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

Multi-scale models of whole cells: progress and challenges

Whole-cell modeling is "the ultimate goal" of computational systems biology and "a grand challenge for 21st century" (Tomita, Trends in Biotechnology, 2001, 19(6), 205-10). These complex, highly detailed models account for the activity of every molecule in a cell and serve as comprehensive knowledgebases for the modeled system. Their scope and utility far surpass those of other systems models. In fact, whole-cell models (WCMs) are an amalgam of several types of "system" models. The models are simulated using a hybrid modeling method where the appropriate mathematical methods for each biological process are used to simulate their behavior. Given the complexity of the models, the process of developing and curating these models is labor-intensive and to date only a handful of these models have been developed. While whole-cell models provide valuable and novel biological insights, and to date have identified some novel biological phenomena, their most important contribution has been to highlight the discrepancy between available data and observations that are used for the parametrization and validation of complex biological models. Another realization has been that current whole-cell modeling simulators are slow and to run models that mimic more complex (e.g., multi-cellular) biosystems, those need to be executed in an accelerated fashion on high-performance computing platforms. In this manuscript, we review the progress of whole-cell modeling to date and discuss some of the ways that they can be improved.

Integrating multiple regulations on enzyme activity: the case of phosphoenolpyruvate carboxykinases

Data on protein post-translational modifications (PTMs) increased exponentially in the last years due to the refinement of mass spectrometry techniques and the development of databases to store and share datasets. Nevertheless, these data per se do not create comprehensive biochemical knowledge. Complementary studies on protein biochemistry are necessary to fully understand the function of these PTMs at the molecular level and beyond, for example, designing rational metabolic engineering strategies to improve crops. Phosphoenolpyruvate carboxykinases (PEPCKs) are critical enzymes for plant metabolism with diverse roles in plant development and growth. Multiple lines of evidence showed the complex regulation of PEPCKs, including PTMs. Herein, we present PEPCKs as an example of the integration of combined mechanisms modulating enzyme activity and metabolic pathways. PEPCK studies strongly advanced after the production of the recombinant enzyme and the establishment of standardized biochemical assays. Finally, we discuss emerging open questions for future research and the challenges in integrating all available data into functional biochemical models.

Parameter Identification in Metabolic Reaction Networks by Means of Multiple Steady-State Measurements

In this work, we investigate some theoretical aspects related to the estimation approach proposed by Liebermeister and Klipp, 2006, in which general rate laws, derived from standardized enzymatic mechanisms, are exploited to kinetically describe the fluxes of a metabolic reaction network, and multiple metabolic steady-state measurements are exploited to estimate the unknown kinetic parameters. Further mathematical details are deeply investigated, and necessary conditions on the amount of information required to solve the identification problem are given. Moreover, theoretical results for the parameter identifiability are provided, and symmetrical and modular properties of the proposed approach are highlighted when the global identification problem is decoupled into smaller and simpler identification problems related to the single reactions of the network. Among the advantages of the proposed innovative approach are (i) non-restrictive conditions to guarantee the solvability of the parameter estimation problem, (ii) the unburden of the usual computational complexity for such identification problems, and (iii) the ease of obtaining the required number of measurements, which are actually steady-state data, experimentally easier to obtain with respect to the time-dependent ones. A simple example concludes the paper, highlighting the mentioned advantages of the method and the implementation of the related theoretical result.

CAMSAP2 organizes a γ-tubulin-independent microtubule nucleation centre through phase separation

Microtubules are dynamic polymers consisting of αβ-tubulin heterodimers. The initial polymerization process, called microtubule nucleation, occurs spontaneously via αβ-tubulin. Since a large energy barrier prevents microtubule nucleation in cells, the γ-tubulin ring complex is recruited to the centrosome to overcome the nucleation barrier. However, a considerable number of microtubules can polymerize independently of the centrosome in various cell types. Here, we present evidence that the minus-end-binding calmodulin-regulated spectrin-associated protein 2 (CAMSAP2) serves as a strong nucleator for microtubule formation by significantly reducing the nucleation barrier. CAMSAP2 co-condensates with αβ-tubulin via a phase separation process, producing plenty of nucleation intermediates. Microtubules then radiate from the co-condensates, resulting in aster-like structure formation. CAMSAP2 localizes at the co-condensates and decorates the radiating microtubule lattices to some extent. Taken together, these in vitro findings suggest that CAMSAP2 supports microtubule nucleation and growth by organizing a nucleation centre as well as by stabilizing microtubule intermediates and growing microtubules.

Cells are able to hold their shape thanks to tube-like structures called microtubules that are made of hundreds of tubulin proteins. Microtubules are responsible for maintaining the uneven distribution of molecules throughout the cell, a phenomenon known as polarity that allows cells to differentiate into different types with various roles.

A protein complex called the γ-tubulin ring complex (γ-TuRC) is necessary for microtubules to form. This protein helps bind the tubulin proteins together and stabilises microtubules. However, recent research has found that in highly polarized cells such as neurons, which have highly specialised regions, microtubules can form without γ-TuRC. Searching for the proteins that could be filling in for γ-TuRC in these cells some evidence has suggested that a group known as CAMSAPs may be involved, but it is not known how.

To characterize the role of CAMSAPs, Imasaki, Kikkawa et al. studied how one of these proteins, CAMSAP2, interacts with tubulins. To do this, they reconstituted both CAMSAP2 and tubulins using recombinant biotechnology and mixed them in solution. These experiments showed that CAMSAP2 can help form microtubules by bringing together their constituent proteins so that they can bind to each other more easily. Once microtubules start to form, CAMSAP2 continues to bind to them, stabilizing them and enabling them to grow to full size.

