System estimation from metabolic time-series data

Insights into yeast response to chemotherapeutic agent through time series genome-scale metabolic models

Organism-specific genome-scale metabolic models (GSMMs) can unveil molecular mechanisms within cells and are commonly used in diverse applications, from synthetic biology, biotechnology, and systems biology to metabolic engineering. There are limited studies incorporating time-series transcriptomics in GSMM simulations. Yeast is an easy-to-manipulate model organism for tumor research. Here, a novel approach (TS-GSMM) was proposed to integrate time-series transcriptomics with GSMMs to narrow down the feasible solution space of all possible flux distributions and attain time-series flux samples. The flux samples were clustered using machine learning techniques, and the clusters' functional analysis was performed using reaction set enrichment analysis. A time series transcriptomics response of Yeast cells to a chemotherapeutic reagent-doxorubicin-was mapped onto a Yeast GSMM. Eleven flux clusters were obtained with our approach, and pathway dynamics were displayed. Induction of fluxes related to bicarbonate formation and transport, ergosterol and spermidine transport, and ATP production were captured. Integrating time-series transcriptomics data with GSMMs is a promising approach to reveal pathway dynamics without any kinetic modeling and detects pathways that cannot be identified through transcriptomics-only analysis. The codes are available at https://github.com/karabekmez/TS-GSMM.

Surrogate modeling and control of medical digital twins

The vision of personalized medicine is to identify interventions that maintain or restore a person's health based on their individual biology. Medical digital twins, computational models that integrate a wide range of health-related data about a person and can be dynamically updated, are a key technology that can help guide medical decisions. Such medical digital twin models can be high-dimensional, multi-scale, and stochastic. To be practical for healthcare applications, they often need to be simplified into low-dimensional surrogate models that can be used for optimal design of interventions. This paper introduces surrogate modeling algorithms for the purpose of optimal control applications. As a use case, we focus on agent-based models (ABMs), a common model type in biomedicine for which there are no readily available optimal control algorithms. By deriving surrogate models that are based on systems of ordinary differential equations, we show how optimal control methods can be employed to compute effective interventions, which can then be lifted back to a given ABM. The relevance of the methods introduced here extends beyond medical digital twins to other complex dynamical systems.

Towards inferring absolute concentrations from relative abundance in time-course GC-MS metabolomics data

Metabolomics, the large-scale study of metabolites, has significant appeal as a source of information for metabolic modeling and other scientific applications. One common approach for measuring metabolomics data is gas chromatography-mass spectrometry (GC-MS). However, GC-MS metabolomics data are typically reported as relative abundances, precluding their use with approaches and tools where absolute concentrations are necessary. While chemical standards can be used to help provide quantification, their use is time-consuming, expensive, or even impossible due to their limited availability. The ability to infer absolute concentrations from GC-MS metabolomics data without chemical standards would have significant value. We hypothesized that when analyzing time-course metabolomics datasets, the mass balances of metabolism and other biological information could provide sufficient information towards inference of absolute concentrations. To demonstrate this, we developed and characterized MetaboPAC, a computational framework that uses two approaches-one based on kinetic equations and another using biological heuristics-to predict the most likely response factors that allow translation between relative abundances and absolute concentrations. When used to analyze noiseless synthetic data generated from multiple types of kinetic rate laws, MetaboPAC performs significantly better than negative control approaches when 20% of kinetic terms are known a priori. Under conditions of lower sampling frequency and high noise, MetaboPAC is still able to provide significant inference of concentrations in 3 of 4 models studied. This provides a starting point for leveraging biological knowledge to extract concentration information from time-course intracellular GC-MS metabolomics datasets, particularly for systems that are well-studied and have partially known kinetic structures.

Bayesian Inference of Stochastic Dynamic Models Using Early-Rejection Methods Based on Sequential Stochastic Simulations

Stochastic modelling is an important method to investigate the functions of noise in a wide range of biological systems. However, the parameter inference for stochastic models is still a challenging problem partially due to the large computing time required for stochastic simulations. To address this issue, we propose a novel early-rejection method by using sequential stochastic simulations. We first show that a large number of stochastic simulations are required to obtain reliable inference results. Instead of generating a large number of simulations for each parameter sample, we propose to generate these simulations in a number of stages. The simulation process will go to the next stage only if the accuracy of simulations at the current stage satisfies a given error criterion. We propose a formula to determine the error criterion and use a stochastic differential equation model to examine the effects of different criteria. Three biochemical network models are used to evaluate the efficiency and accuracy of the proposed method. Numerical results suggest the proposed early-rejection method achieves substantial improvement in the efficiency for the inference of stochastic models.

