RNA structure prediction

RNA functions

• RNA functions as– mRNA– rRNA– tRNA– Nuclear export– Spliceosome– Regulatory molecules (RNAi)– Enzymes– Virus– Retrotransposons– Medicine

Base pairs

• C-G stronger than U-A• Non-standard G-U

CC

C

N

N C

O

C

CC

C

O

N C

O

N

cytosine

Uracyl

N

CC

C

NC

NC

N

N

O

N

CC

C

NC

NC

N

N

Adenine

Guanine

PYRIMIDINES PURINES

H donor acceptor

• Base-pairs are usually coplanar

• are almost always stacked

• stems – continuous stacks

• 3D structure of a stack is a helix

hairpin

Stacking

Predictable structures

Hard-to-predict structures

• Pseudoknots, kissing hairpins, hairpin-bulge

Secondary structure notations

Tertiary structure

RNAi

Structure representation

Main approaches to RNA secondary structure prediction

• Energy minimization – dynamic programming approach– does not require prior sequence alignment– require estimation of energy terms

contributing to secondary structure

• Comparative sequence analysis– Using sequence alignment to find conserved

residues and covariant base pairs.– most trusted

Dotplot

Think!

• Make a dotplot of an RNA molecule– Sequence : GGGAAAUCC

• What is the secondary structure?

Dynamic programming approach

• Nussinov algorithm

Dynamic programming approach

a) i,j is paired E(i,j) = E(i+1,j-1) + (ri,rj)b) i is unpaired E(i,j) = E(i+1,j) c) j is unpaired E(i,j) = E(i,j-1)d) bifurcation E(i,j) = E(i,k)+E(k+1,j)

i+1 j-1 i+1 j ji j-1i j i

ik k+1

a) b) c) d)

Let E(i,j) = minimum energy for subchain starting at i and ending at j(ri,rj) = energy of pair ri, rj (rj = base at position j)

RNA secondary structure algorithm

• Given: RNA sequence x1,x2,x3,x4,x5,x6,…,xL

• Initialization: E(i, i-1) = 0 for i = 2 to LE(i, i) = 0 for i = 1 to L

Recursion: for n = 2 to L # iteration over length

E(i,j) = min { E(i+1, j), E(i, j-1), E(i+1, j-1)+ (ri,rj) ,

min i<k<j {E(i,k)+E(k+1, j)} }

• Cost: O(n3)

ExampleLet (ri,rj) = -1 if ri,rj form a base pair and 0 otherwise Input : GGAAAUCC

G G A A A U C C

G 0

G 0 0

A 0 0

A 0 0

A 0 0

U 0 0

C 0 0

C 0 0

E(i,j) = lowestenergy conformation for subchain from i to j

i

j

Here we should have min energy for AAAUC

Example-continued

G G A A A U C C

G 0 0

G 0 0 0

A 0 0 0

A 0 0 0

A 0 0 -1

U 0 0 0

C 0 0 0

C 0 0

GGA (i=2, j=3)min { 0,

0,0+ (GA)

} = 0

AAU (i=5, j=6)min { 0,

0,0+ (AU)

} = -1

-1

0

i

j

Recovering the structure from the DP table

• Complexity O(n3)• Main difference to sequence alignment – we are

tracing back a tree-like structure not a single optimal path (bifurcation introduces branch points).

• Method 1: Leave pointers as you compute the table: for each element of the table store (at most two) pointers to the subsequences used in the solution.

• Method 2: Recover history based on numerical values in the table.– Stacking – check value along diagonal– Bifurcation - find k such that E(i,k)+E(k+1,j) =

E(i,j)

More realistic energy function

Stacking energies

Even more realistic energy function

Loops have destabilizing effect structure (d) should have lower energy that (b).

Destabilizing contribution of loops should depend on the loop length (k).

Stacking has additional stabilizing contribution .

(k) (k)

(k)

More realistic energy function requires slightly more involved

recurrenceE(i,j) = min{ E(i+1,j), E(i,j-1), min{E(i,k)+E(k+1,j), L(i,j)} where

L(i,j) = {(ri,rj) + (j-i-1) if L(i,j) is a hairpin loop;

(ri,rj) + ij-1if hairpin

mink{(ri,rj) + (k)+E(i+k+1,j-1)} if i-bulge

mink {(ri,rj) + (k)+E(i+1,j-k-1)} if j-bulge

mink1,k2{(ri,rj) + (k1+k2)+E(i+k1+1,j-k2-1)} if internal loop

}Extra “min” gives O(n4) algorithm

Covariance method

In a correct multiple alignment RNAs, conserved base pairs are often revealed by the presence of frequent correlated compensatory mutations.

Two boxed positions are covarying to maintain Watson-Crick complementary. This covariation implies a base pair which may then be extended in both directions.

GCCUUCGGGCGACUUCGGUCGGCUUCGGCC

Alignment

Quantities measure of pairwise sequence covariation

Mutual information Mij between two aligned columns i, j

Mij = i,j fxixj log2 (fxixj/fxi fxj)Where fxixj frequency of the pair (observed)

fxi frequency of nucleotide xi at position iObservations:

0 <= Mij <=2

i,j uncorrelated Mij = 0

MI: examples

A

A

C

G

U

U

G

C

fAi = .5fCi = .25fGi = .25fUj = .5fCj = .25fGj = .25

fAU = .5fCG = .25fGC = .25

Mij = xixj fxixj log2 (fxixj/fxi fxj) =

.5 log2 (.5/(.5*.5))+2*.25 log2 (.25/(.25*.25))=

.5 *1 +.5*2 = 1.5A

A

A

A

U

U

U

UMij = 1 log 1 = 0

U

A

C

G

A

U

G

C

Mij = 4*.25 log 4 = 2

i j

Other methods

• HMMs• Stochastic context free grammars

Conclusion

• RNA secondary structure prediction– Single sequence:

• Dot-plot• Nussinov dynamic programming• Energy function

– Covariance analysis• Mutual information• Hidden Markov Models • SCFGs

Finding “most probable structure”

• S – structure then, E(S) free energy of S p(S) = exp(-E(S)/kT)/Q Q = x exp(-E(x)/kT) ) partition function• Problem: computing Q• Method to compute Q – dynamic

programming (similar as presented before but scores are replaced with probabilities and min energy with sum of probabilities).

tRNA

Answer

http://ludwig-sun2.unil.ch/~bsondere/nussinov/form.html#CYK)