Reconstructing metabolic flux vectors from extreme pathways: defining the alpha-spectrum

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.

Differential Expression Analysis Utilizing Condition-Specific Metabolic Pathways

Pathway analysis is ubiquitous in biological data analysis due to the ability to integrate small simultaneous changes in functionally related components. While pathways are often defined based on either manual curation or network topological properties, an attractive alternative is to generate pathways around specific functions, in which metabolism can be defined as the production and consumption of specific metabolites. In this work, we present an algorithm, termed MetPath, that calculates pathways for condition-specific production and consumption of specific metabolites. We demonstrate that these pathways have several useful properties. Pathways calculated in this manner (1) take into account the condition-specific metabolic role of a gene product, (2) are localized around defined metabolic functions, and (3) quantitatively weigh the importance of expression to a function based on the flux contribution of the gene product. We demonstrate how these pathways elucidate network interactions between genes across different growth conditions and between cell types. Furthermore, the calculated pathways compare favorably to manually curated pathways in predicting the expression correlation between genes. To facilitate the use of these pathways, we have generated a large compendium of pathways under different growth conditions for E. coli. The MetPath algorithm provides a useful tool for metabolic network-based statistical analyses of high-throughput data.

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.

Finding MEMo: minimum sets of elementary flux modes

Metabolic network reconstructions are widely used in computational systems biology for in silico studies of cellular metabolism. A common approach to analyse these models are elementary flux modes (EFMs), which correspond to minimal functional units in the network. Already for medium-sized networks, it is often impossible to compute the set of all EFMs, due to their huge number. From a practical point of view, this might also not be necessary because a subset of EFMs may already be sufficient to answer relevant biological questions. In this article, we study MEMos or minimum sets of EFMs that can generate all possible steady-state behaviours of a metabolic network. The number of EFMs in a MEMo may be by several orders of magnitude smaller than the total number of EFMs. Using MEMos, we can compute generating sets of EFMs in metabolic networks where the whole set of EFMs is too large to be enumerated.

Extreme pathway analysis reveals the organizing rules of metabolic regulation

Cellular systems shift metabolic states by adjusting gene expression and enzyme activities to adapt to physiological and environmental changes. Biochemical and genetic studies are identifying how metabolic regulation affects the selection of metabolic phenotypes. However, how metabolism influences its regulatory architecture still remains unexplored. We present a new method of extreme pathway analysis (the minimal set of conically independent metabolic pathways) to deduce regulatory structures from pure pathway information. Applying our method to metabolic networks of human red blood cells and Escherichia coli, we shed light on how metabolic regulation are organized by showing which reactions within metabolic networks are more prone to transcriptional or allosteric regulation. Applied to a human genome-scale metabolic system, our method detects disease-associated reactions. Thus, our study deepens the understanding of the organizing principle of cellular metabolic regulation and may contribute to metabolic engineering, synthetic biology, and disease treatment.

redGEM: Systematic reduction and analysis of genome-scale metabolic reconstructions for development of consistent core metabolic models

Genome-scale metabolic reconstructions have proven to be valuable resources in enhancing our understanding of metabolic networks as they encapsulate all known metabolic capabilities of the organisms from genes to proteins to their functions. However the complexity of these large metabolic networks often hinders their utility in various practical applications. Although reduced models are commonly used for modeling and in integrating experimental data, they are often inconsistent across different studies and laboratories due to different criteria and detail, which can compromise transferability of the findings and also integration of experimental data from different groups. In this study, we have developed a systematic semi-automatic approach to reduce genome-scale models into core models in a consistent and logical manner focusing on the central metabolism or subsystems of interest. The method minimizes the loss of information using an approach that combines graph-based search and optimization methods. The resulting core models are shown to be able to capture key properties of the genome-scale models and preserve consistency in terms of biomass and by-product yields, flux and concentration variability and gene essentiality. The development of these “consistently-reduced” models will help to clarify and facilitate integration of different experimental data to draw new understanding that can be directly extendable to genome-scale models.

