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Phylogenetic trees III Maximum Parsimony Gerhard Jger Words, Bones, Genes, Tools February 28, 2018 Gerhard Jger Maximum Parsimony WBGT 1 / 30 Background Background Gerhard Jger Maximum Parsimony WBGT 2 / 30 Background estimates

  1. Phylogenetic trees III Maximum Parsimony Gerhard Jäger Words, Bones, Genes, Tools February 28, 2018 Gerhard Jäger Maximum Parsimony WBGT 1 / 30

  2. Background Background Gerhard Jäger Maximum Parsimony WBGT 2 / 30

  3. Background estimates complete scenario (or distribution over scenarios) for each WBGT Maximum Parsimony Gerhard Jäger least in theory, that is...) fjnds the tree that best explains the observed variation in the data (at character character-based tree estimation Character-based tree estimation the history of individual characters distances are black boxes — we get a tree, but we learn nothing about no solid statistical justifjcation for those algorithms Neighbor Joining and UPGMA produce good but sub-optimal trees same rate very strong theoretical assumptions - e.g., all characters evolve at the distance-based tree estimation has several drawbacks: 3 / 30

  4. Parsimony Parsimony Gerhard Jäger Maximum Parsimony WBGT 4 / 30

  5. Parsimony Parsimony of a tree background reading: Ewens and Grant (2005), 15.6 suppose a character matrix and a tree are given parsimony score: minimal number of mutations that has to be assumed to explain the character values at the tips, given the tree Gerhard Jäger Maximum Parsimony WBGT 5 / 30

  6. Parsimony Parsimony of a tree WBGT Maximum Parsimony Gerhard Jäger 6 / 30 Kopf kop head tête testa cap "head" "head" "head" "head" "head" "head"

  7. Parsimony Parsimony of a tree WBGT Maximum Parsimony Gerhard Jäger 6 / 30 Kopf kop head tête testa cap "head" "head" "head" "head" "head" "head"

  8. Parsimony Parsimony of a tree WBGT Maximum Parsimony Gerhard Jäger 6 / 30 ? ? ? "head" ? ? Kopf kop head tête testa cap "head" "head" "head" "head" "head" "head"

  9. Parsimony Parsimony of a tree WBGT Maximum Parsimony Gerhard Jäger 6 / 30 *kop testa "head" "head" Kopf kop head tête testa cap "head" "head" "head" "head" "head" "head"

  10. Parsimony Parsimony of a tree WBGT Maximum Parsimony Gerhard Jäger 6 / 30 *kaput- "head" *haubud- caput "head" "head" *kop testa "head" "head" Kopf kop head tête testa cap "head" "head" "head" "head" "head" "head"

  11. Parsimony Parsimony reconstruction WBGT Maximum Parsimony Gerhard Jäger 7 / 30 B Parsimony = 2 B B C A B C A A C B

  12. Parsimony Parsimony reconstruction WBGT Maximum Parsimony Gerhard Jäger 7 / 30 A Parsimony = 3 A B C A B C A A C B

  13. Parsimony Parsimony reconstruction WBGT Maximum Parsimony Gerhard Jäger 7 / 30 C Parsimony = 3 A C C A B C A A C B

  14. Parsimony Weight matrix WBGT Maximum Parsimony Gerhard Jäger Weighted parsimony reconstruction 8 / 30 B Weighted Parsimony = 3 A B C B B A 0 1 2 C 1 0 2 B A C 2 2 0 A A B C C B

  15. Parsimony Weight matrix WBGT Maximum Parsimony Gerhard Jäger Weighted parsimony reconstruction 8 / 30 A Weighted Parsimony = 4 A B C A B A 0 1 2 C 1 0 2 B A C 2 2 0 A A B C C B

  16. Parsimony Weight matrix WBGT Maximum Parsimony Gerhard Jäger Weighted parsimony reconstruction 8 / 30 C Weighted Parsimony = 5 A B C A C A 0 1 2 C 1 0 2 B A C 2 2 0 A A B C C B

