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Network modularity, currency metabolites and graph Network modularity, currency metabolites representa- tions of and graph representations of metabolism metabolism PETTER HOLME metabolic Petter Holme networks, intro networks KTH,

  1. what systems? Network What systems can be analyzed with complex network modularity, currency methods? metabolites and graph representa- items are, naturally, coupled pairwise MAYBE? tions of metabolism the network is relatively sparse (the average degree is PETTER HOLME constant) ALMOST TRUE metabolic there is a dynamic system on the network TRUE! networks, intro the time scale of this dynamics is faster than the networks dynamics of network evolution TRUE! (physicist style?) modularity & currency metabolites subnetwork hierarchies

  2. what systems? Network What systems can be analyzed with complex network modularity, currency methods? metabolites and graph representa- items are, naturally, coupled pairwise MAYBE? tions of metabolism the network is relatively sparse (the average degree is PETTER HOLME constant) ALMOST TRUE metabolic there is a dynamic system on the network TRUE! networks, intro the time scale of this dynamics is faster than the networks dynamics of network evolution TRUE! (physicist style?) modularity & currency metabolites subnetwork hierarchies

  3. what systems? Network What systems can be analyzed with complex network modularity, currency methods? metabolites and graph representa- items are, naturally, coupled pairwise MAYBE? tions of metabolism the network is relatively sparse (the average degree is PETTER HOLME constant) ALMOST TRUE metabolic there is a dynamic system on the network TRUE! networks, intro the time scale of this dynamics is faster than the networks dynamics of network evolution TRUE! (physicist style?) modularity & currency metabolites subnetwork hierarchies

  4. representations Network ATP + NADH ↔ ADP + NADPH modularity, currency metabolites and graph representa- tions of metabolism PETTER HOLME metabolic networks, intro NADH kinase networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  5. representations Network ATP + NADH ↔ ADP + NADPH modularity, currency metabolites and graph representa- tions of metabolism PETTER HOLME metabolic networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies vertices: substrates edges: between products / substrates on different sides

  6. representations Network ATP + NADH ↔ ADP + NADPH modularity, currency metabolites and graph representa- tions of metabolism PETTER HOLME metabolic networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies vertices: substrates edges: between products / substrates on the same side

  7. representations Network ATP + NADH ↔ ADP + NADPH modularity, currency metabolites and graph representa- tions of metabolism PETTER HOLME metabolic networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies vertices: substrates edges: between all products / substrates

  8. representations Network ATP + NADH ↔ ADP + NADPH modularity, currency metabolites and graph representa- tions of metabolism PETTER HOLME NADH kinase metabolic networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies vertices: substrates + enzymes (reactions) edges: between substance / reaction vertices