Reduced models are used commonly to understand the metabolism of organisms and to integrate experimental data for many different studies such as physiology, fluxomics and metabolomics. Without consistent or clear criteria on how these reduced models are actually developed, it is difficult to ensure that they reflect the detailed knowledge that is kept in genome scale metabolic network models (GEMs). The redGEM algorithm presented here allows us to systematically develop consistently reduced metabolic models from their genome-scale counterparts. We applied redGEM for the construction of a core model for E. coli central carbon metabolism. We constructed the core model irJO1366 based on the latest genome-scale E. coli metabolic reconstruction (iJO1366). irJO1366 contains the central carbon pathways and other immediate pathways that must be connected to them for consistency with the iJO1366. irJO1366 can be used to understand metabolism of the organism and also to provide guidance for metabolic engineering purposes. The algorithm is also designed to be modular so that heterologous reactions or pathways can be appended to the core model akin to a “plug-and-play”, synthetic biology approach. The algorithm is applicable to any compartmentalized or non-compartmentalized GEM.

New views on the selection acting on genetic polymorphism in central metabolic genes

Studies of the polymorphism of central metabolic genes as a source of fitness variation in natural populations date back to the discovery of allozymes in the 1960s. The unique features of these genes and their enzymes and our knowledge base greatly facilitates the systems-level study of this group. The expectation that pathway flux control is central to understanding the molecular evolution of genes is discussed, as well as studies that attempt to place gene-specific molecular evolution and polymorphism into a context of pathway and network architecture. There is an increasingly complex picture of the metabolic genes assuming additional roles beyond their textbook anabolic and catabolic reactions. In particular, this review emphasizes the potential role of these genes as part of the energy-sensing machinery. It is underscored that the concentrations of key cellular metabolites are the reflections of cellular energy status and nutritional input. These metabolites are the top-down signaling messengers that set signaling through signaling pathways that are involved in energy economy. I propose that the polymorphisms in central metabolic genes shift metabolite concentrations and in that fashion act as genetic modifiers of the energy-state coupling to the transcriptional networks that affect physiological trade-offs with significant fitness consequences.

A principal components method constrained by elementary flux modes: analysis of flux data sets

Background

Non-negative linear combinations of elementary flux modes (EMs) describe all feasible reaction flux distributions for a given metabolic network under the quasi steady state assumption. However, only a small subset of EMs contribute to the physiological state of a given cell.

Results

In this paper, a method is proposed that identifies the subset of EMs that best explain the physiological state captured in reaction flux data, referred to as principal EMs (PEMs), given a pre-specified universe of EM candidates. The method avoids the evaluation of all possible combinations of EMs by using a branch and bound approach which is computationally very efficient. The performance of the method is assessed using simulated and experimental data of Pichia pastoris and experimental fluxome data of Saccharomyces cerevisiae. The proposed method is benchmarked against principal component analysis (PCA), commonly used to study the structure of metabolic flux data sets.

Conclusions

The overall results show that the proposed method is computationally very effective in identifying the subset of PEMs within a large set of EM candidates (cases with ~100 and ~1000 EMs were studied). In contrast to the principal components in PCA, the identified PEMs have a biological meaning enabling identification of the key active pathways in a cell as well as the conditions under which the pathways are activated. This method clearly outperforms PCA in the interpretability of flux data providing additional insights into the underlying regulatory mechanisms.

Electronic supplementary material

The online version of this article (doi:10.1186/s12859-016-1063-0) contains supplementary material, which is available to authorized users.

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.

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.

Minimal metabolic pathway structure is consistent with associated biomolecular interactions

Pathways are a universal paradigm for functionally describing cellular processes. Even though advances in high-throughput data generation have transformed biology, the core of our biological understanding, and hence data interpretation, is still predicated on human-defined pathways. Here, we introduce an unbiased, pathway structure for genome-scale metabolic networks defined based on principles of parsimony that do not mimic canonical human-defined textbook pathways. Instead, these minimal pathways better describe multiple independent pathway-associated biomolecular interaction datasets suggesting a functional organization for metabolism based on parsimonious use of cellular components. We use the inherent predictive capability of these pathways to experimentally discover novel transcriptional regulatory interactions in Escherichia coli metabolism for three transcription factors, effectively doubling the known regulatory roles for Nac and MntR. This study suggests an underlying and fundamental principle in the evolutionary selection of pathway structures; namely, that pathways may be minimal, independent, and segregated.

