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Minimal Cut Sets and Its Application to Study Metabolic Pathway Structures

Minimal Cut Sets and Its Application to Study Metabolic Pathway Structures

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  • 1. Minimal Cut Sets and Its Application to Study Metabolic Pathway Structures Nguyen Vu-Ngoc Tung1,3 Beurton-Aimar Marie1 Colombié Sophie2 1Laboratoire Bordelais de Recherche en Informatique, UMR 5800 2 INRA Bordeaux Aquitaine, Fruit Biology and Pathology BP 81. 3 Faculty of Science and Technology, Hoa Sen University. Nice’13 Thematic Research School, 2013 Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 1 / 22
  • 2. Motivation One of the major current challenges in Systems Biology is how to understand complex structure of metabolic networks. Metabolic pathways are specific subsets into a metabolic network identified as functional processes of cells. Elementary Flux Modes (EFMs) are minimal sets of reactions that represent feasible pathways under steady state condition (Schuster,2000) . Minimal Cut Sets (MCSs) are minimal sets of reactions that inhibit the production of a certain objective reaction. Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 2 / 22
  • 3. Motivation One of the major current challenges in Systems Biology is how to understand complex structure of metabolic networks. Metabolic pathways are specific subsets into a metabolic network identified as functional processes of cells. Elementary Flux Modes (EFMs) are minimal sets of reactions that represent feasible pathways under steady state condition (Schuster,2000) . Minimal Cut Sets (MCSs) are minimal sets of reactions that inhibit the production of a certain objective reaction. Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 2 / 22
  • 4. Motivation One of the major current challenges in Systems Biology is how to understand complex structure of metabolic networks. Metabolic pathways are specific subsets into a metabolic network identified as functional processes of cells. Elementary Flux Modes (EFMs) are minimal sets of reactions that represent feasible pathways under steady state condition (Schuster,2000) . Minimal Cut Sets (MCSs) are minimal sets of reactions that inhibit the production of a certain objective reaction. Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 2 / 22
  • 5. Motivation One of the major current challenges in Systems Biology is how to understand complex structure of metabolic networks. Metabolic pathways are specific subsets into a metabolic network identified as functional processes of cells. Elementary Flux Modes (EFMs) are minimal sets of reactions that represent feasible pathways under steady state condition (Schuster,2000) . Minimal Cut Sets (MCSs) are minimal sets of reactions that inhibit the production of a certain objective reaction. Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 2 / 22
  • 6. Outline 1 Context 2 Integrated Approach Elementary Flux Modes Analysis Minimal Cut Sets Analysis 3 Graph Cut Sets Definitions and Notations Algorithms MCSs in Metabolic Networks 4 Application Metabolic Network Description Computing Tools Results and Discussion 5 Conclusion and Perspective Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 3 / 22
  • 7. Context Context Metabolic Pathways Analysis Identifying pathways involved in specific production. Discovering how to increase the yield of a product, to channel a product into desired pathways or in functional reconstruction from genomic data (Schuster,1999). Predicting key aspects of network functionality, robustness and gene regulation from network structure (Stelling,2002). Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 4 / 22
  • 8. Context Context Metabolic Pathways Analysis Identifying pathways involved in specific production. Discovering how to increase the yield of a product, to channel a product into desired pathways or in functional reconstruction from genomic data (Schuster,1999). Predicting key aspects of network functionality, robustness and gene regulation from network structure (Stelling,2002). Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 4 / 22
  • 9. Context Context Metabolic Pathways Analysis Identifying pathways involved in specific production. Discovering how to increase the yield of a product, to channel a product into desired pathways or in functional reconstruction from genomic data (Schuster,1999). Predicting key aspects of network functionality, robustness and gene regulation from network structure (Stelling,2002). Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 4 / 22
