Statistical Alignment: Computational Properties, Homology Testing and Goodness-of-Fit

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Statistical Alignment Computational Properties,
Homology Testing and Goodness-of-Fit

• J. Hein, C. Wiuf, B. Knudsen, M.B. Moller and G.
  Wibling

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Main Objective of the paper

• To show how to accelerate the statistical
  alignment algorithms several orders of magnitude
  using the model of insertion and deletions by
  Thorne, Kishino, and Felsenstein in 1991 (TKF91
  model).
• To propose a new homology test based on the
  model.
• To describe a goodness-of-fit test that allows
  testing the proposed insertion-deletion process
  inherent to the model.

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Why isnt statistical alignment popular?

• Computationally VERY SLOW
• Authors of the paper accelerated the statistical
  alignment algorithms several orders of magnitude
  compared with the TKF91 algorithm.
• Lack of user-friendly software?
• Usually written in Fortran or C, or the compiled
  program only works in UNIX environment, but most
  biologists dont know much about it.
• Authors of the paper have provided a web
  interface to the program

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parsimony and similarity alignments

• Parsimony strategy minimizing the distance
• For example

• Similarity strategy maximizing the similarity
  score
• For example BLAST

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TKF91 model of substitutions

• continuous time Markov model on the state space
  of nucleotides or amino acids
• Rate matrix Q is specified
• Describes the intensity of different substitution
  events over an infinitesimal time period.
• Probability that i has changed to j after time t
  is

• The process is assumed to be time reversible

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TKF91 model of the indel process

• Can be view as a Markov model with all sequences
  as possible states
• indel part of the model
• links connecting the letters of the sequences
• each has a mortal link on the right
• left end has an immortal link
• For example ? A ? G ? G ?
• If the type of the nucleotide is ignored, can be
  represented as ? ? ? ?

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TKF91 model

• mortal link can give birth to a new mortal link
  or die out
• immortal link can also give birth but would not
  die
• Therefore, the rates can be written as
• ? A ? G ? G ?
• I0 S1 I1/D1 S2 I2/D2 S3 I3/D3
• where I is the birth rate
• D is the death rate, DgtI
• S is the substitution rate

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TKF91 model

• To calculate the probability of a particular
  alignment
• s(1) ? A ? T ? -
• s(2) ? C ? T ? G ?
• P(s(1), s(2), alignment) (p1)(?A?P1 ?PAC)(?T
  ?P2 ? PTT ??G)

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Calculating the probability of two sequences

• Without conditioning on the alignment, it is
  necessary to sum over all alignments weighted
  with their probabilities according to the TKF91
  process.
• Confine likelihood calculations to a band close
  to the similarity based alignment allows an
  efficient numerical optimization algorithm for
  finding the maximum likelihood estimate
• The recursions originally presented by Thorne,
  Kishino and Felsenstein can be simplified.