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Dynamic Programming (cont’d)

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  1. Dynamic Programming (cont’d) CS 498 SS Saurabh Sinha

  2. Spliced Alignment • Begins by selecting either all putative exons between potential acceptor and donor sites or by finding all substrings similar to the target protein (as in the Exon Chaining Problem). • This set is further filtered in a such a way that attempt to retain all true exons, with some false ones. • Then find the chain of exons such that the sequence similarity to the target protein sequence is maximized

  3. Spliced Alignment Problem: Formulation • Input: Genomic sequences G, target sequence T, and a set of candidate exons (blocks) B. • Output: A chain of exons Γ such that the global alignment score between Γ* and T is maximized Γ* - concatenation of all exons from chain Γ

  4. The DAG • Vertices: One vertex for each block in B • Directed edge connecting non-overlapping blocks • Label of vertex = string of block it represents • A path through the DAG spells out the string obtained by concatenating that particular chain of blocks • Weight of a path is the score of the optimal alignment between the string it spells out and the target sequence

  5. Dynamic programming • Genomic sequence G = g1g2…gn • Target sequence T = t1t2…tm • As usual, we want to find the optimal alignment score of the i-prefix of G and the j-prefix of T • Problem is, there are many i-prefixes possible (since multiple blocks may include position i)

  6. Idea • Find the optimal alignment score of the i-prefix of G and the j-prefix of T assuming that this alignment uses a particular block B at position i • S(i, j, B) • For every block B that includes i

  7. Recurrence If i is not the starting vertex of block B: • S(i, j, B) = max { S(i – 1, j, B) – indel penalty S(i, j – 1, B) – indel penalty S(i – 1, j – 1, B) + δ(gi, tj) } If i is the starting vertex of block B: • S(i, j, B) = max { S(i, j – 1, B) – indel penalty maxall blocks B’ preceding block B S(end(B’), j, B’) – indel penalty maxall blocks B’ preceding block B S(end(B’), j – 1, B’) + δ(gi, tj) }

  8. RNA secondary structure prediction

  9. RNA • RNA is similar to DNA chemically. It is usually only a single strand. T(hyamine) is replaced by U(racil) • Some forms of RNA can form secondary structures by “pairing up” with itself. This can have change its properties dramatically. DNA and RNA can pair with each other. tRNA linear and 3D view: http://www.cgl.ucsf.edu/home/glasfeld/tutorial/trna/trna.gif

  10. RNA • There’s more to RNA than mRNA • RNA can adopt interesting non-linear structures, and catalyze reactions • tRNAs (transfer RNAs) are the “adapters” that implement translation

  11. Secondary structure • Several interesting RNAs have a conserved secondary structure (resulting from base-pairing interactions) • Sometimes, the sequence itself may not be conserved for the function to be retained • It is important to tell what the secondary structure is going to be, for homology detection

  12. Conserved secondary structure N-Y A A N-N’ N-N’ R N-N’ N-N’ N-N’ N-N’ N-N’ / N Consensus binding site for R17 phage coat protein. N = A/C/G/U, N’ is a complementary base pairing to N, Y is C/U, R is A/G Source: DEKM

  13. Basics of secondary structure • G-C pairing: three bonds (strong) • A-U pairing: two bonds (weaker) • Base pairs are approximately coplanar

  14. Basics of secondary structure

  15. Basics of secondary structure • G-C pairing: three bonds (strong) • A-U pairing: two bonds (weaker) • Base pairs are approximately coplanar • Base pairs are stacked onto other base pairs (arranged side by side): “stems”

  16. Secondary structure elements loop at the end of a stem stem loop … on both sides of stem single stranded bases within a stem … only on one side of stem Loop: single stranded subsequences bounded by base pairs

  17. Non-canonical base pairs • G-C and A-U are the canonical base pairs • G-U is also possible, almost as stable

  18. Nesting • Base pairs almost always occur in a nested fashion • If positions i and j are paired, and positions i’ and j’ are paired, then these two base-pairings are said to be nested if: • i < i’ < j’ < j OR i’ < i < j < j’ • Non-nested base pairing: pseudoknot

  19. Pseudoknot (9, 18) (2, 11) NOT NESTED 9 18 11 2

  20. Pseudoknot problems • Pseudoknots are not handled by the algorithms we shall see • Pseudoknots do occur in many important RNAs • But the total number of pseudoknotted base pairs is typically relatively small

  21. Secondary structure predictionApproach 1. • Find the secondary structure with most base pairs. • Nussinov’s algorithm • Recursive: finds best structure for small subsequences, and works its way outwards to larger subsequences

  22. Nussinov’s algorithm: idea • There are only four possible ways of getting the best structure for subsequence (i,j) from the best structures of the smaller subsequences (1) Add unpaired position i onto best structure for subsequence (i+1,j) i+1 j i

  23. Nussinov’s algorithm: idea • There are only four possible ways of getting the best structure for subsequence (i,j) from the best structures of the smaller subsequences (2) Add unpaired position j onto best structure for subsequence (i,j-1) i j-1 j

  24. Nussinov’s algorithm: idea • There are only four possible ways of getting the best structure for subsequence (i,j) from the best structures of the smaller subsequences (3) Add (i,j) pair onto best structure for subsequence (i+1,j-1) i+1 j-1 i j

  25. Nussinov’s algorithm: idea • There are only four possible ways of getting the best structure for subsequence (i,j) from the best structures of the smaller subsequences (4)Combine two optimal substructures (i,k) and (k+1,j) i k k+1 j

  26. Nussinov RNA folding algorithm • Given a sequence s of length L with symbols s1 … sL. Let (i,j) = 1 if si and sj are a complementary base pair, and 0 otherwise. • We recursively calculate scores g(i,j) which are the maximal number of base pairs that can be formed for subsequence si…sj. • Dynamic programming

