More Alignments

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More Alignments
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Outline

• We have learned
• The basic DP algorithm for alignment
• The general gap penalty
• The affined gap penalty
• Now let us study several other models of
  alignments
• Local alignment (to compare substrings)
• Fit alignment (to embed one sequence to
  another)
• Linear space alignment (to reduce space
  requirement)
• These models are suitable in different
  circumstances

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Mouse, Human, Chimpanzee
Mouse to Human
Chimpanzee to Human
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Mouse v.s. Human
Chromosome X of Mouse to Human
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Local Alignment

• Typically, when people do alignment, theyre
  actually finding good local alignments.
• Given two sequences S and T
• Find subregions of S and T for which theres
  enough sequence similarity that theyre likely to
  have come from the homologous model.
• (That is, find subregions with greatest score.)
• AATTAG-CCGATGAC
• TGGAGGCTGATATA
• This is done by exactly the same sort of
  procedure as before, except we add only one
  change.

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Warm-up suffix alignment

• Suppose we only get the free gap at the
  prefixes of the alignment.
• AATTAG-CCGAT
• TGGAGGCTGAT
• That is, we choose two suffixes, and align them
  together optimally.
• Let Di,j denote the optimal suffix alignment
  alignment score of s1..i, t1..j.

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Last column

• Consider the last column of this optimal suffix
  alignment. Four cases arise
• Case 1 si v.s. tj
• Case 2 si v.s.
• Case 3 tj v.s.
• Case 4 an empty alignment
• Case 4 is the only new case comparing to the
  basic alignment.

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DP algorithm for suffix alignment

• Di-1, j-1 f(si,
  tj)
• Di,j max Di-1, j indel
• Di, j-1 indel
• 0

How to backtrace?
How to do local alignment?
Answer will be here
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Local Alignment

• The optimal local alignment is maxi,j Di,j
• Suppose the optimal local alignment aligns
  Si..i and Tj..j together, then it is the
  best suffix alignment of S1..i and T1..j.
• Moreover, any local alignment is a suffix
  alignment of two prefixes, and vice versa.
• DP algorithm as before, but backtrack from the
  maximum element in the matrix.
• When extended to affined gap, this is the
  classical Smith-Waterman algorithm.

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A Little History

• The algorithm was first proposed by Temple Smith
  and Michael Waterman in 1981. Cited more than
  3000 times till now (2008).
• The global alignment algorithm was called the
  Needleman-Wunsch algorithm, which was published
  in 1970.

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A quick 1-slide catch-up

• We want to find local alignments.
• We build the matrix that consists of the score of
  the best local alignment (which might be the
  empty one) that ends with si and tj.
• That works by traditional dynamic programming.
• To incorporate more complicated gap penalties, we
  may need extra matrices, as for global alignment.
• After O(nm) time, we have the matrix. Find the
  highest entry, and backtrack until the score is
  zero.
• Thats the optimal local alignment.

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An important side note

• Its very straightforward to not just find one
  local alignment of S and T, but many of them.
• This is crucially important when several
  subregions of S and T are evolutionarily
  conserved.
• One key example, which will come to later in the
  term, is gene finding the coding parts of genes
  are conserved, and the other parts, not so much.
• But for now, think about that O(nm) runtime.

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Is this runtime good enough?

• No!
• Genbank is 1011 letters long.
• To fill in the DP matrix takes 1022 time.
• Thats no good.
• Local alignments must be computed heuristically
  to avoid hideous runtimes
• This is done by filtration, as we will learn in
  the future.

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Fit Alignment

• Given sequence S and T. Find a global alignment
  between S and a substring of T, maximizing the
  alignment score.
• Deleting the prefix of T is free, deleting the
  suffix of T is free.

S
T
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Linear Space Alignment

• Why linear space?
• Computer RAM used to be very expensive in 80s.
• Prediction The cost for 128 kilobytes of memory
  will fall below U100 in the near future.
• Creative Computing magazine. December 1981, page
  6.
• Even today, keeping everything in the L2 cache or
  within one page of the RAM will speed up the
  computation.
• We have learned the linear space if only
  alignment score, instead of the alignment, is
  required.
• We will learn how to get slightly more than the
  alignment score

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Linear Space Alignment

• We want to compute one more piece of information
• the j such that S1..m/2 aligns with T1..j in
  the optimal alignment.
• This is only slightly more than the alignment
  score.
• Can this be done in linear space?
• So what?

m/2
S
T
j
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Compute the j
C A T T G
A
T
T
G
A
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How to find the midpoint?

• Keep two columns of alignment scores.
• Each column depends only on the previous column.
• For columns past the middle of the DP matrix,
  keep track of which position in column n/2 the
  path to a point (i,j) in the DP matrix went
  through.
• This is exactly equivalent to knowing, in the
  optimal alignment of s1si and t1tj, of what
  fraction of T was aligned to the first n/2
  letters of S.
• And it only requires two rows of pointers, as
  well.

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Compute the j
C A T T G
A
T
T
G
A
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Algorithm

• Use the previous idea to compute j such that the
  optimal alignment can be divided into two parts
• S1..m/2 v.s. T1..j
• Sm/21..m v.s. Tj1..n
• Then we use the same algorithm to recursively
  compute the optimal alignments of these two
  parts.
• Return the concatenation of these two optimal
  alignments.

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Time Complexity

• T(m,n) mn T(m/2,j)T(m/2,n-j) 2 mn

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Making it practical

• This algorithm is pain in the neck to implement
  for affined gap penalty.
• But for the linear gap penalty, it is all right.
• And well do this in our assignment.