A novel methodology to estimate metabolic flux distributions in constraint-based models

Quite generally, constraint-based metabolic flux analysis describes the space of viable flux configurations for a metabolic network as a high-dimensional polytope defined by the linear constraints that enforce the balancing of production and consumption fluxes for each chemical species in the system. In some cases, the complexity of the solution space can be reduced by performing an additional optimization, while in other cases, knowing the range of variability of fluxes over the polytope provides a sufficient characterization of the allowed configurations. There are cases, however, in which the thorough information encoded in the individual distributions of viable fluxes over the polytope is required. Obtaining such distributions is known to be a highly challenging computational task when the dimensionality of the polytope is sufficiently large, and the problem of developing cost-effective ad hoc algorithms has recently seen a major surge of interest. Here, we propose a method that allows us to perform the required computation heuristically in a time scaling linearly with the number of reactions in the network, overcoming some limitations of similar techniques employed in recent years. As a case study, we apply it to the analysis of the human red blood cell metabolic network, whose solution space can be sampled by different exact techniques, like Hit-and-Run Monte Carlo (scaling roughly like the third power of the system size). Remarkably accurate estimates for the true distributions of viable reaction fluxes are obtained, suggesting that, although further improvements are desirable, our method enhances our ability to analyze the space of allowed configurations for large biochemical reaction networks.

Metabolic flux is a determinant of the evolutionary rates of enzyme-encoding genes

Relationships between evolutionary rates and gene properties on a genomic, functional, pathway, or system level are being explored to unravel the principles of the evolutionary process. In particular, functional network properties have been analyzed to recognize the constraints they may impose on the evolutionary fate of genes. Here we took as a case study the core metabolic network in human erythrocytes and we analyzed the relationship between the evolutionary rates of its genes and the metabolic flux distribution throughout it. We found that metabolic flux correlates with the ratio of nonsynonymous to synonymous substitution rates. Genes encoding enzymes that carry high fluxes have been more constrained in their evolution, while purifying selection is more relaxed in genes encoding enzymes carrying low metabolic fluxes. These results demonstrate the importance of considering the dynamical functioning of gene networks when assessing the action of selection on system-level properties.

Signaling network prediction by the Ontology Fingerprint enhanced Bayesian network

BACKGROUND

Despite large amounts of available genomic and proteomic data, predicting the structure and response of signaling networks is still a significant challenge. While statistical method such as Bayesian network has been explored to meet this challenge, employing existing biological knowledge for network prediction is difficult. The objective of this study is to develop a novel approach that integrates prior biological knowledge in the form of the Ontology Fingerprint to infer cell-type-specific signaling networks via data-driven Bayesian network learning; and to further use the trained model to predict cellular responses.

RESULTS

We applied our novel approach to address the Predictive Signaling Network Modeling challenge of the fourth (2009) Dialog for Reverse Engineering Assessment's and Methods (DREAM4) competition. The challenge results showed that our method accurately captured signal transduction of a network of protein kinases and phosphoproteins in that the predicted protein phosphorylation levels under all experimental conditions were highly correlated (R2 = 0.93) with the observed results. Based on the evaluation of the DREAM4 organizer, our team was ranked as one of the top five best performers in predicting network structure and protein phosphorylation activity under test conditions.

CONCLUSIONS

Bayesian network can be used to simulate the propagation of signals in cellular systems. Incorporating the Ontology Fingerprint as prior biological knowledge allows us to efficiently infer concise signaling network structure and to accurately predict cellular responses.

Optimal flux spaces of genome-scale stoichiometric models are determined by a few subnetworks

The metabolism of organisms can be studied with comprehensive stoichiometric models of their metabolic networks. Flux balance analysis (FBA) calculates optimal metabolic performance of stoichiometric models. However, detailed biological interpretation of FBA is limited because, in general, a huge number of flux patterns give rise to the same optimal performance. The complete description of the resulting optimal solution spaces was thus far a computationally intractable problem. Here we present CoPE-FBA: Comprehensive Polyhedra Enumeration Flux Balance Analysis, a computational method that solves this problem. CoPE-FBA indicates that the thousands to millions of optimal flux patterns result from a combinatorial explosion of flux patterns in just a few metabolic sub-networks. The entire optimal solution space can now be compactly described in terms of the topology of these sub-networks. CoPE-FBA simplifies the biological interpretation of stoichiometric models of metabolism, and provides a profound understanding of metabolic flexibility in optimal states.

