Detecting Subtle Sequence Signals: a Gibbs Sampling Strategy for Multiple Alignment

1
Detecting Subtle Sequence Signals a Gibbs
Sampling Strategy for Multiple Alignment

• Lawrence, Altschul, Boguski, Liu, Neuwald,
  Wootton, Science, 1993

2
Motif Finding Problem

• Biological description given a set of sequences,
  find the motif shared by all or most sequences,
  while its starting position in each sequence is
  unknown
• Assumption
• Each motif appears exactly once in one sequence
• The motif has fixed length

3
Generative Model

• Suppose the sequences are aligned, the aligned
  regions are generated from a motif model
  (position-specific multinomial distribution)
• The unaligned regions are generated from a
  background model

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Notations

• Set of symbols
• Sequences S S1, S2, , SN
• Starting positions of motifs A a1, a2, , aN
• Motif model ( ) qij P(symbol at the i-th
  position j)
• Background model pj P(symbol j)
• Count of symbols in each column cij count of
  symbol, j, in the i-th column in the aligned
  region

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Motif Finding Problem

• Problem find starting positions and model
  parameters simultaneously to maximize the
  posterior probability

• This is equivalent to maximizing the likelihood
  by Bayes Theorem, assuming uniform prior
  distribution

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Equivalent Scoring Function

• Maximize the log-odds ratio

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Sampling and Optimization

• To maximize a function, f(x)
• Brute force method try all possible x
• Sample method sample x from probability
  distribution p(x) f(x)
• Idea suppose xmax is argmax of f(x), then it is
  also argmax of p(x), thus we have a high
  probability of selecting xmax

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Detailed Balance of Markov Chain

• To sample from a probability distribution p(x),
  we set up a Markov chain s.t. each state
  represents a value of x and for any two states, x
  and y, the transitional probabilities satisfy

• This would then imply

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Gibbs Sampling

• Idea a joint distribution may be hard to sample
  from, but it may be easy to sample from the
  conditional distributions where all variables are
  fixed except one
• To sample from p(x1, x2, xn), let each state of
  the Markov chain represent (x1, x2, xn), the
  probability of moving to a state (x1, x2, xn)
  is p(xi x1, xi-1,xi1,xn). Then the detailed
  balance is satisfied.

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Gibbs Sampling
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Gibbs Sampling in Motif Finding

• We should sample from joint distribution of A and
  ?, but ? has a high dimension, and thus this is
  inefficient.
• Instead, we draw sample from

• It can be shown that to find the conditional
  probability, p(azA,S) is roughly equivalent to
  compute the estimator of ? from A, then compute
  the probability of generating az according to
  this ?.

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Estimator of ?

• Given an alignment A, i.e. the starting positions
  of motifs, ? can be estimated by its MLE with
  smoothing (alternatively, Dirichlet prior with
  parameter bj)

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Algorithm

• Randomly initialize A0
• Repeat
• (1) randomly choose a sequence z from S
• A At \ az
• compute ?t estimator of ? given S and A
• (2) sample az according to P(az x), which is
  proportional to Qx/Px
• update At1 A union x
• Select At that maximizes F

Qx the probability of generating x according to
?t Px the probability of generating x
according to the background model