How important is thermodynamics for identifying elementary flux modes?

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.

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.

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

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

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

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

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

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Ecmtool computes all ECMs from a reconstructed metabolic network

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ECM enumeration enables metabolic characterization of larger networks than ever

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

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

Answer Set Programming for Computing Constraints-Based Elementary Flux Modes: Application to Escherichia coli Core Metabolism

Elementary Flux Modes (EFMs) provide a rigorous basis to systematically characterize the steady state, cellular phenotypes, as well as metabolic network robustness and fragility. However, the number of EFMs typically grows exponentially with the size of the metabolic network, leading to excessive computational demands, and unfortunately, a large fraction of these EFMs are not biologically feasible due to system constraints. This combinatorial explosion often prevents the complete analysis of genome-scale metabolic models. Traditionally, EFMs are computed by the double description method, an efficient algorithm based on matrix calculation; however, only a few constraints can be integrated into this computation. They must be monotonic with regard to the set inclusion of the supports; otherwise, they must be treated in post-processing and thus do not save computational time. We present aspefm, a hybrid computational tool based on Answer Set Programming (ASP) and Linear Programming (LP) that permits the computation of EFMs while implementing many different types of constraints. We apply our methodology to the Escherichia coli core model, which contains 226×106 EFMs. In considering transcriptional and environmental regulation, thermodynamic constraints, and resource usage considerations, the solution space is reduced to 1118 EFMs that can be computed directly with aspefm. The solution set, for E. coli growth on O2 gradients spanning fully aerobic to anaerobic, can be further reduced to four optimal EFMs using post-processing and Pareto front analysis.

Dynamic metabolic resource allocation based on the maximum entropy principle

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

Thermodynamic Limits and Optimality of Microbial Growth

Understanding microbial growth with the use of mathematical models has a long history that dates back to the pioneering work of Jacques Monod in the 1940s. Monod’s famous growth law expressed microbial growth rate as a simple function of the limiting nutrient concentration. However, to explain growth laws from underlying principles is extremely challenging. In the second half of the 20th century, numerous experimental approaches aimed at precisely measuring heat production during microbial growth to determine the entropy balance in a growing cell and to quantify the exported entropy. This has led to the development of thermodynamic theories of microbial growth, which have generated fundamental understanding and identified the principal limitations of the growth process. Although these approaches ignored metabolic details and instead considered microbial metabolism as a black box, modern theories heavily rely on genomic resources to describe and model metabolism in great detail to explain microbial growth. Interestingly, however, thermodynamic constraints are often included in modern modeling approaches only in a rather superficial fashion, and it appears that recent modeling approaches and classical theories are rather disconnected fields. To stimulate a closer interaction between these fields, we here review various theoretical approaches that aim at describing microbial growth based on thermodynamics and outline the resulting thermodynamic limits and optimality principles. We start with classical black box models of cellular growth, and continue with recent metabolic modeling approaches that include thermodynamics, before we place these models in the context of fundamental considerations based on non-equilibrium statistical mechanics. We conclude by identifying conceptual overlaps between the fields and suggest how the various types of theories and models can be integrated. We outline how concepts from one approach may help to inform or constrain another, and we demonstrate how genome-scale models can be used to infer key black box parameters, such as the energy of formation or the degree of reduction of biomass. Such integration will allow understanding to what extent microbes can be viewed as thermodynamic machines, and how close they operate to theoretical optima.

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.

Integrated thermodynamic analysis of electron bifurcating [FeFe]-hydrogenase to inform anaerobic metabolism and H2 production

Electron bifurcating, [FeFe]-hydrogenases are recently described members of the hydrogenase family and catalyze a combination of exergonic and endergonic electron exchanges between three carriers (2 ferredoxin_red^- + NAD(P)H + 3 H⁺ = 2 ferredoxin_ox + NAD(P)⁺ + 2 H₂). A thermodynamic analysis of the bifurcating, [FeFe]-hydrogenase reaction, using electron path-independent variables, quantified potential biological roles of the reaction without requiring enzyme details. The bifurcating [FeFe]-hydrogenase reaction, like all bifurcating reactions, can be written as a sum of two non-bifurcating reactions. Therefore, the thermodynamic properties of the bifurcating reaction can never exceed the properties of the individual, non-bifurcating, reactions. The bifurcating [FeFe]-hydrogenase reaction has three competitive properties: 1) enabling NAD(P)H-driven proton reduction at pH₂ higher than the concurrent operation of the two, non-bifurcating reactions, 2) oxidation of NAD(P)H and ferredoxin simultaneously in a 1:1 ratio, both are produced during typical glucose fermentations, and 3) enhanced energy conservation (~10 kJ mol^-1 H₂) relative to concurrent operation of the two, non-bifurcating reactions. Our analysis demonstrated ferredoxin E°' largely determines the sensitivity of the bifurcating reaction to pH₂, modulation of the reduced/oxidized electron carrier ratios contributed less to equilibria shifts. Hydrogenase thermodynamics data were integrated with typical and non-typical glycolysis pathways to evaluate achieving the 'Thauer limit' (4 H₂ per glucose) as a function of temperature and pH₂. For instance, the bifurcating [FeFe]-hydrogenase reaction permits the Thauer limit at 60 °C if pH ₂ ≤ ~10 mbar. The results also predict Archaea, expressing a non-typical glycolysis pathway, would not benefit from a bifurcating [FeFe]-hydrogenase reaction; interestingly, no Archaea have been observed experimentally with a [FeFe]-hydrogenase enzyme.

