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CSE 5290: Algorithms for Bioinformatics Fall 2009

Understand the differences between local and global alignment problems in bioinformatics, their applications, and algorithms. Learn to compute optimal alignments and scoring matrices for gene sequences.

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CSE 5290: Algorithms for Bioinformatics Fall 2009

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  1. CSE 5290:Algorithms for Bioinformatics Fall 2009 Suprakash Datta datta@cs.yorku.ca Office: CSEB 3043 Phone: 416-736-2100 ext 77875 Course page: http://www.cs.yorku.ca/course/5290 CSE 5290, Fall 2009

  2. Next Local and global sequencealignment Some of the following slides are based on slides by the authors of our text. CSE 5290, Fall 2009

  3. Local vs. Global Alignment • The Global Alignment Problem tries to find the longest path between vertices (0,0) and (n,m) in the edit graph. • The Local Alignment Problem tries to find the longest path among paths between arbitrary vertices (i,j) and (i’, j’) in the edit graph. CSE 5290, Fall 2009

  4. Local vs. Global Alignment • The Global Alignment Problem tries to find the longest path between vertices (0,0) and (n,m) in the edit graph. • The Local Alignment Problem tries to find the longest path among paths between arbitrary vertices (i,j) and (i’, j’) in the edit graph. • In the edit graph with negatively-scored edges, Local Alignment may score higher than Global Alignment CSE 5290, Fall 2009

  5. Local vs. Global Alignment (cont’d) • Global Alignment • Local Alignment—better alignment to find conserved segment --T—-CC-C-AGT—-TATGT-CAGGGGACACG—A-GCATGCAGA-GAC | || | || | | | ||| || | | | | |||| | AATTGCCGCC-GTCGT-T-TTCAG----CA-GTTATG—T-CAGAT--C tccCAGTTATGTCAGgggacacgagcatgcagagac |||||||||||| aattgccgccgtcgttttcagCAGTTATGTCAGatc CSE 5290, Fall 2009

  6. Compute a “mini” Global Alignment to get Local Local Alignment: Example Local alignment Global alignment CSE 5290, Fall 2009

  7. Local Alignments: Why? • Two genes in different species may be similar over short conserved regions and dissimilar over remaining regions. • Example: • Homeobox genes have a short region called the homeodomain that is highly conserved between species. • A global alignment would not find the homeodomain because it would try to align the ENTIRE sequence CSE 5290, Fall 2009

  8. The Local Alignment Problem • Goal: Find the best local alignment between two strings • Input : Strings v, w and scoring matrix δ • Output : Alignment of substrings of v and w whose alignment score is maximum among all possible alignment of all possible substrings CSE 5290, Fall 2009

  9. The Problem with this Problem • Long run time O(n4): - In the grid of size n x n there are ~n2 vertices (i,j) that may serve as a source. - For each such vertex computing alignments from (i,j) to (i’,j’) takes O(n2) time. • This can be remedied by giving free rides CSE 5290, Fall 2009

  10. Compute a “mini” Global Alignment to get Local Local Alignment: Example Local alignment Global alignment CSE 5290, Fall 2009

  11. Local Alignment: Example CSE 5290, Fall 2009

  12. Local Alignment: Example CSE 5290, Fall 2009

  13. Local Alignment: Example CSE 5290, Fall 2009

  14. Local Alignment: Example CSE 5290, Fall 2009

  15. Local Alignment: Example CSE 5290, Fall 2009

  16. Local Alignment: Running Time • Long run time O(n4): - In the grid of size n x n there are ~n2 vertices (i,j) that may serve as a source. - For each such vertex computing alignments from (i,j) to (i’,j’) takes O(n2) time. • This can be remedied by giving free rides CSE 5290, Fall 2009

  17. Local Alignment: Free Rides Yeah, a free ride! Vertex (0,0) The dashed edges represent the free rides from (0,0) to every other node. CSE 5290, Fall 2009

  18. Notice there is only this change from the original recurrence of a Global Alignment The Local Alignment Recurrence • The largest value of si,j over the whole edit graph is the score of the best local alignment. • The recurrence: 0 si,j = max si-1,j-1 + δ(vi, wj) s i-1,j + δ(vi, -) s i,j-1 + δ(-, wj) { CSE 5290, Fall 2009