These results shed light on how polarity is established in cells such as neurons, muscle cells, and epithelial cells. Additionally, the ability to observe intermediate structures during microtubule formation can provide insights into the processes that these structures are involved in.

Model Balancing: A Search for In-Vivo Kinetic Constants and Consistent Metabolic States

Enzyme kinetic constants in vivo are largely unknown, which limits the construction of large metabolic models. Given measured metabolic fluxes, metabolite concentrations, and enzyme concentrations, these constants may be inferred by model fitting, but the estimation problems are hard to solve if models are large. Here we show how consistent kinetic constants, metabolite concentrations, and enzyme concentrations can be determined from data if metabolic fluxes are known. The estimation method, called model balancing, can handle models with a wide range of rate laws and accounts for thermodynamic constraints between fluxes, kinetic constants, and metabolite concentrations. It can be used to estimate in-vivo kinetic constants, to complete and adjust available data, and to construct plausible metabolic states with predefined flux distributions. By omitting one term from the log posterior-a term for penalising low enzyme concentrations-we obtain a convex optimality problem with a unique local optimum. As a demonstrative case, we balance a model of E. coli central metabolism with artificial or experimental data and obtain a physically and biologically plausible parameterisation of reaction kinetics in E. coli central metabolism. The example shows what information about kinetic constants can be obtained from omics data and reveals practical limits to estimating in-vivo kinetic constants. While noise-free omics data allow for a reasonable reconstruction of in-vivo kcat and KM values, prediction from noisy omics data are worse. Hence, adjusting kinetic constants and omics data to obtain consistent metabolic models is the main application of model balancing.

A Multi-Scale Approach to Modeling E. coli Chemotaxis

The degree to which we can understand the multi-scale organization of cellular life is tied to how well our models can represent this organization and the processes that drive its evolution. This paper uses Vivarium-an engine for composing heterogeneous computational biology models into integrated, multi-scale simulations. Vivarium's approach is demonstrated by combining several sub-models of biophysical processes into a model of chemotactic E. coli that exchange molecules with their environment, express the genes required for chemotaxis, swim, grow, and divide. This model is developed incrementally, highlighting cross-compartment mechanisms that link E. coli to its environment, with models for: (1) metabolism and transport, with transport moving nutrients across the membrane boundary and metabolism converting them to useful metabolites, (2) transcription, translation, complexation, and degradation, with stochastic mechanisms that read real gene sequence data and consume base pairs and ATP to make proteins and complexes, and (3) the activity of flagella and chemoreceptors, which together support navigation in the environment.

Bayesian Data Integration Questions Classic Study on Protease Self-Digest Kinetics

We combine Bayesian data integration with kinetic modeling to rigorously identify reaction mechanisms. This approach forces models to be consistent not only with kinetic measurements but with all available information. We revisit a classic study on trypsin self-digest acceleration by colloidal silica. Bayesian data integration reveals that the mechanism suggested in that study is inconsistent with its presented data. We propose an improved hypothesis. However, the detailed mechanism of the surface reaction cannot be inferred from the available data.

Parameter balancing: consistent parameter sets for kinetic metabolic models

Summary

Measured kinetic constants are key input data for metabolic models, but they are often uncertain, inconsistent and incomplete. Parameter balancing translates such data into complete and consistent parameter sets while accounting for predefined ranges and physical constraints. Based on Bayesian regression, it determines a most plausible parameter set as well as uncertainty ranges for all model parameters. Our tools for parameter balancing support standard model and data formats and enable an easy customization of prior distributions and constraints for biochemical constants. Modellers can balance kinetic constants, thermodynamic data and metabolomic data to obtain thermodynamically consistent metabolic states that comply with user-defined flux directions.

Availability and implementation

An online tool for parameter balancing, a stand-alone Python command line tool, a Python package and a Matlab toolbox (which uses the CPLEX solver) are freely available at www.parameterbalancing.net.

FRET Analysis of Ionic Strength Sensors in the Hofmeister Series of Salt Solutions Using Fluorescence Lifetime Measurements

Living cells are complex, crowded, and dynamic and continually respond to environmental and intracellular stimuli. They also have heterogeneous ionic strength with compartmentalized variations in both intracellular concentrations and types of ions. These challenges would benefit from the development of quantitative, noninvasive approaches for mapping the heterogeneous ionic strength fluctuations in living cells. Here, we investigated a class of recently developed ionic strength sensors that consists of mCerulean3 (a cyan fluorescent protein) and mCitrine (a yellow fluorescent protein) tethered via a linker made of two charged α-helices and a flexible loop. The two helices are designed to bear opposite charges, which is hypothesized to increase the ionic screening and therefore a larger intermolecular distance. In these protein constructs, mCerulean3 and mCitrine act as a donor-acceptor pair undergoing Förster resonance energy transfer (FRET) that is dependent on both the linker amino acids and the environmental ionic strength. Using time-resolved fluorescence of the donor (mCerulean3), we determined the sensitivity of the energy transfer efficiencies and the donor-acceptor distances of these sensors at variable concentrations of the Hofmeister series of salts (KCl, LiCl, NaCl, NaBr, NaI, Na₂SO₄). As controls, similar measurements were carried out on the FRET-incapable, enzymatically cleaved counterparts of these sensors as well as a construct designed with two electrostatically neutral α-helices (E6G2). Our results show that the energy transfer efficiencies of these sensors are sensitive to both the linker amino acid sequence and the environmental ionic strength, whereas the sensitivity of these sensors to the identity of the dissolved ions of the Hofmeister series of salts seems limited. We also developed a theoretical framework to explain the observed trends as a function of the ionic strength in terms of the Debye screening of the electrostatic interaction between the two charged α-helices in the linker region. These controlled solution studies represent an important step toward the development of rationally designed FRET-based environmental sensors while offering different models for calculating the energy transfer efficiency using time-resolved fluorescence that is compatible with future in vivo studies.

