Modes and cuts in metabolic networks: Complexity and algorithms

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.

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

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

On the 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.

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

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

The topology of genome-scale metabolic reconstructions unravels independent modules and high network flexibility

The topology of metabolic networks is recognisably modular with modules weakly connected apart from sharing a pool of currency metabolites. Here, we defined modules as sets of reversible reactions isolated from the rest of metabolism by irreversible reactions except for the exchange of currency metabolites. Our approach identifies topologically independent modules under specific conditions associated with different metabolic functions. As case studies, the E.coli iJO1366 and Human Recon 2.2 genome-scale metabolic models were split in 103 and 321 modules respectively, displaying significant correlation patterns in expression data. Finally, we addressed a fundamental question about the metabolic flexibility conferred by reversible reactions: “Of all Directed Topologies (DTs) defined by fixing directions to all reversible reactions, how many are capable of carrying flux through all reactions?”. Enumeration of the DTs for iJO1366 model was performed using an efficient depth-first search algorithm, rejecting infeasible DTs based on mass-imbalanced and loopy flux patterns. We found the direction of 79% of reversible reactions must be defined before all directions in the network can be fixed, granting a high degree of flexibility.

Genome-scale metabolic reconstructions represent all biochemical reactions that an organism can accomplish. These reconstructions are complex and often difficult to study in great detail. A way to overcome this limitation is to focus on specific pathways or subsystems. We present a novel method to identify metabolic modules based on the network topology. The method relies on reaction directions and ignores currency metabolites, which artificially connect distant metabolic reactions. In this way, topologically independent modules are built, where inputs and outputs are controlled by irreversible reactions. The method is automatic and unbiased, and, the result is a set of condition specific modules with defined metabolic functions. As a proof-of-concept we generated biologically relevant modules for the E.coli and Human genome-scale metabolic reconstructions supported by transcriptomic data. Finally, we applied the novel approach to study the network flexibility conferred by reversible reactions. In the case of the E. coli model, we found that the direction of 79% of structurally reversible reactions (those not directionally constrained by surrounding irreversible reactions) must be fixed to determine all the reaction directions in the network. Therefore, reversible reactions operate practically independent of each other.

Combining Kinetic and Constraint-Based Modelling to Better Understand Metabolism Dynamics

To understand the phenotypic capabilities of organisms, it is useful to characterise cellular metabolism through the analysis of its pathways. Dynamic mathematical modelling of metabolic networks is of high interest as it provides the time evolution of the metabolic components. However, it also has limitations, such as the necessary mechanistic details and kinetic parameters are not always available. On the other hand, large metabolic networks exhibit a complex topological structure which can be studied rather efficiently in their stationary regime by constraint-based methods. These methods produce useful predictions on pathway operations. In this review, we present both modelling techniques and we show how they bring complementary views of metabolism. In particular, we show on a simple example how both approaches can be used in conjunction to shed some light on the dynamics of metabolic networks.

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.

Boosting the extraction of elementary flux modes in genome-scale metabolic networks using the linear programming approach

MOTIVATION

Elementary flux modes (EFMs) are a key tool for analyzing genome-scale metabolic networks, and several methods have been proposed to compute them. Among them, those based on solving linear programming (LP) problems are known to be very efficient if the main interest lies in computing large enough sets of EFMs.

RESULTS

Here, we propose a new method called EFM-Ta that boosts the efficiency rate by analyzing the information provided by the LP solver. We base our method on a further study of the final tableau of the simplex method. By performing additional elementary steps and avoiding trivial solutions consisting of two cycles, we obtain many more EFMs for each LP problem posed, improving the efficiency rate of previously proposed methods by more than one order of magnitude.

AVAILABILITY AND IMPLEMENTATION

Software is freely available at https://github.com/biogacop/Boost_LP_EFM.

CONTACT

fguil@um.es.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

SWIFTCORE: a tool for the context-specific reconstruction of genome-scale metabolic networks

BACKGROUND

High-throughput omics technologies have enabled the comprehensive reconstructions of genome-scale metabolic networks for many organisms. However, only a subset of reactions is active in each cell which differs from tissue to tissue or from patient to patient. Reconstructing a subnetwork of the generic metabolic network from a provided set of context-specific active reactions is a demanding computational task.

RESULTS

We propose SWIFTCC and SWIFTCORE as effective methods for flux consistency checking and the context-specific reconstruction of genome-scale metabolic networks which consistently outperform the previous approaches.

CONCLUSIONS

We have derived an approximate greedy algorithm which efficiently scales to increasingly large metabolic networks. SWIFTCORE is freely available for non-commercial use in the GitHub repository at https://mtefagh.github.io/swiftcore/.