Diverse classes of constraints enable broader applicability of a linear programming-based dynamic metabolic modeling framework

Current metabolic modeling tools suffer from a variety of limitations, from scalability to simplifying assumptions, that preclude their use in many applications. We recently created a modeling framework, Linear Kinetics-Dynamic Flux Balance Analysis (LK-DFBA), that addresses a key gap: capturing metabolite dynamics and regulation while retaining a potentially scalable linear programming structure. Key to this framework's success are the linear kinetics and regulatory constraints imposed on the system. However, while the linearity of these constraints reduces computational complexity, it may not accurately capture the behavior of many biochemical systems. Here, we developed three new classes of LK-DFBA constraints to better model interactions between metabolites and the reactions they regulate. We tested these new approaches on several synthetic and biological systems, and also performed the first-ever comparison of LK-DFBA predictions to experimental data. We found that no single constraint approach was optimal across all systems examined, and systems with the same topological structure but different parameters were often best modeled by different types of constraints. However, we did find that when genetic perturbations were implemented in the systems, the optimal constraint approach typically remained the same as for the wild-type regardless of the model topology or parameterization, indicating that just a single wild-type dataset could allow identification of the ideal constraint to enable model predictivity for a given system. These results suggest that the availability of multiple constraint approaches will allow LK-DFBA to model a wider range of metabolic systems.

Dynamic Flux Balance Analysis to Evaluate the Strain Production Performance on Shikimic Acid Production in Escherichia coli

Flux balance analysis (FBA) is used to improve the microbial production of useful compounds. However, a large gap often exists between the FBA solution and the experimental yield, because of growth and byproducts. FBA has been extended to dynamic FBA (dFBA), which is applicable to time-varying processes, such as batch or fed-batch cultures, and has significantly contributed to metabolic and cultural engineering applications. On the other hand, the performance of the experimental strains has not been fully evaluated. In this study, we applied dFBA to the production of shikimic acid from glucose in Escherichia coli, to evaluate the production performance of the strain as a case study. The experimental data of glucose consumption and cell growth were used as FBA constraints. Bi-level FBA optimization with maximized growth and shikimic acid production were the objective functions. Results suggest that the shikimic acid concentration in the high-shikimic-acid-producing strain constructed in the experiment reached up to 84% of the maximum value by simulation. Thus, this method can be used to evaluate the performance of strains and estimate the milestones of strain improvement.

Separable Nonlinear Least-Squares Parameter Estimation for Complex Dynamic Systems

Nonlinear dynamic models are widely used for characterizing processes that govern complex biological pathway systems. Over the past decade, validation and further development of these models became possible due to data collected via high-throughput experiments using methods from molecular biology. While these data are very beneficial, they are typically incomplete and noisy, which renders the inference of parameter values for complex dynamic models challenging. Fortunately, many biological systems have embedded linear mathematical features, which may be exploited, thereby improving fits and leading to better convergence of optimization algorithms. In this paper, we explore options of inference for dynamic models using a novel method of separable nonlinear least-squares optimization and compare its performance to the traditional nonlinear least-squares method. The numerical results from extensive simulations suggest that the proposed approach is at least as accurate as the traditional nonlinear least-squares, but usually superior, while also enjoying a substantial reduction in computational time.

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

BACKGROUND

The 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.

RESULTS

In 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.

CONCLUSIONS

LK-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.

PARAMETER AND UNCERTAINTY ESTIMATION FOR DYNAMICAL SYSTEMS USING SURROGATE STOCHASTIC PROCESSES

Inference on unknown quantities in dynamical systems via observational data is essential for providing meaningful insight, furnishing accurate predictions, enabling robust control, and establishing appropriate designs for future experiments. Merging mathematical theory with empirical measurements in a statistically coherent way is critical and challenges abound, e.g., ill-posedness of the parameter estimation problem, proper regularization and incorporation of prior knowledge, and computational limitations. To address these issues, we propose a new method for learning parameterized dynamical systems from data. We first customize and fit a surrogate stochastic process directly to observational data, front-loading with statistical learning to respect prior knowledge (e.g., smoothness), cope with challenging data features like heteroskedasticity, heavy tails, and censoring. Then, samples of the stochastic process are used as "surrogate data" and point estimates are computed via ordinary point estimation methods in a modular fashion. Attractive features of this two-step approach include modularity and trivial parallelizability. We demonstrate its advantages on a predator-prey simulation study and on a real-world application involving within-host influenza virus infection data paired with a viral kinetic model, with comparisons to a more conventional Markov chain Monte Carlo (MCMC) based Bayesian approach.

