A kinetic model of Escherichia coli core metabolism satisfying multiple sets of mutant flux data

In contrast to stoichiometric-based models, the development of large-scale kinetic models of metabolism has been hindered by the challenge of identifying kinetic parameter values and kinetic rate laws applicable to a wide range of environmental and/or genetic perturbations. The recently introduced ensemble modeling (EM) procedure provides a promising remedy to address these challenges by decomposing metabolic reactions into elementary reaction steps and incorporating all phenotypic observations, upon perturbation, in its model parameterization scheme. Here, we present a kinetic model of Escherichia coli core metabolism that satisfies the fluxomic data for wild-type and seven mutant strains by making use of the EM concepts. This model encompasses 138 reactions, 93 metabolites and 60 substrate-level regulatory interactions accounting for glycolysis/gluconeogenesis, pentose phosphate pathway, TCA cycle, major pyruvate metabolism, anaplerotic reactions and a number of reactions in other parts of the metabolism. Parameterization is performed using a formal optimization approach that minimizes the discrepancies between model predictions and flux measurements. The predicted fluxes by the model are within the uncertainty range of experimental flux data for 78% of the reactions (with measured fluxes) for both the wild-type and seven mutant strains. The remaining flux predictions are mostly within three standard deviations of reported ranges. Converting the EM-based parameters into a Michaelis–Menten equivalent formalism revealed that 35% of K_m and 77% of k_cat parameters are within uncertainty range of the literature-reported values. The predicted metabolite concentrations by the model are also within uncertainty ranges of metabolomic data for 68% of the metabolites. A leave-one-out cross-validation test to evaluate the flux prediction performance of the model showed that metabolic fluxes for the mutants located in the proximity of mutations used for training the model can be predicted more accurately. The constructed model and the parameterization procedure presented in this study pave the way for the construction of larger-scale kinetic models with more narrowly distributed parameter values as new metabolomic/fluxomic data sets are becoming available for E. coli and other organisms.     

Systematic Construction of Kinetic Models from Genome-Scale Metabolic Networks

The quantitative effects of environmental and genetic perturbations on metabolism can be studied in silico using kinetic models. We present a strategy for large-scale model construction based on a logical layering of data such as reaction fluxes, metabolite concentrations, and kinetic constants. The resulting models contain realistic standard rate laws and plausible parameters, adhere to the laws of thermodynamics, and reproduce a predefined steady state. These features have not been simultaneously achieved by previous workflows. We demonstrate the advantages and limitations of the workflow by translating the yeast consensus metabolic network into a kinetic model. Despite crudely selected data, the model shows realistic control behaviour, a stable dynamic, and realistic response to perturbations in extracellular glucose concentrations. The paper concludes by outlining how new data can continuously be fed into the workflow and how iterative model building can assist in directing experiments.     

Assessing the impact of substrate-level enzyme regulations limiting ethanol titer in Clostridium thermocellum using a core kinetic model

Clostridium thermocellum is a promising candidate for consolidated bioprocessing because it can directly ferment cellulose to ethanol. Despite significant efforts, achieved yields and titers fall below industrially relevant targets. This implies that there still exist unknown enzymatic, regulatory, and/or possibly thermodynamic bottlenecks that can throttle back metabolic flow. By (i) elucidating internal metabolic fluxes in wild-type C. thermocellum grown on cellobiose via ¹³C-metabolic flux analysis (¹³C-MFA), (ii) parameterizing a core kinetic model, and (iii) subsequently deploying an ensemble-docking workflow for discovering substrate-level regulations, this paper aims to reveal some of these factors and expand our knowledgebase governing C. thermocellum metabolism. Generated ¹³C labeling data were used with ¹³C-MFA to generate a wild-type flux distribution for the metabolic network. Notably, flux elucidation through MFA alluded to serine generation via the mercaptopyruvate pathway. Using the elucidated flux distributions in conjunction with batch fermentation process yield data for various mutant strains, we constructed a kinetic model of C. thermocellum core metabolism (i.e. k-ctherm138). Subsequently, we used the parameterized kinetic model to explore the effect of removing substrate-level regulations on ethanol yield and titer. Upon exploring all possible simultaneous (up to four) regulation removals we identified combinations that lead to many-fold model predicted improvement in ethanol titer. In addition, by coupling a systematic method for identifying putative competitive inhibitory mechanisms using K-FIT kinetic parameterization with the ensemble-docking workflow, we flagged 67 putative substrate-level inhibition mechanisms across central carbon metabolism supported by both kinetic formalism and docking analysis.     

Channeling in native microbial pathways: Implications and challenges for metabolic engineering

Intracellular enzymes can be organized into a variety of assemblies, shuttling intermediates from one active site to the next. Eukaryotic compartmentalization within mitochondria and peroxisomes and substrate tunneling within multi-enzyme complexes have been well recognized. Intriguingly, the central pathways in prokaryotes may also form extensive channels, including the heavily branched glycolysis pathway. In vivo channeling through cascade enzymes is difficult to directly measure, but can be inferred from in vitro tests, reaction thermodynamics, transport/reaction modeling, analysis of molecular diffusion and protein interactions, or steady state/dynamic isotopic labeling. Channeling presents challenges but also opportunities for metabolic engineering applications. It rigidifies fluxes in native pathways by trapping or excluding metabolites for bioconversions, causing substrate catabolite repressions or inferior efficiencies in engineered pathways. Channeling is an overlooked regulatory mechanism used to control flux responses under environmental/genetic perturbations. The heterogeneous distribution of intracellular enzymes also confounds kinetic modeling and multiple-omics analyses. Understanding the scope and mechanisms of channeling in central pathways may improve our interpretation of robust fluxomic topology throughout metabolic networks and lead to better design and engineering of heterologous pathways.     

Optimization-driven identification of genetic perturbations accelerates the convergence of model parameters in ensemble modeling of metabolic networks

The ensemble modeling (EM) approach has shown promise in capturing kinetic and regulatory effects in the modeling of metabolic networks. Efficacy of the EM procedure relies on the identification of model parameterizations that adequately describe all observed metabolic phenotypes upon perturbation. In this study, we propose an optimization-based algorithm for the systematic identification of genetic/enzyme perturbations to maximally reduce the number of models retained in the ensemble after each round of model screening. The key premise here is to design perturbations that will maximally scatter the predicted steady-state fluxes over the ensemble parameterizations. We demonstrate the applicability of this procedure for an Escherichia coli metabolic model of central metabolism by successively identifying single, double, and triple enzyme perturbations that cause the maximum degree of flux separation between models in the ensemble. Results revealed that optimal perturbations are not always located close to reaction(s) whose fluxes are measured, especially when multiple perturbations are considered. In addition, there appears to be a maximum number of simultaneous perturbations beyond which no appreciable increase in the divergence of flux predictions is achieved. Overall, this study provides a systematic way of optimally designing genetic perturbations for populating the ensemble of models with relevant model parameterizations.     

Kinetic modeling of microbial growth,enzyme activity,and gene deletions: An integrated model of β-glucosidase function in Cellvibrio japonicus

