Building phylogenetic trees

1
Building phylogenetic trees

• Topics in Computational Biology
• 25.2.2004
• Pia Laine

2
Contents

• Phylogeny
• Phylogenetic trees
• How to make a phylogenetic tree from pairwise
  distances
• UPGMA method ( an example)
• Neighbor-Joining method ( an example)
• Comparison of methods
• Conclusion

3
Phylogeny

• Phylogeny is the evolution of related
  species/genes
• Phylogenetic tree diagram showing evolutionary
  lineages of species/genes
• The history of genes or species may be very
  different
• Genes can be homologous or analogous, but still
  remind each other
• Homologous sequences can be devided into two
  parts
• Orthologous sequences diverged by specification
  from a common ancestor
• Paralogous sequences evolved by gene dublication
  within species
• Analogous sequences may appear and function very
  similarly, but they do not have a common ancestor
• WHEN WE WANT TO EXPLORE EVOLUTIONARY
  RELATIONSHIPS, WE NEED TO HANDLE ORTHOLOGOUS
  SEQUENCES

4
Phylogenetic trees

• WHY construct a phylogenetic tree?
• to understand lineage of various species
• to understand how various functions evolved
• to inform multiple alignments
• Trees can be rooted (a common ancestor in known)
  or unrooted
• Leaves are the terminal nodes that correspond to
  the observed sequences of genes or species (A, B,
  C, D)
• Internal nodes are hypothetical ancestral nodes
• All trees will be assumed to be binary, meaning
  that an edge that branches splits into two
  daughter edges
• Each edge has a certain amount of evolutionary
  divergence associated to it, defined by some
  measure of distance between sequences, or from a
  model of substitution of residues over the course
  of evolution

5
Phylogenetic trees

• Different ways to represent a phylogenetic tree
  (illustrated by Treeview)

6
Different algorithms used to infer phylogeny from
sequence data

• Distance methods
• Parsimony
• Likelihood
• Probabilistic methods
• Phylogenetic invariants

7
Route from the molecular sequences to the
phylogenetic tree

• Distance methods
• Select a set of related (orthologous) nucleotide
  or amino acid sequences
• Perform multiple sequence alignment (Clustal
  series widely used)
• Calculate pairwise distances of the sequence
  using chosen evolution model of substitution
  (Distances between sequences describe the
  evolution the smaller distances are the closer
  they are related)
• Select the most suitable algorithm to infer
  phylogeny
• View the tree with a certain program (Treeview,
  NJPlot,..)

8
Making a tree from pairwise distances

• Distances dij between each pair of sequences i
  and j are calculated in the given dataset
• Different ways defining distances
• For nucleotide sequences
• Jukes-Cantor, Kimura-2-parameter K2P, HKY
  (Hasegawa-Kishino-Yano), F84, Tamura-Nei, General
  time-reversible model, General 12-parameter model
• For amino acid sequences
• PAM-matrices, BLOSUM-matrices

9
Distance matrix methods

• UPGMA
• Algorithm introduced by Sokal and Michener 1958
• Neighbor-Joining
• Algorithm introduced by Saitou and Nei 1987
• Modified by Studier and Keppler 1988

10
Clustering method UPGMA

• UPGMA Unweighted pair group method using
  arithmetic averages
• Simple method
• It works by clustering the sequences, at each
  stage connecting two clusters and finally
  creating a new node on a tree
• Method assumes equal rate of evolutionary change
  along branches ? Molecular clock assumption

11
UPGMA
A
C
B
D

• UPGMA produces a rooted tree
• Branch lengths satisfy a molecular clock
• ? The divergence of sequences is assumed to occur
  at the same constant rate at all points in the
  tree
• Trees that are clocklike are rooted and the total
  branch length from the root up to any leaf is
  equal
• Trees are often referred to be ultrametric
• A distance measures are ultrametric if either all
  three distances are equal
• dij dik djk or two of them are equal and one
  is smaller djk lt dij dik
• ? UPGMA is guaranteed to build the correct tree
  if distances are ultrametric
• Method can be used for reconstructing phylogenies
  if evolutionary rates are assumed to be same in
  all lineages ? criticism in the phylogeny
  literature
• Suitable for the species closely related
• Running time O(n2)

