Graph-theoretical identification of pathways for biochemical reactions

A rigorous method for identifying biochemical reaction or metabolic pathways through its systematic synthesis has been established. The current method for synthesizing networks of metabolic pathways follows the general framework of a highly exacting combinatorial method. The method is capable of generating not only all combinatorially independent, feasible reaction networks only once, but also those combinations of independent pathways. A case study involving the conversion of glucose to pyruvate with 14 elementary reactions illustrates the efficiency and efficacy of the method. All the results have been obtained with a PC (Pentium-III 550 MHz, 256 MB RAM) within 1 s.     

Dynamic modeling of complex biological systems: a link between metabolic and macroscopic description

In this study, a class of dynamic models based on metabolic reaction pathways is analyzed, showing that systems with complex intracellular reaction networks can be represented by macroscopic reactions relating extracellular components only. Based on rigorous assumptions, the model reduction procedure is systematic and allows an equivalent 'input-output' representation of the system to be derived. The procedure is illustrated with a few examples.     

High quality visualization of biochemical pathways in BioPath

Biochemical reactions form large and complex networks. Comprehensible visual representations of these networks help biochemists understand the relationships between the chemical components. Typically pathway diagrams are manually produced drawings. Because of the steady progress of knowledge and the complex relationships in these networks, automatic visualizations are necessary. Bio-Path is a system for the exploration and automatic visualization of biochemical pathways. It has been developed to obtain an electronic version of the well-known Boehringer Biochemical Pathways poster and offers new possibilities to find information and to navigate through pathways. BioPath has a specific database containing reactions and a hierarchical clustering of reactions and reaction networks. One feature is the automatic generation of pathways from the database and their high quality visualization. This paper states the requirements for the visualization of biochemical pathways, presents a layout algorithm and shows how BioPath can be used to explore biochemical reaction networks.     

Construction of an E. Coli genome-scale atom mapping model for MFA calculations

Metabolic flux analysis (MFA) has so far been restricted to lumped networks lacking many important pathways, partly due to the difficulty in automatically generating isotope mapping matrices for genome-scale metabolic networks. Here we introduce a procedure that uses a compound matching algorithm based on the graph theoretical concept of pattern recognition along with relevant reaction information to automatically generate genome-scale atom mappings which trace the path of atoms from reactants to products for every reaction. The procedure is applied to the iAF1260 metabolic reconstruction of Escherichia coli yielding the genome-scale isotope mapping model imPR90068. This model maps 90,068 non-hydrogen atoms that span all 2,077 reactions present in iAF1260 (previous largest mapping model included 238 reactions). The expanded scope of the isotope mapping model allows the complete tracking of labeled atoms through pathways such as cofactor and prosthetic group biosynthesis and histidine metabolism. An EMU representation of imPR90068 is also constructed and made available.     

Topological analysis of mass-balanced signaling networks: a framework to obtain network properties including crosstalk

Signal transduction networks have only been studied at a small scale because large-scale reconstructions and suitable in silico analysis methods have not been available. Since reconstructions of large signaling networks are progressing well there is now a need to develop a framework for analysing structural properties of signaling networks. One such framework is presented here, one that is based on systemically independent pathways and a mass-balanced representation of signaling events. This approach was applied to a prototypic signaling network and it allowed for: (1) a systemic analysis of all possible input/output relationships, (2) a quantitative evaluation of network crosstalk, or the interconnectivity of systemically independent pathways, (3) a measure of the redundancy in the signaling network, (4) the participation of reactions in signaling pathways, and (5) the calculation of correlated reaction sets. These properties emerge from network structure and can only be derived and studied within a defined mathematical framework. The calculations presented are the first of their kind for a signaling network, while similar analysis has been extensively performed for prototypic and genome-scale metabolic networks. This approach does not yet account for dynamic concentration profiles. Due to the scalability of the stoichiometric formalism used, the results presented for the prototypic signaling network can be obtained for large signaling networks once their reconstruction is completed.     

Gradient directed regularization for sparse Gaussian concentration graphs, with applications to inference of genetic networks

Large-scale microarray gene expression data provide the possibility of constructing genetic networks or biological pathways. Gaussian graphical models have been suggested to provide an effective method for constructing such genetic networks. However, most of the available methods for constructing Gaussian graphs do not account for the sparsity of the networks and are computationally more demanding or infeasible, especially in the settings of high dimension and low sample size. We introduce a threshold gradient descent (TGD) regularization procedure for estimating the sparse precision matrix in the setting of Gaussian graphical models and demonstrate its application to identifying genetic networks. Such a procedure is computationally feasible and can easily incorporate prior biological knowledge about the network structure. Simulation results indicate that the proposed method yields a better estimate of the precision matrix than the procedures that fail to account for the sparsity of the graphs. We also present the results on inference of a gene network for isoprenoid biosynthesis in Arabidopsis thaliana. These results demonstrate that the proposed procedure can indeed identify biologically meaningful genetic networks based on microarray gene expression data.     

