PhyloGibbs: Siddharthan et al PLoS Comp Bio 2005

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PhyloGibbs Siddharthan et alPLoS Comp Bio 2005

• Presented by
• Saurabh Sinha

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PhyloGibbs

• Motif finding combined with model of evolution
• Neutral evolution selection operating on
  binding sites
• Gibbs sampling approach

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Recall Gibbs sampling

• Motif described by start positions of sites X
  (x1,x2,xN) where xi is the location of the site
  in ith sequence
• Motif model ? is the PWM from which each site is
  sampled
• Goal to sample from posterior P(?, X data)
• In reality, sample from Pr(X ?, data)
• Sample X using Gibbs sampling sample xi from
  conditional distribution Pr(xi X - xi)

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PhyloGibbs

• Input is a multiple alignment of each sequence
  (from several species)
• Site is replaced by window of fixed length on
  multiple alignment

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Basics placing motif windows
ttttCGTGAT-GCGTCGtttttttttt gagaCGTGATcGCGTCGagaat
atata cccc-------------CCAAGATCAGAccc aaata-------
-----CCAACATCAGAaaa
Multiple alignment from DIALIGN Vertically
aligned caps are evolutionarily related bases.
3 legal windows, 1 illegal window All possible
legal windows identified in preprocessing.
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Motif windows

• A legal window does not have gaps
• A legal window can span one species or multiple
  species
• If spanning multiple species, the sites in the
  window are assumed to be evolutionarily related
  binding sites

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Sampling probability

• For a single-species motif window, sampling
  probability is
• For a window spanning multiple species, sampling
  probability is

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Evolutionary model

• is given by evolutionary model

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Evolutionary model

• Evolving binding site must bind the same protein
• All bases mutate at a fixed rate ?
• When a base at position i of a binding site is
  mutated to letter ?, probability that selection
  will fix this mutation is given by the PWM
  component w?i.
• Recall that under strong selection s, probability
  of fixation to a new allele (starting copy
  number1), is s.
• Then, probability that a base at position i will
  mutate from ? from ? in time t is given by
• where q e-?t

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Evolutionary model

• As q-gt0 (i.e., large t), this becomes w?i
• so the equilibrium distribution is wi
• The model is multiplicative, i.e.,

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Evolutionary model

• For a star topology,

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Configurations

• Given a set of sequences, a configuration is a
  set of (legally placed) motif windows
• AND a color assigned to each window
• Could have multiple windows assigned with the
  same color
• Different colors represent different
  transcription factors
• Goal is to find multiple types of motifs
  together, by using different colors
• Color 0 is background (random) sequence

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Sampling

• The algorithm samples configurations from their
  posterior probability distribution
• Prior on configurations
• n(C) colored windows in C

avoid too many sites/windws
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Probabilities

• P(SC) is the probability that
• for each color in C,
• all windows of color C were sampled from the same
  PWM,
• though that PWM is unknown

P(S?CB) is for background windows (color
0) P(Sc) is for all windows of color c ? C
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Probabilities

• P(Sc) is the probability that all windows of
  color c were sampled from the same PWM, though
  that PWM is unknown
• Integration of all PWMs (for single species case)

?
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Probabilities

• P(Sc) is the probability that all windows of
  color c were sampled from the same PWM, though
  that PWM is unknown
• Integration of all PWMs (for single species case)

• Dirichlet prior
• is usually 1, so
• ?() is a factorial

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Probabilities

• What about Pr(Sc) in the multiple species case?
• Well come to this later

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Sampling

• We have to sample configuration C from the
  posterior distribution P(CS)
• Monte Carlo Markov Chain (MCMC)
• Take the current configuration C and move
  probabilistically to a new configuration C such
  that detailed balance holds
• P(CS)P(C --gt C) P(CS)P(C --gt C)

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Move set

• What kinds of moves are allowed ?
• Moves are
• shift one or more colored windows, or
• add or remove a colored window, or
• recolor a randomly chosen window from current
  configuration

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Move set

• A move takes current config C, constructs the
  set X of all C ? X that differ from C by a
  single change
• It then chooses one of the C as per
  P(CS)/P(CS) ?C?XP(CS)
• Window shift move takes a single window and
  resamples its position
• Gibbs sampling sample a joint prob. distr. by
  resampling one variable at a time, while keeping
  others fixed.

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Summarizing the samples

• After many moves, each configuration will be
  sampled as per P(CS)
• Wish to report all relevant features that are
  shared by configurations with high posterior
  probability
• Would like to identify groups of windows that
  with high probability share a color

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Anneal and track

• Simulated annealing to find maximum likelihood
  configuration C
• sample from P(CS)?
• slowly increase ? over time
• This provides a reference set of windows that
  will be tracked

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Tracking

• Fix the reference configuration C
• Continue sampling from P(CS), and for each
  window w in C, track the probability p(w,c) that
  window w belongs to the same motif (color) as the
  windows in color c of reference configuration C

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One final point

• What is Pr(Sc) in the multiple species case ?