Kinetic modeling of Stickland reactions-coupled methanogenesis for a methanogenic culture

Studying amino acid catabolism-coupled methanogenesis is the important standpoints to decipher the metabolic behavior of a methanogenic culture. l-Glycine and l-alanine are acted as sole carbon and nitrogen sources for acidogenic bacteria. One amino acid is oxidized and another one is reduced for acetate production via pyruvate by oxidative deamination process in the Stickland reactions. Herein, we have developed a kinetic model for the Stickland reactions-coupled methanogenesis (SRCM) and simulated objectively to maximize the rate of methane production. We collected the metabolic information from enzyme kinetic parameters for amino acid catabolism of Clostridium acetobutylicum ATCC 824 and methanogenesis of Methanosarcina acetivorans C2A. The SRCM model of this study consisted of 18 reactions and 61 metabolites with enzyme kinetic parameters derived experimental data. The internal or external metabolic flux rate of this system found to control the acidogenesis and methanogenesis in a methanogenic culture. Using the SRCM model, flux distributions were calculated for each reaction and metabolite in order to maximize the methane production rate from the glycine–alanine pair. Results of this study, we demonstrated the metabolic behavior, metabolite pairing while mutually interact, and advantages of syntrophic metabolism of amino acid-directed methane production in a methanogenic starter culture.

DMPy: a Python package for automated mathematical model construction of large-scale metabolic systems

Background

Models of metabolism are often used in biotechnology and pharmaceutical research to identify drug targets or increase the direct production of valuable compounds. Due to the complexity of large metabolic systems, a number of conclusions have been drawn using mathematical methods with simplifying assumptions. For example, constraint-based models describe changes of internal concentrations that occur much quicker than alterations in cell physiology. Thus, metabolite concentrations and reaction fluxes are fixed to constant values. This greatly reduces the mathematical complexity, while providing a reasonably good description of the system in steady state. However, without a large number of constraints, many different flux sets can describe the optimal model and we obtain no information on how metabolite levels dynamically change. Thus, to accurately determine what is taking place within the cell, finer quality data and more detailed models need to be constructed.

Results

In this paper we present a computational framework, DMPy, that uses a network scheme as input to automatically search for kinetic rates and produce a mathematical model that describes temporal changes of metabolite fluxes. The parameter search utilises several online databases to find measured reaction parameters. From this, we take advantage of previous modelling efforts, such as Parameter Balancing, to produce an initial mathematical model of a metabolic pathway. We analyse the effect of parameter uncertainty on model dynamics and test how recent flux-based model reduction techniques alter system properties. To our knowledge this is the first time such analysis has been performed on large models of metabolism. Our results highlight that good estimates of at least 80% of the reaction rates are required to accurately model metabolic systems. Furthermore, reducing the size of the model by grouping reactions together based on fluxes alters the resulting system dynamics.

Conclusion

The presented pipeline automates the modelling process for large metabolic networks. From this, users can simulate their pathway of interest and obtain a better understanding of how altering conditions influences cellular dynamics. By testing the effects of different parameterisations we are also able to provide suggestions to help construct more accurate models of complete metabolic systems in the future.

Electronic supplementary material

The online version of this article (10.1186/s12918-018-0584-8) contains supplementary material, which is available to authorized users.

Formulation, construction and analysis of kinetic models of metabolism: A review of modelling frameworks

Kinetic models are critical to predict the dynamic behaviour of metabolic networks. Mechanistic kinetic models for large networks remain uncommon due to the difficulty of fitting their parameters. Recent modelling frameworks promise new ways to overcome this obstacle while retaining predictive capabilities. In this review, we present an overview of the relevant mathematical frameworks for kinetic formulation, construction and analysis. Starting with kinetic formalisms, we next review statistical methods for parameter inference, as well as recent computational frameworks applied to the construction and analysis of kinetic models. Finally, we discuss opportunities and limitations hindering the development of larger kinetic reconstructions.

Towards in vivo estimation of reaction kinetics using high-throughput metabolomics data: a maximum likelihood approach

Background

High-throughput assays such as mass spectrometry have opened up the possibility for large-scale in vivo measurements of the metabolome. This data could potentially be used to estimate kinetic parameters for many metabolic reactions. However, high-throughput in vivo measurements have special properties that are not taken into account in existing methods for estimating kinetic parameters, including significant relative errors in measurements of metabolite concentrations and reaction rates, and reactions with multiple substrates and products, which are sometimes reversible. A new method is needed to estimate kinetic parameters taking into account these factors.

Results

A new method, InVEst (In Vivo Estimation), is described for estimating reaction kinetic parameters, which addresses the specific challenges of in vivo data. InVEst uses maximum likelihood estimation based on a model where all measurements have relative errors. Simulations show that InVEst produces accurate estimates for a reversible enzymatic reaction with multiple reactants and products, that estimated parameters can be used to predict the effects of genetic variants, and that InVEst is more accurate than general least squares and graphic methods on data with relative errors. InVEst uses the bootstrap method to evaluate the accuracy of its estimates.

Conclusions

InVEst addresses several challenges of in vivo data, which are not taken into account by existing methods. When data have relative errors, InVEst produces more accurate and robust estimates. InVEst also provides useful information about estimation accuracy using bootstrapping. It has potential applications of quantifying the effects of genetic variants, inference of the target of a mutation or drug treatment and improving flux estimation.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-015-0214-7) contains supplementary material, which is available to authorized users.

