1. Lecture 21 RNA Secondary Structure Prediction
CS5263 Bioinformatics
Lecture 21
RNA Secondary Structure Prediction

2. Road map Biological roles for RNA What’s “secondary structure”?
How is it represented?
Why is it important?
How to predict?

3. Central dogma DNA RNA Protein The flow of genetic information
transcription
translation
DNA
RNA
Protein
Replication

4. Classical Roles for RNA
mRNA - Message RNA
tRNA - Transfer RNA (~61 kinds, ~ 75nt)
rRNA - Ribosomal RNA (~4 kinds, 120-5k nt)
RNA
Protein
Ribosome

5. Classical Roles for RNA
mRNA
tRNA
rRNA
Ribosome

6. “Semi-classical” RNA
snRNA - small nuclear RNA (splicing: U1, etc, nt)
RNaseP - tRNA processing (~300 nt)
SRP - signal recognition particle; membrane targeting (~ nt)
tmRNA - resetting stalled ribosomes, destroy aberrant mRNA
Telomerase - ( nt)
snoRNA - small nucleolar RNA (many varieties; nt)

7. New Roles for RNA Riboswitch: an mRNA regulates its own activity
siRNA (Nobel prize 2006, Fire & Mello)
microRNAs
saRNA: small activating RNA
Hundreds of families
Rfam release 1, 1/2003: 25 families, 55k instances
Rfam release 7, 3/2005: 503 families, 300k instances

8. Example: Riboswitch

9. Non-coding RNAs 1% of DNA codes for
Dramatic discoveries in last 5 years
100s of new families
Many roles: regulation, transport, stability, catalysis, …
1% of DNA codes for
protein, but 30% of it is copied into RNA, i.e.
ncRNA >> mRNA

10. Take-home message
RNAs play many important roles in the cell beyond the classical roles
Many of which yet to be discovered
RNA functions are determined by structures

11. RNA structure Primary: sequence Secondary: base-pairing
Tertiary: 3D shape

12. RNA base-pairing Watson-Crick Pairing “Wobble Pair” G – U ~1kcal/mole
C-G ~3kcal/mole
A-U ~2kcal/mole
“Wobble Pair” G – U ~1kcal/mole
Non-canonical Pairs

13. tRNA structure

14. Secondary structure prediction
Given: CAUUUGUGUACCU….
Goal:
How can we compute that?

15. Terminology Hairpin Loops Interior loops Stems Multi-branched loop
Bulge loop

16. Pseudoknot 5’ 5 10 15 20 25 30 35 40 45 3’
5’-
ucgacuguaaaaaagcgggcgacuuucagucgcucuuuuugucgcgcgc
-3’ 10 20 30 40
Makes structure prediction hard. Not considered in most algorithms.

17. The Nussinov algorithm
Goal: maximizing the number of base-pairs
Idea: Dynamic programming
Loop matching
Nussinov, Pieczenik, Griggs, Kleitman ’78
Too simple for accurate prediction, but stepping-stone for later algorithms

18. The Nussinov algorithmC U
Problem:
Find the RNA structure with the maximum (weighted) number of nested pairings
Nested: no pseudoknot
ACCACGCUUAAGACACCUAGCUUGUGUCCUGGAGGUCUAUAAGUCAGACCGCGAGAGGGAAGACUCGUAUAAGCG

19. The Nussinov algorithm
Given sequence X = x1…xN,
Define DP matrix: F(i, j) = maximum number of base-pairs if xi…xj folds optimally
Matrix is symmetric, so let i < j

20. The Nussinov algorithm
Can be summarized into two cases:
(i, j) paired: optimal score is 1 + F(i+1, j-1)
(i, j) unpaired: optimal score is
maxk F(i, k) + F(k+1, j)
a number of other ways to summarize, all equivalent

21. The Nussinov algorithm
F(i, i) = 0
F(i+1, j-1) + S(xi, xj)
F(i, j) = max
maxk F(i, k) + F(k+1, j)
S(xi, xj) = 1 if xi, xj can form a base-pair, and 0 otherwise
Generalize: S(A, U) = 2, S(C, G) = 3, S(G, U) = 1
Or other types of scores (later)
F(1, N) gives the optimal score for the whole seq

