Building Phylogenies

Distance-Based Methods

Methods

• Distance-based• Parsimony• Maximum likelihood

Distance Matrices

a 0

b 6 0

c 7 3 0

d 14 10 9 0

a b c d

a

b

c

d

1 2 3 4 50 6 7 8

Distance matrix is additive if there is a tree that fits it exactly

Ultrametric Matrices

a 0

b 2 0

c 6 6 0

d 10 10 10 0

a b c d

a

b

c

d

1 2 3 4 50

Additive + molecular clock assumption

Methods

• Fitch - Margoliash• UPGMA• Neighbor-joining• Many others

Least squares trees

• Minimize

over all trees

• Choice of weights wij :

– Uniform: wij 1

– Fitch-Margoliash: wij 1/Dij2

– Others . . .

ji

ijijij dDwQ 2

Sarich's (1969) immunological distances

Least squares tree for Sarich’s data

Clustering Methods

• E.g., UPGMA and Neighbor-Joining• A cluster is a set of taxa• Interspecies distances translate into

intercluster distances• Clusters are repeatedly merged

– “Closest” clusters merged first– Distances are recomputed after

merging

UPGMA• Unweighted pair group method using arithmetic

averages

• The distance between clusters Ci and Cj is

• After merging Ci and Cj to create cluster Ck define distance from k to every other cluster r as

ji CqCppq

ji

ij DCC

D,

1

ji

jjriir

krCC

CDCDD

UPGMA: Initialization

1.Assign each sequence i to its own cluster Ci

2.Define one leaf (tip) of tree for each sequence and place it at height 0

UPGMA: Iteration

1.Choose the two clusters i and j with smallest Dij

2.Create a new cluster k, where Ck = Ci Cj

3.Compute Dkr for all r.4.Define a new node k with children i and j,

and place it at height Dij /2.5.Add k to the current clusters and delete i

and j Let i and j be the remaining clusters.

Place root at height Dij /2

Repeat until only two clusters remain:

UPGMA Example

UPGMA tree for Sarich’s data

A pitfall of UPGMA

• The algorithm produces an ultrametric tree: the distance from the root to any leaf is the same

• UPGMA assumes a constant molecular clock: all species accumulate mutations (evolve) at the same rate.

UPGMA fails when molecular clock assumption doesn’t

hold

Neighbor Joining

• Saitou and Nei, Molecular Biology and Evolution 4 (1987)

• Idea: Find a pair of leaves that are close to each other but far from other leaves– Implicitly finds a pair of neighboring leaves

• Advantages: – Works well for additive and other nonadditive

matrices– Does not have the molecular clock assumption

Long branches must be handled carefully!

0.1

0.1

0.1

0.4 0.4

and are closer to each other than to or . Obvious approach produces incorrect clusters!

Compensating for long edges

Introduce “correction terms”

ji

iji Dn

u2

1

jiijij uuDD

“Corrected” distances:

Distances are reduced for pairs that are far away from all other species: They may be close to each other.

Average dist. to other taxa

Neighbor-joining

1. Choose i, j such that Dij ui uj is minimum2. Define a new leaf k whose distances to i and j are

3. Compute the distance from k to every other leaf r

4. Delete i and j

ijijjk

jiijik

uuDd

uuDd

21

21

21

21

ijjrirkr DDDD 21

Repeat the following until only two leaves remain:

Connect the 2 remaining leaves by a branch of length Dij

NJ tree for Sarich’s data

Computing distance matrices

• Based on sequence alignment• Various possibilities:

– Distance = average number of differences– Try different PAM matrices; distance =

index of matrix that gives highest score– Feng and Doolitle: Based on alignment

scores – roughly ratio to max possible score (see text)

• Read, e.g., PHYLIP documentation:http://evolution.genetics.washington.edu/phylip/general.html

Distance correction

• The amount of evolutionary change is not linearly related to time

• Over a long period of time, a series of substitutions may bring us back to where we started

• Percentage difference may underestimate evolutionary time

Jukes-Cantor Model

Correcting for multiple substitutions in the JC model

dt

34

1ln43

Many other models!