From Ernst Haeckel, 1891

The Tree of Life

Classical approach considers morphological features number of legs, lengths of legs, etc.

Modern approach considers molecular features gene sequences protein sequences

Use of molecular data provides objective criteria for constructing phylogenetic trees

Phylogenetic Analysis

Phylogenetic analysis is based on homologous sequences in different species (e.g., globins)

Sequences can be homologous for different reasons: orthologs -- sequences diverged after a speciation event

paralogs -- sequences diverged after a duplication event

xenologs -- sequences diverged after horizontal transfer (e.g., by virus)

Phylogenetic Analysis

A tree is a collection of nodes and edges with no cycles (i.e. there is no path from a node to itself)

Tree topology refers to the “shape” of the tree

Tree Terminology

tree not a tree

topologically equivalent

A tree is a collection of nodes and edges with no cycles (i.e. there is no path from a node to itself)

Classification of nodes (in the context of phylogenetic trees) root – (a single distinguished node) represents the common

ancestor internal nodes – represent intermediate ancestors in the course of

evolution leaves – (the non-branching nodes) represent the species for which

the tree is built

Tree Terminology

tree not a tree

Rooted Trees internal nodes have 3 edges (1 for parent, 2 for children) a special node (the root) has 2 edges the leaves (the given taxa) have one edge

Unrooted trees – same as above but do not have root node

Tree Terminology

Classification of nodes (in the context of phylogenetic trees) root – (a single distinguished node) represents the common ancestor internal nodes – represent ancestors in the course of evolution leaves – (the non-branching nodes) represent the species for which

the tree is built

When the root node is not specified the tree is unrooted

Tree Terminology

Three Leaf Nodes

Only one unrooted tree is possible

Four Leaf Nodes

AA

D

C

B

D

BC

Three different unrooted trees are possible

A

B

C

D

A

B

C

Counting Trees How many trees are there that have n leaf nodes (or taxa)?

How many trees are there that have n leaf nodes (or taxa)?

NR = Number of possible rooted trees

=

NU = Number of possible unrooted trees

=

)32(7531)!2(2

)!32(2

nn

nn

)52(7531)!3(2

)!52(3

nn

nn

Counting Trees

n Unrooted Rooted3 1 3

4 3 15

5 15 105

6 105 945

7 945 10395

8 10395 135135

9 135135 2027025

10 2027025 34459425

11 34459425 654729075

12 65729075 1.375*10-10

Tree Explosion

The number of possible rooted trees for 15 different taxa is

213,458,046,767,875

Assuming a computer can create a tree in 10-9 seconds, it would take 2.47 days of computation time to create them.

For 20 taxa, there are 8,200,794,532,637,891,559,337 possible trees and the same computer would take 259,867 years to generate this many trees!

Tree Explosion

Distance-based UPGMA – Unweighted Pair-Group Mathod with Arithmetic Means Fitch-Margoliash (FM) Neighbor-Joining

Character-based Maximum parsimony algorithm

Algorithms

Distance-based algorithms expect as input a matrix of distances (dij) between each pair of sequences

Distance data can be generated from the available sequences and models of base substitution

Jukes-Cantor model

p – fraction of mismatches

Kimura model

P – fraction of transitions

Q – fraction transversions

Distance Data

)3

41ln(

4

3pdij

)21

1ln(4

1)

21

1ln(2

1

QQPdij

UPGMA Algorithm

Main idea: Group the taxa into clusters and repeatedly merge the closest two clusters until one cluster remains

Algorithm Add a leaf to the tree for each taxon Initially make each taxon be its own cluster Find the closest clusters and connect with node in the tree

(place new node at equal distance from the clusters) Repeat previous step until all clusters are connected

UPGMA Algorithm

x4

x2

x3

x5

x1

x3 x5x1 x2 x4

root

The algorithm needs to compute distance between clusters

The distance between clusters Ci and Cj is defined to be the average distance between all pairs of taxa in Ci and Cj

UPGMA Clustering

ji CqCpji

ji qpdCC

CCd,

),(||||

1),(

The algorithm needs to compute distance between clusters

The distance between clusters Ci and Cj is defined to be the average distance between all pairs of taxa in Ci and Cj

