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

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.

EFMlrs: a Python package for elementary flux mode enumeration via lexicographic reverse search

Background

Elementary flux mode (EFM) analysis is a well-established, yet computationally challenging approach to characterize metabolic networks. Standard algorithms require huge amounts of memory and lack scalability which limits their application to single servers and consequently limits a comprehensive analysis to medium-scale networks. Recently, Avis et al. developed mplrs—a parallel version of the lexicographic reverse search (lrs) algorithm, which, in principle, enables an EFM analysis on high-performance computing environments (Avis and Jordan. mplrs: a scalable parallel vertex/facet enumeration code. arXiv:1511.06487, 2017). Here we test its applicability for EFM enumeration.

Results

We developed EFMlrs, a Python package that gives users access to the enumeration capabilities of mplrs. EFMlrs uses COBRApy to process metabolic models from sbml files, performs loss-free compressions of the stoichiometric matrix, and generates suitable inputs for mplrs as well as efmtool, providing support not only for our proposed new method for EFM enumeration but also for already established tools. By leveraging COBRApy, EFMlrs also allows the application of additional reaction boundaries and seamlessly integrates into existing workflows.

Conclusion

We show that due to mplrs’s properties, the algorithm is perfectly suited for high-performance computing (HPC) and thus offers new possibilities for the unbiased analysis of substantially larger metabolic models via EFM analyses. EFMlrs is an open-source program that comes together with a designated workflow and can be easily installed via pip.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12859-021-04417-9.

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.

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

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

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.

Comparison of pathway analysis and constraint-based methods for cell factory design

BACKGROUND

Computational strain optimisation methods (CSOMs) have been successfully used to exploit genome-scale metabolic models, yielding strategies useful for allowing compound overproduction in metabolic cell factories. Minimal cut sets are particularly interesting since their definition allows searching for intervention strategies that impose strong growth-coupling phenotypes, and are not subject to optimality bias when compared with simulation-based CSOMs. However, since both types of methods have different underlying principles, they also imply different ways to formulate metabolic engineering problems, posing an obstacle when comparing their outputs.

RESULTS

In this work, we perform an in-depth analysis of potential strategies that can be obtained with both methods, providing a critical comparison of performance, robustness, predicted phenotypes as well as strategy structure and size. To this end, we devised a pipeline including enumeration of strategies from evolutionary algorithms (EA) and minimal cut sets (MCS), filtering and flux analysis of predicted mutants to optimize the production of succinic acid in Saccharomyces cerevisiae. We additionally attempt to generalize problem formulations for MCS enumeration within the context of growth-coupled product synthesis. Strategies from evolutionary algorithms show the best compromise between acceptable growth rates and compound overproduction. However, constrained MCSs lead to a larger variety of phenotypes with several degrees of growth-coupling with production flux. The latter have proven useful in revealing the importance, in silico, of the gamma-aminobutyric acid shunt and manipulation of cofactor pools in growth-coupled designs for succinate production, mechanisms which have also been touted as potentially useful for metabolic engineering.

CONCLUSIONS

The two main groups of CSOMs are valuable for finding growth-coupled mutants. Despite the limitations in maximum growth rates and large strategy sizes, MCSs help uncover novel mechanisms for compound overproduction and thus, analyzing outputs from both methods provides a richer overview on strategies that can be potentially carried over in vivo.

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

Motivation

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

Results

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

Availability and Implementation

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

Contact

hyunseob.song@pnnl.gov.

Supplementary information

Supplementary data are available at Bioinformatics online.

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

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

Direct calculation of minimal cut sets involving a specific reaction knock-out

MOTIVATION

The concept of Minimal Cut Sets (MCSs) is used in metabolic network modeling to describe minimal groups of reactions or genes whose simultaneous deletion eliminates the capability of the network to perform a specific task. Previous work showed that MCSs where closely related to Elementary Flux Modes (EFMs) in a particular dual problem, opening up the possibility to use the tools developed for computing EFMs to compute MCSs. Until recently, however, there existed no method to compute an EFM with some specific characteristic, meaning that, in the case of MCSs, the only strategy to obtain them was to enumerate them using, for example, the standard K-shortest EFMs algorithm.

RESULTS

In this work, we adapt the recently developed theory to compute EFMs satisfying several constraints to the calculation of MCSs involving a specific reaction knock-out. Importantly, we emphasize that not all the EFMs in the dual problem correspond to real MCSs, and propose a new formulation capable of correctly identifying the MCS wanted. Furthermore, this formulation brings interesting insights about the relationship between the primal and the dual problem of the MCS computation.

AVAILABILITY AND IMPLEMENTATION

A Matlab-Cplex implementation of the proposed algorithm is available as a supplementary material

CONTACT

fplanes@ceit.es

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.