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

Free-energy transduction mechanisms shape the flux space of metabolic networks

The transduction of free energy in metabolic networks represents a thermodynamic mechanism by which the free energy derived from nutrients is converted to drive nonspontaneous, energy-requiring metabolic reactions. This transduction is typically observed in processes that generate energy-rich molecules such as ATP and NAD(P)H, which, in turn, power specific reactions, particularly biosynthetic reactions. This property establishes a pivotal connection between the intricate topology of metabolic networks and their ability to redirect energy for functional purposes. The present study proposes a dedicated framework aimed at exploring the relationship between free-energy dissipation, network topology, and metabolic objectives. The starting point is that, regardless of the network topology, nonequilibrium chemostatting conditions impose stringent thermodynamic constraints on the feasible flux steady states to satisfy energy and entropy balances. An analysis of randomly sampled reaction networks shows that the network topology imposes additional constraints that restrict the accessible flux solution space, depending on key structural features such as the reaction's molecularity, reaction cycles, and conservation laws. Notably, topologies featuring multimolecular reactions that implement free-energy transduction mechanisms tend to extend the accessible flux domains, facilitating the achievement of metabolic objectives such as anabolic flux maximization or flux rerouting capacity. This approach is applied to a coarse-grained model of carbohydrate metabolism, highlighting the structural requirements for optimal biomass yield.

A hierarchy of metabolite exchanges in metabolic models of microbial species and communities

The metabolic network of an organism can be analyzed as a constraint-based model. This analysis can be biased, optimizing an objective such as growth rate, or unbiased, aiming to describe the full feasible space of metabolic fluxes through pathway analysis or random flux sampling. In particular, pathway analysis can decompose the flux space into fundamental and formally defined metabolic pathways. Unbiased methods scale poorly with network size due to combinatorial explosion, but a promising approach to improve scalability is to focus on metabolic subnetworks, e.g., cells' metabolite exchanges with each other and the environment, rather than the full metabolic networks. Here, we applied pathway enumeration and flux sampling to metabolite exchanges in microbial species and a microbial community, using models ranging from central carbon metabolism to genome-scale and focusing on pathway definitions that allow direct targeting of subnetworks such as metabolite exchanges (elementary conversion modes, elementary flux patterns, and minimal pathways). Enumerating growth-supporting metabolite exchanges, we found that metabolite exchanges from different pathway definitions were related through a hierarchy, and we show that this hierarchical relationship between pathways holds for metabolic networks and subnetworks more generally. Metabolite exchange frequencies, defined as the fraction of pathways in which each metabolite was exchanged, were similar across pathway definitions, with a few specific exchanges explaining large differences in pathway counts. This indicates that biological interpretation of predicted metabolite exchanges is robust to the choice of pathway definition, and it suggests strategies for more scalable pathway analysis. Our results also signal wider biological implications, facilitating detailed and interpretable analysis of metabolite exchanges and other subnetworks in fields such as metabolic engineering and synthetic biology.

Creating new-to-nature carbon fixation: A guide

Synthetic biology aims at designing new biological functions from first principles. These new designs allow to expand the natural solution space and overcome the limitations of naturally evolved systems. One example is synthetic CO2-fixation pathways that promise to provide more efficient ways for the capture and conversion of CO2 than natural pathways, such as the Calvin Benson Bassham (CBB) cycle of photosynthesis. In this review, we provide a practical guideline for the design and realization of such new-to-nature CO2-fixation pathways. We introduce the concept of "synthetic CO2-fixation", and give a general overview over the enzymology and topology of synthetic pathways, before we derive general principles for their design from their eight naturally evolved analogs. We provide a comprehensive summary of synthetic carbon-assimilation pathways and derive a step-by-step, practical guide from the theoretical design to their practical implementation, before ending with an outlook on new developments in the field.

A mathematically rigorous algorithm to define, compute and assess relevance of the probable dissociation constants in characterizing a biochemical network

Metabolism results from enzymatic- and non-enzymatic interactions of several molecules, is easily parameterized with the dissociation constant and occurs via biochemical networks. The dissociation constant is an empirically determined parameter and cannot be used directly to investigate in silico models of biochemical networks. Here, we develop and present an algorithm to define, compute and assess the relevance of the probable dissociation constant for every reaction of a biochemical network. The reactants and reactions of this network are modelled by a stoichiometry number matrix. The algorithm computes the null space and then serially generates subspaces by combinatorially summing the spanning vectors that are non-trivial and unique. This is done until the terms of each row either monotonically diverge or form an alternating sequence whose terms can be partitioned into subsets with almost the same number of oppositely signed terms. For a selected null space-generated subspace the algorithm utilizes several statistical and mathematical descriptors to select and bin terms from each row into distinct outcome-specific subsets. The terms of each subset are summed, mapped to the real-valued open interval [Formula: see text] and used to populate a reaction-specific outcome vector. The p1-norm for this vector is then the probable dissociation constant for this reaction. These steps are continued until every reaction of a modelled network is unambiguously annotated. The assertions presented are complemented by computational studies of a biochemical network for aerobic glycolysis. The fundamental premise of this work is that every row of a null space-generated subspace is a valid reaction and can therefore, be modelled as a reaction-specific sequence vector with a dimension that corresponds to the cardinality of the subspace after excluding all trivial- and redundant-vectors. A major finding of this study is that the row-wise sum or the sum of the terms contained in each reaction-specific sequence vector is mapped unambiguously to a positive real number. This means that the probable dissociation constants, for all reactions, can be directly computed from the stoichiometry number matrix and are suitable indicators of outcome for every reaction of the modelled biochemical network. Additionally, we find that the unambiguous annotation for a biochemical network will require a minimum number of iterations and will determine computational complexity.

