Finding the optimal pairwise alignment

1
Finding the optimal pairwise alignment

• We are interested in finding the alignment of two
  sequences that maximizes the similarity score
  given an arbitrary cost matrix. This is also
  called the sum-of-pairs optimization.
• Given two sequences of length m and n we can use
  the Smith Waterman algorithm to find the optimal
  alignment in O(mn) time and space (using a
  dynamic programming algorithm).

2
Expected accuracy of alignment

• The dynamic programming formulation allows us to
  find the optimal alignment defined by a scoring
  matrix and gap penalties.
• We now look at a different formulation of
  alignment that allows us to compute the most
  accurate one instead of the optimal one.

3
Posterior probability of xi aligned to yj

• Let A be the set of all alignments of sequences x
  and y, and define P(ax,y) to be the probability
  that alignment a (of x and y) is the true
  alignment a.
• We define the posterior probability of the ith
  residue of x (xi) aligning to the jth residue of
  y (yj) in the true alignment (a) of x and y as

Do et. al., Genome Research, 2005
4
Expected accuracy of alignment

• We can define the expected accuracy of an
  alignment a as
• The maximum expected accuracy alignment can be
  obtained by the same dynamic programming
  algorithm

Do et. al., Genome Research, 2005
5
Example for expected accuracy

• True alignment
• AC_CG
• ACCCA
• Expected accuracy(11011)/41
• Estimated alignment
• ACC_G
• ACCCA
• Expected accuracy(110.101) 0.75

6
Estimating posterior probabilities

• If correct posterior probabilities can be
  computed then we can compute the correct
  alignment. Now it remains to estimate these
  probabilities from the data
• Probcons estimate probabilities from pairwise
  HMM using forward and backward recursions
• Probalign use partition function posterior
  probabilities

7
Estimating posterior probabilities

• We are interested in estimating posterior
  probabilities for two sequences x and y.
• By generating an ensemble A(n,x,y) of n
  alignments of x and y we can estimate P(xiyj) by
  counting the number of times xi is aligned to
  yj.. Note that this means we are assigning equal
  weights to all alignments in the ensemble.

8
Generating ensemble of alignments

• We use stochastic backtracking to generate a
  given number of optimal and suboptimal
  alignments.
• At every step in the traceback we assign a
  probability to each of the three possible
  positions.