Analyzing Evolutionary Trees

Transcription

1 Analyzing Evolutionary Trees Katherine St. John Lehman College and the Graduate Center City University of New York Katherine St. John City University of New York 1

2 Overview Talk Outline

3 Talk Outline Overview Constructing Trees

4 Talk Outline Overview Constructing Trees Comparing Reconstruction Methods Case Study: Quartet-Based Methods

5 Talk Outline Overview Constructing Trees Comparing Reconstruction Methods Case Study: Quartet-Based Methods Evaluating the Results Comparing, Clustering, and Visualizing Katherine St. John City University of New York 2

6 Talk Outline Overview Constructing Trees Comparing Reconstruction Methods Case Study: Quartet-Based Methods Evaluating the Results Comparing, Clustering, and Visualizing Katherine St. John City University of New York 3

7 Goal: Reconstruct the Evolutionary History (

8 Goal: Reconstruct the Evolutionary History ( The evolutionary process not only determines relationships among taxa, but allows prediction of structural, physiological, and biochemical properties. Katherine St. John City University of New York 4

9 Process for Reconstruction: Input Data Start with information about the taxa. For example: Morphological Characters

10 Process for Reconstruction: Input Data Start with information about the taxa. For example: Morphological Characters Biomolecular Sequences A GTTAGAAGGCGGCCAGCGAC... B CATTTGTCCTAACTTGACGG... C CAAGAGGCCACTGCAGAATC... D CCGACTTCCAACCTCATGCG... E ATGGGGCACGATGGATATCG... F TACAAATACGCGCAAGTTCG... Katherine St. John City University of New York 5

11 Process for Reconstruction

12 Process for Reconstruction Input Data A GTTAGAAGGC... B CATTTGTCCT... C CAAGAGGCCA... D CCGACTTCCA... E ATGGGGCACG... F TACAAATACG...

13 Process for Reconstruction Input Data A GTTAGAAGGC... B CATTTGTCCT... C CAAGAGGCCA... D CCGACTTCCA... E ATGGGGCACG... F TACAAATACG... Reconstruction Algorithms Maximum Parsimony Maximum Likelihood Distance Methods: NJ, Quartet-Based, Fast Convering,.

14 Process for Reconstruction Input Data Reconstruction Algorithms Output Tree A GTTAGAAGGC... B CATTTGTCCT... C CAAGAGGCCA... D CCGACTTCCA... E ATGGGGCACG... F TACAAATACG... Maximum Parsimony Maximum Likelihood Distance Methods: NJ, Quartet-Based, Fast Convering,. Katherine St. John City University of New York 6

15 Applications In addition to finding the evolutionary history of species, phylogeny is also used for:

16 Applications In addition to finding the evolutionary history of species, phylogeny is also used for: drug discovery: used to determine structural and biochemical properties of potential drugs

17 Applications In addition to finding the evolutionary history of species, phylogeny is also used for: drug discovery: used to determine structural and biochemical properties of potential drugs determining origins of HIV infection

18 Applications In addition to finding the evolutionary history of species, phylogeny is also used for: drug discovery: used to determine structural and biochemical properties of potential drugs determining origins of HIV infection origin of other virus and bacteria strains Katherine St. John City University of New York 7

19 Talk Outline Overview Constructing Trees Comparing Reconstruction Methods Case Study: Quartet-Based Methods Evaluating the Results Comparing, Clustering, and Visualizing Katherine St. John City University of New York 8

20 Process for Reconstruction Input Data A GTTAGAAGGC... B CATTTGTCCT... C CAAGAGGCCA... D CCGACTTCCA... E ATGGGGCACG... F TACAAATACG... Reconstruction Algorithms Maximum Parsimony Maximum Likelihood Distance Methods: NJ, Quartet-Based, Fast Convering,. Output Tree Katherine St. John City University of New York 9

21 Algorithms for Reconstruction Most optimization criteria are hard:

22 Algorithms for Reconstruction Most optimization criteria are hard: Maximum Parsimony: (NP-hard: Foulds & Graham 82) find the tree that can explain the observed sequences with a minimal number of substitutions.

23 Algorithms for Reconstruction Most optimization criteria are hard: Maximum Parsimony: (NP-hard: Foulds & Graham 82) find the tree that can explain the observed sequences with a minimal number of substitutions. Maximum Likelihood Estimation: (NP-hard: Roch 06) find the tree with the maximum likelihood: P(data tree).

