Modeling of uncertainties in biochemical reactions

Mathematical modeling is an indispensable tool for research and development in biotechnology and bioengineering. The formulation of kinetic models of biochemical networks depends on knowledge of the kinetic properties of the enzymes of the individual reactions. However, kinetic data acquired from experimental observations bring along uncertainties due to various experimental conditions and measurement methods. In this contribution, we propose a novel way to model the uncertainty in the enzyme kinetics and to predict quantitatively the responses of metabolic reactions to the changes in enzyme activities under uncertainty. The proposed methodology accounts explicitly for mechanistic properties of enzymes and physico‐chemical and thermodynamic constraints, and is based on formalism from systems theory and metabolic control analysis. We achieve this by observing that kinetic responses of metabolic reactions depend: (i) on the distribution of the enzymes among their free form and all reactive states; (ii) on the equilibrium displacements of the overall reaction and that of the individual enzymatic steps; and (iii) on the net fluxes through the enzyme. Relying on this observation, we develop a novel, efficient Monte Carlo sampling procedure to generate all states within a metabolic reaction that satisfy imposed constrains. Thus, we derive the statistics of the expected responses of the metabolic reactions to changes in enzyme levels and activities, in the levels of metabolites, and in the values of the kinetic parameters. We present aspects of the proposed framework through an example of the fundamental three‐step reversible enzymatic reaction mechanism. We demonstrate that the equilibrium displacements of the individual enzymatic steps have an important influence on kinetic responses of the enzyme. Furthermore, we derive the conditions that must be satisfied by a reversible three‐step enzymatic reaction operating far away from the equilibrium in order to respond to changes in metabolite levels according to the irreversible Michelis–Menten kinetics. The efficient sampling procedure allows easy, scalable, implementation of this methodology to modeling of large‐scale biochemical networks. Biotechnol. Bioeng. 2011;108: 413–423. © 2010 Wiley Periodicals, Inc.     

Control analysis of metabolic networks. 2. Total differentials and general formulation of the connectivity relations

The mathematical background of the connectivity relations of metabolic control theory is analysed. The connectivity relations are shown to reflect general properties of total differentials of reaction rate vi, flux J, and metabolite concentration Xj. Connectivity relations hold for any metabolic network in which all vi are homogeneous functions of enzyme concentration Ei. This notion allows established algebraic methods to be used for the formulation of connectivity relations for metabolic systems in which numerous constraints are imposed on metabolite concentrations. A general procedure to derive connectivity relations for such metabolic systems is given. To encourage a broader audience to apply control theory to physiological systems, an easy-to-use graphical procedure is derived for formulating connectivity relations for biochemical systems in which no metabolite is involved in more than one constraint.     

Physiologically Shrinking the Solution Space of a Saccharomyces cerevisiae Genome-Scale Model Suggests the Role of the Metabolic Network in Shaping Gene Expression Noise

Sampling the solution space of genome-scale models is generally conducted to determine the feasible region for metabolic flux distribution. Because the region for actual metabolic states resides only in a small fraction of the entire space, it is necessary to shrink the solution space to improve the predictive power of a model. A common strategy is to constrain models by integrating extra datasets such as high-throughput datasets and C¹³-labeled flux datasets. However, studies refining these approaches by performing a meta-analysis of massive experimental metabolic flux measurements, which are closely linked to cellular phenotypes, are limited. In the present study, experimentally identified metabolic flux data from 96 published reports were systematically reviewed. Several strong associations among metabolic flux phenotypes were observed. These phenotype-phenotype associations at the flux level were quantified and integrated into a Saccharomyces cerevisiae genome-scale model as extra physiological constraints. By sampling the shrunken solution space of the model, the metabolic flux fluctuation level, which is an intrinsic trait of metabolic reactions determined by the network, was estimated and utilized to explore its relationship to gene expression noise. Although no correlation was observed in all enzyme-coding genes, a relationship between metabolic flux fluctuation and expression noise of genes associated with enzyme-dosage sensitive reactions was detected, suggesting that the metabolic network plays a role in shaping gene expression noise. Such correlation was mainly attributed to the genes corresponding to non-essential reactions, rather than essential ones. This was at least partially, due to regulations underlying the flux phenotype-phenotype associations. Altogether, this study proposes a new approach in shrinking the solution space of a genome-scale model, of which sampling provides new insights into gene expression noise.     

