Metabolic Network Analysis

Transcription

1 Metabolic Network nalysis Overview -- modelling chemical reaction networks -- Levels of modelling Lecture II: Modelling chemical reaction networks dr. Sander Hille Various views on reaction networks as graphs Fluxes and stoichiometry (large) system of ODEs (Ordinary Differential Equations) Snellius, Niels ohrweg, room 0 () Sander Hille Types of modelling Types of modelling What to model? Why? How? our focus Qualitative Quantitative e aware of purpose of mathematical modelling: What questions should be answered by the model and its subsequent analysis and/or simulation? Predictive Fair level of realistic detail targeted at providing detailed insight in changes in behaviour or optimal control In silicon version of reality used to predict development of a system with appropriate accuracy all relevant processes must be known in detail Determines the type of model used, level of detail, or complexity, of the model Explorative ( Toy models ) (Highly) simplified system view targeted at understanding particular aspects of the system Fair level of realistic detail targeted at discovering detailed realistic structure of the system () Sander Hille () Sander Hille

2 (5) Sander Hille Types of modelling Our questions concerning metabolic networks: How can we best intervene in an organism s metabolism in order to have a desired effect on steady production of particular metabolites? How is metabolism organised? Does it have a modular structure? How did this structure evolve? Why? Why is it compartmentised in eukaryots? How is metabolism effectively controlled? (6) Sander Hille 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 Goal: to obtain an unbiased -- objective -- hierarchy in the system, derived (solely) from its intrinsic structure our focus Levels of modelling -- a traditional hierarchy -- Last week we encountered a traditional hierarchy :. ellular level / level of the full organism Levels of modelling -- a traditional hierarchy -- Last week we encountered a traditional hierarchy :. Pathways atabolism nabolism Nutrients atabolism nabolism Structural components Energy. Sector view, including (some) internal processes TP DP. Modules of (high-level) chemical reactions (e.g. glycolysis, alvin cycle ) Nutrients atabolism nabolism Structural components Glucose G6P F6P Intermediary metabolites atabolism nabolism. Pathways 5. Detailed chemistry (7) Sander Hille (8) Sander Hille

3 Overview of modelling approaches -- large chemical reaction networks -- Overview of modelling approaches -- large chemical reaction networks -- (fter R. Steuer, Phytochemistry 68 (007), 9-5) (fter R. Steuer, Phytochemistry 68 (007), 9-5) System size Level of detail System size Level of detail Network nalysis Stoichiometric nalysis Structural kinetic models Detailed kinetic models Network nalysis Stoichiometric nalysis Structural kinetic models Detailed kinetic models Global saturation onvergence to steady state oncentration(s) Structural properties No kinetic parameters Statistics on structure Understanding evolution of networks Steady state analysis No kinetic parameters Implicit system of ODEs dmissible flux distributions nalysis onvergence of dynamics to steady No kinetic state parameters Global saturation Implicit system of ODEs Possible dynamics: bifurcation analysis oncentration(s) nalysis of dynamics Needs kinetic parameters Explicit system of ODEs Detailed dynamics dmissible flux cone Feedback strength time Feedback strength dmissible flux Our cone Starting point focus in the lectures time (9) Sander Hille (9) Sander Hille hemical reaction networks viewed as mathematical graphs 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 ). vertices edges TP DP Other examples: graph Glucose G6P F6P omplete graphs Disconnected graph Graphs with cycle and with self-loops (0) Sander Hille () Sander Hille

4 graph G=(V,E) can be represented by a matrix, the adjacency matrix of G, in the following manner:. Label the vertices by natural numbers,,..., n. The adjacency matrix is the n n matrix with coefficients bipartite graph is a graph G=(V,E) such that the set of vertices is the disjoint union of two subsets V and V, such that there are no edges connecting vertices within each of these subsets. V V Example: Note: symmetric! V V bipartite graph In a bipartite graph the vertices can be coloured in such a way that no two vertices of the same colour are connected through an edge. () Sander Hille () Sander Hille bipartite graph is a graph G=(V,E) such that the set of vertices is the disjoint union of two subsets V and V, such that there are no edges connecting vertices within each of these subsets. 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 Two paths from v 0 to v of length directed graph Not a bipartite graph path of length n from v 0 V to v V is a sequence of arrows in, a,, a n such that a starts in v 0, a n ends in v and the end point of a i is the starting point of a i+. () Sander Hille (5) Sander Hille

5 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: Note: asymmetric! The concept of a bipartite graph can be applied to directed graphs also 6 7 bipartite directed graph Directed bipartite graphs have a specially structured adjaceny matrix Hence may be coded more efficiently (6) Sander Hille (7) Sander Hille 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 : Substrate graph: (directed graph) rrow between substrate / products when connected through a reaction + also mbigious R mbiguity in substrate graphs may be circumvented by using hypergraph notation of chemical reaction networks hypergraph (8) Sander Hille (9) Sander Hille 5

6 of chemical reaction networks nother ambiguity is present even in bipartite graphs associated to chemical reactions hemical reaction: (unidirectional) (Simplified) adjacency matrix : + Incorporates multiplicity Multiplicity is not represented in the graph Stoichiometric matrix use weighted edges of chemical reaction networks Multiple chemical reactions: (unidirectional) ipartite graph: Reaction graph: rrow from a reaction to another when the endpoint uses a product of the first as a substrate : R: R: + D + E D R R R E D R (0) Sander Hille () Sander Hille Network nalysis Network nalysis Summarising: hemical reaction network Summarising: hemical reaction network Detailed network graph (ipartite directed graph) Detailed network graph (ipartite directed graph) Substrate graph Reaction graph Substrate graph Reaction graph May use nodes of different shapes instead of collours to distinguish compounds from reaction: Network nalysis is the term used in the literature for studying the properties of these graphs. Network statistics () Sander Hille () Sander Hille 6

7 Network nalysis N: total number of nodes (vertices) Nodes degree or connectivity k: number of neighbours of a particular node P(k): degree distribution (frequency of nodes of nodes degree k) Mean path length l ij : length of shortest path from node i to node j <l>: mean path length: Network nalysis lustering coefficient k i : number of neighbours of node i n i : number of edges connecting the k i neighbours of node i to each other i : clustering coefficient of node i <>: average clustering coefficient connectedness among the neighbours of node i small world network : <l> depends logarithmically on N (k): average clustering coefficient of all nodes that are connected to k neighbours (i.e of nodes degree k) () Sander Hille () Sander Hille Network nalysis lustering coefficient (continued) When ( power law ) then the network has a hierarchical structure When then the network is scale-free Network nalysis Statistics from some real life networks (for undirected substrate graphs with currency metabolites removed ranging over 6 rcheae, acteria, 5 Eukaryotes) Metabolic networks are scale-free a: rchaeoglubus fulgidus (rcheae) b: Escherichia coli (acteria) c: aenorhabidtis elegans (Eukaryotes) d: verage over all researched spieces (Jeong ea. Nature 07 (000), pp.65 65) (5) Sander Hille (6) Sander Hille 7

8 Network nalysis Statistics from some real life networks (for undirected substrate graphs with currency metabolites removed ranging over 6 rcheae, acteria, 5 Eukaryotes) Fluxes and stoichiometry rcheae (6) acteria () Eukaryotes (5) (Ravasz, Somera, Mongru, Oltvai, arabasi, Science 97 (00), pp ) (7) Sander Hille (8) Sander Hille 8