Advanced stoichiometric analysis of metabolic networks of mammalian systems

Metabolic engineering tools have been widely applied to living organisms to gain a comprehensive understanding about cellular networks and to improve cellular properties. Metabolic flux analysis (MFA), flux balance analysis (FBA), and metabolic pathway analysis (MPA) are among the most popular tools in stoichiometric network analysis. Although application of these tools into well-known microbial systems is extensive in the literature, various barriers prevent them from being utilized in mammalian cells. Limited experimental data, complex regulatory mechanisms, and the requirement of more complex nutrient media are some major obstacles in mammalian cell systems. However, mammalian cells have been used to produce therapeutic proteins, to characterize disease states or related abnormal metabolic conditions, and to analyze the toxicological effects of some medicinally important drugs. Therefore, there is a growing need for extending metabolic engineering principles to mammalian cells in order to understand their underlying metabolic functions. In this review article, advanced metabolic engineering tools developed for stoichiometric analysis including MFA, FBA, and MPA are described. Applications of these tools in mammalian cells are discussed in detail, and the challenges and opportunities are highlighted.

Network reduction in metabolic pathway analysis: elucidation of the key pathways involved in the photoautotrophic growth of the green alga Chlamydomonas reinhardtii

Metabolic pathway analysis aims at discovering and analyzing meaningful routes and their interactions in metabolic networks. A major difficulty in applying this technique lies in the decomposition of metabolic flux distributions into elementary mode or extreme pathway activity patterns, which in general is not unique. We propose a network reduction approach based on the decomposition of a set of flux vectors representing adaptive microbial metabolic behavior in bioreactors into a minimal set of shared pathways. Several optimality criteria from the literature were compared in order to select the most appropriate objective function. We further analyze photoautotrophic metabolism of the green alga Chlamydomonas reinhardtii growing in a photobioreactor under maximal growth rate conditions. Key pathways involved in its adaptive metabolic response to changes in light influx are identified and discussed using an energetic approach.

Elementary mode analysis: a useful metabolic pathway analysis tool for reprograming microbial metabolic pathways

Elementary mode analysis is a useful metabolic pathway analysis tool to characterize cellular metabolism. It can identify all feasible metabolic pathways known as elementary modes that are inherent to a metabolic network. Each elementary mode contains a minimal and unique set of enzymatic reactions that can support cellular functions at steady state. Knowledge of all these pathway options enables systematic characterization of cellular phenotypes, analysis of metabolic network properties (e.g. structure, regulation, robustness, and fragility), phenotypic behavior discovery, and rational strain design for metabolic engineering application. This chapter focuses on the application of elementary mode analysis to reprogram microbial metabolic pathways for rational strain design and the metabolic pathway evolution of designed strains.

Metabolic response of perfused livers to various oxygenation conditions

Isolated liver perfusion systems have been used to characterize intrinsic metabolic changes in liver as a result of various perturbations, including systemic injury, hepatotoxin exposure, and warm ischemia. Most of these studies were done using hyperoxic conditions (95% O(2)) but without the use of oxygen carriers in the perfusate. Prior literature data do not clearly establish the impact of oxygenation, and in particular that of adding oxygen carriers to the perfusate, on the metabolic functions of the liver. Therefore, herein the effects of oxygen delivery in the perfusion system on liver metabolism were investigated by comparing three modes of oxygenation. Rat livers were perfused via the portal and hepatic veins at a constant flow rate of 3 mL/min/g liver in a recirculating perfusion system. In the first group, the perfusate was equilibrated in a membrane oxygenator with room air (21% O(2)) before entering the liver. In the second group, the perfusate was equilibrated with a 95% O(2)/5% CO(2) gas mixture. In the third group, the perfusate was supplemented with washed bovine red blood cells (RBCs) at 10% hematocrit and also equilibrated with the 95% O(2)/5% CO(2) gas mixture. Oxygen and CO(2) gradients across the liver were measured periodically with a blood gas analyzer. The rate of change in the concentration of major metabolites in the perfusate was measured over time. Net extracellular fluxes were calculated from these measurements and applied to a stoichiometric-based optimization problem to determine the intracellular fluxes and active pathways in the perfused livers. Livers perfused with RBCs consumed oxygen at twice the rate observed using hyperoxic (95% O(2)) perfusate without RBCs, and also produced more urea and ketone bodies. At the flow rate used, the oxygen supply in perfusate without RBCs was just sufficient to meet the average oxygen demand of the liver but would be insufficient if it increased above baseline, as is often the case in response to environmental perturbations. Metabolic pathway analysis suggests that significant anaerobic glycolysis occurred in the absence of RBCs even using hyperoxic perfusate. Conversely, when RBCs were used, glucose production from lactate and glutamate, as well as pathways related to energy metabolism were upregulated. RBCs also reversed an increase in PPP fluxes induced by the use of hyperoxic perfusate alone. In conclusion, the use of oxygen carriers is required to investigate the effect of various perturbations on liver metabolism.

