1. . 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 17.1-17.3, Setubal&Meidanis 6.1

2. 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. 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. 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. 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. 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. 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. 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. 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. 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. 11 Fitch’s Algorithm – Step 1 # of changes = # union operations T T CT T C T A G T AGT GT

12. 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. 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. 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. 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. 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. 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. 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. 19 Exploring the Space of Trees taxa (n) # trees 4 15 5 105 6 945 8 135,135 10 30,405,375

20. 20 Does This Implies Big MP is Hard? taxa (n) # trees 4 15 5 105 6 945 8 135,135 10 30,405,375 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. 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.