Understanding the complex growth and metabolic dynamics in microorganisms requires advanced kinetic models containing both metabolic reactions and enzymatic regulation to predict phenotypic behaviors under different conditions and perturbations. Most current kinetic models lack gene expression dynamics and are separately calibrated to distinct media, which consequently makes them unable to account for genetic perturbations or multiple substrates. This challenge limits our ability to gain a comprehensive understanding of microbial processes towards advanced metabolic optimizations that are desired for many biotechnology applications. Here, we present an integrated computational and experimental approach for the development and optimization of mechanistic kinetic models for microbial growth and metabolic and enzymatic dynamics. Our approach integrates growth dynamics, gene expression, protein secretion, and gene-deletion phenotypes. We applied this methodology to build a dynamic model of the growth kinetics in batch culture of the bacterium Cellvibrio japonicus grown using either cellobiose or glucose media. The model parameters were inferred from an experimental data set using an evolutionary computation method. The resulting model was able to explain the growth dynamics of C. japonicus using either cellobiose or glucose media and was also able to accurately predict the metabolite concentrations in the wild-type strain as well as in β-glucosidase gene deletion mutant strains. We validated the model by correctly predicting the non-diauxic growth and metabolite consumptions of the wild-type strain in a mixed medium containing both cellobiose and glucose, made further predictions of mutant strains growth phenotypes when using cellobiose and glucose media, and demonstrated the utility of the model for designing industrially-useful strains. Importantly, the model is able to explain the role of the different β-glucosidases and their behavior under genetic perturbations. This integrated approach can be extended to other metabolic pathways to produce mechanistic models for the comprehensive understanding of enzymatic functions in multiple substrates.     

ΔFBA—Predicting metabolic flux alterations using genome-scale metabolic models and differential transcriptomic data

Genome-scale metabolic models (GEMs) provide a powerful framework for simulating the entire set of biochemical reactions in a cell using a constraint-based modeling strategy called flux balance analysis (FBA). FBA relies on an assumed metabolic objective for generating metabolic fluxes using GEMs. But, the most appropriate metabolic objective is not always obvious for a given condition and is likely context-specific, which often complicate the estimation of metabolic flux alterations between conditions. Here, we propose a new method, called ΔFBA (deltaFBA), that integrates differential gene expression data to evaluate directly metabolic flux differences between two conditions. Notably, ΔFBA does not require specifying the cellular objective. Rather, ΔFBA seeks to maximize the consistency and minimize inconsistency between the predicted flux differences and differential gene expression. We showcased the performance of ΔFBA through several case studies involving the prediction of metabolic alterations caused by genetic and environmental perturbations in Escherichia coli and caused by Type-2 diabetes in human muscle. Importantly, in comparison to existing methods, ΔFBA gives a more accurate prediction of flux differences.     

A General Framework for Thermodynamically Consistent Parameterization and Efficient Sampling of Enzymatic Reactions

Kinetic models provide the means to understand and predict the dynamic behaviour of enzymes upon different perturbations. Despite their obvious advantages, classical parameterizations require large amounts of data to fit their parameters. Particularly, enzymes displaying complex reaction and regulatory (allosteric) mechanisms require a great number of parameters and are therefore often represented by approximate formulae, thereby facilitating the fitting but ignoring many real kinetic behaviours. Here, we show that full exploration of the plausible kinetic space for any enzyme can be achieved using sampling strategies provided a thermodynamically feasible parameterization is used. To this end, we developed a General Reaction Assembly and Sampling Platform (GRASP) capable of consistently parameterizing and sampling accurate kinetic models using minimal reference data. The former integrates the generalized MWC model and the elementary reaction formalism. By formulating the appropriate thermodynamic constraints, our framework enables parameterization of any oligomeric enzyme kinetics without sacrificing complexity or using simplifying assumptions. This thermodynamically safe parameterization relies on the definition of a reference state upon which feasible parameter sets can be efficiently sampled. Uniform sampling of the kinetics space enabled dissecting enzyme catalysis and revealing the impact of thermodynamics on reaction kinetics. Our analysis distinguished three reaction elasticity regions for common biochemical reactions: a steep linear region (0> ΔG_r >-2 kJ/mol), a transition region (-2> ΔG_r >-20 kJ/mol) and a constant elasticity region (ΔG_r <-20 kJ/mol). We also applied this framework to model more complex kinetic behaviours such as the monomeric cooperativity of the mammalian glucokinase and the ultrasensitive response of the phosphoenolpyruvate carboxylase of Escherichia coli. In both cases, our approach described appropriately not only the kinetic behaviour of these enzymes, but it also provided insights about the particular features underpinning the observed kinetics. Overall, this framework will enable systematic parameterization and sampling of enzymatic reactions.     