24 Algorithms for Reconstruction Most optimization criteria are hard: Maximum Parsimony: (NP-hard: Foulds & Graham 82) find the tree that can explain the observed sequences with a minimal number of substitutions. Maximum Likelihood Estimation: (NP-hard: Roch 06) find the tree with the maximum likelihood: P(data tree). For both, the best known algorithms require an exponential number of trees to be checked.

25 Algorithms for Reconstruction Most optimization criteria are hard: Maximum Parsimony: (NP-hard: Foulds & Graham 82) find the tree that can explain the observed sequences with a minimal number of substitutions. Maximum Likelihood Estimation: (NP-hard: Roch 06) find the tree with the maximum likelihood: P(data tree). For both, the best known algorithms require an exponential number of trees to be checked. This is not feasible for more than 20 taxa. Katherine St. John City University of New York 10

26 Approximating Trees Exact answers are often wanted, but hard to find.

27 Approximating Trees Exact answers are often wanted, but hard to find. But approximate is often good enough:

28 Approximating Trees Exact answers are often wanted, but hard to find. But approximate is often good enough: drug design: predicting function via similarity

29 Approximating Trees Exact answers are often wanted, but hard to find. But approximate is often good enough: drug design: predicting function via similarity sequence alignment: guide trees for alignment

30 Approximating Trees Exact answers are often wanted, but hard to find. But approximate is often good enough: drug design: predicting function via similarity sequence alignment: guide trees for alignment use as priors or starting points for expensive searches Katherine St. John City University of New York 11

31 Approximation Algorithms Since calculating the exact answer is hard, algorithms that estimate the answer have been developed.

32 Approximation Algorithms Since calculating the exact answer is hard, algorithms that estimate the answer have been developed. Heuristics for maximum parsimony and maximum likelihood estimation (use clever ways to limit the number of trees checked, while still sampling much of tree-space )

33 Approximation Algorithms Since calculating the exact answer is hard, algorithms that estimate the answer have been developed. Heuristics for maximum parsimony and maximum likelihood estimation (use clever ways to limit the number of trees checked, while still sampling much of tree-space ) Polynomial-time methods, based on the distance between taxa Katherine St. John City University of New York 12

34 Distance-Based Methods These methods calculate the distance between taxa: B D A C F E B D A C F E and then determine the tree using the distance matrix.

35 Distance-Based Methods These methods calculate the distance between taxa: B D A C F E B D A C F E and then determine the tree using the distance matrix. One way to calculate distance is to take differences divided by the length (the normalized Hamming distance). Katherine St. John City University of New York 13

36 Distance-Based Methods Popular distance based methods include

37 Distance-Based Methods Popular distance based methods include Neighbor Joining (Saitou & Nei 87) which repeatedly joins the nearest neighbors to build a tree, and

38 Distance-Based Methods Popular distance based methods include Neighbor Joining (Saitou & Nei 87) which repeatedly joins the nearest neighbors to build a tree, and Quartet-based methods that decide the topology for every 4 taxa and then assemble them to form a tree (Berry et al. 1999, 2000, 2001).

39 Distance-Based Methods Popular distance based methods include Neighbor Joining (Saitou & Nei 87) which repeatedly joins the nearest neighbors to build a tree, and Quartet-based methods that decide the topology for every 4 taxa and then assemble them to form a tree (Berry et al. 1999, 2000, 2001). Many of these methods have good performance empirically, and some can be proven to have nice accuracy properties. Katherine St. John City University of New York 14

40 Testing Methods Empirically How accurate are the methods at reconstructing trees?

41 Testing Methods Empirically How accurate are the methods at reconstructing trees? In biological applications, the true, historical tree is almost never known, which makes assessing the quality of phylogenetic reconstruction methods problematic.