Shortcut when combining Ci and Cj to form new cluster Ck

UPGMA Clustering

ji CqCpji

ji qpdCC

CCd,

),(||||

1),(

||||

),(||),(||),(

ji

ljjliilk CC

CCdCCCdCCCd

UPGMA Example

Assume the following distance matrix

x1 x2 x3 x4 x5

x1 - 16 6 16 6

x2 16 - 16 8 16

x3 6 16 - 16 2

x4 16 8 16 - 16

x5 6 16 2 16 -Closest Pair is {x3, x5} so cluster them, C1 = {x3,C5}

Compute the distance from C1 to the rest

d(C1,x1) = 1/2 (d(x3,x1) + d(x5,x1) ) = 6

d(C1,x2) = 1/2 (d(x3,x2) + d(x5,x2) ) = 16

d(C1,x4) = 1/2 (d(x3,x4) + d(x5,x4) ) = 16

Add new node for x3, x5 at height d(x3,x5) / 2 = 1

x3 x5

11

UPGMA

x1 x2 x4 C1

x1 - 16 16 6

x2 16 - 8 16

x4 16 8 - 16

C1 6 16 16 -

Closest Pair is {x1, C1} so cluster them, C2 = {x1,C1}

Compute the distances from C2 to the

d(C2,x2) = 1/3 (d(x1,x2) + d(x3,x2) +d(x5,x2) ) = 16

d(C2,x4) = 1/3 (d(x1,x4) + d(x3,x4) +d(x5,x4) ) = 16

Add new node for x1, C1 at height d(x1,C1) / 2 = 3

The updated distance matrix – C1 replaced x3, x5

x1

3 2

x3 x5

11

UPGMA

Closest Pair is {x2, x4} so cluster them, C3 = {x2,x4}

Compute the distances from C3 to the rest

d(C3,C2) = 1/6 (d(x2,x1) + d(x2,x3) +d(x2,x5) +

d(x4,x1) + d(x4,x3) +d(x4,x5)) = 16

Add new node for x2, x4 at height d(x2,x4) / 2 = 4

The updated distance matrix – C2 replaced x1, C1

x2 x4 C2

x2 - 8 16

x4 8 - 16

C2 16 16 -

x3 x5

1

x1

3 2

1

x2 x4

4 4

UPGMA

Closest Pair is {C2, C3} so cluster them, C4 = {C2,C3}

Add new node for C2, C3 at height d(C2,C4) / 2 = 8

The updated distance matrix – C3 replaced x2, x4

C2 C3

C2 - 16

C3 16 -

x3 x5

1

x1

3 2

1

x2 x4

4 4

45

root

UPGMA

Done!

Double-check if original distances between taxa are preserved (not guaranteed)

UPGMA Summary

Distance-based algorithm that produces rooted trees

Assumes that all species evolve at the same rate

(molecular clock hypothesis)

Implication of molecular clock hypothesis is that

distance from root to any taxon is the same

Final tree may not preserve original

distances between the taxa

x3 x5

1

x1

3 2

1

x2 x4

4 4

45

root

Fitch-Margoliash (FM) Algorithm

FM Algorithm

Similar to UPGMA but removes molecular clock assumption

(i.e. distance from an internal node to leaves differs)

Produces unrooted trees

Algorithm (similar to UPGMA) Add a leaf to the tree for each taxon Initially make each taxon be its own cluster Find the closest clusters and connect with node in the tree (place new node at equal distance from the clusters

at distance given by 3-point formula) Repeat previous step until all clusters are connected

Given three taxa i, j, k with distances d(i, j), d(i, k), d(j, k)

where should the interior node m be placed to connect the

taxa and preserve the distances?

i

j

k

m

)),(),(),((2

1),( jidkjdkidkmd

FM and 3-point formula

),(),(),(),(),( jidkmdkjdkmdkid ),(),(),( jidjmdmid

),(),(),( kmdkidmid

),(),(),( kmdkjdjmd

Given three taxa i, j, k with distances d(i, j), d(i, k), d(j, k)

where should the interior node m be placed to connect the

taxa and preserve the distances?

i

j

k

m

FM and 3-point formula

)),(),(),((2

1),( jidkjdkidkmd

)),(),(),((2

1),( jkdijdikdimd

)),(),(),((2

1),( kidjkdjidjmd

Algorithm (similar to UPGMA) Add a leaf to the tree for each taxon Initially make each taxon be its own cluster Find the closest clusters and connect with node in the tree

(place new at distance given by 3-point formula, where the points are clusters of tax and we use the distance between

clusters) Repeat previous step until all clusters are connected

FM Algorithm

x4

x2

x3

x5

x1

x3

x5

x1

x2

x4

Apply the FM algorithm to the following distance matrix:

