineering and synthetic biology, and for pursuing investigations of the design principles of cellular and multi-cellular organisms.

Metabolomics analysis: Finding out metabolic building blocks

In this paper we propose a new methodology for the analysis of metabolic networks. We use the notion of strongly connected components of a graph, called in this context metabolic building blocks. Every strongly connected component is contracted to a single node in such a way that the resulting graph is a directed acyclic graph, called a metabolic DAG, with a considerably reduced number of nodes. The property of being a directed acyclic graph brings out a background graph topology that reveals the connectivity of the metabolic network, as well as bridges, isolated nodes and cut nodes. Altogether, it becomes a key information for the discovery of functional metabolic relations. Our methodology has been applied to the glycolysis and the purine metabolic pathways for all organisms in the KEGG database, although it is general enough to work on any database. As expected, using the metabolic DAGs formalism, a considerable reduction on the size of the metabolic networks has been obtained, specially in the case of the purine pathway due to its relative larger size. As a proof of concept, from the information captured by a metabolic DAG and its corresponding metabolic building blocks, we obtain the core of the glycolysis pathway and the core of the purine metabolism pathway and detect some essential metabolic building blocks that reveal the key reactions in both pathways. Finally, the application of our methodology to the glycolysis pathway and the purine metabolism pathway reproduce the tree of life for the whole set of the organisms represented in the KEGG database which supports the utility of this research.

Metabolic Network Modeling of Microbial Interactions in Natural and Engineered Environmental Systems

We review approaches to characterize metabolic interactions within microbial communities using Stoichiometric Metabolic Network (SMN) models for applications in environmental and industrial biotechnology. SMN models are computational tools used to evaluate the metabolic engineering potential of various organisms. They have successfully been applied to design and optimize the microbial production of antibiotics, alcohols and amino acids by single strains. To date however, such models have been rarely applied to analyze and control the metabolism of more complex microbial communities. This is largely attributed to the diversity of microbial community functions, metabolisms, and interactions. Here, we firstly review different types of microbial interaction and describe their relevance for natural and engineered environmental processes. Next, we provide a general description of the essential methods of the SMN modeling workflow including the steps of network reconstruction, simulation through Flux Balance Analysis (FBA), experimental data gathering, and model calibration. Then we broadly describe and compare four approaches to model microbial interactions using metabolic networks, i.e., (i) lumped networks, (ii) compartment per guild networks, (iii) bi-level optimization simulations, and (iv) dynamic-SMN methods. These approaches can be used to integrate and analyze diverse microbial physiology, ecology and molecular community data. All of them (except the lumped approach) are suitable for incorporating species abundance data but so far they have been used only to model simple communities of two to eight different species. Interactions based on substrate exchange and competition can be directly modeled using the above approaches. However, interactions based on metabolic feedbacks, such as product inhibition and synthropy require extensions to current models, incorporating gene regulation and compounding accumulation mechanisms. SMN models of microbial interactions can be used to analyze complex “omics” data and to infer and optimize metabolic processes. Thereby, SMN models are suitable to capitalize on advances in high-throughput molecular and metabolic data generation. SMN models are starting to be applied to describe microbial interactions during wastewater treatment, in-situ bioremediation, microalgae blooms methanogenic fermentation, and bioplastic production. Despite their current challenges, we envisage that SMN models have future potential for the design and development of novel growth media, biochemical pathways and synthetic microbial associations.

Which sets of elementary flux modes form thermodynamically feasible flux distributions?

Elementary flux modes (EFMs) are non-decomposable steady-state fluxes through metabolic networks. Every possible flux through a network can be described as a superposition of EFMs. The definition of EFMs is based on the stoichiometry of the network, and it has been shown previously that not all EFMs are thermodynamically feasible. These infeasible EFMs cannot contribute to a biologically meaningful flux distribution. In this work, we show that a set of thermodynamically feasible EFMs need not be thermodynamically consistent. We use first principles of thermodynamics to define the feasibility of a flux distribution and present a method to compute the largest thermodynamically consistent sets (LTCSs) of EFMs. An LTCS contains the maximum number of EFMs that can be combined to form a thermodynamically feasible flux distribution. As a case study we analyze all LTCSs found in Escherichia coli when grown on glucose and show that only one LTCS shows the required phenotypical properties. Using our method, we find that in our E. coli model < 10% of all EFMs are thermodynamically relevant.

