Recap: Properties of Trees. Rooting an unrooted tree. Questions trees can address: Data for phylogeny reconstruction. Rooted vs unrooted trees:

Transcription

1 Pairwise sequence alignment (global and local) Recap: Properties of rees Multiple sequence alignment global local ubstitution matrices atabase ing L equence statistics Leaf nodes contemporary taxa Internal nodes - ancestral taxa opology relationships between species ranch lengths degree of change Evolutionary tree reconstruction Gene Finding Protein structure prediction RN structure prediction omputational genomics If the mutation rate is constant in all lineages (molecular clock hypothesis), the branch lengths are proportional to time. Rooted vs unrooted trees: Rooting an unrooted tree root: common ancestor uman Mouse hicken almon arp Zebrafish rout almon Unrooted trees give no information about the order of speciation events Molecular clock If the mutation rate is constant in all lineages (molecular clock hypothesis), the root can be inferred mathematically Outgroup dd a taxon whose distance from all taxa in the data set is greater than distances within the data set (but not too distant). e.g., given a set of dog, fox and wolf data, use a bear as an outgroup. ata for phylogeny reconstruction Questions trees can address: Morphology ehavior iochemistry Molecular and sequence data Which taxa are most closely related? What is the ancestral state? Where is rapid change occuring When did lineages diverge? haracter data: shared characteristics istance data: difference between species 1

2 Evolutionary ree Reconstruction Maximum Parsimony Given observations of similarities or differences between k species, find the hypothesis (tree) that best explains the data with respect to some criterion: Maximum parsimony (character data) Minimum evolution (distance data) Maximum Likelihood (character data) Parsimony score: minimum number of mutations needed to explain data ssumptions election dominates -> Few changes No multiple substitutions -> ites are independent haracter vs distance data Multiple equence lignment as haracter ata nts entipds Each column (or site) is one character. ees nts entipds ees nts n11 n12 n22 n1 n2 n ~~~~LEKQELLKQWEVLKQNIPRLFLIIE ~~~MLEKQELLKQWEVLKQNIPRLFLILE ~~~MLERQELLKQWEVLKQNIPGRLFLIIE ~~~~~~~~~~ELLKQWEVLKQNIPGLFLIIE ees R R nts G R entipds G ees nts entipedes uman Rabbit Pig hicken Multiple equence lignment as istance ata ~~~~LEKQELLKQWEVLKQNIPLRLFLIIE ~~~MLEKQELLKQWEVLKQNIPLRLFLILE ~~~MLERQELLKQWEVLKQNIPGLRLFLIIE ~~~~~~~~~~ELLKQWEVLKQNIPGLLFLIIE Finding the optimal tree Given k taxa, onsider all trees with k leaves core each tree with respect to chosen evolutionary model. elect highest scoring tree(s) uman Rabbit Pig hicken uman Rabbit Pig 0 6 hicken 0 Phylogeny reconstruction is NP-complete: Except in special cases when the data obeys specific constraints, the only way to find the best tree is to consider all trees. 2

3 oday ow many unrooted trees with k leaves? ow many evolutionary trees are there? Given a tree and a multiple alignment, how to determine the parsimony score. hree taxa Four axa k E(k) (k) 1 4 Five taxa 7 1 Number of unrooted trees for k taxa he number of trees gets big fast E( k) = E( k 1) + 2 = 2k ( k) = E( k 1) ( k 1) = (2k )! ( k) = k 2 ( k )! k 1 i= (2i ) Number of leaves Number of unrooted binary trees ,027, x x x ow big is that? Number of leaves Number of unrooted binary trees x x x Exhaustive (<12 taxa) ge of the universe (seconds): 4.42 x iameter of the universe: 2.70 x Number of stars in the universe: (Phylogeny reconstruction is NP-complete.)

4 2. ranch-and-bound (<18 taxa) Method Exhaustive Result ime (k) ypical k 12 Parsimony score is nondecreasing as you add edges ow do you find a pretty good tree? Method Exhaustive ranch and bound Result ime (k) (k) ypical k euristic earch for optimal trees by finding good trees and then rearranging them in the hopes of finding an even better tree euristic Global optimum uboptimal island of trees ranch swapping Nearest-neighbour interchange (NNI) tarting trees reespace 4

5 ranch swapping ranch swapping ubtree pruning and regrafting (PR) ree-bisection reconnection (R) oday Method Exhaustive ranch and bound Result ime (k) (k) ypical k ow many evolutionary trees are there? Given a tree and a multiple alignment, how to determine the parsimony score. euristic uboptimal You choose You choose Finding the most parsimonious tree Inferring ancestral sequences and computing the parsimony score Given k taxa and n characters (e.g., columns in an M), For each topology, t, with k leaves score(t) = 0 For each of the n characters Find the optimal labeling of internal nodes score(t) = score(t) + count_mutations (1) (2) () (4) _ Parsimony score: 4 _

6 etermining the parsimony score of a tree etermining the parsimony score of a tree Input: M: k taxa, n columns, aka characters or sites. ree:. n assignment of the sequences in the M to the leaves of. Output: core: he minimum number of mutations, over all possible ancestral sequences, required to explain the data he ancestral sequences that minimize the score (sometimes.) Fitch s algorithm Input: tree, leaf labels Output: minimum number of mutations required to explain leaf labels oes not determine the ancestral sequences! urbin et al., p 17. Fitch s algorithm Root tree arbitrarily; Global = 0. ORE (i) If i is a leaf, return {label(i)} Else R(l) = ORE (left(i)) R(r) = ORE (right(i)) If R(r) R(l) = Ø R(i) = R(r) U R(l) // No label avoids mutation = +1 // Pass all labels up tree Else R(i) = R(r) R(l) // hoose label that avoids // mutation Final score = Finding the most parsimonious tree Given k taxa and n characters (e.g., columns in an M), For each topology, t, with k leaves score(t) = 0 For each of the n characters Find the optimal labeling of internal nodes score(t) = score(t) + count_mutations Not all columns are informative! Informative sites: olumns that distinguish alternate trees Informative sites 4 G G G G G G 1 2 I 6