Dynamic Flux Estimation Research Articles

A new strategy for dynamic metabolic flux estimation by integrating transient metabolome data into genome-scale metabolic models.

Metabolic flux analysis (MFA) is a powerful tool for studying microbial cell physiology. Isotope-based MFA is the accepted standard for studying metabolic fluxes under steady-state conditions. However, its application under dynamic extracellular conditions is limited due to lack of proper techniques, such as rapid sampling and quenching, high cost and laborious execution. Here, we propose a new strategy to tackle this through incorporating dynamic metabolite abundance data into genome-scale metabolic models (GEM). First, a dummy extracellular pool concept is proposed for each dynamically changing metabolite, which represents a "sink" or "source", with corresponding dummy reactions coded into the GEM model. The dynamic model (expressed as differential equations) is then transformed into a quasi-steady-state model (expressed as linear equations), which can be easily solved by constraining the GEM model with the dynamic metabolite quantification data. For this, common linear-programming optimization algorithms were utilized to estimate the dynamic fluxes. Dynamic high-accuracy metabolite abundance data were obtained through the Isotope Dilution Mass Spectrometry (IDMS) method and high-speed sampling-quenching, and it was demonstrated that the newly proposed strategy could be successfully applied to obtain intracellular dynamic fluxes of Aspergillus niger under regimes of single and periodic extracellular glucose pulses. The applicability of the new method was also tested on dynamic fluxes estimation in a glucose pulse-response study of Saccharomyces cerevisiae. The proposed method provides a powerful tool to investigate cell physiology under dynamic conditions, especially relevant for bioprocess scale-up to industrial-scale bioreactors.

LK-DFBA: a linear programming-based modeling strategy for capturing dynamics and metabolite-dependent regulation in metabolism

BackgroundThe systems-scale analysis of cellular metabolites, “metabolomics,” provides data ideal for applications in metabolic engineering. However, many of the computational tools for strain design are built around Flux Balance Analysis (FBA), which makes assumptions that preclude direct integration of metabolomics data into the underlying models. Finding a way to retain the advantages of FBA’s linear structure while relaxing some of its assumptions could allow us to account for metabolite levels and metabolite-dependent regulation in strain design tools built from FBA, improving the accuracy of predictions made by these tools. We designed, implemented, and characterized a modeling strategy based on Dynamic FBA (DFBA), called Linear Kinetics-Dynamic Flux Balance Analysis (LK-DFBA), to satisfy these specifications. Our strategy adds constraints describing the dynamics and regulation of metabolism that are strictly linear. We evaluated LK-DFBA against alternative modeling frameworks using simulated noisy data from a small in silico model and a larger model of central carbon metabolism in E. coli, and compared each framework’s ability to recapitulate the original system.ResultsIn the smaller model, we found that we could use regression from a dynamic flux estimation (DFE) with an optional non-linear parameter optimization to reproduce metabolite concentration dynamic trends more effectively than an ordinary differential equation model with generalized mass action rate laws when tested under realistic data sampling frequency and noise levels. We observed detrimental effects across all tested modeling approaches when metabolite time course data were missing, but found these effects to be smaller for LK-DFBA in most cases. With the E. coli model, we produced qualitatively reasonable results with similar properties to the smaller model and explored two different parameterization structures that yield trade-offs in computation time and accuracy.ConclusionsLK-DFBA allows for calculation of metabolite concentrations and considers metabolite-dependent regulation while still retaining many computational advantages of FBA. This provides the proof-of-principle for a new metabolic modeling framework with the potential to create genome-scale dynamic models and the potential to be applied in strain engineering tools that currently use FBA.

Identification of microbiota dynamics using robust parameter estimation methods

The compositions of in-host microbial communities (microbiota) play a significant role in host health, and a better understanding of the microbiota’s role in a host’s transition from health to disease or vice versa could lead to novel medical treatments. One of the first steps toward this understanding is modeling interaction dynamics of the microbiota, which can be exceedingly challenging given the complexity of the dynamics and difficulties in collecting sufficient data. Methods such as principal differential analysis, dynamic flux estimation, and others have been developed to overcome these challenges. Despite their advantages, these methods are still vastly underutilized in fields such as mathematical biology, and one potential reason for this is their sophisticated implementation. While this paper focuses on applying principal differential analysis to microbiota data, we also provide comprehensive details regarding the derivation and numerics of this method and include a functional implementation for readers’ benefit. For further validation of these methods, we demonstrate the feasibility of principal differential analysis using simulation studies and then apply the method to intestinal and vaginal microbiota data. In working with these data, we capture experimentally confirmed dynamics while also revealing potential new insights into the system dynamics.