Decomposition of metabolic network into functional modules based on the global connectivity structure of reaction graph

MOTIVATION: Metabolic networks are organized in a modular, hierarchical manner. Methods for a rational decomposition of the metabolic network into relatively independent functional subsets are essential to better understand the modularity and organization principle of a large-scale, genome-wide network. Network decomposition is also necessary for functional analysis of metabolism by pathway analysis methods that are often hampered by the problem of combinatorial explosion due to the complexity of metabolic network. Decomposition methods proposed in literature are mainly based on the connection degree of metabolites. To obtain a more reasonable decomposition, the global connectivity structure of metabolic networks should be taken into account. RESULTS: In this work, we use a reaction graph representation of a metabolic network for the identification of its global connectivity structure and for decomposition. A bow-tie connectivity structure similar to that previously discovered for metabolite graph is found also to exist in the reaction graph. Based on this bow-tie structure, a new decomposition method is proposed, which uses a distance definition derived from the path length between two reactions. An hierarchical classification tree is first constructed from the distance matrix among the reactions in the giant strong component of the bow-tie structure. These reactions are then grouped into different subsets based on the hierarchical tree. Reactions in the IN and OUT subsets of the bow-tie structure are subsequently placed in the corresponding subsets according to a 'majority rule'. Compared with the decomposition methods proposed in literature, ours is based on combined properties of the global network structure and local reaction connectivity rather than, primarily, on the connection degree of metabolites. The method is applied to decompose the metabolic network of Escherichia coli. Eleven subsets are obtained. More detailed investigations of the subsets show that reactions in the same subset are really functionally related. The rational decomposition of metabolic networks, and subsequent studies of the subsets, make it more amenable to understand the inherent organization and functionality of metabolic networks at the modular level. SUPPLEMENTARY INFORMATION: http://genome.gbf.de/bioinformatics/     

Analysis of metabolic network based on conservation of molecular structure

Recent work has revealed much about chemical reactions inside hundreds of organisms as well as universal characteristics of metabolic networks, which shed light on the evolution of the networks. However, characteristics of individual metabolites have been neglected. For example, some carbohydrates have structures that are decomposed into small molecules by metabolic reactions, but coenzymes such as ATP are mostly preserved. Such differences in metabolite characteristics are important for understanding the universal characteristics of metabolic networks. To quantify the structure conservation of metabolites, we defined the "structure conservation index" (SCI) for each metabolite as the fraction of metabolite atoms restored to their original positions through metabolic reactions. As expected, coenzymes and coenzyme-like metabolites that have reaction loops in the network show a higher SCI. Using the index, we found that the sum of metabolic fluxes is negatively correlated with the structure preservation of metabolite. Also, we found that each reaction path around high SCI metabolites changes independently, while changes in reaction paths involving low SCI metabolites coincide through evolution processes. These correlations may provide a clue to universal properties of metabolic networks.     

Identification of distributed metabolic objectives in the hypermetabolic liver by flux and energy balance analysis

A methodology for inferring distributed metabolic objectives from time series flux data is developed by combining metabolic flux analysis, pathway identification, free energy balances, and nested optimization. This methodology is used to investigate the metabolic response of the rat liver to burn injury-induced whole body inflammation. Gibbs free energy changes were computed for stoichiometrically balanced sequences of reactions, or pathways, rather than individual reactions, to account for energetic coupling between reactions. Systematic enumeration of pathways proceeded by elementary flux mode (EFM) analysis. Together with stoichiometric balances and external metabolite flux measurements, the DeltaG(PATH)(o) criterion provided sufficient constraints to solve a series of nested optimization problems on the metabolic goal functions and associated flux distributions of fasted livers during the first-week time course of burn injury. The optimization results suggest that there is a consistent metabolic goal function for the liver that is insensitive to the changing metabolic burdens experienced by the liver during the first-week time course. As defined by the goal function coefficients, the global metabolic objective was to distribute the metabolic resources between amino acid metabolism and ketone body synthesis. These findings point to a role for the time-invariant structure of the metabolic reaction network, expressed as stoichiometric and thermodynamic constraints, in shaping the cellular metabolic objective.     

