Bayesian modeling of complex metabolic pathways

Many chronic diseases are the result of a complex sequence of biochemical reactions involving exposures to various environmental agents, metabolized by a number of different genes. Routine epidemiologic analyses of such associations have tended to rely on standard contingency table or logistic regression methods, typically focusing on one variable at a time or pairwise combinations. We consider two statistical alternatives to this approach, one based on Bayesian model averaging, one based on pharmacokinetic modeling of the biochemical pathways. These approaches are illustrated using data from a case-control study of colorectal polyps in relation to tobacco smoking and consumption of well done red meat, both viewed as sources of heterocyclic amines and polycyclic aromatic hydrocarbons. The new analyses are structured in a manner that attempts to take advantage of prior knowledge of the metabolism of these classes of compounds and the various genes that regulate these pathways.     

A mathematical model of metabolic insulin signaling pathways

We develop a mathematical model that explicitly represents many of the known signaling components mediating translocation of the insulin-responsive glucose transporter GLUT4 to gain insight into the complexities of metabolic insulin signaling pathways. A novel mechanistic model of postreceptor events including phosphorylation of insulin receptor substrate-1, activation of phosphatidylinositol 3-kinase, and subsequent activation of downstream kinases Akt and protein kinase C-zeta is coupled with previously validated subsystem models of insulin receptor binding, receptor recycling, and GLUT4 translocation. A system of differential equations is defined by the structure of the model. Rate constants and model parameters are constrained by published experimental data. Model simulations of insulin dose-response experiments agree with published experimental data and also generate expected qualitative behaviors such as sequential signal amplification and increased sensitivity of downstream components. We examined the consequences of incorporating feedback pathways as well as representing pathological conditions, such as increased levels of protein tyrosine phosphatases, to illustrate the utility of our model for exploring molecular mechanisms. We conclude that mathematical modeling of signal transduction pathways is a useful approach for gaining insight into the complexities of metabolic insulin signaling.     

An integrated modeling-experimental strategy for the analysis of metabolic pathways

An inherent problem in studying the behavior of a metabolic pathway is the impossibility of developing a complete, detailed model that includes all the cellular processes that have an impact on the set of fluxes in such a pathway. Lacking this, one requires some means of modeling the interactions between a metabolic pathway and other cellular processes for the purpose of analyzing pathway characteristics within the cell (e.g., determining sensitivity coefficients for various steps in the pathway) with a minimal amount of time and effort. A general framework is developed for studying these issues in a rigorous manner. Using this framework, detailed knowledge about a metabolic pathway (i.e., a set of rate expressions for steps in the pathway) can be combined with the results from a relatively simple set of experiments in order to obtain estimates for the sensitivity of the pathway to enzyme activities, inhibition constants, and other parameters that determine the pathway's behavior, while accounting for the pathway's interaction with the rest of the cellular metabolism. A model system representing amino acid production is used to illustrate the problem and to provide results based on computational experiments. The modeling strategy described here should be useful in genetic design to improve pathway fluxes and metabolic network selectivity.     

Experimental and mathematical approaches to modeling plant metabolic networks

To support their sessile and autotrophic lifestyle higher plants have evolved elaborate networks of metabolic pathways. Dynamic changes in these metabolic networks are among the developmental forces underlying the functional differentiation of organs, tissues and specialized cell types. They are also important in the various interactions of a plant with its environment. Further complexity is added by the extensive compartmentation of the various interconnected metabolic pathways in plants. Thus, although being used widely for assessing the control of metabolic flux in microbes, mathematical modeling approaches that require steady-state approximations are of limited utility for understanding complex plant metabolic networks. However, considerable progress has been made when manageable metabolic subsystems were studied. In this article, we will explain in general terms and using simple examples the concepts underlying stoichiometric modeling (metabolic flux analysis and metabolic pathway analysis) and kinetic approaches to modeling (including metabolic control analysis as a special case). Selected studies demonstrating the prospects of these approaches, or combinations of them, for understanding the control of flux through particular plant pathways are discussed. We argue that iterative cycles of (dry) mathematical modeling and (wet) laboratory testing will become increasingly important for simulating the distribution of flux in plant metabolic networks and deriving rational experimental designs for metabolic engineering efforts.     

