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.     

Bringing metabolic networks to life: integration of kinetic, metabolic, and proteomic data

Background

Translating a known metabolic network into a dynamic model requires reasonable guesses of all enzyme parameters. In Bayesian parameter estimation, model parameters are described by a posterior probability distribution, which scores the potential parameter sets, showing how well each of them agrees with the data and with the prior assumptions made.

Results

We compute posterior distributions of kinetic parameters within a Bayesian framework, based on integration of kinetic, thermodynamic, metabolic, and proteomic data. The structure of the metabolic system (i.e., stoichiometries and enzyme regulation) needs to be known, and the reactions are modelled by convenience kinetics with thermodynamically independent parameters. The parameter posterior is computed in two separate steps: a first posterior summarises the available data on enzyme kinetic parameters; an improved second posterior is obtained by integrating metabolic fluxes, concentrations, and enzyme concentrations for one or more steady states. The data can be heterogenous, incomplete, and uncertain, and the posterior is approximated by a multivariate log-normal distribution. We apply the method to a model of the threonine synthesis pathway: the integration of metabolic data has little effect on the marginal posterior distributions of individual model parameters. Nevertheless, it leads to strong correlations between the parameters in the joint posterior distribution, which greatly improve the model predictions by the following Monte-Carlo simulations.

Conclusion

We present a standardised method to translate metabolic networks into dynamic models. To determine the model parameters, evidence from various experimental data is combined and weighted using Bayesian parameter estimation. The resulting posterior parameter distribution describes a statistical ensemble of parameter sets; the parameter variances and correlations can account for missing knowledge, measurement uncertainties, or biological variability. The posterior distribution can be used to sample model instances and to obtain probabilistic statements about the model's dynamic behaviour.     

Dynamic modeling of lactic acid fermentation metabolism with Lactococcus lactis

A dynamic model of lactic acid fermentation using Lactococcus lactis was constructed, and a metabolic flux analysis (MFA) and metabolic control analysis (MCA) were performed to reveal an intensive metabolic understanding of lactic acid bacteria (LAB). The parameter estimation was conducted with COPASI software to construct a more accurate metabolic model. The experimental data used in the parameter estimation were obtained from an LC-MS/ MS analysis and time-course simulation study. The MFA results were a reasonable explanation of the experimental data. Through the parameter estimation, the metabolic system of lactic acid bacteria can be thoroughly understood through comparisons with the original parameters. The coefficients derived from the MCA indicated that the reaction rate of L-lactate dehydrogenase was activated by fructose 1,6-bisphosphate and pyruvate, and pyruvate appeared to be a stronger activator of L-lactate dehydrogenase than fructose 1,6-bisphosphate. Additionally, pyruvate acted as an inhibitor to pyruvate kinase and the phosphotransferase system. Glucose 6-phosphate and phosphoenolpyruvate showed activation effects on pyruvate kinase. Hexose transporter was the strongest effector on the flux through L-lactate dehydrogenase. The concentration control coefficient (CCC) showed similar results to the flux control coefficient (FCC).     

Solving the Differential Biochemical Jacobian from Metabolomics Covariance Data

High-throughput molecular analysis has become an integral part in organismal systems biology. In contrast, due to a missing systematic linkage of the data with functional and predictive theoretical models of the underlying metabolic network the understanding of the resulting complex data sets is lacking far behind. Here, we present a biomathematical method addressing this problem by using metabolomics data for the inverse calculation of a biochemical Jacobian matrix, thereby linking computer-based genome-scale metabolic reconstruction and in vivo metabolic dynamics. The incongruity of metabolome coverage by typical metabolite profiling approaches and genome-scale metabolic reconstruction was solved by the design of superpathways to define a metabolic interaction matrix. A differential biochemical Jacobian was calculated using an approach which links this metabolic interaction matrix and the covariance of metabolomics data satisfying a Lyapunov equation. The predictions of the differential Jacobian from real metabolomic data were found to be correct by testing the corresponding enzymatic activities. Moreover it is demonstrated that the predictions of the biochemical Jacobian matrix allow for the design of parameter optimization strategies for ODE-based kinetic models of the system. The presented concept combines dynamic modelling strategies with large-scale steady state profiling approaches without the explicit knowledge of individual kinetic parameters. In summary, the presented strategy allows for the identification of regulatory key processes in the biochemical network directly from metabolomics data and is a fundamental achievement for the functional interpretation of metabolomics data.     