Consistent Estimation of Gibbs Energy Using Component Contributions

Standard Gibbs energies of reactions are increasingly being used in metabolic modeling for applying thermodynamic constraints on reaction rates, metabolite concentrations and kinetic parameters. The increasing scope and diversity of metabolic models has led scientists to look for genome-scale solutions that can estimate the standard Gibbs energy of all the reactions in metabolism. Group contribution methods greatly increase coverage, albeit at the price of decreased precision. We present here a way to combine the estimations of group contribution with the more accurate reactant contributions by decomposing each reaction into two parts and applying one of the methods on each of them. This method gives priority to the reactant contributions over group contributions while guaranteeing that all estimations will be consistent, i.e. will not violate the first law of thermodynamics. We show that there is a significant increase in the accuracy of our estimations compared to standard group contribution. Specifically, our cross-validation results show an 80% reduction in the median absolute residual for reactions that can be derived by reactant contributions only. We provide the full framework and source code for deriving estimates of standard reaction Gibbs energy, as well as confidence intervals, and believe this will facilitate the wide use of thermodynamic data for a better understanding of metabolism.     

A computational analysis of protein interactions in metabolic networks reveals novel enzyme pairs potentially involved in metabolic channeling

Protein-protein interactions are operative at almost every level of cell structure and function as, for example, formation of sub-cellular organelles, packaging of chromatin, muscle contraction, signal transduction, and regulation of gene expression. Public databases of reported protein-protein interactions comprise hundreds of thousands interactions, and this number is steadily growing. Elucidating the implications of protein-protein interactions for the regulation of the underlying cellular or extra-cellular reaction network remains a great challenge for computational biochemistry. In this work, we have undertaken a systematic and comprehensive computational analysis of reported enzyme-enzyme interactions in the metabolic networks of the model organisms Escherichia coli and Saccharomyces cerevisiae. We grouped all enzyme pairs according to the topological distance that the catalyzed reactions have in the metabolic network and performed a statistical analysis of reported enzyme-enzyme interactions within these groups. We found a higher frequency of reported enzyme-enzyme interactions within the group of enzymes catalyzing reactions that are adjacent in the network, i.e. sharing at least one metabolite. As some of these interacting enzymes have already been implicated in metabolic channeling our analysis may provide a useful screening for candidates of this phenomenon. To check for a possible regulatory role of interactions between enzymes catalyzing non-neighboring reactions, we determined potentially regulatory enzymes using connectivity in the network and absolute change of Gibbs free energy. Indeed a higher portion of reported interactions pertain to such potentially regulatory enzymes.     

Optimal re-design of primary metabolism in Escherichia coli using linlog kinetics

This paper examines the validity of the linlog approach, which was recently developed in our laboratory, by comparison of two different kinetic models for the metabolic network of Escherichia coli. The first model is a complete mechanistic model; the second is an approximative model in which linlog kinetics are applied. The parameters of the linlog model (elasticities) are derived from the mechanistic model. Three different optimization cases are examined. In all cases, the objective is to calculate the enzyme levels that maximize a certain flux while keeping the total amount of enzyme constant and preventing large changes of metabolite concentrations. For an average variation of metabolite levels of 10% and individual changes of a factor 2, the predicted enzyme levels, metabolite concentrations and fluxes of both models are highly similar. This similarity holds for changes in enzyme level of a factor 4-6 and for changes in fluxes up to a factor 6. In all three cases, the predicted optimal enzyme levels could neither have been found by intuition-based approaches, nor on basis of flux control coefficients. This demonstrates that kinetic models are essential tools in Metabolic Engineering. In this respect, the linlog approach is a valuable extension of MCA, since it allows construction of kinetic models, based on MCA parameters, that can be used for constrained optimization problems and are valid for large changes of metabolite and enzyme levels.     