  9. representations Network different representations, give different information modularity, currency metabolites most common representation = substance and graph representa- graphs—following atoms, the number of conversions tions of metabolism between two molecules are small ⇔ the graph distance is PETTER small HOLME metabolic networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  10. representations Network different representations, give different information modularity, currency metabolites most common representation = substance and graph representa- graphs—following atoms, the number of conversions tions of metabolism between two molecules are small ⇔ the graph distance is PETTER small HOLME metabolic networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  11. representations Network different representations, give different information modularity, currency metabolites most common representation = substance and graph representa- graphs—following atoms, the number of conversions tions of metabolism between two molecules are small ⇔ the graph distance is PETTER small HOLME metabolic networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  12. the dogmas of network science Network real networks have both structure and randomness modularity, currency metabolites the network structure relates to the function of the network and graph representa- tions of metabolism PETTER HOLME metabolic networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  13. the dogmas of network science Network real networks have both structure and randomness modularity, currency metabolites the network structure relates to the function of the network and graph representa- tions of metabolism PETTER HOLME metabolic networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  14. the dogmas of network science Network real networks have both structure and randomness modularity, currency metabolites the network structure relates to the function of the network and graph representa- tions of metabolism PETTER HOLME metabolic networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  15. what is network structure? Network how the network differs a random network modularity, currency metabolites to be more precise: how the network differs from a null and graph representa- model tions of metabolism PETTER HOLME metabolic networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  16. what is network structure? Network how the network differs a random network modularity, currency metabolites to be more precise: how the network differs from a null and graph representa- model tions of metabolism PETTER HOLME metabolic networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  17. what is network structure? Network how the network differs a random network modularity, currency metabolites to be more precise: how the network differs from a null and graph representa- model tions of metabolism PETTER HOLME metabolic networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  18. network null-models Network Network structures are relative . . . modularity, currency metabolites . . . so one has to be clear about what to compare with . . . and graph representa- a null model tions of metabolism Null model 1: random graphs (Poisson random graphs, PETTER Erd˝ os-R´ enyi graphs) HOLME Null model 2: random graphs constrained to the set of metabolic networks, degrees of the original graph intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  19. network null-models Network Network structures are relative . . . modularity, currency metabolites . . . so one has to be clear about what to compare with . . . and graph representa- a null model tions of metabolism Null model 1: random graphs (Poisson random graphs, PETTER Erd˝ os-R´ enyi graphs) HOLME Null model 2: random graphs constrained to the set of metabolic networks, degrees of the original graph intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  20. network null-models Network Network structures are relative . . . modularity, currency metabolites . . . so one has to be clear about what to compare with . . . and graph representa- a null model tions of metabolism Null model 1: random graphs (Poisson random graphs, PETTER Erd˝ os-R´ enyi graphs) HOLME Null model 2: random graphs constrained to the set of metabolic networks, degrees of the original graph intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  21. network null-models Network Network structures are relative . . . modularity, currency metabolites . . . so one has to be clear about what to compare with . . . and graph representa- a null model tions of metabolism Null model 1: random graphs (Poisson random graphs, PETTER Erd˝ os-R´ enyi graphs) HOLME Null model 2: random graphs constrained to the set of metabolic networks, degrees of the original graph intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  22. random graphs Network modularity, currency metabolites and graph representa- tions of metabolism PETTER HOLME metabolic networks, intro for each pair of vertices, networks (physicist style?) with probability p , add an edge modularity & currency metabolites subnetwork hierarchies

  23. random graphs Network modularity, currency metabolites and graph representa- tions of metabolism PETTER HOLME metabolic networks, intro for each pair of vertices, networks (physicist style?) with probability p , add an edge modularity & currency metabolites subnetwork hierarchies

  24. random graphs Network modularity, currency metabolites and graph representa- tions of metabolism PETTER HOLME metabolic networks, intro for each pair of vertices, networks (physicist style?) with probability p , add an edge modularity & currency metabolites subnetwork hierarchies

  25. random graphs Network modularity, currency metabolites and graph representa- tions of metabolism PETTER HOLME metabolic networks, intro for each pair of vertices, networks (physicist style?) with probability p , add an edge modularity & currency metabolites subnetwork hierarchies

  26. random graphs Network modularity, currency metabolites and graph representa- tions of metabolism PETTER HOLME metabolic networks, intro for each pair of vertices, networks (physicist style?) with probability p , add an edge modularity & currency metabolites subnetwork hierarchies

  27. random graphs Network modularity, currency metabolites and graph representa- tions of metabolism PETTER HOLME metabolic networks, intro for each pair of vertices, networks (physicist style?) with probability p , add an edge modularity & currency metabolites subnetwork hierarchies

  28. random graphs Network modularity, currency metabolites and graph representa- tions of metabolism PETTER HOLME metabolic networks, intro for each pair of vertices, networks (physicist style?) with probability p , add an edge modularity & currency metabolites subnetwork hierarchies

  29. random graphs Network modularity, currency metabolites and graph representa- tions of metabolism PETTER HOLME metabolic networks, intro for each pair of vertices, networks (physicist style?) with probability p , add an edge modularity & currency metabolites subnetwork hierarchies

  30. random graphs Network modularity, currency metabolites and graph representa- tions of metabolism PETTER HOLME metabolic networks, intro for each pair of vertices, networks (physicist style?) with probability p , add an edge modularity & currency metabolites subnetwork hierarchies

  31. random graphs Network modularity, currency metabolites and graph representa- tions of metabolism PETTER HOLME metabolic networks, intro for each pair of vertices, networks (physicist style?) with probability p , add an edge modularity & currency metabolites subnetwork hierarchies

  32. random graphs Network modularity, currency metabolites and graph representa- tions of metabolism PETTER HOLME metabolic networks, intro for each pair of vertices, networks (physicist style?) with probability p , add an edge modularity & currency metabolites subnetwork hierarchies

