Class%202:%20Basic%20Sequence%20Alignment

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Class 2 Basic Sequence Alignment
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Sequence Comparison

• Much of bioinformatics involves sequences
• DNA sequences
• RNA sequences
• Protein sequences
• We can think of these sequences as strings of
  letters
• DNA RNA alphabet of 4 letters
• Protein alphabet of 20 letters

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Sequence Comparison (cont)

• Finding similarity between sequences is important
  for many biological questions
• For example
• Find genes/proteins with common origin
• Allows to predict function structure
• Locate common subsequences in genes/proteins
• Identify common motifs
• Locate sequences that might overlap
• Help in sequence assembly

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Sequence Alignment

• Input two sequences over the same alphabet
• Output an alignment of the two sequences
• Example
• GCGCATGGATTGAGCGA
• TGCGCCATTGATGACCA
• A possible alignment
• -GCGC-ATGGATTGAGCGA
• TGCGCCATTGAT-GACC-A

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Alignments

• -GCGC-ATGGATTGAGCGA
• TGCGCCATTGAT-GACC-A
• Three elements
• Perfect matches
• Mismatches
• Insertions deletions (indel)

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Choosing Alignments

• There are many possible alignments
• For example, compare
• -GCGC-ATGGATTGAGCGA
• TGCGCCATTGAT-GACC-A
• to
• ------GCGCATGGATTGAGCGA
• TGCGCC----ATTGATGACCA--
• Which one is better?

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Scoring Alignments

• Rough intuition
• Similar sequences evolved from a common ancestor
• Evolution changed the sequences from this
  ancestral sequence by mutations
• Replacements one letter replaced by another
• Deletion deletion of a letter
• Insertion insertion of a letter
• Scoring of sequence similarity should examine how
  many operations took place

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Simple Scoring Rule

• Score each position independently
• Match 1
• Mismatch -1
• Indel -2
• Score of an alignment is sum of positional scores

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Example

• Example
• -GCGC-ATGGATTGAGCGA
• TGCGCCATTGAT-GACC-A
• Score (1x13) (-1x2) (-2x4) 3
• ------GCGCATGGATTGAGCGA
• TGCGCC----ATTGATGACCA--
• Score (1x5) (-1x6) (-2x11) -23

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More General Scores

• The choice of 1,-1, and -2 scores was quite
  arbitrary
• Depending on the context, some changes are more
  plausible than others
• Exchange of an amino-acid by one with similar
  properties (size, charge, etc.)
• vs.
• Exchange of an amino-acid by one with opposite
  properties

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Additive Scoring Rules

• We define a scoring function by specifying a
  function
• ?(x,y) is the score of replacing x by y
• ?(x,-) is the score of deleting x
• ?(-,x) is the score of inserting x
• The score of an alignment is the sum of position
  scores

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Edit Distance

• The edit distance between two sequences is the
  cost of the cheapest set of edit operations
  needed to transform one sequence into the other
• Computing edit distance between two sequences
  almost equivalent to finding the alignment that
  minimizes the distance

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Computing Edit Distance

• How can we compute the edit distance??
• If s n and t m, there are more than
  alignments
• The additive form of the score allows to perform
  dynamic programming to compute edit distance
  efficiently

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Recursive Argument

• Suppose we have two sequencess1..i1 and
  t1..j1
• The best alignment must be in one of three cases
• 1. Last position is (si1,tj1 )
• 2. Last position is (si1,-)
• 3. Last position is (-, tj 1 )

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Recursive Argument

• Suppose we have two sequencess1..i1 and
  t1..j1
• The best alignment must be in one of three cases
• 1. Last position is (si1,tj 1 )
• 2. Last position is (si 1,-)
• 3. Last position is (-, tj 1 )

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Recursive Argument

• Suppose we have two sequencess1..i1 and
  t1..j1
• The best alignment must be in one of three cases
• 1. Last position is (si1,tj 1 )
• 2. Last position is (si 1,-)
• 3. Last position is (-, tj 1 )

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Recursive Argument

• Define the notation
• Using the recursive argument, we get the
  following recurrence for V

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Recursive Argument

• Of course, we also need to handle the base cases
  in the recursion

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Dynamic Programming Algorithm
We fill the matrix using the recurrence rule
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Dynamic Programming Algorithm
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Reconstructing the Best Alignment

• To reconstruct the best alignment, we record
  which case in the recursive rule maximized the
  score

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Reconstructing the Best Alignment

• We now trace back the path the corresponds to the
  best alignment

AAAC AG-C
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Reconstructing the Best Alignment

• Sometimes, more than one alignment has the best
  score

AAAC A-GC
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Complexity

• Space O(mn)
• Time O(mn)
• Filling the matrix O(mn)
• Backtrace O(mn)

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Space Complexity

• In real-life applications, n and m can be very
  large
• The space requirements of O(mn) can be too
  demanding
• If m n 1000 we need 1MB space
• If m n 10000, we need 100MB space
• We can afford to perform extra computation to
  save space
• Looping over million operations takes less than
  seconds on modern workstations
• Can we trade off space with time?

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Why Do We Need So Much Space?

• To find d(s,t), need O(n) space
• Need to compute Vi,m
• Can fill in V, column by column, storing only
  two columns in memory

• Note however
• This trick fails when we need to reconstruct
  the sequence
• Trace back information eats up all the memory

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Space Efficient Version Outline

• Idea perform divide and conquer
• Find position (n/2, j) at which the best
  alignment crosses s midpoint

s
t
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Finding the Midpoint

• Suppose s1,n and t1,m are given
• We can write the score of the best alignment that
  goes through j as
• d(s1,n/2,t1,j) d(sn/21,n,tj1,m)
• Thus, we need to compute these two quantities for
  all values of j

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Finding the Midpoint (cont)

• Define
• Fi,j d(s1,i,t1,j)
• Bi,j d(si1,n,tj1,m)
• Fi,j Bi,j score of best alignment through
  (i,j)
• We compute Fi,j as we did before
• We compute Bi,j in exactly the same manner,
  going backward from Bn,m

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Time Complexity Analysis

• Finding mid-point cmn (c - a constant)
• Recursive sub-problems of sizes (n/2,j) and
  (n/2,m-j-1)
• T(m,n) cmn T(j,n/2) T(m-j-1, n/2)
• Lemma T(m,n) ? 2cmn
• Time complexity is linear in size of the problem
• At worse, twice the time of regular solution.

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Local Alignment

• Consider now a different question
• Can we find similar substring of s and t
• Formally, given s1..n and t1..m find i,j,k,
  and l such that d(si..j,tk..l) is maximal

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Local Alignment

• As before, we use dynamic programming
• We now want to setVi,j to record the best
  alignment of a suffix of s1..i and a suffix of
  t1..j
• How should we change the recurrence rule?

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Local Alignment

• New option
• We can start a new match instead of extend
  previous alignment

Alignment of empty suffixes
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Local Alignment Example
s TAATA t TACTAA
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Local Alignment Example
s TAATA t TACTAA
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Local Alignment Example
s TAATA t TACTAA
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Local Alignment Example
s TAATA t TACTAA
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Sequence Alignment

• We saw two variants of sequence alignment
• Global alignment
• Local alignment
• Other variants
• Finding best overlap (exercise)
• All are based on the same basic idea of dynamic
  programming