22. How to fill in the DP matrix?
F(i+1, j-1) + S(xi, xj)
F(i, j) = max
maxk F(i, k) + F(k+1, j)
(i, j) i
i+1
How to fill in the DP matrix?
j–1 j

23. How to fill in the DP matrix?
F(i+1, j-1) + S(xi, xj)
F(i, j) = max
maxk F(i, k) + F(k+1, j)
How to fill in the DP matrix?
j – i = 1

24. How to fill in the DP matrix?
F(i+1, j-1) + S(xi, xj)
F(i, j) = max
maxk F(i, k) + F(k+1, j)
How to fill in the DP matrix?
j – i = 2

25. How to fill in the DP matrix?
F(i+1, j-1) + S(xi, xj)
F(i, j) = max
maxk F(i, k) + F(k+1, j)
How to fill in the DP matrix?
j – i = 3

26. How to fill in the DP matrix?
F(i+1, j-1) + S(xi, xj)
F(i, j) = max
maxk F(i, k) + F(k+1, j)
How to fill in the DP matrix?
j – i = N - 1

27. Minimum Loop length Sharp turns unlikely
Let minimum length of hairpin loop be 1
F(i, j) = 0 for j – i < 2
U  A
G  C
C  G G C

28. Algorithm Initialization: F(i, i) = 0; for i = 1 to N
Iteration:
For L = 1 to N-1
For i = 1 to N – l
j = min(i + L, N)
F(i+1, j -1) + s(xi, xj)
F(i, j) = max
max{ i  k < j } F(i, k) + F(k+1, j)
Termination:
Best score is given by F(1, N)
(Need to trace back; refer to the Durbin book)

29. Complexity Time complexity: O(N3) Memory: O(N2) For L = 1 to N-1
For i = 1 to N – l
j = min(i + L, N)
F(i+1, j -1) + s(xi, xj)
F(i, j) = max
max{ i  k < j } F(i, k) + F(k+1, j)
Time complexity: O(N3)
Memory: O(N2)

30. Example RNA sequence: GGGAAAUCC Only count # of base-pairs
G-C = 1
G-U = 1
Minimum hairpin loop length = 1

31. G G G A A A U C C
G G G A A A U C C

32. G G G A A A U C C 1
G G G A A A U C C

33. G G G A A A U C C 1
G G G A A A U C C

34. G G G A A A U C C 1
G G G A A A U C C

35. AAA G  U G  C AA AA A  U A  U G  C G G  C G G G G A A A U C C 11 2 3
G  U
G  C
AAA
G G G A A A U C C
A  U
G  C G
A  U G
G  C AA AA

36. AAA G  U G  C AA AA A  U A  U G  C G G  C G G G G A A A U C C 11 2 3
G  U
G  C
AAA
G G G A A A U C C
A  U
G  C G
A  U G
G  C AA AA

37. AAA G  U G  C AA AA A  U A  U G  C G G  C G G G G A A A U C C 11 2 3
G  U
G  C
AAA
G G G A A A U C C
A  U
G  C G
A  U G
G  C AA AA

38. AAA G  U G  C AA AA A  U A  U G  C G G  C G G G G A A A U C C 11 2 3
G  U
G  C
AAA
G G G A A A U C C
A  U
G  C G
A  U G
G  C AA AA

39. Energy minimization For L = 1 to N-1 For i = 1 to N – l
j = min(i + L, N);
E(i+1, j -1) + e(xi, xj)
E(i, j) = min
min{ i  k < j } E(i, k) + E(k+1, j)
e(xi, xj) represents the energy for xi base pair with xj
Energy are negative values. Therefore minimization rather than maximize.
More complex energy rules: energy depends on neighboring bases

40. Terminology Hairpin Loops Interior loops Stems Multi-branched loop
Bulge loop

41. The Zuker algorithm – main ideas
Instead of base pairs, pairs of base pairs (more accurate)
Separate score for bulges
Separate score for different-size & composition of loops
Separate score for interactions between stem & beginning of loop
Use additional matrix to remember current state. similar to affine-gap alignment.

42. Two popular implementation
mFold by Zuker
RNAfold in the Vienna package (Hofacker)
Includes several useful utilities, such as structure comparison, searching, base-paring probability from partition functions, etc.

