Constructing Evolutionary/Phylogenetic Trees

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1 Constructing Evolutionary/Phylogenetic Trees 2 broad categories: istance-based methods Ultrametric Additive: UPGMA Transformed istance Neighbor-Joining Character-based Maximum Parsimony Maximum Likelihood Bayesian Methods 3/31/05 1

2 Ultrametric An ultrametric tree: decreasing internal node labels distance between two nodes is label of least common ancestor. An ultrametric distance matrix: Symmetric matrix such that for every i, j, k, there is tie for maximum of (i,j), (j,k), (i,k) ij, ik jk i j k 3/31/05 2

3 Ultrametric: Assumptions Molecular Clock Hypothesis, Zuckerkandl & Pauling, 1962: Accepted point mutations in amino acid sequence of a protein occurs at a constant rate. Varies from protein to protein Varies from one part of a protein to another 3/31/05 3

4 Ultrametric: Example A B C E F G H A B C E F G H A E B,,F,H C,G 3/31/05 4

5 Ultrametric: Example A B C E F G H 5 A B C E F G H A C,G B F E H 3/31/05 5

6 Ultrametric: istances Computed A B C E F G H 5 A B C 2 E F G H A C,G B F E H 3/31/05 6

7 Unweighted pair-group method with arithmetic means (UPGMA) A B C AB C B d AB C d (AB)C C d AC d BC d (AB) d C d A d B d C d (AB)C = (d AC + d BC ) /2 A d AB /2 B The result is an ultrametric tree. 3/31/05 7

8 Transformed istance Method UPGMA makes errors when rate constancy among lineages does not hold. Remedy: introduce an outgroup & make corrections n ko ' = Now apply UPGMA ij ij 2 io jo + k = 1 n 3/31/05 8

9 Example: Start with an alignment 3/31/05 9

10 Example: Compute distances between sequences 3/31/05 10

11 Example: Tree using UPGMA 3/31/05 11

12 Additive-istance Trees Additive distance trees are edge-weighted trees, with distance between leaf nodes are exactly equal to length of path between nodes. A B C A B A C B 1 2 C 3/31/05 12

13 Unrooted Trees on 4 Taxa A C B A C A B B C 3/31/05 13

14 Four-Point Condition If the true tree is as shown below, then 1. d AB + d C < d AC + d B, and 2. d AB + d C < d A + d BC A C B 3/31/05 14

15 Saitou & Nei: Neighbor-Joining Method Start with a star topology. Find the pair to separate such that the total length of the tree is minimized. The pair is then replaced by its arithmetic mean, and the process is repeated. S 12 = ( n 2) n ( k 2k k = 3 ( n ) 2) 3 i j n ij 3/31/05 15

16 Neighbor-Joining 1 2 n n = = n j i ij n k k k n n S ) ( 1 ) ( 2) 2( 1 2 3/31/05 16

17 Example (Unrooted Tree) 3/31/05 17

18 Example (Rooted Tree) 3/31/05 18

19 Bootstrapping: Select columns 3/31/05 19

20 Constructing Evolutionary/Phylogenetic Trees 2 broad categories: istance-based methods Ultrametric Additive: UPGMA Transformed istance Neighbor-Joining Character-based Maximum Parsimony Maximum Likelihood Bayesian Methods 3/31/05 20

21 Character-based Methods Input: characters, morphological features, sequences, etc. Output: phylogenetic tree that provides the history of what features changed. [Perfect Phylogeny Problem] one leaf/object, 1 edge per character, path changed traits A B C B E E A C 3/31/05 21

22 Example Perfect phylogeny does not always exist A B C E A B C E B E 5 A C 3/31/05 22

23 Maximum Parsimony Minimize the total number of mutations implied by the evolutionary history 3/31/05 23

24 Examples of Character ata Characters/Sites A Sequences B A A G A G T T C A C A G C C G T T C T A G A T A T C C A E A G A G A T C C T 3/31/05 24

25 Maximum Parsimony Method: Example Sequence s Characters/Sites A A G A G T T C A 2 A G C C G T T C T 3 A G A T A T C C A 4 A G A G A T C C T 3/31/05 25

26 Unrooted Trees on 4 Taxa A C B A C A B B C 3/31/05 26

27 A A G A G T T C A 2 A G C C G T T C T 3 A G A T A T C C A 4 A G A G A T C C T 3/31/05 27

28 Inferring nucleotides on internal nodes 3/31/05 28

29 Searching for the Maximum Parsimony Tree: Exhaustive Search 3/31/05 29

30 Searching for the Maximum Parsimony Tree: Branch-&-Bound 3/31/05 30

31 Probabilistic Models of Evolution Assuming a model of substitution, Pr{S i (t+ ) = Y S i (t) = X}, Using this formula it is possible to compute the likelihood that data is generated by a given phylogenetic tree T under a model of substitution. Now find the tree with the maximum likelihood. Y X Time elapsed? Prob of change along edge? Pr{S i (t+ ) = Y S i (t) = X} Prob of data? Product of prob for all edges 3/31/05 31

