Constructing Evolutionary Trees
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1 Constructing Evolutionary Trees 0-0
2
3 HIV Evolutionary Tree SIVs (monkeys)! HIV (human)! human infection! human HIV/M human HIV/M chimpanzee SIV chimpanzee SIV human HIV/N human HIV/N chimpanzee SIV chimpanzee SIV chimpanzee SIV chimpanzee SIV chimpanzee SIV chimpanzee SIV chimpanzee SIV human HIV/O human HIV/O chimpanzee SIV chimpanzee SIV red-capped manabey SIV drill SIV vervet monkey SIV tantalus monkey SIV sooty mangabey SIV human HIV/A human HIV/B sooty mangabey SIV Sykes s monkey SIV greater spot-nosed monkey SIV De Brazzas monkey SIV
4 HIV Evolutionary Tree SIVs (monkeys)! HIV (human)! human infection! But how did biologists generate this? human HIV/M human HIV/M chimpanzee SIV chimpanzee SIV human HIV/N human HIV/N chimpanzee SIV chimpanzee SIV chimpanzee SIV chimpanzee SIV chimpanzee SIV chimpanzee SIV chimpanzee SIV human HIV/O human HIV/O chimpanzee SIV chimpanzee SIV red-capped manabey SIV drill SIV vervet monkey SIV tantalus monkey SIV sooty mangabey SIV human HIV/A human HIV/B sooty mangabey SIV Sykes s monkey SIV greater spot-nosed monkey SIV De Brazzas monkey SIV
5 Constructing a Distance Matrix SPECIES ALIGNMENT Chimp Human Seal Whale ACGTAGGCCT ATGTAAGACT TCGAGAGCAC TCGAAAGCAT
6 Constructing a Distance Matrix D i,j = number of differing symbols between i-th and j-th species. SPECIES ALIGNMENT DISTANCE MATRIX Chimp Human Seal Whale Chimp ACGTAGGCCT Human ATGTAAGACT Seal TCGAGAGCAC Whale TCGAAAGCAT 4 5 0
7 Constructing a Distance Matrix D i,j = number of differing symbols between i-th and j-th species. SPECIES ALIGNMENT DISTANCE MATRIX Chimp Human Seal Whale Chimp ACGTAGGCCT Human ATGTAAGACT Seal TCGAGAGCAC Whale TCGAAAGCAT 4 5 0
8 bacteria LIFE Trees archaebacteria EUKARYOTES protoctists PLANTS green algae fungi ANIMALS Tree: Connected network containing no cycles. mosses sponges ferns cnidarians flowering! seed plants non-flowering! seed plants flatworms VERTEBRATES lophophorates echinoderms rotifers roundworms ARTHROPODS TETRAPODS AMNIOTES cartilaginous! fish bony fish amphibians segmented! worms mollusks crustaceans insects chelicerates mammals turtles snakes! crocodiles! & lizards & birds
9 bacteria LIFE Trees archaebacteria EUKARYOTES protoctists PLANTS green algae fungi ANIMALS Tree: Connected network containing no cycles. mosses sponges flowering! seed plants non-flowering! seed plants ferns cnidarians flatworms Leaves (degree = ): present-day species lophophorates rotifers roundworms VERTEBRATES echinoderms ARTHROPODS TETRAPODS AMNIOTES cartilaginous! fish bony fish amphibians segmented! worms mollusks crustaceans insects chelicerates mammals turtles snakes! crocodiles! & lizards & birds
10 bacteria LIFE Trees archaebacteria EUKARYOTES protoctists PLANTS green algae fungi ANIMALS Tree: Connected network containing no cycles. mosses sponges flowering! seed plants non-flowering! seed plants ferns cnidarians flatworms Leaves (degree = ): present-day species lophophorates rotifers roundworms TETRAPODS AMNIOTES VERTEBRATES cartilaginous! fish bony fish amphibians echinoderms segmented! worms mollusks crustaceans insects ARTHROPODS chelicerates Internal nodes (degree ): ancestral species mammals turtles snakes! crocodiles! & lizards & birds
11 bacteria LIFE Trees archaebacteria EUKARYOTES protoctists PLANTS green algae fungi ANIMALS Tree: Connected network containing no cycles. mosses sponges flowering! seed plants non-flowering! seed plants ferns cnidarians flatworms Leaves (degree = ): present-day species lophophorates rotifers roundworms TETRAPODS AMNIOTES VERTEBRATES cartilaginous! fish bony fish amphibians echinoderms segmented! worms mollusks crustaceans insects ARTHROPODS chelicerates Internal nodes (degree ): ancestral species turtles snakes! crocodiles! & lizards & birds mammals Exercise: Design a Tree struct with a few methods.