  33. random graphs Network modularity, currency metabolites and graph representa- tions of metabolism PETTER HOLME metabolic networks, intro for each pair of vertices, networks (physicist style?) with probability p , add an edge modularity & currency metabolites subnetwork hierarchies

  34. random rewiring Network modularity, currency metabolites and graph representa- tions of metabolism PETTER HOLME metabolic networks, intro start from the original graph networks (physicist style?) choose edge pairs, and swap them modularity & currency metabolites subnetwork hierarchies

  35. random rewiring Network modularity, currency metabolites and graph representa- tions of metabolism PETTER HOLME metabolic networks, intro start from the original graph networks (physicist style?) choose edge pairs, and swap them modularity & currency metabolites subnetwork hierarchies

  36. random rewiring Network modularity, currency metabolites and graph representa- tions of metabolism PETTER HOLME metabolic networks, intro start from the original graph networks (physicist style?) choose edge pairs, and swap them modularity & currency metabolites subnetwork hierarchies

  37. random rewiring Network modularity, currency metabolites and graph representa- tions of metabolism PETTER HOLME metabolic networks, intro start from the original graph networks (physicist style?) choose edge pairs, and swap them modularity & currency metabolites subnetwork hierarchies

  38. random rewiring Network modularity, currency metabolites and graph representa- tions of metabolism PETTER HOLME metabolic networks, intro start from the original graph networks (physicist style?) choose edge pairs, and swap them modularity & currency metabolites subnetwork hierarchies

  39. degree distribution Network modularity, currency metabolites and graph representa- tions of metabolism PETTER HOLME metabolic networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  40. network structure of metabolism Network Power-law degree distributions. modularity, currency metabolites Increasing average degree. and graph representa- Network modularity? tions of metabolism PETTER HOLME metabolic networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  41. network structure of metabolism Network Power-law degree distributions. modularity, currency metabolites Increasing average degree. and graph representa- Network modularity? tions of metabolism PETTER HOLME metabolic networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  42. network structure of metabolism Network Power-law degree distributions. modularity, currency metabolites Increasing average degree. and graph representa- Network modularity? tions of metabolism PETTER HOLME metabolic networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  43. network structure of metabolism Network Power-law degree distributions. modularity, currency metabolites Increasing average degree. and graph representa- Network modularity? tions of metabolism PETTER HOLME metabolic networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  44. network structure of metabolism Network modularity, currency metabolites and graph representa- tions of metabolism PETTER HOLME metabolic networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  45. network structure of metabolism Network modularity, currency metabolites and graph representa- tions of metabolism PETTER HOLME metabolic networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  46. modularity (of a partition) Network modularity, Definition currency metabolites 2  and graph    representa-     �  �    Q =   tions of  e ii −  e ij  (1)           metabolism            i j PETTER HOLME the sum is over the a partition into clusters and e ij is the metabolic networks, fraction of edges that leads between vertices of cluster i and j intro 2 networks (physicist style?) 1 modularity & currency Q ≈ 0 . 48 metabolites subnetwork hierarchies

  47. modularity (of a graph) Network modularity, Definition currency metabolites and graph ˆ Q ( G ) = partitions Q max (2) representa- tions of metabolism PETTER (null model: random graphs) HOLME metabolic Definition networks, intro � � ˆ Q ( G ) = partitions Q − max partitions Q max (3) networks (physicist style?) modularity & (null model: random graphs with the same degree sequence) currency metabolites subnetwork hierarchies

  48. modularity (of a graph) Network modularity, Definition currency metabolites and graph ˆ Q ( G ) = partitions Q max (2) representa- tions of metabolism PETTER (null model: random graphs) HOLME metabolic Definition networks, intro � � ˆ Q ( G ) = partitions Q − max partitions Q max (3) networks (physicist style?) modularity & (null model: random graphs with the same degree sequence) currency metabolites subnetwork hierarchies

  49. modularity (the biological idea) Network modularity, currency metabolites and graph representa- tions of metabolism PETTER HOLME (Bhattacharyya et al. , 2006, Annu. Rev. Biochem. 75 , pp. 655–80) metabolic networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  50. modularity (the biological idea) Network modularity, currency metabolites and graph representa- tions of metabolism PETTER HOLME metabolic networks, intro networks (physicist style?) modularity & currency metabolites (Bhattacharyya et al. , 2006, Annu. Rev. Biochem. 75 , pp. subnetwork hierarchies 655–80)