Nonparametric dynamic modeling

Challenging as it typically is, the estimation of parameter values seems to be an unavoidable step in the design and implementation of any dynamic model. Here, we demonstrate that it is possible to set up, diagnose, and simulate dynamic models without the need to estimate parameter values, if the situation is favorable. Specifically, it is possible to establish nonparametric models for nonlinear compartment models, including metabolic pathway models, if sufficiently many high-quality time series data are available that describe the biological phenomenon under investigation in an appropriate and representative manner. The proposed nonparametric strategy is a variant of the method of Dynamic Flux Estimation (DFE), which permits the estimation of numerical flux profiles from metabolic time series data. However, instead of attempting to formulate these numerical profiles as explicit functions and to optimize their parameter values, as it is done in DFE, the metabolite and flux profiles are used here directly as a scaffold for a library from which values are interpolated and retrieved for the simulation of the differential equations describing the model. Beyond simulations, the proposed methods render it possible to determine steady states from non-steady state data, perform sensitivity analyses, and estimate the Jacobian of the system at a steady state.

Identification of Metabolic Pathway Systems.

The estimation of parameters in even moderately large biological systems is a significant challenge. This challenge is greatly exacerbated if the mathematical formats of appropriate process descriptions are unknown. To address this challenge, the method of dynamic flux estimation (DFE) was proposed for the analysis of metabolic time series data. Under ideal conditions, the first phase of DFE yields numerical representations of all fluxes within a metabolic pathway system, either as values at each time point or as plots against their substrates and modulators. However, this numerical result does not reveal the mathematical format of each flux. Thus, the second phase of DFE selects functional formats that are consistent with the numerical trends obtained from the first phase. While greatly facilitating metabolic data analysis, DFE is only directly applicable if the pathway system contains as many dependent variables as fluxes. Because most actual systems contain more fluxes than metabolite pools, this requirement is seldom satisfied. Auxiliary methods have been proposed to alleviate this issue, but they are not general. Here we propose strategies that extend DFE toward general, slightly underdetermined pathway systems.

New insights into the complex regulation of the glycolytic pathway in Lactococcus lactis. I. Construction and diagnosis of a comprehensive dynamic model

This article and the companion paper use computational systems modeling to decipher the complex coordination of regulatory signals controlling the glycolytic pathway in the dairy bacterium Lactococcus lactis. In this first article, the development of a comprehensive kinetic dynamic model is described. The model is based on in vivo NMR data that consist of concentration trends in key glycolytic metabolites and cofactors. The model structure and parameter values are identified with a customized optimization strategy that uses as its core the method of dynamic flux estimation. For the first time, a dynamic model with a single parameter set fits all available glycolytic time course data under anaerobic operation. The model captures observations that had not been addressed so far and suggests the existence of regulatory effects that had been observed in other species, but not in L. lactis. The companion paper uses this model to analyze details of the dynamic control of glycolysis under aerobic and anaerobic conditions.

Effective Estimation of Dynamic Metabolic Fluxes Using (13)C Labeling and Piecewise Affine Approximation: From Theory to Practical Applicability.

The design of microbial production processes relies on rational choices for metabolic engineering of the production host and the process conditions. These require a systematic and quantitative understanding of cellular regulation. Therefore, a novel method for dynamic flux identification using quantitative metabolomics and 13C labeling to identify piecewise-affine (PWA) flux functions has been described recently. Obtaining flux estimates nevertheless still required frequent manual reinitalization to obtain a good reproduction of the experimental data and, moreover, did not optimize on all observables simultaneously (metabolites and isotopomer concentrations). In our contribution we focus on measures to achieve faster and robust dynamic flux estimation which leads to a high dimensional parameter estimation problem. Specifically, we address the following challenges within the PWA problem formulation: (1) Fast selection of sufficient domains for the PWA flux functions, (2) Control of over-fitting in the concentration space using shape-prescriptive modeling and (3) robust and efficient implementation of the parameter estimation using the hybrid implicit filtering algorithm. With the improvements we significantly speed up the convergence by efficiently exploiting that the optimization problem is partly linear. This allows application to larger-scale metabolic networks and demonstrates that the proposed approach is not purely theoretical, but also applicable in practice.