  • 10. Integrated Approach Elementary Flux Modes Analysis Analyze Reliability of Metabolite Production Elementary Flux Modes Analysis Constraint-based approach (Schuster,1994). Identifying all genetically independent pathways (Trinh,2009). Being unique and non-decomposable set of reactions. Selecting groups of reactions which interact together and respecting the well-known steady-state mass balancing equation. Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 5 / 22
  • 11. Integrated Approach Elementary Flux Modes Analysis Analyze Reliability of Metabolite Production Elementary Flux Modes Analysis Constraint-based approach (Schuster,1994). Identifying all genetically independent pathways (Trinh,2009). Being unique and non-decomposable set of reactions. Selecting groups of reactions which interact together and respecting the well-known steady-state mass balancing equation. Steady-State Mass Balancing Assumption dS dt = Nv (1) S is a vector of concentration values. N is the stoichiometric matrix of m metabolites × r reactions. v is the r-dimensional (flux) vector of the reaction rates. At the steady state: Nv = 0. Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 5 / 22
  • 12. Integrated Approach Minimal Cut Sets Analysis Analyze Fragility of Metabolic Networks Mininal Cut Set Analysis Finding all sets of reactions able to eliminate a given objective functioning. A Minimal Cut Set (MCS) is a unique and minimal set of reactions (Klamt,2004). EFMs and MCSs complement each other in a duality based relationship (Klamt,2005;Ballerstein,2012). Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 6 / 22
  • 13. Graph Cut Sets Definitions and Notations Cut Sets definitions Notations Let G = (V, E) be an undirected graph with n = |V|, m = |E|. Cut Sets A cut C = {S, S} where S ∪ S = V(G) and S ∩ S = ∅. ∀u, v ∈ V, the set δ(S) = {(u, v) ∈ E ∧ S ⊂ V : u ∈ S, v ∈ S} is a cut set since removal from G disconnects G into more than one subgraphs. The size of a cut set is |δ(S)| in unweighted graphs. Otherwise, the cut set size equals to sum of the weights of the edges in δ(S). A minimum cut set is a cut set of a certain minimum size. Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 7 / 22
  • 14. Graph Cut Sets Definitions and Notations Cut Set Definitions Minimal Cut Set Example Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 8 / 22
  • 15. Graph Cut Sets Definitions and Notations Cut Set Definitions Minimal Cut Set Example s-t Cut Set Definition A cut s − t of an undirected graph G is simply a cut C = {S, S} with s ∈ S and t ∈ S. Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 8 / 22
  • 16. Graph Cut Sets Definitions and Notations Cut Set Definitions Example s-t cut set betwen s = a and t = d. MCSs in Directed Graphs In directed graphs, cut sets are defined similarly. The MCS value: summing all the weights of all the crossed edges (between the two subsets) coming out S. Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 9 / 22
  • 17. Graph Cut Sets Algorithms Algorithms to Compute MCS Finding one MCS Originated from the well-known max flow theorem (Elias,1956;Ford,1956). Gomory and Hu (1961) introduced a tree structure to find minimum s − t cuts for all pairs of s and t. Improving by Hao Orlin in 1992. The first deterministic minimum cut algorithm: Nagamochi and Ibaraki (NI)(1992): O(|V||E| + |V|2log|V|. Stoer and Wagner (1997) simplified NI and implemented it in JGraphT library. Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 10 / 22
  • 18. Graph Cut Sets Algorithms Algorithms to Compute MCSs Finding all MCSs Applied in the field of reliability engineering (Ariyoshi,1972; Arunkumar:1979). Constructing a binary relation associated with an optimal maximum flow (Curet,2002). Definition of MCS in Metabolic Network Context A set of reactions is called a cut set (with respect to a defined objective reaction) if after the removal of these reactions from the network no feasible balanced flux distribution involves the objective reaction (Klamt,2004). Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 11 / 22
  • 19. Graph Cut Sets MCSs in Metabolic Networks Klamt’s algorithm Main Ideas In small networks it is relatively easy to calculate the MCSs but ..... Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 12 / 22