  51. modularity and robustness Network robustness—sensitivity of a relevant dynamic system on modularity, currency the network to perturbations metabolites and graph perturbations = changes in: representa- tions of metabolism network topology PETTER concentrations HOLME delays in periodic input metabolic in spreading of harmful things (like disease), modularity networks, intro increases robustness networks (physicist in other systems where a flow is needed throughout the style?) networks, modularity might decrease robustness modularity & currency metabolites subnetwork hierarchies

  52. modularity and robustness Network robustness—sensitivity of a relevant dynamic system on modularity, currency the network to perturbations metabolites and graph perturbations = changes in: representa- tions of metabolism network topology PETTER concentrations HOLME delays in periodic input metabolic in spreading of harmful things (like disease), modularity networks, intro increases robustness networks (physicist in other systems where a flow is needed throughout the style?) networks, modularity might decrease robustness modularity & currency metabolites subnetwork hierarchies

  53. modularity and robustness Network robustness—sensitivity of a relevant dynamic system on modularity, currency the network to perturbations metabolites and graph perturbations = changes in: representa- tions of metabolism network topology PETTER concentrations HOLME delays in periodic input metabolic in spreading of harmful things (like disease), modularity networks, intro increases robustness networks (physicist in other systems where a flow is needed throughout the style?) networks, modularity might decrease robustness modularity & currency metabolites subnetwork hierarchies

  54. modularity and robustness Network robustness—sensitivity of a relevant dynamic system on modularity, currency the network to perturbations metabolites and graph perturbations = changes in: representa- tions of metabolism network topology PETTER concentrations HOLME delays in periodic input metabolic in spreading of harmful things (like disease), modularity networks, intro increases robustness networks (physicist in other systems where a flow is needed throughout the style?) networks, modularity might decrease robustness modularity & currency metabolites subnetwork hierarchies

  55. modularity and robustness Network robustness—sensitivity of a relevant dynamic system on modularity, currency the network to perturbations metabolites and graph perturbations = changes in: representa- tions of metabolism network topology PETTER concentrations HOLME delays in periodic input metabolic in spreading of harmful things (like disease), modularity networks, intro increases robustness networks (physicist in other systems where a flow is needed throughout the style?) networks, modularity might decrease robustness modularity & currency metabolites subnetwork hierarchies

  56. modularity and robustness Network robustness—sensitivity of a relevant dynamic system on modularity, currency the network to perturbations metabolites and graph perturbations = changes in: representa- tions of metabolism network topology PETTER concentrations HOLME delays in periodic input metabolic in spreading of harmful things (like disease), modularity networks, intro increases robustness networks (physicist in other systems where a flow is needed throughout the style?) networks, modularity might decrease robustness modularity & currency metabolites subnetwork hierarchies

  57. modularity and robustness Network robustness—sensitivity of a relevant dynamic system on modularity, currency the network to perturbations metabolites and graph perturbations = changes in: representa- tions of metabolism network topology PETTER concentrations HOLME delays in periodic input metabolic in spreading of harmful things (like disease), modularity networks, intro increases robustness networks (physicist in other systems where a flow is needed throughout the style?) networks, modularity might decrease robustness modularity & currency metabolites subnetwork hierarchies

  58. modularity and robustness Network robustness—sensitivity of a relevant dynamic system on modularity, currency the network to perturbations metabolites and graph perturbations = changes in: representa- tions of metabolism network topology PETTER concentrations HOLME delays in periodic input metabolic in spreading of harmful things (like disease), modularity networks, intro increases robustness networks (physicist in other systems where a flow is needed throughout the style?) networks, modularity might decrease robustness modularity & currency metabolites subnetwork hierarchies

  59. modularity and robustness Network modularity, currency metabolites modular redundancy and graph representa- tions of metabolism PETTER HOLME metabolic networks, intro networks (physicist style?) modularity & currency (Kitano, 2004, Nat. Rev. Genet. 5 , pp. 826–837.) metabolites subnetwork hierarchies