Application of one-step method to parameter estimation in ODE models

In this paper, we study application of Le Cam's one-step method to parameter estimation in ordinary differential equation models. This computationally simple technique can serve as an alternative to numerical evaluation of the popular non-linear least squares estimator, which typically requires the use of a multistep iterative algorithm and repetitive numerical integration of the ordinary differential equation system. The one-step method starts from a preliminary n -consistent estimator of the parameter of interest and next turns it into an asymptotic (as the sample size n→∞) equivalent of the least squares estimator through a numerically straightforward procedure. We demonstrate performance of the one-step estimator via extensive simulations and real data examples. The method enables the researcher to obtain both point and interval estimates. The preliminary n -consistent estimator that we use depends on non-parametric smoothing, and we provide a data-driven methodology for choosing its tuning parameter and support it by theory. An easy implementation scheme of the one-step method for practical use is pointed out.

The three-legged stool of understanding metabolism: integrating metabolomics with biochemical genetics and computational modeling

Traditional biochemical research has resulted in a good understanding of many aspects of metabolism. However, this reductionist approach is time consuming and requires substantial resources, thus raising the question whether modern metabolomics and genomics should take over and replace the targeted experiments of old. We proffer that such a replacement is neither feasible not desirable and propose instead the tight integration of modern, system-wide omics with traditional experimental bench science and dedicated computational approaches. This integration is an important prerequisite toward the optimal acquisition of knowledge regarding metabolism and physiology in health and disease. The commentary describes advantages and drawbacks of current approaches to assessing metabolism and highlights the challenges to be overcome as we strive to achieve a deeper level of metabolic understanding in the future.

The best models of metabolism

Biochemical systems are among of the oldest application areas of mathematical modeling. Spanning a time period of over one hundred years, the repertoire of options for structuring a model and for formulating reactions has been constantly growing, and yet, it is still unclear whether or to what degree some models are better than others and how the modeler is to choose among them. In fact, the variety of options has become overwhelming and difficult to maneuver for novices and experts alike. This review outlines the metabolic model design process and discusses the numerous choices for modeling frameworks and mathematical representations. It tries to be inclusive, even though it cannot be complete, and introduces the various modeling options in a manner that is as unbiased as that is feasible. However, the review does end with personal recommendations for the choices of default models. WIREs Syst Biol Med 2017, 9:e1391. doi: 10.1002/wsbm.1391 For further resources related to this article, please visit the WIREs website.

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.

Engineering Microbial Metabolite Dynamics and Heterogeneity

As yields for biological chemical production in microorganisms approach their theoretical maximum, metabolic engineering requires new tools, and approaches for improvements beyond what traditional strategies can achieve. Engineering metabolite dynamics and metabolite heterogeneity is necessary to achieve further improvements in product titers, productivities, and yields. Metabolite dynamics, the ensemble change in metabolite concentration over time, arise from the need for microbes to adapt their metabolism in response to the extracellular environment and are important for controlling growth and productivity in industrial fermentations. Metabolite heterogeneity, the cell-to-cell variation in a metabolite concentration in an isoclonal population, has a significant impact on ensemble productivity. Recent advances in single cell analysis enable a more complete understanding of the processes driving metabolite heterogeneity and reveal metabolic engineering targets. The authors present an overview of the mechanistic origins of metabolite dynamics and heterogeneity, why they are important, their potential effects in chemical production processes, and tools and strategies for engineering metabolite dynamics and heterogeneity. The authors emphasize that the ability to control metabolite dynamics and heterogeneity will bring new avenues of engineering to increase productivity of microbial strains.

Stepwise inference of likely dynamic flux distributions from metabolic time series data

MOTIVATION

Most metabolic pathways contain more reactions than metabolites and therefore have a wide stoichiometric matrix that corresponds to infinitely many possible flux distributions that are perfectly compatible with the dynamics of the metabolites in a given dataset. This under-determinedness poses a challenge for the quantitative characterization of flux distributions from time series data and thus for the design of adequate, predictive models. Here we propose a method that reduces the degrees of freedom in a stepwise manner and leads to a dynamic flux distribution that is, in a statistical sense, likely to be close to the true distribution.