12
Algorithm UPGMA

• Initialisation
• Assign each sequence i in dataset to its own
  cluster
• Define one leaf of T for each sequence, and
  place at height zero
• Iteration
• Find the two clusters i and j for which dij is
  the smallest (pick randomly if several equal
  distances)
• Define a new cluster ij by Cij Ci U Cj.
  Cluster ij has nij ni nj members ( initially
  ni 1 )
• Connect i and j on the tree to a new node v
• The branch lengths from new node to i and j are
• placed at height

13
Algorithm UPGMA (cont.)

• Iteration (cont.)
• Compute the distances between the new cluster
  and the remaining clusters by using
• Add ij to the current clusters and remove i and
  j
• Termination
• When only two clusters i and j remain, place the
  root at height

14
An example UPGMA (1)

• Distance matrix (arbitrary)
• for four items (sequences)
• A, B, C and D
• Actually distances are not ultrametric, because
  three distances are not equal
• dij ? dik ? djk or two of them are not equal and
  one is smaller djk lt dij ? dik

Step 1. Find the smallest distance, dij, between
two clusters ? A and C, where dij is 7
15
An example UPGMA (2)

• Step 2. Define new cluster ij, which has nij
  ni nj
• members (initially ni 1)
• New cluster ? A and C
• nAC nA nC2
• Step 3. Connect A and C on the tree to a new
  node v1
• Step 4. The branch lengths from new node v1 to A
  and C

3,5
A
C
3,5
16
An example UPGMA (3)

• Step 5. Compute the distances between the new
  cluster AC and the remaining clusters (B and D)
• Step 6. Delete the columns and rows of the
  distance matrix that correspond to clusters A and
  C, and add a column and a row for cluster AC

?New distance matrix
17
An example UPGMA (4)

• 2nd iteration process
• Step 1. Find the two sequences i and j for which
  dij
• is the smallest (randomly if several equal
  distances)
• ?AC-B
• Step 2. Define new cluster (ij), which has nij
  ni nj
• members ( initially ni 1 ) New cluster ? AC and
  B
• nACB nAC nB 2 1 3
• Step 3. Connect AC and B on the tree to a new
  node v2
• Step 4. The branch lengths from new node v2 to AC
  and B
• ?

3,5
A
C
3,5
B
4,25
18
An example UPGMA (5)

• Step 5. Compute the distances between the new
  cluster and the remaining cluster (D)
• Step 6. Delete the columns and rows of the
  distance matrix that correspond to clusters AC
  and B, and add a column and a row for cluster ACB

?New distance matrix
19
An example UPGMA (6)

• Termination
• Only two clusters (ACB and D) remaining
• Place the root height

Original distance matrix and final phylogenetic
tree(including the branch lengths)
3,5
A
0,75
C
1,92
3,5
B
4,25
D
6,17
20
Neighbor-Joining (N-J)
D
B

• Another algorithm that works by clustering the
  sequences
• Does not assume molecular clock
• N-J trees are unrooted
• N-J assumes additivity
• Def. Edge lengths are said to be additive if the
  distance between any pair of leaves is the sum of
  lengths of the edges on the path connecting them
• Method uses an approximate algorithm, where the
  tree is built by finding a pair of neighboring
  leaves i and j that minimize the length of the
  tree. Finally neighboring leaves are joined.
• Running time O(n2)

A
C
21
Algorithm Neighbor-Joining

• Initialisation
• Define T to be the set of leaf nodes, one for
  each given sequence
• Iteration
• Compute for each sequence,
  where n is the number of sequences in the
  distance matrix
• Pick a pair i and j (for which dij ui uj is
  the smallest (pick randomly if several equal)
• Join items i and j with a new node v
• Compute the branch lengths from a new node v to
  items i and j
• Compute the distances between new node v and
  remaining items
• Remove i and j from the distance matrix and
  replace them by new node v
• Termination
• When only two items i and j remain, add the
  remaining edge between i and j, with length dij

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An example N-J (1)
Step 1. Compute for each row in distance
matrix Step 2. Compute (the lower-diagonal
matrix) and choose the smallest (most
negative)
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An example N-J (2)