  60. modularity and robustness Network distributed redundancy (A. Wagner): modularity, currency metabolites In distributed robustness, many parts of a system and graph representa- contribute to its function, but all of these parts have tions of metabolism different roles. When one part fails or is changed PETTER through mutations, other parts can compensate for HOLME this failure, but not simply by standing in for the failed metabolic part. networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  61. currency metabolites Network e modularity, t P a currency − v metabolites u 3 r e and graph e y P t p d a e e representa- − l P y r t t P o 6 e a a tions of h − − n r r , c e 6 1 e e e metabolism 6 y d c c o e l e e e e l g y y h t a PETTER s s a s s l l p r P g g o o v o o e HOLME s − c c t t u c P P c c o 3 r u u y u u − − h y l l l , g g r r g 1 3 2 p p f f metabolic networks, ATP intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  62. currency metabolites Network Wagner & Fell, 2001 Schuster et al. , 2002 Ma & Zeng, 2003 modularity, currency ATP ATP ATP metabolites and graph ADP ADP ADP representa- tions of NADP NADP NADP metabolism NADPH NADPH PETTER HOLME NAD NAD metabolic NADH networks, intro P i P i networks H 2 O H 2 O (physicist style?) H + modularity & PP i currency metabolites CMP subnetwork CO 2 hierarchies O 2 NH 3

  63. currency metabolites (a definition) Network currency metabolites have high degree modularity, 1 currency metabolites they make not meaningful shortcuts 2 and graph representa- i.e. tie together distant parts of the network 3 tions of metabolism i.e. tie different modules together 4 PETTER HOLME . . . let’s turn this around to a definition . . . metabolic networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  64. currency metabolites (a definition) Network currency metabolites have high degree modularity, 1 currency metabolites they make not meaningful shortcuts 2 and graph representa- i.e. tie together distant parts of the network 3 tions of metabolism i.e. tie different modules together 4 PETTER HOLME . . . let’s turn this around to a definition . . . metabolic networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  65. currency metabolites (a definition) Network currency metabolites have high degree modularity, 1 currency metabolites they make not meaningful shortcuts 2 and graph representa- i.e. tie together distant parts of the network 3 tions of metabolism i.e. tie different modules together 4 PETTER HOLME . . . let’s turn this around to a definition . . . metabolic networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  66. currency metabolites (a definition) Network currency metabolites have high degree modularity, 1 currency metabolites they make not meaningful shortcuts 2 and graph representa- i.e. tie together distant parts of the network 3 tions of metabolism i.e. tie different modules together 4 PETTER HOLME . . . let’s turn this around to a definition . . . metabolic networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  67. currency metabolites (a definition) Network currency metabolites have high degree modularity, 1 currency metabolites they make not meaningful shortcuts 2 and graph representa- i.e. tie together distant parts of the network 3 tions of metabolism i.e. tie different modules together 4 PETTER HOLME . . . let’s turn this around to a definition . . . metabolic networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  68. currency metabolites (a definition) Network currency metabolites have high degree modularity, 1 currency metabolites they make not meaningful shortcuts 2 and graph representa- i.e. tie together distant parts of the network 3 tions of metabolism i.e. tie different modules together 4 PETTER HOLME . . . let’s turn this around to a definition . . . metabolic networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  69. currency metabolites (a definition) Network modularity, Definition currency metabolites Remove vertices in order of (currently) highest degree. The set and graph representa- of removed vertices that gives the network the highest tions of metabolism modularity is the set of currency metabolites. PETTER HOLME metabolic networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  70. human currency metabolites Network 0.16 modularity, currency metabolites 0.14 and graph representa- 0.12 ∆ Q tions of metabolism 0.10 currency metabolites PETTER HOLME commodity metabolites 0.08 metabolic 0 10 20 30 40 networks, intro number of deleted vertices networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  71. detected currency metabolites Network Wagner & Fell, 2001 Schuster et al. , 2002 Ma & Zeng, 2003 modularity, currency ATP ATP ATP metabolites and graph ADP ADP ADP representa- tions of NADP NADP NADP metabolism NADPH NADPH PETTER HOLME NAD NAD metabolic NADH networks, intro P i P i networks H 2 O H 2 O (physicist style?) H + modularity & PP i currency metabolites CMP subnetwork CO 2 hierarchies O 2 NH 3