  19. Power of ZERO: there is only this change from the original recurrence of a Global Alignment - since there is only one “free ride” edge entering into every vertex The Local Alignment Recurrence • The largest value of si,j over the whole edit graph is the score of the best local alignment. • The recurrence: 0 si,j = max si-1,j-1 + δ(vi, wj) s i-1,j + δ(vi, -) s i,j-1 + δ(-, wj) { CSE 5290, Fall 2009

  20. Scoring Indels: Naive Approach • A fixed penalty σis given to every indel: • -σ for 1 indel, • -2σ for 2 consecutive indels • -3σ for 3 consecutive indels, etc. Can be too severe penalty for a series of 100 consecutive indels CSE 5290, Fall 2009

  21. This is more likely. This is less likely. Affine Gap Penalties • In nature, a series of k indels often come as a single event rather than a series of k single nucleotide events: ATA__GC ATATTGC ATAG_GC AT_GTGC Normal scoring would give the same score for both alignments CSE 5290, Fall 2009

  22. Accounting for Gaps • Gaps- contiguous sequence of spaces in one of the rows • Score for a gap of length x is: -(ρ +σx) where ρ >0 is the penalty for introducing a gap: gap opening penalty ρ will be large relative to σ: gap extension penalty because you do not want to add too much of a penalty for extending the gap. CSE 5290, Fall 2009

  23. Affine Gap Penalties • Gap penalties: • -ρ-σ when there is 1 indel • -ρ-2σ when there are 2 indels • -ρ-3σ when there are 3 indels, etc. • -ρ- x·σ (-gap opening - x gap extensions) • Somehow reduced penalties (as compared to naïve scoring) are given to runs of horizontal and vertical edges CSE 5290, Fall 2009

  24. Affine Gap Penalties and Edit Graph To reflect affine gap penalties we have to add “long” horizontal and vertical edges to the edit graph. Each such edge of length x should have weight - - x * CSE 5290, Fall 2009

  25. Adding “Affine Penalty” Edges to the Edit Graph There are many such edges! Adding them to the graph increases the running time of the alignment algorithm by a factor of n (where n is the number of vertices) So the complexity increases from O(n2) to O(n3) CSE 5290, Fall 2009

  26. Manhattan in 3 Layers ρ δ δ σ δ ρ δ δ σ CSE 5290, Fall 2009

  27. Affine Gap Penalties and 3 Layer Manhattan Grid • The three recurrences for the scoring algorithm creates a 3-layered graph. • The top level creates/extends gaps in the sequence w. • The bottom level creates/extends gaps in sequence v. • The middle level extends matches and mismatches. CSE 5290, Fall 2009

  28. Switching between 3 Layers • Levels: • The main level is for diagonal edges • The lower level is for horizontal edges • The upper level is for vertical edges • A jumping penalty is assigned to moving from the main level to either the upper level or the lower level (-r- s) • There is a gap extension penalty for each continuation on a level other than the main level (-s) CSE 5290, Fall 2009

  29. The 3-leveled Manhattan Grid Gaps in w Matches/Mismatches Gaps in v CSE 5290, Fall 2009

  30. Affine Gap Penalty Recurrences Continue Gap in w (deletion) si,j = s i-1,j - σ max s i-1,j –(ρ+σ) si,j = s i,j-1 - σ max s i,j-1 –(ρ+σ) si,j = si-1,j-1 + δ(vi, wj) max s i,j s i,j Start Gap in w (deletion): from middle Continue Gap in v (insertion) Start Gap in v (insertion):from middle Match or Mismatch End deletion: from top End insertion: from bottom CSE 5290, Fall 2009

  31. Next • Multiple Alignments CSE 5290, Fall 2009

  32. Multiple Alignment versus Pairwise Alignment • Up until now we have only tried to align two sequences. CSE 5290, Fall 2009

  33. Multiple Alignment versus Pairwise Alignment • Up until now we have only tried to align two sequences. • What about more than two? And what for? CSE 5290, Fall 2009

  34. Multiple Alignment versus Pairwise Alignment • Up until now we have only tried to align two sequences. • What about more than two? And what for? • A faint similarity between two sequences becomes significant if present in many • Multiple alignments can reveal subtle similarities that pairwise alignments do not reveal CSE 5290, Fall 2009

  35. Generalizing the Notion of Pairwise Alignment • Alignment of 2 sequences is represented as a 2-row matrix • In a similar way, we represent alignment of 3 sequences as a 3-row matrix A T _ G C G _ A _ C G T _ A A T C A C _ A • Score: more conserved columns, better alignment CSE 5290, Fall 2009