Quantitative determinants of aerobic glycolysis identify flux through the enzyme GAPDH as a limiting step

Aerobic glycolysis or the Warburg Effect (WE) is characterized by the increased metabolism of glucose to lactate. It remains unknown what quantitative changes to the activity of metabolism are necessary and sufficient for this phenotype. We developed a computational model of glycolysis and an integrated analysis using metabolic control analysis (MCA), metabolomics data, and statistical simulations. We identified and confirmed a novel mode of regulation specific to aerobic glycolysis where flux through GAPDH, the enzyme separating lower and upper glycolysis, is the rate-limiting step in the pathway and the levels of fructose (1,6) bisphosphate (FBP), are predictive of the rate and control points in glycolysis. Strikingly, negative flux control was found and confirmed for several steps thought to be rate-limiting in glycolysis. Together, these findings enumerate the biochemical determinants of the WE and suggest strategies for identifying the contexts in which agents that target glycolysis might be most effective.

DOI:http://dx.doi.org/10.7554/eLife.03342.001

Cells generate energy from a sugar called glucose via a process called glycolysis. This process involves many enzymes that catalyze 10 different chemical reactions, and it essentially converts glucose step-by-step into a simpler chemical called pyruvate.

Pyruvate is then normally transported into structures within the cell called mitochondria, where it is further broken down using oxygen to release more energy. However, in cells that are rapidly dividing, pyruvate is converted into another chemical called lactate—which releases energy more quickly, but releases less energy overall. Cancer cells often convert most of their glucose into lactate, rather than breaking down pyruvate in their mitochondria: an observation known as the ‘Warburg effect’. And while many factors affect how a cell releases energy from pyruvate, it remains unclear what regulates which of these biochemical processes is most common in a living cell.

In this study, Shestov et al. have developed a computational model for the process of glycolysis and used this to investigate the causes of the Warburg Effect. The model was based on the known characteristics of the enzymes and chemical reactions involved at each step. It predicted that the activity of the enzyme called GAPDH, which carries out the sixth step in glycolysis, in many cases affects how much lactate is produced. This suggests that this enzyme represents a bottleneck in the pathway.

Next, Shestov et al. performed experiments where they used drugs to block different stages of the glycolysis pathway, and confirmed that the GAPDH enzyme is important for regulating this pathway in living cancer cells too. In these treated cells, the levels of a chemical called fructose-1,6-biphosphate (which is made in a step in the pathway between glucose and pyruvate) were either very high or very low. Shestov et al. proposed that the flow of chemicals through the glycolysis pathway is controlled by the GAPDH enzyme when the chemicals used by the enzymes upstream of GAPDH in the pathway (which includes fructose-1,6-biphosphate) are plentiful. However, if these chemicals are limited, other enzymes that are involved in earlier steps of the pathway regulate the process instead.

The findings of Shestov et al. reveal that the regulation of glycolysis is more complex than previously thought, and is also very different when cells are undergoing the Warburg Effect. In the future, these findings might help to identify the types of cancer that could be effectively treated using drugs that target the glycolysis process, which are currently being tested in pre-clinical studies.

DOI:http://dx.doi.org/10.7554/eLife.03342.002

A transcriptomics-based kinetic model for ethylene biosynthesis in tomato (Solanum lycopersicum) fruit: development, validation and exploration of novel regulatory mechanisms

The gaseous plant hormone ethylene is involved in many physiological processes including climacteric fruit ripening, in which it is a key determinant of fruit quality. A detailed model that describes ethylene biochemistry dynamics is missing. Often, kinetic modeling is used to describe metabolic networks or signaling cascades, mostly ignoring the link with transcriptomic data. We have constructed an elegant kinetic model that describes the transfer of genetic information into abundance and metabolic activity of proteins for the entire ethylene biosynthesis pathway during fruit development and ripening of tomato (Solanum lycopersicum). Our model was calibrated against a vast amount of transcriptomic, proteomic and metabolic data and showed good descriptive qualities. Subsequently it was validated successfully against several ripening mutants previously described in the literature. The model was used as a predictive tool to evaluate novel and existing hypotheses regarding the regulation of ethylene biosynthesis. This bottom-up kinetic network model was used to indicate that a side-branch of the ethylene pathway, the formation of the dead-end product 1-(malonylamino)-1-aminocyclopropane-1-carboxylic acid (MACC), might have a strong effect on eventual ethylene production. Furthermore, our in silico analyses indicated potential (post-) translational regulation of the ethylene-forming enzyme ACC oxidase.

Kinetic models in industrial biotechnology - Improving cell factory performance

An increasing number of industrial bioprocesses capitalize on living cells by using them as cell factories that convert sugars into chemicals. These processes range from the production of bulk chemicals in yeasts and bacteria to the synthesis of therapeutic proteins in mammalian cell lines. One of the tools in the continuous search for improved performance of such production systems is the development and application of mathematical models. To be of value for industrial biotechnology, mathematical models should be able to assist in the rational design of cell factory properties or in the production processes in which they are utilized. Kinetic models are particularly suitable towards this end because they are capable of representing the complex biochemistry of cells in a more complete way compared to most other types of models. They can, at least in principle, be used to in detail understand, predict, and evaluate the effects of adding, removing, or modifying molecular components of a cell factory and for supporting the design of the bioreactor or fermentation process. However, several challenges still remain before kinetic modeling will reach the degree of maturity required for routine application in industry. Here we review the current status of kinetic cell factory modeling. Emphasis is on modeling methodology concepts, including model network structure, kinetic rate expressions, parameter estimation, optimization methods, identifiability analysis, model reduction, and model validation, but several applications of kinetic models for the improvement of cell factories are also discussed.

Why measure enzyme activities in the era of systems biology?