Microbial Pathway Thermodynamics: Stoichiometric Models Unveil Anabolic and Catabolic Processes

The biotechnological exploitation of microorganisms enables the use of metabolism for the production of economically valuable substances, such as drugs or food. It is, thus, unsurprising that the investigation of microbial metabolism and its regulation has been an active research field for many decades. As a result, several theories and techniques were developed that allow for the prediction of metabolic fluxes and yields as biotechnologically relevant output parameters. One important approach is to derive macrochemical equations that describe the overall metabolic conversion of an organism and basically treat microbial metabolism as a black box. The opposite approach is to include all known metabolic reactions of an organism to assemble a genome-scale metabolic model. Interestingly, both approaches are rather successful at characterizing and predicting the expected product yield. Over the years, macrochemical equations especially have been extensively characterized in terms of their thermodynamic properties. However, a common challenge when characterizing microbial metabolism by a single equation is to split this equation into two, describing the two modes of metabolism, anabolism and catabolism. Here, we present strategies to systematically identify separate equations for anabolism and catabolism. Based on metabolic models, we systematically identify all theoretically possible catabolic routes and determine their thermodynamic efficiency. We then show how anabolic routes can be derived, and we use these to approximate biomass yield. Finally, we challenge the view of metabolism as a linear energy converter, in which the free energy gradient of catabolism drives the anabolic reactions.

Generalized metabolic flux analysis framework provides mechanism-based predictions of ophthalmic complications in type 2 diabetes patients

Chronic metabolic diseases arise from changes in metabolic fluxes through biomolecular pathways and gene networks accumulated over the lifetime of an individual. While clinical and biochemical profiles present just real-time snapshots of the patients' health, efficient computation models of the pathological disturbance of biomolecular processes are required to achieve individualized mechanistic insights into disease progression. Here, we describe the Generalized metabolic flux analysis (GMFA) for addressing this gap. Suitably grouping individual metabolites/fluxes into pools simplifies the analysis of the resulting more coarse-grain network. We also map non-metabolic clinical modalities onto the network with additional edges. Instead of using the time coordinate, the system status (metabolite concentrations and fluxes) is quantified as function of a generalized extent variable (a coordinate in the space of generalized metabolites) that represents the system's coordinate along its evolution path and evaluates the degree of change between any two states on that path. We applied GMFA to analyze Type 2 Diabetes Mellitus (T2DM) patients from two cohorts: EVAS (289 patients from Singapore) and NHANES (517) from the USA. Personalized systems biology models (digital twins) were constructed. We deduced disease dynamics from the individually parameterized metabolic network and predicted the evolution path of the metabolic health state. For each patient, we obtained an individual description of disease dynamics and predict an evolution path of the metabolic health state. Our predictive models achieve an ROC-AUC in the range 0.79-0.95 (sensitivity 80-92%, specificity 62-94%) in identifying phenotypes at the baseline and predicting future development of diabetic retinopathy and cataract progression among T2DM patients within 3 years from the baseline. The GMFA method is a step towards realizing the ultimate goal to develop practical predictive computational models for diagnostics based on systems biology. This tool has potential use in chronic disease management in medical practice.

Supplementary Information

The online version contains supplementary material available at 10.1007/s13755-023-00218-x.

ReDirection: an R-package to compute the probable dissociation constant for every reaction of a user-defined biochemical network