42 Testing Methods Empirically How accurate are the methods at reconstructing trees? In biological applications, the true, historical tree is almost never known, which makes assessing the quality of phylogenetic reconstruction methods problematic. (an exception: Hillis 92 created an evolutionary tree in the laboratory)

43 Testing Methods Empirically How accurate are the methods at reconstructing trees? In biological applications, the true, historical tree is almost never known, which makes assessing the quality of phylogenetic reconstruction methods problematic. (an exception: Hillis 92 created an evolutionary tree in the laboratory) Simulation is used instead to evaluate methods, given a model of evolution. Katherine St. John City University of New York 15

44 Simulation Studies 1. Construct a model tree.

45 Simulation Studies 1. Construct a model tree. 2. Evolve sequences down the tree. A GTTAGAAGGCGGCCA... B CATTTGTCCTAACTT... C CAAGAGGCCACTGCA... D CCGACTTCCAACCTC... E ATGGGGCACGATGGA... F TACAAATACGCGCAA...

46 Simulation Studies 1. Construct a model tree. 2. Evolve sequences down the tree. 3. Reconstruct the tree using method. A GTTAGAAGGCGGCCA... B CATTTGTCCTAACTT... C CAAGAGGCCACTGCA... D CCGACTTCCAACCTC... E ATGGGGCACGATGGA... F TACAAATACGCGCAA...

47 Simulation Studies 1. Construct a model tree. 2. Evolve sequences down the tree. 3. Reconstruct the tree using method. A GTTAGAAGGCGGCCA... B CATTTGTCCTAACTT... C CAAGAGGCCACTGCA... D CCGACTTCCAACCTC... E ATGGGGCACGATGGA... F TACAAATACGCGCAA Evaluate the accuracy of the constructed tree. Katherine St. John City University of New York 16

48 Simulation Studies 1. Construct a model tree. 2. Evolve sequences down the tree. 3. Reconstruct the tree using method. A GTTAGAAGGCGGCCA... B CATTTGTCCTAACTT... C CAAGAGGCCACTGCA... D CCGACTTCCAACCTC... E ATGGGGCACGATGGA... F TACAAATACGCGCAA Evaluate the accuracy of the constructed tree. Katherine St. John City University of New York 17

49 Simulating Data: Choosing Trees Usually chosen from a random distribution on trees: Uniform, or Yule-Harding (birth-death trees)

50 Simulating Data: Choosing Trees Usually chosen from a random distribution on trees: Uniform, or Yule-Harding (birth-death trees) Can view this as two different random processes:

51 Simulating Data: Choosing Trees Usually chosen from a random distribution on trees: Uniform, or Yule-Harding (birth-death trees) Can view this as two different random processes: generate the tree shape, and then

52 Simulating Data: Choosing Trees Usually chosen from a random distribution on trees: Uniform, or Yule-Harding (birth-death trees) Can view this as two different random processes: generate the tree shape, and then assign weights or branch lengths to the shape. Katherine St. John City University of New York 18

53 Simulation Studies 1. Construct a model tree. 2. Evolve sequences down the tree. 3. Reconstruct the tree using method. A GTTAGAAGGCGGCCA... B CATTTGTCCTAACTT... C CAAGAGGCCACTGCA... D CCGACTTCCAACCTC... E ATGGGGCACGATGGA... F TACAAATACGCGCAA Evaluate the accuracy of the constructed tree. Katherine St. John City University of New York 19

54 Simulating Data: Evolving Sequences The Jukes-Cantor (JC) model is the simplest Markov model of biomolecular sequence evolution.

55 Simulating Data: Evolving Sequences The Jukes-Cantor (JC) model is the simplest Markov model of biomolecular sequence evolution. A DNA sequence (a string over {A, C, T, G}) at the root evolves down a rooted binary tree T. AACGT Katherine St. John City University of New York 20

56 Simulating Data: Evolving Sequences The Jukes-Cantor (JC) model is the simplest Markov model of biomolecular sequence evolution. A DNA sequence (a string over {A, C, T, G}) at the root evolves down a rooted binary tree T. AACGT 0 1 AACGA AACGT Katherine St. John City University of New York 21

57 Simulating Data: Evolving Sequences The Jukes-Cantor (JC) model is the simplest Markov model of biomolecular sequence evolution. A DNA sequence (a string over {A, C, T, G}) at the root evolves down a rooted binary tree T. AACGT 0 1 AACGA AACGT ACCCT GACGT AACGA GGCGT Katherine St. John City University of New York 22