Towards scaling elementary flux mode computation

While elementary flux mode (EFM) analysis is now recognized as a cornerstone computational technique for cellular pathway analysis and engineering, EFM application to genome-scale models remains computationally prohibitive. This article provides a review of aspects of EFM computation that elucidates bottlenecks in scaling EFM computation. First, algorithms for computing EFMs are reviewed. Next, the impact of redundant constraints, sensitivity to constraint ordering and network compression are evaluated. Then, the advantages and limitations of recent parallelization and GPU-based efforts are highlighted. The article then reviews alternative pathway analysis approaches that aim to reduce the EFM solution space. Despite advances in EFM computation, our review concludes that continued scaling of EFM computation is necessary to apply EFM to genome-scale models. Further, our review concludes that pathway analysis methods that target specific pathway properties can provide powerful alternatives to EFM analysis.

Flux tope analysis: studying the coordination of reaction directions in metabolic networks

Motivation

Elementary flux mode (EFM) analysis allows an unbiased description of metabolic networks in terms of minimal pathways (involving a minimal set of reactions). To date, the enumeration of EFMs is impracticable in genome-scale metabolic models. In a complementary approach, we introduce the concept of a flux tope (FT), involving a maximal set of reactions (with fixed directions), which allows one to study the coordination of reaction directions in metabolic networks and opens a new way for EFM enumeration.

Results

A FT is a (nontrivial) subset of the flux cone specified by fixing the directions of all reversible reactions. In a consistent metabolic network (without unused reactions), every FT contains a 'maximal pathway', carrying flux in all reactions. This decomposition of the flux cone into FTs allows the enumeration of EFMs (of individual FTs) without increasing the problem dimension by reaction splitting. To develop a mathematical framework for FT analysis, we build on the concepts of sign vectors and hyperplane arrangements. Thereby, we observe that FT analysis can be applied also to flux optimization problems involving additional (inhomogeneous) linear constraints. For the enumeration of FTs, we adapt the reverse search algorithm and provide an efficient implementation. We demonstrate that (biomass-optimal) FTs can be enumerated in genome-scale metabolic models of B.cuenoti and E.coli, and we use FTs to enumerate EFMs in models of M.genitalium and B.cuenoti.

Availability and implementation

The source code is freely available at https://github.com/mpgerstl/FTA.

Supplementary information

Supplementary data are available at Bioinformatics online.

Computing Minimum Reaction Modifications in a Boolean Metabolic Network

In metabolic network modification, we newly add enzymes or/and knock-out genes to maximize the biomass production with minimum side-effect. Although this problem has been studied for various problem settings via mathematical models including flux balance analysis, elementary mode, and Boolean models, some important problem settings still remain to be studied. In this paper, we consider the Boolean Reaction Modification (BRM) problem, where a host metabolic network and a reference metabolic network are given in the Boolean model. The host network initially produces some toxic compounds and cannot produce some necessary compounds, but the reference network can produce the necessary compounds, and we should minimize the total number of removed reactions from the host network and added reactions from the reference network so that the toxic compounds are not producible, but the necessary compounds are producible in the resulting host network. We developed integer linear programming (ILP)-based methods for BRM, and compared them with OptStrain and SimOptStrain. The results show that our method performed better for reducing the total number of added and removed reactions, while OptStrain and SimOptStrain performed better for optimizing the production of the target compound. Our developed software is freely available at "http://sunflower.kuicr.kyoto-u.ac.jp/~rogi/solBRM/solBRM.html ".

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.

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.

The MONGOOSE Rational Arithmetic Toolbox

The modeling of metabolic networks has seen a rapid expansion following the complete sequencing of thousands of genomes. The constraint-based modeling framework has emerged as one of the most popular approaches to reconstructing and analyzing genome-scale metabolic models. Its main assumption is that of a quasi-steady-state, requiring that the production of each internal metabolite be balanced by its consumption. However, due to the multiscale nature of the models, the large number of reactions and metabolites, and the use of floating-point arithmetic for the stoichiometric coefficients, ensuring that this assumption holds can be challenging.The MONGOOSE toolbox addresses this problem by using rational arithmetic, thus ensuring that models are analyzed in a reproducible manner and consistently with modeling assumptions. In this chapter we present a protocol for the complete analysis of a metabolic network model using the MONGOOSE toolbox, via its newly developed GUI, and describe how it can be used as a model-checking platform both during and after the model construction process.