RankAggreg, an R package for weighted rank aggregation

Background

Determining the parameters of a mathematical model from quantitative measurements is the main bottleneck of modelling biological systems. Parameter values can be estimated from steady-state data or from dynamic data. The nature of suitable data for these two types of estimation is rather different. For instance, estimations of parameter values in pathway models, such as kinetic orders, rate constants, flux control coefficients or elasticities, from steady-state data are generally based on experiments that measure how a biochemical system responds to small perturbations around the steady state. In contrast, parameter estimation from dynamic data requires time series measurements for all dependent variables. Almost no literature has so far discussed the combined use of both steady-state and transient data for estimating parameter values of biochemical systems.

Results

In this study we introduce a constrained optimization method for estimating parameter values of biochemical pathway models using steady-state information and transient measurements. The constraints are derived from the flux connectivity relationships of the system at the steady state. Two case studies demonstrate the estimation results with and without flux connectivity constraints. The unconstrained optimal estimates from dynamic data may fit the experiments well, but they do not necessarily maintain the connectivity relationships. As a consequence, individual fluxes may be misrepresented, which may cause problems in later extrapolations. By contrast, the constrained estimation accounting for flux connectivity information reduces this misrepresentation and thereby yields improved model parameters.

Conclusion

The method combines transient metabolic profiles and steady-state information and leads to the formulation of an inverse parameter estimation task as a constrained optimization problem. Parameter estimation and model selection are simultaneously carried out on the constrained optimization problem and yield realistic model parameters that are more likely to hold up in extrapolations with the model.     

Cybernetic models based on lumped elementary modes accurately predict strain‐specific metabolic function

In a recent article, Song and Ramkrishna (Song and Ramkrishna [2010]. Biotechnol Bioeng 106(2):271–284) proposed a lumped hybrid cybernetic model (L‐HCM) towards extracting maximum information about metabolic function from a minimum of data. This approach views the total uptake flux as distributed among lumped elementary modes (L‐EMs) so as to maximize a prescribed metabolic objective such as growth or uptake rate. L‐EM is computed as a weighted average of EMs where the weights are related to the yields of vital products (i.e., biomass and ATP). In this article, we further enhance the predictive power of L‐HCMs through modifications in lumping weights with additional parameters that can be tuned with data viewed to be critical. The resulting model is able to make predictions of diverse metabolic behaviors varying greatly with strain types as evidenced from case studies of anaerobic growth of various Escherichia coli strains. Incorporation of the new lumping formula into L‐HCM remarkably improves model predictions with a few critical data, thus presenting L‐HCM as a dynamic tool as being not only qualitatively correct but also quantitatively accurate. Biotechnol. Bioeng. 2011; 108:127–140. © 2010 Wiley Periodicals, Inc.     

A Method for Estimating the Predictive Power in a Model of a Biological System with Low Sensitivity to Parameters

The ambiguity of parameter estimates for the model of a biological system may be due to low sensitivity of the model to perturbations of input data (parameters), which mathematically reflects biological mechanisms of robustness. We developed a novel method for estimating the predictive power of a model with the ambiguity of parameter estimates. The predictions are understood as a correct reproduction of the system behavior by the model when changing input data and parameters. The method is based on the relative sensitivity analysis of the fitted model to stiff parameters of the predicted model. The application principles of our approach are demonstrated using a model for the formation of the mRNA expression pattern of the hb gene in the Drosophila embryo and its ability to predict the hb pattern in the Kr null mutant. The nonlinear nature of the system is simulated by a saturating sigmoid function, which is the cause of low sensitivity. Our method allows us to estimate the predictive power of the model and uncover the causes of poor predictions, as well as choose the relevant level of the model detail in terms of predictions.     