58 Simulating Data: Evolving Sequences The Jukes-Cantor (JC) model is the simplest Markov model of biomolecular sequence evolution. A DNA sequence (a string over {A, C, T, G}) at the root evolves down a rooted binary tree T. AACGT 0 1 AACGA AACGT ACCCT GACGT AACGA GGCGT GACGT AACGT GACGT GGCGA Katherine St. John City University of New York 23

59 Simulating Data: Evolving Sequences The Jukes-Cantor (JC) model is the simplest Markov model of biomolecular sequence evolution. A DNA sequence (a string over {A, C, T, G}) at the root evolves down a rooted binary tree T. AACGT 0 1 AACGA AACGT ACCCT GACGT AACGA GGCGT GACGT AACGT GACGT GGCGA Katherine St. John City University of New York 24

60 Simulating Data: Evolving Sequences The Jukes-Cantor (JC) model is the simplest Markov model of biomolecular sequence evolution. A DNA sequence (a string over {A, C, T, G}) at the root evolves down a rooted binary tree T. {ACCCT, GACGT, AACGT, GACGT, GGCGA} Katherine St. John City University of New York 25

61 Simulating Data: Evolving Sequences The Jukes-Cantor (JC) model is the simplest Markov model of biomolecular sequence evolution. A DNA sequence (a string over {A, C, T, G}) at the root evolves down a rooted binary tree T. The assumptions of the model are: 1. the sites (i.e., the positions within the sequences) evolve independently and identically 2. if a site changes state it changes with equal probability to each of the remaining states, and 3. the number of changes of each site on an edge e is a Poisson random variable with expectation λ(e) (this is also called the length of the edge e). Katherine St. John City University of New York 26

62 Simulation Studies 1. Construct a model tree. 2. Evolve sequences down the tree. 3. Reconstruct the tree using method. A GTTAGAAGGCGGCCA... B CATTTGTCCTAACTT... C CAAGAGGCCACTGCA... D CCGACTTCCAACCTC... E ATGGGGCACGATGGA... F TACAAATACGCGCAA Evaluate the accuracy of the constructed tree. Katherine St. John City University of New York 27

63 Simulation Studies 1. Construct a model tree. 2. Evolve sequences down the tree. 3. Reconstruct the tree using method. A GTTAGAAGGCGGCCA... B CATTTGTCCTAACTT... C CAAGAGGCCACTGCA... D CCGACTTCCAACCTC... E ATGGGGCACGATGGA... F TACAAATACGCGCAA Evaluate the accuracy of the constructed tree. Katherine St. John City University of New York 28

64 Evaluating Accuracy To compare reconstructed tree to model tree, the Robinson-Foulds Score is often used: False Positives + False Negatives total edges b a c d e f... b c d a f e

65 Evaluating Accuracy To compare reconstructed tree to model tree, the Robinson-Foulds Score is often used: False Positives + False Negatives total edges b a c d e f... b c d a f e If there are many possible answers, choose the one with the best parsimony score: the sum of the number of site changes acrosss the edges in the tree. Katherine St. John City University of New York 29

66 Talk Outline Overview Constructing Trees Comparing Reconstruction Methods Case Study: Quartet-Based Methods Evaluating the Results Comparing, Clustering, and Visualizing Katherine St. John City University of New York 30

67 Case Study: Quartet Methods A quartet is an unrooted binary tree on four taxa: a b c d {ab cd} a c b d {ac bd} a d b c {ad bc}

68 Case Study: Quartet Methods A quartet is an unrooted binary tree on four taxa: b a d c c a d b d a c b {ab cd} {ac bd} {ad bc} Let Q(T ) = all quartets that agree with T. [Erdős et al. 1997]: T can be reconstructed from Q(T ) in polynomial time. Katherine St. John City University of New York 31

69 Case Study: Quartet Methods Quartet-based methods operate in two phases:

70 Case Study: Quartet Methods Quartet-based methods operate in two phases: Construct quartets on all four taxa sets.

71 Case Study: Quartet Methods Quartet-based methods operate in two phases: Construct quartets on all four taxa sets. Combine these quartets into a tree.

72 Case Study: Quartet Methods Quartet-based methods operate in two phases: Construct quartets on all four taxa sets. Combine these quartets into a tree. Running time: For most optimizations, determining a quartet is fast.

73 Case Study: Quartet Methods Quartet-based methods operate in two phases: Construct quartets on all four taxa sets. Combine these quartets into a tree. Running time: For most optimizations, determining a quartet is fast. There are Θ(n 4 ) quartets, giving Ω(n 4 ) running time.

