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Summarizing Multiple Gene Trees Using Cluster Networks Regula Rupp, Daniel H. Huson MIEP, June 2008 Overview Trees, Clusters and Cluster Networks Hardwired vs. Softwired Networks Lowest Single Ancestor (LSA) LSA Consensus vs.

  1. Solutions:  No single “root” -> Add full set to the clusters  Leaves with more than one in-edge  Internal nodes with multiple in-edges and multiple out- edges -> If in-degree >1, insert new edge: C MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 31 7

  2. Solutions:  No single “root” -> Add full set to the clusters  Leaves with more than one in-edge  Internal nodes with multiple in-edges and multiple out- edges -> If in-degree >1, insert new edge: -> C C MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 32 7

  3. ABCDEF ABCDE BCDE DEF BC AB DE B A C F D E MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 33 8

  4. ABCDEF ABCDE BCDE DEF BC AB DE A C F D E B MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 34 8

  5. ABCDEF ABCDE BCDE DEF BC AB A C DE F B D E MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 35 7 8

  6. Solutions:  No single “root” -> Add full set to the clusters  Leaves with more than one in-edge  Internal nodes with multiple in-edges and multiple out-edges -> If in-degree >1, insert new edge  Clusters represented by nodes instead of edges MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 36 9

  7. Solutions:  No single “root” -> Add full set to the clusters  Leaves with more than one in-edge  Internal nodes with multiple in-edges and multiple out-edges -> If in-degree >1, insert new edge  Clusters represented by nodes instead of edges -> Represent every cluster by its in-edge (which is unique now!) MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 37 9

  8. ABCDEF ABCDE BCDE DEF BC AB A C DE F B D E MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 38 10

  9. ABCDE DEF BCDE AB BC F A C DE A C E F D B B D E MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 39 10

  10. Reticulation edges ABCDE DEF BCDE AB BC F A C DE A C E F D B B D E MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 40 10

  11. Result: cluster network MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 41 11

  12. Result: cluster network A cluster network consists of a rooted directed acyclic graph together with a leaf-labeling and 3 additional properties: MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 42 11

  13. Result: cluster network A cluster network consists of a rooted directed acyclic graph together with a leaf-labeling and 3 additional properties: - Uniqueness MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 43 11

  14. Result: cluster network A cluster network consists of a rooted directed acyclic graph together with a leaf-labeling and 3 additional properties: - Uniqueness - Nestedness MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 44 11

  15. Result: cluster network A cluster network consists of a rooted directed acyclic graph together with a leaf-labeling and 3 additional properties: - Uniqueness - Nestedness - Reducedness MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 45 11

  16. Uniqueness A C F B D E MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 46 12

  17. Uniqueness {B,C,D,E} A C F B D E MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 47 12

  18. Uniqueness {B,C,D,E} A C F B D E MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 48 12

  19. Nestedness A C F B D E MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 49 13

  20. Nestedness {B,C} ⊂ {B,C,D,E} A C F B D E MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 50 13

  21. Nestedness {B,C} ⊂ {B,C,D,E} A C F B D E MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 51 13

  22. Reducedness A C F B D E MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 52 14

  23. Reducedness A C F B D E MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 53 14

  24. Reducedness X A C F B D E MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 54 14

  25. Displaying clusters in cluster networks MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 55 15

  26. Displaying clusters in cluster networks Every non-reticulation edge e in a cluster network defines a cluster, namely the set of labels of all nodes below e. MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 56 15

  27. Displaying clusters in cluster networks Every non-reticulation edge e in a cluster network defines a cluster, namely the set of labels of all nodes below e. We call this the „hardwired interpretation“. MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 57 15

  28. Displaying clusters in cluster networks Every non-reticulation edge e in a cluster network defines a cluster, namely the set of labels of all nodes below e. We call this the „hardwired interpretation“. In contrast we define the „softwired interpretation“ where we may switch reticulation edges on or off. MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 58 15

  29. Hardwired / Softwired T2: T1: A B C D E F A B C D E F MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 59 16

  30. Hardwired / Softwired T2: T1: A B C D E F A B C D E F Hardwired: A B C D E F MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 60 16

  31. Hardwired / Softwired T2: T1: A B C D E F A B C D E F Hardwired: A B C D E F A B C D E F MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 61 16

  32. Hardwired / Softwired T2: T1: A B C D E F A B C D E F Hardwired: A B C D E F A B C D E F MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 62 16

  33. Hardwired / Softwired T2: T1: A B C D E F A B C D E F Hardwired: Softwired: A B C D E F A B C D E F MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 63 16

  34. Hardwired / Softwired  Cluster network, “Hardwired”: blue edges always on  Reticulate network, “Softwired”: For any reticulation, any blue edge can be on or off MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 64 17

  35. Hardwired / Softwired  Cluster network, “Hardwired”: blue edges always on – More reticulations, “looks complicated”  Reticulate network, “Softwired”: For any reticulation, any blue edge can be on or off MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 65 17

  36. Hardwired / Softwired  Cluster network, “Hardwired”: blue edges always on – More reticulations, “looks complicated”  Reticulate network, “Softwired”: For any reticulation, any blue edge can be on or off – Number of reticulations can be minimized MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 66 17