Information about the abundance and biological activities of proteins is essential to reveal how genes affect phenotypes. Over the past decade, mass spectrometry (MS)-based proteomics has revolutionized the identification and quantification of proteins, and the detection of post-translational modifications. Interpretation of proteomics data depends on information about the biological activities of proteins, which has created a bottleneck in research. This review focuses on enzymes in central metabolism. We examine the methods used for measuring enzyme activities, and discuss how these methods provide information about the kinetic and regulatory properties of enzymes, their turnover, and how this information can be integrated into metabolic models. We also discuss how robotized assays could enable the genetic networks that control enzyme abundance to be analyzed.

Weighting schemes in metabolic graphs for identifying biochemical routes

Metabolism forms an integral part of all cells and its study is important to understand the functioning of the system, to understand alterations that occur in disease state and hence for subsequent applications in drug discovery. Reconstruction of genome-scale metabolic graphs from genomics and other molecular or biochemical data is now feasible. Few methods have also been reported for inferring biochemical pathways from these networks. However, given the large scale and complex inter-connections in the networks, the problem of identifying biochemical routes is not trivial and some questions still remain open. In particular, how a given path is altered in perturbed conditions remains a difficult problem, warranting development of improved methods. Here we report a comparison of 6 different weighting schemes to derive node and edge weights for a metabolic graph, weights reflecting various kinetic, thermodynamic parameters as well as abundances inferred from transcriptome data. Using a network of 50 nodes and 107 edges of carbohydrate metabolism, we show that kinetic parameter derived weighting schemes [Formula: see text] fare best. However, these are limited by their extent of availability, highlighting the usefulness of omics data under such conditions. Interestingly, transcriptome derived weights yield paths with best scores, but are inadequate to discriminate the theoretical paths. The method is tested on a system of Escherichia coli stress response. The approach illustrated here is generic in nature and can be used in the analysis for metabolic network from any species and perhaps more importantly for comparing condition-specific networks.

New types of experimental data shape the use of enzyme kinetics for dynamic network modeling

Since the publication of Leonor Michaelis and Maude Menten's paper on the reaction kinetics of the enzyme invertase in 1913, molecular biology has evolved tremendously. New measurement techniques allow in vivo characterization of the whole genome, proteome or transcriptome of cells, whereas the classical enzyme essay only allows determination of the two Michaelis-Menten parameters V and K(m). Nevertheless, Michaelis-Menten kinetics are still commonly used, not only in the in vitro context of enzyme characterization but also as a rate law for enzymatic reactions in larger biochemical reaction networks. In this review, we give an overview of the historical development of kinetic rate laws originating from Michaelis-Menten kinetics over the past 100 years. Furthermore, we briefly summarize the experimental techniques used for the characterization of enzymes, and discuss web resources that systematically store kinetic parameters and related information. Finally, describe the novel opportunities that arise from using these data in dynamic mathematical modeling. In this framework, traditional in vitro approaches may be combined with modern genome-scale measurements to foster thorough understanding of the underlying complex mechanisms.

Towards kinetic modeling of genome-scale metabolic networks without sacrificing stoichiometric, thermodynamic and physiological constraints

Mathematical modeling is an essential tool for the comprehensive understanding of cell metabolism and its interactions with the environmental and process conditions. Recent developments in the construction and analysis of stoichiometric models made it possible to define limits on steady-state metabolic behavior using flux balance analysis. However, detailed information on enzyme kinetics and enzyme regulation is needed to formulate kinetic models that can accurately capture the dynamic metabolic responses. The use of mechanistic enzyme kinetics is a difficult task due to uncertainty in the kinetic properties of enzymes. Therefore, the majority of recent works considered only mass action kinetics for reactions in metabolic networks. Herein, we applied the optimization and risk analysis of complex living entities (ORACLE) framework and constructed a large-scale mechanistic kinetic model of optimally grown Escherichia coli. We investigated the complex interplay between stoichiometry, thermodynamics, and kinetics in determining the flexibility and capabilities of metabolism. Our results indicate that enzyme saturation is a necessary consideration in modeling metabolic networks and it extends the feasible ranges of metabolic fluxes and metabolite concentrations. Our results further suggest that enzymes in metabolic networks have evolved to function at different saturation states to ensure greater flexibility and robustness of cellular metabolism.

A cooperative strategy for parameter estimation in large scale systems biology models

Background

Mathematical models play a key role in systems biology: they summarize the currently available knowledge in a way that allows to make experimentally verifiable predictions. Model calibration consists of finding the parameters that give the best fit to a set of experimental data, which entails minimizing a cost function that measures the goodness of this fit. Most mathematical models in systems biology present three characteristics which make this problem very difficult to solve: they are highly non-linear, they have a large number of parameters to be estimated, and the information content of the available experimental data is frequently scarce. Hence, there is a need for global optimization methods capable of solving this problem efficiently.

Results

A new approach for parameter estimation of large scale models, called Cooperative Enhanced Scatter Search (CeSS), is presented. Its key feature is the cooperation between different programs (“threads”) that run in parallel in different processors. Each thread implements a state of the art metaheuristic, the enhanced Scatter Search algorithm (eSS). Cooperation, meaning information sharing between threads, modifies the systemic properties of the algorithm and allows to speed up performance. Two parameter estimation problems involving models related with the central carbon metabolism of E. coli which include different regulatory levels (metabolic and transcriptional) are used as case studies. The performance and capabilities of the method are also evaluated using benchmark problems of large-scale global optimization, with excellent results.