An algorithm for efficient identification of branched metabolic pathways

This article presents a new graph-based algorithm for identifying branched metabolic pathways in multi-genome scale metabolic data. The term branched is used to refer to metabolic pathways between compounds that consist of multiple pathways that interact biochemically. A branched pathway may produce a target compound through a combination of linear pathways that split compounds into smaller ones, work in parallel with many compounds, and join compounds into larger ones. While branched metabolic pathways predominate in metabolic networks, most previous work has focused on identifying linear metabolic pathways. The ability to automatically identify branched pathways is important in applications that require a deeper understanding of metabolism, such as metabolic engineering and drug target identification. The algorithm presented in this article utilizes explicit atom tracking to identify linear metabolic pathways and then merges them together into branched metabolic pathways. We provide results on several well-characterized metabolic pathways that demonstrate that the new merging approach can efficiently find biologically relevant branched metabolic pathways.     

Determination of the mechanism of retrotransfer by mechanistic mathematical modeling.

Two mathematical models to elucidate the mechanism of retromobilization (or retrotransfer), that is, the ability of conjugative plasmids to mobilize genes into the cell containing the conjugative plasmid, were developed. This study deals with retromobilization of nonconjugative plasmids (Tra-Mob+). Plasmid transfer was modeled by two mass action models. The first is based on the hypothesis that retromobilization of the Tra-Mob+ vector occurs in one step, by means of the pilus formed by the Tra+ plasmid in the original host. In the second model, retromobilization is considered to be a two-step process involving two transfer events. The first step involves the transfer of the Tra+ plasmid from the recipient cell to the donor of the nonconjugative vector, and during the second encounter the nonconjugative vector is mobilized toward the recipient. Since the relationships between the number of transconjugants and the number of recipients for the two models are different, filter matings were performed for short time periods with different initial densities of the recipient population. Comparison of the numbers of transconjugants with the results of the mathematical equations confirmed the hypothesis that retromobilization is a one-step conjugation process.     

A flexible state-space approach for the modeling of metabolic networks I: development of mathematical methods

We introduce a novel, flexible, optimization-based mathematical framework for the modeling of arbitrarily complex metabolic networks: topological metabolic analysis (TMA). The framework is adapted from state-space approaches used by Manousiouthakis and co-workers for the representation of complex heat- and mass-exchanger networks. We offer a thorough discussion of the mathematics and general theory underlying the framework, and discuss certain mathematical advantages of our modeling representation in comparison with other commonly used techniques (MFA and FBA). We employ a novel aggregate objective function for use with our basic constraint model, including a generalized least-squares term (for fitting available experimental measurements) and a linear design term (for representing biological or engineering goals). Using a case-study taken from recent literature (McKinlay et al., 2007), we demonstrate (among other benefits) the ability of this objective to identify alternate distinct-yet-equally optimal solutions for a given modeling problem. We also show that these solutions, obtained using only external metabolite uptake and secretion measurements, provide useful biological insights and compare favorably with solutions obtained on the basis of (13)C isotope-tracing data.     

Application of a generalized MWC model for the mathematical simulation of metabolic pathways regulated by allosteric enzymes

In our effort to elucidate the systems biology of the model organism, Escherichia coli, we have developed a mathematical model that simulates the allosteric regulation for threonine biosynthesis pathway starting from aspartate. To achieve this goal, we used kMech, a Cellerator language extension that describes enzyme mechanisms for the mathematical modeling of metabolic pathways. These mechanisms are converted by Cellerator into ordinary differential equations (ODEs) solvable by Mathematica. In this paper, we describe a more flexible model in Cellerator, which generalizes the Monod, Wyman, Changeux (MWC) model for enzyme allosteric regulation to allow for multiple substrate, activator and inhibitor binding sites. Furthermore, we have developed a model that describes the behavior of the bifunctional allosteric enzyme aspartate kinase I-homoserine dehydrogenase I (AKI-HDHI). This model predicts the partition of enzyme activities in the steady state which paves the way for a more generalized prediction of the behavior of bifunctional enzymes.     