A review of parameters and heuristics for guiding metabolic pathfinding

Recent developments in metabolic engineering have led to the successful biosynthesis of valuable products, such as the precursor of the antimalarial compound, artemisinin, and opioid precursor, thebaine. Synthesizing these traditionally plant-derived compounds in genetically modified yeast cells introduces the possibility of significantly reducing the total time and resources required for their production, and in turn, allows these valuable compounds to become cheaper and more readily available. Most biosynthesis pathways used in metabolic engineering applications have been discovered manually, requiring a tedious search of existing literature and metabolic databases. However, the recent rapid development of available metabolic information has enabled the development of automated approaches for identifying novel pathways. Computer-assisted pathfinding has the potential to save biochemists time in the initial discovery steps of metabolic engineering. In this paper, we review the parameters and heuristics used to guide the search in recent pathfinding algorithms. These parameters and heuristics capture information on the metabolic network structure, compound structures, reaction features, and organism-specificity of pathways. No one metabolic pathfinding algorithm or search parameter stands out as the best to use broadly for solving the pathfinding problem, as each method and parameter has its own strengths and shortcomings. As assisted pathfinding approaches continue to become more sophisticated, the development of better methods for visualizing pathway results and integrating these results into existing metabolic engineering practices is also important for encouraging wider use of these pathfinding methods.

Identification of bifurcation transitions in biological regulatory networks using Answer-Set Programming

Background

Numerous cellular differentiation processes can be captured using discrete qualitative models of biological regulatory networks. These models describe the temporal evolution of the state of the network subject to different competing transitions, potentially leading the system to different attractors. This paper focusses on the formal identification of states and transitions that are crucial for preserving or pre-empting the reachability of a given behaviour.

Methods

In the context of non-deterministic automata networks, we propose a static identification of so-called bifurcations, i.e., transitions after which a given goal is no longer reachable. Such transitions are naturally good candidates for controlling the occurrence of the goal, notably by modulating their propensity. Our method combines Answer-Set Programming with static analysis of reachability properties to provide an under-approximation of all the existing bifurcations.

Results

We illustrate our discrete bifurcation analysis on several models of biological systems, for which we identify transitions which impact the reachability of given long-term behaviour. In particular, we apply our implementation on a regulatory network among hundreds of biological species, supporting the scalability of our approach.

Conclusions

Our method allows a formal and scalable identification of transitions which are responsible for the lost of capability to reach a given state. It can be applied to any asynchronous automata networks, which encompass Boolean and multi-valued models. An implementation is provided as part of the Pint software, available at http://loicpauleve.name/pint.

Electronic supplementary material

The online version of this article (doi:10.1186/s13015-017-0110-3) contains supplementary material, which is available to authorized users.

Computational complexity of algorithms for sequence comparison, short-read assembly and genome alignment

A multitude of algorithms for sequence comparison, short-read assembly and whole-genome alignment have been developed in the general context of molecular biology, to support technology development for high-throughput sequencing, numerous applications in genome biology and fundamental research on comparative genomics. The computational complexity of these algorithms has been previously reported in original research papers, yet this often neglected property has not been reviewed previously in a systematic manner and for a wider audience. We provide a review of space and time complexity of key sequence analysis algorithms and highlight their properties in a comprehensive manner, in order to identify potential opportunities for further research in algorithm or data structure optimization. The complexity aspect is poised to become pivotal as we will be facing challenges related to the continuous increase of genomic data on unprecedented scales and complexity in the foreseeable future, when robust biological simulation at the cell level and above becomes a reality.

How important is thermodynamics for identifying elementary flux modes?

We present a method for computing thermodynamically feasible elementary flux modes (tEFMs) using equilibrium constants without need of internal metabolite concentrations. The method is compared with the method based on a binary distinction between reversible and irreversible reactions. When all reactions are reversible, adding the constraints based on equilibrium constants reduces the number of elementary flux modes (EFMs) by a factor of two. Declaring in advance some reactions as irreversible, based on reliable biochemical expertise, can in general reduce the number of EFMs by a greater factor. But, even in this case, computing tEFMs can rule out some EFMs which are biochemically irrelevant. We applied our method to two published models described with binary distinction: the monosaccharide metabolism and the central carbon metabolism of Chinese hamster ovary cells. The results show that the binary distinction is in good agreement with biochemical observations. Moreover, the suppression of the EFMs that are not consistent with the equilibrium constants appears to be biologically relevant.

