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Project 3: Dynamic Programming
Optimal String Alignment
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Development of a Dynamic
Programming Algorithm
1. Characterize the structure of a solution
2. Recursively define the value of a solution
3. Compute the value of a solution in a bottom-up
fashion(using a table)
4. Construct a solution from computed information
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Example: Optimal String Alignment
Images taken from http://www.sbc.su.se/~per/molbioinfo2001/dynprog/dynamic.html
• Want to find optimal (best) alignment between two strings where
matches count as 1, mismatches count as 0, and “gaps” count as 0.
Example:
G A A T T C A G T T A
G G A T C G A
G _ A A T T C A G T T A
G G _ A _ T C _ G _ _ A
Score == 6
• Notice that every alignment can start in exactly one of three
ways:
(1) Two non-gap characters
(2) Non-gap above, gap below
(3) Gap above, non-gap below
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Example: Optimal String Alignment
• Notice in an optimal alignment of (s1,s2,…,sn) with (t1,t2,…,tm)
one of these must be true:
(1) S2..n must be optimally aligned with T2..m
(2) S2..n must be optimally aligned with T1..m
(3) S1..n must be optimally aligned with T2..m
• The score of each is then
(1) Score(S2..n, T2..m) + Score(S1,T1)
(2) Score(S2..n, T1..m) + Score(S1,gap)
(3) Score(S1..n, T2..m) + Score(T1,gap)
• We want to select the maximum of these, giving
Score(S1..n, T1..m) =
max(Score(S2..n, T2..m) + Score(S1,T1),
Only gives
score of optimal,
Score(S2..n, T1..m) + Score(S1,gap),
not actual alignment
Score(S1..n, T2..m) + Score(gap,T1))
base case given by scoring function
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Example: Optimal String Alignment
• Let’s build a table, M, of values such that M[i][j] is the optimal
score for S1..i aligned with T1..j.
• If we have this, M[n][m] is the overall optimal score
• M is zero-indexed to allow for beginning gap
• Notice by our recurrence relation,
M[i][j] = max(M[i-1][j-1] + Score(Si,Tj),
M[i][j-1] + Score(gap,Tj),
M[i-1][j] + Score(Si,gap))
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Example: Optimal String Alignment
Base Case
M[i][0] = Score(Si,gap)
M[0][j] = Score(gap,Sj)
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Example: Optimal String Alignment
M[i][j] = max(M[i-1][j-1] + Score(Si,Tj),
M[i][j-1] + Score(gap,Tj),
M[i-1][j] + Score(Si,gap))
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Example: Optimal String Alignment
M[i][j] = max(M[i-1][j-1] + Score(Si,Tj),
M[i][j-1] + Score(gap,Tj),
M[i-1][j] + Score(Si,gap))
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Example: Optimal String Alignment
M[i][j] = max(M[i-1][j-1] + Score(Si,Tj),
M[i][j-1] + Score(gap,Tj),
M[i-1][j] + Score(Si,gap))
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Example: Optimal String Alignment
M[i][j] = max(M[i-1][j-1] + Score(Si,Tj),
M[i][j-1] + Score(gap,Tj),
M[i-1][j] + Score(Si,gap))
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Example: Optimal String Alignment
M[i][j] = max(M[i-1][j-1] + Score(Si,Tj),
M[i][j-1] + Score(gap,Tj),
M[i-1][j] + Score(Si,gap))
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Example: Optimal String Alignment
• Still need to find actual alignment that results in maximum score
• Can be done by tracing back from optimal value to find where it
must have originated
representing earlier optimal alignment
G _ A A T T C A G T T A
G G _ A _ T C _ G _ _ A