Principal elementary mode analysis (PEMA)

Principal component analysis (PCA) has been widely applied in fluxomics to compress data into a few latent structures in order to simplify the identification of metabolic patterns. These latent structures lack a direct biological interpretation due to the intrinsic constraints associated with a PCA model. Here we introduce a new method that significantly improves the interpretability of the principal components with a direct link to metabolic pathways. This method, called principal elementary mode analysis (PEMA), establishes a bridge between a PCA-like model, aimed at explaining the maximum variance in flux data, and the set of elementary modes (EMs) of a metabolic network. It provides an easy way to identify metabolic patterns in large fluxomics datasets in terms of the simplest pathways of the organism metabolism. The results using a real metabolic model of Escherichia coli show the ability of PEMA to identify the EMs that generated the different simulated flux distributions. Actual flux data of E. coli and Pichia pastoris cultures confirm the results observed in the simulated study, providing a biologically meaningful model to explain flux data of both organisms in terms of the EM activation. The PEMA toolbox is freely available for non-commercial purposes on http://mseg.webs.upv.es.

Robustness analysis of elementary flux modes generated by column generation

Elementary flux modes (EFMs) are vectors defined from a metabolic reaction network, giving the connections between substrates and products. EFMs-based metabolic flux analysis (MFA) estimates the flux over each EFM from external flux measurements through least-squares data fitting. The measurements used in the data fitting are subject to errors. A robust optimization problem includes information on errors and gives a way to examine the sensitivity of the solution of the EFMs-based MFA to these errors. In general, formulating a robust optimization problem may make the problem significantly harder. We show that in the case of the EFMs-based MFA, when the errors are only in measurements and bounded by an interval, the robust problem can be stated as a convex quadratic programming (QP) problem. We have previously shown how the data fitting problem may be solved in a column-generation framework. In this paper, we show how column generation may be applied also to the robust problem, thereby avoiding explicit enumeration of EFMs. Furthermore, the option to indicate intervals on metabolites that are not measured is introduced in this column generation framework. The robustness of the data is evaluated in a case-study, which indicates that the solutions of our non-robust problems are in fact near-optimal also when robustness is considered, implying that the errors in measurement do not have a large impact on the optimal solution. Furthermore, we showed that the addition of intervals on unmeasured metabolites resulted in a change in the optimal solution.

Designing minimal microbial strains of desired functionality using a genetic algorithm

Background

The rational, in silico prediction of gene-knockouts to turn organisms into efficient cell factories is an essential and computationally challenging task in metabolic engineering. Elementary flux mode analysis in combination with constraint minimal cut sets is a particularly powerful method to identify optimal engineering targets, which will force an organism into the desired metabolic state. Given an engineering objective, it is theoretically possible, although computationally impractical, to find the best minimal intervention strategies.

Results

We developed a genetic algorithm (GA-MCS) to quickly find many (near) optimal intervention strategies while overcoming the above mentioned computational burden. We tested our algorithm on Escherichia coli metabolic networks of three different sizes to find intervention strategies satisfying three different engineering objectives.

Conclusions

We show that GA-MCS finds all practically relevant targets for any (non)-linear engineering objective. Our algorithm also found solutions comparable to previously published results. We show that for large networks optimal solutions are found within a fraction of the time used for a complete enumeration.

In Silico Constraint-Based Strain Optimization Methods: the Quest for Optimal Cell Factories

Shifting from chemical to biotechnological processes is one of the cornerstones of 21st century industry. The production of a great range of chemicals via biotechnological means is a key challenge on the way toward a bio-based economy. However, this shift is occurring at a pace slower than initially expected. The development of efficient cell factories that allow for competitive production yields is of paramount importance for this leap to happen. Constraint-based models of metabolism, together with in silico strain design algorithms, promise to reveal insights into the best genetic design strategies, a step further toward achieving that goal. In this work, a thorough analysis of the main in silico constraint-based strain design strategies and algorithms is presented, their application in real-world case studies is analyzed, and a path for the future is discussed.

PreProPath: An Uncertainty-Aware Algorithm for Identifying Predictable Profitable Pathways in Biochemical Networks

Pathway analysis is a powerful approach to enable rational design or redesign of biochemical networks for optimizing metabolic engineering and synthetic biology objectives such as production of desired chemicals or biomolecules from specific nutrients. While experimental methods can be quite successful, computational approaches can enhance discovery and guide experimentation by efficiently exploring very large design spaces. We present a computational algorithm, Predictably Profitable Path (PreProPath), to identify target pathways best suited for engineering modifications. The algorithm utilizes uncertainties about the metabolic networks operating state inherent in the underdetermined linear equations representing the stoichiometric model. Flux Variability Analysis is used to determine the operational flux range. PreProPath identifies a path that is predictable in behavior, exhibiting small flux ranges, and profitable, containing the least restrictive flux-limiting reaction in the network. The algorithm is computationally efficient because it does not require enumeration of pathways. The results of case studies show that PreProPath can efficiently analyze variances in metabolic states and model uncertainties to suggest pathway engineering strategies that have been previously supported by experimental data.

