Mathematical iology: Metabolic Network nalysis Interdisciplinary lecture series for ioinformatics, Mathematics, iology and/or Life Science & Technology dr. Sander Hille shille@math.leidenuniv.nl http://pub.math.leidenuniv.nl/~hillesc Snellius, Niels ohrweg 1, room 40 Modeling (bio-)chemical reaction networks () Sander Hille 1
iochemistry One aspect that distinguishes bio(logical) chemistry from general chemistry: The chemical compounds used by biological systems represent only a tiny fraction of all possible chemical carbon-based compounds with molecular mass in the same range: estimated: ~ 10 3 10 4 out of ~ 10 60 (h.m. Dobson (004). hemical space and biology, Nature 43, 84-88) (3) Sander Hille Types of modelling What to model? Why? How? e aware of purpose of mathematical modelling: What questions should be answered by the model and its subsequent analysis and/or simulation? Determines the type of model used, level of detail, or complexity, of the model (4) Sander Hille Spring semester 015
Types of modelling our focus Predictive Explorative Qualitative Fair level of realistic detail targeted at providing detailed insight in changes in behaviour or optimal control ( Toy models ) (Highly) simplified system view targeted at understanding particular aspects of the system Quantitative In silicon version of reality used to predict development of a system with appropriate accuracy all relevant processes must be known in detail Fair level of realistic detail targeted at discovering detailed realistic structure of the system (5) Sander Hille Spring semester 015 Overview of modelling approaches -- large chemical reaction networks -- (fter R. Steuer, Phytochemistry 68 (007), 139-151) System size Level of detail Network nalysis Stoichiometric nalysis Structural kinetic models Detailed kinetic models Global saturation onvergence to steady state oncentration(s) dmissible flux cone Feedback strength time (6) Sander Hille 3
Levels of modelling -- defining a hierarchy -- hierarchical organisation (of metabolism) has been defined based on man-made concepts in order to better understand functioning of metabolism traditional hierarchy (found in most textbooks) In contrast, graph theoretical results allow to introduce a network-based hierarchy. Network-based approach our focus Goal: to obtain an unbiased -- objective -- hierarchy in the system, derived (solely) from its intrinsic structure (7) Sander Hille Metabolism Metabolism: The set of chemical reactions that occur in living organisms in order to maintain life. Nutrients atabolism nabolism Structural components sugars cellulose proteins fats proteins (e.g. enzymes) membranes RN / DN organelles (8) Sander Hille 4
Metabolism Metabolism: The set of chemical reactions that occur in living organisms in order to maintain life. Energy Nutrients atabolism nabolism Structural components sugars cellulose proteins fats (9) Sander Hille Intermediary metabolites pyruvate acetyl coenzyme proteins (e.g. enzymes) membranes RN / DN organelles Metabolism -- detailed process view -- currency metabolites TP / DP ND + / NDH atabolism Uptake Extracellular & intracellular digestion conversion Oxidation nabolism Precursor production ctivation into reactive forms ssembly into complex molecules Nutrients: e.g. starch, cellulose, proteins, amino acids monosaccharides glycerol fatty acids Pyruvate monosacharides 1. acetyl-o. oxaloacetate 3. α oxoglutarate amino acids monosaccharides terpenoids nucleotides proteins polysaccharides lipids nucleic acids (10) Sander Hille 5
Metabolism -- urrency metabolites -- DP / TP: denosine Di-(Tri-)Phosphate Function: Transport of energy within cells e.g. for metabolism; phosphate donor 3x phosphate adenosine ND + / NDH: Nicotinamide denosine Dinucleotide Function: Transfer of electrons from one compound to another in redox reactions ND + : oxidizing agent, accepts electrons NDH: reducing agent, donates electrons (11) Sander Hille Levels of modelling -- a traditional hierarchy -- traditional hierarchy : 1. ellular level / level of the full organism Nutrients atabolism nabolism Structural components Energy. Sector view, including (some) internal processes Nutrients atabolism nabolism Structural components Intermediary metabolites atabolism nabolism 3. Pathways (1) Sander Hille 6
traditional hierarchy : Levels of modelling -- a traditional hierarchy -- 3. Pathways atabolism nabolism TP DP 4. Modules of (high-level) chemical reactions (e.g. glycolysis, alvin cycle ) Glucose G6P F6P 5. Detailed chemistry (13) Sander Hille Examples of (parts of) metabolic networks (14) Sander Hille 7
Metabolic networks -- example: photosynthesis -- ell with chloroplasts ross section (15) Sander Hille Metabolic networks -- example: photosynthesis -- (Photo: Kristian Peters) (16) Sander Hille hloroplasts in a moss, Plagiomnium affine 8