Integrated stoichiometric, thermodynamic and kinetic modelling of steady state metabolism

The quantitative analysis of biochemical reactions and metabolites is at frontier of biological sciences. The recent availability of high-throughput technology data sets in biology has paved the way for new modelling approaches at various levels of complexity including the metabolome of a cell or an organism. Understanding the metabolism of a single cell and multi-cell organism will provide the knowledge for the rational design of growth conditions to produce commercially valuable reagents in biotechnology. Here, we demonstrate how equations representing steady state mass conservation, energy conservation, the second law of thermodynamics, and reversible enzyme kinetics can be formulated as a single system of linear equalities and inequalities, in addition to linear equalities on exponential variables. Even though the feasible set is non-convex, the reformulation is exact and amenable to large-scale numerical analysis, a prerequisite for computationally feasible genome scale modelling. Integrating flux, concentration and kinetic variables in a unified constraint-based formulation is aimed at increasing the quantitative predictive capacity of flux balance analysis. Incorporation of experimental and theoretical bounds on thermodynamic and kinetic variables ensures that the predicted steady state fluxes are both thermodynamically and biochemically feasible. The resulting in silico predictions are tested against fluxomic data for central metabolism in Escherichia coli and compare favourably with in silico prediction by flux balance analysis.     

A kinetic model describes metabolic response to perturbations and distribution of flux control in the benzenoid network of Petunia hybrida

In recent years there has been much interest in the genetic enhancement of plant metabolism; however, attempts at genetic modification are often unsuccessful due to an incomplete understanding of network dynamics and their regulatory properties. Kinetic modeling of plant metabolic networks can provide predictive information on network control and response to genetic perturbations, which allow estimation of flux at any concentration of intermediate or enzyme in the system. In this research, a kinetic model of the benzenoid network was developed to simulate whole network responses to different concentrations of supplied phenylalanine (Phe) in petunia flowers and capture flux redistributions caused by genetic manipulations. Kinetic parameters were obtained by network decomposition and non‐linear least squares optimization of data from petunia flowers supplied with either 75 or 150 mm ²H₅‐Phe. A single set of kinetic parameters simultaneously accommodated labeling and pool size data obtained for all endogenous and emitted volatiles at the two concentrations of supplied ²H₅‐Phe. The generated kinetic model was validated using flowers from transgenic petunia plants in which benzyl CoA:benzyl alcohol/phenylethanol benzoyltransferase (BPBT) was down‐regulated via RNAi. The determined in vivo kinetic parameters were used for metabolic control analysis, in which flux control coefficients were calculated for fluxes around the key branch point at Phe and revealed that phenylacetaldehyde synthase activity is the primary controlling factor for the phenylacetaldehyde branch of the benzenoid network. In contrast, control of flux through the β‐oxidative and non‐β‐oxidative pathways is highly distributed.     

Monte-Carlo Modeling of the Central Carbon Metabolism of Lactococcus lactis: Insights into Metabolic Regulation

Metabolic pathways are complex dynamic systems whose response to perturbations and environmental challenges are governed by multiple interdependencies between enzyme properties, reactions rates, and substrate levels. Understanding the dynamics arising from such a network can be greatly enhanced by the construction of a computational model that embodies the properties of the respective system. Such models aim to incorporate mechanistic details of cellular interactions to mimic the temporal behavior of the biochemical reaction system and usually require substantial knowledge of kinetic parameters to allow meaningful conclusions. Several approaches have been suggested to overcome the severe data requirements of kinetic modeling, including the use of approximative kinetics and Monte-Carlo sampling of reaction parameters. In this work, we employ a probabilistic approach to study the response of a complex metabolic system, the central metabolism of the lactic acid bacterium Lactococcus lactis, subject to perturbations and brief periods of starvation. Supplementing existing methodologies, we show that it is possible to acquire a detailed understanding of the control properties of a corresponding metabolic pathway model that is directly based on experimental observations. In particular, we delineate the role of enzymatic regulation to maintain metabolic stability and metabolic recovery after periods of starvation. It is shown that the feedforward activation of the pyruvate kinase by fructose-1,6-bisphosphate qualitatively alters the bifurcation structure of the corresponding pathway model, indicating a crucial role of enzymatic regulation to prevent metabolic collapse for low external concentrations of glucose. We argue that similar probabilistic methodologies will help our understanding of dynamic properties of small-, medium- and large-scale metabolic networks models.     