  • 20. Graph Cut Sets MCSs in Metabolic Networks Klamt’s algorithm Main Ideas In small networks it is relatively easy to calculate the MCSs but ..... For larger networks, we need a systematic computation scheme. The algorithm needs to guarantee: MCSs are real cut sets. MCSs are minimal. All MCSs are found. Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 12 / 22
  • 21. Graph Cut Sets MCSs in Metabolic Networks Klamt’s algorithm Main Ideas In small networks it is relatively easy to calculate the MCSs but ..... For larger networks, we need a systematic computation scheme. The algorithm needs to guarantee: MCSs are real cut sets. MCSs are minimal. All MCSs are found. Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 12 / 22
  • 22. Graph Cut Sets MCSs in Metabolic Networks Algorithm for computing MCSs Preparatory phase 1 Calculate the EFMs in the given network. 2 Define the objective reaction obR. 3 Choose all EFMs where reaction obR is non-zero and store it in the binary array efm_obR. 4 Initialize the arrays mcs and precutsets as follows: Append {j} to mcs if reaction {j} is essential, otherwise to precutsets. {j} is essential if efm_obR[i][j] = 1 for each EFM_i Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 13 / 22
  • 23. Graph Cut Sets MCSs in Metabolic Networks Algorithm for computing MCSs Main phase 1 FOR i=2 TO MAX_CUTSETSIZE 1 new_precutsets = []; 2 FOR j = 1 TO r 1 Remove all sets from precutsets where reaction j participates. 2 Find all sets of reactions in precutsets that do not cover any EFM in efm_obR where reaction j participates. Combine each of these sets with reaction j and store the new preliminary cut sets in temp_precutsets. 3 Drop all temp_precutsets which are a superset of any of the already determined minimal cut sets stored in mcs. 4 Find all retained temp_precutsets which do now cover all EFMs and append them to mcs. Append all others to new_precutsets 3 IF isempty(new_precutsets) BREAK; ELSE precutsets = new_precutsets; 2 return mcs; Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 13 / 22
  • 24. Graph Cut Sets MCSs in Metabolic Networks Algorithm for computing MCSs Simple Example efm: [{R1, R2, objR}, {R3, objR}] mcs: [{objR}, {R1, R3}, {R2, R3}] Source: reused a simple example from M. Bader. Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 13 / 22
  • 25. Application Metabolic Network Description Application to 5 Networks Purpose To verify the hypothesis: MCS computing can provide a smaller number of solutions than EFMs. Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 14 / 22
  • 26. Application Metabolic Network Description Application to 5 Networks Purpose To verify the hypothesis: MCS computing can provide a smaller number of solutions than EFMs. Data 3 networks to model energetic metabolism of mitochondria into 3 tissues: muscle, liver, yeast, approx. 40 reactions (Pérès Sabine, PhD thesis, 2005). 2 networks to model the central metabolism of heterotrophic plant cells, approx. 80 reactionsa including several biological pathways: glycolysis, Pentose Phosphate pathway, Starch and Sucrose synthesis and degradation. a described more detail in Beurton-Aimar M. et al., BMC Sys. Bio., 2011. Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 14 / 22
  • 27. Application Metabolic Network Description Mitochondrial Network ACoA Cit HB AcylCoA Carnitine AcylCarnitine ATP Pi− HB_ext T10 Citulline CarbmoylP Orni AACoA HMGCoA AA 2 AA_ext T11 Intermenbran space Matrix NAD H Pyr NAD NADH R7 Mal Fum Suc−CoA Suc Akg Isocit R12 OAA ATP ADP NAD NADH NADH ADPATP FAD FADH2 NAD NADH ASP Glu NAD NAD NADH Glutamine NH3 2 ATP 2 ADP R21 Suc−CoA Suc 8 7 NAD 7 FAD 7 NADH 7 FADH2 FADH2 FAD ADP NADH2 R2 R1 R3 R22 R6 R15 R14 R13 R11 R10 R9 R8 R24 R25 R17 R16 R26 R27 R30 R31 R28 H H Pi2− R5 T8 Citru_ext Ornit_ext H_ext T6 Mal Pyr_ext + H_ext Akg Mal Mal Cit + H Glu + HAsp MalFum NH3 R23 H H_ext Akg T21Orni_ext ATP_ext ADP_ext H_ext Pi_ext T5 T4 Akg_extPi2−_ext Mal_ext T7 Pi2−_ext Glu_ext + H Asp_ext T12 T13 Fum_extMal_ext Glutamine_ext H_ext Glu_ext T20 Carnitine_ext T3 AcylCarnitine_ext Mal_ext Akg_ext Cit_ext + H_ext Mal_ext T1 T2 H T19 Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 15 / 22
  • 28. Application Metabolic Network Description Metabolic Network of Heterotrophic Plant Cell Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 16 / 22
  • 29. Application Computing Tools Computing Tools for EFMs and MCSs CellNetAnalyzer CellNetAnalyzer (CNA)a derived from Metatoolb Package for MATLAB containing several modules to visualize networks and to analyze their structures. CNA enables users to compute both EFMs and MCSs. Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 17 / 22
