Molecular dating methods continued

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Molecular dating methods continued
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Why should we expect a clock?

• Under neutral evolution but that is too fast for
  most (all?) data sets
• If there is reasonable constancy of population
  size, mutation rate, and patterns of selection
• Perhaps all we can hope is that rates of
  evolution will change slowly and/or rarely

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How do we test for clock like evolution?

• Relative rates tests
• Likelihood ratio tests

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The likelihood approach

• Consider two models of evolution
• The usual model
• The same model but
• A root is specified
• The summed branch lengths from any node to all
  descendants of that node are the same
• Do a likelihood ratio test with n-2 degrees of
  freedom

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If a clock model is not rejected

• Calculate rates and then extrapolate from known
  to unknown pairwise distances

DOA 0.4 DAB 0.1 TOA 90 TAB (0.1/0.4)
x 90 22.5 Ma
O
A
B
0.05
0.05
0.2
0.15
90
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90 Ma
O
90
22.5
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Should obtain confidence intervals around date
estimates

• Look at the curvature of the likelihood surface
  (PAML)
• Use bootstrapping (parametric or non-parametric)

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Calibrating the tree

• How does one attach a date to an internal node?
  How old is the fossil? Where does a fossil fit
  on the tree?

F (90 Ma)
O
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What does that tell us?
F (90 Ma)
O
This node is at least 90 Ma
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What else?
This node is less than 90 Ma
F
O
This node is at least 90 Ma
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The lineage leading to F could have been missed
F
O
This node is at least 90 Ma
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General issues

• Fossils generally provide only minimal ages
• The age is attached to the node below the lowest
  place on the tree that the fossil could attach
• Maximal or absolute ages can only be asserted
  when there is lots of fossil data
• Geological events can sometimes be used to obtain
  minimal ages

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What if a clock is rejected?

• Until recently three (bad) choices
• Give-up on molecular dating
• Go ahead and use molecular dating anyway
• Delete extra-fast or extra-slow taxa

• Now one has several options
• Assume local clocks
• NPRS
• PL
• Model-based methods (Bayesian)

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Local clock
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Local clock
We assign branches to rate categories but force a
single rate per category (R1-c) Find the set of
rates that maximize the likelihood Can do a
likelihood ratio test against a strict clock (df
c-1)
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Non-Parametric Rate-Smoothing(NPRS Sanderson
1998)
d1
Node k
a
d2
Adjust times so as to minimize overall roughness

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Penalized Likelihood(Sanderson 2001)

• Semi-parametric likelihood approach
• The observed data are branch-lengths

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Penalized Likelihood

• Given the duration of a branch and its rate of
  evolution, we can calculate the probability
  (Poisson) of a given branch length
• For a set of branching times and rates we can
  calculate the likelihood of obtaining this set of
  branch lengths

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Penalized likelihood

• What would be the maximum likelihood solution?
• A different rate on each branch
• To prevent this we penalize the likelihood by
  subtracting the roughness function (same as NPRS)
  adjusted by a smoothing parameter, ?
• Adjust the degree of penalization using ?
• How do you pick a value of ? ?

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Penalized Likelihood

• Selects optimal value of ? using
  cross-validation pick the value that minimizes
  the errors made in predicting terminal branch
  lengths

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Penalized Likelihood

• More flexible than NPRS
• More difficult to implement
• Worth trying for non-clock-like data

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Bayesian dating

• Two main approaches
• Local clocks (a certain number of changes of rate
  are permitted and their position is kept track of
  during MCMC)
• Autocorrelation of rates (evolutionary rates tend
  to change gradually) - somewhat analogous to NPRS
  and PL

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Thorne et al. (2001) methodMol. Biol. Evol.
18352-361

• Most widely used. See http//www.plant.ch/bayesi
  andating1.4.pdf
• Each node has a rate. A nodes rate will tend to
  be correlated with the ancestral node
• The rate change (as a function of time) via
  Brownian motion
• The rate of a node is normally distributed with a
  mean equal to the ancestral node and a variance
  estimated from the data