Dynamic Programming for Sequence alignment

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Dynamic Programming for Sequence alignment

• Neha Jain
• Lecturer
• School of Biotechnology
• Devi Ahilya University, Indore

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Sequence alignment

• Sequence alignment is the procedure of comparing
  two (pair-wise alignment) or more multiple
  sequences by searching for a series of individual
  characters or patterns that are in the same order
  in the sequences.
• There are two types of alignment local and
  global.
• In Global alignment, an attempt is made to align
  the entire sequence. If two sequences have
  approximately the same length and are quite
  similar, they are suitable for the global
  alignment.
• Local alignment concentrates on finding
  stretches of sequences with high level of matches.

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Interpretation of sequence alignment

• Sequence alignment is useful for discovering
  structural, functional and evolutionary
  information.
• Sequences that are very much alike may have
  similar secondary and 3D structure, similar
  function and likely a common ancestral sequence.
  It is extremely unlikely that such sequences
  obtained similarity by chance
• Large scale genome studies revealed existence of
  horizontal transfer of genes and other sequences
  between species, which may cause similarity
  between some sequences in very distant species.

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Methods of sequence alignment

• Dot matrix analysis- Starting from the first
  character in second sequence, one moves across
  the page keeping in the first row and placing a
  dot in many column where the character in A is
  the same. The process is continued until all
  possible comparisons between both the sequences
  are made. Any region of similarity is revealed by
  a diagonal row of dots
• The dynamic programming (DP) algorithm- The
  method compares every pair of characters in the
  two sequences and generates an alignment, which
  is the best or optimal.
• Word or k-tuple methods BLAST is the best
  example to deal with k-tuple.

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Pairwise Sequence Alignment

• The Aim given two sequences and scoring system
  find the best alignment
• Points to remember
• 1) Should consider all possible Pairs
• 2) Take the best score found
• 3) There may be more than one best alignment

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Finding the best alignment is hard!!

• How to get optimal alignment?
• The number of possible alignments is large.
• If both sequences have the same length there is
  one possible for complete alignment with no gap.
• More complicated when gaps are allowed
• It is not good idea to go over all alignments
• Solution Dynamic Programming Algorithm

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Dynamic Programming

• General optimization method
• Proposed by Richard Bellman of Princeton
  University in 1950s. The word dynamic was chosen
  by Bellman to capture the time-varying aspect of
  the problems, and because it sounded impressive.
   The word programming referred to the use of the
  method to find an optimal program
• Extensively used in sequence alignment and other
  computational problems
• Applied to biological sequences by Needleman and
  Wunsch

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Dynamic Programming

• Original problem is broken into smaller sub
  problems and then solved
• Pieces of larger problem have a sequential
  dependency
• 4th piece can be solved using solution of the 3rd
  piece, the 3rd piece can be solved by using
  solution of the 2nd piece and so on

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Dynamic Programming

• First solve all the subproblems
• Store each intermediate solution in a table along
  with a score
• Uses an m x n matrix of scores where m and n
  are the lengths of sequences being aligned.
• Can be used for
• Local Alignment (Smith-Waterman Algorithm)
• Global Alignment (Needleman-Wunsch Algorithm)

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Formal description of dynamic programming
algorithm

• This diagram indicates the moves that are
  possible to reach a certain position (i,j)
  starting from the previous row and column at
  position (i -1, j-1) or from any position in the
  same row or column
• Diagonal move with no gap penalties or move from
  any other position from column j or row i, with a
  gap penalty that depends on the size of the gap

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Dynamic Programming

• Sequence alignment has an optimal-substructure
  property
• As a result DP makes it easier to consider all
  possible alignments
• DP algorithms solve optimization problems by
  dividing the problem into independent
  subproblems.
• Each subproblem is then only solved once, and the
  answer is stored in a table, thus avoiding the
  work of recomputing the solution.

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Dynamic Programming

• With sequence alignment, the subproblems can be
  thought of as the alignment of the prefixes of
  the two sequences to a certain point.
• DP matrix is computed.
• The optimal alignment score for any particular
  point in the matrix is built upon the optimal
  alignment that has been computed to that point.

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Dynamic Programming

• Advantage The method is guaranteed to give a
  global optimum given the choice of parameters
  the scoring matrix and gap penalty with no
  approximation
• A disadvantage Many alignment may give the same
  optimal score. And none of these correspond to
  the biologically correct alignment

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Dynamic Programming

• Comparison of a- ß- chains of chicken
  hemoglobin, Fitch Smith found 17 optimal
  alignments, only one of which was correct
  biologically (1317 alignments were 5 of
  optimal score)
• Another bad news The time required to align two
  sequences of length n m is proportional to n
  x m.
• This makes DP unsuitable for use in searching a
  sequence DB for a match to a probe sequence

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Dynamic Programming

• Steps Involved
• Initialization
• Matrix Fill (scoring)
• Traceback (alignment)

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Gap Penalties..????