A mixed-integer linear programming approach to the reduction of genome-scale metabolic networks

Background

Constraint-based analysis has become a widely used method to study metabolic networks. While some of the associated algorithms can be applied to genome-scale network reconstructions with several thousands of reactions, others are limited to small or medium-sized models. In 2015, Erdrich et al. introduced a method called NetworkReducer, which reduces large metabolic networks to smaller subnetworks, while preserving a set of biological requirements that can be specified by the user. Already in 2001, Burgard et al. developed a mixed-integer linear programming (MILP) approach for computing minimal reaction sets under a given growth requirement.

Results

Here we present an MILP approach for computing minimum subnetworks with the given properties. The minimality (with respect to the number of active reactions) is not guaranteed by NetworkReducer, while the method by Burgard et al. does not allow specifying the different biological requirements. Our procedure is about 5-10 times faster than NetworkReducer and can enumerate all minimum subnetworks in case there exist several ones. This allows identifying common reactions that are present in all subnetworks, and reactions appearing in alternative pathways.

Conclusions

Applying complex analysis methods to genome-scale metabolic networks is often not possible in practice. Thus it may become necessary to reduce the size of the network while keeping important functionalities. We propose a MILP solution to this problem. Compared to previous work, our approach is more efficient and allows computing not only one, but even all minimum subnetworks satisfying the required properties.

Electronic supplementary material

The online version of this article (doi:10.1186/s12859-016-1412-z) contains supplementary material, which is available to authorized users.

PSAMM: A Portable System for the Analysis of Metabolic Models

The genome-scale models of metabolic networks have been broadly applied in phenotype prediction, evolutionary reconstruction, community functional analysis, and metabolic engineering. Despite the development of tools that support individual steps along the modeling procedure, it is still difficult to associate mathematical simulation results with the annotation and biological interpretation of metabolic models. In order to solve this problem, here we developed a Portable System for the Analysis of Metabolic Models (PSAMM), a new open-source software package that supports the integration of heterogeneous metadata in model annotations and provides a user-friendly interface for the analysis of metabolic models. PSAMM is independent of paid software environments like MATLAB, and all its dependencies are freely available for academic users. Compared to existing tools, PSAMM significantly reduced the running time of constraint-based analysis and enabled flexible settings of simulation parameters using simple one-line commands. The integration of heterogeneous, model-specific annotation information in PSAMM is achieved with a novel format of YAML-based model representation, which has several advantages, such as providing a modular organization of model components and simulation settings, enabling model version tracking, and permitting the integration of multiple simulation problems. PSAMM also includes a number of quality checking procedures to examine stoichiometric balance and to identify blocked reactions. Applying PSAMM to 57 models collected from current literature, we demonstrated how the software can be used for managing and simulating metabolic models. We identified a number of common inconsistencies in existing models and constructed an updated model repository to document the resolution of these inconsistencies.

The broad application of genome-scale metabolic modeling has made it a useful technique for tackling fundamental questions in biological research and engineering. Today over 100 models have been constructed for organisms that carry out a diverse array of metabolic activities spanning all three kingdoms of life. These models, however, have been curated independently following different conventions. The maintenance of model consistency has been challenging due to the lack of consensus in model representation and the absence of integrated modeling software for associating mathematical simulations with the annotation and biological interpretation of metabolic models. To solve this problem, we developed a new software package, PSAMM, and a new model format that incorporates heterogeneous, model-specific annotation information into modular representations of model definitions and simulation settings. PSAMM provides significant advances in standardizing the workflow of model annotation and consistency checking. Compared to existing tools, PSAMM supports more flexible configurations and is more efficient in running constraint-based simulations. All functions of PSAMM are freely available for academic users and can be downloaded from a public Git repository (https://zhanglab.github.io/psamm/) under the GNU General Public License.

Computing smallest intervention strategies for multiple metabolic networks in a boolean model

This article considers the problem whereby, given two metabolic networks N1 and N2, a set of source compounds, and a set of target compounds, we must find the minimum set of reactions whose removal (knockout) ensures that the target compounds are not producible in N1 but are producible in N2. Similar studies exist for the problem of finding the minimum knockout with the smallest side effect for a single network. However, if technologies of external perturbations are advanced in the near future, it may be important to develop methods of computing the minimum knockout for multiple networks (MKMN). Flux balance analysis (FBA) is efficient if a well-polished model is available. However, that is not always the case. Therefore, in this article, we study MKMN in Boolean models and an elementary mode (EM)-based model. Integer linear programming (ILP)-based methods are developed for these models, since MKMN is NP-complete for both the Boolean model and the EM-based model. Computer experiments are conducted with metabolic networks of clostridium perfringens SM101 and bifidobacterium longum DJO10A, respectively known as bad bacteria and good bacteria for the human intestine. The results show that larger networks are more likely to have MKMN solutions. However, solving for these larger networks takes a very long time, and often the computation cannot be completed. This is reasonable, because small networks do not have many alternative pathways, making it difficult to satisfy the MKMN condition, whereas in large networks the number of candidate solutions explodes. Our developed software minFvskO is available online.