Projection to latent pathways (PLP): a constrained projection to latent variables (PLS) method for elementary flux modes discrimination

Background

Elementary flux modes (EFM) are unique and non-decomposable sets of metabolic reactions able to operate coherently in steady-state. A metabolic network has in general a very high number of EFM reflecting the typical functional redundancy of biological systems. However, most of these EFM are either thermodynamically unfeasible or inactive at pre-set environmental conditions.

Results

Here we present a new algorithm that discriminates the "active" set of EFM on the basis of dynamic envirome data. The algorithm merges together two well-known methods: projection to latent structures (PLS) and EFM analysis, and is therefore termed projection to latent pathways (PLP). PLP has two concomitant goals: (1) maximisation of correlation between EFM weighting factors and measured envirome data and (2) minimisation of redundancy by eliminating EFM with low correlation with the envirome.

Conclusions

Overall, our results demonstrate that PLP slightly outperforms PLS in terms of predictive power. But more importantly, PLP is able to discriminate the subset of EFM with highest correlation with the envirome, thus providing in-depth knowledge of how the environment controls core cellular functions. This offers a significant advantage over PLS since its abstract structure cannot be associated with the underlying biological structure.

Metabolic network analysis of perfused livers under fed and fasted states: incorporating thermodynamic and futile-cycle-associated regulatory constraints

Isolated liver perfusion systems have been extensively used to characterize intrinsic metabolic changes in liver under various conditions, including systemic injury, hepatotoxin exposure, and warm ischemia. Most of these studies were performed utilizing fasted animals prior to perfusion so that a simplified metabolic network could be used in order to determine intracellular fluxes. However, fasting induced metabolic alterations might interfere with disease related changes. Therefore, there is a need to develop a "unified" metabolic flux analysis approach that could be similarly applied to both fed and fasted states. In this study we explored a methodology based on elementary mode analysis in order to determine intracellular fluxes and active pathways simultaneously. In order to decrease the solution space, thermodynamic constraints, and enzymatic regulatory properties for the formation of futile cycles were further considered in the model, resulting in a mixed integer quadratic programming problem. Given the published experimental observations describing the perfused livers under fed and fasted states, the proposed approach successfully determined that gluconeogenesis, glycogenolysis and fatty acid oxidation were active in both states. However, fasting increased the fluxes in gluconeogenic reactions whereas it decreased fluxes associated with glycogenolysis, TCA cycle, fatty acid oxidation and electron transport reactions. This analysis further identified that more pathways were found to be active in fed state while their weight values were relatively lower compared to fasted state. Glucose, lactate, glutamine, glutamate and ketone bodies were also found to be important external metabolites whose extracellular fluxes should be used in the hepatic metabolic network analysis. In conclusion, the mathematical formulation explored in this study is an attractive tool to analyze the metabolic network of perfused livers under various disease conditions. This approach could be simultaneously applied to both fasted and fed data sets.

Decomposing flux distributions into elementary flux modes in genome-scale metabolic networks

MOTIVATION

Elementary flux mode (EFM) is a fundamental concept as well as a useful tool in metabolic pathway analysis. One important role of EFMs is that every flux distribution can be decomposed into a set of EFMs and a number of methods to study flux distributions originated from it. Yet finding such decompositions requires the complete set of EFMs, which is intractable in genome-scale metabolic networks due to combinatorial explosion.

RESULTS

In this article, we proposed an algorithm to decompose flux distributions into EFMs in genome-scale networks. It is an iterative scheme of a mixed integer linear program. Unlike previous optimization models to find pathways, any feasible solutions can become EFMs in our algorithm. This advantage enables the algorithm to approximate the EFM of largest contribution to an objective reaction in a flux distribution. Our algorithm is able to find EFMs of flux distributions with complex structures, closer to the realistic case in which a cell is subject to various constraints. A case of Escherichia coli growth in the Lysogeny broth (LB) medium containing various carbon sources was studied. Essential metabolites and their syntheses were located. Information on the contribution of each carbon source not obvious from the apparent flux distribution was also revealed. Our work further confirms the utility of finding EFMs by optimization models in genome-scale metabolic networks.

CONTACT

joshua.chan@connect.polyu.hk

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Cell functional enviromics: unravelling the function of environmental factors

Background

While functional genomics, focused on gene functions and gene-gene interactions, has become a very active field of research in molecular biology, equivalent methodologies embracing the environment and gene-environment interactions are relatively less developed. Understanding the function of environmental factors is, however, of paramount importance given the complex, interactive nature of environmental and genetic factors across multiple time scales.