Thermodynamically based profiling of drug metabolism and drug-drug metabolic interactions: a case study of acetaminophen and ethanol toxic interaction

Drug-drug metabolic interactions can result in unwanted side effects, including reduced drug efficacy and formation of toxic metabolic intermediates. In this work, thermodynamic constraints on non-equilibrium metabolite concentrations are used to reveal the biochemical interactions between the metabolic pathways of ethanol and acetaminophen (N-acetyl-p-aminophenol), two drugs known to interact unfavorably. It is known that many reactions of these pathways are coupled to the central energy metabolic reactions through a number of metabolites and the cellular redox potential. Based on these observations, a metabolic network model has been constructed and a database of thermodynamic properties for all participating metabolites and reactions has been compiled. Constraint-based computational analysis of the feasible metabolite concentrations reveals that the non-toxic pathways for APAP metabolism and the pathway for detoxifying N-acetyl-p-benzoquinoneimine (NAPQI) are inhibited by network interactions with ethanol metabolism. These results point to the potential utility of thermodynamically based profiling of metabolic network interactions in screening of drug candidates and analysis of potential toxicity.     

Dynamic simulation and metabolic re-design of a branched pathway using linlog kinetics

This paper presents a new mathematical framework for modeling of in vivo dynamics and for metabolic re-design: the linlog approach. This approach is an extension of metabolic control analysis (MCA), valid for large changes of enzyme and metabolite levels. Furthermore, the presented framework combines MCA with kinetic modeling, thereby also combining the merits of both approaches. The linlog framework includes general expressions giving the steady-state fluxes and metabolite concentrations as a function of enzyme levels and extracellular concentrations, and a metabolic design equation that allows direct calculation of required enzyme levels for a desired steady state when control and response coefficients are available. Expressions giving control coefficients as a function of the enzyme levels are also derived. The validity of the linlog approximation in metabolic modeling is demonstrated by application of linlog kinetics to a branched pathway with moiety conservation, reversible reactions and allosteric interactions. Results show that the linlog approximation is able to describe the non-linear dynamics of this pathway very well for concentration changes up to a factor 20. Also the metabolic design equation was tested successfully.     

Metabolic flux control analysis of branch points: an improved approach to obtain flux control coefficients from large perturbation data

An overview of published approaches for the metabolic flux control analysis of branch points revealed that often not all fundamental constraints on the flux control coefficients have been taken into account. This has led to contradictory statements in literature on the minimum number of large perturbation experiments required to estimate the complete set of flux control coefficients C(J) for a metabolic branch point. An improved calculation procedure, based on approximate Lin-log reaction kinetics, is proposed, providing explicit analytical solutions of steady state fluxes and metabolite concentrations as a function of large changes in enzyme levels. The obtained solutions allow direct calculation of elasticity ratios from experimental data and subsequently all C(J)-values from the unique relation between elasticity ratio's and flux control coefficients. This procedure ensures that the obtained C(J)-values satisfy all fundamental constraints. From these it follows that for a three enzyme branch point only one characterised or two uncharacterised large flux perturbations are sufficient to obtain all C(J)- values. The improved calculation procedure is illustrated with four experimental cases.     

Time hierarchy, equilibrium and non-equilibrium in metabolic systems.

A metabolic system consists of cooperating biochemical reactions. The motion is described by differential equations in the metabolites. The right-hand sides of these equations are linear combinations of the velocities of the individual reactions. These velocities depend in a non-linear manner on the metabolite concentrations (according to the law of mass action). A characteristic "metabolic" time may be defined for the motion of the whole system. It scales the essential metabolic events whose evolution time is comparable to this metabolite time unit. The constituent reactions of the metabolic system have an individual characteristic time which need not coincide with the general metabolic time. The individual time characterises the approach to the individual equilibrium of the isolated undisturbed reaction. According to the ratio of these two time scales, a single reaction may be fast, or slow, or essential, as compared with the metabolic events. Characteristic time of a single reaction and its steady-state deviation from equilibrium are closely related. It can be shown that the relative deviation from equilibrium of a reaction within the metabolic network is of the same numerical order as the ratio between individual time to metabolic time. The interaction of many reactions with different characteristic times introduces a time hierarchy into the system. This can be made transparent by appropriate scaling and by linear transformation of the system. The subsystem of fast cooperating reactions (dehydrogenases, phosphotransferases) attains a state which is near to the individual equilibrium and reestablishes this state after perturbation. The equilibration is fast; an ultrarapid phase of cofactor equilibrium can be distinguished from the fast phase of substrate equilibrium (exchange of metabolic material between different pathways). During the slower metabolic phase these near-equilibria manifest themselves as stoichiometric linkage between unrelated metabolites. The latter cease to be independent variables and combine to metabolic pools. It can be strictly shown that the essential variables at the metabolic time scale are carrier pools and the degree of occupancy of these carriers by metabolic groups. Chemically different types of carrier pools may be functionally linked together by fast reactions. A consequence of such an arrangement of reactions are distance effects: Changes at one end of a metabolic map may be directly conveyed to other pathways via stoichiometric linkage brought about by fast equilibration of cofactor reactions.     