RESULTS

We applied the proposed method to the lignin biosynthesis pathway in switchgrass. The system consists of 16 metabolites and 23 enzymatic reactions. It has seven degrees of freedom and therefore admits a large space of dynamic flux distributions that all fit a set of metabolic time series data equally well. The proposed method reduces this space in a systematic and biologically reasonable manner and converges to a likely dynamic flux distribution in just a few iterations. The estimated solution and the true flux distribution, which is known in this case, show excellent agreement and thereby lend support to the method.

AVAILABILITY AND IMPLEMENTATION

The computational model was implemented in MATLAB (version R2014a, The MathWorks, Natick, MA). The source code is available at https://github.gatech.edu/VoitLab/Stepwise-Inference-of-Likely-Dynamic-Flux-Distributions and www.bst.bme.gatech.edu/research.php .

CONTACT

mojdeh@gatech.edu or eberhard.voit@bme.gatech.edu.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

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.

Effects of Storage Time on Glycolysis in Donated Human Blood Units

Background: Donated blood is typically stored before transfusions. During storage, the metabolism of red blood cells changes, possibly causing storage lesions. The changes are storage time dependent and exhibit donor-specific variations. It is necessary to uncover and characterize the responsible molecular mechanisms accounting for such biochemical changes, qualitatively and quantitatively; Study Design and Methods: Based on the integration of metabolic time series data, kinetic models, and a stoichiometric model of the glycolytic pathway, a customized inference method was developed and used to quantify the dynamic changes in glycolytic fluxes during the storage of donated blood units. The method provides a proof of principle for the feasibility of inferences regarding flux characteristics from metabolomics data; Results: Several glycolytic reaction steps change substantially during storage time and vary among different fluxes and donors. The quantification of these storage time effects, which are possibly irreversible, allows for predictions of the transfusion outcome of individual blood units; Conclusion: The improved mechanistic understanding of blood storage, obtained from this computational study, may aid the identification of blood units that age quickly or more slowly during storage, and may ultimately improve transfusion management in clinics.

New insights into the complex regulation of the glycolytic pathway in Lactococcus lactis. II. Inference of the precisely timed control system regulating glycolysis

The dairy bacterium Lactococcus lactis has to master a complicated task. It must control its essentially linear glycolytic pathway in such a fashion that, when the substrate, glucose, runs out, it retains enough phosphoenolpyruvate and fructose-1,6-bisphosphate to be able to restart glycolysis as soon as new glucose becomes available. Although glycolysis is arguably the best-studied metabolic pathway, its details in L. lactis are still unclear, and it is, in particular, not understood how the bacterium manages the stop-and-start task. The primary purpose of this paper and its companion is a clarification of some of the details of the governing regulatory strategies with which L. lactis manages to retain the necessary metabolites for the restart of glycolysis after periods of starvation. The paper furthermore discusses how the bacterium changes these strategies when it is subjected to aerobic rather than anaerobic conditions.

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.

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.

Challenges of Inversely Estimating Jacobian from Metabolomics Data

Inferring dynamics of metabolic networks directly from metabolomics data provides a promising way to elucidate the underlying mechanisms of biological systems, as reported in our previous studies (Weckwerth, 2011; Sun and Weckwerth, 2012; Nägele et al., 2014) by a differential Jacobian approach. The Jacobian is solved from an overdetermined system of equations as JC + CJ^T = −2D, called Lyapunov Equation in its generic form,¹ where J is the Jacobian, C is the covariance matrix of metabolomics data, and D is the fluctuation matrix. Lyapunov Equation can be further simplified as the linear form Ax = b. Frequently, this linear equation system is ill-conditioned, i.e., a small variation in the right side b results in a big change in the solution x, thus making the solution unstable and error-prone. At the same time, inaccurate estimation of covariance matrix and uncertainties in the fluctuation matrix bring biases to the solution x. Here, we first reviewed common approaches to circumvent the ill-conditioned problems, including total least squares, Tikhonov regularization, and truncated singular value decomposition. Then, we benchmarked these methods on several in silico kinetic models with small to large perturbations on the covariance and fluctuation matrices. The results identified that the accuracy of the reverse Jacobian is mainly dependent on the condition number of A, the perturbation amplitude of C, and the stiffness of the kinetic models. Our research contributes a systematical comparison of methods to inversely solve Jacobian from metabolomics data.

