Lecture 6A – Introduction to Trees & Optimality Criteria

Branches: n-taxa -> 2n-3 branches 1, 2, 4, 6, & 7 are external (leaves)3 & 5 are internal branches (edges)

Nodes A – E are terminals x, y, & z are internal (vertices)

If we break branch 3, we have two sub-trees (A,B) and (C,(D,E)).

((A,B),C,(D,E)).

Newick Format

Rooting – The tree is an unrooted tree.

Also note that there is free rotation around nodes:

The Scope of the Problem

Taxa Unrooted Trees3 14 35 156 1057 9458 10,3959 135,13510 2.027 X 106

22 3 X 1023

50 3 X 1074

100 2 X 1082

1000 2 X 102,860

10 mil 5 X 1068,667,340

II. Optimality Criteria

A. Parsimony

First, the score of a tree (i.e., its length) for the entire data set is given by:

li is the length of character i when optimized on tree t.

wi is the weight we assign to character i.

The Fitch Algorithm: state sets and accumulated lengths.

We erect a state set at each terminal node and assign an accumulated length of zero to terminal nodes. This is the minimum number of changes in the daughter subtree.

The Fitch Algorithm: state sets and accumulated lengths.

1 – Form the intersection of the state sets of the two daughter nodes. If the intersection is non-empty, assign the set for the internal node equal to the intersection. The accumulated length of the internal node is the sum of those of the daughter nodes.

2 – If the intersection is empty, we assign the union of the two daughter nodes to the state set for the internal node. The accumulated length is the sum of those of the daughter nodes plus one.

empty

Union:0+0+1=1

non-empty

Intersection:0+0=0

empty

Union:1+0+1=2 So li = 2

Sankoff Algorithm – Character-state vectors and step-matrices

Step Matrix – define ci,j

A CG TA -- 41 4C 4 --4 1G 1 4-- 4T 4 14 --

Step one: Fill in the character-state vectors for terminal nodes.

Each cell is the sk(i)

Step two: Fill in vectors for other nodes, descending tree.

s1(A) = cAG + cAA = 1 + 0 = 1,

s1(C) = cCG + cCA = 4 + 4 = 8,

s1(G) = cGG + cGA = 0 + 1 = 1,

s1(T) = cTG + cTA = 4 + 4 = 8

Node 1: Node 2:

s2(A) = 4 + 4 = 8

s2(C) = 0 + 0 = 0

s2(G) = 4 + 4 = 8

s2(T) = 1 + 1 = 2

For nodes below, we must calculate the cost for each possible state assignment for daughter nodes.

s3(A) = min[s1A + cAj] + min[s2A + cAj]

s3(C) = min[s1C + cCj] + min[s2C + cCj]

s3(G) = min[s1G + cGj] + min[s2G + cGj]

s3(T) = min[s1T + cTj] + min[s2T + cTj]

So we fill in the character-state vector for node 3.

From daughter node 1From step matrix

= min[1,12,2,12] + min[8,4,9,6] = 1+4 = 5

5

= min [5,8,5,9] + min[12,0,12,3] = 5+0 = 5

5

= min [2,12,1,12] + min[9,4,8,6] = 1+4 = 5

5

= min [5,9,5,8] + min[12,1,12,2] = 5+1 = 6

6

Points to note:1) Two types of weighting are possible: weighting of transformations within characters

(which we demonstrated with the step matrix) and weighting among characters, which are reflected in the weighted sum of lengths across characters.

2) One can’t compare tree lengths across weighting schemes. In the first example, with all transformations having the same cost, the length of the character on this tree was 2.

In the second, with a 4:1 step matrix to weight transversions, the length was 5.