  • 30. Application Computing Tools Computing Tools for EFMs and MCSs CellNetAnalyzer CellNetAnalyzer (CNA)a derived from Metatoolb Package for MATLAB containing several modules to visualize networks and to analyze their structures. CNA enables users to compute both EFMs and MCSs. Problem: Taking more than 10 days to obtain MCSs of PCA with CNA (running on a linux server). a http://www.mpi-magdeburg.mpg.de/projects/cna/cna.html b http://pinguin.biologie.uni-jena.de/bioinformatik/networks/ Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 17 / 22
  • 31. Application Computing Tools Computing Tools for EFMs and MCSs Efmtool A new implementation to compute EFMs in Java (Terzer,2008) a. Supporting multi-threading and seems to be robust to compute large networks. Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 18 / 22
  • 32. Application Computing Tools Computing Tools for EFMs and MCSs Efmtool A new implementation to compute EFMs in Java (Terzer,2008) a. Supporting multi-threading and seems to be robust to compute large networks. Problem: open source but not easy to use (many parameters) and lacks of manuals. a http://www.csb.ethz.ch/tools/efmtool/ Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 18 / 22
  • 33. Application Computing Tools Computing Tools for EFMs and MCSs Efmtool A new implementation to compute EFMs in Java (Terzer,2008) a. Supporting multi-threading and seems to be robust to compute large networks. Problem: open source but not easy to use (many parameters) and lacks of manuals. a http://www.csb.ethz.ch/tools/efmtool/ regEfmtool Written by C. Jungreuthmayer (Jung,2012)a. Containing several scripts clearly documented. New available operations: possibility to define genetics constraints as logical rules to compute EFMs. a http: //www.biotec.boku.ac.at/regulatoryelementaryfluxmode.html Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 18 / 22
  • 34. Application Results and Discussion Results of Computation of 5 Networks Tissues Nb.React Nb. Int.Meta Nb.EFMs Nb.MCSs Muscle 37 31 3, 253 (17.7) 42, 534 (10.2) Liver 44 36 2, 307 (16.7) 47, 203 (11.4) Yeast 40 34 4, 627 (15.3) 90, 318 (11.6) PCA 78 55 114, 614 (37.7) 93, 009 (11.1) PCC 89 50 9, 319, 997 (33.1) 2, 815, 375(11.8) Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 19 / 22
  • 35. Application Results and Discussion Results of Computation of 5 Networks Tissues Nb.React Nb. Int.Meta Nb.EFMs Nb.MCSs Muscle 37 31 3, 253 (17.7) 42, 534 (10.2) Liver 44 36 2, 307 (16.7) 47, 203 (11.4) Yeast 40 34 4, 627 (15.3) 90, 318 (11.6) PCA 78 55 114, 614 (37.7) 93, 009 (11.1) PCC 89 50 9, 319, 997 (33.1) 2, 815, 375(11.8) Correlation between Number of EFMs and MCSs The size of the EFMs set is a measure of the network robustness (Stelling 2004). Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 19 / 22
  • 36. Application Results and Discussion Results of Computation of 5 Networks Tissues Nb.React Nb. Int.Meta Nb.EFMs Nb.MCSs Muscle 37 31 3, 253 (17.7) 42, 534 (10.2) Liver 44 36 2, 307 (16.7) 47, 203 (11.4) Yeast 40 34 4, 627 (15.3) 90, 318 (11.6) PCA 78 55 114, 614 (37.7) 93, 009 (11.1) PCC 89 50 9, 319, 997 (33.1) 2, 815, 375(11.8) Correlation between Number of EFMs and MCSs The size of the EFMs set is a measure of the network robustness (Stelling 2004). Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 19 / 22
  • 37. Application Results and Discussion Results of Computation of 5 Networks Tissues Nb.React Nb. Int.Meta Nb.EFMs Nb.MCSs Muscle 37 31 3, 253 (17.7) 42, 534 (10.2) Liver 44 36 2, 307 (16.7) 47, 203 (11.4) Yeast 40 34 4, 627 (15.3) 90, 318 (11.6) PCA 78 55 114, 614 (37.7) 93, 009 (11.1) PCC 89 50 9, 319, 997 (33.1) 2, 815, 375(11.8) Correlation between Number of EFMs and MCSs The size of the EFMs set is a measure of the network robustness (Stelling 2004). But no obvious relationship between the number of reactions (or internal metabolites) and of EFMs. Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 19 / 22
