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Pairwise Sequence Alignment (I) (Lecture for CS498-CXZ Algorithms in Bioinformatics) Sept. 22, 2005 ChengXiang Zhai Department of Computer Science University of Illinois, Urbana-Champaign Many slides are taken/adapted from http://www.bioalgorithms.info/slides.htmhttp://www.bioalgorithms.info/slides.htm

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Comparing Genes in Two Genomes Small islands of similarity corresponding to similarities between exons Such comparisons are quite common in biology research

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Alignment of sequences is one of the most basic and most important problems in bioinformatics…

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Outline Defining the problem of alignment The longest common subsequence problem Dynamic programming algorithms for alignment

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Aligning Two Strings Given the strings: v = ATGTTAT w = ATCGTAC One possible alignment of the strings: AT_GTTAT_ ATCGT_A_C 1 st row – string v with with space symbols “-” inserted 2 nd row – string w with with space symbols “-” inserted

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Aligning Two Strings (cont’d) Another way to represent each row shows the number of symbols of the sequence present up to a given position. For example the above sequences can be represented as: 0 1 2 2 3 4 5 6 7 7 0 1 2 3 4 5 5 6 6 7 AT_GTTAT_ ATCGT_A_C

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Alignment Matrix Both rows of the alignment can be represented in the resulting matrix: 0 1 2 2 3 4 5 6 7 7 0 1 2 3 4 5 5 6 6 7 AT_GTTAT_ ATCGT_A_C 01223456770122345677 01234556670123455667

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Alignment as a Path in the Edit Graph 0 1 2 2 3 4 5 6 7 7 A T _ G T T A T _ A T _ G T T A T _ A T C G T _ A _ C A T C G T _ A _ C 0 1 2 3 4 5 5 6 6 7 (0,0), (1,1)

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Alignment as a Path in the Edit Graph 0 1 2 2 3 4 5 6 7 7 A T _ G T T A T _ A T _ G T T A T _ A T C G T _ A _ C A T C G T _ A _ C 0 1 2 3 4 5 5 6 6 7 (0,0), (1,1), (2,2)

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Alignment as a Path in the Edit Graph 0 1 2 2 3 4 5 6 7 7 A T _ G T T A T _ A T _ G T T A T _ A T C G T _ A _ C A T C G T _ A _ C 0 1 2 3 4 5 5 6 6 7 (0,0), (1,1), (2,2), (2,3), (3,4)

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Alignment as a Path in the Edit Graph 0 1 2 2 3 4 5 6 7 7 A T _ G T T A T _ A T _ G T T A T _ A T C G T _ A _ C A T C G T _ A _ C 0 1 2 3 4 5 5 6 6 7 (0,0), (1,1), (2,2), (2,3), (3,4), (4,5), (5,5), (6,6), (7,6), (7,7) - End Result -

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Alignment as a Path in the Edit Graph Every path in the edit graph corresponds to an alignment:

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How to Score an Alignment? Simplest –Every match scores 1 –Every mismatch scores 0 –An alignment is scored based on the number of common symbols –Lead to the longest common subsequence problem More sophisticated –? –To be covered later

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Alignments in Edit Graph (cont’d) and represent indels in v and w Score 0. represent exact matches. Score 1.

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Alignments in Edit Graph (cont’d) The score of the alignment path in the graph is 5.

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The Longest Common Subsequence (LCS) Problem Find the longest subsequence common to two strings. Input: Two strings, v and w. Output: The longest common subsequence of v and w. A subsequence is not necessarily consecutive v = ATGTTAT w = ATCGTAC v = AT GTTAT | | | | | “ATGTA” w = ATCGT AC Longest common subsequence Best alignment

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How to solve the LCS problem efficiently?

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Brute Force Approach Enumerate all the sequences up to length min(|v|,|w|) For each one, check to see if it is a subsequence of v and w Very expensive…. (How many sequences do we have to enumerate? )

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The Idea of Dynamic Programming Think of an alignment as a path in an edit graph We only need to keep track of the best alignment (i.e., the longest common subsequence) Score a longer alignment based on shorter alignments

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Alignment as a Path in the Edit Graph 0122345677 0122345677 v= AT_GTTAT_ w= ATCGT_A_C 0123455667 0123455667 (0,0), (1,1), (2,2), (2,3), (3,4), (4,5), (5,5), (6,6), (7,6), (7,7) Use each cell to store the best alignment so far…

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Alignment: Dynamic Programming Use this scoring algorithm s i,j = s i-1, j-1 +1 if v i = w j max s i-1, j s i, j-1

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Dynamic Programming Example There are no matches in the beginning of the sequence Label column i=1 to be all zero, and row j=1 to be all zero

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Dynamic Programming Example S i,j = S i-1, j-1 max S i-1, j S i, j-1 value from NW +1, if v i = w j value from North (top) value from West (left) Keep track of the best alignment score and the path contributing to it

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Alignment: Backtracking Arrows show where the score originated from. if from the top if from the left if v i = w j

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Dynamic Programming Example Continuing with the scoring algorithm gives this result.

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LCS Algorithm 1. LCS(v,w) 2. for i 1 to n 3. S i,0 0 4. for j 1 to m 5. S 0,j 0 6. for i 1 to n 7. for j 1 to m 8. s i-1,j 9. s i,j max s i,j-1 10. s i-1,j-1 + 1, if v i = w j 11. “ “ if s i,j = s i-1,j b i,j “ “ if s i,j = s i,j-1 “ “ if s i,j = s i-1,j-1 + 1 return (s n,m, b)

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Now What? LCS(v,w) created the alignment grid Now we need a way to read the best alignment of v and w Follow the arrows backwards from the (|v|,|w|) cell

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LCS Runtime To create the nxm matrix of best scores from vertex (0,0) to all other vertices, it takes O(nm) time. Why O(nm)? The pseudocode consists of a nested “for” loop inside of another “for” loop to set up a nxm matrix.

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How do we improve the scoring of alignments? Can we still find an alignment efficiently? We’ll talk about these later…

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The LCS Recurrence Revisited The formula can be rewritten by adding zero to the edges that come from an indel, since the penalty of indels are 0: s i-1, j-1 +1 if v i = w j s i,j = max s i-1, j + 0 s i, j-1 + 0 Insertion/deletion score Matching score

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What You Should Know How an alignment corresponds to a path in an edit graph How the LCS problem corresponds to alignment with a simple scoring method How the dynamic programming algorithm solves the LCS problem (= simple alignment)

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