Phylogenetic Trees - Parsimony Tutorial #11

Size: px

Start display at page:

Download "Phylogenetic Trees - Parsimony Tutorial #11"

Audra Cook
5 years ago
Views:

1 Phylogenetic rees Parsimony utorial #11 Ilan ronau.. Based on original slides of Ydo Wexler & Dan eiger Phylogenetic econstruction he input: a speciescharacters matrix he ouput: a tree with n leaves corresponding to the input species characters n species Each character represents some observable trait. Each character taes values from a finite set. 2 1

Most Parsimonious ree Parsimonyscore: Number of characterchanges (mutation along the evolutionary tree (tree containing labels on internal vertice Example: Score = 4 Score = 3 0 AAA 0 0 AAA 1 1 AAA 1

2 Most Parsimonious ree Parsimonyscore: Number of characterchanges (mutation along the evolutionary tree (tree containing labels on internal vertice Example: Score = 4 Score = 3 0 AAA 0 0 AAA 1 1 AAA 1 0 AAA 2 AA AA AAA A Most parsimonious tree: ree with minimal parsimony score 1 AAA 0 0 AA 1 AA AAA Minimal Evolution Principle AA A 3 Small vs. Large Parsimony We brea the problem into two: 1. Small parsimony: iven the topology find the best assignment to internal nodes 2. Large parsimony: Find the topology which gives best score Large parsimony is NPhard We ll show solution to small parsimony (Fitch and Sanoff s algorithm Input to small parsimony: tree with characterstate assignments to leaves Example: Aardvar Bison Chimp Dog Elephant A: CAA B: CAACA C: CA D: CAC E: CA 4 2

3 Fitch s Algorithm Execute independently for each character: 1. Bottomup phase: Determine set of possible states for each internal node 2. opdown phase: Pic states for each internal node Dynamic Programming framewor 1 2 Aardvar Bison Chimp Dog CAA CA CAACA CAC Elephant CA 5 Fitch s Algorithm Bottomup phase Determine set of possible states for each internal node Initialization: i = {s i } for all leaves Do a postorder (from leaves to root) traversal of tree Determine i of internal node i with children, : i C if C A A φ Parsimonyscore = # union operations score = 3 6 3

4 Fitch s Algorithm opdown phase Pic states for each internal node Pic arbitrary state in root for the root Do preorder (from root to leave traversal of tree Determine s of internal node with parent i: si if si s arbitrary state C C A A Complexity: O(mn) #states score = 3 #characters #taxa/nodes 7 Weighted Parsimony Sanoff s algorithm Each mutation a b costs differently S(a,b). 1. Bottomup phase: Determine i ( cost of optimal stateassignment for subtree of i, when it is assigned state s. 2. opdown phase: Pic optimal states for each internal node Same as algorithm for optimal lifted tree alignment (utorial #4) 8 4

5 Sanoff s Algorithm Bottomup phase Determine i ( for each internal node 0 if si = s Initialization: i ( Do a postorder (from leaves to root) traversal of tree Determine i of internal node i with children, : { ( s') + S( } + min { ( s') S( } ( = min s' s' i + Natural generalization For nonbinary trees C A emember pointers s s 9 Sanoff s Algorithm opdown phase Pic states for each internal node Select minimal cost character for root (s minimizing root () Do preorder (from root to leave traversal of tree: For internal node, with parent i, select state that produced minimal cost at i (use pointers ept in 1 st stage) min i ( = min s { ( s') + S( } ' { ( s') + S( } + s' Complexity: O(m 2 n) C A #states #characters #taxa/nodes 10 5

6 Fitch s Algorithm as special case of Sanoff s algorithm Unweighted parsimony: 0 if a = b S ( a, b) 1 Sanoff s algorithm: i ( cost of optimal subtree of i, when it is assigned state s Fitch s algorithm: Score(i) cost of optimal stateassignment for subtree of i i set of optimal stateassignment for subtree of i We need to show that: tricy part 1. here is an optimal tree which assigns node i with state from i. 2. Fitch s bottomup recursive formula for i. is correct: i if φ Chec for yourselves 11 Fitch s Algorithm as special case of Sanoff s algorithm 0 if a = b Unweighted parsimony: S ( a, b) 1 Score(i) cost of optimal stateassignment for subtree of i i set of optimal stateassignment for subtree of i We need to show that: 1. here is an optimal tree which assigns node i with state from i. rivially true for the root Assume (to the contrary) that in an optimal assignment, some node is assigned s Why is this not the case for the weighted version? i root Parsimonyscore is integer s (s ) Score()+1 By switching from s to some s we do not raise the parsimonyscore 12 6

7 Exploring the Space of rees We saw how to find optimal stateassignment for a given tree topology We need to explore space of topologies iven n sequences there are (2n3)!! possible rooted trees and (2n5)!! possible unrooted trees 2 n!! = n 2 n ( n 2)! taxa (n) # rooted trees # unrooted trees ,135 10, ,459,425 2,027, Exploring the Space of rees Possible solutions: 1. Heuristic solutions for traveling through topologyspace 2. Find (basic) topology using distancebased methods (NJ) Notice another problem: We obtain stateassignments to taxa using multiple alignment We obtain optimal MA using topology of phylogenetic tree (e.g. CLUSAL) Solution: Again, use some initial topology (via NJ) A A A C C C C 1,C 2,, C m 14 7

Parsimony-Based Approaches to Inferring Phylogenetic Trees

Parsimony-Based Approaches to Inferring Phylogenetic Trees BMI/CS 576 www.biostat.wisc.edu/bmi576.html Mark Craven craven@biostat.wisc.edu Fall 0 Phylogenetic tree approaches! three general types! distance: