Phylogenetic trees

OUTLINE

Phylogeny UPGMA Neighbor Joining Method

Phylogeny

Understanding life through time, over long periods of past time, the connections between all groups of organisms

as understood by ancestor/descendant relationships,

Tree of life.

Phylogeny

Phylogeny

Rooted and Unrooted trees:

Phylogeny

Rooted and Unrooted trees:– Most phylogenetic methods produce unrooted trees,

because they detect differences between sequences, but have no means to orient residue changes relatively to time.

Phylogeny

Rooted and Unrooted trees:– Two means to root an unrooted tree :

The outgroup method : include in the analysis a group of sequences known a priori to be external to the group under study; the root is by necessity on the branch joining the outgroup to other sequences.

Make the molecular clock hypothesis : all lineages are supposed to have evolved with the same speed since divergence from their common ancestor. Root the tree at the midway point between the two most distant taxa in the tree, as determined by branch lengths. The root is at the equidistant point from all tree leaves.

Phylogeny

Rooted and Unrooted trees:– Two means to root an unrooted tree :

Phylogeny

Orthology / Paralogy:

Phylogeny

Species Tree and Gene Tree:

Evolutionary relationship between seven eukaryotes

E gene tree for Na+-K+ ion pump membrane protein family members

Phylogeny

Species Tree and Gene Tree:

Phylogeny

Additive Tree:A distance matrix corresponding to a tree is called additive,– THEOREM: D is additive if and only if:

For every four indices i,j,k,l, the maximum and median of the three pairwise sums are identical:

Dij+Dkl < Dik+Djl = Dil+Djk

UPGMA

Building Phylogenetic Trees by UPGMA:– Unweighted Pair – Group Method using arithmetic

Averages,– Assume constant mutation rate,– The two sequences with with the shortest

evolutionary distance between them are assumed to have been the last two diverge, and represented by the most racent internal node.

UPGMA

Building Phylogenetic Trees by UPGMA:– The distance between two clusters:

Assume we have N sequences, Cluster X has NX sequences, cluster Y has NY sequences,

dXY : the evlotionary distance between X and Y

YjXiij

YXXY d

NNd

,

1

UPGMA

Building Phylogenetic Trees by UPGMA:– When cluster X and Y are combined to make a new

cluster Z: No need to use sequence – sequence distances, Calculate the distance of each cluster (such as W) to the new

cluster Z

YX

YWYXWXZW NN

dNdNd

UPGMA

Building Phylogenetic Trees by UPGMA:– Example:

The distance matrix

UPGMA


UPGMA


A – D becomes a new cluster lets say V, We have to modify the distance matrix, What are the distances between:

– V and B,– V and C,– V and E,– V and F.

UPGMA



– V and B (Calculate),

611

6*16*1

DA

DBDABAVB NN

dNdNd

UPGMA



– V and C (Calculate),

811

8*18*1

DA

DCDACAVC NN

dNdNd

UPGMA



– V and E (Calculate),

211

2*12*1

DA

DEDAEAVE NN

dNdNd

UPGMA



– V and F (Calculate),

611

6*16*1

DA

DFDAFAVF NN

dNdNd

UPGMA


New matrix:

UPGMA


Cluster according to min distance:

UPGMA


V – E becomes a new cluster lets say W, We have to modify the distance matrix, What are the distances between:

– W and B,– W and C,– W and F.

UPGMA



– W and B (Calculate),

612

6*16*2

EV

EBEVBVWB NN

dNdNd

UPGMA



– W and C (Calculate),

812

8*18*2

EV

ECEVCVWC NN

dNdNd

UPGMA



– W and F (Calculate),

612

6*16*2

EV

EFEVFVWF NN

dNdNd

UPGMA


New matrix:

UPGMA



UPGMA


F – B becomes a new cluster lets say X, We have to modify the distance matrix, What are the distance between:

– W and X.

UPGMA


What are the distance between: W and X (Calculate).

6)666666(*2*3

1

)(1

1

,

EFEBDFDBAFABXW

XjWiij

XWWX

ddddddNN

dNN

d

UPGMA


New matrix:

UPGMA



UPGMA


X – W becomes a new cluster lets say Y, We have to modify the distance matrix, What are the distance between:

– Y and C.

UPGMA


What are the distance between: Y and C (Calculate).

8)88888(*1*5

1

)(1

1

,

FCBCECDCACCY

CjYiij

CYYC

dddddNN

dNN

d

UPGMA


New matrix:

UPGMA



Fitch-Margoliash Method:

Building Phylogenetic Trees by Fitch-Margoliash:– Do not make the assumption of constant mutation

rate,– Assume that the distances are additive.


Building Phylogenetic Trees by Fitch-Margoliash:– The distances dij:

32

31

21

bbd

bbd

bbd

BC

AC

AB


Building Phylogenetic Trees by Fitch-Margoliash:– The branch lengths:

)(2

1

)(2

1

)(2

1

3

2

1

ABBCAC

ACBCAB

BCACAB

dddb

dddb

dddb


Building Phylogenetic Trees by Fitch-Margoliash:– The distances between clusters are defined as

UPGMA:

YjXiij

YXXY d

NNd

,

1

YX

YWYXWXZW NN

dNdNd


Building Phylogenetic Trees by Fitch-Margoliash:– Another Example:

A B C D E

A 22 39 39 41

B 41 41 43

C 18 20

D 10

E



D and E are the closest sequences

A B C D E

A 22 39 39 41

B 41 41 43

C 18 20

D 10

E



Name {A, B, C} as W,

A B C D E

A 22 39 39 41

B 41 41 43

C 18 20

D 10

E



Distance between W and D:

A B C D E

A 22 39 39 41

B 41 41 43

C 18 20

D 10

E33)184139(*1*3

1

)(1

1

,

CDBDADDW

DjWiij

DWWD

dddNN

dNN

d



Distance between W and E:

A B C D E

A 22 39 39 41

B 41 41 43

C 18 20

D 10

E35)204341(*1*3

1

)(1

1

,

CEBEAEEW

EjWiij

EWWE

dddNN

dNN

d



Branches a, b and c:

29433

6410

4)351033(2

1)(

2

1

c

b

ddda WEDEWD



Update the distance matrix:

A B C D E

A 22 39 39 41

B 41 41 43

C 18 20

D 10

E

A B C {D,E}

A 22 39 40

B 41 42

C 19

{D,E}



{D,E} and C are the closest sequences

A B C {D,E}

A 22 39 40

B 41 42

C 19

{D,E}



Name {A, B} as W:

A B C {D,E}

A 22 39 40

B 41 42

C 19

{D,E}



Distance between W and C:

40)4139(*1*2

1

)(1

1

,

BCACCW

CjWiij

CWWC

ddNN

dNN

d A B C D E

A 22 39 39 41

B 41 41 43

C 18 20

D 10

E



Distance between W and {D,E} (name {D,E} as X):

41)43414139(*2*2

1

)(1

1

,

BEBDAEADXW

XjWiij

XWWX

ddddNN

dNN

d A B C D E

A 22 39 39 41

B 41 41 43

C 18 20

D 10

E



Distance between C and {D,E} (name {D,E} as X):

19)2018(*2*1

1

)(1

1

,

CECDXC

XjCiij

XCCX

ddNN

dNN

d A B C D E

A 22 39 39 41

B 41 41 43

C 18 20

D 10

E




31940

10919

9)411940(2

1)(

2

1

c

b

ddda WXCXWC



Update the distance matrix:

A B C {D,E}

A 22 39 40

B 41 42

C 19

{D,E}

A B {C,D,E}

A 22 39.5

B 41.5

{C,D,E}



Now we are in thee trivial case of 3 sequences (remember the previous example):

A B {C,D,E}

A 22 39.5

B 41.5

{C,D,E}



FINAL TREE:

The Neighbor-Joining Method:

Building Phylogenetic Trees by Neighbor-Joining:– The true tree will be that for which the total branch

length, S, is shortest,– Neighbors: a pair of nodes that are seperated by just

one other node,


Building Phylogenetic Trees by Neighbor-Joining:– Algorithm (Given a distance matrix):

Iterate Until 2 Nodes are left:– For each node find

– Choose pair (i, j) with smallest – Mege two nodes i and j with a new internal node Y, and

find branch lengths by

– Update the distance matrix using


Building Phylogenetic Trees by Neighbor-Joining:– Example:

References

M. Zvelebil, J. O. Baum, “Understanding Bioinformatics”, 2008, Garland Science

Andreas D. Baxevanis, B.F. Francis Ouellette, “Bioinformatics: A practical guide to the analysis of genes and proteins”, 2001, Wiley.

Barbara Resch, “Hidden Markov Models - A Tutorial for the Course Computational Intelligence”, 2010.

Phylogenetic trees

Education

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