Dynamic Programming (cont’d)

CS 466Saurabh Sinha

RNA secondary structure prediction

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.

http://www.cgl.ucsf.edu/home/glasfeld/tutorial/trna/trna.giftRNA linear and 3D view:

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

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

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 coatprotein. N = A/C/G/U,N’ is a complementarybase pairing to N,Y is C/U, R is A/G

Source: DEKM

Basics of secondary structure

• G-C pairing: three bonds (strong)• A-U pairing: two bonds (weaker)• Base pairs are approximately coplanar

Basics of secondary structure

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”

Secondary structure elements

Loop: single stranded subsequences bounded by base pairs

loop at the endof a stem

stem loop

single strandedbases withina stem

… only on oneside of stem

… on bothsides of stem

Non-canonical base pairs

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

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

Pseudoknot

2 11

9 18(9, 18)(2, 11)NOT NESTED

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

Secondary structure prediction

• 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

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)

ii+1 j

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)

jj-1i

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)

ji+1 j-1

i

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

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

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)

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

Traceback

Push (1,L) onto stackRepeat until stack is empty:pop (i,j)if i >= j continueelse 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)