Design Principles as a Guide for Constraint Based and Dynamic Modeling: Towards an Integrative Workflow

During the last 10 years, systems biology has matured from a fuzzy concept combining omics, mathematical modeling and computers into a scientific field on its own right. In spite of its incredible potential, the multilevel complexity of its objects of study makes it very difficult to establish a reliable connection between data and models. The great number of degrees of freedom often results in situations, where many different models can explain/fit all available datasets. This has resulted in a shift of paradigm from the initially dominant, maybe naive, idea of inferring the system out of a number of datasets to the application of different techniques that reduce the degrees of freedom before any data set is analyzed. There is a wide variety of techniques available, each of them can contribute a piece of the puzzle and include different kinds of experimental information. But the challenge that remains is their meaningful integration. Here we show some theoretical results that enable some of the main modeling approaches to be applied sequentially in a complementary manner, and how this workflow can benefit from evolutionary reasoning to keep the complexity of the problem in check. As a proof of concept, we show how the synergies between these modeling techniques can provide insight into some well studied problems: Ammonia assimilation in bacteria and an unbranched linear pathway with end-product inhibition.

Dynamics of the Heat Stress Response of Ceramides with Different Fatty-Acyl Chain Lengths in Baker's Yeast

The article demonstrates that computational modeling has the capacity to convert metabolic snapshots, taken sequentially over time, into a description of cellular, dynamic strategies. The specific application is a detailed analysis of a set of actions with which Saccharomyces cerevisiae responds to heat stress. Using time dependent metabolic concentration data, we use a combination of mathematical modeling, reverse engineering, and optimization to infer dynamic changes in enzyme activities within the sphingolipid pathway. The details of the sphingolipid responses to heat stress are important, because they guide some of the longer-term alterations in gene expression, with which the cells adapt to the increased temperature. The analysis indicates that all enzyme activities in the system are affected and that the shapes of the time trends in activities depend on the fatty-acyl CoA chain lengths of the different ceramide species in the system.

Improved metabolite profile smoothing for flux estimation

As genome-scale metabolic models become more sophisticated and dynamic, one significant challenge in using these models is to effectively integrate increasingly prevalent systems-scale metabolite profiling data into them. One common data processing step when integrating metabolite data is to smooth experimental time course measurements: the smoothed profiles can be used to estimate metabolite accumulation (derivatives), and thus the flux distribution of the metabolic model. However, this smoothing step is susceptible to the (often significant) noise in experimental measurements, limiting the accuracy of downstream model predictions. Here, we present several improvements to current approaches for smoothing metabolite time course data using defined functions. First, we use a biologically-inspired mathematical model function taken from transcriptional profiling and clustering literature that captures the dynamics of many biologically relevant transient processes. We demonstrate that it is competitive with, and often superior to, previously described fitting schemas, and may serve as an effective single option for data smoothing in metabolic flux applications. We also implement a resampling-based approach to buffer out sensitivity to specific data sets and allow for more accurate fitting of noisy data. We found that this method, as well as the addition of parameter space constraints, yielded improved estimates of concentrations and derivatives (fluxes) in previously described fitting functions. These methods have the potential to improve the accuracy of existing and future dynamic metabolic models by allowing for the more effective integration of metabolite profiling data.

MP-GeneticSynth: inferring biological network regulations from time series

MP-GeneticSynth is a Java tool for discovering the logic and regulation mechanisms responsible for observed biological dynamics in terms of finite difference recurrent equations. The software makes use of: (i) metabolic P systems as a modeling framework, (ii) an evolutionary approach to discover flux regulation functions as linear combinations of given primitive functions, (iii) a suitable reformulation of the least squares method to estimate function parameters considering simultaneously all the reactions involved in complex dynamics. The tool is available as a plugin for the virtual laboratory MetaPlab. It has graphical and interactive interfaces for data preparation, a priori knowledge integration, and flux regulator analysis. Availability and implementation: Source code, binaries, documentation (including quick start guide and videos) and case studies are freely available at http://mplab.sci.univr.it/plugins/mpgs/index.html.

150 years of the mass action law

This year we celebrate the 150th anniversary of the law of mass action. This law is often assumed to have been “there” forever, but it has its own history, background, and a definite starting point. The law has had an impact on chemistry, biochemistry, biomathematics, and systems biology that is difficult to overestimate. It is easily recognized that it is the direct basis for computational enzyme kinetics, ecological systems models, and models for the spread of diseases. The article reviews the explicit and implicit role of the law of mass action in systems biology and reveals how the original, more general formulation of the law emerged one hundred years later ab initio as a very general, canonical representation of biological processes.

