Models of molecular evolution

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Models of molecular evolution
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Steps in evaluating one tree

• Pick a set of branch lengths
• Calculate the ln-likelihood of character pattern
  i across all possible histories
• Add the LnL of each character pattern up to get
  the overall likelihood
• Adjust branch-lengths, substitution parameters,
  etc. so as to maximize LnL
• The result is the trees Likelihood score

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Typical Simplifying Assumptions

• Stationarity
• Reversibility
• Site independence
• Markovian process (no memory)

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The simplest model of molecular evolution
Jukes-Cantor
Instantaneous rate matrix (Q-matrix)
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Calculating probabilities of change

• To convert the Q matrix into a matrix giving the
  probability of starting at state i and ending in
  state j, t time units later uses the formula

P(t) eQt
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The simplest model of molecular evolution
Jukes-Cantor
Substitution probability matrix (P-matrix)
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More complicated (realistic) models for DNA

• Allow deviation from equiprobable base
  frequencies
• HKY85 F81GTR
• Allow two substitution types (ti and tv)
• K2P HKY85
• Allow for six substitution types
• GTR

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Relationship among models
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Accommodating rate heterogeneity

• Allow different subsets of sites to have
  different rates
• Invariant-sites model
• Some characters assigned a rate of 0, remaining
  characters analyzed as usual
• Proportion of invariant characters estimated by
  ML
• Discrete approximation to a gamma-distribution

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Summary of the Gamma correction

• The gamma function has a scale parameter and
  shape parameter (?) scale parameter 1/?
• ? represents variation in rates
• Very high values all characters have rate 1
• Low values ( 0.5) most characters change little
• Value of 0 every character has its own rate
• Estimate the value of ? that maximizes L

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Discrete approximation

• Divide the distribution into N equal sets
• Assign the median rate for the set to all sites
  in the set
• Empirically it performs well with only four rate
  categories
• Adding categories does not add parameters

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Objective criterion for choosing a model of
molecular evolution

• Pick a more complex model (one with extra
  parameters) only if the gain in likelihood is
  more than would be expected
• Likelihood ratio test Twice the ratio of the
  likelihoods under the two models follows the
  chi-square distribution with the number of
  parameters equal to the number of extra free
  parameters in the more complex model.

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Relationship between MP and ML

• One argument - MP is inherently nonparametric ?
  No direct comparison possible
• MP is an ML model that makes particular
  assumptions

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The Goldman (1990) model(see Lewis 1998 for more)

• We force all branch lengths to be equal
• The Likelihood for a character only includes the
  set of ancestral states that maximizes the
  likelihood

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Why use MP

• The model is clearly less realistic, but
• We can do more thorough searches and data
  exploration (computational efficiency)
• Robust results will usually still be supported

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Why use ML

• The model (assumptions) are explicit
• We can statistically compare alternative models
• We can conduct parametric statistical tests
  (under the assumption that we have used the
  correct model)
• But, even the most complex model is still
  unrealistically simple