74 Case Study: Quartet Methods Quartet-based methods operate in two phases: Construct quartets on all four taxa sets. Combine these quartets into a tree. Running time: For most optimizations, determining a quartet is fast. There are Θ(n 4 ) quartets, giving Ω(n 4 ) running time. In practice, the input quality is insufficient to ensure that all quartets are accurately inferred.

75 Case Study: Quartet Methods Quartet-based methods operate in two phases: Construct quartets on all four taxa sets. Combine these quartets into a tree. Running time: For most optimizations, determining a quartet is fast. There are Θ(n 4 ) quartets, giving Ω(n 4 ) running time. In practice, the input quality is insufficient to ensure that all quartets are accurately inferred. Quartet methods have to handle incorrect quartets. Katherine St. John City University of New York 32

76 Popular Quartet Methods Q or Buneman Method [Berry & Gascuel 97, Buneman 71]: Only add edges that agree with all input quartets. Doesn t tolerate errors outputs conservative, but unresolved tree.

77 Popular Quartet Methods Q or Buneman Method [Berry & Gascuel 97, Buneman 71]: Only add edges that agree with all input quartets. Doesn t tolerate errors outputs conservative, but unresolved tree. Quartet Cleaning (QC) [Berry et al. 1999]: Add edges with a small number of errors proportional to q e. Many variants: all handle a small number of errors.

78 Popular Quartet Methods Q or Buneman Method [Berry & Gascuel 97, Buneman 71]: Only add edges that agree with all input quartets. Doesn t tolerate errors outputs conservative, but unresolved tree. Quartet Cleaning (QC) [Berry et al. 1999]: Add edges with a small number of errors proportional to q e. Many variants: all handle a small number of errors. Quartet Puzzling [Strimmer & von Haeseler 1996]: Order taxa randomly, greedily add edges, repeat 1000 times. Output majority tree. Most popular with biologists. Katherine St. John City University of New York 33

79 Standard Method: Neighbor Joining (NJ) [Saitou & Nei 1987]: very popular and fast: O(n 3 ). Based on the distance between nodes, join neighboring leaves, replace them by their parent, calculate distances to this node, and repeat. This process eventually returns a binary (fully resolved) tree. Joining the leaves with the minimal distance does not suffice, so subtract the averaged distances to compensate for long edges. Experimental work shows that NJ trees are reasonably accurate, given a rate of evolution is neither too low nor too high. Katherine St. John City University of New York 34

80 Our Study [St. John, Moret, Warnow, & Vawter: SODA01; J Algorithms 03] A detailed, large-scale experimental study of quartet methods and NJ under the Jukes-Cantor model of evolution:

81 Our Study [St. John, Moret, Warnow, & Vawter: SODA01; J Algorithms 03] A detailed, large-scale experimental study of quartet methods and NJ under the Jukes-Cantor model of evolution: Our results indicate that NJ always outperforms the quartet-based methods we examined, in terms of both accuracy and speed.

82 Our Study [St. John, Moret, Warnow, & Vawter: SODA01; J Algorithms 03] A detailed, large-scale experimental study of quartet methods and NJ under the Jukes-Cantor model of evolution: Our results indicate that NJ always outperforms the quartet-based methods we examined, in terms of both accuracy and speed. Give new theory about convergence rates of quartet-based methods which helps explain our observations. Katherine St. John City University of New York 35

83 Experimental Design: Parameter Space In all, our study used 16,000 datasets and required many months of computation on the two clusters. Taxa: 5, 10, 20, expected evolutionary rates: from to per tree edge. For each, we generated 100 tree shapes, grouped into 10 runs of 10 trials. Sequence lengths: 500, 2,000, 8,000, and 32,000. Katherine St. John City University of New York 36

84 Measuring Accuracy: Quartets and Edges Topological accuracy is a more demanding criterion than quartet accuracy. % tree edges * % quartets (a) seq. length 500 % tree edges ** % quartets (b) seq. length 2000 λ e * (Percent of true tree edges recovered by Quartet Puzzling for 40 taxa and two sequence lengths) Katherine St. John City University of New York 37

85 Sensitivity to Input Quality Methods that estimate quartets and then combine them into a single tree can be greatly affected by the quality of the input quartets. % edges % quartets Q NJ % edges % quartets QCNJ Katherine St. John City University of New York 38

86 Running Times NJ was clearly the fastest method tested.

87 Running Times NJ was clearly the fastest method tested. QCML and QP were by far the slowest of the methods tested, slow enough that running them on more than a fifty taxa appears infeasible at present.