Conclusions

The cooperative CeSS strategy is a general purpose technique that can be applied to any model calibration problem. Its capability has been demonstrated by calibrating two large-scale models of different characteristics, improving the performance of previously existing methods in both cases. The cooperative metaheuristic presented here can be easily extended to incorporate other global and local search solvers and specific structural information for particular classes of problems.

Parameter estimation using meta-heuristics in systems biology: a comprehensive review

This paper gives a comprehensive review of the application of meta-heuristics to optimization problems in systems biology, mainly focussing on the parameter estimation problem (also called the inverse problem or model calibration). It is intended for either the system biologist who wishes to learn more about the various optimization techniques available and/or the meta-heuristic optimizer who is interested in applying such techniques to problems in systems biology. First, the parameter estimation problems emerging from different areas of systems biology are described from the point of view of machine learning. Brief descriptions of various meta-heuristics developed for these problems follow, along with outlines of their advantages and disadvantages. Several important issues in applying meta-heuristics to the systems biology modelling problem are addressed, including the reliability and identifiability of model parameters, optimal design of experiments, and so on. Finally, we highlight some possible future research directions in this field.

eQuilibrator--the biochemical thermodynamics calculator

The laws of thermodynamics constrain the action of biochemical systems. However, thermodynamic data on biochemical compounds can be difficult to find and is cumbersome to perform calculations with manually. Even simple thermodynamic questions like ‘how much Gibbs energy is released by ATP hydrolysis at pH 5?’ are complicated excessively by the search for accurate data. To address this problem, eQuilibrator couples a comprehensive and accurate database of thermodynamic properties of biochemical compounds and reactions with a simple and powerful online search and calculation interface. The web interface to eQuilibrator (http://equilibrator.weizmann.ac.il) enables easy calculation of Gibbs energies of compounds and reactions given arbitrary pH, ionic strength and metabolite concentrations. The eQuilibrator code is open-source and all thermodynamic source data are freely downloadable in standard formats. Here we describe the database characteristics and implementation and demonstrate its use.

Gaussian graphical modeling reconstructs pathway reactions from high-throughput metabolomics data

Background

With the advent of high-throughput targeted metabolic profiling techniques, the question of how to interpret and analyze the resulting vast amount of data becomes more and more important. In this work we address the reconstruction of metabolic reactions from cross-sectional metabolomics data, that is without the requirement for time-resolved measurements or specific system perturbations. Previous studies in this area mainly focused on Pearson correlation coefficients, which however are generally incapable of distinguishing between direct and indirect metabolic interactions.

Results

In our new approach we propose the application of a Gaussian graphical model (GGM), an undirected probabilistic graphical model estimating the conditional dependence between variables. GGMs are based on partial correlation coefficients, that is pairwise Pearson correlation coefficients conditioned against the correlation with all other metabolites. We first demonstrate the general validity of the method and its advantages over regular correlation networks with computer-simulated reaction systems. Then we estimate a GGM on data from a large human population cohort, covering 1020 fasting blood serum samples with 151 quantified metabolites. The GGM is much sparser than the correlation network, shows a modular structure with respect to metabolite classes, and is stable to the choice of samples in the data set. On the example of human fatty acid metabolism, we demonstrate for the first time that high partial correlation coefficients generally correspond to known metabolic reactions. This feature is evaluated both manually by investigating specific pairs of high-scoring metabolites, and then systematically on a literature-curated model of fatty acid synthesis and degradation. Our method detects many known reactions along with possibly novel pathway interactions, representing candidates for further experimental examination.

Conclusions

In summary, we demonstrate strong signatures of intracellular pathways in blood serum data, and provide a valuable tool for the unbiased reconstruction of metabolic reactions from large-scale metabolomics data sets.

Genome-scale metabolic networks

During the last decade, models have been developed to characterize cellular metabolism at the level of an entire metabolic network. The main concept that underlies whole-network metabolic modeling is the identification and mathematical definition of constraints. Here, we review large-scale metabolic network modeling, in particular, stoichiometric- and constraint-based approaches. Although many such models have been reconstructed, few networks have been extensively validated and tested experimentally, and we focus on these. We describe how metabolic networks can be represented using stoichiometric matrices and well-defined constraints on metabolic fluxes. We then discuss relatively successful approaches, including flux balance analysis (FBA), pathway analysis, and common extensions or modifications to these approaches. Finally, we describe techniques for integrating these approaches with models of other biological processes.

Thermodynamically consistent Bayesian analysis of closed biochemical reaction systems

Background

Estimating the rate constants of a biochemical reaction system with known stoichiometry from noisy time series measurements of molecular concentrations is an important step for building predictive models of cellular function. Inference techniques currently available in the literature may produce rate constant values that defy necessary constraints imposed by the fundamental laws of thermodynamics. As a result, these techniques may lead to biochemical reaction systems whose concentration dynamics could not possibly occur in nature. Therefore, development of a thermodynamically consistent approach for estimating the rate constants of a biochemical reaction system is highly desirable.

Results

We introduce a Bayesian analysis approach for computing thermodynamically consistent estimates of the rate constants of a closed biochemical reaction system with known stoichiometry given experimental data. Our method employs an appropriately designed prior probability density function that effectively integrates fundamental biophysical and thermodynamic knowledge into the inference problem. Moreover, it takes into account experimental strategies for collecting informative observations of molecular concentrations through perturbations. The proposed method employs a maximization-expectation-maximization algorithm that provides thermodynamically feasible estimates of the rate constant values and computes appropriate measures of estimation accuracy. We demonstrate various aspects of the proposed method on synthetic data obtained by simulating a subset of a well-known model of the EGF/ERK signaling pathway, and examine its robustness under conditions that violate key assumptions. Software, coded in MATLAB^®, which implements all Bayesian analysis techniques discussed in this paper, is available free of charge at http://www.cis.jhu.edu/~goutsias/CSS%20lab/software.html.