Parameter estimation in a generalized discrete-time model of density dependence

Flexible discrete-time per-capita-growth-rate models accommodating a variety of density-dependent relationships offer parsimonious explanations for the variation of population abundance through time. However, the accuracy of standard approaches to parameter estimation and confidence interval construction for such models has not been explored in a generalized setting or with consideration of limited sample sizes typical for ecology. Here, we use simulated data to quantify the relative effects of sample size, population perturbations, and environmental stochasticity on statistical inference. We focus on the key parameters that inform population dynamic predictions in a generalized Beverton–Holt model. We find that reliable parameter estimation requires data spanning ranges where both low and high density dependence act. However, the asymptotic distribution of the likelihood ratio test statistic can be fairly accurate for constructing confidence regions even when point estimation is poor. Consideration of the joint profile likelihood surface is shown to be useful for assessing reliability of point estimates and dynamical population predictions. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Predicting invasions in Australia by a Neotropical shrub under climate change: the challenge of novel climates and parameter estimation

Aim  To test how well species distributions and abundance can be predicted following invasion and climate change when using only species distribution and abundance data to estimate parameters.
Location  Models were developed for the species' native range in the Americas and applied to Australia.
Methods  We developed a predictive model for an invasive neotropical shrub ( Parkinsonia aculeata) using a popular ecophysiological bioclimatic modelling technique (CLIMEX) fitted against distribution and abundance data in the Americas. The effect of uncertainty in model parameter estimates on predictions in Australia was tested. Alternative data sources were used when model predictions were sensitive to uncertainty in parameter estimates. The resulting best-fit model was run under two climate change scenarios.
Results  Of the 19 parameters used, 9 could not be fitted using data from the native range. However, only parameters that lowered temperature or increased moisture requirements for growth noticeably altered the model prediction in Australia. Differences in predictions were dramatic, and reflect climates in Australia that were not represented in the Americas (novel climates). However, these poorly fitted parameters could be fitted post hoc using alternative data sources prior to predicting responses to climate change.
Conclusions  Novel climates prevented the development of a predictive model which relied only on native-range distribution and abundance data because certain parameters could not be fitted. In fact, predictions were more sensitive to parameter uncertainty than to climate change scenarios. Where uncertainty in parameter estimates affected predictions, it could be addressed through the inclusion of alternative data sources. However, this may not always be possible, for example in the absence of post-invasion data.     

Reducing the allowable kinetic space by constructing ensemble of dynamic models with the same steady-state flux

Dynamic models of metabolism are instrumental for gaining insight and predicting possible outcomes of perturbations. Current approaches start from the selection of lumped enzyme kinetics and determine the parameters within a large parametric space. However, kinetic parameters are often unknown and obtaining these parameters requires detailed characterization of enzyme kinetics. In many cases, only steady-state fluxes are measured or estimated, but these data have not been utilized to construct dynamic models. Here, we extend the previously developed Ensemble Modeling methodology by allowing various kinetic rate expressions and employing a more efficient solution method for steady states. We show that anchoring the dynamic models to the same flux reduces the allowable parameter space significantly such that sampling of high dimensional kinetic parameters becomes meaningful. The methodology enables examination of the properties of the model's structure, including multiple steady states. Screening of models based on limited steady-state fluxes or metabolite profiles reduces the parameter space further and the remaining models become increasingly predictive. We use both succinate overproduction and central carbon metabolism in Escherichia coli as examples to demonstrate these results.     

Dynamic flux balance analysis of diauxic growth in Escherichia coli

Flux Balance Analysis (FBA) has been used in the past to analyze microbial metabolic networks. Typically, FBA is used to study the metabolic flux at a particular steady state of the system. However, there are many situations where the reprogramming of the metabolic network is important. Therefore, the dynamics of these metabolic networks have to be studied. In this paper, we have extended FBA to account for dynamics and present two different formulations for dynamic FBA. These two approaches were used in the analysis of diauxic growth in Escherichia coli. Dynamic FBA was used to simulate the batch growth of E. coli on glucose, and the predictions were found to qualitatively match experimental data. The dynamic FBA formalism was also used to study the sensitivity to the objective function. It was found that an instantaneous objective function resulted in better predictions than a terminal-type objective function. The constraints that govern the growth at different phases in the batch culture were also identified. Therefore, dynamic FBA provides a framework for analyzing the transience of metabolism due to metabolic reprogramming and for obtaining insights for the design of metabolic networks.     