Results

Here, we propose a systems biology framework, where the function of environmental factors is set at its core. We set forth a "reverse" functional analysis approach, whereby cellular functions are reconstructed from the analysis of dynamic envirome data. Our results show these data sets can be mapped to less than 20 core cellular functions in a typical mammalian cell culture, while explaining over 90% of flux data variance. A functional enviromics map can be created, which provides a template for manipulating the environmental factors to induce a desired phenotypic trait.

Conclusion

Our results support the feasibility of cellular function reconstruction guided by the analysis and manipulation of dynamic envirome data.

Pathway analysis of liver metabolism under stressed condition

Pathway analysis is a useful tool which reveals important metabolic network properties. However, the big challenge is to propose an objective function for estimating active pathways, which represent the actual state of network. In order to provide weight values for all possible pathways within the metabolic network, this study presents different approaches, considering the structural and physiological properties of the metabolic network, aiming at a unique decomposition of the flux vector into pathways. These methods were used to analyze the hepatic metabolism considering available data sets obtained from the perfused livers of fasted rats receiving burn injury. Utilizing unique decomposition techniques and different fluxes revealed that higher weights were always attributed to short pathways. Specific pathways, including pyruvate, glutamate and oxaloacetate pools, and urea production from arginine, were found to be important or essential in all methods and experimental conditions. Moreover the pathways, including serine production from glycine and conversion between acetoacetate and B-OH-butyrate, were assigned higher weights. Pathway analysis was also used to identify the main sources for the production of certain products in the hepatic metabolic network to gain a better understanding of the effects of burn injury on liver metabolism.

Genome-scale metabolic networks

During the last decade, models have been developed to characterize cellular metabolism at the level of an entire metabolic network. The main concept that underlies whole-network metabolic modeling is the identification and mathematical definition of constraints. Here, we review large-scale metabolic network modeling, in particular, stoichiometric- and constraint-based approaches. Although many such models have been reconstructed, few networks have been extensively validated and tested experimentally, and we focus on these. We describe how metabolic networks can be represented using stoichiometric matrices and well-defined constraints on metabolic fluxes. We then discuss relatively successful approaches, including flux balance analysis (FBA), pathway analysis, and common extensions or modifications to these approaches. Finally, we describe techniques for integrating these approaches with models of other biological processes.

Computationally efficient flux variability analysis

BACKGROUND

Flux variability analysis is often used to determine robustness of metabolic models in various simulation conditions. However, its use has been somehow limited by the long computation time compared to other constraint-based modeling methods.

RESULTS

We present an open source implementation of flux variability analysis called fastFVA. This efficient implementation makes large-scale flux variability analysis feasible and tractable allowing more complex biological questions regarding network flexibility and robustness to be addressed.

CONCLUSIONS

Networks involving thousands of biochemical reactions can be analyzed within seconds, greatly expanding the utility of flux variability analysis in systems biology.

Which metabolic pathways generate and characterize the flux space? A comparison among elementary modes, extreme pathways and minimal generators

Important efforts are being done to systematically identify the relevant pathways in a metabolic network. Unsurprisingly, there is not a unique set of network-based pathways to be tagged as relevant, and at least four related concepts have been proposed: extreme currents, elementary modes, extreme pathways, and minimal generators. Basically, there are two properties that these sets of pathways can hold: they can generate the flux space--if every feasible flux distribution can be represented as a nonnegative combination of flux through them--or they can comprise all the nondecomposable pathways in the network. The four concepts fulfill the first property, but only the elementary modes fulfill the second one. This subtle difference has been a source of errors and misunderstandings. This paper attempts to clarify the intricate relationship between the network-based pathways performing a comparison among them.

Computational modeling of mammalian signaling networks

One of the most exciting developments in signal transduction research has been the proliferation of studies in which a biological discovery was initiated by computational modeling. In this study, we review the major efforts that enable such studies. First, we describe the experimental technologies that are generally used to identify the molecular components and interactions in, and dynamic behavior exhibited by, a network of interest. Next, we review the mathematical approaches that are used to model signaling network behavior. Finally, we focus on three specific instances of 'model-driven discovery': cases in which computational modeling of a signaling network has led to new insights that have been verified experimentally.

