Distance Based Methods for estimating phylogenetic trees

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Distance Based Methodsfor estimating
phylogenetic trees
Phylogenetics Workhop, 16-18 August 2006
Barbara Holland
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Overview

• How do we get distance data?
• Observed vs. actual distances
• Correcting for hidden changes
• Not all distances are tree-like
• Tree building clustering methods
• UPGMA
• Neighbor-joining
• Tree building optimality criteria
• Least Squares

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What do edge lengths represent?

• In some trees edges represent time, in which case
  all modern sequences should be the same distance
  from the root.
• Sometimes edge lengths represent the product µt
  of the rate of change µ and time t in which case
  different tips can be different distances from
  the root provided that the rate has changed
  across the tree.

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Distance matrices

• There are many ways of building phylogenetic
  trees, one family of methods uses a distance
  matrix as a starting point.
• A distance matrix is a table that indicates
  pairwise dissimilarity, for instance...

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Properties of distances

• d(x,x) 0
• d(x,y) d(y,x)
• d(x,y) d(y,z) gt d(x,z) (the triangle
  inequality)
• The distances used in phylogenetics always have
  the first two properties but sometimes not the
  third.

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I want to build a tree - will any old distances
do?

• Not all distances will be suitable for building
  trees.
• Tree-building methods do not discriminate, they
  will return a tree regardless of whether you give
  them roadmap distances or distances based on a
  sequence alignment.
• Some distances are perfectly tree-like.

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Perfectly tree-like distances
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Perfectly tree-like distances
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Perfectly tree-like distances
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Perfectly tree-like distances
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Perfectly tree-like distances
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Perfectly tree-like distances
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The 4-Point Condition

• Distances that fit exactly on a tree can be
  characterised by a condition on any quartet i, j,
  k, l (i.e. it must hold true for any 4 taxa).
• We write d(x,y) for the distance between x and y.
• Given 4 taxa i, j, k, l, of the 3 sums
• d(i,j) d(k,l)
• d(i,k) d(j,l)
• d(i,l) d(j,k)
• The largest two are equal.
• Distances with this property are called additive,
  because the weights on the paths along the tree
  add up to the values in the distance matrix.

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Why is this true of tree-like distances?
i
k
i
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i
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j
j
l
l
l
d(i,k)d(j,l)
d(i,j)d(k,l)
d(i,l)d(j,k)
lt

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Clock-like distances

• An even stricter condition on distances is that
  they fit on a clock-like tree.
• Distances with this property are called
  ultrametric.

time
d(i,k) d(j,k) gt d(i,j)
i
j
k
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Where do we get distances from?

• Distances can be derived from Multiple Sequence
  Alignments (MSAs).
• The most basic distance is just a count of the
  number of sites which differ between two
  sequences divided by the sequence length. These
  are sometimes known as p-distances.

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Other sources of distances

• Immunological data
• Similarity between proteins A and B can be
  assessed by how well the immune system responds
  to B after already having seen A.
• DNA/DNA hybridization
• more similar DNA hybrids "melt" at higher
  temperatures
• Fragment length polymorphism
• Chop DNA up using restriction enzymes.
• Amplify some fragments usign PCR
• Run the fragments out on an electrophoretic gel
• Compare profiles of different genomes
• BLAST scores

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Observed distances usually underestimate the true
number of changes
ATTTGCGATA
Actual Changes 2 Observed Changes 2
ATTTGCGGTA
ATCTGCGATA
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• Parallel changes
• Reversals
• Superimposed changes

ATTCGCGATA
Actual Changes 4 Observed Changes 2
ATTTGCGGTA
ATCTGCGATA
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• Parallel changes
• Reversals
• Superimposed changes

ATTTGCGATA
Actual Changes 4 Observed Changes 2
ATTCGCGATA
ATTTGCGGTA
ATCTGCGATA
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• Parallel changes
• Reversals
• Superimposed changes

ATTTGCGATA
Actual Changes 3 Observed Changes 2
ATTTGCGTTA
ATTTGCGGTA
ATCTGCGATA
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Correcting for hidden changes

• Given a statistical model of how point mutations
  occur it is possible to estimate the true genetic
  distance from the observed distance.

