1. Dynamic Programming (cont’d)
CS 466
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:
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 change its properties dramatically.
tRNA linear and 3D view:

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’ R / 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
single stranded
bases within
a stem
… on both
sides of 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 2 11

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 prediction Approach 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 prediction Approach 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. Neighboring base pairs: stack Single bulges OK in stacking
Source: DEKM
U U
A A
G-C A
U-A
A-U
C-G
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. 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))
s[j] is external base
(a base not in any loop)
s[j] pairs with s[i],
for some i < j

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)