Biochemical networks integrate enzyme-mediated substrate conversions with non-enzymatic complex formation and disassembly to accomplish complex biochemical and physiological functions. The choice of parameters and constraints used in most of these studies is numerically motivated and network-specific. Although sound in theory, the outcomes that result depart significantly from the intracellular milieu and are less likely to retain relevance in a clinical setting. There is a need for a computational tool which is biochemically relevant, mathematically rigorous, and unbiased, and can ascribe functionality to and generate potentially testable hypotheses for a user-defined biochemical network. Here, we present "ReDirection," an R-package which computes the probable dissociation constant for every reaction of a biochemical network directly from a null space-generated subspace of the stoichiometry number matrix of the modeled network. "ReDirection" delineates this subspace by excluding all trivial and redundant or duplicate occurrences of non-trivial vectors, combinatorially summing the vectors that remain and verifying that the upper or lower bounds of the sequence of terms formed by each row of this subspace belong to the open real-valued intervals - ∞ , - 1 or 1 , ∞ or whether the number of terms that are differently signed are almost equal. "ReDirection" iterates these steps until these bounds are consistent and unambiguous for all reactions of the modeled biochemical network. Thereafter, "ReDirection" filters the terms from each row of this subspace, bins them to outcome-specific subsets, sums and maps this to an outcome-specific reaction vector, and computes the p1-norm, which is the probable dissociation constant for a reaction. "ReDirection" works on first principles, does not discriminate between enzymatic and non-enzymatic reactions, offers a biochemically relevant and mathematically rigorous environment to explore user-defined biochemical networks under baseline and perturbed conditions, and can be used to address empirically intractable biochemical problems. The utility and relevance of "ReDirection" are highlighted by numerical studies on stoichiometric number models of biochemical networks of galactose metabolism and heme and cholesterol biosynthesis. "ReDirection" is freely available and accessible from the comprehensive R archive network (CRAN) with the URL (https://cran.r-project.org/package=ReDirection).

Modeling the metabolic dynamics at the genome-scale by optimized yield analysis

The hybrid cybernetic model (HCM) approach is a dynamic modeling framework that integrates enzyme synthesis and activity regulation. It has been widely applied in bioreaction engineering, particularly in the simulation of microbial growth in different mixtures of carbon sources. In a HCM, the metabolic network is decomposed into elementary flux modes (EFMs), whereby the network can be reduced into a few pathways by yield analysis. However, applying the HCM approach on conventional genome-scale metabolic models (GEMs) is still a challenge due to the high computational demands. Here, we present a HCM strategy that introduced an optimized yield analysis algorithm (opt-yield-FBA) to simulate metabolic dynamics at the genome-scale without the need for EFMs calculation. The opt-yield-FBA is a flux-balance analysis (FBA) based method that can calculate optimal yield solutions and yield space for GEM. With the opt-yield-FBA algorithm, the HCM strategy can be applied to get the yield spaces and avoid the computational burden of EFMs, and it can therefore be applied for developing dynamic models for genome-scale metabolic networks. Here, we illustrate the strategy by applying the concept to simulate the dynamics of microbial communities.

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.

CNApy: a CellNetAnalyzer GUI in Python for analyzing and designing metabolic networks

SUMMARY

Constraint-based reconstruction and analysis (COBRA) is a widely used modeling framework for analyzing and designing metabolic networks. Here, we present CNApy, an open-source cross-platform desktop application written in Python, which offers a state-of-the-art graphical front-end for the intuitive analysis of metabolic networks with COBRA methods. While the basic look-and-feel of CNApy is similar to the user interface of the MATLAB toolbox CellNetAnalyzer, it provides various enhanced features by using components of the powerful Qt library. CNApy supports a number of standard and advanced COBRA techniques and further functionalities can be easily embedded in its GUI facilitating modular extension in the future.

AVAILABILITY AND IMPLEMENTATION

CNApy can be installed via conda and its source code is freely available at https://github.com/cnapy-org/CNApy under the Apache 2 license.

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.

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.

Study of the bioenergetics to identify the novel pathways as a drug target against Mycobacterium tuberculosis using Petri net

Tuberculosis is one of the life-threatening diseases globally, caused by the bacteria Mycobacterium tuberculosis. In order to control this epidemic globally, there is an urgent need to discover new drugs with novel mechanism of action that can help in shortening the duration of treatment for both drug resistant and drug sensitive tuberculosis. Mycobacterium essentially depends on oxidative phosphorylation for its growth and establishment of pathogenesis. This pathway is unique in Mycobacterium tuberculosis as compared to host due to the differences in some of the enzyme complexes carrying electron transfer. Hence, it serves as an important drug target area. The uncouplers which inhibit adenosine triphosphate synthesis, could play a vital role in serving as antimycobacterial agents and thus could help in eradicating this deadly disease. In this article, the bioenergetics of Mycobacterium tuberculosis are studied with and without uncouplers using Petri net. Petri net is among the most widely used mathematical and computational tools to model and study the complex biochemical networks. We first represented the bioenergetic pathway as a Petri net which is then validated and analyzed using invariant analysis techniques of Petri net. The valid mathematical models presented here are capable to explain the molecular mechanism of uncouplers and the processes occurring within the electron transport chain of Mycobacterium tuberculosis. The results explained the net behavior in agreement with the biological results and also suggested some possible processes and pathways to be studied as a drug target for developing antimycobacterials.

