CMSC423 Bioinformatic Algorithms Databases and Tools Lecture 10 inexact alignment dynamic programming gapped alignment CMSC423 Fall 2008 1 Intuition What is the best way to align strings S1 and S2 just look at last character for now what is it aligned to S1 n S2 m AG C GTAG GTCAG A S1 n S2 m S1 n S2 m CMSC423 Fall 2008 2 The recurrences AG C GTAG GTCAG AScore i j is the maximum of 1 Score i 1 j 1 Value S1 i S2 j AG C G AG C G GTCAG GTCAT 2 Score i 1 j Value S1 i S1 i aligned to gap AG C GT GTCAG3 Score i j 1 Value S2 j S2 j aligned to gap AG C GTCA CMSC423 Fall 2008 3 The dynamic programming table Score i j is the maximum of 1 Score i 1 j 1 Value S1 i S2 j S1 i 1 S2 j 1 aligned 2 Score i 1 j Value S1 i S1 i aligned to gap 3 Score i j 1 Value S2 j S2 j aligned to gap G T C A G A 0 2 4 6 8 10 14 A 2 4 6 8 CMSC423 Fall 2008 G 4 8 6 4 C 6 6 4 16 G 8 T A G 10 12 14 Value A A 10 Value A G 5 Value A 2 Note we only look at 3 adjacent boxes 4 How do you output the result Goal produce the nice string with gaps that is shown in the examples Idea create the string backwards starting from the right As you follow backtrack pointers if you follow diagonal pointer add characters to both output strings aligned versions of original strings if you move up add gap character to string represented on the y axis add string character to string represented on x axis if you move left gap goes in string on x axis and character in string on y axis When you reach 0 0 output the two aligned strings CMSC423 Fall 2008 5 Local vs global alignment Can we change the algorithm to allow S1 to be a substring of S2 ACAGTTGACCCGTGCAT TG CC G Key idea gaps at the end of S2 are free Simply change the first row in the DP table to 0s Answer is no longer Score n m rather the largest value in the last row CMSC423 Fall 2008 6 Sub string alignment C G T 0 2 4 6 A 0 G 0 C 0 10 8 6 G 0 8 20 18 T 0 A 0 18 30 28 G 0 26 AGCGTAG CGT CMSC423 Fall 2008 7 Local alignment What if we just want a region of similarity ACAGTTGACCCGTGCAT GTCATG CC GAGATCG First row and column set to 0s Allow alignment to start anywhere Score i j max 0 case 1 case 2 case 3 Answer is location in matrix with highest score CMSC423 Fall 2008 8 Local alignment C T C G T C 0 0 0 0 0 0 0 A 0 G 0 C 0 G 0 T 0 A 0 G 0 0 10 20 30 AGCGTAG CTCGTC CMSC423 Fall 2008 9 Various flavors of alignment Alignment problem also called edit distance how many changes do you have to make to a string to convert it into another one Edit distance also called Levenshtein distance Local alignment Smith Waterman Global alignment Needleman Wunsch CMSC423 Fall 2008 10 Gap penalties 11 How much do we pay for gaps In the edit distance alignment framework Cost n gaps in a row n Cost gap This doesn t work for e g RNA DNA alignments ACAGTTCGACTAGAGGACCTAGACCACTCTGT TTCGA TAGACCAC Affine gap penalties Cost n gaps in a row Cost gap open n Cost gap Gap opening penalty is high gap extension penalty is low once we start a gap we might as well pile more gaps on top CMSC423 Fall 2008 12 Dynamic programming solution Traditional 1 table approach doesn t work anymore Instead use 4 tables V stores value of best alignment between S1 1 i S2 1 j G best alignment between S1 1 i S2 1 j s t S1 i aligned with S2 j E best alignment between S1 1 i S2 1 j s t alignment ends with gap in S1 F best alignment between S1 1 i S2 1 j s t alignment ends with gap in S2 V i j max E i j F i j G i j As in traditional approach find box in V matrix where V i j is maximal CMSC423 Fall 2008 13 Affine gap recurrences V i j max E i j F i j G i j G i j V i 1 j 1 Value S1 i S2 j irrespective how we got here hence use of V S1 i and S2 j are matched E i j max E i j 1 V i j 1 GapOpen GapExtend either we add a gap in S1 to an existing one E GapExtend or we add a gap in S1 when there was none V GapOpenGapExtend F i j max F i 1 j V i 1 j GapOpen GapExtend either we add a gap in S2 to an existing one F GapExtend or we add a gap in S2 when there was none V GapOpenGapExtend CMSC423 Fall 2008 14 Running times All these algorithms run in O mn quadratic time Note this is significantly worse than exact matching Next we ll talk about speed up opportunities BTW how much space is needed If we only need to find the best score not the exact alignment as well O min m n If we need to find the best alignment elegant divide and conquer algorithm leads to linear space solution CMSC423 Fall 2008 15
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