RADICAL PHYLOGENETIC INVERSION Mike Hendy Institute of Fundamental Sciences Massey University Palmerston North New Zealand November 2010 Mike Hendy RADICAL PHYLOGENETIC INVERSION
Acknowledgements ◮ David Penny, Massey University ◮ Mike Steel, Canterbury University ◮ Peter Waddell, University of South Carolina ◮ Andreas Dress, Universit¨ at Bielefeld Mike Hendy RADICAL PHYLOGENETIC INVERSION
Local to Global ◮ X = { x 0 , x 1 , · · · , x n } is a set of n + 1 taxa, and T is an X –tree (the leaves represent the taxa). Example X = { x 0 , x 1 , x 2 , x 3 } : x 1 x 2 ❅ � ❅ � � ❅ x 0 x 3 � ❅ Mike Hendy RADICAL PHYLOGENETIC INVERSION
Local to Global ◮ X = { x 0 , x 1 , · · · , x n } is a set of n + 1 taxa, and T is an X –tree (the leaves represent the taxa). Example X = { x 0 , x 1 , x 2 , x 3 } : x 1 x 2 ATGGTAT ACGGTGT ❅ � ❅ � ❅ � ❅ � � ❅ � ❅ x 0 x 3 � ❅ � ❅ ACGGTTT ACGATGT ◮ Nucleotide sequences evolve from an ancestral sequence, under a model of nucleotide substitution governed by local stochastic matrices M e on each edge of T . Mike Hendy RADICAL PHYLOGENETIC INVERSION
Local to Global ◮ X = { x 0 , x 1 , · · · , x n } is a set of n + 1 taxa, and T is an X –tree (the leaves represent the taxa). Example X = { x 0 , x 1 , x 2 , x 3 } : x 1 x 2 ATGGTAT ACGGTGT ❅ � ❅ � ❅ � ❅ � � ❅ � ❅ x 0 x 3 � ❅ � ❅ ACGGTTT ACGATGT ◮ Nucleotide sequences evolve from an ancestral sequence, under a model of nucleotide substitution governed by local stochastic matrices M e on each edge of T . ◮ We (usually) can only observe are the global data, the aligned homologous sequences for the taxa x i ∈ X . ATGGTAT ACGGTGT ? ACGGTTT ACGATGT Mike Hendy RADICAL PHYLOGENETIC INVERSION
Local to Global ◮ X = { x 0 , x 1 , · · · , x n } is a set of n + 1 taxa, and T is an X –tree (the leaves represent the taxa). Example X = { x 0 , x 1 , x 2 , x 3 } : x 1 x 2 ATGGTAT ACGGTGT ❅ � ❅ � ❅ � ❅ � � ❅ � ❅ x 0 x 3 � ❅ � ❅ ACGGTTT ACGATGT ◮ Nucleotide sequences evolve from an ancestral sequence, under a model of nucleotide substitution governed by local stochastic matrices M e on each edge of T . ◮ We (usually) can only observe are the global data, the aligned homologous sequences for the taxa x i ∈ X . ATGGTAT ACGGTGT ? ACGGTTT ACGATGT ◮ The inverse problem of phylogenetics, deduce the local structure of T , and (if possible) the stochastic matrices M e , from the global data (aligned sequences). Mike Hendy RADICAL PHYLOGENETIC INVERSION