Parameter estimation of dynamic biological network models using integrated fluxes

Background

Parameter estimation is often the bottlenecking step in biological system modeling. For ordinary differential equation (ODE) models, the challenge in this estimation has been attributed to not only the lack of parameter identifiability, but also computational issues such as finding globally optimal parameter estimates over highly multidimensional search space. Recent methods using incremental estimation approach could alleviate the computational difficulty by performing the parameter estimation one-reaction-at-a-time. However, incremental estimation strategies usually require data smoothing and are known to produce biased parameter estimates.

Results

In this article, we presented a new parameter estimation method called integrated flux parameter estimation (IFPE). We employed the integral form of the ODE such that we could compute the integral of reaction fluxes from time-series concentration data without data smoothing. Here, we formulated the parameter estimation as a nested optimization problem. In the outer optimization, we performed a minimization of model prediction errors over parameters associated with a subset of reactions labeled as independent. The dimension of the independent reaction subset was equal to the degrees of freedom in the calculation of integrated fluxes (IF) from concentration data. We selected the independent reactions such that given their IF values, the IFs of the remaining (dependent) reactions could be uniquely determined. Meanwhile, in the inner optimization, we estimated the model parameters associated with the dependent reactions, one-reaction-at-a-time, by minimizing the dependent IF prediction errors. We demonstrated the performance of the IFPE method using two case studies: a generalized mass action model of a branched pathway and a lin-log ODE model of Lactococcus lactis glycolytic pathway.

Conclusions

The IFPE significantly outperformed standard simultaneous parameter estimation in terms of computational efficiency and scaling. In comparison to incremental parameter estimation (IPE) method, the IFPE produced parameter estimates with significantly lower bias and did not require time-series data smoothing. The advantages of IFPE over the IPE however came at the cost of a small increase in the computational time.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-014-0127-x) contains supplementary material, which is available to authorized users.

Approximate Bayesian computation schemes for parameter inference of discrete stochastic models using simulated likelihood density

BACKGROUND

Mathematical modeling is an important tool in systems biology to study the dynamic property of complex biological systems. However, one of the major challenges in systems biology is how to infer unknown parameters in mathematical models based on the experimental data sets, in particular, when the data are sparse and the regulatory network is stochastic.

RESULTS

To address this issue, this work proposed a new algorithm to estimate parameters in stochastic models using simulated likelihood density in the framework of approximate Bayesian computation. Two stochastic models were used to demonstrate the efficiency and effectiveness of the proposed method. In addition, we designed another algorithm based on a novel objective function to measure the accuracy of stochastic simulations.

CONCLUSIONS

Simulation results suggest that the usage of simulated likelihood density improves the accuracy of estimates substantially. When the error is measured at each observation time point individually, the estimated parameters have better accuracy than those obtained by a published method in which the error is measured using simulations over the entire observation time period.

Prediction model for glucose metabolism based on lipid metabolism

OBJECTIVES

We developed a robust, long-term clinical prediction model to predict conditions leading to early diabetes using laboratory values other than blood glucose and insulin levels. Our model protects against missing data and noise that occur during long-term analysis.

METHODS

RESULTS of a 75-g oral glucose tolerance test (OGTT) were divided into three groups: diabetes, impaired glucose tolerance (IGT), and normal (n = 114, 235, and 325, respectively). For glucose metabolic and lipid metabolic parameters, near 30-day mean values and 10-year integrated values were compared. The relation between high-density lipoprotein cholesterol (HDL-C) and variations in HbA1c was analyzed in 158 patients. We also constructed a state space model consisting of an observation model (HDL-C and HbA1c) and an internal model (disorders of lipid metabolism and glucose metabolism) and applied this model to 116 cases.

RESULTS

The root mean square error between the observed HbA1c and predicted HbA1c was 0.25.

CONCLUSIONS

In the observation model, HDL-C levels were useful for prediction of increases in HbA1c. Even with numerous missing values over time, as occurs in clinical practice, clinically valid predictions can be made using this state space model.

