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

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Detecting Subtle Sequence Signals a Gibbs
Sampling Strategy for Multiple Alignment

• Lawrence et al. 1993

Presented By Manish Agrawal Slides adapted from
Prof Sinhas notes.
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A motif model

• To define a motif, lets say we know where the
  motif starts in the sequence
• The motif start positions in their sequences can
  be represented as s (s1,s2,s3,,st)

Genes regulated by same transcription factor
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Motifs Matrices and Consensus

• Line up the patterns by their start indexes
• s (s1, s2, , st)
• Construct position weight matrix with
  frequencies of each nucleotide in columns
• Consensus nucleotide in each position has the
  highest frequency in column

• a
  G g t a c T t
• C c A t a c g t
• Alignment a c g t T A g t
• a c g t C c A t
• C c g t a c g G
• 
  _________________
• A 3 0 1 0 3 1 1 0
• Matrix C 2 4 0 0 1 4 0 0
• G 0 1 4 0 0 0 3 1
• T 0 0 0 5 1 0 1 4
• _________________
• Consensus A C G T A C G T

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Motif Finding Problem(Simplified)

• 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.

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Generative Model

• Suppose the sequences are aligned, the aligned
  regions are generated from a motif model.
• Motif model is a PWM. A PWM is a
  position-specific multinomial distribution.
• For each position i (from 1 to W), a multinomial
  distribution on amino acids, consisting of
  variables qi1, qi2,..,qi20
• The unaligned regions are generated from a
  background model p1,p2, , p20

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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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Probability of data given model
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Scoring Function

• Maximize the log-odds ratio
• Is greater than zero if the data is a better
  match to the motif model than to the background
  model

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Scoring function

• A particular alignment A gives us the
• counts cij.
• In the scoring function F, use

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Scoring function

• Thus, given an alignment A, we can calculate the
  scoring function F
• We need to find A that maximizes this scoring
  function, which is a log-odds score

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Optimization and Sampling

• 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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Markov Chain Sampling

• 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 to maximize F

• Gibbs sampling is a special type of Markov chain
  sampling algorithm
• Our goal is to find the optimal A (a1,aN)
• The Markov chain we construct will only have
  transitions from A to alignments A that differ
  from A in only one of the ai
• In round-robin order, pick one of the ai to
  replace
• Consider all A formed by replacing ai with some
  other starting position ai in sequence Si
• Move to one of these A probabilistically
• Iterate the last three steps

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Algorithm

• Randomly initialize A0
• Repeat
• (1) randomly choose a sequence z from S
• A At \ az compute ?t from A
• (2) sample az according to P(az x), which is
  proportional to Qx/Px update At1 A ? 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
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Algorithm
Current solution At
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Algorithm
Choose one az to replace
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Algorithm
x
For each candidate site x in sequence z,
calculate Qx and Px Probabilities of sampling x
from motif model and background model resp.
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Algorithm
x
Among all possible candidates, choose one (say
x) with probability proportional to Qx/Px
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Algorithm
x
Set At1 A ? x
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Algorithm
x
Repeat
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Local optima

• The algorithm may not find the global or true
  maximum of the scoring function
• Once At contains many similar substrings,
  others matching these will be chosen with higher
  probability
• Algorithm will get locked into a local
  optimum
• all neighbors have poorer scores, hence low
  chance of moving out of this solution

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Phase shifts

• After every M iterations, compare the current At
  with alignments obtained by shifting every
  aligned substring ai by some amount, either to
  left or right

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Phase shift
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Phase shift
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Pattern Width

• The algorithm described so far requires pattern
  width(W) to be input.
• We can modify the algorithm so that it executes
  for a range of plausible widths.
• The function F is not immediately useful for this
  purpose as its optimal value always increases
  with increasing W.

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Pattern Width

• Another function based on the incomplete-data
  log-probability ratio G can be used.
• Dividing G by the number of free parameters
  needed to specify the pattern (19W in the case of
  proteins) produced a statistic useful for
  choosing pattern width. This quantity can be
  called information per parameter.

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Examples

• The algorithm was applied to locate
  helix-turn-helix (HTH) motif, which represent a
  large class of sequence-specific DNA binding
  structures involved in numerous cases of gene
  regulation.
• Detection and alignment of HTH motifs is a well
  recognized problem because of the great sequence
  variation.

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HTH Motif
Complete Sequences
Non-site seq
Random seq
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Convergence behavior of Gibbs Sampling Algorithm
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Time complexity analysis

• For a typical protein sequence, it was found
  that, for a single pattern width, each input
  sequence needs to be sampled fewer than T 100
  times before convergence.
• LW multiplications are performed in Step2 of the
  algorithm.
• Total multiplications to execute the algorithm
  TNLavgW
• Linear Time complexity has been observed in
  applications

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Motif finding

• The Gibbs sampling algorithm was originally
  applied to find motifs in amino acid sequences
• Protein motifs represent common sequence patterns
  in proteins, that are related to certain
  structure and function of the protein
• Gibbs sampling is extensively used to find motifs
  in DNA sequence, i.e., transcription factor
  binding sites

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Thank You