  17. Parsimony Dynamic Programming (Sankofg Algorithm) WBGT Maximum Parsimony Gerhard Jäger 9 / 30 � wp ( mother , s ) s ′ ∈ states ( w ( s, s ′ ) + wp ( d, s ′ )) min = d ∈ daughters B C A A C B 0 ∞ ∞ ∞ ∞ ∞ 0 ∞ 0 ∞ ∞ ∞ 0 ∞ ∞ ∞ 0 0

  18. Parsimony Dynamic Programming (Sankofg Algorithm) WBGT Maximum Parsimony Gerhard Jäger 9 / 30 � wp ( mother , s ) s ′ ∈ states ( w ( s, s ′ ) + wp ( d, s ′ )) min = d ∈ daughters 0 2 4 4 0 4 B C A A C B 0 ∞ ∞ ∞ ∞ ∞ 0 ∞ 0 ∞ ∞ ∞ 0 ∞ ∞ ∞ 0 0

  19. Parsimony Dynamic Programming (Sankofg Algorithm) WBGT Maximum Parsimony Gerhard Jäger 9 / 30 � wp ( mother , s ) s ′ ∈ states ( w ( s, s ′ ) + wp ( d, s ′ )) min = d ∈ daughters 1 1 4 3 2 2 0 2 4 4 0 4 B C A A C B 0 ∞ ∞ ∞ ∞ ∞ 0 ∞ 0 ∞ ∞ ∞ 0 ∞ ∞ ∞ 0 0

  20. Parsimony Dynamic Programming (Sankofg Algorithm) WBGT Maximum Parsimony Gerhard Jäger 9 / 30 � wp ( mother , s ) s ′ ∈ states ( w ( s, s ′ ) + wp ( d, s ′ )) min = d ∈ daughters 3 5 4 1 1 4 3 2 2 0 2 4 4 0 4 B C A A C B 0 ∞ ∞ ∞ ∞ ∞ 0 ∞ 0 ∞ ∞ ∞ 0 ∞ ∞ ∞ 0 0

  21. Parsimony Dynamic Programming (Sankofg Algorithm) WBGT Maximum Parsimony Gerhard Jäger 10 / 30 � wp ( mother , s ) s ′ ∈ states ( w ( s, s ′ ) + wp ( d, s ′ )) min = d ∈ daughters 3 5 4 1 1 4 3 2 2 0 2 4 4 0 4 B C A A C B 0 ∞ ∞ ∞ ∞ ∞ 0 ∞ 0 ∞ ∞ ∞ 0 ∞ ∞ ∞ 0 0

  22. Parsimony Dynamic Programming (Sankofg Algorithm) WBGT Maximum Parsimony Gerhard Jäger 10 / 30 � wp ( mother , s ) s ′ ∈ states ( w ( s, s ′ ) + wp ( d, s ′ )) min = d ∈ daughters 3 5 4 1 1 4 3 2 2 0 2 4 4 0 4 B C A A C B 0 ∞ ∞ ∞ ∞ ∞ 0 ∞ 0 ∞ ∞ ∞ 0 ∞ ∞ ∞ 0 0

  23. Parsimony Dynamic Programming (Sankofg Algorithm) WBGT Maximum Parsimony Gerhard Jäger 10 / 30 � wp ( mother , s ) s ′ ∈ states ( w ( s, s ′ ) + wp ( d, s ′ )) min = d ∈ daughters 3 5 4 1 1 4 3 2 2 0 2 4 4 0 4 B C A A C B 0 ∞ ∞ ∞ ∞ ∞ 0 ∞ 0 ∞ ∞ ∞ 0 ∞ ∞ ∞ 0 0

  24. Parsimony Dynamic Programming (Sankofg Algorithm) WBGT Maximum Parsimony Gerhard Jäger 10 / 30 � wp ( mother , s ) s ′ ∈ states ( w ( s, s ′ ) + wp ( d, s ′ )) min = d ∈ daughters 3 5 4 1 1 4 3 2 2 0 2 4 4 0 4 B C A A C B 0 ∞ ∞ ∞ ∞ ∞ 0 ∞ 0 ∞ ∞ ∞ 0 ∞ ∞ ∞ 0 0