Metabolic networks -- example: glycolysis -- (http://www.biologyclass.net/pathways.html) (17) Sander Hille Metabolic networks -- some characteristics -- lmost all reactions are catalyzed by enzymes TP DP Glucose G6P F6P Example of an enzyme Hexokinase Phosphoglucose isomerase Enzymes Some reactions do not (hardly) occur in their absence lthough chemically: Principle of microscopic reversability Large number of (substrate) molecules rystal structure of hexokinase (18) Sander Hille 9
hemical reaction networks viewed as mathematical graphs TP DP Glucose G6P F6P (19) Sander Hille Graph representations -- some basic mathematical terminology -- n undirected graph G is an ordered pair (V,E) of a finite collection of vertices V (or nodes ) together with a set E of two-point subsets of V, the edges (or lines ). edges vertices Other examples: graph omplete graphs Disconnected graph Graphs with cycle and with self-loops (0) Sander Hille 10
Graph representations -- some basic mathematical terminology -- graph G=(V,E) can be represented by a matrix, the adjacency matrix of G, in the following manner: 1. Label the vertices by natural numbers 1,,..., n. The adjacency matrix is the n n matrix with coefficients Example: 4 Note: symmetric! 1 1 3 (1) Sander Hille Graph representations -- some basic mathematical terminology -- bipartite graph is a graph G=(V,E) such that the set of vertices is the disjoint union of two subsets V 1 and V, such that there are no edges connecting vertices within each of these subsets. V V V 1 V 1 bipartite graph In a bipartite graph the vertices can be colloured in such a way that no two vertices of the same collour are connected through an edge. () Sander Hille 11
Graph representations -- some basic mathematical terminology -- bipartite graph is a graph G=(V,E) such that the set of vertices is the disjoint union of two subsets V 1 and V, such that there are no edges connecting vertices within each of these subsets. Not a bipartite graph (3) Sander Hille Graph representations -- some basic mathematical terminology -- directed graph G is an ordered pair (V,) of a collection of vertices V (or nodes ) together with a set V V of ordered pairs of vertices, called arrows (or directed edges, arcs ). v 0 v 1 Two paths from v 0 to v 1 of length 4 directed graph path of length n from v 0 V to v 1 V is a sequence of arrows in, a 1,, a n such that a 1 starts in v 0, a n ends in v 1 and the end point of a i is the starting point of a i+1. (4) Sander Hille 1
Graph representations -- some basic mathematical terminology -- n adjacency matrix can be defined for a directed graph G=(V,) similarly to that for an undirected graph: The adjacency matrix of a directed graph with n vertices is the n n matrix with coefficients Example: 4 Note: asymmetric! 1 1 3 (5) Sander Hille Graph representations -- some basic mathematical terminology -- The concept of a bipartite graph can be applied to directed graphs also 1 bipartite directed graph 3 6 7 4 5 1 3 4 5 1 3 4 5 6 7 0 6 7 0 Directed bipartite graphs have a specially structured adjaceny matrix Hence may be coded more efficiently 1 3 4 5 6 7 (6) Sander Hille 13
Graph representations of chemical reaction networks ipartite directed graphs can be used to model chemical reaction networks: hemical reaction: (unidirectional) : + ipartite graph: (directed graph) Substrate / product hemical reaction mbigious also R Substrate graph: (directed graph) rrow between substrate / products when connected through a reaction (7) Sander Hille Graph representations of chemical reaction networks mbiguity in substrate graphs may be circumvented by using hypergraph notation hypergraph (8) Sander Hille 14
Graph representations of chemical reaction networks nother ambiguity is present even in bipartite graphs associated to chemical reactions hemical reaction: (unidirectional) : + Multiplicity is not represented in the graph use weighted edges (Simplified) adjacency matrix Stoichiometric matrix (9) Sander Hille Incorporates multiplicity Graph representations of chemical reaction networks Multiple chemical reactions: (unidirectional) : R: R3: + D + E D ipartite graph: R E D R3 Reaction graph: rrow from a reaction to another when the endpoint uses a product of the first as a substrate R3 R (30) Sander Hille 15
Network nalysis Summarising: hemical reaction network Detailed network graph (ipartite directed graph) Substrate graph Reaction graph May use nodes of different shapes instead of collours to distinguish compounds from reaction: (31) Sander Hille Summarising: Network nalysis hemical reaction network Detailed network graph (ipartite directed graph) Substrate graph Reaction graph Network nalysis is the term used in the literature for studying the properties of these graphs. Network statistics (3) Sander Hille 16