Derivative processes for modelling metabolic fluxes

Motivation: One of the challenging questions in modelling biological systems is to characterize the functional forms of the processes that control and orchestrate molecular and cellular phenotypes. Recently proposed methods for the analysis of metabolic pathways, for example, dynamic flux estimation, can only provide estimates of the underlying fluxes at discrete time points but fail to capture the complete temporal behaviour. To describe the dynamic variation of the fluxes, we additionally require the assumption of specific functional forms that can capture the temporal behaviour. However, it also remains unclear how to address the noise which might be present in experimentally measured metabolite concentrations.Results: Here we propose a novel approach to modelling metabolic fluxes: derivative processes that are based on multiple-output Gaussian processes (MGPs), which are a flexible non-parametric Bayesian modelling technique. The main advantages that follow from MGPs approach include the natural non-parametric representation of the fluxes and ability to impute the missing data in between the measurements. Our derivative process approach allows us to model changes in metabolite derivative concentrations and to characterize the temporal behaviour of metabolic fluxes from time course data. Because the derivative of a Gaussian process is itself a Gaussian process, we can readily link metabolite concentrations to metabolic fluxes and vice versa. Here we discuss how this can be implemented in an MGP framework and illustrate its application to simple models, including nitrogen metabolism in Escherichia coli.Availability and implementation: R code is available from the authors upon request.Contact: j.norkunaite@imperial.ac.uk; m.stumpf@imperial.ac.ukSupplementary information: Supplementary data are available at Bioinformatics online.

Characterizability of metabolic pathway systems from time series data

Over the past decade, the biomathematical community has devoted substantial effort to the complicated challenge of estimating parameter values for biological systems models. An even more difficult issue is the characterization of functional forms for the processes that govern these systems. Most parameter estimation approaches tacitly assume that these forms are known or can be assumed with some validity. However, this assumption is not always true. The recently proposed method of Dynamic Flux Estimation (DFE) addresses this problem in a genuinely novel fashion for metabolic pathway systems. Specifically, DFE allows the characterization of fluxes within such systems through an analysis of metabolic time series data. Its main drawback is the fact that DFE can only directly be applied if the pathway system contains as many metabolites as unknown fluxes. This situation is unfortunately rare. To overcome this roadblock, earlier work in this field had proposed strategies for augmenting the set of unknown fluxes with independent kinetic information, which however is not always available. Employing Moore–Penrose pseudo-inverse methods of linear algebra, the present article discusses an approach for characterizing fluxes from metabolic time series data that is applicable even if the pathway system is underdetermined and contains more fluxes than metabolites. Intriguingly, this approach is independent of a specific modeling framework and unaffected by noise in the experimental time series data. The results reveal whether any fluxes may be characterized and, if so, which subset is characterizable. They also help with the identification of fluxes that, if they could be determined independently, would allow the application of DFE.

Incremental parameter estimation of kinetic metabolic network models

BackgroundAn efficient and reliable parameter estimation method is essential for the creation of biological models using ordinary differential equation (ODE). Most of the existing estimation methods involve finding the global minimum of data fitting residuals over the entire parameter space simultaneously. Unfortunately, the associated computational requirement often becomes prohibitively high due to the large number of parameters and the lack of complete parameter identifiability (i.e. not all parameters can be uniquely identified).ResultsIn this work, an incremental approach was applied to the parameter estimation of ODE models from concentration time profiles. Particularly, the method was developed to address a commonly encountered circumstance in the modeling of metabolic networks, where the number of metabolic fluxes (reaction rates) exceeds that of metabolites (chemical species). Here, the minimization of model residuals was performed over a subset of the parameter space that is associated with the degrees of freedom in the dynamic flux estimation from the concentration time-slopes. The efficacy of this method was demonstrated using two generalized mass action (GMA) models, where the method significantly outperformed single-step estimations. In addition, an extension of the estimation method to handle missing data is also presented.ConclusionsThe proposed incremental estimation method is able to tackle the issue on the lack of complete parameter identifiability and to significantly reduce the computational efforts in estimating model parameters, which will facilitate kinetic modeling of genome-scale cellular metabolism in the future.

Estimation of dynamic flux profiles from metabolic time series data

BackgroundAdvances in modern high-throughput techniques of molecular biology have enabled top-down approaches for the estimation of parameter values in metabolic systems, based on time series data. Special among them is the recent method of dynamic flux estimation (DFE), which uses such data not only for parameter estimation but also for the identification of functional forms of the processes governing a metabolic system. DFE furthermore provides diagnostic tools for the evaluation of model validity and of the quality of a model fit beyond residual errors. Unfortunately, DFE works only when the data are more or less complete and the system contains as many independent fluxes as metabolites. These drawbacks may be ameliorated with other types of estimation and information. However, such supplementations incur their own limitations. In particular, assumptions must be made regarding the functional forms of some processes and detailed kinetic information must be available, in addition to the time series data.ResultsThe authors propose here a systematic approach that supplements DFE and overcomes some of its shortcomings. Like DFE, the approach is model-free and requires only minimal assumptions. If sufficient time series data are available, the approach allows the determination of a subset of fluxes that enables the subsequent applicability of DFE to the rest of the flux system. The authors demonstrate the procedure with three artificial pathway systems exhibiting distinct characteristics and with actual data of the trehalose pathway in Saccharomyces cerevisiae.ConclusionsThe results demonstrate that the proposed method successfully complements DFE under various situations and without a priori assumptions regarding the model representation. The proposed method also permits an examination of whether at all, to what degree, or within what range the available time series data can be validly represented in a particular functional format of a flux within a pathway system. Based on these results, further experiments may be designed to generate data points that genuinely add new information to the structure identification and parameter estimation tasks at hand.