Personalized Metabolic Analysis of Diseases

The metabolic wiring of patient cells is altered drastically in many diseases, including cancer. Understanding the nature of such changes may pave the way for new therapeutic opportunities as well as the development of personalized treatment strategies for patients. In this paper, we propose an algorithm called Metabolitics, which allows systems-level analysis of changes in the biochemical network of cells in disease states. It enables the study of a disease at both reaction- and pathway-level granularities for a detailed and summarized view of disease etiology. Metabolitics employs flux variability analysis with a dynamically built objective function based on biofluid metabolomics measurements in a personalized manner. Moreover, Metabolitics builds supervised classification models to discriminate between patients and healthy subjects based on the computed metabolic network changes. The use of Metabolitics is demonstrated for three distinct diseases, namely, breast cancer, Crohn's disease, and colorectal cancer. Our results show that the constructed supervised learning models successfully differentiate patients from healthy individuals by an average f1-score of 88 percent. Besides, in addition to the confirmation of previously reported breast cancer-associated pathways, we discovered that Biotin Metabolism along with Arginine and Proline Metabolism is subject to a significant increase in flux capacity, which have not been reported before.

Addressing uncertainty in genome-scale metabolic model reconstruction and analysis

The reconstruction and analysis of genome-scale metabolic models constitutes a powerful systems biology approach, with applications ranging from basic understanding of genotype-phenotype mapping to solving biomedical and environmental problems. However, the biological insight obtained from these models is limited by multiple heterogeneous sources of uncertainty, which are often difficult to quantify. Here we review the major sources of uncertainty and survey existing approaches developed for representing and addressing them. A unified formal characterization of these uncertainties through probabilistic approaches and ensemble modeling will facilitate convergence towards consistent reconstruction pipelines, improved data integration algorithms, and more accurate assessment of predictive capacity.

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.

NetFlow: A tool for isolating carbon flows in genome-scale metabolic networks

Genome-scale stoichiometric models (GSMs) have been widely utilized to predict and understand cellular metabolism. GSMs and the flux predictions resulting from them have proven indispensable to fields ranging from metabolic engineering to human disease. Nonetheless, it is challenging to parse these flux predictions due to the inherent size and complexity of the GSMs. Several previous approaches have reduced this complexity by identifying key pathways contained within the genome-scale flux predictions. However, a reduction method that overlays carbon atom transitions on stoichiometry and flux predictions is lacking. To fill this gap, we developed NetFlow, an algorithm that leverages genome-scale carbon mapping to extract and quantitatively distinguish biologically relevant metabolic pathways from a given genome-scale flux prediction. NetFlow extends prior approaches by utilizing both full carbon mapping and context-specific flux predictions. Thus, NetFlow is uniquely able to quantitatively distinguish between biologically relevant pathways of carbon flow within the given flux map. NetFlow simulates ¹³C isotope labeling experiments to calculate the extent of carbon exchange, or carbon yield, between every metabolite in the given GSM. Based on the carbon yield, the carbon flow to or from any metabolite or between any pair of metabolites of interest can be isolated and readily visualized. The resulting pathways are much easier to interpret, which enables an in-depth mechanistic understanding of the metabolic phenotype of interest. Here, we first demonstrate NetFlow with a simple network. We then depict the utility of NetFlow on a model of central carbon metabolism in E. coli. Specifically, we isolated the production pathway for succinate synthesis in this model and the metabolic mechanism driving the predicted increase in succinate yield in a double knockout of E. coli. Finally, we describe the application of NetFlow to a GSM of lycopene-producing E. coli, which enabled the rapid identification of the mechanisms behind the measured increases in lycopene production following single, double, and triple knockouts.

Opportunities at the Interface of Network Science and Metabolic Modeling

Metabolism plays a central role in cell physiology because it provides the molecular machinery for growth. At the genome-scale, metabolism is made up of thousands of reactions interacting with one another. Untangling this complexity is key to understand how cells respond to genetic, environmental, or therapeutic perturbations. Here we discuss the roles of two complementary strategies for the analysis of genome-scale metabolic models: Flux Balance Analysis (FBA) and network science. While FBA estimates metabolic flux on the basis of an optimization principle, network approaches reveal emergent properties of the global metabolic connectivity. We highlight how the integration of both approaches promises to deliver insights on the structure and function of metabolic systems with wide-ranging implications in discovery science, precision medicine and industrial biotechnology.

Engineering Escherichia coli for the utilization of ethylene glycol

Background

A considerable challenge in the development of bioprocesses for producing chemicals and fuels has been the high cost of feedstocks relative to oil prices, making it difficult for these processes to compete with their conventional petrochemical counterparts. Hence, in the absence of high oil prices in the near future, there has been a shift in the industry to produce higher value compounds such as fragrances for cosmetics. Yet, there is still a need to address climate change and develop biotechnological approaches for producing large market, lower value chemicals and fuels.