Global to Local - Pathset Group The path Π ij is the set of edges of T x 1 x 2 ❅ � ❅ � connecting the leaves x i , x j . The pathset � ❅ x 0 x 3 � ❅ group of T is the group generated by the paths Π 01 , Π 02 , · · · Π 0 n under disjoint union. ◮ Mike Hendy RADICAL PHYLOGENETIC INVERSION
Global to Local - Pathset Group The path Π ij is the set of edges of T x 1 x 2 ❅ � ❅ � connecting the leaves x i , x j . The pathset � ❅ x 0 x 3 � ❅ group of T is the group generated by the paths Π 01 , Π 02 , · · · Π 0 n under disjoint union. ◮ ◮ The 8 pathsets of the pathset group of T 23 : x 1 x 2 x 1 x 2 ❅ � ❅ � ❅ � ❅ � � � � � x 0 x 0 Π ∅ Π 01 Π 02 Π 12 x 1 � x 2 x 1 x 2 ❅ ❅ � ❅ � ❅ � � ❅ ❅ ❅ � ❅ � ❅ ❅ ❅ � ❅ x 0 x 3 x 3 x 3 x 0 x 3 Π 03 Π 13 Π 23 Π 0123 Mike Hendy RADICAL PHYLOGENETIC INVERSION
Global to Local - Pathset Group The path Π ij is the set of edges of T x 1 x 2 ❅ � ❅ � connecting the leaves x i , x j . The pathset � ❅ x 0 x 3 � ❅ group of T is the group generated by the paths Π 01 , Π 02 , · · · Π 0 n under disjoint union. ◮ ◮ The 8 pathsets of the pathset group of T 23 : x 1 x 2 x 1 x 2 ❅ � ❅ � ❅ � ❅ � � � � � x 0 x 0 Π ∅ Π 01 Π 02 Π 12 x 1 � x 2 x 1 x 2 ❅ ❅ � ❅ � ❅ � � ❅ ❅ ❅ � ❅ � ❅ ❅ ❅ � ❅ x 0 x 3 x 3 x 3 x 0 x 3 Π 03 Π 13 Π 23 Π 0123 x 1 x 2 ◮ The pathset Π 0123 is the disjoint union ❅ � ❅ � � ❅ of Π 01 , Π 02 and Π 03 . � ❅ x 0 x 3 Π 0123 Mike Hendy RADICAL PHYLOGENETIC INVERSION
Global to Local - Pathsets ◮ The edge e 23 belongs to the four pathsets Π B where | B ∩ { 2 , 3 }| is an odd number. x 1 x 2 x 1 x 2 ❅ � ❅ � ❅ � ❅ � � � � � x 0 x 0 Π ∅ Π 01 Π 02 Π 12 x 1 � x 2 x 1 x 2 ❅ ❅ � ❅ � ❅ � � ❅ ❅ ❅ � ❅ � ❅ ❅ ❅ � ❅ x 0 x 3 x 3 x 3 x 0 x 3 Π 03 Π 13 Π 23 Π 0123 Mike Hendy RADICAL PHYLOGENETIC INVERSION
Global to Local - Pathsets ◮ The edge e 23 belongs to the four pathsets Π B where | B ∩ { 2 , 3 }| is an odd number. x 1 x 2 x 1 x 2 ❅ � ❅ � ❅ � ❅ � � � � � x 0 x 0 Π ∅ Π 01 Π 02 Π 12 x 1 � x 2 x 1 x 2 ❅ ❅ � ❅ � ❅ � � ❅ ❅ ❅ � ❅ � ❅ ❅ ❅ � ❅ x 0 x 3 x 3 x 3 x 0 x 3 Π 03 Π 13 Π 23 Π 0123 ◮ In general edge e A in Π B iff | A ∩ B | is odd. Mike Hendy RADICAL PHYLOGENETIC INVERSION
Global to Local - Pathsets ◮ The edge e 23 belongs to the four pathsets Π B where | B ∩ { 2 , 3 }| is an odd number. x 1 x 2 x 1 x 2 ❅ � ❅ � ❅ � ❅ � � � � � x 0 x 0 Π ∅ Π 01 Π 02 Π 12 x 1 � x 2 x 1 x 2 ❅ ❅ � ❅ � ❅ � � ❅ ❅ ❅ � ❅ � ❅ ❅ ❅ � ❅ x 0 x 3 x 3 x 3 x 0 x 3 Π 03 Π 13 Π 23 Π 0123 ◮ In general edge e A in Π B iff | A ∩ B | is odd. ◮ An X –tree has 2 | X |− 1 = 2 n pathsets. Mike Hendy RADICAL PHYLOGENETIC INVERSION
