Bioinformatics Algorithms and Data Structures

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Bioinformatics Algorithms and Data Structures

• Chapter 11 sections4-7
• Lecturer Dr. Rose
• Slides by Dr. Rose
• February 4 6, 2003

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Edit Graphs

• Key idea weighted edit graph
• Defn. Given strings S1 and S2 of lengths n and m,
  respectively, a weighted edit graph has (n1) by
  (m1) nodes, labelled (i,j) , 0 ? i ? n, 0 ? j ?
  m. The edges edge weights are problem specific.

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Edit Graphs

• Example edit distance problem
• The weighted graph for the edit distance problem
  has directed edges from node (i, j) to the nodes
  (i 1, j) , (i, j 1) , and (i 1, j 1),
  provided they exist.
• The weight of the directed edges to nodes (i 1,
  j) , (i, j 1) is 1.
• The weight of the directed edge to (i 1, j
  1) is t(i 1, j 1).
• Figure 11.4 in the textbook shows an edit graph.

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Edit Graphs

• Thm. An edit transcript for strings S1 and S2 has
  the minimum number of edit operations ? it
  corresponds to a shortest path from 0,0 to n,m in
  the edit graph.
• Cor. The set of all shortest paths from 0,0 to
  n,m in the edit graph specifies all optimal edit
  transcript of S1 to S2.

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Weight Edit Distance

• There are two ways of assigning weight or costs
  to calculate edit distance
• By edit operation
• By alphabet, i.e., different costs for different
  characters
• Our initial approach was to assign weight by edit
  operation, i.e., 1 for insert, delete, replace,
  and 0 for match.
• We can generalize our approach by assigning the
  weight d for an insertion or deletion, r for a
  replacement, and e for a match.

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Weight Edit Distance

• QWhat values for d, r, and e have we been
  using?
• A d 1 r 1, and e 0.
• Q What would happen if r gt 2d?
• A Replacements would never occur.
• Defn. The operation-weight distance problem
  entails finding an edit transcript transforming
  S1 to S2 with the minimum total operation weight.

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Weight Edit Distance

• Q What changes should we make to the definition
  of edit distance, D(i,j), to reflect operation
  weight?
• We have to specify an operation-specific
  definition.
• The base conditions become
• D(i,0) i d. Why?
• D(0,j) j d. Why?

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Weight Edit Distance

• The general recurrence becomes
• D(i,j) minD(i,j-1) d, D(i-1,j) d,
  D(i-1,j-1) t(i,j)
• Where t(i,j) e if S1(i) S2 (j) o/w t(i,j)
  r
• Q Why?
• A the cost of
• Delete (from i-1,j) is d
• Insert (from i,j-1) is d
• Match (from i-1,j-1) is e
• Replace (from i-1,j-1) is r

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Weight Edit Distance

• The alternative to operation-weight edit distance
  is alphabet-weight edit distance.
• Idea different characters have different cost.
• Q How would we modify the edit distance
  function, D(i,j), to support alphabet-weight edit
  distance?
• A Let weight(x) denote the weight associated
  with character x for all x in the alphabet.
• Then D(i,0) ?weight(S1(i))
• And D(0,j) ?weight(S2(j))
• Q what about the general recurrence D(i,j)?

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Weight Edit Distance

• A D(i,j) minD(i,j-1) weight(S2(j)),
  D(i-1,j) weight(S1(i)), D(i-1,j-1) t(i,j)
• Where t(i,j) weight(S2(j)), if S1(i) ? S2(j),
  o/w 0.
• Note for proteins, edit distance usually refers
  to alphabet-weight edit distance.
• As the text mentions the weights are usually
  derived from the PAM matrices of Dayhoff or the
  BLOSUM matrices of Henikoff.
• Edit distance for DNA strings is usually either
  unweighted or operation-weighted edit distance.

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String Similarity

• The relatedness of two strings can be expressed
  in terms of similarity.
• This similarity is usually expressed in terms of
  alignment rather than in terms of edit distance.
• Defn. Let S be the alphabet for strings S1 and
  S2. Let S? be S with the additional character -
  denoting space. Let s(x,y) denote the value
  obtained by aligning character x with character y.