Results

In this work, we study ethylene glycol (EG), a novel feedstock that we believe has promise to address this challenge. We engineer Escherichia coli (E. coli) to consume EG and examine glycolate production as a case study for chemical production. Using a combination of modeling and experimental studies, we identify oxygen concentration as an important metabolic valve in the assimilation and use of EG as a substrate. Two oxygen-based strategies are thus developed and tested in fed-batch bioreactors. Ultimately, the best glycolate production strategy employed a target respiratory quotient leading to the highest observed fermentation performance. With this strategy, a glycolate titer of 10.4 g/L was reached after 112 h of production time in a fed-batch bioreactor. Correspondingly, a yield of 0.8 g/g from EG and productivity of 0.1 g/L h were measured during the production stage. Our modeling and experimental results clearly suggest that oxygen concentration is an important factor in the assimilation and use of EG as a substrate. Finally, our use of metabolic modeling also sheds light on the intracellular distribution through central metabolism, implicating flux to 2-phosphoglycerate as the primary route for EG assimilation.

Conclusion

Overall, our work suggests that EG could provide a renewable starting material for commercial biosynthesis of fuels and chemicals that may achieve economic parity with petrochemical feedstocks while sequestering carbon dioxide.

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.

Environmental flexibility does not explain metabolic robustness

Cells show remarkable resilience against genetic and environmental perturbations. However, its evolutionary origin remains obscure. In order to leverage methods of systems biology for examining cellular robustness, a computationally accessible way of quantification is needed. Here, we present an unbiased metric of structural robustness in genome-scale metabolic models based on concepts prevalent in reliability engineering and fault analysis. The probability of failure (PoF) is defined as the (weighted) portion of all possible combinations of loss-of-function mutations that disable network functionality. It can be exactly determined if all essential reactions, synthetic lethal pairs of reactions, synthetic lethal triplets of reactions etc. are known. In theory, these minimal cut sets (MCSs) can be calculated for any network, but for large models the problem remains computationally intractable. Herein, we demonstrate that even at the genome scale only the lowest-cardinality MCSs are required to efficiently approximate the PoF with reasonable accuracy. Building on an improved theoretical understanding, we analysed the robustness of 489 E. coli, Shigella, Salmonella, and fungal genome-scale metabolic models (GSMMs). In contrast to the popular "congruence theory", which explains the origin of genetic robustness as a byproduct of selection for environmental flexibility, we found no correlation between network robustness and the diversity of growth-supporting environments. On the contrary, our analysis indicates that amino acid synthesis rather than carbon metabolism dominates metabolic robustness.

Speeding up the core algorithm for the dual calculation of minimal cut sets in large metabolic networks

Background

The concept of minimal cut sets (MCS) has become an important mathematical framework for analyzing and (re)designing metabolic networks. However, the calculation of MCS in genome-scale metabolic models is a complex computational problem. The development of duality-based algorithms in the last years allowed the enumeration of thousands of MCS in genome-scale networks by solving mixed-integer linear problems (MILP). A recent advancement in this field was the introduction of the MCS² approach. In contrast to the Farkas-lemma-based dual system used in earlier studies, the MCS² approach employs a more condensed representation of the dual system based on the nullspace of the stoichiometric matrix, which, due to its reduced dimension, holds promise to further enhance MCS computations.

Results

In this work, we introduce several new variants and modifications of duality-based MCS algorithms and benchmark their effects on the overall performance. As one major result, we generalize the original MCS² approach (which was limited to blocking the operation of certain target reactions) to the most general case of MCS computations with arbitrary target and desired regions. Building upon these developments, we introduce a new MILP variant which allows maximal flexibility in the formulation of MCS problems and fully leverages the reduced size of the nullspace-based dual system. With a comprehensive set of benchmarks, we show that the MILP with the nullspace-based dual system outperforms the MILP with the Farkas-lemma-based dual system speeding up MCS computation with an averaged factor of approximately 2.5. We furthermore present several simplifications in the formulation of constraints, mainly related to binary variables, which further enhance the performance of MCS-related MILP. However, the benchmarks also reveal that some highly condensed formulations of constraints, especially on reversible reactions, may lead to worse behavior when compared to variants with a larger number of (more explicit) constraints and involved variables.

Conclusions

Our results further enhance the algorithmic toolbox for MCS calculations and are of general importance for theoretical developments as well as for practical applications of the MCS framework.

