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Modeling of Biological Networks Modeling Biological Networks Dr. Carlo Cosentino School of Computer and Biomedical Engineering Department of Experimental and Clinical Medicine Università degli Studi Magna Graecia Catanzaro, Italy [email protected] http://bioingegneria.unicz.it/~cosentino Dr. Carlo Cosentino1 Carnegie Mellon University, Pittsburgh, 2008 Outline Ó Classification of biological networks Ó Modeling metabolic networks Ó Modeling gene regulatory networks Ó Inferring gene regulatory networks Dr. Carlo Cosentino2 Carnegie Mellon University, Pittsburgh, 2008 Types of Biological Network Ó Several different kinds of biological network can be distinguished at the molecular level Ô Gene regulatory Ô Metabolic Ô Signal transduction Ô Protein–protein interaction Ó Moreover other networks can be considered as we move to different description levels, e.g. Ô Immunological Ô Ecological Ó Here we will focus exclusively on molecular processes that take place within the cell Dr. Carlo Cosentino3 Carnegie Mellon University, Pittsburgh, 2008 Goals Ó A major challenge consists in identifying with reasonable accuracy the complex macromolecular interactions at the gene, metabolite and protein levels Ó Once identified, the network model can be used to Ô simulate the process it represents Ô predict the features of its dynamical behavior Ô extrapolate cellular phenotypes Dr. Carlo Cosentino4 Carnegie Mellon University, Pittsburgh, 2008 Graphs Ó A very useful formal tool for describing and visualizing biological networks is represented by graphs Ô A graph, or undirected graph, is an ordered pair G=(V,E), where V is the set of the vertices, or nodes, and E is the set of unordered pairs of distinct vertices, called edges or lines Ô For each edge {u,v}, the nodes u and v are said to be adjacent Ô We have a directed graph, or digraph, if E is a set of ordered pairs Ô In digraphs, the in–degree, kin, (out–degree, kout) of a node is the number of edges incident to (from) that node Barabasi et al, Nature Review Genetics 101(5), 101–114 , 2004 Dr. Carlo Cosentino5 Carnegie Mellon University, Pittsburgh, 2008 Topological Characteristics Ô The degree distribution , P(k), gives the probability that a selected node has exactly k links Ô It allows us to distinguish between different classes of networks (see next slide) Ô The clustering coefficient of a node I, CI, measures the aggregation of its adjacents (number of “triangles” passing through node I) Ô C(k) is the average clustering coefficient of all nodes with k links Barabasi et al, Nature Review Genetics 101(5), 101–114 , 2004 Dr. Carlo Cosentino6 Carnegie Mellon University, Pittsburgh, 2008 Erdös–Rényi Random Networks Ó The Erdös–Rényi model of a random network starts with N nodes and connects each pair of nodes with probability p Ó The degree follows a Poisson distribution, thus many nodes have the same number of links (close to the average degree <k> Ó The tail decreases exponentially, which indicates that nodes with k very different from the average are rare Ó The clustering coefficient is independent of a node’s degree Barabasi et al, Nature Review Genetics 101(5), 101–114 , 2004 Dr. Carlo Cosentino7 Carnegie Mellon University, Pittsburgh, 2008 Scale–Free Networks Ó Scale–free networks are characterized by a power–law degree distribution Ó The probability that a node has k links follows P(k)~k-γ, where γ is the degree exponent Ó The probability that a node is highly connected is statistically more significant than in a random graph Ó In the Barabási–Albert model, at each time point a node with M links is added to the network, which connects to an already existing node I with probability Πi=ki/Σjkj Ó The underlying mechanism is that nodes with many links have higher probability of getting more (this is also referred to as preferential attachment) Barabasi et al, Nature Review Genetics 101(5), 101–114 , 2004 Dr. Carlo Cosentino8 Carnegie Mellon University, Pittsburgh, 2008 Hierarchical Networks Ó A hierarchical structure arises in systems that combine modularity and scale–free topology Ó The hierarchical model is based on the replication of a small cluster of four nodes (the central ones) Ó The external nodes of the replicas are linked to the central node of the original cluster Ó The resulting network has a power–law degree distribution, thus it is scale–free Ó The average clustering coefficient scales with the degree following C(k )~k -1 Barabasi et al, Nature Review Genetics 101(5), 101–114 , 2004 Dr. Carlo Cosentino9 Carnegie Mellon University, Pittsburgh, 2008 Graphs of Biological Networks Ó Depending on the kind of biological network, the edges and nodes of the graph have different meaning Ô Metabolic network Õnodes: metabolic product, edge: a reaction transforming A into B Ô Transcriptional regulation network (protein–DNA) Õnodes: genes and proteins, edge: a TF regulates a gene Ô Protein – protein network Õnodes: proteins, edge: interaction between proteins Ô Gene regulatory networks (functional association network) Õnodes: genes, edge: expressions of A and B are correlated Dr. Carlo