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Correcting under a simple model

• The Jukes-Cantor model states that all states
  A,C,G,T and all changes between states, e.g.
  A?C, are equally likely.

u/3
As a mathematical conviencence imagine we have a
rate 4u/3 of change to a random state, this
includes the possibility of a state changing to
itself.
u/3
u/3
u/3
u/3
u/3
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A Poisson process

• The probability of no change at a site over time
  t is e-4/3ut
• The probability of at least one event is then 1-
  e-4/3ut
• The probability of at least one event that leads
  to a different state from the one we started at
  is ¾(1- e-4/3ut) as one time out of four we will
  mutate to the same base we started with.
• The expected observed distance d given a true
  genetic distance of ut is d ¾(1- e-4/3ut)
• Inverting this formula gives our correction D
  ut -3/4 ln (1-4/3d)

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Correcting for hidden changes

• Correction for hidden changes has been shown
  (both theoretically and by simulation studies) to
  improve accuracy.
• However, this is not universally true.
• If data is clock-like then corrections will not
  change the relative size of the distances
• However, the more complicated the model is the
  larger the variance (error) of the distances will
  become.

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Under the Jukes-Cantor model where all point
mutations are equally likely the correction is
Dactual ¾ ln(1 4/3dobserved)
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An interesting observation

• Uncorrected distances always obey the triangle
  inequality d(x,y) d(y,z) gt d(x,z)
• But corrected distance do not.
• E.g. if sequences a and b differ at 10 / 100
  sites and sequences b and c differ at a different
  10 / 100 sites the uncorrected distances are
  d(a,b) d(b,c) 0.1, d(a,c) 0.2 and the
  corrected distances (under the JC model) are
  D(a,b) D(b,c) 0.107, D(a,c) 0.233

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Tree building - UPGMA

• UPGMA works by progressively clustering the most
  similar taxa until all the taxa form a rooted
  clock-like tree.
• Find the smallest entry in the distance matrix,
  say d(x,y).
• Form a new internal node, z, that is a parent to
  x and y and set the edge lengths from z to x and
  z to y to half d(x,y).
• Update the distance matrix by setting the
  distances from the new node z to all the other
  taxa to be the average distance between groups x
  and y.
• REPEAT until all groups have been joined.

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What precisely is meant by the average distance?

• If we a joining two groups i and j that already
  have ni and nj members we update the distances
  using

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Step 1 Find the smallest entry in the distance
matrix


d(i,j)

A

B

C

D

E

F

A
-






B

2

-





C

4

4

-




D

4

4

2

-



E

7

7

7

7

-


F

5

5

5

5

6

-

G

8

8

8

8

9

5



Step 2 - Cluster taxa A and B, form a new
internal node I Calculate the lengths of the new
edges d(A,I)d(B,I)1/2 d(A,B)1
B
A
A
Step 3 Update the distance matrix d(C,I)
½(d(A,C) d(B,C)) 4 etc...
1
1
B
G
I
C


D
C
F
E
D
F
E

G
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Step 1 Find the smallest entry in the distance
matrix


d(i,j)

I (AB)

C

D

E

F

I (AB)

-





C

4

-




D

4

2

-



E

7

7

7

-


F

5

5

5

6

-

G

8

8

8

9

5


Step 2 - Cluster taxa C and D, form a new
internal node II Calculate the lengths of the new
edges d(C,II)d(D,II)1/2 d(C,D)1
A
B
B
D
A
C
Step 3 Update the distance matrix d(I,II)1/2(d(
I,C)d(I,D)) 4 d(E,II) ½(d(E,C)
d(E,D)) 7 etc...
1
1
1
1
1
1
C
I

I
II

D
E
E

F

F
G
G
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And so on...
A
A
B
B
C
A
B
D
A
C
D
B
G
C
D
I

I
I
II
II
C

F
III
E
E
D
E

E
F

F

F
G
G
G
F
B
C
A
D
E
G
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B
C
F
A
B
C
D
A
D
E
1
1
1
1
II
I
2.5
II
1
II
1
I
I
3.4
III
3.8
0.5
0.9
III
III
IV
V
IV
IV
V
E
0.4
VI


G

G
...until we have a rooted tree. But, is it the
right tree?
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UPGMA is not consistent for additive distances


d(i,j)

A

B

C

D

E

F

The tree that matches the distances is not
recovered by UPGMA.
A

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B

2

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C

4

4

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D

4

4

2

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E

7

7

7

7

-


F

5

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G

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5



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Inconsistency

• When a method is given perfect data but still
  gets the wrong tree it is said to be
  inconsistent.
• UPGMA is inconsistent for data that isnt
  ultrametric (clock-like).
• Next well look at a method that is consistent
  for any additive data.