MEMO: A Method for Computing Metabolic Modules for Cell-Free Production Systems

Cell-free bioproduction systems represent a promising alternative to classical microbial fermentation processes to synthesize value-added products from biological feedstocks. An essential step for establishing cell-free production systems is the identification of suitable metabolic modules with defined properties. Here we present MEMO, a novel computational approach to find smallest metabolic modules with specified stoichiometric and thermodynamic constraints supporting the design of cell-free systems in various regards. In particular, one key challenge for a sustained operation of cell-free systems is the regeneration of utilized cofactors (such as ATP and NAD(P)H). Given a production pathway with certain cofactor requirements, MEMO can be used to compute smallest regeneration modules that recover these cofactors with required stoichiometries. MEMO incorporates the stoichiometric and thermodynamic constraints in a single mixed-integer linear program, which can then be solved to find smallest suitable modules from a given reaction database. We illustrate the applicability of MEMO by calculating regeneration modules for the recently published synthetic CETCH cycle for in vitro carbon dioxide fixation. We demonstrate that MEMO is very flexible in taking into account the diverse constraints of the CETCH cycle (e.g., regeneration of 1 ATP, 4 NADPH and of 1 acetyl-group without net production of CO₂ and with permitted side production of malate) and is able to determine multiple solutions in reasonable time in two large reaction databases (MetaCyc and BiGG). The most promising regeneration modules found utilize glycerol as substrate and require only 8 enzymatic steps. It is also shown that some of these modules are robust against spontaneous loss of cofactors (e.g., oxidation of NAD(P)H or hydrolysis of ATP). Furthermore, we demonstrate that MEMO can also find cell-free production systems with integrated product synthesis and cofactor regeneration. Overall, MEMO provides a powerful method for finding metabolic modules and for designing cell-free production systems as one particular application.

Automatic construction of metabolic models with enzyme constraints

Background

In order to improve the accuracy of constraint-based metabolic models, several approaches have been developed which intend to integrate additional biological information. Two of these methods, MOMENT and GECKO, incorporate enzymatic (kcat) parameters and enzyme mass constraints to further constrain the space of feasible metabolic flux distributions. While both methods have been proven to deliver useful extensions of metabolic models, they may considerably increase size and complexity of the models and there is currently no tool available to fully automate generation and calibration of such enzyme-constrained models from given stoichiometric models.

Results

In this work we present three major developments. We first conceived short MOMENT (sMOMENT), a simplified version of the MOMENT approach, which yields the same predictions as MOMENT but requires significantly fewer variables and enables direct inclusion of the relevant enzyme constraints in the standard representation of a constraint-based model. When measurements of enzyme concentrations are available, these can be included as well leading in the extreme case, where all enzyme concentrations are known, to a model representation that is analogous to the GECKO approach. Second, we developed the AutoPACMEN toolbox which allows an almost fully automated creation of sMOMENT-enhanced stoichiometric metabolic models. In particular, this includes the automatic read-out and processing of relevant enzymatic data from different databases and the reconfiguration of the stoichiometric model with embedded enzymatic constraints. Additionally, tools have been developed to adjust (kcat and enzyme pool) parameters of sMOMENT models based on given flux data. We finally applied the new sMOMENT approach and the AutoPACMEN toolbox to generate an enzyme-constrained version of the E. coli genome-scale model iJO1366 and analyze its key properties and differences with the standard model. In particular, we show that the enzyme constraints improve flux predictions (e.g., explaining overflow metabolism and other metabolic switches) and demonstrate, for the first time, that these constraints can markedly change the spectrum of metabolic engineering strategies for different target products.

Conclusions

The methodological and tool developments presented herein pave the way for a simplified and routine construction and analysis of enzyme-constrained metabolic models.

The number of active metabolic pathways is bounded by the number of cellular constraints at maximal metabolic rates

Growth rate is a near-universal selective pressure across microbial species. High growth rates require hundreds of metabolic enzymes, each with different nonlinear kinetics, to be precisely tuned within the bounds set by physicochemical constraints. Yet, the metabolic behaviour of many species is characterized by simple relations between growth rate, enzyme expression levels and metabolic rates. We asked if this simplicity could be the outcome of optimisation by evolution. Indeed, when the growth rate is maximized-in a static environment under mass-conservation and enzyme expression constraints-we prove mathematically that the resulting optimal metabolic flux distribution is described by a limited number of subnetworks, known as Elementary Flux Modes (EFMs). We show that, because EFMs are the minimal subnetworks leading to growth, a small active number automatically leads to the simple relations that are measured. We find that the maximal number of flux-carrying EFMs is determined only by the number of imposed constraints on enzyme expression, not by the size, kinetics or topology of the network. This minimal-EFM extremum principle is illustrated in a graphical framework, which explains qualitative changes in microbial behaviours, such as overflow metabolism and co-consumption, and provides a method for identification of the enzyme expression constraints that limit growth under the prevalent conditions. The extremum principle applies to all microorganisms that are selected for maximal growth rates under protein concentration constraints, for example the solvent capacities of cytosol, membrane or periplasmic space.

Creation and analysis of biochemical constraint-based models using the COBRA Toolbox v.3.0

Constraint-based reconstruction and analysis (COBRA) provides a molecular mechanistic framework for integrative analysis of experimental molecular systems biology data and quantitative prediction of physicochemically and biochemically feasible phenotypic states. The COBRA Toolbox is a comprehensive desktop software suite of interoperable COBRA methods. It has found widespread application in biology, biomedicine, and biotechnology because its functions can be flexibly combined to implement tailored COBRA protocols for any biochemical network. This protocol is an update to the COBRA Toolbox v.1.0 and v.2.0. Version 3.0 includes new methods for quality-controlled reconstruction, modeling, topological analysis, strain and experimental design, and network visualization, as well as network integration of chemoinformatic, metabolomic, transcriptomic, proteomic, and thermochemical data. New multi-lingual code integration also enables an expansion in COBRA application scope via high-precision, high-performance, and nonlinear numerical optimization solvers for multi-scale, multi-cellular, and reaction kinetic modeling, respectively. This protocol provides an overview of all these new features and can be adapted to generate and analyze constraint-based models in a wide variety of scenarios. The COBRA Toolbox v.3.0 provides an unparalleled depth of COBRA methods.

