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

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

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

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

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

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

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

Next-generation, genome-scale metabolic models allow to study the reallocation of cellular resources upon changing environmental conditions, by not only modeling flux distributions, but also expression profiles of the catalyzing proteome. In particular, they do no longer assume a fixed biomass composition. Methods to identify optimal solutions in such comprehensive models exist, however, an unbiased understanding of all feasible allocations is missing so far. Here we develop new concepts, called elementary growth modes and vectors, that provide a generalized definition of minimal pathways, thereby extending classical elementary flux modes (used in traditional models with a fixed biomass composition). The new concepts provide an understanding of all possible flux distributions and of all possible biomass compositions. In other words, elementary growth modes and vectors are the unique functional units in any comprehensive model of cellular growth. As an example, we show that lipid accumulation upon nitrogen starvation is a consequence of resource allocation and does not require active regulation. Our work puts current approaches on a theoretical basis and allows to seamlessly transfer existing workflows (e.g. for the design of cell factories) to next-generation metabolic models.

Thermodynamic Approaches in Flux Analysis

Networks of reactions inside the cell are constrained by the laws of mass and energy balance. Constrained-based modelling (CBM) is the most used method to describe the mass balance of metabolic network. The main key concepts in CBM are stoichiometric analysis such as elementary flux mode analysis or flux balance analysis. Some of these methods have focused on adding thermodynamics constraints to eliminate non-physical fluxes or inconsistencies in the metabolic system. Here, we review the main different approaches and how they tackle the different class of problems.

Seeing the wood for the trees: a forest of methods for optimization and omic-network integration in metabolic modelling

Metabolic modelling has entered a mature phase with dozens of methods and software implementations available to the practitioner and the theoretician. It is not easy for a modeller to be able to see the wood (or the forest) for the trees. Driven by this analogy, we here present a 'forest' of principal methods used for constraint-based modelling in systems biology. This provides a tree-based view of methods available to prospective modellers, also available in interactive version at http://modellingmetabolism.net, where it will be kept updated with new methods after the publication of the present manuscript. Our updated classification of existing methods and tools highlights the most promising in the different branches, with the aim to develop a vision of how existing methods could hybridize and become more complex. We then provide the first hands-on tutorial for multi-objective optimization of metabolic models in R. We finally discuss the implementation of multi-view machine learning approaches in poly-omic integration. Throughout this work, we demonstrate the optimization of trade-offs between multiple metabolic objectives, with a focus on omic data integration through machine learning. We anticipate that the combination of a survey, a perspective on multi-view machine learning and a step-by-step R tutorial should be of interest for both the beginner and the advanced user.

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.

OptMDFpathway: Identification of metabolic pathways with maximal thermodynamic driving force and its application for analyzing the endogenous CO2 fixation potential of Escherichia coli

Constraint-based modeling techniques have become a standard tool for the in silico analysis of metabolic networks. To further improve their accuracy, recent methodological developments focused on integration of thermodynamic information in metabolic models to assess the feasibility of flux distributions by thermodynamic driving forces. Here we present OptMDFpathway, a method that extends the recently proposed framework of Max-min Driving Force (MDF) for thermodynamic pathway analysis. Given a metabolic network model, OptMDFpathway identifies both the optimal MDF for a desired phenotypic behavior as well as the respective pathway itself that supports the optimal driving force. OptMDFpathway is formulated as a mixed-integer linear program and is applicable to genome-scale metabolic networks. As an important theoretical result, we also show that there exists always at least one elementary mode in the network that reaches the maximal MDF. We employed our new approach to systematically identify all substrate-product combinations in Escherichia coli where product synthesis allows for concomitant net CO₂ assimilation via thermodynamically feasible pathways. Although biomass synthesis cannot be coupled to net CO₂ fixation in E. coli we found that as many as 145 of the 949 cytosolic carbon metabolites contained in the genome-scale model iJO1366 enable net CO₂ incorporation along thermodynamically feasible pathways with glycerol as substrate and 34 with glucose. The most promising products in terms of carbon assimilation yield and thermodynamic driving forces are orotate, aspartate and the C4-metabolites of the tricarboxylic acid cycle. We also identified thermodynamic bottlenecks frequently limiting the maximal driving force of the CO₂-fixing pathways. Our results indicate that heterotrophic organisms like E. coli hold a possibly underestimated potential for CO₂ assimilation which may complement existing biotechnological approaches for capturing CO₂. Furthermore, we envision that the developed OptMDFpathway approach can be used for many other applications within the framework of constrained-based modeling and for rational design of metabolic networks.