Elementary mode analysis: a useful metabolic pathway analysis tool for characterizing cellular metabolism

Elementary mode analysis is a useful metabolic pathway analysis tool to identify the structure of a metabolic network that links the cellular phenotype to the corresponding genotype. The analysis can decompose the intricate metabolic network comprised of highly interconnected reactions into uniquely organized pathways. These pathways consisting of a minimal set of enzymes that can support steady state operation of cellular metabolism represent independent cellular physiological states. Such pathway definition provides a rigorous basis to systematically characterize cellular phenotypes, metabolic network regulation, robustness, and fragility that facilitate understanding of cell physiology and implementation of metabolic engineering strategies. This mini-review aims to overview the development and application of elementary mode analysis as a metabolic pathway analysis tool in studying cell physiology and as a basis of metabolic engineering.

Decomposition of complex microbial behaviors into resource-based stress responses

MOTIVATION

Highly redundant metabolic networks and experimental data from cultures likely adapting simultaneously to multiple stresses can complicate the analysis of cellular behaviors. It is proposed that the explicit consideration of these factors is critical to understanding the competitive basis of microbial strategies.

RESULTS

Wide ranging, seemingly unrelated Escherichia coli physiological fluxes can be simply and accurately described as linear combinations of a few ecologically relevant stress adaptations. These strategies were identified by decomposing the central metabolism of E.coli into elementary modes (mathematically defined biochemical pathways) and assessing the resource investment cost-benefit properties for each pathway. The approach capitalizes on the inherent tradeoffs related to investing finite resources like nitrogen into different pathway enzymes when the pathways have varying metabolic efficiencies. The subset of ecologically competitive pathways represented 0.02% of the total permissible pathways. The biological relevance of the assembled strategies was tested against 10 000 randomly constructed pathway subsets. None of the randomly assembled collections were able to describe all of the considered experimental data as accurately as the cost-based subset. The results suggest these metabolic strategies are biologically significant. The current descriptions were compared with linear programming (LP)-based flux descriptions using the Euclidean distance metric. The current study's pathway subset described the experimental fluxes with better accuracy than the LP results without having to test multiple objective functions or constraints and while providing additional ecological insight into microbial behavior. The assembled pathways seem to represent a generalized set of strategies that can describe a wide range of microbial responses and hint at evolutionary processes where a handful of successful metabolic strategies are utilized simultaneously in different combinations to adapt to diverse conditions.

Algebraic method for the analysis of signaling crosstalk

A unified mathematical description that expresses the characteristics of whole systems is necessary for an understanding of signal transduction cascades. In this study we explore an algebraic method, named extreme signaling flow, enhanced from the concept of extreme pathway, to analyze signal transduction systems. This method enables us to represent the long-term potentiation (LTP) and the long-term depression (LTD) of hippocampal neuronal plasticity in an integrated simulation model. The model is validated by comparing the results of redundancy, reaction participation, and in silico knockout analysis with biological knowledge available from the literature. The following properties are assumed in these computational analyses: (1) LTP is fault-tolerant under network modification, (2) protein kinase C and MAPK have numerous routes to LTP induction, (3) calcium-calmodulin kinase II has a few routes to LTP induction, and (4) calcineurin has many routes to LTD induction. These results demonstrate that our approach produces an integrated framework for analyzing properties of large-scale systems with complicated signal transduction.

Identification of flux regulation coefficients from elementary flux modes: A systems biology tool for analysis of metabolic networks

Within a metabolic network, the elementary flux modes enables a unique description of different operations of the network. Thus, the metabolic fluxes can be specified as convex combinations of the elementary flux modes. Here, we describe an approach to identify the set of elementary flux modes that operates in a given metabolic network through the use of measurements of macroscopic fluxes, that is, fluxes in and out of the cell. Besides enabling estimation of the metabolic fluxes, the parameters of the linear combinations of the elementary flux modes provide valuable physiological information; we call these parameters flux regulation coefficients (FRCs). These coefficients indicate which enzyme subsets are important at different growth conditions. We demonstrate how FRCs can be used to map the operation of the metabolic network of the yeast Saccharomyces sp. under different growth conditions.