12 Trees Most Recent Ancestor! TIME! Present Day! Rooted tree: one node is designated as the root (most recent common ancestor)
13 Trees mathoverflow.net! Unrooted tree: no node is designated as the root (we haven t inferred the location of most recent ancestor).
14 Distance-Based Phylogeny Distance-Based Phylogeny Problem: Construct an evolutionary tree from a distance matrix. Input: A distance matrix. Output: The unrooted tree corresponding to this distance matrix. Think: Is this problem clearly stated?
15 Distance-Based Phylogeny Distance-Based Phylogeny Problem: Construct an evolutionary tree from a distance matrix. Input: A distance matrix. Output: The unrooted tree corresponding to this distance matrix. Think: Is this problem clearly stated? We haven t stated what corresponding to means, so this isn t a computational problem!
16 Fitting a Tree to a Matrix Chimp Human Seal Whale Chimp Human Seal Whale Exercise: Find a tree fitting this distance matrix.
17 Fitting a Tree to a Matrix Chimp Human Seal Whale Chimp Human Seal Whale Chimp! 3 Seal! Human! 0 Whale!
18 Fitting a Tree to a Matrix Chimp Human Seal Whale Chimp Human Seal Whale Chimp! 3 Seal! Human! 0 Whale!
19 Fitting a Tree to a Matrix Chimp Human Seal Whale Chimp Human Seal Whale Chimp! 3 Seal! Human! 0 Whale!
20 Fitting a Tree to a Matrix Chimp Human Seal Whale Chimp Human Seal Whale Chimp! 3 Seal! Human! 0 Whale!
21 Fitting a Tree to a Matrix Chimp Human Seal Whale Chimp Human Seal Whale Chimp! 3 Seal! Human! 0 Whale!
22 Fitting a Tree to a Matrix Chimp Human Seal Whale Chimp Human Seal Whale Chimp! 3 Seal! Human! 0 Whale!
23 Fitting a Tree to a Matrix Chimp Human Seal Whale Chimp Human Seal Whale Chimp! 3 Seal! Human! 0 Whale!
24 Return to Distance-Based Phylogeny Distance-Based Phylogeny Problem: Construct an evolutionary tree from a distance matrix. Input: A distance matrix. Output: The unrooted tree fitting this distance matrix. Think: Is this problem clearly formulated?
25 Return to Distance-Based Phylogeny Exercise: Design an algorithm (pseudocode) that will construct a tree fitting any input distance matrix (mammal matrix reproduced below for reference). Chimp Human Seal Whale Chimp Human Seal Whale 4 5 0
26 Return to Distance-Based Phylogeny Exercise: What tree does your method construct for the following matrix? i j k l i 0 3 j k 0 3 l 3 3 0
27 Return to Distance-Based Phylogeny Exercise: Find the tree fitting this distance matrix. i j k l i j k l 3 5 0
28 Sometimes, No Tree Fits Exercise: Find the tree fitting this distance matrix. i j k l i j k l NO Solution! Additive matrix: distance matrix such that there exists an unrooted tree fitting it.
29 Sometimes, More Than One Tree Fits Chimp Human Seal Whale Chimp Human Seal Whale Chimp! 3 Seal! Human! 0 Whale!
30 Sometimes, More Than One Tree Fits Chimp Human Seal Whale Chimp Human Seal Whale Seal! Chimp! Whale! Human!
31 Which Tree is Better? Chimp! 3 Seal! Human! 0 Whale! 0.5 Seal! Chimp! Whale! Human!
32 Which Tree is Better? Chimp! 3 Seal! Human! 0 Whale! 0.5 Seal! Chimp! 3.5 # incoming edges =! # incoming edges =! 0 Whale! Human!
33 Which Tree is Better? Chimp! 3 Seal! Human! 0 Whale! Degree: number of edges touching a node.
34 Which Tree is Better? Chimp! 3 Seal! Human! 0 Whale! Degree: number of edges touching a node. Simple tree: tree with no nodes of degree.
35 Which Tree is Better? Chimp! 3 Seal! Human! 0 Whale! Degree: number of edges touching a node. Simple tree: tree with no nodes of degree. Theorem: There is a unique simple tree fitting an additive matrix.