Analysis of mitochondrial metabolism in situ: Combining stable isotope labeling with selective permeabilization

To date, it is well-established that mitochondrial dysfunction does not only play a vital role in cancer but also in other pathological conditions such as neurodegenerative diseases and inflammation. An important tool for the analysis of cellular metabolism is the application of stable isotope labeled substrates, which allow for the tracing of atoms throughout metabolic networks. While such analyses yield very detailed information about intracellular fluxes, the determination of compartment specific fluxes is far more challenging. Most approaches for the deconvolution of compartmented metabolism use computational models whereas experimental methods are rare. Here, we developed an experimental setup based on selective permeabilization of the cytosolic membrane that allows for the administration of stable isotope labeled substrates directly to mitochondria. We demonstrate how this approach can be used to infer metabolic changes in mitochondria induced by either chemical or genetic perturbations and give an outlook on its potential applications.     

Flux Imbalance Analysis and the Sensitivity of Cellular Growth to Changes in Metabolite Pools

Stoichiometric models of metabolism, such as flux balance analysis (FBA), are classically applied to predicting steady state rates - or fluxes - of metabolic reactions in genome-scale metabolic networks. Here we revisit the central assumption of FBA, i.e. that intracellular metabolites are at steady state, and show that deviations from flux balance (i.e. flux imbalances) are informative of some features of in vivo metabolite concentrations. Mathematically, the sensitivity of FBA to these flux imbalances is captured by a native feature of linear optimization, the dual problem, and its corresponding variables, known as shadow prices. First, using recently published data on chemostat growth of Saccharomyces cerevisae under different nutrient limitations, we show that shadow prices anticorrelate with experimentally measured degrees of growth limitation of intracellular metabolites. We next hypothesize that metabolites which are limiting for growth (and thus have very negative shadow price) cannot vary dramatically in an uncontrolled way, and must respond rapidly to perturbations. Using a collection of published datasets monitoring the time-dependent metabolomic response of Escherichia coli to carbon and nitrogen perturbations, we test this hypothesis and find that metabolites with negative shadow price indeed show lower temporal variation following a perturbation than metabolites with zero shadow price. Finally, we illustrate the broader applicability of flux imbalance analysis to other constraint-based methods. In particular, we explore the biological significance of shadow prices in a constraint-based method for integrating gene expression data with a stoichiometric model. In this case, shadow prices point to metabolites that should rise or drop in concentration in order to increase consistency between flux predictions and gene expression data. In general, these results suggest that the sensitivity of metabolic optima to violations of the steady state constraints carries biologically significant information on the processes that control intracellular metabolites in the cell.     

Parameter optimization and sensitivity analysis for large kinetic models using a real-coded genetic algorithm

Dynamic modeling is a powerful tool for predicting changes in metabolic regulation. However, a large number of input parameters, including kinetic constants and initial metabolite concentrations, are required to construct a kinetic model. Therefore, it is important not only to optimize the kinetic parameters, but also to investigate the effects of their perturbations on the overall system. We investigated the efficiency of the use of a real-coded genetic algorithm (RCGA) for parameter optimization and sensitivity analysis in the case of a large kinetic model involving glycolysis and the pentose phosphate pathway in Escherichia coli K-12. Sensitivity analysis of the kinetic model using an RCGA demonstrated that the input parameter values had different effects on model outputs. The results showed highly influential parameters in the model and their allowable ranges for maintaining metabolite-level stability. Furthermore, it was revealed that changes in these influential parameters may complement one another. This study presents an efficient approach based on the use of an RCGA for optimizing and analyzing parameters in large kinetic models.     