B C D E

A .31 1.01 .75 1.03

B - 1.00 .69 .90

C - - .61 .42

D - - - .37

A and B are closest; temporarily group C-D-E and compute d(A, B), d(A, C-D-E), d(B, C-D-E) to apply 3-point formula

d(A,C-D-E) = 1/3(1.01+.75+1.03) = .93

d(B,C-D-E) = 1/3(1.00+.69+.90) = .863

d(A, B) = .31 only used to helpus group A, B

By 3-point formula:

d(C-D-E,X) = 1/2(d(C-D-E,A) + d(C-D-E,B) – d(A,B))

d(B, X) = 1/2(d(B,A) + d(B,C-D-E) – d(A,C-D-E))

d(A, X) = 1/2(d(A,B) + d(A,C-D-E) – d(B,C-D-E))

C-D-E.7415

A

.1215

.1885

B

X

A and B are combined in a cluster for the rest of the algorithm, so need to recompute the distances from A-B to other clusters:

d(A-B,C) = 1/2(1.01 + 1.00) = 1.005

d(A-B,D) = 1/2(.75 +.69) = .72

d(A-B, E) = 1/2(1.03 + .90) = .965

The updated table is:

C D E

A-B 1.005 .72 .965

C - .61 .42

D - - .37

The partial tree so far is:

A

.1215

.1885

B

Based on the updated table

C D E

A-B 1.005 .72 .965

C - .61 .42

D - - .37

D and E are closest; temporarily group A-B-C and compute d(D, E), d(D, A-B-C), d(E, A-B-C) to apply 3-point formula

d(D,A-B-C) = 1/3(.75+.69+.61) = .683

d(E,A-B-C) = 1/3(1.03+.90+.42) = .783

d(D, E) = .37

only used to helpus group D, E

.135

.548

.235E

D

A-B-C YBy 3-point formula:

d(A-B-C,Y) = 1/2(d(A-B-C, D) + d(A-B-C,E) – d(D,E))

d(D, Y) = 1/2(d(D,E) + d(D,A-B-C) – d(E,A-B-C))

d(E, Y) = 1/2(d(E,D) + d(E,A-B-C) – d(D,A-B-C))

The partial tree so far is:

D and E are combined in a cluster for the rest of the algorithm, so need to recompute the distances from D-E to other clusters:

d(A-B,D-E) = 1/4 (.75+1.03+.69+90) = .8425

d(A-B,C) = 1/2(1.01 + 1.00) = 1.005

d(C,D-E) = 1/2 (.61+.42) = .515

.135

.235E

DA

.1215

.1885

B

The updated table is now:C D-E

A-B 1.005 .8425

C - .515

Based on the updated table

C D-E

A-B 1.005 .8425

C - .515

There are only three clusters, so just apply the 3-point formula

d(A-B,Z) = 1/2(d(A-B, D-E) + d(A-B,C) – d(D-E,C))

d(D-E,Z) = 1/2(d(D-E,A-B) + d(D-E,-C) – d(A-B,C))

d(C, Y) = 1/2(d(C,A-B) + d(C,D-E) – d(A-B,D-E))

A-B

.33875

.17625

.66625

C

D-E

Z

Now we need to expand the clusters A-B, D-E

We also need to compute the values for a and b:

The negative value for b is a cause for concern about the quality of the data. If we are confident of our data and since .00875 is close to 0, b would be set to 0.

A-B

.33875

.17625

.66625

C

D-E

Z

.33875

CA

.1215

.1885

B

a

.135

.235

E

DbZ

d(A-B, Z) = 1/2 (d(A,Z) + d(B, Z)) = 1/2 (.1885+a + .1215+a) = .66625

a = .51125

d(D-E, Z) = 1/2 (d(D,Z) + d(E, Z)) = 1/2 (.235+b + .135+b) = .17265

b = -.00875

FM Summary

Distance-based algorithm that produces unrooted trees

Removes the assumption of molecular clock, but does not give information about the root (common ancestor)

To detect the root could introduce an extra taxon (outgroup) that is more distantly related to the given taxa

Neighbor-Joining (NJ) Algorithm

NJ Algorithm

Similar to FM (also removes molecular clock assumption)

but more sophisticated in how it selects clusters to join

Produces unrooted trees

Algorithm (similar to FM) Add a leaf to the tree for each taxon Initially make each taxon be its own cluster Find the closest clusters (using more sophisticated criterion) (place new node at distance given by a variant of 3-point formula) Repeat previous step until all clusters are connected

Suppose that you are given n taxa x1, x2, x3, …, xn, and suppose that you have some tree that fits the distance data

NJ “closeness” Criterion

observation: d(x1,x2) + d(xi,xj) < d(x1,xi) + d(x2,xj)

x2

x1

x4x5

x3

x6

yz

(right side includes yz twice, left does not)