88 Running Times NJ was clearly the fastest method tested. QCML and QP were by far the slowest of the methods tested, slow enough that running them on more than a fifty taxa appears infeasible at present. With default settings, QP takes more than 200 days of computation to analyze ten runs of ten trials each for a single set of parameters on 80 taxa with a sequence length of 500. (30 minutes for NJ). Katherine St. John City University of New York 39

89 Quartet Study Results Quartet quality impacts performance significantly. Maximizing the number of quartets correct doesn t yield better trees.

90 Quartet Study Results Quartet quality impacts performance significantly. Maximizing the number of quartets correct doesn t yield better trees. Clear linear order of accuracy for the methods (except under very low rates of evolution): NJ, QP, QC and Q.

91 Quartet Study Results Quartet quality impacts performance significantly. Maximizing the number of quartets correct doesn t yield better trees. Clear linear order of accuracy for the methods (except under very low rates of evolution): NJ, QP, QC and Q. Study led to the development of new methods that weight quartets and have better accuracy on shorter sequences. Katherine St. John City University of New York 40

92 Talk Outline Overview Constructing Trees Comparing Reconstruction Methods Case Study: Quartet-Based Methods Evaluating the Results Comparing, Clustering, and Visualizing Katherine St. John City University of New York 41

93 Analyzing & Visualizing Sets of Trees Amenta & Klingner, InfoVis 02 Hillis, Heath, & St. John, Sys. Biol. 05 Katherine St. John City University of New York 42

94 Evaluating the Results Often, a search will result in many (often thousands) of trees with the same score.

95 Evaluating the Results Often, a search will result in many (often thousands) of trees with the same score. Input Data A GTTAGAAGGC... B CATTTGTCCT... C CAAGAGGCCA... D CCGACTTCCA... E ATGGGGCACG... F TACAAATACG... Reconstruction Algorithms Maximum Parsimony Maximum Likelihood Distance Methods: NJ, Quartet-Based, Fast Convering,. Output Tree Katherine St. John City University of New York 43

96 Evaluating the Results Often, a search, will result in many (often thousands) of trees with the same score. Input Data A GTTAGAAGGC... B CATTTGTCCT... C CAAGAGGCCA... D CCGACTTCCA... E ATGGGGCACG... F TACAAATACG... Reconstruction Algorithms Maximum Parsimony Maximum Likelihood Output Trees Katherine St. John City University of New York 44

97 Summarizing Trees Input Trees Consensus Method Output Trees Strict Consensus Majority-rule Katherine St. John City University of New York 45

98 Visualizing Sets of Trees Efficiency is important for real-time visualization. Katherine St. John City University of New York 46

99 Strict Consensus Tree Input trees Strict Consensus s 0 s 1 s 2 s 3 s 4 s 0 s 1 s 2 s 3 s 4 s 0 s 1 s 2 s 4 s 3 s 0 s 1 s 2 s 3 s 4 s 1 s 2 s 0 s 3 s 4 s 2 s 3 s 0 s 1 s 4 s 2 s 4 s 0 s 1 s 3 s 1 s 2 s 3 s 0 s 4 s 1 s 2 s 3 s 0 s 4 s 2 s 3 s 4 s 0 s 1 Katherine St. John City University of New York 47

100 Majority-rule Tree Input trees Majority-rule Tree s 0 s 1 s 2 s 3 s 4 s 0 s 1 s 2 s 3 s 4 s 0 s 1 s 2 s 4 s 3 s 0 s 1 s 2 s 3 s 4 Includes splits found in a majority of trees Can be 2/3 majority, etc. Katherine St. John City University of New York 48

101 Running Time Majority-rule Consensus: O((n/w)(nt + lg x)): Margush & McMorris, (x = total number of splits, nt) Implemented in PHYLIP, Felsenstein et al. O(n 2 + nt 2 ): Wareham, O(nt): Amenta, Clarke, & S, WABI 03. Strict Consensus: O(nt): Day, 1985 (assuming w = O(lg n)) Katherine St. John City University of New York 49