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String Similarity

• Defn. The value of alignment A is defined as

Where S1 and S2 denote strings after the
insertion of spaces and their length is denoted
by l.
If s(x,y) is greater than or equal to zero if x
y match and negative if they mismatch, then we
look for the alignment with the largest score
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String Similarity

• Example S a, g, c, t. Let s(x,y) be defined
  by

Q What is the value of the following
alignment? a t a - a c t g t g t a g a c - g t
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String Similarity

• Defn. Given a scoring matrix over S?, define the
  similarity of two strings S1 and S2 as the
  value of the alignment A that maximizes the total
  alignment value of S1 and S2 .
• This also defines the optimal alignment value of
  the strings S1 and S2.

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Computing Similarity

• Q How can we compute the optimal alignment value
  of the strings S1 and S2?
• A Use dynamic programming.
• Defn. Let V(i,j) denote the value of the optimal
  alignment of prefixes S11..i and S21..j.
• If strings S1 and S2 have lengths n and m,
  respectively, then the value of the optimal
  alignment of these strings is given by V(n,m).
• Q What do you guess the time complexity will be?
• A O(n,m)

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Computing Similarity
The optimal alignment value relation is defined
similar to the edit distance relation. Base
Conditions

• Define the general recurrence relation as
• V(i,j) maxV(i - 1, j - 1) s(S1(i), S2(j)),
  V(i - 1, j ) s(S1(i),_), V(i, j - 1) s(_,
  S2(j))

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Computing Similarity

• V(i,j) maxV(i - 1, j - 1) s(S1(i), S2(j)),
  V(i - 1, j ) s(S1(i),_), V(i, j - 1) s(_,
  S2(j))
• Q What does this recurrence relation say?
• A The optimal alignment of the prefixes S11..i
  and S21..j is the maximum of
• The optimal alignment of S11..i-1 and
  S21..j-1 extended by aligning S1(i) and S2(j).
• The optimal alignment of S11..i-1 and S21..j
  extended by aligning S1(i) with a space.
• The optimal alignment of S11..i and S21..j-1
  extended by aligning a space with S2(j).

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Longest Common Subsequence

• Defn. A subsequence of a string S, is a subset of
  characters arranged in their original relative
  order.
• Example
• S interdepartmentaladministratorstaskforce
• subsequence gt idiots
• interdepartmentaladministratorstaskforce
• Obviously every substring of S is also a
  subsequence of S.
• Defn. a common subsequence of two strings is a
  subsequence that appears in both strings.

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Longest Common Subsequence

• Defn. The longest subsequence problem entails
  finding the longest common subsequence (lcs) of
  two strings.
• Thm. The optimal alignment of A forms a longest
  common subsequence, if a scoring scheme is use in
  which each matching pair of characters scores a 1
  and a mismatch or space scores 0.

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Alignment Graphs

• Like distance, similarity can be viewed as a path
  problem the graph that is analogous to the edit
  graph (section 11.4) is called an alignment
  graph.
• Defn. An alignment graph is a DAG similar to an
  edit graph in which the edge weights correspond
  to costs for aligning specific character pairs.
• The optimal alignment corresponds to the longest
  path, in terms of sum of edge costs, from 0,0 to
  n,m of the dynamic programming table.
• The longest paths (optimal alignments) can be
  found in O(nm).

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End-Space Free Alignment

• End-space free alignment an alignment variant
  in which leading and trailing spaces contribute
  zero weight.
• Example
• e x a m p l e - h e c o u l d a - - - h a d a - -
  b e e r
• - - - - - - - - h e w o u l d n t a s h o t h i s
  d e a r
• The first eight spaces are free.
• This encourages (biases towards)
• Alignment of one string inside the other or
• Alignment of the prefix of one string with the
  suffix of the other

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End-Space Free Alignment

• Q When should interior or prefix/suffix matching
  be preferred?
• A When it matches the nature of the problem
  being modeled.
• An example is shotgun sequence assembly Explain!
  
• Start with a large collection of partially
  overlapping substrings that come from multiple
  copies of one original, but unknown string.
• Use comparisons of pairs of substrings to infer
  the original string.

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End-Space Free Alignment

• Q Would you expect substrings that overlap in
  the original string to show significant
  alignment?
• A Perhaps. In any case, with some slop for
  sequencing errors, either
• one string would align inside the other or
• the prefix of one string would align with the
  suffix of the other
• In contrast, a significant alignment of randomly
  selected substrings from this collection is
  unlikely.
• An End-Space Free Alignment would detect this
  difference and score overlapping substrings
  higher.