32 Computing Maximum Likelihood Tree 3/31/05 32

33 3/31/05 33

34 Models of Nucleotide Substitution Jukes-Cantor 1-3α α α α α 1-3α α α α α 1-3α α α α α 1-3α Kimura 3ST 1-α-β-γ α β γ α 1- α- β-γ γ β β γ 1- α- β-γ α γ β α 1- α- β-γ PAM Matrix for Amino Acids 3/31/05 34

35 PHYLIP s Characterbased Methods 3/31/05 35

36 PHYLIP s istancebased Methods 3/31/05 36

37 Bootstrap: Estimating confidence level of a phylogenetic hypothesis 3/31/05 37

38 Tree Evaluation Methods Bootstrap Skewness Test (Randomized Trees) Permutation Test (Randomized Characters) Parametric bootstrap Likelihood Ratio Tests 3/31/05 38

39 Computing Evolutionary Relationships: Basic Assumptions Evolutionary divergences are strictly bifurcating, i.e., observed data can be represented by a tree. [Exceptions: transfer of genetic material between organisms] Sampling of individuals from a group is enough to determine the relationships Each individual evolves independently. 3/31/05 39

40 UPGMA(fast) Comparisons assumes rate constancy; rarely used. Produces ultrametric trees Additive Methods (fast) If four-point condition is satisfied, works well Quality depends on quality of distance data oes not consider multiple substitutions at site Long sequences & small distances, small errors. Produces additive distance trees 3/31/05 40

41 Additive Metrics: Four-Point Condition If the true tree is as shown below, then 1. d AB + d C < d AC + d B, and 2. d AB + d C < d A + d BC A C B 3/31/05 41

42 Ultrametric vs Additive Metrics Check whether A is an additive distance matrix. Check whether U is an ultrametric matrix COMMENT: This is just a reduction. It does not mean that one can build a tree for the same data! 3/31/05 42

43 Comparisons Maximum Parsimony Character-based methods trusted more. MP assumes simplest explanation is always the best! No explicit assumptions, except that a tree that requires fewer substitutions is better Faster evolution on long branches may give rise to homoplasies, and MP may go wrong. Performance depends on the number of informative sites, and is usually not so good with finding clades. If more than one tree, builds consensus trees. 3/31/05 43

44 Comparisons Maximum Likelihood Uses all sites, unlike MP method. Makes assumptions on the rate and pattern of substitution. Relatively insensitive to violations of assumptions Not very robust (if some sequences very divergent). SLOW! Computationally intensive. Optimum usually cannot be found, since the search space is too large. Fast heuristics exist. Simulations show that it s better than MP and ME. 3/31/05 44

45 Phylogenetic Software PHYLIP, WEBPHYLIP, PhyloBLAST (large set of programs, command-line) PAUP (point-&-click) (MP-based) PUZZLE, TREE-PUZZLE, PAML, MOLPHY (ML-based) MrBAYES (Bayesian Methods) MACCLAE (MP?) LAMARC 3/31/05 45

46 Recipes to minimize errors in phylogenetic analysis Use large amounts of data. Randomize order, if needed. Exclude unreliable data (for e.g., when alignment is not known for sure) Exclude fast-evolving sequences or sites (3 rd codon positions), or only use it for close relationships. Most methods will incorrectly group sequences with similar base composition (Additive methods are robust in the presence of such sequences) Check validity of independence assumptions (e.g. changes on either side of a hairpin structure) Use all methods to look at data. 3/31/05 46

47 Alignments Inputs for phylogenetic analysis usually is a multiple sequence alignment. Programs such as CLUSTALW, produce good alignments, but not good trees. Aligning according to secondary or tertiary trees are better for phylogenetic analysis. Which alignment method is better for which phylogenetic analysis method? OPEN! 3/31/05 47

48 Other Heuristics Branch Swapping to modify existing trees Quartet Puzzling: rapid tree searching 3/31/05 48

Constructing Evolutionary/Phylogenetic Trees

Constructing Evolutionary/Phylogenetic Trees 2 broad categories: Distance-based methods Ultrametric Additive: UPGMA Transformed Distance Neighbor-Joining Character-based Maximum Parsimony Maximum Likelihood