102 Our Algorithm Randomized algorithm for majority-rule consensus tree with expected running time O(nt) (n = number of leaves, t = number of trees). Optimal, since reading t trees on n taxa requires Ω(nt) time. Implementation in tree-set visualization software module for Mesquite (W&D Maddison). Katherine St. John City University of New York 50

103 Basic Idea Traverse all input trees in post-order, counting occurrences of splits in a hash table. Build majority-rule tree on splits which appear more than t 2 times. Key issue: Representation of splits Katherine St. John City University of New York 51

104 Distances Between Sets of Trees Katherine St. John City University of New York 52

105 Distances Between Trees Robinson-Foulds distance: # of edges that occur in only one tree. Calculate in O(n) time using Day s Algorithm (1985). Extends naturally to weighted trees. Katherine St. John City University of New York 53

106 Other Natural Metrics Tree-bisection-reconnect (TBR): A C D B E G F A C D B E G F A B A B C D E G F C D G E F TBR is NP-hard. (Allen & Steel 01) Many attempts, but no approximations with provable bounds. Katherine St. John City University of New York 54

107 Other Natural Metrics Subtree-prune-regraft (SPR): A B A B C C G D F D E E G F C D A E B G F NP-hard for rooted trees (Bordewich & Semple 05). 5-approximation for rooted trees (Bonet, Amenta, Mahindru, & S. 06). Katherine St. John City University of New York 55

108 Our Contribution Nicotiana Campanula Scaevola Stokesia Dimorphotheca Senecio Gerbera Gazania Echinops Felicia Tagetes Chromolaena Blennosperma Coreopsis Vernonia Cacosmia Cichorium Achillea Carthamnus Flaveria Piptocarpa Helianthus Tragopogon Chrysanthemum Eupatorium Lactuca Barnadesia Dasyphyllum Nicotiana Campanula Scaevola Stokesia Dimorphotheca Senecio Gazania Gerbera Echinops Felicia Tagetes Chromolaena Blennosperma Coreopsis Vernonia Cacosmia Cichorium Achillea Carthamnus Flaveria Piptocarpa Helianthus Tragopogon Chrysanthemum Eupatorium Lactuca Barnadesia Dasyphyllum Tree-Bisection-Reconnection (TBR): counter-example to past approximation algorithms

109 Our Contribution Nicotiana Campanula Scaevola Stokesia Dimorphotheca Senecio Gerbera Gazania Echinops Felicia Tagetes Chromolaena Blennosperma Coreopsis Vernonia Cacosmia Cichorium Achillea Carthamnus Flaveria Piptocarpa Helianthus Tragopogon Chrysanthemum Eupatorium Lactuca Barnadesia Dasyphyllum Nicotiana Campanula Scaevola Stokesia Dimorphotheca Senecio Gazania Gerbera Echinops Felicia Tagetes Chromolaena Blennosperma Coreopsis Vernonia Cacosmia Cichorium Achillea Carthamnus Flaveria Piptocarpa Helianthus Tragopogon Chrysanthemum Eupatorium Lactuca Barnadesia Dasyphyllum Tree-Bisection-Reconnection (TBR): counter-example to past approximation algorithms Subtree-Prune-Regraft (SPR): First approximation algorithm for rooted SPR Katherine St. John City University of New York 56

110 Approximation Algorithm Outline For each sibling pair (a,b) in the first tree, find in the second and do: Case 1: a b 1 subtree between a & b in T 2 : Cut the subtrees in T 2. N := N + 1 Case 2: a b 2 subtrees between a & b in T 2 : Cut a & b in both. N := N + 2. Case 3: a & b are in separate components for T 2 : Cut a & b in both. N := N + 2. a b Case 4: b is a singleton in T 2 : Cut b in T 1. Note that N doesn t change. a b Case 5: a b The sibling pair occurs in both: Replace by a new node in both. Note that N doesn t change. (Hein et al. version of the algorithm; Rodrigues et al. varies slightly in Cases 1 & 2.) Katherine St. John City University of New York 57

111 Accounting: how good is your estimate? (All based on having a maximum agreement forest for the distance metric.)

112 Accounting: how good is your estimate? (All based on having a maximum agreement forest for the distance metric.) Look at each edge that should be cut (called a link) and see how many extra edges were cut for it. Charge that link the number of edges cut for it.