Fast-SL: an efficient algorithm to identify synthetic lethal sets in metabolic networks

MOTIVATION

Synthetic lethal sets are sets of reactions/genes where only the simultaneous removal of all reactions/genes in the set abolishes growth of an organism. Previous approaches to identify synthetic lethal genes in genome-scale metabolic networks have built on the framework of flux balance analysis (FBA), extending it either to exhaustively analyze all possible combinations of genes or formulate the problem as a bi-level mixed integer linear programming (MILP) problem. We here propose an algorithm, Fast-SL, which surmounts the computational complexity of previous approaches by iteratively reducing the search space for synthetic lethals, resulting in a substantial reduction in running time, even for higher order synthetic lethals.

RESULTS

We performed synthetic reaction and gene lethality analysis, using Fast-SL, for genome-scale metabolic networks of Escherichia coli, Salmonella enterica Typhimurium and Mycobacterium tuberculosis. Fast-SL also rigorously identifies synthetic lethal gene deletions, uncovering synthetic lethal triplets that were not reported previously. We confirm that the triple lethal gene sets obtained for the three organisms have a precise match with the results obtained through exhaustive enumeration of lethals performed on a computer cluster. We also parallelized our algorithm, enabling the identification of synthetic lethal gene quadruplets for all three organisms in under 6 h. Overall, Fast-SL enables an efficient enumeration of higher order synthetic lethals in metabolic networks, which may help uncover previously unknown genetic interactions and combinatorial drug targets.

AVAILABILITY AND IMPLEMENTATION

The MATLAB implementation of the algorithm, compatible with COBRA toolbox v2.0, is available at https://github.com/RamanLab/FastSL CONTACT: kraman@iitm.ac.in

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

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.

Computing Elementary Flux Modes Involving a Set of Target Reactions

Elementary flux mode (EM) computation is an important tool in the constraint-based analysis of genome-scale metabolic networks. Due to the combinatorial complexity of these networks, as well as the advances in the level of detail to which they can be reconstructed, an exhaustive enumeration of all EMs is often not practical. Therefore, in recent years interest has shifted towards searching EMs with specific properties. We present a novel method that allows computing EMs containing a given set of target reactions. This generalizes previous algorithms where the set of target reactions consists of a single reaction. In the one-reaction case, our method compares favorably to the previous approaches. In addition, we present several applications of our algorithm for computing EMs containing two target reactions in genome-scale metabolic networks. A software tool implementing the algorithms described in this paper is available at https://sourceforge.net/projects/caefm.

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.

An exact arithmetic toolbox for a consistent and reproducible structural analysis of metabolic network models

Constraint-based models are currently the only methodology that allows the study of metabolism at the whole-genome scale. Flux balance analysis is commonly used to analyse constraint-based models. Curiously, the results of this analysis vary with the software being run, a situation that we show can be remedied by using exact rather than floating-point arithmetic. Here we introduce MONGOOSE, a toolbox for analysing the structure of constraint-based metabolic models in exact arithmetic. We apply MONGOOSE to the analysis of 98 existing metabolic network models and find that the biomass reaction is surprisingly blocked (unable to sustain non-zero flux) in nearly half of them. We propose a principled approach for unblocking these reactions and extend it to the problems of identifying essential and synthetic lethal reactions and minimal media. Our structural insights enable a systematic study of constraint-based metabolic models, yielding a deeper understanding of their possibilities and limitations.

Current tools to analyse constraint-based models of metabolic networks have limited accuracy due to their use of floating-point arithmetic. Here the authors present MONGOOSE, a new computational tool that analyses such models in exact arithmetic, providing improved accuracy and reproducibility.

Fast reconstruction of compact context-specific metabolic network models

Systemic approaches to the study of a biological cell or tissue rely increasingly on the use of context-specific metabolic network models. The reconstruction of such a model from high-throughput data can routinely involve large numbers of tests under different conditions and extensive parameter tuning, which calls for fast algorithms. We present fastcore, a generic algorithm for reconstructing context-specific metabolic network models from global genome-wide metabolic network models such as Recon X. fastcore takes as input a core set of reactions that are known to be active in the context of interest (e.g., cell or tissue), and it searches for a flux consistent subnetwork of the global network that contains all reactions from the core set and a minimal set of additional reactions. Our key observation is that a minimal consistent reconstruction can be defined via a set of sparse modes of the global network, and fastcore iteratively computes such a set via a series of linear programs. Experiments on liver data demonstrate speedups of several orders of magnitude, and significantly more compact reconstructions, over a rival method. Given its simplicity and its excellent performance, fastcore can form the backbone of many future metabolic network reconstruction algorithms.

