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Sequence Alignment

Explore the importance of sequence alignment in bioinformatics, scoring methods, computation of edit distance, and dynamic programming algorithms for optimal alignment. Learn about amino acid sequences and alignment techniques for DNA, RNA, and protein sequences.

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Sequence Alignment

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  1. Sequence Alignment .

  2. Sequences Much of bioinformatics involves sequences • DNA sequences • RNA sequences • Protein sequences We can think of these sequences as strings of letters • DNA & RNA: alphabet of 4 letters • Protein: alphabet of 20 letters

  3. Glycine (G, GLY) Alanine (A, ALA) Valine (V, VAL) Leucine (L, LEU) Isoleucine (I, ILE) Phenylalanine (F, PHE) Proline (P, PRO) Serine (S, SER) Threonine (T, THR) Cysteine (C, CYS) Methionine (M, MET) Tryptophan (W, TRP) Tyrosine (T, TYR) Asparagine (N, ASN) Glutamine (Q, GLN) Aspartic acid (D, ASP) Glutamic Acid (E, GLU) Lysine (K, LYS) Arginine (R, ARG) Histidine (H, HIS) START: AUG STOP: UAA, UAG, UGA 20 Amino Acids

  4. g1 g2 Sequence Comparison • Finding similarity between sequences is important for many biological questions For example: • Find similar proteins • Allows to predict function & structure • Locate similar subsequences in DNA • Allows to identify (e.g) regulatory elements • Locate DNA sequences that might overlap • Helps in sequence assembly

  5. Sequence Alignment Input: two sequences over the same alphabet Output: an alignment of the two sequences Example: • GCGCATGGATTGAGCGA • TGCGCCATTGATGACCA A possible alignment: -GCGC-ATGGATTGAGCGA TGCGCCATTGAT-GACC-A

  6. Alignments -GCGC-ATGGATTGAGCGA TGCGCCATTGAT-GACC-A Three elements: • Perfect matches • Mismatches • Insertions & deletions (indel)

  7. Choosing Alignments There are many possible alignments For example, compare: -GCGC-ATGGATTGAGCGA TGCGCCATTGAT-GACC-A to ------GCGCATGGATTGAGCGA TGCGCC----ATTGATGACCA-- Which one is better?

  8. Scoring Alignments Rough intuition: • Similar sequences evolved from a common ancestor • Evolution changed the sequences from this ancestral sequence by mutations: • Replacements: one letter replaced by another • Deletion: deletion of a letter • Insertion: insertion of a letter • Scoring of sequence similarity should examine how many operations took place

  9. Simple Scoring Rule Score each position independently: • Match: +1 • Mismatch : -1 • Indel -2 Score of an alignment is sum of positional scores

  10. Example Example: -GCGC-ATGGATTGAGCGA TGCGCCATTGAT-GACC-A Score: (+1x13) + (-1x2) + (-2x4) = 3 ------GCGCATGGATTGAGCGA TGCGCC----ATTGATGACCA-- Score: (+1x5) + (-1x6) + (-2x11) = -23

  11. More General Scores • The choice of +1,-1, and -2 scores was quite arbitrary • Depending on the context, some changes are more plausible than others • Exchange of an amino-acid by one with similar properties (size, charge, etc.) vs. • Exchange of an amino-acid by one with opposite properties

  12. Additive Scoring Rules • We define a scoring function by specifying a function • (x,y) is the score of replacing x by y • (x,-) is the score of deleting x • (-,x) is the score of inserting x • The score of an alignment is the sum of position scores

  13. Edit Distance • The edit distance between two sequences is the “cost” of the “cheapest” set of edit operations needed to transform one sequence into the other • Computing edit distance between two sequences almost equivalent to finding the alignment that minimizes the distance

  14. Computing Edit Distance • How can we compute the edit distance?? • If |s| = n and |t| = m, there are more than alignments • The additive form of the score allows to perform dynamic programming to compute edit distance efficiently

  15. Recursive Argument • Suppose we have two sequences:s[1..n+1] and t[1..m+1] The best alignment must be in one of three cases: 1. Last position is (s[n+1],t[m +1] ) 2. Last position is (s[n +1],-) 3. Last position is (-, t[m +1] )

  16. Recursive Argument • Suppose we have two sequences:s[1..n+1] and t[1..m+1] The best alignment must be in one of three cases: 1. Last position is (s[n+1],t[m +1] ) 2. Last position is (s[n +1],-) 3. Last position is (-, t[m +1] )