  27. Recursion • Starting with all subsequences of length 2, to length L • g(i,j) = max of • g(i+1, j) • g(i,j-1) • g(i+1,j-1) + (i,j) • maxi < k < j [g(i,k) + g(k+1,j)] • Initialization • g(i,i-1) = 0 • g(i,i) = 0 O(n2) ? No. O(n3)

  28. Traceback • As usual in sequence alignment ? • Optimal sequence alignment is a linear path in the dynamic programming table • Optimal secondary structure can have “bifurcations” • Traceback uses a pushdown stack

  29. Traceback Push (1,L) onto stack Repeat until stack is empty: pop (i,j) if i >= j continue else if g(i+1,j) = g(i,j) push (i+1,j) else if g(i,j-1) = g(i,j) push (i,j-1) else if g(i+1,j-1) + (i,j) = g(i,j) record (i,j) base pair push (i+1,j-1) else for k = i+1 to j-1, if g(i,k)+g(k+1,j) g(i,j) push (k+1,j) push (i,k) break (for loop)

  30. Secondary structure predictionApproach 2 • Based on minimization of ∆G, the equilibrium free energy, rather than maximization of number of base pairs • Better fit to real (experimental) ∆G • Energy of stem is sum of “stacking” contributions from the interface between neighboring base pairs

  31. Source: DEKM U U A A G-C G-C A G-C U-A A-U C-G A-U A A 4nt loop +5.9 -1.1 terminal mismatch of hairpin -2.9 stack 1nt bulge +3.3 -2.9 stack (special case) -1.8 stack -0.9 stack -1.8 stack -2.1 stack dangle -0.3 • Neighboring base pairs: stack • Single bulges OK in stacking • Longer bulges: no stacking term • Loop destabilisation energy • Loop terminal mismatch energy

  32. hairpin loop: exactly one base pair internal loop: exactly two base pairs bulge: internal loop with one base from each base pair being adjacent multibranched loop: > 2 base pairs in a loop, one base pair is closest to ends of RNA. this is the “exterior” or “closing” base pair all other base pairs are “interior” Source: Martin Tompa’s lecture notes

  33. Energy contributions • eS(i,j): Free energy of stacked pair (i,j) and (i+1,j-1) • eH(i,j): Free energy of a loop closed by (i,j): depends on length of loop, bases at i,j, and bases adjacent to them • eL(i,j,i’,j’): Free energy of an internal loop or bulge, with (i,j) and (i’,j’) being the bordering base pairs. Depends on bases at these positions, and unpaired bases adjacent to them • eM(i,j,i1,j1,…ik,jk): Free energy of a multibranch loop with (i,j) as the closing base pair and i1j1 etc as the internal base pairs

  34. Zuker’s algorithm:Dynamic programming • W(j): FE of optimal structure of s[1..j] • V(i,j): FE of optimal structure of s[i..j] assuming i,j form a base pair • VBI(i,j): FE of optimal structure of s[i..j] assuming i,j closes a bulge or internal loop • VM(i,j): FE of optimal structure of s[i..j] assuming i,j closes a multibranch loop • WM(i,j): used to compute VM

  35. s[j] is external base (a base not in any loop) s[j] pairs with s[i], for some i < j Dynamic programming recurrences • W(j): FE of optimal structure of s[1..j] • W(0) = 0 • W(j) = min( W(j-1), min1<=i<jV(i,j)+W(i-1))

  36. Dynamic programming recurrences • V(i,j): FE of optimal structure of s[i..j] assuming i,j form a base pair • V(i,j) = infinity if i >= j • V(i,j) = min( eH(i,j), eS(i,j) + V(i+1,j-1), VBI(i,j), VM(i,j)) if i < j i,j is exterior base pair of a hairpin loop

  37. Dynamic programming recurrences • V(i,j): FE of optimal structure of s[i..j] assuming i,j form a base pair • V(i,j) = infinity if i >= j • V(i,j) = min( eH(i,j), eS(i,j) + V(i+1,j-1), VBI(i,j), VM(i,j)) if i < j i,j is exterior pair of a stacked pair. i+1,j-1 is therefore a pair too.

  38. Dynamic programming recurrences • V(i,j): FE of optimal structure of s[i..j] assuming i,j form a base pair • V(i,j) = infinity if i >= j • V(i,j) = min( eH(i,j), eS(i,j) + V(i+1,j-1), VBI(i,j), VM(i,j)) if i < j i,j is exterior pair of a bulge or interior loop

  39. Dynamic programming recurrences • V(i,j): FE of optimal structure of s[i..j] assuming i,j form a base pair • V(i,j) = infinity if i >= j • V(i,j) = min( eH(i,j), eS(i,j) + V(i+1,j-1), VBI(i,j), VM(i,j)) if i < j i,j is exterior pair of a multibranch loop

  40. Dynamic programming recurrences • VBI(i,j): FE of optimal structure of s[i..j] assuming i,j closes a bulge or internal loop • VBI(i,j) = min (eL(i,j,i’,j’) + V(i’,j’)) • Slow ! i’,j’ i<i’<j’<j Energy of the bulge

  41. Dynamic programming recurrences • VM(i,j): FE of optimal structure of s[i..j] assuming i,j closes a multibranch loop • VM(i,j) = min (eM(i,j,i1,j1,..,ik,jk) + ∑hV(ih,jh)) • Very slow ! k>=2 i1,j1,…ik,jk Energy of the loop itself

  42. Order of computation • What order to fill the DP table in ? • Increasing order of (j-i) • VBI(i,j) and VM(i,j) before V(i,j)