Avoiding the Enumeration of Infeasible Elementary Flux Modes by Including Transcriptional Regulatory Rules in the Enumeration Process Saves Computational Costs

Despite the significant progress made in recent years, the computation of the complete set of elementary flux modes of large or even genome-scale metabolic networks is still impossible. We introduce a novel approach to speed up the calculation of elementary flux modes by including transcriptional regulatory information into the analysis of metabolic networks. Taking into account gene regulation dramatically reduces the solution space and allows the presented algorithm to constantly eliminate biologically infeasible modes at an early stage of the computation procedure. Thereby, computational costs, such as runtime, memory usage, and disk space, are extremely reduced. Moreover, we show that the application of transcriptional rules identifies non-trivial system-wide effects on metabolism. Using the presented algorithm pushes the size of metabolic networks that can be studied by elementary flux modes to new and much higher limits without the loss of predictive quality. This makes unbiased, system-wide predictions in large scale metabolic networks possible without resorting to any optimization principle.

Predicting genetic engineering targets with Elementary Flux Mode Analysis: a review of four current methods

Elementary flux modes (EFMs) are a well-established tool in metabolic modeling. EFMs are minimal, feasible, steady state pathways through a metabolic network. They are used in various approaches to predict targets for genetic interventions in order to increase production of a molecule of interest via a host cell. Here we give an introduction to the concept of EFMs, present an overview of four methods which use EFMs in order to predict engineering targets and lastly use a toy model and a small-scale metabolic model to demonstrate and compare the capabilities of these methods.

Interplay between constraints, objectives, and optimality for genome-scale stoichiometric models

High-throughput data generation and genome-scale stoichiometric models have greatly facilitated the comprehensive study of metabolic networks. The computation of all feasible metabolic routes with these models, given stoichiometric, thermodynamic, and steady-state constraints, provides important insights into the metabolic capacities of a cell. How the feasible metabolic routes emerge from the interplay between flux constraints, optimality objectives, and the entire metabolic network of a cell is, however, only partially understood. We show how optimal metabolic routes, resulting from flux balance analysis computations, arise out of elementary flux modes, constraints, and optimization objectives. We illustrate our findings with a genome-scale stoichiometric model of Escherichia coli metabolism. In the case of one flux constraint, all feasible optimal flux routes can be derived from elementary flux modes alone. We found up to 120 million of such optimal elementary flux modes. We introduce a new computational method to compute the corner points of the optimal solution space fast and efficiently. Optimal flux routes no longer depend exclusively on elementary flux modes when we impose additional constraints; new optimal metabolic routes arise out of combinations of elementary flux modes. The solution space of feasible metabolic routes shrinks enormously when additional objectives---e.g. those related to pathway expression costs or pathway length---are introduced. In many cases, only a single metabolic route remains that is both feasible and optimal. This paper contributes to reaching a complete topological understanding of the metabolic capacity of organisms in terms of metabolic flux routes, one that is most natural to biochemists and biotechnologists studying and engineering metabolism.

Metabolomics integrated elementary flux mode analysis in large metabolic networks

Elementary flux modes (EFMs) are non-decomposable steady-state pathways in metabolic networks. They characterize phenotypes, quantify robustness or identify engineering targets. An EFM analysis (EFMA) is currently restricted to medium-scale models, as the number of EFMs explodes with the network's size. However, many topologically feasible EFMs are biologically irrelevant. We present thermodynamic EFMA (tEFMA), which calculates only the small(er) subset of thermodynamically feasible EFMs. We integrate network embedded thermodynamics into EFMA and show that we can use the metabolome to identify and remove thermodynamically infeasible EFMs during an EFMA without losing biologically relevant EFMs. Calculating only the thermodynamically feasible EFMs strongly reduces memory consumption and program runtime, allowing the analysis of larger networks. We apply tEFMA to study the central carbon metabolism of E. coli and find that up to 80% of its EFMs are thermodynamically infeasible. Moreover, we identify glutamate dehydrogenase as a bottleneck, when E. coli is grown on glucose and explain its inactivity as a consequence of network embedded thermodynamics. We implemented tEFMA as a Java package which is available for download at https://github.com/mpgerstl/tEFMA.

tEFMA: computing thermodynamically feasible elementary flux modes in metabolic networks

: Elementary flux modes (EFMs) are important structural tools for the analysis of metabolic networks. It is known that many topologically feasible EFMs are biologically irrelevant. Therefore, tools are needed to find the relevant ones. We present thermodynamic tEFM analysis (tEFMA) which uses the cellular metabolome to avoid the enumeration of thermodynamically infeasible EFMs. Specifically, given a metabolic network and a not necessarily complete metabolome, tEFMA efficiently returns the full set of thermodynamically feasible EFMs consistent with the metabolome. Compared with standard approaches, tEFMA strongly reduces the memory consumption and the overall runtime. Thus tEFMA provides a new way to analyze unbiasedly hitherto inaccessible large-scale metabolic networks.