NADP-malic enzyme from plants: a ubiquitous enzyme involved in different metabolic pathways

NADP-malic enzyme (NADP-ME) is a widely distributed enzyme that catalyzes the oxidative decarboxylation of L-malate. Photosynthetic NADP-MEs are found in C4 bundle sheath chloroplasts and in the cytosol of CAM plants, while non-photosynthetic NADP-MEs are either plastidic or cytosolic in various plants. We propose a classification of plant NADP-MEs based on their physiological function and localization and we describe recent advances in the characterization of each isoform. Based on the alignment of amino acid sequences of plant NADP-MEs, we identify putative binding sites for the substrates and analyze the phylogenetic origin of each isoform, revealing several features of the molecular evolution of this ubiquitous enzyme.     

Mathematical modelling of metabolic pathways affected by an enzyme deficiency. A mathematical model of glycolysis in normal and pyruvate-kinase-deficient red blood cells

A mathematical model of glycolysis in human erythrocytes is proposed to study the influence of a pyruvate kinase deficiency on the energy metabolism. The model takes into account the main regulatory properties of the non-equilibrium enzymes and the magnesium-complex formation by the adenine nucleotides and by 2,3-bisphosphoglycerate. In the normal case (no enzyme defect) the calculated flux rates and metabolite concentrations are in a good agreement with experimental data. It is shown that a severe pyruvate kinase deficiency manifested in a tenfold diminished activity of that enzyme leads to a remarkable decrease of the glycolytic flux and the ATP concentration of about 50% of the normal values. On the other hand a lowering of the pyruvate kinase activity to half of the normal value, characteristic for the heterozygotes, gives no significant alterations of the metabolite concentrations and the flux rates compared with the normal case which is in accordance with the lack of clinical symptoms for a metabolic disease of these probands. For three patients with known alterations of their pyruvate kinase mutants the calculated metabolite concentrations and the control characteristics permit estimation of the degree of disorder of the glycolytic pathway. The resulting classification corresponds well to other independent experimental and clinical findings. In particular, the calculation demonstrates that there is no simple correlation between the lowered enzyme activity and the reduced flux rate through the affected pathway.     

A systems biology framework for modeling metabolic enzyme inhibition of Mycobacterium tuberculosis

Background  

Because metabolism is fundamental in sustaining microbial life, drugs that target pathogen-specific metabolic enzymes and pathways can be very effective. In particular, the metabolic challenges faced by intracellular pathogens, such as Mycobacterium tuberculosis, residing in the infected host provide novel opportunities for therapeutic intervention.     

The rationalization of high enzyme concentration in metabolic pathways such as glycolysis

The cellular concentration of enzymes of some major metabolic pathways, such as glycolysis, can approach millimolarity. This concentration of enzyme can catalyze in vitro rates which are 100-fold higher than maximum pathway flux. In an attempt to understand the need for such high enzyme concentration, an artificial metabolic pathway of five enzymes (apropos the central enzymes of glycolysis) has been modeled. Numerical methods were then used to determine the effect of enzyme concentration on: (1) the change in total free metabolite concentration as the pathway changes from low flux to high flux, (2) the time lag (transient time) in the rate of final product formation upon the transition from low flux to high flux. Both the changes in metabolite pool size and the transient time decreased with increased enzyme concentrations. When all enzymatic reactions were assigned Keq of unity, a concentration for each enzyme of 25 microM is sufficient to provide a transient time of 1 sec. When Keq different from unity are introduced, more enzyme is required to provide comparably short transient times. Under the latter condition, a pathway of sufficiently low transient time would require all the enzyme available in mammalian muscle. It is shown that there is little scope for further increases in either enzyme concentration or of catalytic efficiency of independent enzymes. Therefore, an alternative method of increasing efficiency is considered in which enzyme-bound metabolites can serve directly as substrates for subsequent enzymes in a metabolic pathway.