  17. Recursive Argument • Suppose we have two sequences:s[1..n+1] and t[1..m+1] The best alignment must be in one of three cases: 1. Last position is (s[n+1],t[m +1] ) 2. Last position is (s[n +1],-) 3. Last position is (-, t[m +1] )

  18. Recursive Argument Define the notation: • Using our recursive argument, we get the following recurrence for V:

  19. T S Recursive Argument • Of course, we also need to handle the base cases in the recursion: AA - - versus We fill the matrix using the recurrence rule:

  20. T S Dynamic Programming Algorithm We continue to fill the matrix using the recurrence rule

  21. T S +1 -2 -A A- -2 (A- versus -A) Dynamic Programming Algorithm versus

  22. T S Dynamic Programming Algorithm

  23. T S Conclusion: d(AAAC,AGC) = -1 Dynamic Programming Algorithm

  24. T S Reconstructing the Best Alignment • To reconstruct the best alignment, we record which case(s) in the recursive rule maximized the score

  25. T S AAAC AG-C Reconstructing the Best Alignment • We now trace back a path that corresponds to the best alignment

  26. T S AAAC -AGC AAAC AG-C Reconstructing the Best Alignment • Sometimes, more than one alignment has the best score AAAC A-GC

  27. Complexity Space: O(mn) Time: O(mn) • Filling the matrix O(mn) • Backtrace O(m+n)

  28. Space Complexity • In real-life applications, n and m can be very large • The space requirements of O(mn) can be too demanding • If m = n = 1000 we need 1MB space • If m = n = 10000, we need 100MB space • We can afford to perform extra computation to save space • Looping over million operations takes less than seconds on modern workstations • Can we trade off space with time?

  29. Why Do We Need So Much Space? To compute d(s,t), we only need O(n) space • Need to compute V[n,m] • Can fill in V, column by column, only storing the last two columns in memory Note however • This “trick” fails when we need to reconstruct the sequence • Trace back information “eats up” all the memory

  30. A G C 1 2 3 0 0 0 -2 -4 -6 1 -2 1 -1 -3 A 2 -4 -1 0 -2 A 3 -6 -3 -2 -1 A 4 -8 -5 -4 -1 C Why Do We Need So Much Space? To find d(s,t), need O(n) space • Need to compute V[n,m] • Can fill in V, column by column, storing only two columns in memory Note however • This “trick” fails when we need to reconstruct the sequence • Trace back information “eats up” all the memory

  31. t • Construct two alignments • A= s[1,n/2] vs t[1,j] • B= s[n/2+1,n] vs t[j+1,m] s • Return the concatenated alignment AB Space Efficient Version: Outline Input: Sequences s[1,n] and t[1,m] to be aligned. Idea: perform divide and conquer • Find position (n/2, j) at which the best alignment crosses a midpoint

  32. We need to find j that maximizes this score. Such j determines a point (n/2,j) through which the best alignment passes. t • Thus, we need to compute these two quantities for all values of j s Finding the Midpoint • The score of the best alignment that goes through j equals: d(s[1,n/2],t[1,j]) + d(s[n/2+1,n],t[j+1,m])

  33. t s Finding the Midpoint (Algorithm) Define • V[i,j] = d(s[1..i],t[1..j]) (As before) • B[i,j] = d(s[i+1..n],t[j+1..m]) (Symmetrically) • F[i,j] + B[i,j] = score of best alignment through (i,j)

  34. Computing V[i,j] V[i,j] = d(s[1..i],t[1..j]) As before: Requires linear space complexity

  35. Computing B[i,j] B[i,j] = d(s[i+1..n],t[j+1..m]) Symmetrically (replacing i with i+1, and j with j+1): Requires linear space complexity

  36. Local Alignment Consider now a different question: • Can we find similar substring of s and t • Formally, given s[1..n] and t[1..m] find i,j,k, and l such that d(s[i..j],t[k..l]) is maximal

  37. Local Alignment • As before, we use dynamic programming • We now want to setV[i,j] to record the best alignment of a suffix of s[1..i] and a suffix of t[1..j] • How should we change the recurrence rule?

  38. Local Alignment New option: • We can start a new match instead of extend previous alignment Alignment of empty suffixes

  39. Local Alignment Example s =TAATA t =ATCTAA

  40. Local Alignment Example s =TAATA t =TACTAA

  41. Local Alignment Example s =TAATA t =TACTAA

  42. Local Alignment Example s =TAATA t =TACTAA

  43. Sequence Alignment We seen two variants of sequence alignment: • Global alignment • Local alignment Other variants: • Finding best overlap (exercise) All are based on the same basic idea of dynamic programming

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