43. Accuracy 50-70% for sequences up to 300 nt Not perfect, but useful
Possible reasons:
Energy rule not perfect: 5-10% error
Many alternative structures within this error range
Alternative structure do exist
Structure may change in presence of other molecules

44. Comparative structure prediction
Given K homologous aligned RNA sequences:
Human aagacuucggaucuggcgacaccc
Mouse uacacuucggaugacaccaaagug
Worm aggucuucggcacgggcaccauuc
Fly ccaacuucggauuuugcuaccaua
Orc aagccuucggagcgggcguaacuc
If ith and jth positions are always base paired and covary, then they are likely to be paired

45. Mutual information
fab(i,j): # of times the pair a, b are in positions i, j
fa (i): # of times the base a is in positions i
aagacuucggaucuggcgacaccc
uacacuucggaugacaccaaagug
aggucuucggcacgggcaccauuc
ccaacuucggauuuugcuaccaua
aagccuucggagcgggcguaacuc
fgc(3,13) = 3/5
fcg(3,13) = 1/5
fau(3,13) = 1/5
fg(3) = 3/5
fc(3) = 1/5
fa(3) = 1/5
fc(13) = 3/5
fg(13) = 1/5
fu(13) = 1/5

46. Mutual information Also called covariance score
M is high if base a in position i always follow by base b in position j
Does not require a to base-pair with b
Advantage: can detect non-canonical base-pairs
However, M = 0 if no mutation at all, even if perfect base-pairs
aagacuucggaucuggcgacaccc
uacacuucggaugacaccaaagug
aggucuucggcacgggcaccauuc
ccaacuucggauuuugcuaccaua
aagccuucggagcgggcguaacuc
One way to get around is to combine covariance and energy scores

47. Comparative structure prediction
Given a multiple alignment, can infer structure that maximizes the sum of mutual information, by DP
However, alignment is hard, since structure often more important than sequence

48. Comparative structure prediction
In practice:
Get multiple alignment
Find covarying bases – deduce structure
Improve multiple alignment (by hand)
Go to 2
A manual EM process!!

49. Comparative structure prediction
Align then fold
Align and fold
Fold then align

50. Context-free Grammar for RNA Secondary Structure
S = SS | aSu | cSg | uSa | gSc | L
L = aL | cL | gL | uL |  S ag u cg
aaacgg
ugcc S S S L S S
a L L a L 
a c g g a g u g c c c g u

51. Stochastic Context-free Grammar (SCFG)
Probabilistic context-free grammar
Probabilities can be converted into weights
CFG vs SCFG is similar to RG vs HMM
S = SS
S = aSu | uSa | L
S = cSg | gSc | L
S = uSg | gSu | L
L = aL | cL | gL | uL | 
e(xi, xj) + F(i+1, j-1)
F(i, j) = max L(i, j)
maxk (F(i, k) + F(k+1, j))
L(i, j) = 0 2 3 1

52. SCFG Decoding
Decoding: given a grammar (SCFG/HMM) and a sequence, find the best parse (highest probability or score)
CYK algorithm (Viterbi)
The Nussinov and Zuker algorithms are essentially special cases of CYK
CYK and SCFG are also used in other domains (NLP, Compiler, etc).

53. SCFG Evaluation Given a sequence and a SCFG model
Estimate P(seq is generated by model), summing over all possible paths
Inside-outside algorithm
Analogous to forward-background
Inside: bottom-up parsing (P(xi..xj))
Outside: top-down parsing (P(x1..xi-1 xj+1..xN))
Can calculate base-paring probability
Analogous to posterior decoding
Essentially the same idea implemented in the Vienna RNAfold package

54. SCFG Learning Covariance model: similar to profile HMMs
Given a set of sequences with common structures, simultaneously learn SCFG parameters and optimally parse sequences into states
EM on SCFG
Inside-outside algorithm
Efficiency is a bottleneck
Have been successfully applied to predict tRNA genes and structures
tRNAScan

55. Future directions Structure prediction Structural comparison tools
Secondary
Tertiary
Structural comparison tools
Structural alignment
Structure search tools
“RNA-BLAST”
Structural motif finding
“RNA-MEME”