Cosentino10 Carnegie Mellon University, Pittsburgh, 2008 Topology of Biological Networks Ó An extensive commentary has been published by Albert in 2005, reviewing literature on the topology of different kinds of biological networks Albert, Scale–free networks in cell biology, Journal of Cell Science 118(21), 4947–4957, 2005 Ó Experimental evidences are reviewed for metabolic, transcriptional regulatory, signal transduction, functional association networks Ó All of the considered networks approximately exhibit power–law degree distribution, at least for the in– or for the out–degree Ó For instance, transcriptional regulation networks exhibit a scale–free out– degree distribution, signifying the potential of transcription factors to regulate multiple targets Ó On the other hand, their in–degree is a more restricted exponential function, suggesting that combinatorial regulation by several TFs is less frequent Dr. Carlo Cosentino11 Carnegie Mellon University, Pittsburgh, 2008 P–P Interaction Network in Yeast Ó This network is based on yeast two–hybrid experiments Ó Few highly connected nodes (hubs) hold the network together Ó The color of a node indicates the phenotypic effect deriving from removing the corresponding protein Ô red: lethal Ô green: non–lethal Ô orange: slow growth Ô yellow: unknown Barabasi et al, Nature Review Genetics 101(5), 101–114 , 2004 Dr. Carlo Cosentino12 Carnegie Mellon University, Pittsburgh, 2008 Outline Ó Classification of biological networks Ó Modeling metabolic networks Ó Modeling gene regulatory networks Ó Inferring gene regulatory networks Dr. Carlo Cosentino13 Carnegie Mellon University, Pittsburgh, 2008 Metabolic Reactions Ó Living cells require energy and material for Ô building up membranes Ô storing molecules Ô replenishing enzymes Ô replication and repair of DNA Ô movement Ó Metabolic reactions can be divided in two categories Ô Catabolic reactions: breakdown of complex compounds to get energy and building blocks Ô Anabolic reactions: assembling of the compounds used by the cellular mechanisms Dr. Carlo Cosentino14 Carnegie Mellon University, Pittsburgh, 2008 Basic Concepts of Metabolism Ó Historically metabolism is the part of cell functioning that has been studied more thoroughly during the last decades Ó This implies that several well assessed mathematical tools exist for describing this kind of networks Ô Enzyme kinetics investigates the dynamic properties of the individual reactions in isolation Ô Stoichiometric analysis deals with the balance of compound production and degradation at the network level Ô Metabolic control analysis describes the effect of perturbations in the network, in terms of changes of metabolites concentrations Ó Most of the tools used in the quantitative study of metabolic networks can also be applied to other types of networks Dr. Carlo Cosentino15 Carnegie Mellon University, Pittsburgh, 2008 Glycolysis Ó We will exploit the case–study of glycolysis in yeast in order to illustrate the theoretical concepts introduced hereafter Ó The pathway shown below is part of the glycolysis process List of Reactions Ó v1: hexokinase Ó v5: aldolase Ó v2: consumption of glucose–6–phosphate Ó v6: ATP production in lower glycolysis in other pathways Ó v7: ATP consumption in other pathways Ó v3: phosphoglucoisomerase Ó v8: adenylate kinase Ó v4: phosphofructokinase Hynne et al, Full–scale model of glycolysis in Saccharomyces cerevisiae (2001) Biophys. Chem. 94, 121–163 Dr. Carlo Cosentino16 Carnegie Mellon University, Pittsburgh, 2008 ODE Model of Glycolysis Ó The system of ODEs describing the pathway is Dr. Carlo Cosentino17 Carnegie Mellon University, Pittsburgh, 2008 ODE Model with Constant Glucose Ó The kinetic rates as functions of reactants can be derived by applying the models presented in the previous lecture Model Parameters Dr. Carlo Cosentino18 Carnegie Mellon University, Pittsburgh, 2008 Stoichiometric Analysis Ó The basic elements considered in stoichiometric analysis of metabolic networks are Ô The concentrations of the various species Ô The reactions or transport processes affecting such concentrations Ó The stoichiometric coefficients denote the proportion of substrate and product molecules involved in a reaction Ó For instance, if we consider the reaction the stoichiometric coefficients of S1, S2, P are 1,1,-2 respectively Dr. Carlo Cosentino19 Carnegie Mellon University, Pittsburgh, 2008 Stoichiometric Analysis Ó The change of concentrations in time can be described by means of ODEs Ó For the simple reaction above we have Ó This means that the degradation of S1 with rate v is accompanied by the degradation of S2 with the same rate and by the production of P with a double rate Dr. Carlo Cosentino20 Carnegie Mellon University, Pittsburgh, 2008 Stoichiometric Matrix Ó In general, for a system of m substances and r reactions, the system dynamics are described by Ó The number nij is the stoichiometric coefficient of the i-th metabolite in the j-th reaction Ó For the sake of simplicity, we assume that the changes of concentrations are only due to reactions (i.e.
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