Bayesian flux balance analysis applied to a skeletal muscle metabolic model

In this article, the steady state condition for the multi-compartment models for cellular metabolism is considered. The problem is to estimate the reaction and transport fluxes, as well as the concentrations in venous blood when the stoichiometry and bound constraints for the fluxes and the concentrations are given. The problem has been addressed previously by a number of authors, and optimization-based approaches as well as extreme pathway analysis have been proposed. These approaches are briefly discussed here. The main emphasis of this work is a Bayesian statistical approach to the flux balance analysis (FBA). We show how the bound constraints and optimality conditions such as maximizing the oxidative phosphorylation flux can be incorporated into the model in the Bayesian framework by proper construction of the prior densities. We propose an effective Markov chain Monte Carlo (MCMC) scheme to explore the posterior densities, and compare the results with those obtained via the previously studied linear programming (LP) approach. The proposed methodology, which is applied here to a two-compartment model for skeletal muscle metabolism, can be extended to more complex models.

Integration of enzyme activities into metabolic flux distributions by elementary mode analysis

Background

In systems biology, network-based pathway analysis facilitates understanding or designing metabolic systems and enables prediction of metabolic flux distributions. Network-based flux analysis requires considering not only pathway architectures but also the proteome or transcriptome to predict flux distributions, because recombinant microbes significantly change the distribution of gene expressions. The current problem is how to integrate such heterogeneous data to build a network-based model.

Results

To link enzyme activity data to flux distributions of metabolic networks, we have proposed Enzyme Control Flux (ECF), a novel model that integrates enzyme activity into elementary mode analysis (EMA). ECF presents the power-law formula describing how changes in enzyme activities between wild-type and a mutant are related to changes in the elementary mode coefficients (EMCs). To validate the feasibility of ECF, we integrated enzyme activity data into the EMCs of Escherichia coli and Bacillus subtilis wild-type. The ECF model effectively uses an enzyme activity profile to estimate the flux distribution of the mutants and the increase in the number of incorporated enzyme activities decreases the model error of ECF.

Conclusion

The ECF model is a non-mechanistic and static model to link an enzyme activity profile to a metabolic flux distribution by introducing the power-law formula into EMA, suggesting that the change in an enzyme profile rather reflects the change in the flux distribution. The ECF model is highly applicable to the central metabolism in knockout mutants of E. coli and B. subtilis.

Metabolic network properties help assign weights to elementary modes to understand physiological flux distributions

MOTIVATION

Elementary modes (EMs) analysis has been well established. The existing methodologies for assigning weights to EMs cannot be directly applied for large-scale metabolic networks, since the tremendous number of modes would make the computation a time-consuming or even an impossible mission. Therefore, developing more efficient methods to deal with large set of EMs is urgent.

RESULT

We develop a method to evaluate the performance of employing a subset of the elementary modes to reconstruct a real flux distribution by using the relative error between the real flux vector and the reconstructed one as an indicator. We have found a power function relationship between the decrease of relative error and the increase of the number of the selecting EMs, and a logarithmic relationship between the increases of the number of non-zero weighted EMs and that of the number of the selecting EMs. Our discoveries show that it is possible to reconstruct a given flux distribution by a selected subset of EMs from a large metabolic network and furthermore, they help us identify the 'governing modes' to represent the cellular metabolism for such a condition.

An interval approach for dealing with flux distributions and elementary modes activity patterns

This work introduces the use of an interval representation of fluxes. This representation can be useful in two common situations: (a) when fluxes are uncertain due to the lack of accurate measurements and (b) when the flux distribution is partially unknown. In addition, the interval representation can be used for other purposes such as dealing with inconsistency or representing a range of behaviour. Two main problems are addressed. On the one hand, the translation of a metabolic flux distribution into an elementary modes or extreme pathways activity pattern is analysed. In general, there is not a unique solution for this problem but a range of solutions. To represent the whole solution region in an easy way, it is possible to compute the alpha-spectrum (i.e., the range of possible values for each elementary mode or extreme pathway activity). Herein, a method is proposed which, based on the interval representation of fluxes, makes it possible to compute the alpha-spectrum from an uncertain or even partially unknown flux distribution. On the other hand, the concept of the flux-spectrum is introduced as a variant of the metabolic flux analysis methodology that presents some advantages: applicable when measurements are insufficient (underdetermined case), integration of uncertain measurements, inclusion of irreversibility constraints and an alternative procedure to deal with inconsistency. Frequently, when applying metabolic flux analysis the available measurements are insufficient and/or uncertain and the complete flux distribution cannot be uniquely calculated. The method proposed here allows the determination of the ranges of possible values for each non-calculable flux, resulting in a flux region called flux-spectrum. In order to illustrate the proposed methods, the example of the metabolic network of CHO cells cultivated in stirred flasks is used.