Global to Local - Pathsets x 1 x 2 x 1 A = { 2 , 3 } x 2 ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ ❅ � � ❅ � ❅ � ❅ ❅ � � ❅ � ❅ � ❅ � ❅ � ❳❳ ✘ ❅ ❅ ❳❳ ✘ � � ✘✘ ✘✘ ❳ ❳ � � ❅ ❅ � ❅ � ❅ � ❅ � ❅ � � ❅ ❅ � ❅ � ❅ � ❅ � � ❅ ❅ � ❅ � ❅ � ❅ ◮ x 0 � ❅ � ❅ x 3 x 0 x 3 Π 02 + Π 12 + Π 03 + Π 13 Π 01 + Π 12 + Π 03 + Π 13 − Π ∅ − Π 01 − Π 23 − Π 0123 Mike Hendy RADICAL PHYLOGENETIC INVERSION
Global to Local - Pathsets x 1 x 2 x 1 A = { 2 , 3 } x 2 ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ ❅ � � ❅ � ❅ � ❅ ❅ � � ❅ � ❅ � ❅ � ❅ � ❳❳ ✘ ❅ ❅ ❳❳ ✘ � � ✘✘ ✘✘ ❳ ❳ � � ❅ ❅ � ❅ � ❅ � ❅ � ❅ � � ❅ ❅ � ❅ � ❅ � ❅ � � ❅ ❅ � ❅ � ❅ � ❅ ◮ x 0 � ❅ � ❅ x 3 x 0 x 3 Π 02 + Π 12 + Π 03 + Π 13 Π 01 + Π 12 + Π 03 + Π 13 − Π ∅ − Π 01 − Π 23 − Π 0123 x 1 A = { 1 , 2 , 3 } x 2 x 1 A = { 1 , 2 } x 2 ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ ❅ � � ❅ ❅ � � ❅ � ❅ � ❅ ❅ � � ❅ ❅ � � ❅ � ❅ � ❅ � ❅ � ❳❳ ✘ ❳❳ ✘ ❅ ❅ ✘✘ � � ❅ ❅ ✘✘ � � ❳ � � ❅ ❅ � � ❅ ❅ � ❅ � ❅ � ❅ � ❅ � � ❅ ❅ � � ❅ ❅ � ❅ � ❅ � � ❅ ❅ � � ❅ ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ ◮ x 0 x 3 x 0 x 3 Π 01 + Π 02 + Π 03 + Π 0123 Π 01 + Π 02 + Π 13 + Π 23 − Π ∅ − Π 12 − Π 13 − Π 23 − Π ∅ − Π 12 − Π 03 − Π 0123 Mike Hendy RADICAL PHYLOGENETIC INVERSION
Pathsets ◮ Each edge e A ∈ E ( T ) belongs to the 2 | X |− 2 pathsets Π B with | A ∩ B | odd. x 4 x 0 The edge e 124 belongs to ❅ � e 4 the 8 pathsets Π 01 , Π 02 , ❅ � e 1234 e 14 e 124 ❅ � Π 13 , Π 23 , Π 04 , Π 0124 , Π 34 � ❅ e 1 e 2 e 3 and Π 1234 . � ❅ � ❅ x 1 x 2 x 3 Mike Hendy RADICAL PHYLOGENETIC INVERSION
Pathsets ◮ Each edge e A ∈ E ( T ) belongs to the 2 | X |− 2 pathsets Π B with | A ∩ B | odd. x 4 x 0 The edge e 124 belongs to ❅ � e 4 the 8 pathsets Π 01 , Π 02 , ❅ � e 1234 e 14 e 124 ❅ � Π 13 , Π 23 , Π 04 , Π 0124 , Π 34 � ❅ e 1 e 2 e 3 and Π 1234 . � ❅ � ❅ x 1 x 2 x 3 ◮ Each pathset Π B comprises all edges e A ∈ E ( T ) with | A ∩ B | odd. x 4 x 0 The pathset Π 1234 contains ❅ � e 4 the edges e 1 , e 2 , e 3 , e 4 and ❅ � e 1234 e 14 e 124 ❅ � e 124 . � ❅ e 1 e 2 e 3 � ❅ � ❅ x 1 x 2 x 3 Mike Hendy RADICAL PHYLOGENETIC INVERSION
Global to Local - Pathweights ◮ Suppose w : E ( T ) → R is an edge-weighting function. Then for any even-ordered subset A ⊆ X , the weight of the pathset Π A is � w (Π A ) = w ( e B ) . e B ∈ E ( T ): | A ∩ B |≡ 1 (mod 2) Mike Hendy RADICAL PHYLOGENETIC INVERSION
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