Thermodynamic analysis of the pathway for ethanol production from cellobiose in Clostridium thermocellum

Clostridium thermocellum is a candidate for consolidated bioprocessing by carrying out both cellulose solubilization and fermentation. However, despite significant efforts the maximum ethanol titer achieved to date remains below industrially required targets. Several studies have analyzed the impact of increasing ethanol concentration on C. thermocellum's membrane properties, cofactor pool ratios, and altered enzyme regulation. In this study, we explore the extent to which thermodynamic equilibrium limits maximum ethanol titer. We used the max-min driving force (MDF) algorithm (Noor et al., 2014) to identify the range of allowable metabolite concentrations that maintain a negative free energy change for all reaction steps in the pathway from cellobiose to ethanol. To this end, we used a time-series metabolite concentration dataset to flag five reactions (phosphofructokinase (PFK), fructose bisphosphate aldolase (FBA), glyceraldehyde-3-phosphate dehydrogenase (GAPDH), aldehyde dehydrogenase (ALDH) and alcohol dehydrogenase (ADH)) which become thermodynamic bottlenecks under high external ethanol concentrations. Thermodynamic analysis was also deployed in a prospective mode to evaluate genetic interventions which can improve pathway thermodynamics by generating minimal set of reactions or elementary flux modes (EFMs) which possess unique genetic variations while ensuring mass and redox balance with ethanol production. MDF evaluation of all generated (336) EFMs indicated that, i) pyruvate phosphate dikinase (PPDK) has a higher pathway MDF than the malate shunt alternative due to limiting CO₂ concentrations under physiological conditions, and ii) NADPH-dependent glyceraldehyde-3-phosphate dehydrogenase (GAPN) can alleviate thermodynamic bottlenecks at high ethanol concentrations due to cofactor modification and reduction in ATP generation. The combination of ATP linked phosphofructokinase (PFK-ATP) and NADPH linked alcohol dehydrogenase (ADH-NADPH) with NADPH linked aldehyde dehydrogenase (ALDH-NADPH) or ferredoxin: NADP ​+ ​oxidoreductase (NADPH-FNOR) emerges as the best intervention strategy for ethanol production that balances MDF improvements with ATP generation, and appears to functionally reproduce the pathway employed by the ethanologen Thermoanaerobacterium saccharolyticum. Expanding the list of measured intracellular metabolites and improving the quantification accuracy of measurements was found to improve the fidelity of pathway thermodynamics analysis in C. thermocellum. This study demonstrates even before addressing an organism's enzyme kinetics and allosteric regulations, pathway thermodynamics can flag pathway bottlenecks and identify testable strategies for enhancing pathway thermodynamic feasibility and function.

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.

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.

Toward Synthetic Biology Strategies for Adipic Acid Production: An in Silico Tool for Combined Thermodynamics and Stoichiometric Analysis of Metabolic Networks

Adipic acid, a nylon-6,6 precursor, has recently gained popularity in synthetic biology. Here, 16 different production routes to adipic acid were evaluated using a novel tool for network-embedded thermodynamic analysis of elementary flux modes. The tool distinguishes between thermodynamically feasible and infeasible modes under determined metabolite concentrations, allowing the thermodynamic feasibility of theoretical yields to be assessed. Further, patterns that always caused infeasible flux distributions were identified, which will aid the development of tailored strain design. A review of cellular efflux mechanisms revealed that significant accumulation of extracellular product is only possible if coupled with ATP hydrolysis. A stoichiometric analysis demonstrated that the maximum theoretical product carbon yield heavily depends on the metabolic route, ranging from 32 to 99% on glucose and/or palmitate in Escherichia coli and Saccharomyces cerevisiae metabolic models. Equally important, metabolite concentrations appeared to be thermodynamically restricted in several pathways. Consequently, the number of thermodynamically feasible flux distributions was reduced, in some cases even rendering whole pathways infeasible, highlighting the importance of pathway choice. Only routes based on the shikimate pathway were thermodynamically favorable over a large concentration and pH range. The low pH capability of S. cerevisiae shifted the thermodynamic equilibrium of some pathways toward product formation. One identified infeasible-pattern revealed that the reversibility of the mitochondrial malate dehydrogenase contradicted the current state of knowledge, which imposes a major restriction on the metabolism of S. cerevisiae. Finally, the evaluation of industrially relevant constraints revealed that two shikimate pathway-based routes in E. coli were the most robust.

Metabolic enzyme cost explains variable trade-offs between microbial growth rate and yield

Microbes may maximize the number of daughter cells per time or per amount of nutrients consumed. These two strategies correspond, respectively, to the use of enzyme-efficient or substrate-efficient metabolic pathways. In reality, fast growth is often associated with wasteful, yield-inefficient metabolism, and a general thermodynamic trade-off between growth rate and biomass yield has been proposed to explain this. We studied growth rate/yield trade-offs by using a novel modeling framework, Enzyme-Flux Cost Minimization (EFCM) and by assuming that the growth rate depends directly on the enzyme investment per rate of biomass production. In a comprehensive mathematical model of core metabolism in E. coli, we screened all elementary flux modes leading to cell synthesis, characterized them by the growth rates and yields they provide, and studied the shape of the resulting rate/yield Pareto front. By varying the model parameters, we found that the rate/yield trade-off is not universal, but depends on metabolic kinetics and environmental conditions. A prominent trade-off emerges under oxygen-limited growth, where yield-inefficient pathways support a 2-to-3 times higher growth rate than yield-efficient pathways. EFCM can be widely used to predict optimal metabolic states and growth rates under varying nutrient levels, perturbations of enzyme parameters, and single or multiple gene knockouts.

Resource allocation in living organisms

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