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End-Space Free Alignment

• We can deduce candidate neighbor pairs by
• Computing End-Space Free Alignment for every pair
  of substrings.
• High scoring alignments are likely neighbors.
• To compute this
• Use a recurrence for global alignment where
  spaces count.
• Change the definition of V(i,0), V(0,j) to
  address leading spaces V(i,0) V(0,j) 0 for
  all i and j.
• Compute the alignment graph in O(mn) How?

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End-Space Free Alignment

• Unlike global alignment the value of optimal
  alignment is not necessarily in cell (n,m).
• The optimal alignment will now be found in
• A cell in row n, if the last character of S1
  contributes to the value of the alignment but the
  last characters of S2 do not.
• A cell in column m, if the last character of S2
  contributes to the value of the alignment but the
  last characters of S1 do not.
• The optimal alignment will be the cell in row n
  or column m that has the largest value.

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• And now for something completely different
• Approximate Matching

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Approximate Matching

• Basic idea Threshold-hold defined similarity
• Defn. A substring T of T is an approximate
  occurrence of P ? the optimal alignment of P to
  T has value at least ?, the threshold parameter.
• Approach
• Use the standard recurrence for global alignment.
• Do not charge preceding spaces V(i,0) V(0,j)
  0 for all i and j.
• Leave backpointers while computing the table

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Approximate Matching

• Q How can we recognize an approximate occurrence
  of P in T from the table computation?
• A If the length of P is n, then for some j,
  V(n,j) ? ?
• More specifically
• Thm. The approximate occurrence of P in T ends at
  position j of T ? V(n,j) ? ?
• This tells us where in T the approximate
  occurrence ends. Where in T does it start?

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Approximate Matching

• Thm.(version 0) The approximate occurrence of P
  in T ends at position j of T ? V(n,j) ? ?
• This tells us where in T the approximate
  occurrence ends. Where in T does it start?
• We can find the start by following the path from
  cell (n,j) back to (0,k). k is the starting
  position in T.
• Thm.(version 1) Tk..j is an approximate
  occurrence of P in T ? V(n,j) ? ? and there is a
  path of backpointers from (n,j) to (0,k).

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Approximate Matching

• The table computation takes O(nm).
• Consider depending on the threshold d, T may
  contain a great many approximate occurrences of
  P.
• Q Can all approximate occurrences be explicitly
  output in O(nm)?
• A Perhaps not.
• Textbook suggest locating all j s.t. V(n,j) ? ?
  and explicitly outputting a shortest approximate
  occurrence.
• Traverse backpointers from (n,j) until reaching
  (0,k)
• Choose vertical pointers over diagonal pointers
• Choose diagonal pointers over horizontal pointers.

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Approximate Matching

• How does this particular preference produce a
  shortest path?
• Choose vertical pointers over diagonal pointers
• Choose diagonal pointers over horizontal
  pointers.
• Recall
• Horzontal edges correspond to inserting space in
  P, this lengthens the path. Clearly this is to be
  avoided.
• Diagonal edges correspond to matches or
  mismatches.
• Vertical edges correspond to inserting space in T
  .
• There is no obvious reason for choosing diagonal
  over vertical edges, however, some preference
  must be made for tie-breaking.
• Except choosing vertical results in match that is
  shortest in T.

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• Global Alignment vs Local Alignment

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Local Alignment

• So far we have focused on global alignment. This
  makes sense if
• We expect one string to be contained in the other
  or
• We expect the strings to be close related.
• Example comparing amino acid sequences from the
  same protein family.

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Local Alignment

• Local alignment exposes regions of high
  similarity.
• This may be interesting even if we expect the
  strings to be globally dissimilar.
• Can you think of examples?
• Comparing proteins from different protein
  families
• How about searching for lateral gene transfer
  from prokaryotic genomes to eukaryotic genomes?
• Huh????

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Local Alignment

• Local alignment problem. Find maximally similar
  (optimal global alignment) substrings a and b of
  S1 and S2, respectively.
• Example from text S1 pqraxabcstvq, S2
  xyaxbacsll
• a a x a b - c s
• b a x - b a c s
• This global alignment is predicated on
• a score of 2 for a match
• a score of 2 for a mismatch
• a score of 1 for a space
• Resulting in a value of 8.