113 Accounting: how good is your estimate? (All based on having a maximum agreement forest for the distance metric.) Look at each edge that should be cut (called a link) and see how many extra edges were cut for it. Charge that link the number of edges cut for it. Tricky to count! Past attempts for TBR make a subtle mistake that charges non-links. Katherine St. John City University of New York 58

114 Counterexample to Rodrigues et al. 3-Approximation T 1 T a b c d e f g h i j Enlargement of the subtrees between 9 &10: a b c d e f g h i j k l m n o Best Case: Cut the red subtrees; gives TBR distance of 27. Algorithm: Cuts each sibling pair, then subtrees; gives distance of Katherine St. John City University of New York 59

115 5-Approximation With the work of Bordewich & Semple 05, it s now possible to do analysis for rooted SPR.

116 5-Approximation With the work of Bordewich & Semple 05, it s now possible to do analysis for rooted SPR. Slightly modifying the algorithms for TBR, we have a 5-approximation for rooted SPR (very detailed proof).

117 5-Approximation With the work of Bordewich & Semple 05, it s now possible to do analysis for rooted SPR. Slightly modifying the algorithms for TBR, we have a 5-approximation for rooted SPR (very detailed proof). Have an example to show that it can t be better than a 3-approximation, but likely much better in practice.

118 5-Approximation With the work of Bordewich & Semple 05, it s now possible to do analysis for rooted SPR. Slightly modifying the algorithms for TBR, we have a 5-approximation for rooted SPR (very detailed proof). Have an example to show that it can t be better than a 3-approximation, but likely much better in practice. Algorithm runs in linear time. Katherine St. John City University of New York 60

119 Small vs. Large Sets of Trees TreeJuxtaposer compares visually small sets of trees, and Treecomp focuses on comparing, analyzing, and visualizing large sets of phylogenetic trees. Katherine St. John City University of New York 61

120 Comparing Two Trees Will Fisher, UT Katherine St. John City University of New York TreeJuxtaposer, Munzner et al. 62

121 TreeJuxtaposer (Munzner, Guimbretiere, Tasiran, Zhang, & Zhou, SIGGRAPH 2003) Katherine St. John City University of New York 63

122 TreeJuxtaposer Can compare trees of several hundred thousand nodes. Guaranteed visibility: highlighted areas remain visually apparent at all times. Focus+Context navigation technique ( rubber sheet ). Katherine St. John City University of New York 64

123 SequenceJuxtaposer (Slack, Munzner, Hildebrand, & S, 2004) Katherine St. John City University of New York 65

124 SequenceJuxtaposer Accordion drawing guarantees context, visibility, and frame rate. Interactively view a million basepairs. Regular expression searches. Annotation files displayed and navigatable. Katherine St. John City University of New York 66

125 Tree Set Visualization Katherine St. John City University of New York 67

126 Visualizing chains of trees: Color indicates maximum likelihood score Animate feature Applications (Rana dataset from Hillis Lab 500 trees from MrBayes search) Katherine St. John City University of New York 68

127 Summary Constructing Trees: Efficient approximation algorithms for large datasets Comparing Reconstruction Methods: Case Study: Quartet-Based Methods Evaluating the Results: Comparing, Clustering, and Visualizing Katherine St. John City University of New York 69

128 Future Work Efficient heuristics for biological tree differences. Analysis of tree distributions. Dimensionality reduction techniques applied to sequences. Linking TreeJuxtaposer to SequenceJuxtaposer Visualizing & Analyzing Morphological Data Katherine St. John City University of New York 70

129 Acknowledgments National Science Foundation David Hillis Laboratory, U. Texas Wildebeest & Research Clusters at CUNY Mesquite by David and Wayne Maddison treecomp module, many of our students: Left to Right: Back Row: Isaac Boamah (CUNY/UMd), Jeff Klingner (UT/Stanford), Katherine St. John (CUNY), Tamara Munzner (UBC), Silvio Neris (CUNY) Front Row: Denise Edwards (CUNY), Ruchi Kalra (CUNY), Nina Amenta (UT/UCD), Ingrid Montealegre (CUNY), Fred Clarke (CUNY/UCD) Katherine St. John City University of New York 71