Identification of crucial parameters in a mathematical multiscale model of glioblastoma growth

Glioblastomas are highly malignant brain tumours. Mathematical models and their analysis provide a tool to support the understanding of the development of these tumours as well as the design of more effective treatment strategies. We have previously developed a multiscale model of glioblastoma progression that covers processes on the cellular and molecular scale. Here, we present a novel nutrient-dependent multiscale sensitivity analysis of this model that helps to identify those reaction parameters of the molecular interaction network that influence the tumour progression on the cellular scale the most. In particular, those parameters are identified that essentially determine tumour expansion and could be therefore used as potential therapy targets. As indicators for the success of a potential therapy target, a deceleration of the tumour expansion and a reduction of the tumour volume are employed. From the results, it can be concluded that no single parameter variation results in a less aggressive tumour. However, it can be shown that a few combined perturbations of two systematically selected parameters cause a slow-down of the tumour expansion velocity accompanied with a decrease of the tumour volume. Those parameters are primarily linked to the reactions that involve the microRNA-451 and the thereof regulated protein MO25.

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.uk

Supplementary 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.

Mesoscopic modeling as a starting point for computational analyses of cystic fibrosis as a systemic disease

Probably the most prominent expectation associated with systems biology is the computational support of personalized medicine and predictive health. At least some of this anticipated support is envisioned in the form of disease simulators that will take hundreds of personalized biomarker data as input and allow the physician to explore and optimize possible treatment regimens on a computer before the best treatment is applied to the actual patient in a custom-tailored manner. The key prerequisites for such simulators are mathematical and computational models that not only manage the input data and implement the general physiological and pathological principles of organ systems but also integrate the myriads of details that affect their functionality to a significant degree. Obviously, the construction of such models is an overwhelming task that suggests the long-term development of hierarchical or telescopic approaches representing the physiology of organs and their diseases, first coarsely and over time with increased granularity. This article illustrates the rudiments of such a strategy in the context of cystic fibrosis (CF) of the lung. The starting point is a very simplistic, generic model of inflammation, which has been shown to capture the principles of infection, trauma, and sepsis surprisingly well. The adaptation of this model to CF contains as variables healthy and damaged cells, as well as different classes of interacting cytokines and infectious microbes that are affected by mucus formation, which is the hallmark symptom of the disease (Perez-Vilar and Boucher, 2004) [1]. The simple model represents the overall dynamics of the disease progression, including so-called acute pulmonary exacerbations, quite well, but of course does not provide much detail regarding the specific processes underlying the disease. In order to launch the next level of modeling with finer granularity, it is desirable to determine which components of the coarse model contribute most to the disease dynamics. The article introduces for this purpose the concept of module gains or ModGains, which quantify the sensitivity of key disease variables in the higher-level system. In reality, these variables represent complex modules at the next level of granularity, and the computation of ModGains therefore allows an importance ranking of variables that should be replaced with more detailed models. The "hot-swapping" of such detailed modules for former variables is greatly facilitated by the architecture and implementation of the overarching, coarse model structure, which is here formulated with methods of biochemical systems theory (BST). This article is part of a Special Issue entitled: Computational Proteomics, Systems Biology & Clinical Implications. Guest Editor: Yudong Cai.

Modeling cellular compartmentation in one-carbon metabolism

Folate-mediated one-carbon metabolism (FOCM) is associated with risk for numerous pathological states including birth defects, cancers, and chronic diseases. Although the enzymes that constitute the biological pathways have been well described and their interdependency through the shared use of folate cofactors appreciated, the biological mechanisms underlying disease etiologies remain elusive. The FOCM network is highly sensitive to nutritional status of several B-vitamins and numerous penetrant gene variants that alter network outputs, but current computational approaches do not fully capture the dynamics and stochastic noise of the system. Combining the stochastic approach with a rule-based representation will help model the intrinsic noise displayed by FOCM, address the limited flexibility of standard simulation methods for coarse-graining the FOCM-associated biochemical processes, and manage the combinatorial complexity emerging from reactions within FOCM that would otherwise be intractable.

Biochemical Systems Theory: A Review

Biochemical systems theory (BST) is the foundation for a set of analytical andmodeling tools that facilitate the analysis of dynamic biological systems. This paper depicts major developments in BST up to the current state of the art in 2012. It discusses its rationale, describes the typical strategies and methods of designing, diagnosing, analyzing, and utilizing BST models, and reviews areas of application. The paper is intended as a guide for investigators entering the fascinating field of biological systems analysis and as a resource for practitioners and experts.

Incremental parameter estimation of kinetic metabolic network models

BACKGROUND

An 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).

RESULTS

In 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.

CONCLUSIONS

The 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.