  • 38. Application Results and Discussion Results of Computation of 5 Networks Tissues Nb.React Nb. Int.Meta Nb.EFMs Nb.MCSs Muscle 37 31 3, 253 (17.7) 42, 534 (10.2) Liver 44 36 2, 307 (16.7) 47, 203 (11.4) Yeast 40 34 4, 627 (15.3) 90, 318 (11.6) PCA 78 55 114, 614 (37.7) 93, 009 (11.1) PCC 89 50 9, 319, 997 (33.1) 2, 815, 375(11.8) Correlation between Number of EFMs and MCSs The size of the EFMs set is a measure of the network robustness (Stelling 2004). But no obvious relationship between the number of reactions (or internal metabolites) and of EFMs. The number of MCSs is unfortunately not at all lower than the number of EFMs in the 3 mitochondrial networks. Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 19 / 22
  • 39. Application Results and Discussion Results of Computation of 5 Networks Tissues Nb.React Nb. Int.Meta Nb.EFMs Nb.MCSs Muscle 37 31 3, 253 (17.7) 42, 534 (10.2) Liver 44 36 2, 307 (16.7) 47, 203 (11.4) Yeast 40 34 4, 627 (15.3) 90, 318 (11.6) PCA 78 55 114, 614 (37.7) 93, 009 (11.1) PCC 89 50 9, 319, 997 (33.1) 2, 815, 375(11.8) Correlation between Number of EFMs and MCSs The size of the EFMs set is a measure of the network robustness (Stelling 2004). But no obvious relationship between the number of reactions (or internal metabolites) and of EFMs. The number of MCSs is unfortunately not at all lower than the number of EFMs in the 3 mitochondrial networks. When the number of EFMs is huge (PCA, PCC), the number of MCSs begins to be lower than EFM number. Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 19 / 22
  • 40. Application Results and Discussion Results of Computation of 5 Networks Comparison of EFMs and MCSs Length The average length of EFMs increases with the number of reactions while the average length of MCSs remains stable. Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 20 / 22
  • 41. Application Results and Discussion Results of Computation of 5 Networks Comparison of EFMs and MCSs Length The average length of EFMs increases with the number of reactions while the average length of MCSs remains stable. EFM length: when the number of reactions doubles, the length too. For example, the average length of EFMs muscle is 17.7, comparing to the values obtained for the PCA network, 37.7. MCS length: while the number of reactions doubles from mitochondria to plant cell networks, the average length is only 10% point more. Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 20 / 22
  • 42. Application Results and Discussion Results of Computation of 5 Networks Comparison of EFMs and MCSs Length The average length of EFMs increases with the number of reactions while the average length of MCSs remains stable. EFM length: when the number of reactions doubles, the length too. For example, the average length of EFMs muscle is 17.7, comparing to the values obtained for the PCA network, 37.7. MCS length: while the number of reactions doubles from mitochondria to plant cell networks, the average length is only 10% point more. To inhibit a specific functionning, the number of reactions to stop is approximatively the same whatever the network size. Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 20 / 22
  • 43. Conclusion and Perspective Conclusion and Perspective Conclusion Metabolic networks are complex networks - Evidence! To study network architecture of a whole metabolism, automatic tools are necessary. Connexion among several pathways are impossible to manage only by hands. As EFMs computing, computing MCSs could generate huge results. Post treatments like classification are mandatory. Most of available algorithms require large capacity of computing: memory size and processor speed. New machines and types of programming: GPU and algorithm improvements help to solve the problem and allow to analyze networks larger and larger. Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 21 / 22
  • 44. Conclusion and Perspective Conclusion and Perspective Conclusion Metabolic networks are complex networks - Evidence! To study network architecture of a whole metabolism, automatic tools are necessary. Connexion among several pathways are impossible to manage only by hands. As EFMs computing, computing MCSs could generate huge results. Post treatments like classification are mandatory. Most of available algorithms require large capacity of computing: memory size and processor speed. New machines and types of programming: GPU and algorithm improvements help to solve the problem and allow to analyze networks larger and larger. Perspective New technics to analyze results coming from graph theory and data mining have to be implemented to provide tools to do it. Connection with techniques like flux balance analysis is our nextNguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 21 / 22
  • 45. Thanks for your attention! Questions?