Interpreting Metabolomic Profiles using Unbiased Pathway Models

Human disease is heterogeneous, with similar disease phenotypes resulting from distinct combinations of genetic and environmental factors. Small-molecule profiling can address disease heterogeneity by evaluating the underlying biologic state of individuals through non-invasive interrogation of plasma metabolite levels. We analyzed metabolite profiles from an oral glucose tolerance test (OGTT) in 50 individuals, 25 with normal (NGT) and 25 with impaired glucose tolerance (IGT). Our focus was to elucidate underlying biologic processes. Although we initially found little overlap between changed metabolites and preconceived definitions of metabolic pathways, the use of unbiased network approaches identified significant concerted changes. Specifically, we derived a metabolic network with edges drawn between reactant and product nodes in individual reactions and between all substrates of individual enzymes and transporters. We searched for “active modules”—regions of the metabolic network enriched for changes in metabolite levels. Active modules identified relationships among changed metabolites and highlighted the importance of specific solute carriers in metabolite profiles. Furthermore, hierarchical clustering and principal component analysis demonstrated that changed metabolites in OGTT naturally grouped according to the activities of the System A and L amino acid transporters, the osmolyte carrier SLC6A12, and the mitochondrial aspartate-glutamate transporter SLC25A13. Comparison between NGT and IGT groups supported blunted glucose- and/or insulin-stimulated activities in the IGT group. Using unbiased pathway models, we offer evidence supporting the important role of solute carriers in the physiologic response to glucose challenge and conclude that carrier activities are reflected in individual metabolite profiles of perturbation experiments. Given the involvement of transporters in human disease, metabolite profiling may contribute to improved disease classification via the interrogation of specific transporter activities.     

In vivo kinetics of primary metabolism in Saccharomyces cerevisiae studied through prolonged chemostat cultivation

In this study, prolonged chemostat cultivation is applied to investigate in vivo enzyme kinetics of Saccharomyces cerevisiae. S. cerevisiae was grown in carbon-limited aerobic chemostats for 70-95 generations, during which multiple steady states were observed, characterized by constant intracellular fluxes but significant changes in intracellular metabolite concentrations and enzyme capacities. We provide evidence for two relevant kinetic mechanisms for sustaining constant fluxes: in vivo near-equilibrium of reversible reactions and tight regulation of irreversible reactions by coordinated changes of metabolic effectors. Using linear-logarithmic kinetics, we illustrate that these multiple steady-state measurements provide linear constraints between elasticity parameters instead of their absolute values. Upon perturbation by a glucose pulse, glucose uptake and ethanol excretion in prolonged cultures were remarkably lower, compared to a reference culture perturbed at 10 generations. Metabolome measurements during the transient indicate that the differences might be due to a reduced ATP regeneration capacity in prolonged cultures.     

Thermodynamic calculations for biochemical transport and reaction processes in metabolic networks

Thermodynamic analysis of metabolic networks has recently generated increasing interest for its ability to add constraints on metabolic network operation, and to combine metabolic fluxes and metabolite measurements in a mechanistic manner. Concepts for the calculation of the change in Gibbs energy of biochemical reactions have long been established. However, a concept for incorporation of cross-membrane transport in these calculations is still missing, although the theory for calculating thermodynamic properties of transport processes is long known. Here, we have developed two equivalent equations to calculate the change in Gibbs energy of combined transport and reaction processes based on two different ways of treating biochemical thermodynamics. We illustrate the need for these equations by showing that in some cases there is a significant difference between the proposed correct calculation and using an approximative method. With the developed equations, thermodynamic analysis of metabolic networks spanning over multiple physical compartments can now be correctly described.