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Computing Local Alignment

• Q How can local alignment be computed?
• Q Can global alignment be used to find local
  alignment?
• A Not efficiently. Global alignment effectively
  averages out local similarity.
• Use explicit search for local similarity.

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Computing Local Alignment

• Q Assuming S1 and S2 have respective lengths n
  and m, how many pairs of substrings are there?
• A There are O(n2m2) pairs of substrings.
• Q If we wanted to, how could we show there are
  this many substrings?

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Computing Local Alignment

• Observation Computing global alignment for each
  of the O(n2m2) pairs of substrings gt O(nm).
• Surprisingly, we can compute local alignment in
  O(nm) even though there are O(n2m2) pairs of
  substrings.
• Assumption the global alignment of two empty
  strings has value zero.

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Computing Local Alignment

• First consider a restricted version of local
  alignment.
• Defn. The local suffix alignment problem entails
  finding a suffix a of S11..i and a suffix b of
  S21..j s.t. V(a,b) is the maximum over all
  pairs of suffixes of S11..i and S21..j.
• Let v(i,j) denote the value of the optimal suffix
  alignment for the index pair i,j.

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Computing Local Alignment

• Local suffix alignment example
• S1 abcxdex, S2 xxcxdeabc, Score 2 for matches
  and 1 for mismatches or spaces
• v(3,4) 1, how?
• The cs match but there is an additional -
  aligned with x.
• v(4,4) 4, how?
• The cs match and the final xs match
• v(5,4) 3, how?
• Same as v(4,4) but extended with d aligned with
  -

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Computing Local Alignment

• Observation v(i,j) ? 0.
• Q Why is this true?
• A We can always choose a and/or b to be the
  empty string.
• Let v denote the value of optimal local
  alignment for strings of length n and m.
• Thm. v maxv(i,j) i ? n, j ? m

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Computing Local Alignment

• We need to understand why this theorem,
  v maxv(i,j) i ? n, j ? m , is true.
• Proof ?
• v ? maxv(i,j) i ? n, j ? m since any local
  optimal suffix alignment is also a local
  alignment.

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Computing Local Alignment

• ?
• WLOG assume v is derived from the optimal
  solution involving substrings a and b with end
  indices i and j, a and b define the local
  suffix alignment for indices i and j, thus v ?
  v(i,j) ? maxv(i,j) i ? n, j ? m
• From this it is clear that a solution to the
  local suffix alignment problem also solves the
  local alignment problem.

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Computing Local Alignment

• Thm. v(i,j) max0, v(i 1, j - 1) s(S1(i),
  S2 (j)), v(i 1, j) s(S1(i), _), v(i, j -
  1) s(_, S2 (j))
• Where v(i, 0) 0 and v(0, j) 0 for all i,j
• Q What does this recurrence say?
• A The solution to the local alignment problem
  v(i, j) is the larger of
• 0, punt and choose a and b to be empty strings
• v(i 1, j - 1) extended by aligning S1(i) and
  S2 (j)
• v(i 1, j) extended by aligning S1(i) with _
• v(i, j - 1) extended by aligning _ with S2 (j)

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Computing Local Alignment

• Q What is the difference between the equations
  for global alignment and local suffix alignment?
• A There are two differences
• The inclusion of 0 in the local local suffix
  alignment
• The base conditions for local suffix alignment
  v(i,0) 0 and v(0,j) 0 for all i,j.This is
  similar for finding approximate occurrences but
  not for general global alignment.

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Computing Local Alignment

• Approach to computing v
• Compute the table for v(i, j).
• Search the entire table for the largest value,
  let (i, j) denote the cell containing the
  largest value.
• Follow backpointers from cell (i, j) to cell
  (i, j) which has the value zero. This gives the
  optimal local alignment.
• The local optimal alignment substrings are then a
  S1(i.. i and b S2(j.. j

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Computing Local Alignment

• Analysis of computing v
• We know that computing the table to solve v
  takes time O(nm).
• The table contains all optimal local alignments
  for v(i, j). An alignment can be found by
  locating a cell with v and tracing back from it.