• Gaps are due to Insertion or deletion mutations
  in the genes.
• Penalties are given for the gaps.
• Through empirical studies for globular proteins,
  a set of penalty values have been developed that
  appear to suit most alignment purposes.
• They are normally implemented as default values
  in most alignment programs.

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Gap Penalties..????

• Caution-
• Penalty too low- gaps numerous, even non related
  pairs will be aligned.
• If penalties too high- difficult to pair even
  the related ones.
• Another factor to consider is the cost difference
  between opening a gap and extending an existing
  gap. It is known that it is easier to extend a
  gap that has already been started. Thus, gap
  opening should have a much higher penalty than
  gap extension.
• This is based on the rationale that if insertions
  and deletions ever occur, several adjacent
  residues are likely to have been inserted or
  deleted together.
• Affine Gap Penalties- Gap opening penalty should
  always be lower then gap extension penalty..
• Constant Penalty- When gap opening and gap
  extension penalties are same

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Global Alignment Needleman-Wunsch Algorithm

• In global sequence alignment, an attempt to align
  the entirety of two different sequences is made,
  up to and including the ends of the sequence.
• Needleman and Wunsch (1970) were among the first
  to describe a dynamic programming algorithm for
  global sequence alignment.

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Example

• Two sequences TACT, AATC
• Scoring system
• Match 3
• Mismatch -1
• Gap -2

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• Initializing entry (0,0) 0
• Fill the matrix from top left to bottom right
• The score in each entry (i,j) is calculated using
  the three near entries values
• Global alignment score is the bottom right cell
  value
• May find more than one alignment

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4 T 3 C 2 A 1 T 0 -
0 -
1 A
2 A
3 T
4 C
Construct a matrix one sequence (TACT) at the
top another sequence (AATC) at the left

• Entry (i,j)
• i for column, j for row
• alignment of i first letters of one sequence
• with j first letters of another