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Neighbor-joining (NJ)

• NJ works by progressively clustering taxa until
  all the taxa form an unrooted tree.
• Rather than using the distance matrix directly to
  determine which taxa should be clustered at each
  stage, NJ uses the S matrix where
• S(i,j) (N-2)d(i,j) - R(i) - R(j)
• N is the number of taxa.
• R(i) is the sum of the ith row in the distance
  matrix.
• R(j) is the sum of the jth row in the distance
  matrix.
• Find the smallest entry in the S matrix, say
  S(x,y).

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• Form a new internal node, z, that is a parent to
  x and y and calculate the edge lengths from z to
  x and z to y.
• d(x,z) 1/(2(N-2))(N-2)d(x,y) R(x) R(y)
• d(y,z) d(x,y) d(x,z)
• Update the distance matrix
• d(w,z) ½ (d(x,w) d(y,w) d(x,y))
• REPEAT until only two things are left to be
  joined.

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NJ Example
D
S
Step 1
R(cat) 13 R(dog) 15 R(rat) 15 R(cow) 19
e.g. S(cat,dog) (4-2)x3 13 15
-22 S(cat,rat) (4-2)x4 13 15 -20
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NJ Example
D
S
Step 2
Step 1
Cat
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Step 3 d(cat,z) ¼2d(cat,dog) R(cat)
R(dog) ¼ 6 13 15 1 d(dog,z) 3-1
2
z
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Step 4 d(z,rat) ½ d(cat,rat) d(dog,rat)
d(cat,dog) ½ 4 5 3 3 d(z,cow) ½
6 7 3 5
Cat
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z
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Global vs Local methods

• UPGMA and NJ are local construction methods. At
  each step they pick they best pair of taxa to
  cluster, once a decision is made it cannot be
  unmade. This makes these methods very fast.
• There are also global methods for making trees
  based on distances. These evaluate an optimality
  criterion on each possible tree and then pick the
  tree with the best score. Examples of global
  methods for distance data include least squares
  and minimum evolution. Because the number of
  trees grows very quickly with the number of taxa,
  these methods are slow.

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Least Squares

• We would like the path lengths on the tree we
  choose to be as close as possible to the
  corresponding values in the distance matrix.
• With additive data we can always find a tree
  where the path length distances and the distance
  matrix match exactly. However, most data isnt
  perfect...
• We can try and minimise the discrepency between
  the observed distances and the tree distances
  using a least squares approach.

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A family of least squares methods
wij 1 unweighted least squares
(Cavalli-Sforza and Edwards 1967) wij
1/Dij wij 1/Dij2 (Fitch and Margoliash 1967)
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Picking the best weights for a given tree

• The tree distances dij can be represented by the
  equation

where xij,k is an indicator variable that is 1 if
edge k lies on the path from i to j and 0
otherwise. We want to find edge weights ek that
minimise
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The indicator variables can be expressed in
matrix format
1 1 1 0 0 0 0 1 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 0
0 1 0 0 0 1 1 1 0 0 0 0 1 1 0 0 1 0 1 1 0 0 1 0 0
0 0 0 1 0 1 0 1 0 1 1 1 0 0 0 0 0 0 1 1
DAB DAC DAD DAE DBC DBD DBE DCD DCE DDE
e1 e2 e3 e4 e5 e6 e7
B
A
C
e1
e3
e5
e2
e4
D
e
X
e6
e7
D
E
Each row of X corresponds to a path in the
tree We can write D Xe
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Experience the joy of linear algebra

• DXe
• XTD (XTX)e
• e (XTX)-1XTD

This assumes that the weights wij 1
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Minimum evolution

• Uses the least squares method to fit the branch
  lengths for each tree
• BUT uses a different optimality criterion than
  least squares.
• Prefers the tree with the shortest sum of branch
  lengths

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Review

• Observed distances derived from sequence
  alignments will always underestimate the true
  number of mutations. Hence it is ususally a good
  idea to correct for these hidden changes.
• Clustering methods like UPGMA and
  Neighbor-joining are very fast as they only make
  local decisions and never backtrack. These
  methods are often used as a starting point for
  heuristic searches.
• There are also optimality criteria that use
  distances as input, e.g. Least squares and
  minimum evolution.

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Review

• Not all distances can be fit perfectly onto a
  tree.
• Methods can be inconsistent, for example for some
  non-clocklike distances UPGMA is guaranteed to
  recover the wrong tree.
• UPGMA is consistent for clock-like distances and
  NJ is consistant for any additive distances.