Constraints and missing reactions in the urea cycle.

The stoichiometric relations in a series of biochemical reactions are summarized by a stoichiometric number matrix (with a column for each reaction) and a conservation matrix (with a row for each constraint). These two matrices for a series or cycle of biochemical reactions are related because the columns of the stoichiometric number matrix are in the null space of the conservation matrix, and the rows of the transpose of the conservation matrix are in the null space of the transpose of the stoichiometric number matrix. The conservation matrix for a system of biochemical reactions is of interest because it shows the nature of the constraints in addition to the conservation of atoms and groups. Constraints beyond those for the conservation of atoms and groups indicate "missing reactions" that do not occur because the enzymes involved couple reactions that could occur and still conserve atoms and groups. The interpretation of conservation matrices and stoichiometric matrices for a reaction system is complicated by the fact that they are not unique. However, their row-reduced forms are unique, as are their dimensions, which represent the number of reactants and number of independent reactions. Two matrices that look different contain the same information if they have the same row-reduced form. The urea cycle, which involves five enzyme-catalyzed reactions, and its net reaction are discussed in terms of the linear constraints produced by enzyme catalysis. A procedure to obtain a set of conservation equations that will yield the correct net reaction is described.     

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/     

Metabolic design of macroscopic bioreaction models: application to Chinese hamster ovary cells

The aim of this paper is to present a systematic methodology to design macroscopic bioreaction models for cell cultures based upon metabolic networks. The cell culture is seen as a succession of phases. During each phase, a metabolic network represents the set of reactions occurring in the cell. Then, through the use of the elementary flux modes, these metabolic networks are used to derive macroscopic bioreactions linking the extracellular substrates and products. On this basis, as many separate models are obtained as there are phases. Then, a complete model is obtained by smoothly switching from model to model. This is illustrated with batch cultures of Chinese hamster ovary cells.     

A microscale model of bacterial and biofilm dynamics in porous media

A microscale model for the transport and coupled reaction of microbes and chemicals in an idealized two-dimensional porous media has been developed. This model includes the flow, transport, and bioreaction of nutrients, electron acceptors, and microbial cells in a saturated granular porous media. The fluid and chemicals are represented as a continuum, but the bacterial cells and solid granular particles are represented discretely. Bacterial cells can attach to the particle surfaces or be advected in the bulk fluid. The bacterial cells can also be motile and move preferentially via a run and tumble mechanism toward a chemoattractant. The bacteria consume oxygen and nutrients and alter the profiles of these chemicals. Attachment of bacterial cells to the soil matrix and growth of bacteria can change the local permeability. The coupling of mass transport and bioreaction can produce spatial gradients of nutrients and electron acceptor concentrations. We describe a numerical method for the microscale model, show results of a convergence study, and present example simulations of the model system.     

The chemical organization of signaling interactions

MOTIVATION: Cellular chemical signaling pathways form complex networks that are beginning to be studied at the level of chemical kinetics and databases of reactions. Chemical reaction details are traditionally represented as lists of reactions and rates. This does not map readily to the block diagram representation familiar to biologists, and obscures the functional organization of signaling networks. This study examines motifs in signaling chemistry and reports common features that may help to formalize such a mapping between pathway block diagrams and the chemistry. The same motifs may facilitate data representation and provide functional abstraction of the chemistry. RESULTS: I classified 74 interactions between 25 signaling pathways in terms of shared chemical motifs. All interactions in this dataset consist of a few communicating molecules from one set of pathways, and a replicating set of reactions and molecules from another. Each unique combination of interacting pathways duplicates the chemical reaction scheme of this replicating set, but involves different rate constants. Signaling pathways can therefore be described in an object-oriented manner as sets of core reactions with well-defined interfaces between pathways. This generalization lends itself to designing simulators and databases for signaling networks. AVAILABILITY: Software and example models are freely available from http://www.ncbs.res.in/~bhalla/examples/EGFR_example.html.     

Non-linear reduction for kinetic models of metabolic reaction networks

Kinetic models of metabolic networks are essential for predicting and optimizing the transient behavior of cells in culture. However, such models are inherently high dimensional and stiff due to the large number of species and reactions involved and to kinetic rate constants of widely different orders of magnitude. In this paper we address the problem of deriving non-stiff, reduced-order non-linear models of the dominant dynamics of metabolic networks with fast and slow reactions. We present a method, based on singular perturbation analysis, which allows the systematic identification of quasi-steady-state conditions for the fast reactions, and the derivation of explicit non-linear models of the slow dynamics independent of the fast reaction rate expressions. The method is successfully applied to detailed models of metabolism in human erythrocytes and Saccharomyces cerevisiae.     