A small metabolic flux model to identify transient metabolic regulations in Saccharomyces cerevisiae

The understanding of dynamic metabolic regulations is important for physiological studies and strain characterization tasks. The present study combined transient experiments with online metabolic flux analysis (MFA) in order to quantify metabolic regulations, namely carbon catabolite repression of respiration and transient acetic-acid production, in Saccharomyces cerevisiae during aerobic growth on glucose. The aim was to investigate which additional information can be gained from using a small metabolic flux model to study transient growth provoked by shift-up and shift-down experiments, compared to online monitoring alone. The MFA model allowed us to propose new correlations between pathways of the central metabolism. A linear correlation between glycolytic flux and respiratory capacity holds for shift-down and shift-up experiments. This confirmed that respiratory functions were subjected to carbon catabolite repression and suggested that respiratory capacity is controlled by the glycolytic flux rather than the glucose influx. Furthermore, the model showed that control of repression of respiration by the glycolytic flux was a dynamic phenomenon. Co-factor balancing within the MFA model showed that transient acetic-acid production indicated a transient limitation in another part of the central metabolism but not in oxidative phosphorylation. However, at super-critical growth rates and when coupling of anabolism and catabolism is resumed, the limitation shifts to oxidative phosphorylation, with the consequence that ethanol is formed. The online application of small metabolic flux models to transient experiments enhanced the physiological insight into transient growth and opens up the use of transient experiments as an efficient tool to understand dynamic metabolic regulations.     

On the modeling and interpretation of oxygen uptake kinetics from ramp work rate tests

Ramp work rate tests have been used to estimate aerobic parameters in exercise stress testing. Previous studies have suggested an assumption of a linear dynamic system for O2 uptake kinetics. The implication is that model parameters estimated from ramp tests should be similar to those estimated from other dynamic tests. In nine healthy subjects, we found that model parameters used to characterize O2 consumption ramp data were not consistent with those used to characterize step data, when the comparison was made on a subject-to-subject basis. Furthermore the ramp data model parameter values were highly dependent (P less than 0.0001) on the ramp slope. A linear dynamic system interpretation of the ramp data model does not appear to be appropriate, suggesting that caution is needed in the interpretation of ramp data aerobic parameters. The data may be better described by nonlinear or higher order function. Ramp exercise testing is not suitable for assessing dynamic control properties of the cardiorespiratory response to exercise.     

Integrating cybernetic modeling with pathway analysis provides a dynamic, systems-level description of metabolic control

Cybernetic modeling strives to uncover the inbuilt regulatory programs of biological systems and leverage them toward computational prediction of metabolic dynamics. Because of its focus on incorporating the global aims of metabolism, cybernetic modeling provides a systems-oriented approach for describing regulatory inputs and inferring the impact of regulation within biochemical networks. Combining cybernetic control laws with concepts from metabolic pathway analysis has culminated in a systematic strategy for constructing cybernetic models, which was previously lacking. The newly devised framework relies upon the simultaneous application of local controls that maximize the net flux through each elementary flux mode and global controls that modulate the activities of these modes to optimize the overall nutritional state of the cell. The modeling concepts are illustrated using a simple linear pathway and a larger network representing anaerobic E. coli central metabolism. The E. coli model successfully describes the metabolic shift that occurs upon deleting the pta-ackA operon that is responsible for fermentative acetate production. The model also furnishes predictions that are consistent with experimental results obtained from additional knockout strains as well as strains expressing heterologous genes. Because of the stabilizing influence of the included control variables, the resulting cybernetic models are more robust and reliable than their predecessors in simulating the network response to imposed genetic and environmental perturbations.     

Universally sloppy parameter sensitivities in systems biology models

Quantitative computational models play an increasingly important role in modern biology. Such models typically involve many free parameters, and assigning their values is often a substantial obstacle to model development. Directly measuring in vivo biochemical parameters is difficult, and collectively fitting them to other experimental data often yields large parameter uncertainties. Nevertheless, in earlier work we showed in a growth-factor-signaling model that collective fitting could yield well-constrained predictions, even when it left individual parameters very poorly constrained. We also showed that the model had a “sloppy” spectrum of parameter sensitivities, with eigenvalues roughly evenly distributed over many decades. Here we use a collection of models from the literature to test whether such sloppy spectra are common in systems biology. Strikingly, we find that every model we examine has a sloppy spectrum of sensitivities. We also test several consequences of this sloppiness for building predictive models. In particular, sloppiness suggests that collective fits to even large amounts of ideal time-series data will often leave many parameters poorly constrained. Tests over our model collection are consistent with this suggestion. This difficulty with collective fits may seem to argue for direct parameter measurements, but sloppiness also implies that such measurements must be formidably precise and complete to usefully constrain many model predictions. We confirm this implication in our growth-factor-signaling model. Our results suggest that sloppy sensitivity spectra are universal in systems biology models. The prevalence of sloppiness highlights the power of collective fits and suggests that modelers should focus on predictions rather than on parameters.     