• Step 3. Join A and B together with a new node
  v1. Compute the edge lengths, from A to node v
  and from B to node v1
• Step 4. Compute distances between the new node
  v1 and remaining items (C and D)

B
5
v1
3
A
24
An example N-J (3)
New reduced distance matrix

• Step 5. Delete A and B from the distance matrix
  and replace them by new item AB
• Step 6. Continue from step 1, because more than
  two items remain
• Step 1. Compute
• for each row in
• distance matrix
• Step 2 Compute
• and choose
• the smallest (the lower-diagonal matrix)

25
An example N-J (4)

• Step 3 Join v1 and C together with a new node
  v2. Compute the edge lengths, from v1 to node v2
  and from C to node v2
• Step 4 Compute distances between the new node v2
  and remaining items (D)

B
5
v1
v2
1
3
3
A
C
26
An example N-J (5)

• Step 5 Delete AB and C from the distance matrix
  and replace them by ABC
• Step 6 Only two nodes remaining ? connect them

Original distance matrix and final phylogenetic
tree (including the edge lengths)
D
8
B
5
1
3
3
A
C
27
Comparison

• Neighbor-joining
• Unrooted tree, where the direction of evolution
  is unknown
• Suitable for datasets with largely varying rates
  of evolution
• Suitable for large datasets

• UPGMA
• The total branch length from the root up to any
  leaf is equal
• Produces a rooted tree, where the root is
  hypothesized ancestor of the sequences in the
  tree
• Suitable for closely related sequences
• Can be used to infer phylogenies if one can
  assume that evolutionary rates are the same in
  all lineages

D
8
3,5
A
B
5
C
3,5
1
B
3
3
A
C
4,25
D
6,17
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Conclusion

• UPGMA method constructs a rooted phylogenetic
  tree correctly if there is a molecular clock with
  a constant rate of mutation
• UPGMA method is rarely used, because molecular
  clock assumption is not generally true selection
  pressures vary across time periods, genes within
  organisms, organisms, regions within gene
• N-J method produces an unrooted tree without
  molecular clock hypothesis
• N-J method is one of the most popular and widely
  used by molecular evolutionist
• Distance methods are strongly dependent on the
  model of evolution used
• Sequence information is reduced when transforming
  sequence data into distances
• Distance methods are computationaly fast

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Reference

• Durbin, R., Eddy, S., Krogh, A., Mithchison G.
  2003 Biological sequence analysis Probabilistic
  models of proteins and nucleic acid. Campridge
  University Press.
• Li, W. 1997. Molecular Evolution. Sinauer
  Associates, Sunderland, MA.  p. 108
• Felsenstein, J. 2003. Inferring Phylogenies.
  Sinauer Associates, Sunderland, MA. p.147-170

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Examples of phylogeny programs

• Multiple sequence alignment
• Clustal series (W, V) (free, http//www-igbmc.u-st
  rasbg.fr/BioInfo/ClustalX/Top.html )
• Phylogeny packages
• PAUP (http//paup.csit.fsu.edu/ )
• Phylip (free, http//evolution.gs.washington.edu)
  
• MEGA (free, http//www.megasoftware.net)
• Viewing/plotting phylogenetic trees
• Treeview (free, http//taxonomy.zoology.gla.ac.uk/
  rod/treeview.html)
• NJPlot (free, http//pbil.univ-lyon1.fr/software/n
  jplot.html)

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Further reading

• N-J Saitou, N. and M. Nei.1987. The
  neighbor-joining method a new method for
  reconstructing phylogenetic trees. Mol Biol Evol
  4(4) 406-25.
• N-J Studier, J. A., K. J. Keppler, et al. 1988.
  A note on the neighbor-joining algorithm of
  Saitou and Nei The neighbor-joining method a new
  method for reconstructing phylogenetic trees. Mol
  Biol Evol 5(6) 729-31.
• UPGMA Michener, C. D., and R. R. Sokal. 1957. A
  quantative approach to a problem in
  classification. Evolution 11 130-162.
• ClustalW Thompson, J. D., T. J. Gibson, et al.
  1997. The CLUSTAL_X windows interface flexible
  strategies for multiple sequence alignment aided
  by quality analysis tools. Nucleic Acids Res
  25(24) 4876-82.