Hierarchical analysis of dependency in metabolic networks

MOTIVATION: Elucidation of metabolic networks for an increasing number of organisms reveals that even small networks can contain thousands of reactions and chemical species. The intimate connectivity between components complicates their decomposition into biologically meaningful sub-networks. Moreover, traditional higher-order representations of metabolic networks as metabolic pathways, suffers from the lack of rigorous definition, yielding pathways of disparate content and size. RESULTS: We introduce a hierarchical representation that emphasizes the gross organization of metabolic networks in largely independent pathways and sub-systems at several levels of independence. The approach highlights the coupling of different pathways and the shared compounds responsible for those couplings. By assessing our results on Escherichia coli (E.coli metabolic reactions, Genetic Circuits Research Group, University of California, San Diego, http://gcrg.ucsd.edu/organisms/ecoli.html, 'model v 1.01. reactions') against accepted biochemical annotations, we provide the first systematic synopsis of an organism's metabolism. Comparison with operons of E.coli shows that low-level clusters are reflected in genome organization and gene regulation. AVAILABILITY: Source code, data sets and supplementary information are available at http://www.mas.ecp.fr/labo/equipe/gagneur/hierarchy/hierarchy.html     

A simple analogical model for the selection of optical activity and of the most efficient catalysts in the course of molecular evolution

Analogical models, based on the use of simple hydrodynamic devices are proposed to discuss qualitatively and easily the properties of chemical networks of reactions implying a limited supply of a reagent usable in several parallel autocatalytic reaction pathways.This discussion shows that the racemic state can be unstable at the steady state in such systems and that the pathways involving the most efficient catalysts are selected in the course of their chemical evolution.     

Expanding Metabolic Networks: Scopes of Compounds, Robustness, and Evolution

A new method for the mathematical analysis of large metabolic networks is presented. Based on the fact that the occurrence of a metabolic reaction generally requires the existence of other reactions providing its substrates, series of metabolic networks are constructed. In each step of the corresponding expansion process those reactions are incorporated whose substrates are made available by the networks of the previous generations. The method is applied to the set of all metabolic reactions included in the KEGG database. Starting with one or more seed compounds, the expansion results in a final network whose compounds define the scope of the seed. Scopes of all metabolic compounds are calculated and it is shown that large parts of cellular metabolism can be considered as the combined scope of simple building blocks. Analyses of various expansion processes reveal crucial metabolites whose incorporation allows for the increase in network complexity. Among these metabolites are common cofactors such as NAD⁺, ATP, and coenzyme A. We demonstrate that the outcome of network expansion is in general very robust against elimination of single or few reactions. There exist, however, crucial reactions whose elimination results in a dramatic reduction of scope sizes. It is hypothesized that the expansion process displays characteristics of the evolution of metabolism such as the temporal order of the emergence of metabolic pathways. [Reviewing Editor : Dr. David Pollock]     

Sensitivity, principal component and flux analysis applied to signal transduction: the case of epidermal growth factor mediated signaling

MOTIVATION: Novel high-throughput genomic and proteomic tools are allowing the integration of information from a range of biological assays into a single conceptual framework. This framework is often described as a network of biochemical reactions. We present strategies for the analysis of such networks. RESULTS: The direct differential method is described for the systematic evaluation of scaled sensitivity coefficients in reaction networks. Principal component analysis, based on an eigenvalue-eigenvector analysis of the scaled sensitivity coefficient matrix, is applied to rank individual reactions in the network based on their effect on system output. When combined with flux analysis, sensitivity analysis allows model reduction or simplification. Using epidermal growth factor (EGF) mediated signaling and trafficking as an example of signal transduction, we demonstrate that sensitivity analysis quantitatively reveals the dependence of dual-phosphorylated extracellular signal-regulated kinase (ERK) concentration on individual reaction rate constants. It predicts that EGF mediated reactions proceed primarily via an Shc-dependent pathway. Further, it suggests that receptor internalization and endosomal signaling are important features regulating signal output only at low EGF dosages and at later times.     