A Method to Constrain Genome-Scale Models with 13C Labeling Data

Current limitations in quantitatively predicting biological behavior hinder our efforts to engineer biological systems to produce biofuels and other desired chemicals. Here, we present a new method for calculating metabolic fluxes, key targets in metabolic engineering, that incorporates data from ¹³C labeling experiments and genome-scale models. The data from ¹³C labeling experiments provide strong flux constraints that eliminate the need to assume an evolutionary optimization principle such as the growth rate optimization assumption used in Flux Balance Analysis (FBA). This effective constraining is achieved by making the simple but biologically relevant assumption that flux flows from core to peripheral metabolism and does not flow back. The new method is significantly more robust than FBA with respect to errors in genome-scale model reconstruction. Furthermore, it can provide a comprehensive picture of metabolite balancing and predictions for unmeasured extracellular fluxes as constrained by ¹³C labeling data. A comparison shows that the results of this new method are similar to those found through ¹³C Metabolic Flux Analysis (¹³C MFA) for central carbon metabolism but, additionally, it provides flux estimates for peripheral metabolism. The extra validation gained by matching 48 relative labeling measurements is used to identify where and why several existing COnstraint Based Reconstruction and Analysis (COBRA) flux prediction algorithms fail. We demonstrate how to use this knowledge to refine these methods and improve their predictive capabilities. This method provides a reliable base upon which to improve the design of biological systems.     

Cutaneous heat flux models do not reliably predict metabolic rates of marine mammals

Heat flux models have been used to predict metabolic rates of marine mammals, generally by estimating conductive heat transfer through their blubber layer. Recently, Kvadsheim et al. (1997) found that such models tend to overestimate metabolic rates, and that such errors probably result from the asymmetrical distribution of blubber. This problem may be avoided if reliable estimates of heat flux through the skin of the animals are obtained by using models that combine calculations of conductive heat flux through the skin and fur, and convective heat flux from the surface of the animal to the environment. We evaluated this approach based on simultaneous measurements of metabolic rates and of input parameters necessary for heat flux calculations, as obtained from four harp seals (Phoca groenlandica) resting in cold water. Heat flux estimates were made using two free convection models (double-flat-plate and cylindrical geometry) and one forced convection model (single-flat-plate geometry). We found that heat flux estimates generally underestimated metabolic rates, on average by 26-58%, and that small variations in input parameters caused large variations in these estimates. We conclude that cutaneous heat flux models are too inaccurate and sensitive to small errors in input parameters to provide reliable estimates of metabolic rates of marine mammals.     

From Enzyme Kinetics to Metabolic Network Modeling – Visualization Tool for Enhanced Kinetic Analysis of Biochemical Network Models

Model‐based analysis of enzyme kinetics allows the determination of optimal conditions for their use in biocatalysis. For biotransformations or fermentative approaches the modeling of metabolic pathways or complex metabolic networks is necessary to obtain model‐based predictions of steps which limit product formation within the network. To set up adequate kinetic models, relevant mechanistic information about enzyme properties is required and can be taken from in vitro studies with isolated enzymes or from in vivo investigations using stimulus‐response experiments which provide a lot of kinetic information about the metabolic network. But with increasing number of reaction steps and regulatory interdependencies in the network structure the amount of simulation data dramatically increases and the simulation results from the dynamic models become difficult to analyze and interpret. Demonstrated for an Escherichia coli model of the central carbon metabolism, methods for visualization and animation of simulation data were applied and extended to facilitate model analysis and biological interpretation. The dynamic metabolite pool and metabolic flux changes were visualized simultaneously by a software tool. In addition, a new quantification method for enzyme activation/inhibition was proposed, and this information was implemented in the metabolic visualization.