  25. Parsimony Dynamic Programming (Sankofg Algorithm) WBGT Maximum Parsimony Gerhard Jäger 10 / 30 � wp ( mother , s ) s ′ ∈ states ( w ( s, s ′ ) + wp ( d, s ′ )) min = d ∈ daughters 3 5 4 1 1 4 3 2 2 0 2 4 4 0 4 B C A A C B 0 ∞ ∞ ∞ ∞ ∞ 0 ∞ 0 ∞ ∞ ∞ 0 ∞ ∞ ∞ 0 0

  26. Parsimony Searching for the best tree total parsimony score of tree: sum over all characters note: if weight matrix is symmetric, location of the root doesn’t matter Sankofg algorithm effjciently computes parsimony score of a given tree goal: tree which minimizes parsimony score Gerhard Jäger Maximum Parsimony WBGT 11 / 30 no effjcient way to fjnd the optimal tree → heuristic tree search

  27. Searching the tree space Searching the tree space Gerhard Jäger Maximum Parsimony WBGT 12 / 30

  28. Searching the tree space How many rooted tree topologies are there? Gerhard Jäger Maximum Parsimony WBGT 13 / 30 n=2 1 2

  29. Searching the tree space How many rooted tree topologies are there? WBGT Maximum Parsimony Gerhard Jäger 13 / 30 n=2 1 2 n=3 1 3 2 1 2 3 1 2 3 1 3 2

  30. Searching the tree space How many rooted tree topologies are there? WBGT Maximum Parsimony Gerhard Jäger 13 / 30 n=2 1 2 n=3 1 3 2 1 2 3 1 2 3 1 3 2 n=4 1 4 3 2 1 3 4 2 1 3 2 1 3 4 2 1 3 4 2 4

  31. Searching the tree space How many rooted tree topologies are there? WBGT Maximum Parsimony Gerhard Jäger 14 / 30 2 1 3 3 4 15 5 105 6 945 7 10395 8 135135 9 2027025 10 34459425 11 654729075 12 13749310575 13 316234143225 14 7 . 9 e + 12 15 2 . 1 e + 14 16 6 . 1 e + 15 17 1 . 9 e + 17 f (2) = 1 18 6 . 3 e + 18 19 2 . 2 e + 20 20 8 . 2 e + 21 f ( n + 1) = (2 n − 3) f ( n ) 21 3 . 1 e + 23 22 1 . 3 e + 25 23 5 . 6 e + 26 (2 n − 3)! 24 2 . 5 e + 28 f ( n ) = 25 1 . 1 e + 30 26 5 . 8 e + 31 2 n − 2 ( n − 2)! 27 2 . 9 e + 33 28 1 . 5 e + 35 29 8 . 6 e + 36 30 4 . 9 e + 38 31 2 . 9 e + 40 32 1 . 7 e + 42 33 1 . 1 e + 44 34 7 . 2 e + 45 35 4 . 8 e + 47 36 3 . 3 e + 49 37 2 . 3 e + 51 38 1 . 7 e + 53 39 1 . 3 e + 55 40 1 . 0 e + 57

  32. Searching the tree space How many unrooted tree topologies are there? Gerhard Jäger Maximum Parsimony WBGT 15 / 30 n=3 3 2 1

  33. Searching the tree space How many unrooted tree topologies are there? WBGT Maximum Parsimony Gerhard Jäger 15 / 30 n=3 n=4 3 4 2 1 3 2 4 2 3 1 1 2 3 4 1

  34. Searching the tree space How many unrooted tree topologies are there? WBGT Maximum Parsimony Gerhard Jäger 15 / 30 n=5 n=3 n=4 3 4 2 5 3 2 5 5 4 3 4 4 3 2 2 3 2 5 5 4 2 3 4 1 1 1 1 1 1 3 2 4 3 2 4 2 5 2 4 4 5 2 2 3 3 5 3 5 2 3 4 3 4 5 1 1 1 1 1 1 1 2 3 3 4 4 5 5 2 5 2 2 3 4 2 4 2 5 3 4 3 4 5 3 1 1 1 1 1 1

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