From previous slide

NJ “closeness” Criteriond(x1,x2) + d(xi,xj) < d(x1,xi) + d(x2,xj)

d(x1,x2) + d(x3,x4) < d(x1,x3) + d(x2,x4)d(x1,x2) + d(x3,x5) < d(x1,x3) + d(x2,x5)d(x1,x2) + d(x3,x6) < d(x1,x3) + d(x2,x6) … … …

d(x1,x2) + d(x3,xn) < d(x1,x3) + d(x2,xn)

-------------------------------------------------

For a fixed i, say i = 3:

4

),2()3,1()3(4

),3()2,1()3(k kxxdxxdn

k kxxdxxdn

Add d(x3,x1),d(x3,x2) , d(x3,x3), d(x2,x1), d(x2,x2) to both sides

1

),2()3,1()2(1

),3()2,1()2(k kxxdxxdn

k kxxdxxdn

2),1()2()2,1()2( SixxdniSxxdn

iSixxdnSxxdn ),1()2(2)2,1()2(

iSSixxdnSSxxdn 1),1()2(21)2,1()2(

From previous slide, if x1 and x2 are neighbors

Let

Then in general, if xk and xl are neighbors

NJ uses this observation to determine “closeness” and computes the smallest value M(k, l) to determine a cluster

Unlike UPGMA and FM, NJ has a more global view of “closeness” when selecting neighbors

NJ “closeness” Criterion

iSSixxdnSSxxdn 1),1()2(21)2,1()2(

),(),( mkMlkM

lSkSlxkxdnlkM ),()2(),(

If x1 and x2 are neighbors; where should new node y be

NJ new node Placement

x2

x1

x4 x5

x3

y

by 3-point formula

))3,2()3,1()2,1((2/1)1,( xxdxxdxxdxyd

))4,2()4,1()2,1((2/1)1,( xxdxxdxxdxyd

))5,2()5,1()2,1((2/1)1,( xxdxxdxxdxyd

)),2(),1()2,1((2/1)1,( nxxdnxxdxxdxyd … … …

--------------------------------------------------------------

3

),2(3),1()2,1(2/)2()1,()2(k kxxd

k kxxdxxdnxydn

add on right side d(x1,x1 ) + d(x1,x2) - d(x2,x1 ) - d(x2,x2 )

)21)2,1((2/)2()1,()2( SSxxdnxydn

)2

2

2

1)2,1((2/1)1,(

n

S

n

Sxxdxyd

For each pair of nodes xk and xl compute the quantity

Actually, could compute

When xk and xl are replaced by new node y, place y at

From now on Si will always be divided implicitly by (n-2)

NJ mini summary

lSkSlxkxdnlkM ),()2(),(

)22

),((2/1),(

n

Sl

n

Sklxkxdkxyd

22),(),(

n

Sl

n

SklxkxdlkM

)22

)2,1((2/1),(

n

Sk

n

Slxxdlxyd

NJ Algorithm

From the distance matrix compute the criterion matrix

Find the smallest value in M(i, j) – cluster the corresponding pair

Connect taxa xi and xj with a new node y placed at distance

Remove xi and xj and replace with y; update the distance matrix using the 3-point formula

Repeat from beginning

lSkSlxkxdlkM ),(),(

)),((2/1),( jSiSjxixdixyd

)),((2/1),( iSjSjxixdjxyd

)),(),(),((2/1),( jxixdkxjxdkxixdkxyd

Apply the NJ algorithm to the

given distance matrix:x1 x2 x3 x4 x5 x6

x1 - 8 3 14 10 12

x2 8 - 9 10 6 8

x3 3 9 - 15 11 13

x4 14 10 15 - 10 8

x5 10 6 11 10 - 8

X6 12 8 13 8 8 -

First compute Si=sum-of-row / (n-2)

