Parsimony and searching tree-space

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Parsimony and searching tree-space


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The basic idea

• To infer trees we want to find clades (groups)
  that are supported by synapomorpies (shared
  derived traits).
• More simply put, we assume that if species are
  similar it is usually due to common descent
  rather than due to chance.

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Sometimes the data agrees
ACCGCTTA
ACTGCTTA
ACCCCTTA
Time
ACTGCTAA
ACTGCTTA
ACCCCTTA
ACCCCATA
ACCCCTTA ACCCCATA ACTGCTTA ACTGCTAA
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Sometimes not
ACCGCTTA
ACTGCTTA
ACCCCTTA
Time
ACTGCTAA
ACTGCTTC
ACCCCTTC
ACCCCATA
ACCCCTTC ACCCCATA ACTGCTTC ACTGCTAA
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Homoplasy

• When we have two or more characters that cant
  possibly fit on the same tree without requiring
  one character to undergo a parallel change or
  reversal it is called homoplasy.

ACCGCTTA
ACCCCTTA
ACTGCTTA
Time
ACTGCTAA
ACTGCTTC
ACCCCTTC
ACCCCATA
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How can we choose the best tree?

• To decide which tree is best we can use an
  optimality criterion.
• Parsimony is one such criterion.
• It chooses the tree which requires the fewest
  mutations to explain the data.
• The Principle of Parsimony is the general
  scientific principle that accepts the simplest of
  two explanations as preferable.

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S1 ACCCCTTC S2 ACCCCATA
S3 ACTGCTTC S4 ACTGCTAA (1,2),(3,4)
(1,3),(2,4)
1
2
1
3
3
4
2
4
8
S1 ACCCCTTC S2 ACCCCATA
S3 ACTGCTTC S4 ACTGCTAA (1,2),(3,4)
0 (1,3),(2,4) 0
A
A
A
A
A
A
A
A
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S1 ACCCCTTC S2 ACCCCATA
S3 ACTGCTTC S4 ACTGCTAA (1,2),(3,4)
001 (1,3),(2,4) 002
C
C
C
T
C
T
T
T
C
T
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S1 ACCCCTTC S2 ACCCCATA
S3 ACTGCTTC S4 ACTGCTAA (1,2),(3,4)
0011 (1,3),(2,4) 0022
C
C
C
G
C
G
G
G
C
G
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S1 ACCCCTTC S2 ACCCCATA
S3 ACTGCTTC S4 ACTGCTAA (1,2),(3,4)
001101 (1,3),(2,4) 002201
T
A
T
T
T
A
T
T
A
T
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S1 ACCCCTTC S2 ACCCCATA
S3 ACTGCTTC S4 ACTGCTAA (1,2),(3,4)
0011011 (1,3),(2,4) 0022011
T
T
T
T
T
A
T
A
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S1 ACCCCTTC S2 ACCCCATA
S3 ACTGCTTC S4 ACTGCTAA (1,2),(3,4)
00110112 6 (1,3),(2,4) 00220111 7
C
A
C
C
C
A
C
A
A
A
According to the parsimony optimality criterion
we should prefer the tree (1,2),(3,4) over the
tree (1,3),(2,4) as it requires the fewest
mutations.
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Maximum Parsimony

• The parsimony criterion tries to minimise the
  number of mutations required to explain the data
• The Small Parsimony Problem is to compute the
  number of mutations required on a given tree.
• For small examples it is straightforward to see
  how many mutations are needed

Cat
Cat
Rat
Rat
A
G
A
G
G
A
A
A
A
G
G
A
Dog
Dog
Mouse
Mouse
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The Fitch algorithm

• For larger examples we need an algorithm to solve
  the small parsimony problem

f
Site a A b A c C d C e G f G g T h A
g
h
e
a
d
b
c
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The Fitch algorithm

• Label the tips of the tree with the observed
  sequence at the site

G
T
A
G
A
C
A
C
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The Fitch algorithm

• Pick an arbitrary root to work towards

G
T
A
G
A
C
A
C
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The Fitch algorithm

• Work from the tips of the tree towards the root.
  Label each node with the intersection of the
  states of its child nodes.
• If the intersection is empty label the node with
  the union and add one to the cost

G
T
A
A,C,G
A,T
G
C
A
A
C,G
A
C
A
C
Cost 4
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Fitch continued

• The Fitch algorithm also has a second phase that
  allocates states to the internal nodes but it
  does not affect the cost.
• To find the Fitch cost of an alignment for a
  particular tree we just sum the Fitch costs of
  all the sites.

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The large parsimony problem

• The small parsimony problem to find the score
  of a given tree - can be solved in linear time in
  the size of the tree.
• The large parsimony problem is to find the tree
  with minimum score.
• It is known to be NP-Hard.

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How many trees are there?
species unrooted binary tip-labelled trees
4 3
5 3515
6 357105
7 3579945
10 2,027,025
20 2.21020
n (2n-5)!!
An exact search for the best tree, where each
tree is evaluated according to some optimality
criterion such as parsimony quickly becomes
intractable as the number of species increases
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Counting trees
1
1
2
3
1
1
1
2
2
3
1 x 3 3
3
3
4
4
2
4
1
5
2
1
2
5
1
4
2
1
3
2
5
1
2
1 x 3 x 5 15
4
3
5
4
5
3
4
3
3
4
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Search strategies

• Exact search
• possible for small n only
• Branch and Bound
• up to 20 taxa
• Local Search - Heuristics
• pick a good starting tree and use moves within a
  neighbourhood to find a better tree.
• Meta-heuristics
• Genetic algorithms
• Simulated annealing
• The ratchet

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Exact searches

• for small number of taxa (nlt12) it is possible
  to compute the score of every tree
• Branch and Bound searches also guarantee to find
  the optimal solution but use some clever rules to
  avoid having to check all trees. They may be
  effective for up to 25 taxa.

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http//evolution.gs.washington.edu/gs541/2005/lect
ure25.pdf
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No need to evaluate this whole branch of the
search tree, as no tree can have a score better
than 9
http//evolution.gs.washington.edu/gs541/2005/lect
ure25.pdf
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The problem of local optima
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Nearest Neighbor Interchange(NNI)
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Subtree Pruning Regrafting (SPR)
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Tree Bisection Reconnection(TBR)