Flux balance impact degree: a new definition of impact degree to properly treat reversible reactions in metabolic networks

MOTIVATION

Metabolic pathways are complex systems of chemical reactions taking place in every living cell to degrade substrates and synthesize molecules needed for life. Modeling the robustness of these networks with respect to the dysfunction of one or several reactions is important to understand the basic principles of biological network organization, and to identify new drug targets. While several approaches have been proposed for that purpose, they are computationally too intensive to analyze large networks, and do not properly handle reversible reactions.

RESULTS

We propose a new model-the flux balance impact degree-to model the robustness of large metabolic networks with respect to gene knock-out. We formulate the computation of the impact of one or several reaction blocking as linear programs, and propose efficient strategies to solve them. We show that the proposed method better predicts the phenotypic impact of single gene deletions on Escherichia coli than existing methods.

AVAILABILITY

https://sunflower.kuicr.kyoto-u.ac.jp/∼tyoyo/fbid/index.html

Elementary flux modes in a nutshell: properties, calculation and applications

Elementary flux mode (EFM) analysis allows the unbiased decomposition of a metabolic network into minimal functional units, making it a powerful tool for metabolic engineering. While the use of EFM analysis (EFMA) is still limited by the size of the models it can handle, EFMA has been successfully applied to solve real-world metabolic engineering problems. Here we provide a user-oriented introduction to EFMA, provide examples of recent applications, analyze current research strategies to overcome the computational restrictions and give an overview over current approaches, which aim to identify and calculate only biologically relevant EFMs.

regEfmtool: speeding up elementary flux mode calculation using transcriptional regulatory rules in the form of three-state logic

Despite the considerable 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 present regEfmtool which is an extension to efmtool that utilizes transcriptional regulatory networks for the computation of elementary flux modes. The implemented extension significantly decreases the computational costs for the calculation of elementary flux modes, such as runtime, memory usage and disk space by omitting biologically infeasible solutions. Hence, using the presented regEfmtool pushes the size of metabolic networks that can be studied by elementary flux modes to new limits.

Finding minimal generating set for metabolic network with reversible pathways

Elementary flux modes give a mathematical representation of metabolic pathways in metabolic networks satisfying the constraint of non-decomposability. The large cost of their computation shifts attention to computing a minimal generating set which is a conically independent subset of elementary flux modes. When a metabolic network has reversible reactions and also admits a reversible pathway, the minimal generating set is not unique. A theoretical development and computational framework is provided which outline how to compute the minimal generating set in this case. The method is based on combining existing software to compute the minimal generating set for a "pointed cone" together with standard software to compute the Reduced Row Echelon Form.

Minimal cut sets and the use of failure modes in metabolic networks

A minimal cut set is a minimal set of reactions whose inactivation would guarantee a failure in a certain network function or functions. Minimal cut sets (MCSs) were initially developed from the metabolic pathway analysis method (MPA) of elementary modes (EMs); they provide a way of identifying target genes for eliminating a certain objective function from a holistic perspective that takes into account the structure of the whole metabolic network. The concept of MCSs is fairly new and still being explored and developed; the initial concept has developed into a generalized form and its similarity to other network characterizations are discussed. MCSs can be used in conjunction with other constraints-based methods to get a better understanding of the capability of metabolic networks and the interrelationship between metabolites and enzymes/genes. The concept could play an important role in systems biology by contributing to fields such as metabolic and genetic engineering where it could assist in finding ways of producing industrially relevant compounds from renewable resources, not only for economical, but also for sustainability, reasons.

Detecting and investigating substrate cycles in a genome-scale human metabolic network

Substrate cycles, also known as futile cycles, are cyclic metabolic routes that dissipate energy by hydrolysing cofactors such as ATP. They were first described to occur in the muscles of bumblebees and brown adipose tissue in the 1970s. A popular example is the conversion of fructose 6-phosphate to fructose 1,6-bisphosphate and back. In the present study, we analyze a large number of substrate cycles in human metabolism that consume ATP and discuss their statistics. For this purpose, we use two recently published methods (i.e. EFMEvolver and the K-shortest EFM method) to calculate samples of 100,000 and 15,000 substrate cycles, respectively. We find an unexpectedly high number of substrate cycles in human metabolism, with up to 100 reactions per cycle, utilizing reactions from up to six different compartments. An analysis of tissue-specific models of liver and brain metabolism shows that there is selective pressure that acts against the uncontrolled dissipation of energy by avoiding the coexpression of enzymes belonging to the same substrate cycle. This selective force is particularly strong against futile cycles that have a high flux as a result of thermodynamic principles.