Compute

M(1,2) = d(1,2) – S1 – S2 = 8 – 22= -14

M(1,3) = d(1,3) – S1 – S3 = 3 – 24.5= -21.5

M(1,4) = d(1,4) – S1 – S4 = 14 – 26 = -12

M(1,5) = d(1,5) – S1 – S5 = 10 – 23 = -13

M(1,4) = d(1,4) – S1 – S4 = 12 – 24 = -12

and so on …

S1= 11.75 S2=10.25 S3=12.75

S4=14.25 S5=11.25 S6= 12.25

lSkSlxkxdlkM ),(),(

x1 x2 x3 x4 x5 x6

x1 - -14 -21 -12 -13 -12

x2 - -14 -14 -15 -14

x3 - -12 -13 -12

x4 - -15 -18

x5 - -15

X6 -Find min value, i.e. the pair to cluster

From previous slide we need to cluster x1 and x3

Add a new taxon x7 and place it at distance

Recompute distances from x7 to all

others using the 3-point formula

x1

21

x3

x7

1)75.1275.113(2/1)31)3,1((2/1)1,7( SSxxdxxd

2)75.1175.123(2/1)13)3,1((2/1)3,7( SSxxdxxd

d(7,2) = ½(d(1,2) + d(3,2) – d(1,3)) = 7

d(7,4) = ½(d(1,4) + d(3,4) – d(1,3)) = 13

d(7,5) = ½(d(1,5) + d(3,5) – d(1,3)) = 9

d(7,6) = ½(d(1,6) + d(3,6) – d(1,3)) = 11

x2 x4 x5 x6 x7

x2 - 10 6 8 7

x4 10 - 10 8 13

x5 6 10 - 8 9

x6 8 8 8 - 11

x7 7 13 9 11 -

Apply the NJ algorithm to the

new distance matrix:

First compute Si=sum-of-row / (n-2)

Compute

M(2,4) = d(2,4) – S2 – S4 =

M(2,5) = d(2,5) – S2 – S5 =

M(2,6) = d(2,6) – S2 – S6 =

M(2,7) = d(2,7) – S2 – S7 =

and so on …

S2= S4= S5= S6= S7=

lSkSlxkxdlkM ),(),(

x2 x4 x5 x6 x7

x2 - 10 6 8 7

x4 10 - 10 8 13

x5 6 10 - 8 9

x6 8 8 8 - 11

x7 7 13 9 11 -

x2 x4 x5 x6 x7

x2 -

x4 - -

x5 - - -

x6 - - - -

x7 - - - - -

Find min value, i.e. the pair to cluster

From previous slide we need to cluster ? and ??

Add a new taxon x8 and place it at distance

Recompute distances from x8 to all

others using the 3-point formula

x?

??

x??

x8

)???)??,?((2/1)??,8( SSxxdxxd

)???)??,?((2/1)?,8( SSxxdxxd

x? x? x? x8

x? -

x? -

x? -

x6 -

NJ Summary

Distance-based algorithm that produces unrooted trees

Removes the assumption of molecular clock, but does not give information about the root (common ancestor)

Typically performs better than UPGMA and FM – uses a more global criterion to select pairs to cluster

To detect the root could introduce an extra taxon (outgroup) that is more distantly related to the given taxa

Maximum Parsimony (MP)Algorithm

MP Algorithm

Character-based algorithm – does not use distances, but utilizes the character information in sequences

A criticism of distance-based methods is that they do not exploit the structure of the sequences (collapse them to a number – the distance)

Main philosophy is “economy of substitutions” – find the tree that requires the fewest mutations (maximum parsimony)

MP Algorithm

The strategy explore a number of possible trees report the tree with smallest score (most parsimonious)

Need to be able to solve two problems small parsimony problem -- given a candidate tree compute its

parsimony score

large parsimony problem -- generate efficiently viable candidate trees (cannot generate all – tree explosion)

Small Parsimony Problem

Given a candidate tree, compute its parsimony score

Consider a candidate tree for one-site sequences

s1 = A s2 = T s3 = T s4 = G s5 = A

A T T G A

AT

AG

T

AGT

Final Score = 3

Solving Small Parsimony Problem

explore the tree bottom-up (from leaves to interior) for each internal node one level up

if the “labels” at the two child nodes have no symbols in common assign as label at this node the sum of both labels

penalize the tree one unit

if the “labels” at the two child nodes do have

symbols in common, label with common portion

no penalty

AGC

AG

C

AG

GT

G

Solving Small Parsimony Problem

For n-site sequences run the algorithm in parallel for each site and add up the parsimony scores for all sites

Consider a candidate tree for the following sequences

s1 = ATC s2 = ACC s3 = GTA s4 = GCA

ATC ACC GTA GCA

TC

A C

AG

TAC

TCT

A

Final Score = 4

Solving Large Parsimony Problem

Generate efficiently viable candidate trees (cannot try all)

Branch-and-bound approach create a possible tree by some method; calculate its score start building a tree from scratch; discarding trees that cost more

than current best

Solving Large Parsimony Problem

Branch-and-bound approach

http://artedi.ebc.uu.se/course/X3-2004/Phylogeny/Phylogeny-TreeSearch/Phylogeny-Search.html

MP Summary

Character-based algorithm – uses the sequence data

Produces unrooted trees

Economy of substitution – best tree is one that requires fewest number of substitutions

Examines a number of possible trees in search for best one