Ensemble kinetic modeling of metabolic networks from dynamic metabolic profiles

Kinetic modeling of metabolic pathways has important applications in metabolic engineering, but significant challenges still remain. The difficulties faced vary from finding best-fit parameters in a highly multidimensional search space to incomplete parameter identifiability. To meet some of these challenges, an ensemble modeling method is developed for characterizing a subset of kinetic parameters that give statistically equivalent goodness-of-fit to time series concentration data. The method is based on the incremental identification approach, where the parameter estimation is done in a step-wise manner. Numerical efficacy is achieved by reducing the dimensionality of parameter space and using efficient random parameter exploration algorithms. The shift toward using model ensembles, instead of the traditional "best-fit" models, is necessary to directly account for model uncertainty during the application of such models. The performance of the ensemble modeling approach has been demonstrated in the modeling of a generic branched pathway and the trehalose pathway in Saccharomyces cerevisiae using generalized mass action (GMA) kinetics.

Estimation of dynamic flux profiles from metabolic time series data

Background

Advances 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.

Results

The 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.

Conclusions

The 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.

Construction of a genome-scale kinetic model of mycobacterium tuberculosis using generic rate equations

The study of biological systems at the genome scale helps us understand fundamental biological processes that govern the activity of living organisms and regulate their interactions with the environment. Genome-scale metabolic models are usually analysed using constraint-based methods, since detailed rate equations and kinetic parameters are often missing. However, constraint-based analysis is limited in capturing the dynamics of cellular processes. In this paper, we present an approach to build a genome-scale kinetic model of Mycobacterium tuberculosis metabolism using generic rate equations. M. tuberculosis causes tuberculosis which remains one of the largest killer infectious diseases. Using a genetic algorithm, we estimated kinetic parameters for a genome-scale metabolic model of M. tuberculosis based on flux distributions derived from Flux Balance Analysis. Our results show that an excellent agreement with flux values is obtained under several growth conditions, although kinetic parameters may vary in different conditions. Parameter variability analysis indicates that a high degree of redundancy remains present in model parameters, which suggests that the integration of other types of high-throughput datasets will enable the development of better constrained models accounting for a variety of in vivo phenotypes.

An S-System Parameter Estimation Method (SPEM) for biological networks

Advances in experimental biology, coupled with advances in computational power, bring new challenges to the interdisciplinary field of computational biology. One such broad challenge lies in the reverse engineering of gene networks, and goes from determining the structure of static networks, to reconstructing the dynamics of interactions from time series data. Here, we focus our attention on the latter area, and in particular, on parameterizing a dynamic network of oriented interactions between genes. By basing the parameterizing approach on a known power-law relationship model between connected genes (S-system), we are able to account for non-linearity in the network, without compromising the ability to analyze network characteristics. In this article, we introduce the S-System Parameter Estimation Method (SPEM). SPEM, a freely available R software package (http://www.picb.ac.cn/ClinicalGenomicNTW/temp3.html), takes gene expression data in time series and returns the network of interactions as a set of differential equations. The methods, which are presented and tested here, are shown to provide accurate results not only on synthetic data, but more importantly on real and therefore noisy by nature, biological data. In summary, SPEM shows high sensitivity and positive predicted values, as well as free availability and expansibility (because based on open source software). We expect these characteristics to make it a useful and broadly applicable software in the challenging reconstruction of dynamic gene networks.

Steady-state global optimization of metabolic non-linear dynamic models through recasting into power-law canonical models

Background

Design of newly engineered microbial strains for biotechnological purposes would greatly benefit from the development of realistic mathematical models for the processes to be optimized. Such models can then be analyzed and, with the development and application of appropriate optimization techniques, one could identify the modifications that need to be made to the organism in order to achieve the desired biotechnological goal. As appropriate models to perform such an analysis are necessarily non-linear and typically non-convex, finding their global optimum is a challenging task. Canonical modeling techniques, such as Generalized Mass Action (GMA) models based on the power-law formalism, offer a possible solution to this problem because they have a mathematical structure that enables the development of specific algorithms for global optimization.

Results

Based on the GMA canonical representation, we have developed in previous works a highly efficient optimization algorithm and a set of related strategies for understanding the evolution of adaptive responses in cellular metabolism. Here, we explore the possibility of recasting kinetic non-linear models into an equivalent GMA model, so that global optimization on the recast GMA model can be performed. With this technique, optimization is greatly facilitated and the results are transposable to the original non-linear problem. This procedure is straightforward for a particular class of non-linear models known as Saturable and Cooperative (SC) models that extend the power-law formalism to deal with saturation and cooperativity.

Conclusions

Our results show that recasting non-linear kinetic models into GMA models is indeed an appropriate strategy that helps overcoming some of the numerical difficulties that arise during the global optimization task.