  37. Hardwired / Softwired  Cluster network, “Hardwired”: blue edges always on – More reticulations, “looks complicated”  Reticulate network, “Softwired”: For any reticulation, any blue edge can be on or off – Number of reticulations can be minimized – Computationally hard MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 67 17

  38. Hardwired / Softwired  Cluster network, “Hardwired”: blue edges always on – More reticulations, “looks complicated” – Canonical network, computationally easy  Reticulate network, “Softwired”: For any reticulation, any blue edge can be on or off – Number of reticulations can be minimized – Computationally hard MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 68 17

  39. Lowest Single Ancestor: LSA A C D E F B MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 69 18

  40. Lowest Single Ancestor: LSA  In trees: LCA A C D E F B MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 70 18

  41. Lowest Single Ancestor: LSA  In trees: LCA  In cluster networks: LSA = Lowest Single Ancestor: A C D E F B MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 71 18

  42. Lowest Single Ancestor: LSA  In trees: LCA  In cluster networks: LSA = Lowest Single Ancestor: LSA(S) = Lowest node that is on every path from the root to one of the nodes in S. A C D E F B MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 72 18

  43. Lowest Single Ancestor: LSA  In trees: LCA  In cluster networks: LSA = Lowest Single Ancestor: LSA(S) = Lowest node that is on every path from the root to one of the nodes in S. A C D E F B B MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 73 18

  44. Lowest Single Ancestor: LSA  In trees: LCA  In cluster networks: LSA = Lowest Single Ancestor: LSA(S) = Lowest node that is on every path from the root to one of the nodes in S. N A C D E F B B MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 74 18

  45. Lowest Single Ancestor: LSA  In trees: LCA  In cluster networks: LSA = Lowest Single Ancestor: LSA(S) = Lowest node that is on every path from the root to one of the nodes in S. N R A C D E F B B MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 75 18

  46. Lowest Single Ancestor: LSA  In trees: LCA  In cluster networks: LSA = Lowest Single Ancestor: LSA(S) = Lowest node that is on every path from the root to one of the nodes in S. N R A C D E F B B MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 76 18

  47. Lowest Single Ancestor: LSA  In trees: LCA  In cluster networks: LSA = Lowest Single Ancestor: LSA(S) = Lowest node that is on every path from the root to one of the nodes in S. ? LSA{B,N,R} -> N R A C D E F B B MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 77 18

  48. Lowest Single Ancestor: LSA  In trees: LCA  In cluster networks: LSA = Lowest Single Ancestor: LSA(S) = Lowest node that is on every path from the root to one of the nodes in S. ? LSA{B,N,R} -> -> no! N R A C D E F B B MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 78 18

  49. Lowest Single Ancestor: LSA  In trees: LCA  In cluster networks: LSA = Lowest Single Ancestor: LSA(S) = Lowest node that is on every path from the root to one of the nodes in S. LSA{B,N,R} -> N R A C D E F B B MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 79 18

  50. LSA consensus tree LSA of a reticulation A B D E F C MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 80 19

  51. LSA consensus tree LSA of a reticulation R A B D E F C MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 81 19

  52. LSA consensus tree LSA of a reticulation R A B D E F C MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 82 19

  53. LSA consensus tree LSA of a reticulation LSA(R) R A B D E F C MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 83 19

  54. LSA consensus tree LSA of a reticulation A B D E F C MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 84 19

  55. LSA consensus tree LSA of a reticulation A B D E F C MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 85 19

  56. LSA consensus tree LSA of a reticulation A B D E F C MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 86 19

  57. LSA consensus tree LSA of a reticulation A B D E F C MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 87 19

  58. LSA consensus tree LSA of a reticulation -> new consensus method! A B D E F C MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 88 19

  59. LSA consensus tree LSA of a reticulation -> new consensus method! A B D E F C MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 89 19

  60. LSA Consensus MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 90 20

  61. LSA Consensus  New consensus method MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 91 20

  62. LSA Consensus  New consensus method  LSA is lowest node for which all input trees agree that it is an ancestor MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 92 20

  63. LSA Consensus  New consensus method  LSA is lowest node for which all input trees agree that it is an ancestor  Easy to compute MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 93 20

  64. LSA Consensus  New consensus method  LSA is lowest node for which all input trees agree that it is an ancestor  Easy to compute  Different from all other consensus methods (?) MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 94 20

  65. LSA Consensus  New consensus method  LSA is lowest node for which all input trees agree that it is an ancestor  Easy to compute  Different from all other consensus methods (?)  Philippe Gambette: LSA consensus = Adams consensus? MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 95 20

  66. Adams Consensus MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 96 21

  67. Adams Consensus  Sets of trees T1,T2,...,Tn MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 97 21

  68. Adams Consensus  Sets of trees T1,T2,...,Tn  Maximal clusters in Adams Consensus: non-empty intersections of maximal clusters in T1,T2,...,Tn MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 98 21

  69. Adams Consensus  Sets of trees T1,T2,...,Tn  Maximal clusters in Adams Consensus: non-empty intersections of maximal clusters in T1,T2,...,Tn  Restrict trees to maximal clusters of Adams consensus and repeat procedure recursively. MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 99 21

  70. Adams vs. LSA Question: LSA consensus = Adams consensus? MIEP 08 - Summarizing Multiple Gene Trees Using Cluster Networks - Regula Rupp 100 22

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