Conclusions

Our approach provides an attractive statistical methodology for estimating thermodynamically feasible values for the rate constants of a biochemical reaction system from noisy time series observations of molecular concentrations obtained through perturbations. The proposed technique is theoretically sound and computationally feasible, but restricted to quantitative data obtained from closed biochemical reaction systems. This necessitates development of similar techniques for estimating the rate constants of open biochemical reaction systems, which are more realistic models of cellular function.

Parameter balancing in kinetic models of cell metabolism

Kinetic modeling of metabolic pathways has become a major field of systems biology. It combines structural information about metabolic pathways with quantitative enzymatic rate laws. Some of the kinetic constants needed for a model could be collected from ever-growing literature and public web resources, but they are often incomplete, incompatible, or simply not available. We address this lack of information by parameter balancing, a method to complete given sets of kinetic constants. Based on Bayesian parameter estimation, it exploits the thermodynamic dependencies among different biochemical quantities to guess realistic model parameters from available kinetic data. Our algorithm accounts for varying measurement conditions in the input data (pH value and temperature). It can process kinetic constants and state-dependent quantities such as metabolite concentrations or chemical potentials, and uses prior distributions and data augmentation to keep the estimated quantities within plausible ranges. An online service and free software for parameter balancing with models provided in SBML format (Systems Biology Markup Language) is accessible at www.semanticsbml.org. We demonstrate its practical use with a small model of the phosphofructokinase reaction and discuss its possible applications and limitations. In the future, parameter balancing could become an important routine step in the kinetic modeling of large metabolic networks.

Towards a unifying, systems biology understanding of large-scale cellular death and destruction caused by poorly liganded iron: Parkinson's, Huntington's, Alzheimer's, prions, bactericides, chemical toxicology and others as examples

Exposure to a variety of toxins and/or infectious agents leads to disease, degeneration and death, often characterised by circumstances in which cells or tissues do not merely die and cease to function but may be more or less entirely obliterated. It is then legitimate to ask the question as to whether, despite the many kinds of agent involved, there may be at least some unifying mechanisms of such cell death and destruction. I summarise the evidence that in a great many cases, one underlying mechanism, providing major stresses of this type, entails continuing and autocatalytic production (based on positive feedback mechanisms) of hydroxyl radicals via Fenton chemistry involving poorly liganded iron, leading to cell death via apoptosis (probably including via pathways induced by changes in the NF-κB system). While every pathway is in some sense connected to every other one, I highlight the literature evidence suggesting that the degenerative effects of many diseases and toxicological insults converge on iron dysregulation. This highlights specifically the role of iron metabolism, and the detailed speciation of iron, in chemical and other toxicology, and has significant implications for the use of iron chelating substances (probably in partnership with appropriate anti-oxidants) as nutritional or therapeutic agents in inhibiting both the progression of these mainly degenerative diseases and the sequelae of both chronic and acute toxin exposure. The complexity of biochemical networks, especially those involving autocatalytic behaviour and positive feedbacks, means that multiple interventions (e.g. of iron chelators plus antioxidants) are likely to prove most effective. A variety of systems biology approaches, that I summarise, can predict both the mechanisms involved in these cell death pathways and the optimal sites of action for nutritional or pharmacological interventions.

Production of biofuels and biochemicals: in need of an ORACLE

The engineering of cells for the production of fuels and chemicals involves simultaneous optimization of multiple objectives, such as specific productivity, extended substrate range and improved tolerance - all under a great degree of uncertainty. The achievement of these objectives under physiological and process constraints will be impossible without the use of mathematical modeling. However, the limited information and the uncertainty in the available information require new methods for modeling and simulation that will characterize the uncertainty and will quantify, in a statistical sense, the expectations of success of alternative metabolic engineering strategies. We discuss these considerations toward developing a framework for the Optimization and Risk Analysis of Complex Living Entities (ORACLE) - a computational method that integrates available information into a mathematical structure to calculate control coefficients.

Challenges in experimental data integration within genome-scale metabolic models

A report of the meeting "Challenges in experimental data integration within genome-scale metabolic models", Institut Henri Poincaré, Paris, October 10-11 2009, organized by the CNRS-MPG joint program in Systems Biology.

Modular rate laws for enzymatic reactions: thermodynamics, elasticities and implementation

MOTIVATION

Standard rate laws are a key requisite for systematically turning metabolic networks into kinetic models. They should provide simple, general and biochemically plausible formulae for reaction velocities and reaction elasticities. At the same time, they need to respect thermodynamic relations between the kinetic constants and the metabolic fluxes and concentrations.

RESULTS

We present a family of reversible rate laws for reactions with arbitrary stoichiometries and various types of regulation, including mass-action, Michaelis-Menten and uni-uni reversible Hill kinetics as special cases. With a thermodynamically safe parameterization of these rate laws, parameter sets obtained by model fitting, sampling or optimization are guaranteed to lead to consistent chemical equilibrium states. A reformulation using saturation values yields simple formulae for rates and elasticities, which can be easily adjusted to the given stationary flux distributions. Furthermore, this formulation highlights the role of chemical potential differences as thermodynamic driving forces. We compare the modular rate laws to the thermodynamic-kinetic modelling formalism and discuss a simplified rate law in which the reaction rate directly depends on the reaction affinity. For automatic handling of modular rate laws, we propose a standard syntax and semantic annotations for the Systems Biology Markup Language.