Metabolic Pathfinding Using RPAIR Annotation

Metabolic databases contain information about thousands of small molecules and reactions, which can be represented as networks. In the context of metabolic reconstruction, pathways can be inferred by searching optimal paths in such networks. A recurrent problem is the presence of pool metabolites (e.g., water, energy carriers, and cofactors), which are connected to hundreds of reactions, thus establishing irrelevant shortcuts between nodes of the network. One solution to this problem relies on weighted networks to penalize highly connected compounds. A more refined solution takes the chemical structure of reactants into account in order to differentiate between side and main compounds of a reaction. Thanks to an intensive annotation effort at KEGG, decompositions of reactions into reactant pairs (RPAIR) categorized by their role (main, trans, cofac, ligase, and leave) are now available.The goal of this article is to evaluate the impact of RPAIR data on pathfinding in metabolic networks. To this end, we measure the impact of different parameters concerning the construction of the metabolic network: mapping of reactions and reactant pairs onto a graph, use of selected categories of reactant pairs, weighting schemes for compounds and reactions, removal of highly connected metabolites, and reaction directionality. In total, we tested 104 combinations of parameters and identified their optimal values for pathfinding on the basis of 55 reference pathways from three organisms.The best-performing metabolic network combines the biochemical knowledge encoded by KEGG RPAIR with a weighting scheme penalizing highly connected compounds. With this network, we could recover reference pathways from Escherichia coli with an average accuracy of 93% (32 pathways), from Saccharomyces cerevisiae with an average accuracy of 66% (11 pathways), and from humans with an average accuracy of 70% (12 pathways). Our pathfinding approach is available as part of the Network Analysis Tools.     

A modular modulation method for achieving increases in metabolite production

Increasing the production of overproducing strains represents a great challenge. Here, we develop a modular modulation method to determine the key steps for genetic manipulation to increase metabolite production. The method consists of three steps: (i) modularization of the metabolic network into two modules connected by linking metabolites, (ii) change in the activity of the modules using auxiliary rates producing or consuming the linking metabolites in appropriate proportions and (iii) determination of the key modules and steps to increase production. The mathematical formulation of the method in matrix form shows that it may be applied to metabolic networks of any structure and size, with reactions showing any kind of rate laws. The results are valid for any type of conservation relationships in the metabolite concentrations or interactions between modules. The activity of the module may, in principle, be changed by any large factor. The method may be applied recursively or combined with other methods devised to perform fine searches in smaller regions. In practice, it is implemented by integrating to the producer strain heterologous reactions or synthetic pathways producing or consuming the linking metabolites. The new procedure may contribute to develop metabolic engineering into a more systematic practice. © 2015 American Institute of Chemical Engineers Biotechnol. Prog., 31:656–667, 2015     

Reconstruction of metabolic networks from genome data and analysis of their global structure for various organisms

MOTIVATION: Information from fully sequenced genomes makes it possible to reconstruct strain-specific global metabolic network for structural and functional studies. These networks are often very large and complex. To properly understand and analyze the global properties of metabolic networks, methods for rationally representing and quantitatively analyzing their structure are needed. RESULTS: In this work, the metabolic networks of 80 fully sequenced organisms are in silico reconstructed from genome data and an extensively revised bioreaction database. The networks are represented as directed graphs and analyzed by using the 'breadth first searching algorithm to identify the shortest pathway (path length) between any pair of the metabolites. The average path length of the networks are then calculated and compared for all the organisms. Different from previous studies the connections through current metabolites and cofactors are deleted to make the path length analysis physiologically more meaningful. The distribution of the connection degree of these networks is shown to follow the power law, indicating that the overall structure of all the metabolic networks has the characteristics of a small world network. However, clear differences exist in the network structure of the three domains of organisms. Eukaryotes and archaea have a longer average path length than bacteria. AVAILABILITY: The reaction database in excel format and the programs in VBA (Visual Basic for Applications) are available upon request. SUPPLEMENTARY MATERIAL: Bioinformatics Online.     

Modeling mitochondrial function

The mitochondrion represents a unique opportunity to apply mathematical modeling to a complex biological system. Understanding mitochondrial function and control is important since this organelle is critical in energy metabolism as well as playing key roles in biochemical synthesis, redox control/signaling, and apoptosis. A mathematical model, or hypothesis, provides several useful insights including a rigorous test of the consensus view of the operation of a biological process as well as providing methods of testing and creating new hypotheses. The advantages of the mitochondrial system for applying a mathematical model include the relative simplicity and understanding of the matrix reactions, the ability to study the mitochondria as a independent contained organelle, and, most importantly, one can dynamically measure many of the internal reaction intermediates, on line. The developing ability to internally monitor events within the metabolic network, rather than just the inflow and outflow, is extremely useful in creating critical bounds on complex mathematical models using the individual reaction mechanisms available. However, many serious problems remain in creating a working model of mitochondrial function including the incomplete definition of metabolic pathways, the uncertainty of using in vitro enzyme kinetics, as well as regulatory data in the intact system and the unknown chemical activities of relevant molecules in the matrix. Despite these formidable limitations, the advantages of the mitochondrial system make it one of the best defined mammalian metabolic networks that can be used as a model system for understanding the application and use of mathematical models to study biological systems.