Enumerating metabolic pathways for the production of heterologous target chemicals in chassis organisms

Background

We consider the possibility of engineering metabolic pathways in a chassis organism in order to synthesize novel target compounds that are heterologous to the chassis. For this purpose, we model metabolic networks through hypergraphs where reactions are represented by hyperarcs. Each hyperarc represents an enzyme-catalyzed reaction that transforms set of substrates compounds into product compounds. We follow a retrosynthetic approach in order to search in the metabolic space (hypergraphs) for pathways (hyperpaths) linking the target compounds to a source set of compounds.

Results

To select the best pathways to engineer, we have developed an objective function that computes the cost of inserting a heterologous pathway in a given chassis organism. In order to find minimum-cost pathways, we propose in this paper two methods based on steady state analysis and network topology that are to the best of our knowledge, the first to enumerate all possible heterologous pathways linking a target compounds to a source set of compounds. In the context of metabolic engineering, the source set is composed of all naturally produced chassis compounds (endogenuous chassis metabolites) and the target set can be any compound of the chemical space. We also provide an algorithm for identifying precursors which can be supplied to the growth media in order to increase the number of ways to synthesize specific target compounds.

Conclusions

We find the topological approach to be faster by several orders of magnitude than the steady state approach. Yet both methods are generally scalable in time with the number of pathways in the metabolic network. Therefore this work provides a powerful tool for pathway enumeration with direct application to biosynthetic pathway design.

Maximizing output and recognizing autocatalysis in chemical reaction networks is NP-complete

Background

A classical problem in metabolic design is to maximize the production of a desired compound in a given chemical reaction network by appropriately directing the mass flow through the network. Computationally, this problem is addressed as a linear optimization problem over the flux cone. The prior construction of the flux cone is computationally expensive and no polynomial-time algorithms are known.

Results

Here we show that the output maximization problem in chemical reaction networks is NP-complete. This statement remains true even if all reactions are monomolecular or bi-molecular and if only a single molecular species is used as influx. As a corollary we show, furthermore, that the detection of autocatalytic species, i.e., types that can only be produced from the influx material when they are present in the initial reaction mixture, is an NP-complete computational problem.

Conclusions

Hardness results on combinatorial problems and optimization problems are important to guide the development of computational tools for the analysis of metabolic networks in particular and chemical reaction networks in general. Our results indicate that efficient heuristics and approximate algorithms need to be employed for the analysis of large chemical networks since even conceptually simple flow problems are provably intractable.

A systems approach to identifying correlated gene targets for the loss of colour pigmentation in plants

Background

The numerous diverse metabolic pathways by which plant compounds can be produced make it difficult to predict how colour pigmentation is lost for different tissues and plants. This study employs mathematical and in silico methods to identify correlated gene targets for the loss of colour pigmentation in plants from a whole cell perspective based on the full metabolic network of Arabidopsis. This involves extracting a self-contained flavonoid subnetwork from the AraCyc database and calculating feasible metabolic routes or elementary modes (EMs) for it. Those EMs leading to anthocyanin compounds are taken to constitute the anthocyanin biosynthetic pathway (ABP) and their interplay with the rest of the EMs is used to study the minimal cut sets (MCSs), which are different combinations of reactions to block for eliminating colour pigmentation. By relating the reactions to their corresponding genes, the MCSs are used to explore the phenotypic roles of the ABP genes, their relevance to the ABP and the impact their eliminations would have on other processes in the cell.

Results

Simulation and prediction results of the effect of different MCSs for eliminating colour pigmentation correspond with existing experimental observations. Two examples are: i) two MCSs which require the simultaneous suppression of genes DFR and ANS to eliminate colour pigmentation, correspond to observational results of the same genes being co-regulated for eliminating floral pigmentation in Aquilegia and; ii) the impact of another MCS requiring CHS suppression, corresponds to findings where the suppression of the early gene CHS eliminated nearly all flavonoids but did not affect the production of volatile benzenoids responsible for floral scent.

Conclusions

From the various MCSs identified for eliminating colour pigmentation, several correlate to existing experimental observations, indicating that different MCSs are suitable for different plants, different cells, and different conditions and could also be related to regulatory genes. Being able to correlate the predictions with experimental results gives credence to the use of these mathematical and in silico analyses methods in the design of experiments. The methods could be used to prioritize target enzymes for different objectives to achieve desired outcomes, especially for less understood pathways.

