LCS and Extensions to Global and Local Alignment Dr. Nancy Warter-Perez June 26, 2003

LCS and Extensions2 Overview Recursion Recursive solution to hydrophobicity sliding window problem LCS Smith-Waterman Algorithm Extensions to LCS Global Alignment Local Alignment Affine Gap Penalties Programming Workshop 6

June 26, 2003LCS and Extensions3 Project References _alignments.html _alignments.html Computational Molecular Biology – An Algorithmic Approach, Pavel Pevzner Introduction to Computational Biology – Maps, sequences, and genomes, Michael Waterman Algorithms on Strings, Trees, and Sequences – Computer Science and Computational Biology, Dan Gusfield

June 26, 2003LCS and Extensions4 Recursion Problems can be solved iteratively or recursively Recursion is useful in cases where you are building upon a partial solution Consider the hydrophobicity problem

June 26, 2003LCS and Extensions5 Main.cpp #include using namespace std; #include "hydro.h" double hydro[25] = {1.8,0,2.5,-3.5,-3.5,2.8,-0.4,-3.2,4.5,0,-3.9,3.8,1.9,-3.5,0, -1.6,-3.5,-4.5,-0.8,-0.7,0,4.2,-0.9,0,-1.3}; void main () { string seq;int ws, i; cout << "This program will compute the hydrophobicity of an sequence of amino acids.\n"; cout > seq; for(i = 0; i < seq.size(); i++) if((seq.data()[i] >= 'a') && (seq.data()[i] <= 'z')) seq.at(i) = seq.data()[i] - 32; cout > ws; compute_hydro(seq, ws); }

June 26, 2003LCS and Extensions6 Hydro.cpp #include using namespace std; #include "hydro.h" void print_hydro(string seq, int ws, int i, double sum); void compute_hydro(string seq, int ws) { cout << "\n\nThe hydrophocity values are:" << endl; print_hydro(seq, ws, seq.size()-1, 0); } void print_hydro(string seq, int ws, int i, double sum) { if(i == -1) return; if(i > seq.size() - ws) sum += hydro[seq.data()[i] - 'A']; else sum = sum - hydro[seq.data()[i+ws] - 'A'] + hydro[seq.data()[i] - 'A']; print_hydro(seq, ws, i-1, sum); if (i <= seq.size() - ws) cout << "Hydrophocity value:\t" << sum/ws << endl; }

June 26, 2003LCS and Extensions7 hydro.h extern double hydro[25]; void compute_hydro(string seq, int ws);

June 26, 2003LCS and Extensions8 Dynamic Programming Applied to optimization problems Useful when Problem can be recursively divided into sub-problems Sub-problems are not independent

June 26, 2003LCS and Extensions9 Longest Common Subsequence (LCS) Problem Reference: Pevzner Can have insertion and deletions but no substitutions (no mismatches) Ex: V: ATCTGAT W:TGCATA LCS:TCTA

June 26, 2003LCS and Extensions10 LCS Problem (cont.) Similarity score s i-1,j s i,j = max { s i,j-1 s i-1,j-1 + 1, if vi = wj On board example: Pevzner Fig 6.1

June 26, 2003LCS and Extensions11 Indels – insertions and deletions (e.g., gaps) alignment of V and W V = rows of similarity matrix (vertical axis) W = columns of similarity matrix (horizontal axis) Space (gap) in W  (UP) insertion Space (gap) in V  (LEFT) deletion Match (no mismatch in LCS) (DIAG)

June 26, 2003LCS and Extensions12 LCS(V,W) Algorithm for i = 0 to n si,0 = 0 for j = 1 to m s0,j = 0 for i = 1 to n for j = 1 to m if vi = wj si,j = si-1,j-1 + 1; bi,j = DIAG else if si-1,j >= si,j-1 si,j = si-1,j; bi,j = UP else si,j = si,j-1; bi,j = LEFT

June 26, 2003LCS and Extensions13 Print-LCS(V,i,j) if i = 0 or j = 0 return if bi,j = DIAG PRINT-LCS(V, i-1, j-1) print vi else if bi,j = UP PRINT-LCS(V, i-1, j) else PRINT-LCS(V, I, j-1)