  36. Alignments = Paths in… • Align 3 sequences: ATGC, AATC,ATGC CSE 5290, Fall 2009

  37. Alignment Paths x coordinate CSE 5290, Fall 2009

  38. Alignment Paths • Align the following 3 sequences: ATGC, AATC,ATGC x coordinate y coordinate CSE 5290, Fall 2009

  39. Alignment Paths x coordinate y coordinate z coordinate • Resulting path in (x,y,z) space: • (0,0,0)(1,1,0)(1,2,1) (2,3,2) (3,3,3) (4,4,4) CSE 5290, Fall 2009

  40. Aligning Three Sequences source • Same strategy as aligning two sequences • Use a 3-D “Manhattan Cube”, with each axis representing a sequence to align • For global alignments, go from source to sink sink CSE 5290, Fall 2009

  41. 2-D vs 3-D Alignment Grid V W 2-D edit graph 3-D edit graph CSE 5290, Fall 2009

  42. 2-D cell versus 2-D Alignment Cell In 2-D, 3 edges in each unit square In 3-D, 7 edges in each unit cube CSE 5290, Fall 2009

  43. Architecture of 3-D Alignment Cell (i-1,j,k-1) (i-1,j-1,k-1) (i-1,j,k) (i-1,j-1,k) (i,j,k-1) (i,j-1,k-1) (i,j,k) (i,j-1,k) CSE 5290, Fall 2009

  44. si-1,j-1,k-1 + (vi, wj, uk) si-1,j-1,k + (vi, wj, _ ) si-1,j,k-1 + (vi, _, uk) si,j-1,k-1 + (_, wj, uk) si-1,j,k + (vi, _ , _) si,j-1,k + (_, wj, _) si,j,k-1 + (_, _, uk) Multiple Alignment: Dynamic Programming cube diagonal: no indels • si,j,k = max • (x, y, z) is an entry in the 3-D scoring matrix face diagonal: one indel edge diagonal: two indels CSE 5290, Fall 2009

  45. Multiple Alignment: Running Time • For 3 sequences of length n, the run time is 7n3; O(n3) • For ksequences, build a k-dimensional Manhattan, with run time (2k-1)(nk); O(2knk) • Conclusion: dynamic programming approach for alignment between two sequences is easily extended to k sequences but it is impractical due to exponential running time CSE 5290, Fall 2009

  46. Multiple Alignment Induces Pairwise Alignments Every multiple alignment induces pairwise alignments x: AC-GCGG-C y: AC-GC-GAG z: GCCGC-GAG Induces: x: ACGCGG-C; x: AC-GCGG-C; y: AC-GCGAG y: ACGC-GAC; z: GCCGC-GAG; z: GCCGCGAG CSE 5290, Fall 2009

  47. Reverse Problem: Constructing Multiple Alignment from Pairwise Alignments Given 3 arbitrary pairwise alignments: x: ACGCTGG-C; x: AC-GCTGG-C; y: AC-GC-GAG y: ACGC--GAC; z: GCCGCA-GAG; z: GCCGCAGAG Q: can we construct a multiple alignment that induces them? A: NOT ALWAYS Pairwise alignments may be inconsistent CSE 5290, Fall 2009

  48. Inferring Multiple Alignment from Pairwise Alignments • From an optimal multiple alignment, we can infer pairwise alignments between all pairs of sequences, but they are not necessarily optimal • It is difficult to infer a ``good” multiple alignment from optimal pairwise alignments between all sequences CSE 5290, Fall 2009

  49. Combining Optimal Pairwise Alignments into Multiple Alignment Can combine pairwise alignments into multiple alignment Can not combine pairwise alignments into multiple alignment CSE 5290, Fall 2009

  50. Profile Representation of Multiple Alignment In the past we were aligning a sequence against a sequence Can we align a sequence against a profile? Can we align a profile against a profile? - A G G C T A T C A C C T G T A G – C T A C C A - - - G C A G – C T A C C A - - - G C A G – C T A T C A C – G G C A G – C T A T C G C – G G A 1 1 .8 C .6 1 .4 1 .6 .2 G 1 .2 .2 .4 1 T .2 1 .6 .2 - .2 .8 .4 .8 .4 CSE 5290, Fall 2009

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