Literature mining supports a next-generation modeling approach to predict cellular byproduct secretion

The metabolic byproducts secreted by growing cells can be easily measured and provide a window into the state of a cell; they have been essential to the development of microbiology, cancer biology, and biotechnology. Progress in computational modeling of cells has made it possible to predict metabolic byproduct secretion with bottom-up reconstructions of metabolic networks. However, owing to a lack of data, it has not been possible to validate these predictions across a wide range of strains and conditions. Through literature mining, we were able to generate a database of Escherichia coli strains and their experimentally measured byproduct secretions. We simulated these strains in six historical genome-scale models of E. coli, and we report that the predictive power of the models has increased as they have expanded in size and scope. The latest genome-scale model of metabolism correctly predicts byproduct secretion for 35/89 (39%) of designs. The next-generation genome-scale model of metabolism and gene expression (ME-model) correctly predicts byproduct secretion for 40/89 (45%) of designs, and we show that ME-model predictions could be further improved through kinetic parameterization. We analyze the failure modes of these simulations and discuss opportunities to improve prediction of byproduct secretion.     

Local sensitivity analysis and metabolic control analysis of the biological part of the BTEX bioremediation model

Environmental distribution and bioremediation of hydrocarbon pollutants is described in the literature with complex mathematical models. Better understanding and easier model application require detailed model analysis. In this work, local sensitivity analysis of the kinetic parameters and metabolic control analysis of the biological part of the integrated BTEX bioremediation model were performed. Local sensitivity analysis revealed that the dissolved oxygen concentration (S _O) and particulate iron (III) oxide concentration (S _Fe) were the most sensitive to both positive and negative parameter value perturbations. In the case of model reactions, aerobic growth (r1) and aerobic growth on acetate (r13) were observed to be the most sensitive. The elasticity, flux control, and concentration control coefficients were estimated by applying the metabolic control analysis methodology. Metabolic control analysis revealed a positive effect of ammonium on all analysed model reactions. The results also indicated the importance of perturbation of the enzyme level catalysing iron reduction on acetate on model fluxes, as well as the importance of enzyme level catalysing aerobic growth on model metabolite concentration. These results can be used in planning optimal operating strategy for BTEX bioremediation.     

Interrogating the effect of enzyme kinetics on metabolism using differentiable constraint-based models

Metabolic models are typically characterized by a large number of parameters. Traditionally, metabolic control analysis is applied to differential equation-based models to investigate the sensitivity of predictions to parameters. A corresponding theory for constraint-based models is lacking, due to their formulation as optimization problems. Here, we show that optimal solutions of optimization problems can be efficiently differentiated using constrained optimization duality and implicit differentiation. We use this to calculate the sensitivities of predicted reaction fluxes and enzyme concentrations to turnover numbers in an enzyme-constrained metabolic model of Escherichia coli. The sensitivities quantitatively identify rate limiting enzymes and are mathematically precise, unlike current finite difference based approaches used for sensitivity analysis. Further, efficient differentiation of constraint-based models unlocks the ability to use gradient information for parameter estimation. We demonstrate this by improving, genome-wide, the state-of-the-art turnover number estimates for E. coli. Finally, we show that this technique can be generalized to arbitrarily complex models. By differentiating the optimal solution of a model incorporating both thermodynamic and kinetic rate equations, the effect of metabolite concentrations on biomass growth can be elucidated. We benchmark these metabolite sensitivities against a large experimental gene knockdown study, and find good alignment between the predicted sensitivities and in vivo metabolome changes. In sum, we demonstrate several applications of differentiating optimal solutions of constraint-based metabolic models, and show how it connects to classic metabolic control analysis.