Designing Metabolic Division of Labor in Microbial Communities

Microbes face a trade-off between being metabolically independent and relying on neighboring organisms for the supply of some essential metabolites. This balance of conflicting strategies affects microbial community structure and dynamics, with important implications for microbiome research and synthetic ecology. A "gedanken" (thought) experiment to investigate this trade-off would involve monitoring the rise of mutual dependence as the number of metabolic reactions allowed in an organism is increasingly constrained. The expectation is that below a certain number of reactions, no individual organism would be able to grow in isolation and cross-feeding partnerships and division of labor would emerge. We implemented this idealized experiment using in silico genome-scale models. In particular, we used mixed-integer linear programming to identify trade-off solutions in communities of Escherichia coli strains. The strategies that we found revealed a large space of opportunities in nuanced and nonintuitive metabolic division of labor, including, for example, splitting the tricarboxylic acid (TCA) cycle into two separate halves. The systematic computation of possible solutions in division of labor for 1-, 2-, and 3-strain consortia resulted in a rich and complex landscape. This landscape displayed a nonlinear boundary, indicating that the loss of an intracellular reaction was not necessarily compensated for by a single imported metabolite. Different regions in this landscape were associated with specific solutions and patterns of exchanged metabolites. Our approach also predicts the existence of regions in this landscape where independent bacteria are viable but are outcompeted by cross-feeding pairs, providing a possible incentive for the rise of division of labor. IMPORTANCE Understanding how microbes assemble into communities is a fundamental open issue in biology, relevant to human health, metabolic engineering, and environmental sustainability. A possible mechanism for interactions of microbes is through cross-feeding, i.e., the exchange of small molecules. These metabolic exchanges may allow different microbes to specialize in distinct tasks and evolve division of labor. To systematically explore the space of possible strategies for division of labor, we applied advanced optimization algorithms to computational models of cellular metabolism. Specifically, we searched for communities able to survive under constraints (such as a limited number of reactions) that would not be sustainable by individual species. We found that predicted consortia partition metabolic pathways in ways that would be difficult to identify manually, possibly providing a competitive advantage over individual organisms. In addition to helping understand diversity in natural microbial communities, our approach could assist in the design of synthetic consortia.

RedCom: A strategy for reduced metabolic modeling of complex microbial communities and its application for analyzing experimental datasets from anaerobic digestion

Constraint-based modeling (CBM) is increasingly used to analyze the metabolism of complex microbial communities involved in ecology, biomedicine, and various biotechnological processes. While CBM is an established framework for studying the metabolism of single species with linear stoichiometric models, CBM of communities with balanced growth is more complicated, not only due to the larger size of the multi-species metabolic network but also because of the bilinear nature of the resulting community models. Moreover, the solution space of these community models often contains biologically unrealistic solutions, which, even with model linearization and under application of certain objective functions, cannot easily be excluded. Here we present RedCom, a new approach to build reduced community models in which the metabolisms of the participating organisms are represented by net conversions computed from the respective single-species networks. By discarding (single-species) net conversions that violate a minimality criterion in the exchange fluxes, it is ensured that unrealistic solutions in the community model are excluded where a species altruistically synthesizes large amounts of byproducts (instead of biomass) to fulfill the requirements of other species. We employed the RedCom approach for modeling communities of up to nine organisms involved in typical degradation steps of anaerobic digestion in biogas plants. Compared to full (bilinear and linearized) community models, we found that the reduced community models obtained with RedCom are not only much smaller but allow, also in the largest model with nine species, extensive calculations required to fully characterize the solution space and to reveal key properties of communities with maximum methane yield and production rates. Furthermore, the predictive power of the reduced community models is significantly larger because they predict much smaller ranges of feasible community compositions and exchange fluxes still being consistent with measurements obtained from enrichment cultures. For an enrichment culture for growth on ethanol, we also used metaproteomic data to further constrain the solution space of the community models. Both model and proteomic data indicated a dominance of acetoclastic methanogens (Methanosarcinales) and Desulfovibrionales being the least abundant group in this microbial community.

Microbial communities are involved in many fundamental processes in nature, health and biotechnology. The elucidation of interdependencies between the involved players of microbial communities and how the interactions shape the composition, behavior and characteristic features of the consortium has become an important branch of microbiology research. Many communities are based on the exchange of metabolites between the species and constraint-based metabolic modeling has become an important approach for a formal description and quantitative analysis of these metabolic dependencies. However, the complexity of the models rises quickly with a growing number of organisms and the space of predicted feasible behaviors often includes unrealistic solutions. Here we present RedCom, a new approach to build reduced stoichiometric models of balanced microbial communities based on net conversions of the single-species models. We demonstrate the applicability of our RedCom approach by modeling communities of up to nine organisms involved in degradation steps of anaerobic digestion in biogas plants. As one of the first studies in this field, we compare simulation results from the community models with experimental data of laboratory-scale biogas reactors for growth on ethanol and glucose-cellulose media. The results also demonstrate a higher predictive power of the RedCom vs. the full models.

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.

Dynamic control of endogenous metabolism with combinatorial logic circuits

Controlling gene expression during a bioprocess enables real-time metabolic control, coordinated cellular responses, and staging order-of-operations. Achieving this with small molecule inducers is impractical at scale and dynamic circuits are difficult to design. Here, we show that the same set of sensors can be integrated by different combinatorial logic circuits to vary when genes are turned on and off during growth. Three Escherichia coli sensors that respond to the consumption of feedstock (glucose), dissolved oxygen, and by-product accumulation (acetate) are constructed and optimized. By integrating these sensors, logic circuits implement temporal control over an 18-h period. The circuit outputs are used to regulate endogenous enzymes at the transcriptional and post-translational level using CRISPRi and targeted proteolysis, respectively. As a demonstration, two circuits are designed to control acetate production by matching their dynamics to when endogenous genes are expressed (pta or poxB) and respond by turning off the corresponding gene. This work demonstrates how simple circuits can be implemented to enable customizable dynamic gene regulation.

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.

A mathematical framework for yield (vs. rate) optimization in constraint-based modeling and applications in metabolic engineering

BACKGROUND

The optimization of metabolic rates (as linear objective functions) represents the methodical core of flux-balance analysis techniques which have become a standard tool for the study of genome-scale metabolic models. Besides (growth and synthesis) rates, metabolic yields are key parameters for the characterization of biochemical transformation processes, especially in the context of biotechnological applications. However, yields are ratios of rates, and hence the optimization of yields (as nonlinear objective functions) under arbitrary linear constraints is not possible with current flux-balance analysis techniques. Despite the fundamental importance of yields in constraint-based modeling, a comprehensive mathematical framework for yield optimization is still missing.

RESULTS

We present a mathematical theory that allows one to systematically compute and analyze yield-optimal solutions of metabolic models under arbitrary linear constraints. In particular, we formulate yield optimization as a linear-fractional program. For practical computations, we transform the linear-fractional yield optimization problem to a (higher-dimensional) linear problem. Its solutions determine the solutions of the original problem and can be used to predict yield-optimal flux distributions in genome-scale metabolic models. For the theoretical analysis, we consider the linear-fractional problem directly. Most importantly, we show that the yield-optimal solution set (like the rate-optimal solution set) is determined by (yield-optimal) elementary flux vectors of the underlying metabolic model. However, yield- and rate-optimal solutions may differ from each other, and hence optimal (biomass or product) yields are not necessarily obtained at solutions with optimal (growth or synthesis) rates. Moreover, we discuss phase planes/production envelopes and yield spaces, in particular, we prove that yield spaces are convex and provide algorithms for their computation. We illustrate our findings by a small example and demonstrate their relevance for metabolic engineering with realistic models of E. coli.

CONCLUSIONS

We develop a comprehensive mathematical framework for yield optimization in metabolic models. Our theory is particularly useful for the study and rational modification of cell factories designed under given yield and/or rate requirements.

Linear programming model can explain respiration of fermentation products

Many differentiated cells rely primarily on mitochondrial oxidative phosphorylation for generating energy in the form of ATP needed for cellular metabolism. In contrast most tumor cells instead rely on aerobic glycolysis leading to lactate to about the same extent as on respiration. Warburg found that cancer cells to support oxidative phosphorylation, tend to ferment glucose or other energy source into lactate even in the presence of sufficient oxygen, which is an inefficient way to generate ATP. This effect also occurs in striated muscle cells, activated lymphocytes and microglia, endothelial cells and several mammalian cell types, a phenomenon termed the "Warburg effect". The effect is paradoxical at first glance because the ATP production rate of aerobic glycolysis is much slower than that of respiration and the energy demands are better to be met by pure oxidative phosphorylation. We tackle this question by building a minimal model including three combined reactions. The new aspect in extension to earlier models is that we take into account the possible uptake and oxidation of the fermentation products. We examine the case where the cell can allocate protein on several enzymes in a varying distribution and model this by a linear programming problem in which the objective is to maximize the ATP production rate under different combinations of constraints on enzymes. Depending on the cost of reactions and limitation of the substrates, this leads to pure respiration, pure fermentation, and a mixture of respiration and fermentation. The model predicts that fermentation products are only oxidized when glucose is scarce or its uptake is severely limited.

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.