Molecular phylogenetics 3

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Molecular phylogenetics 3

• Level 3 Molecular Evolution and Bioinformatics
• Jim Provan

Page and Holmes Sections 6.5-6
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Maximum likelihood

• Principle of likelihood suggests that the
  explanation that makes the observed outcome most
  probable is preferred
• More formally
• LD Pr (D H)
• In a phylogenetic context
• D is the set of sequences being compared
• H is a phylogenetic tree
• The tree that makes the data the most probable
  evolutionary outcome is the maximum likelihood
  estimate of the phylogeny

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Models, data and hypotheses

• Maximum likelihood requires three elements
• A model of sequence evolution
• A tree
• A data set
• ML methods of tree building must solve two
  problems
• For a given tree topology, what set of branch
  lengths makes the observed data most likely
• Which tree has the greatest likelihood

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Models, data and hypotheses

• Suppose we have two sequences, 1 and 2, separated
  by an average of d substitutions per site
• d mt
• Given a model of substitution for each site we
  can compare the probability Pij(d) that two
  sequences separated by d would have nucleotides i
  and j
• For example, if sequence 1 had nucleotide A then
  PAG(d) is the probability that sequence 2 has a G
  in the corresponding position
• The log likelihood of obtaining the observed
  sequences is the sum of the log likelihoods of
  each individual site

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Models, data and hypotheses

• What model?
• Transition/transversion ratio
• Base composition
• Variation in rate across sites
• In all but simplest models (e.g. Jukes-Cantor),
  differences in transition / transversion rates
  can be taken into account
• Keeping other parameters constant, it is possible
  to calculate ML estimates of individual parameters

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Likelihood ratio tests

• We can test alternative hypotheses concerning the
  same data using a likelihood ratio test
• Likelihood ratio statistic (D) is the ratio of
  the alternative hypothesis (H1) to the null
  hypothesis (H0)
• Because likelihoods are often very small, it is
  more convenient to use log likelihoods
• D log L1 log L0
• where
• L1 is the maximum likelihood of the alternative
  hypothesis H1
• L0 is the maximum likelihood of the null
  hypothesis H0
• Can be used to test various hypotheses such as
  whether a particular model of evolution is valid,
  whether a molecular clock adequately describes
  the data or whether one phylogenetic hypothesis
  is better than another

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Testing models

• A model can be tested to measure how well it fits
  the observed data by comparing likelihood a tree
  and a model confers on the data (Ltree) with
  theoretical best (Lmax)
• Likelihood ratio test can be performed to test
  the adequacy of the HKY85 model to describe the
  hominid mtDNA data set

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Testing rate variation

• If sequences are evolving at different rates,
  then an ultrametric tree will give a poor
  representation of relationships between taxa
• 2D log Lno clock log Lclock

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Comparing phylogenetic hypotheses

• If two trees are not significantly different then
  the sum of these likelihood differences

will not be significantly different from zero
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Objections to likelihood

• Requires an explicit model of evolution
• This is a strength, since it makes us aware of
  the assumptions being made
• However, dependence on a model raises question of
  which model to use
• Computationally expensive
• Finding the best combination of model and tree is
  technically difficult
• Computing likelihood is also time consuming and
  it may be that there is more than one maximal
  likelihood value for a given tree
• Suggested that likelihood is better for testing
  models rather than as an all-purpose phylogenetic
  tool

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Splits

• In the above example, the split gorilla,
  orang-utan, gibbon,human, chimp can be
  written as 00011 in binary notation, or 3 in
  decimal notation
• One advantage is that we can refer to any split
  by a single number

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Spectral analysis

• Provides a means of visualising support for each
  split
• In simple terms, consists of plotting the
  frequencies of each split in the data set
• Straightforward if there is two states for each
  character

Human G T C A T C A T C C 1 1 0 1 1 0 1 1
0 1 Chimp A T T A C C A T T C 0 1 1 1 0 0
1 1 1 1 Gorilla G T T G T T A T T A 1 1 1
0 1 1 1 1 1 0 Orang-utan A C C A C T C C C
A 0 0 0 1 0 1 0 0 0 0 Gibbon A C C G C
C C C C A 0 0 0 0 0 0 0 0 0 0 5 7
6 11 5 12 7 7 6 3
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Spectral analysis
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Spectral analysis

• Since all splits cannot coexist in the same tree,
  some method is needed to decide which splits to
  use to construct the tree
• Five trivial splits will be in every tree
• One possible solution is to choose the two
  mutually compatible, non-trivial splits which
  have the best support
• In this case, the best non-trivial split is
  Orang-utan, Gibbon
• The next best supported split is Human, Chimp,
  which is compatible with this split
• This gives the basic topology Human, Chimp,
  Gorilla, Orang-utan, Gibbon
• Problems with spectral analysis
• Computationally expensive (half a million splits
  for 20 sequences)
• Potential for more than two character states

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Split decomposition
1 2 3 4 5 6 7 8 9 Human T C C T T A A A
A Chimp T T C T A T A A A Gorilla T T A C A A T
A A Orang-utan C C A C A A A T A Gibbon C C A C
A A A A T
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Split decomposition