. Comput. Genomics, Lecture 5b Character Based Methods for Reconstructing Phylogenetic Trees: Maximum Parsimony Based on presentations by Dan Geiger, Shlomo Moran, and Ido Wexler. Modified by Benny Chor. References: Durbin et al 7.4, Gusfield , Setubal&Meidanis 6.1

2 Phylogenetic Trees - Reminder Leaves represent objects (genes, species) being compared Internal nodes are hypothetical ancestral objects In a rooted tree, path from root to a node corresponds to a path in evolutionary time An unrooted tree specifies relationships among objects, but not evolutionary time

3 Parsimony Based Approch Input: Character data (aligned sequences) Goal/Output: A labeled tree (labeled internal nodes) that “explains” the data with a minimal number of changes across edges

4 Parsimony: An Example Various trees that could explain the phylogeny of the following four sequences: AAG, AAA, GGA, AGA. For example, AAA AGAAGA AAG GGA AAA AGAAGA AGA AAA AAG GGA Parsimony prefers the second tree to the first, because it requires less substitution events (three vs. four changes).

5 Big and Small Parsimony Usually the approaches to finding a maximum parsimony tree have two separate components:  A search through the space of trees (BIG parsimony)  Given a specific tree topology, find an assignment of “ancestral labels” to internal nodes as to the minimize the total number of changes across tree edges (small parsimony)

6 Formally: Big Parsimony Input: Character data (aligned sequences) Goal/Output: A labeled tree (labeled internal nodes) that minimizes number of changes across edges (over all trees and internal labelings).

7 Formally: Small Parsimony Input: Character data (aligned sequences) and a tree with sequences at leaves. Goal/Output: A labeling of internal nodes that minimizes number of changes across edges (over all internal labelings).

8 Big, Small, and Weighted Parsimony  Small parsimony has a linear time solution (Fitch’ algorithm). BIG parsimony is NP hard: An easy reduction from vertex cover, that will be shown soon (on the board).  Weighted small parsimony also has a linear time solution (Sankoff’s algorithm, dynamic programming).

9 Small Parsimony: Fitch’s Algorithm  Traverse tree “up”, from leaves to root, finding sets of possible ancestral states (labels) for each internal node.  Traverse tree “down”, from root to leaves, determining ancestral states (labels) for internal nodes.  Key observation: Different sites are independent. Can solve one site at a time.

10 Fitch’s Algorithm – Step 1 Do a post-order (from leaves to root) traversal of tree Find out possible states R i of internal node i with children j and k

11 Fitch’s Algorithm – Step 1 # of changes = # union operations T T CT T C T A G T AGT GT

12 Fitch’s Algorithm – Step 2 Do a pre-order (from root to leaves) traversal of tree Select state r j of internal node j with parent i

13 Fitch’s Algorithm – Step 2 T T CT T C T A G T AGT GT T T CT T C T A G T AGT GT T T CT T C T A G T AGT GT T T CT T C T A G T AGT GT T T CTCT T C T A G T AGT GT T T CTCT T C T A G T AGT GTGT

14 Weighted Version Instead of assuming all state changes are unit cost (  equally likely), use different costs S(a,b) for different changes 1 st step of algorithm is to propagate costs up through tree

15 Weighted Version of Fitch’s Algorithm Want to determine min. cost R i (a) of assigning character a to node i for leaves:

16 Weighted Version of Fitch’s Algorithm want to determine min. cost R i (a) of assigning character a to node i for internal nodes: a b i j k

17 Weighted Version of Fitch’s Algorithm – Step 2 do a pre-order (from root to leaves) traversal of tree select minimal cost character for root For each internal node j, select character that produced minimal cost at parent i

18 Big Parsimony: Exploring the Space of Trees We’ve considered small parsimony: How to find the minimum number of changes for a given tree topology To solve big parsimony, need some search procedure for exploring the space of tree topologies There are unrooted trees on n leaves

19 Exploring the Space of Trees taxa (n) # trees , ,405,375

20 Does This Implies Big MP is Hard? taxa (n) # trees , ,405,375 Not necessarily: There could be some smarter way to zoom directly to best topology. But: We will show hardness of Big MP by a (simple) reduction from vertex cover (VC).

21 Big MP is NP Hard ! First, define VC and VC for triangle free graphs. Then… 1. You will show a poly time reduction from VC to VC for triangle free graphs as part of home assignment (easy). 2. In class, I will show a poly time reduction from VC for triangle free graphs to Big MP (old style, white board proof). This establishes NP hardness of Big MP.