When analyzing metabolic networks, one often searches for metabolic pathways with certain (desired) properties, for example, conversion routes that maximize the yield of a product from a given substrate. While those problems can be solved with established methods of constraint-based modeling, no algorithm is currently available for genome-scale models to identify the pathway that has the highest possible thermodynamic driving force among all solutions with predefined stoichiometric properties. This gap is closed with our new approach OptMDFpathway which is based on the recently introduced concept of Max-min Driving Force (MDF). OptMDFpathway offers various applications, especially in the context of metabolic design of cell factories. To demonstrate the power and usefulness of OptMDFpathway, we employed it to analyze the endogenous CO₂ fixation potential of Escherichia coli. While E. coli cannot assimilate CO₂ into biomass, net CO₂ fixation can take place along linear pathways from substrate to product and we show that thermodynamically feasible pathways with net CO₂ assimilation exist for 145 (34) products when choosing glycerol (glucose) as substrate. Our results indicate that heterotrophic organisms like E. coli hold a possibly underestimated potential for CO₂ assimilation which may complement existing biotechnological approaches for capturing CO₂.

From elementary flux modes to elementary flux vectors: Metabolic pathway analysis with arbitrary linear flux constraints

Elementary flux modes (EFMs) emerged as a formal concept to describe metabolic pathways and have become an established tool for constraint-based modeling and metabolic network analysis. EFMs are characteristic (support-minimal) vectors of the flux cone that contains all feasible steady-state flux vectors of a given metabolic network. EFMs account for (homogeneous) linear constraints arising from reaction irreversibilities and the assumption of steady state; however, other (inhomogeneous) linear constraints, such as minimal and maximal reaction rates frequently used by other constraint-based techniques (such as flux balance analysis [FBA]), cannot be directly integrated. These additional constraints further restrict the space of feasible flux vectors and turn the flux cone into a general flux polyhedron in which the concept of EFMs is not directly applicable anymore. For this reason, there has been a conceptual gap between EFM-based (pathway) analysis methods and linear optimization (FBA) techniques, as they operate on different geometric objects. One approach to overcome these limitations was proposed ten years ago and is based on the concept of elementary flux vectors (EFVs). Only recently has the community started to recognize the potential of EFVs for metabolic network analysis. In fact, EFVs exactly represent the conceptual development required to generalize the idea of EFMs from flux cones to flux polyhedra. This work aims to present a concise theoretical and practical introduction to EFVs that is accessible to a broad audience. We highlight the close relationship between EFMs and EFVs and demonstrate that almost all applications of EFMs (in flux cones) are possible for EFVs (in flux polyhedra) as well. In fact, certain properties can only be studied with EFVs. Thus, we conclude that EFVs provide a powerful and unifying framework for constraint-based modeling of metabolic networks.

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.

Mathematical modelling of clostridial acetone-butanol-ethanol fermentation

Clostridial acetone-butanol-ethanol (ABE) fermentation features a remarkable shift in the cellular metabolic activity from acid formation, acidogenesis, to the production of industrial-relevant solvents, solventogensis. In recent decades, mathematical models have been employed to elucidate the complex interlinked regulation and conditions that determine these two distinct metabolic states and govern the transition between them. In this review, we discuss these models with a focus on the mechanisms controlling intra- and extracellular changes between acidogenesis and solventogenesis. In particular, we critically evaluate underlying model assumptions and predictions in the light of current experimental knowledge. Towards this end, we briefly introduce key ideas and assumptions applied in the discussed modelling approaches, but waive a comprehensive mathematical presentation. We distinguish between structural and dynamical models, which will be discussed in their chronological order to illustrate how new biological information facilitates the ‘evolution’ of mathematical models. Mathematical models and their analysis have significantly contributed to our knowledge of ABE fermentation and the underlying regulatory network which spans all levels of biological organization. However, the ties between the different levels of cellular regulation are not well understood. Furthermore, contradictory experimental and theoretical results challenge our current notion of ABE metabolic network structure. Thus, clostridial ABE fermentation still poses theoretical as well as experimental challenges which are best approached in close collaboration between modellers and experimentalists.