Analysis of complex metabolic behavior through pathway decomposition

Background

Understanding complex systems through decomposition into simple interacting components is a pervasive paradigm throughout modern science and engineering. For cellular metabolism, complexity can be reduced by decomposition into pathways with particular biochemical functions, and the concept of elementary flux modes provides a systematic way for organizing metabolic networks into such pathways. While decomposition using elementary flux modes has proven to be a powerful tool for understanding and manipulating cellular metabolism, its utility, however, is severely limited since the number of modes in a network increases exponentially with its size.

Results

Here, we present a new method for decomposition of metabolic flux distributions into elementary flux modes. Our method can easily operate on large, genome-scale networks since it does not require all relevant modes of the metabolic network to be generated. We illustrate the utility of our method for metabolic engineering of Escherichia coli and for understanding the survival of Mycobacterium tuberculosis (MTB) during infection.

Conclusions

Our method can achieve computational time improvements exceeding 2000-fold and requires only several seconds to generate elementary mode decompositions on genome-scale networks. These improvements arise from not having to generate all relevant elementary modes prior to initiating the decomposition. The decompositions from our method are useful for understanding complex flux distributions and debugging genome-scale models.

Parallelization of Nullspace Algorithm for the computation of metabolic pathways

Elementary mode analysis is a useful metabolic pathway analysis tool in understanding and analyzing cellular metabolism, since elementary modes can represent metabolic pathways with unique and minimal sets of enzyme-catalyzed reactions of a metabolic network under steady state conditions. However, computation of the elementary modes of a genome- scale metabolic network with 100-1000 reactions is very expensive and sometimes not feasible with the commonly used serial Nullspace Algorithm. In this work, we develop a distributed memory parallelization of the Nullspace Algorithm to handle efficiently the computation of the elementary modes of a large metabolic network. We give an implementation in C++ language with the support of MPI library functions for the parallel communication. Our proposed algorithm is accompanied with an analysis of the complexity and identification of major bottlenecks during computation of all possible pathways of a large metabolic network. The algorithm includes methods to achieve load balancing among the compute-nodes and specific communication patterns to reduce the communication overhead and improve efficiency.

Tools-4-Metatool (T4M): online suite of web-tools to process stoichiometric network analysis data from Metatool

Tools-4-Metatool (T4M) is a suite of web-tools, implemented in PERL, which analyses, parses, and manipulates files related to Metatool. Its main goal is to assist the work with Metatool. T4M has two major sets of tools: Analysis and Compare. Analysis visualizes the results of Metatool (convex basis, elementary flux modes, and enzyme subsets) and facilitates the study of metabolic networks. It is composed of five tools: MDigraph, MetaMatrix, CBGraph, EMGraph, and SortEM. Compare was developed to compare different Metatool results from different networks. This set consists of: Compara and ComparaSub which compare network subsets providing outputs in different formats and ComparaEM that seeks for identical elementary modes in two metabolic networks. The suite T4M also includes one script that generates Metatool input: CBasis2Metatool, based on a Metatool output file that is filtered by a list of convex basis' metabolites. Finally, the utility CheckMIn checks the consistency of the Metatool input file. T4M is available at http://solea.quim.ucm.es/t4m.

An application programming interface for CellNetAnalyzer

CellNetAnalyzer (CNA) is a MATLAB toolbox providing computational methods for studying structure and function of metabolic and cellular signaling networks. In order to allow non-experts to use these methods easily, CNA provides GUI-based interactive network maps as a means of parameter input and result visualization. However, with the availability of high-throughput data, there is a need to make CNA's functionality also accessible in batch mode for automatic data processing. Furthermore, as some algorithms of CNA are of general relevance for network analysis it would be desirable if they could be called as sub-routines by other applications. For this purpose, we developed an API (application programming interface) for CNA allowing users (i) to access the content of network models in CNA, (ii) to use CNA's network analysis capabilities independent of the GUI, and (iii) to interact with the GUI to facilitate the development of graphical plugins. Here we describe the organization of network projects in CNA and the application of the new API functions to these projects. This includes the creation of network projects from scratch, loading and saving of projects and scenarios, and the application of the actual analysis methods. Furthermore, API functions for the import/export of metabolic models in SBML format and for accessing the GUI are described. Lastly, two example applications demonstrate the use and versatile applicability of CNA's API. CNA is freely available for academic use and can be downloaded from http://www.mpi-magdeburg.mpg.de/projects/cna/cna.html.