AVAILABILITY

An online tool for inserting the rate laws into SBML models is freely available at www.semanticsbml.org.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Multiscale models of angiogenesis

Vascular disease, cancer, stroke, neurodegeneration, diabetes, inflammation, asthma, obesity, arthritis--the list of conditions that involve angiogenesis reads like main chapters in a book on pathology. Angiogenesis, the growth of capillaries from preexisting vessels, also occurs in normal physiology, in response to exercise or in the process of wound healing.Why and when is angiogenesis prevalent? What controls the process? How can we intelligently control it? These are the key questions driving researchers in fields as diverse as cell biology, oncology, cardiology, neurology, biomathematics, systems biology, and biomedical engineering. As bioengineers, we approach angiogenesis as a complex, interconnected system of events occurring in sequence and in parallel, on multiple levels, triggered by a main stimulus, e.g., hypoxia.

Methods for the inference of biological pathways and networks

In this chapter, we discuss a number of approaches to network inference from large-scale functional genomics data. Our goal is to describe current methods that can be used to infer predictive networks. At present, one of the most effective methods to produce networks with predictive value is the Bayesian network approach. This approach was initially instantiated by Friedman et al. and further refined by Eric Schadt and his research group. The Bayesian network approach has the virtue of identifying predictive relationships between genes from a combination of expression and eQTL data. However, the approach does not provide a mechanistic bases for predictive relationships and is ultimately hampered by an inability to model feedback. A challenge for the future is to produce networks that are both predictive and provide mechanistic understanding. To do so, the methods described in several chapters of this book will need to be integrated. Other chapters of this book describe a number of methods to identify or predict network components such as physical interactions. At the end of this chapter, we speculate that some of the approaches from other chapters could be integrated and used to "annotate" the edges of the Bayesian networks. This would take the Bayesian networks one step closer to providing mechanistic "explanations" for the relationships between the network nodes.

Bidirectionality and compartmentation of metabolic fluxes are revealed in the dynamics of isotopomer networks

Isotope labeling is one of the few methods of revealing the in vivo bidirectionality and compartmentalization of metabolic fluxes within metabolic networks. We argue that a shift from steady state to dynamic isotopomer analysis is required to deal with these cellular complexities and provide a review of dynamic studies of compartmentalized energy fluxes in eukaryotic cells including cardiac muscle, plants, and astrocytes. Knowledge of complex metabolic behaviour on a molecular level is prerequisite for the intelligent design of genetically modified organisms able to realize their potential of revolutionizing food, energy, and pharmaceutical production. We describe techniques to explore the bidirectionality and compartmentalization of metabolic fluxes using information contained in the isotopic transient, and discuss the integration of kinetic models with MFA. The flux parameters of an example metabolic network were optimized to examine the compartmentalization of metabolites and and the bidirectionality of fluxes in the TCA cycle of Saccharomyces uvarum for steady-state respiratory growth.

Use of randomized sampling for analysis of metabolic networks

Genome-scale metabolic network reconstructions in microorganisms have been formulated and studied for about 8 years. The constraint-based approach has shown great promise in analyzing the systemic properties of these network reconstructions. Notably, constraint-based models have been used successfully to predict the phenotypic effects of knock-outs and for metabolic engineering. The inherent uncertainty in both parameters and variables of large-scale models is significant and is well suited to study by Monte Carlo sampling of the solution space. These techniques have been applied extensively to the reaction rate (flux) space of networks, with more recent work focusing on dynamic/kinetic properties. Monte Carlo sampling as an analysis tool has many advantages, including the ability to work with missing data, the ability to apply post-processing techniques, and the ability to quantify uncertainty and to optimize experiments to reduce uncertainty. We present an overview of this emerging area of research in systems biology.

Getting to grips with the plant metabolic network

Research into plant metabolism has a long history, and analytical approaches of ever-increasing breadth and sophistication have been brought to bear. We now have access to vast repositories of data concerning enzymology and regulatory features of enzymes, as well as large-scale datasets containing profiling information of transcripts, protein and metabolite levels. Nevertheless, despite this wealth of data, we remain some way off from being able to rationally engineer plant metabolism or even to predict metabolic responses. Within the past 18 months, rapid progress has been made, with several highly informative plant network interrogations being discussed in the literature. In the present review we will appraise the current state of the art regarding plant metabolic network analysis and attempt to outline what the necessary steps are in order to further our understanding of network regulation.

Bringing metabolic networks to life: convenience rate law and thermodynamic constraints

Background

Translating a known metabolic network into a dynamic model requires rate laws for all chemical reactions. The mathematical expressions depend on the underlying enzymatic mechanism; they can become quite involved and may contain a large number of parameters. Rate laws and enzyme parameters are still unknown for most enzymes.

Results

We introduce a simple and general rate law called "convenience kinetics". It can be derived from a simple random-order enzyme mechanism. Thermodynamic laws can impose dependencies on the kinetic parameters. Hence, to facilitate model fitting and parameter optimisation for large networks, we introduce thermodynamically independent system parameters: their values can be varied independently, without violating thermodynamical constraints. We achieve this by expressing the equilibrium constants either by Gibbs free energies of formation or by a set of independent equilibrium constants. The remaining system parameters are mean turnover rates, generalised Michaelis-Menten constants, and constants for inhibition and activation. All parameters correspond to molecular energies, for instance, binding energies between reactants and enzyme.

Conclusion

Convenience kinetics can be used to translate a biochemical network – manually or automatically - into a dynamical model with plausible biological properties. It implements enzyme saturation and regulation by activators and inhibitors, covers all possible reaction stoichiometries, and can be specified by a small number of parameters. Its mathematical form makes it especially suitable for parameter estimation and optimisation. Parameter estimates can be easily computed from a least-squares fit to Michaelis-Menten values, turnover rates, equilibrium constants, and other quantities that are routinely measured in enzyme assays and stored in kinetic databases.