36 Reformulating Distance-Based Phylogeny Distance-Based Phylogeny Problem: Construct an evolutionary tree from a distance matrix. Input: A distance matrix. Output: The simple tree fitting this distance matrix (if this matrix is additive).
37 Reformulating Distance-Based Phylogeny Distance-Based Phylogeny Problem: Construct an evolutionary tree from a distance matrix. Input: A distance matrix. Output: The simple tree fitting this distance matrix (if this matrix is additive). But what should we do about non-additive matrices?
38 Reformulating Distance-Based Phylogeny Distance-Based Phylogeny Problem: Construct an evolutionary tree from a distance matrix. Input: A distance matrix. Output: The simple tree fitting this distance matrix (if this matrix is additive). But what should we do about non-additive matrices? Heuristic: A quick, practical method that does not necessarily solve a given problem.
39 Modeling Speciations Researchers often assume that all internal nodes correspond to speciations, where one species splits into two.
40 Modeling Speciations Squirrel Monkey! Orangutan! Unrooted binary tree: every node has degree or 3. Chimpanzee! Bonobo! Baboon! Human! Gorilla!
41 Modeling Speciations Rooted binary tree: an unrooted binary tree with a root (of degree ) on one of its edges. Squirrel Monkey! Baboon! Orangutan! Gorilla! Chimpanzee! Bonobo! Human!
42 Ultrametric Trees 33 3 Molecular clock: assigns ages to each node in the tree (age of leaves = 0) Squirrel Monkey! Baboon! Orangutan! Gorilla! Chimpanzee! Bonobo! Human!
43 Ultrametric Trees edge weights: correspond to difference in ages on the nodes the edge connects Squirrel Monkey! Baboon! Orangutan! Gorilla! Chimpanzee! Bonobo! Human!
44 Ultrametric Trees Ultrametric tree: distance from root to any leaf is the same (i.e., age of root) Squirrel Monkey! Baboon! Orangutan! Gorilla! Chimpanzee! Bonobo! Human!
45 UPGMA: A Clustering Heuristic. Form a cluster for each present-day species, each containing a single leaf. i j k l i j k l i 0 j 0 k 0 l 0
46 UPGMA: A Clustering Heuristic. Find the two closest clusters C and C according to the average distance D avg (C, C ) = Σ i in C, j in C D i,j / C C where C denotes the number of elements in C. i j k l i j k l i 0 j 0 k 0 l 0
47 UPGMA: A Clustering Heuristic 3. Merge C and C into a single cluster C. i j k l i j k l { k, l } i 0 j 0 k 0 l 0
48 UPGMA: A Clustering Heuristic 4. Form a new node for C and connect to C and C by an edge. Set age of C as D avg (C, C )/. i j k l i j k l { k, l } i 0 j 0 k 0 l 0
49 UPGMA: A Clustering Heuristic 5. Update the distance matrix by computing the average distance between each pair of clusters. i j { k, l } i j { k, l } { k, l } i 0 j 0 k 0 l 0
50 UPGMA: A Clustering Heuristic 6. Iterate until a single cluster contains all species. i j { k, l } i j { k, l } { i, j } i 0 j 0 k 0 l 0
51 UPGMA: A Clustering Heuristic 6. Iterate until a single cluster contains all species. {i, j} { k, l } {i, j} 0 4 { k, l } { i, j }.5.5 i 0 j 0 k 0 l 0
52 UPGMA: A Clustering Heuristic 6. Iterate until a single cluster contains all species. {i, j} { k, l } {i, j} 0 4 { k, l } i 0 j 0 k 0 l 0
53 UPGMA: A Clustering Heuristic 6. Iterate until a single cluster contains all species i 0 j 0 k 0 l 0
54 UPGMA: A Clustering Heuristic UPGMA(D):. Form a cluster for each present-day species, each containing a single leaf.. Find the two closest clusters C and C according to the average distance D avg (C, C ) = Σ i in C, j in C D i,j / C C where C denotes the number of elements in C 3. Merge C and C into a single cluster C. 4. Form a new node for C and connect to C and C by an edge. Set age of C as D avg (C, C )/. 5. Update the distance matrix by computing the average distance between each pair of clusters. 6. Iterate steps -5 until a single cluster contains all species.
55 UPGMA Doesn t Fit a Tree to a Matrix i j k l i j k l i 0 j 0 k 0 l 0
56 UPGMA Doesn t Fit a Tree to a Matrix i j k l i j k l i 0 j 0 k 0 l 0
57 Quick Quiz Exercise: Apply UPGMA to the following matrix. i j k l i j k l 8 0
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