Automated refinement and inference of analytical models for metabolic networks

The reverse engineering of metabolic networks from experimental data is traditionally a labor-intensive task requiring a priori systems knowledge. Using a proven model as a test system, we demonstrate an automated method to simplify this process by modifying an existing or related model-–suggesting nonlinear terms and structural modifications–-or even constructing a new model that agrees with the system's time series observations. In certain cases, this method can identify the full dynamical model from scratch without prior knowledge or structural assumptions. The algorithm selects between multiple candidate models by designing experiments to make their predictions disagree. We performed computational experiments to analyze a nonlinear seven-dimensional model of yeast glycolytic oscillations. This approach corrected mistakes reliably in both approximated and overspecified models. The method performed well to high levels of noise for most states, could identify the correct model de novo, and make better predictions than ordinary parametric regression and neural network models. We identified an invariant quantity in the model, which accurately derived kinetics and the numerical sensitivity coefficients of the system. Finally, we compared the system to dynamic flux estimation and discussed the scaling and application of this methodology to automated experiment design and control in biological systems in real time.

Automated piecewise power-law modeling of biological systems

Recent trends suggest that future biotechnology will increasingly rely on mathematical models of the biological systems under investigation. In particular, metabolic engineering will make wider use of metabolic pathway models in stoichiometric or fully kinetic format. A significant obstacle to the use of pathway models is the identification of suitable process descriptions and their parameters. We recently showed that, at least under favorable conditions, Dynamic Flux Estimation (DFE) permits the numerical characterization of fluxes from sets of metabolic time series data. However, DFE does not prescribe how to convert these numerical results into functional representations. In some cases, Michaelis–Menten rate laws or canonical formats are well suited, in which case the estimation of parameter values is easy. However, in other cases, appropriate functional forms are not evident, and exhaustive searches among all possible candidate models are not feasible. We show here how piecewise power-law functions of one or more variables offer an effective default solution for the almost unbiased representation of uni- and multivariate time series data. The results of an automated algorithm for their determination are piecewise power-law fits, whose accuracy is only limited by the available data. The individual power-law pieces may lead to discontinuities at break points or boundaries between sub-domains. In many practical applications, these boundary gaps do not cause problems. Potential smoothing techniques, based on differential inclusions and Filippov's theory, are discussed in Appendix A.

Estimation of metabolic pathway systems from different data sources

Parameter estimation is the main bottleneck of metabolic pathway modelling. It may be addressed from the bottom up, using information on metabolites, enzymes and modulators, or from the top down, using metabolic time series data, which have become more prevalent in recent years. The authors propose here that it is useful to combine the two strategies and to complement time-series analysis with kinetic information. In particular, the authors investigate how the recent method of dynamic flux estimation (DFE) may be supplemented with other types of estimation. Using the glycolytic pathway in Lactococcus lactis as an illustration example, the authors demonstrate some strategies of such supplementation.

System estimation from metabolic time-series data.

At the center of computational systems biology are mathematical models that capture the dynamics of biological systems and offer novel insights. The bottleneck in the construction of these models is presently the identification of model parameters that make the model consistent with observed data. Dynamic flux estimation (DFE) is a novel methodological framework for estimating parameters for models of metabolic systems from time-series data. DFE consists of two distinct phases, an entirely model-free and assumption-free data analysis and a model-based mathematical characterization of process representations. The model-free phase reveals inconsistencies within the data, and between data and the alleged system topology, while the model-based phase allows quantitative diagnostics of whether--or to what degree--the assumed mathematical formulations are appropriate or in need of improvement. Hallmarks of DFE are the facility to: diagnose data and model consistency; circumvent undue compensation of errors; determine functional representations of fluxes uncontaminated by errors in other fluxes and pinpoint sources of remaining errors. Our results suggest that the proposed approach is more effective and robust than presently available methods for deriving metabolic models from time-series data. Its avoidance of error compensation among process descriptions promises significantly improved extrapolability toward new data or experimental conditions.