June 26, 2003LCS and Extensions14 Classic Papers Needleman, S.B. and Wunsch, C.D. A General Method Applicable to the Search for Similarities in Amino Acid Sequence of Two Proteins. J. Mol. Biol., 48, pp , ( ssicArticlesArchive/needlemanandwunsch1970.pdf) Needleman, S.B. and Wunsch, C.D. A General Method Applicable to the Search for Similarities in Amino Acid Sequence of Two Proteins. J. Mol. Biol., 48, pp , Smith, T.F. and Waterman, M.S. Identification of Common Molecular Subsequences. J. Mol. Biol., 147, pp , 1981.( sicArticlesArchive/smithandwaterman1981.pdf) Smith, T.F. and Waterman, M.S. Identification of Common Molecular Subsequences. J. Mol. Biol., 147, pp , Smith, T.F. The History of the Genetic Sequence Databases. Genomics, 6, pp , ( iclesArchive/smith1990.pdf) Smith, T.F. The History of the Genetic Sequence Databases. Genomics, 6, pp , 1990.

June 26, 2003LCS and Extensions15 Smith-Waterman (1 of 3) Algorithm The two molecular sequences will be A=a 1 a 2... a n, and B=b 1 b 2... b m. A similarity s(a,b) is given between sequence elements a and b. Deletions of length k are given weight W k. To find pairs of segments with high degrees of similarity, we set up a matrix H. First set H k0 = H ol = 0 for 0 <= k <= n and 0 <= l <= m. Preliminary values of H have the interpretation that H i j is the maximum similarity of two segments ending in a i and b j. respectively. These values are obtained from the relationship H ij =max{H i-1,j-1 + s(a i,b j ), max {H i-k,j – W k }, max{H i,j-l - W l }, 0} ( 1 ) k >= 1 l >= 1 1 <= i <= n and 1 <= j <= m.

June 26, 2003LCS and Extensions16 Smith-Waterman (2 of 3) The formula for H ij follows by considering the possibilities for ending the segments at any a i and b j. (1)If a i and b j are associated, the similarity is H i-l,j-l + s(a i,b j ). (2) If a i is at the end of a deletion of length k, the similarity is H i – k, j - W k. (3) If b j is at the end of a deletion of length 1, the similarity is H i,j-l - W l. (typo in paper) (4) Finally, a zero is included to prevent calculated negative similarity, indicating no similarity up to a i and b j.

June 26, 2003LCS and Extensions17 Smith-Waterman (3 of 3) The pair of segments with maximum similarity is found by first locating the maximum element of H. The other matrix elements leading to this maximum value are than sequentially determined with a traceback procedure ending with an element of H equal to zero. This procedure identifies the segments as well as produces the corresponding alignment. The pair of segments with the next best similarity is found by applying the traceback procedure to the second largest element of H not associated with the first traceback.

June 26, 2003LCS and Extensions18 Extend LCS to Global Alignment si-1,j +  (vi, -) si,j= max {si,j-1 +  (-, wj) si-1,j-1 +  (vi, wj)  (vi, -) =  (-, wj) = -  = fixed gap penalty  (vi, wj) = score for match or mismatch – can be fixed, from PAM or BLOSUM

June 26, 2003LCS and Extensions19 Extend to Local Alignment 0(no negative scores) si-1,j +  (vi, -) si,j= max {si,j-1 +  (-, wj) si-1,j-1 +  (vi, wj)  (vi, -) =  (-, wj) = -  = fixed gap penalty  (vi, wj) = score for match or mismatch – can be fixed, from PAM or BLOSUM

June 26, 2003LCS and Extensions20 Discussion on adding affine gap penalties Affine gap penalty Score for a gap of length x -(  +  x) Where  > 0 is the insert gap penalty  > 0 is the extend gap penalty

June 26, 2003LCS and Extensions21 Alignment with Gap Penalties Can apply to global or local (w/ zero) algorithms  si,j= max {  si-1,j -  si-1,j - (  +  )  si,j= max {  si1,j-1 -  si,j-1 - (  +  ) si-1,j-1 +  (vi, wj) si,j= max {  si,j  si,j Note: keeping with traversal order in Figure 6.1,  is replaced by , and  is replaced by 

June 26, 2003LCS and Extensions22 Programming Workshop 6 Implement LCS LCS(V,W) b and s are global matrices Print-LCS(V,i,j) Write a program that uses LCS and Print-LCS. The program should prompt the user for 2 sequences and print the longest common sequence.