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4 T 3 C 2 A 1 T 0 -
0 0 -
1 A
2 A
3 T
4 C
Initialization entry (0,0) 0
Fill the matrix from top left to bottom right
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4 T 3 C 2 A 1 T 0 -
-2 0 0 -
1 A
2 A
3 T
4 C
entry (1,0) entry(0,0) gap score 0 (-2)
-2
T -
Horizontal line gap in the left sequence
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4 T 3 C 2 A 1 T 0 -
-4 -2 0 0 -
1 A
2 A
3 T
4 C
TA - -
entry (2,0) entry(1,0) gap score -2
(-2) -4
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4 T 3 C 2 A 1 T 0 -
-6 -4 -2 0 0 -
1 A
2 A
3 T
4 C
TAC - - -
entry (3,0) entry(2,0) gap score -4
(-2) -6
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4 T 3 C 2 A 1 T 0 -
-8 -6 -4 -2 0 0 -
1 A
2 A
3 T
4 C
TACT - - - -
entry (4,0) entry(3,0) gap score -6
(-2) -8
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4 T 3 C 2 A 1 T 0 -
-8 -6 -4 -2 0 0 -
-2 1 A
-4 2 A
-6 3 T
-8 4 C
- - - - AATC
Vertical line gap in the top sequence
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Global Alignment Needleman-Wunsch Algorithm
For each position, Si,j is defined to be the
maximum score at position i,j i.e. Si,j
MAXIMUM Si-1, j-1 s(ai,bj)
(match/mismatch in the diagonal), Si,j-1 w
(gap in sequence 1), Si-1,j w (gap in
sequence 2)
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4 T 3 C 2 A 1 T 0 -
-8 -6 -4 -2 0 0 -
? -2 1 A
-4 2 A
-6 3 T
-8 4 C
Three options
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4 T 3 C 2 A 1 T 0 -
-8 -6 -4 -2 0 0 -
-1 -2 1 A
-4 2 A
-6 3 T
-8 4 C
First option Entry(0,0) mismatch score
0(-1) -1
T A
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4 T 3 C 2 A 1 T 0 -
-8 -6 -4 -2 0 0 -
-4 -2 1 A
-4 2 A
-6 3 T
-8 4 C
Second option Entry(1,0) gap score -2(-2)
-4
T - - A
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4 T 3 C 2 A 1 T 0 -
-8 -6 -4 -2 0 0 -
-4 -2 1 A
-4 2 A
-6 3 T
-8 4 C
Third option Entry(0,1) gap score -2(-2)
-4
- T A -
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4 T 3 C 2 A 1 T 0 -
-8 -6 -4 -2 0 0 -
-1 -2 1 A
-4 2 A
-6 3 T
-8 4 C
Choosing the option with the maximal score
T A
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4 T 3 C 2 A 1 T 0 -
-8 -6 -4 -2 0 0 -
? -1 -2 1 A
-4 2 A
-6 3 T
-8 4 C
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4 T 3 C 2 A 1 T 0 -
-8 -6 -4 -2 0 0 -
? -1 -2 1 A
-4 2 A
-6 3 T
-8 4 C
First option Entry(1,0) match score -2(3)
1
TA -A
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4 T 3 C 2 A 1 T 0 -
-8 -6 -4 -2 0 0 -
? -1 -2 1 A
-4 2 A
-6 3 T
-8 4 C
Second option Entry(2,0) gap score -4(-2)
-6
TA - - - A
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4 T 3 C 2 A 1 T 0 -
-8 -6 -4 -2 0 0 -
? -1 -2 1 A
-4 2 A
-6 3 T
-8 4 C
Third option Entry(1,1) gap score -1(-2)
-3
TA A -
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4 T 3 C 2 A 1 T 0 -
-8 -6 -4 -2 0 0 -
1 -1 -2 1 A
-4 2 A
-6 3 T
-8 4 C
Choosing the option with the maximal score
T A - A
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4 T 3 C 2 A 1 T 0 -
-8 -6 -4 -2 0 0 -
-3 -1 1 -1 -2 1 A
-4 2 A
-6 3 T
-8 4 C
TACT - A - -
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4 T 3 C 2 A 1 T 0 -
-8 -6 -4 -2 0 0 -
-3 -1 1 -1 -2 1 A
-3 -4 2 A
-6 3 T
-8 4 C
T - AA
- T AA
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4 T 3 C 2 A 1 T 0 -
-8 -6 -4 -2 0 0 -
-3 -1 1 -1 -2 1 A
0 2 -3 -4 2 A
3 T
4 C
TAC -AA
TACAA -
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4 T 3 C 2 A 1 T 0 -
-8 -6 -4 -2 0 0 -
-3 -1 1 -1 -2 1 A
-2 0 2 -3 -4 2 A
3 1 0 -1 -6 3 T
1 3 -2 -3 -8 4 C
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4 T 3 C 2 A 1 T 0 -
-8 -6 -4 -2 0 0 -
-3 -1 1 -1 -2 1 A
-2 0 2 -3 -4 2 A
3 1 0 -1 -6 3 T
1 3 -2 -3 -8 4 C
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4 T 3 C 2 A 1 T 0 -
-2 0 0 -
1 -1 1 A
0 2 2 A
3 0 3 T
1 3 4 C
Three possible of alignments
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4 T 3 C 2 A 1 T 0 -
-2 0 0 -
1 1 A
0 2 A
3 3 T
1 4 C
T A C T - A A T C
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4 T 3 C 2 A 1 T 0 -
0 0 -
-1 1 A
2 2 A
0 3 T
1 3 4 C
T A - C T A A T C -
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4 T 3 C 2 A 1 T 0 -
0 0 -
-1 1 A
0 2 2 A
3 3 T
1 4 C
T A C T A A - T C
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Local Alignment Algorithm

• Algorithm of Smith Waterman (1981)
• Makes an optimal alignment of the best segment of
  similarity between two sequences
• Sequences that are not highly similar as a whole,
  but contain regions that are highly similar
• Use when one sequence is short and the other is
  very long (e.g. database)
• Can return a number of highly aligned segments

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Does a Local Alignment program always produce a
Local Alignment and a Global Alignment program
always produces a Global Alignment?

• Although a Computer program that is based on the
  Smith waterman local alignment algorithm is used
  for producing an optimal alignment, this does not
  assure that a local alignment will be produced.
• The scoring matrix or match/mismatch scores and
  gap penalties chosen also influence whether or
  not a local alignment is obtained.
• Similar is the case with Needleman-Wunsch
  algorithm.

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• IF the matched regions are long and cover most of
  the sequences and depends on the presence of many
  gaps, the alignment is global.
• A local alignment will tends to be shorter and
  not include many gaps.

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Tools based on Dynamic programming

• Global Alignment-
• GAP- No penalties for terminal gaps, thus suits
  for unequal length sequences.
• Local Alignment-
• SIM, SSEARCH and LALIGN

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Multiple Sequence Alignment

• It is theoretically possible to use dynamic
  programming to align any number of sequences for
  the pair wise alignment
• The amount of computing time increases
  exponentially as the number of sequences
  increases
• Therefore full dynamic programming cannot be
  applied for datasets having more then ten
  sequences
• So heuristic method is used for MSA.

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Thank you