AVAILABILITY AND IMPLEMENTATION

https://github.com/mpgerstl/tEFMA CONTACT: : christian.jungreuthmayer@boku.ac.at or juergen.zanghellini@boku.ac.at

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

TreeEFM: calculating elementary flux modes using linear optimization in a tree-based algorithm

MOTIVATION

Elementary flux modes (EFMs) analysis constitutes a fundamental tool in systems biology. However, the efficient calculation of EFMs in genome-scale metabolic networks (GSMNs) is still a challenge. We present a novel algorithm that uses a linear programming-based tree search and efficiently enumerates a subset of EFMs in GSMNs.

RESULTS

Our approach is compared with the EFMEvolver approach, demonstrating a significant improvement in computation time. We also validate the usefulness of our new approach by studying the acetate overflow metabolism in the Escherichia coli bacteria. To do so, we computed 1 million EFMs for each energetic amino acid and then analysed the relevance of each energetic amino acid based on gene/protein expression data and the obtained EFMs. We found good agreement between previous experiments and the conclusions reached using EFMs. Finally, we also analysed the performance of our approach when applied to large GSMNs.

AVAILABILITY AND IMPLEMENTATION

The stand-alone software TreeEFM is implemented in C++ and interacts with the open-source linear solver COIN-OR Linear program Solver (CLP).

On dynamically generating relevant elementary flux modes in a metabolic network using optimization

Elementary flux modes (EFMs) are pathways through a metabolic reaction network that connect external substrates to products. Using EFMs, a metabolic network can be transformed into its macroscopic counterpart, in which the internal metabolites have been eliminated and only external metabolites remain. In EFMs-based metabolic flux analysis (MFA) experimentally determined external fluxes are used to estimate the flux of each EFM. It is in general prohibitive to enumerate all EFMs for complex networks, since the number of EFMs increases rapidly with network complexity. In this work we present an optimization-based method that dynamically generates a subset of EFMs and solves the EFMs-based MFA problem simultaneously. The obtained subset contains EFMs that contribute to the optimal solution of the EFMs-based MFA problem. The usefulness of our method was examined in a case-study using data from a Chinese hamster ovary cell culture and two networks of varied complexity. It was demonstrated that the EFMs-based MFA problem could be solved at a low computational cost, even for the more complex network. Additionally, only a fraction of the total number of EFMs was needed to compute the optimal solution.

Estimating biological elementary flux modes that decompose a flux distribution by the minimal branching property

MOTIVATION

Elementary flux mode (EFM) is a useful tool in constraint-based modeling of metabolic networks. The property that every flux distribution can be decomposed as a weighted sum of EFMs allows certain applications of EFMs to studying flux distributions. The existence of biologically infeasible EFMs and the non-uniqueness of the decomposition, however, undermine the applicability of such methods. Efforts have been made to find biologically feasible EFMs by incorporating information from transcriptional regulation and thermodynamics. Yet, no attempt has been made to distinguish biologically feasible EFMs by considering their graphical properties. A previous study on the transcriptional regulation of metabolic genes found that distinct branches at a branch point metabolite usually belong to distinct metabolic pathways. This suggests an intuitive property of biologically feasible EFMs, i.e. minimal branching.

RESULTS

We developed the concept of minimal branching EFM and derived the minimal branching decomposition (MBD) to decompose flux distributions. Testing in the core Escherichia coli metabolic network indicated that MBD can distinguish branches at branch points and greatly reduced the solution space in which the decomposition is often unique. An experimental flux distribution from a previous study on mouse cardiomyocyte was decomposed using MBD. Comparison with decomposition by a minimum number of EFMs showed that MBD found EFMs more consistent with established biological knowledge, which facilitates interpretation. Comparison of the methods applied to a complex flux distribution in Lactococcus lactis similarly showed the advantages of MBD. The minimal branching EFM concept underlying MBD should be useful in other applications.

CONTACT

sinhu@bio.dtu.dk or p.ji@polyu.edu.hk

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.