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# String Matching

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1. String Matching String matching: definition of the problem (text,pattern) depends on what we have: text or patterns • Exact matching: • The patterns ---> Data structures for the patterns • 1 pattern ---> The algorithm depends on |p| and || • k patterns ---> The algorithm depends on k, |p| and || • Extensions • Regular Expressions • The text ----> Data structure for the text (suffix tree, ...) • Approximate matching: • Dynamic programming • Sequence alignment (pairwise and multiple) • Sequence assembly: hash algorithm • Probabilistic search: Hidden Markov Models

2. 2.2 Pairwise alignment Given two DNA sequences A (a1a2...an) and B (b1b2...bm) from the alphabet {a,c,t,g} we say that A* and B* from {a,c,t,g,-} are aligned iff • A* and B* become A and B if gaps ( – ) are removed. • |A*|=|B*| • For all i, it is not possible that ai = bi = - MALIG (an example) How many alignments of two sequences exist? Which is the best alignment?

3. 2.2 Number of alignments b1 b2 b3 a1 a2 a3 Given two DNA sequences A (a1a2...an) and B (b1b2...bm) there are: #(a1a2...an ,b1b2...bm) = #(a1a2...an-1 ,b1b2...bm) those that end with (an,-) + #(a1a2...an ,b1b2...bm-1) those that end with (-,bm) + #(a1a2...an-1 ,b1b2...bm-1) those that end with (an,bm) #(a1,b1)

4. 2.2 Number of alignments b1 b2 b3 a1 a2 a3 Given two DNA sequences A (a1a2...an) and B (b1b2...bm) there are: #(a1a2...an ,b1b2...bm) = #(a1a2...an-1 ,b1b2...bm) those that end with (an,-) + #(a1a2...an ,b1b2...bm-1) those that end with (-,bm) + #(a1a2...an-1 ,b1b2...bm-1) those that end with (an,bm) 1 1 1 1 1 1 1

5. 2.2 Number of alignments b1 b2 b3 a1 a2 a3 Given two DNA sequences A (a1a2...an) and B (b1b2...bm) there are: #(a1a2...an ,b1b2...bm) = #(a1a2...an-1 ,b1b2...bm) those that end with (an,-) + #(a1a2...an ,b1b2...bm-1) those that end with (-,bm) + #(a1a2...an-1 ,b1b2...bm-1) those that end with (an,bm) 1 1 1 1 1 1 1 3 ? ?

6. 2.2 Number of alignments b1 b2 b3 a1 a2 a3 Given two DNA sequences A (a1a2...an) and B (b1b2...bm) there are: #(a1a2...an ,b1b2...bm) = #(a1a2...an-1 ,b1b2...bm) those that end with (an,-) + #(a1a2...an ,b1b2...bm-1) those that end with (-,bm) + #(a1a2...an-1 ,b1b2...bm-1) those that end with (an,bm) 1 1 1 1 1 1 1 3 5 7 5 7 ?

7. 2.2 Number of alignments b1 b2 b3 a1 a2 a3 Given two DNA sequences A (a1a2...an) and B (b1b2...bm) then: #(a1a2...an ,b1b2...bm) = #(a1a2...an-1 ,b1b2...bm) those that end with ( an , -) + #(a1a2...an ,b1b2...bm-1) those that end with ( - , bm) + #(a1a2...an-1 ,b1b2...bm-1) those that end with ( an , bm) 1 1 1 1 1 1 1 3 5 7 • 25 • 25 63 5 7 But, what is the assymptotic value?

8. 2.2 Assymptotic value K=n = ( ) > Σ ( ) ( ) 2n n n k=0 n k k As #(a1a2...an ,b1b2...bn) and n! ~ nn e-n (Stirling approximation) then #(a1a2...an ,b1b2...bn) > 22n

9. 2.2 Best alignment • Match: favorable • Mismatch: unfavorable • Gap: worst case How can an alignment be scored? catcactactgacgactatcgtagcgcggctatacatctacgccaa- ctac-t-gtgtagatcgccgg c- tgactgc--acgactatcgt- attgcggctacacactacgcacaactactgtatgtcgc-cgg---- * * *** * * ** * ******* * * **** **** ******* * **** ** * *** Then we assign a score for each case, for example 1,-1,-2. How can the best alignment be found?

10. 2.2 Best alignment C T A C T A C T A C G T A C T G A The cell contains the score of the best alignment of AC and CTACT.

11. Best alignment accaccacaccacaacgagcata… acctgagcgatat a c c . . t Given the maximum score, how can the best alignment be found? • Quadratic cost in space and time • Up to 10,000 bps sequences in length Download alggen tool

12. 2.2 Some slides revisited We have developed the theory according to the following principles: 1) Both sequences have a similar length (global). 2) The model of gaps is linear If there are k consecutive gaps the penalty scores k(-2).

13. 2.2 Semiglobal pairwise alignment Assume that we have sequences with different length S1 S2 It is meaningless to introduce gaps until both sequences have similar length …. The most probable alignment should be Initial gaps Final gaps How can these alignments be found?

14. 2.2 Semiglobal pairwise alignment Note that Initial gaps Final gaps C T A C T A C T A C G T A C T

15. 2.2 Semiglobal pairwise alignment Given a cell C T A C T A C T A C G T A C T 0 0 0 0 0 0 0 0 0 0 0 0 0 The cell contains the score of the best alignment of CTA with the empty sequence.

16. 2.2 Semiglobal pairwise alignment C T A C T A C T A C G T 0 0 0 0 0 0 0… A 1 C 2 T 3 C T A C T A C T A C G T 0 0 0 0 0 0 0… A C T The contribution of the initial gaps is disregarded, then but, what happens with the final gaps?

17. 2.2 Semiglobal pairwise alignment … by checking the last row for the best score. C T A C T A C T A C G T 0 0 0 0 0 0 0… A 1 C 2 T 3 How does the algorithm search for the best alignment? Practice with the alggen tool.

18. 2.2 Affine-gap model score Given the following alignments that have the same score … a g t a c c c c g t a g a g t - c c - - g t a - a g t a c c c c g t a g a g t - c - c - g t a - a g t a c c c c g t a g a g t - c - - c g t a - a g t a c c c c g t a g a g t - - c c - g t a - a g t a c c c c g t a g a g t - - - c c g t a - a g t a c c c c g t a g a g t - - c - c g t a - Which is the most reliable case from a biological point of view?

19. 2.2 Affine-gap model score By scoring the opening gapsgreaterthan the extension gaps, for instance, -10 and -0.5. Then, how can we distinguish between consecutive gaps and separated gaps? a g t a c c c c g t a g a g t - - - c c g t a - a g t a c c c c g t a g a g t - - c - c g t a - Then, the penalty of k consecutive gaps becomes OG + (k-1) EG which is an affine-gap function. How is the best alignment found?.

20. 2.2 Affine-gap model score C T A C T A C T A C G T A C T G A Smallest arrows: refer to the introduction of an opening gap. Largest arrows: refer to the introduction of an extension gap. But from which cell do the largest arrows originate?

21. 2.2 Local alignment Given two sequences, we can consider the alignments of all their substrings… …how can the best of them be found? Two questions arise: - how can the alignments be compared? - how can the best one be selected?

22. 2.2 Local alignment accaccacaccacaacgagcata… acctgagcgatat a c c . . t Given a path Imagine the graph of the scores: can the best subalignments be detected? … It suffices to compare the value of each cell with zero!