  72. modularity and robustness Network modularity, x currency metabolites vi and graph viii representa- commodity metabolites currency metabolites ix tions of iii i metabolism PETTER ii iv HOLME xi v metabolic networks, intro vii networks (physicist style?) i. nitrogen metabolism ii. amino acid metabolism / protein synthesis iii. sugar metabolism iv. citric acid cycle / porphyrin synthesis v. fatty acid metabolism vi. steroid metabolism modularity & vii. metabolism of xenobiotics viii. tyrosine and tryptamine metabolism ix. DNA metabolism / glycolysis currency x. nucleoside / nucleotide metabolism xi. glycerophospholipid and arachidonic acid metabolism metabolites subnetwork hierarchies

  73. different organisms Network organism samples nodes edges c. m. modularity modularity, currency animals 5 1621 4662 6.2 0.157 metabolites and graph plants 1 1561 4302 1 0.144 representa- tions of fungi 2 1281 3654 1.5 0.150 metabolism bacteria 99 1050 2739 1.7 0.140 PETTER HOLME metabolic networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  74. conclusions Network proposed graph based definition of currency metabolites modularity, currency metabolites metabolic networks are modular, but not so much and graph representa- the reason they are not more modular might be give tions of metabolism robustness PETTER HOLME metabolic (Huss & Holme, 2007, IET Syst. Biol. 1 , pp. 280–5.) networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  75. conclusions Network proposed graph based definition of currency metabolites modularity, currency metabolites metabolic networks are modular, but not so much and graph representa- the reason they are not more modular might be give tions of metabolism robustness PETTER HOLME metabolic (Huss & Holme, 2007, IET Syst. Biol. 1 , pp. 280–5.) networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  76. conclusions Network proposed graph based definition of currency metabolites modularity, currency metabolites metabolic networks are modular, but not so much and graph representa- the reason they are not more modular might be give tions of metabolism robustness PETTER HOLME metabolic (Huss & Holme, 2007, IET Syst. Biol. 1 , pp. 280–5.) networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  77. conclusions Network proposed graph based definition of currency metabolites modularity, currency metabolites metabolic networks are modular, but not so much and graph representa- the reason they are not more modular might be give tions of metabolism robustness PETTER HOLME metabolic (Huss & Holme, 2007, IET Syst. Biol. 1 , pp. 280–5.) networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  78. conclusions Network proposed graph based definition of currency metabolites modularity, currency metabolites metabolic networks are modular, but not so much and graph representa- the reason they are not more modular might be give tions of metabolism robustness PETTER HOLME metabolic (Huss & Holme, 2007, IET Syst. Biol. 1 , pp. 280–5.) networks, intro networks (physicist style?) modularity & currency metabolites subnetwork hierarchies

  79. subnetwork hierarchies Network P . Holme, M. Huss & H. Jeong, 2003, Bioinformatics 19 , pp. modularity, currency 532–8. metabolites and graph representa- Start with a directed bipartite networks, with both reaction tions of metabolism and substance vertices (keep the currency metabolites). PETTER HOLME Iteratively remove the reaction vertices with the highest betweenness (fraction of shortest paths passing through a metabolic networks, vertex). intro networks Study the dendrogram of this process. (physicist style?) modularity & 0 currency h 0 metabolites max subnetwork hierarchies h S 2 h 0 S 1 h 0 ( ) ( ) h

  80. subnetwork hierarchies Network P . Holme, M. Huss & H. Jeong, 2003, Bioinformatics 19 , pp. modularity, currency 532–8. metabolites and graph representa- Start with a directed bipartite networks, with both reaction tions of metabolism and substance vertices (keep the currency metabolites). PETTER HOLME Iteratively remove the reaction vertices with the highest betweenness (fraction of shortest paths passing through a metabolic networks, vertex). intro networks Study the dendrogram of this process. (physicist style?) modularity & 0 currency h 0 metabolites max subnetwork hierarchies h S 2 h 0 S 1 h 0 ( ) ( ) h

  81. subnetwork hierarchies Network P . Holme, M. Huss & H. Jeong, 2003, Bioinformatics 19 , pp. modularity, currency 532–8. metabolites and graph representa- Start with a directed bipartite networks, with both reaction tions of metabolism and substance vertices (keep the currency metabolites). PETTER HOLME Iteratively remove the reaction vertices with the highest betweenness (fraction of shortest paths passing through a metabolic networks, vertex). intro networks Study the dendrogram of this process. (physicist style?) modularity & 0 currency h 0 metabolites max subnetwork hierarchies h S 2 h 0 S 1 h 0 ( ) ( ) h

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