PDF 2

RNA secondary structure
prediction methods
• Covariation using mutual information
– requires reliable multiple sequence/structure
alignment of many RNAs
• Stochastic context free grammars (generalization
of HMMs)
• Energy minimization methods
FOCUS of talk on energy minimization, but first
briefly mention other two methods.
Peter Clote
July 2007

Stochastic context free grammars
Context free grammar for
L= {well-balanced parenthesis
expressions}
consists of rules
S λ | (S) | SS
Key notions:
λ is empty word
Terminal symbols: ‘(‘, ‘)’
Variables: S
Peter Clote
July 2007

Derivation
S (S) (SS) ((S)S) (( )S)
(( )(S)) (( )( ))
Leftmost derivation (apply rule to leftmost
occurring variable in each step)
One-one association between leftmost
derivations and parse trees
Peter Clote
July 2007

Context free grammar for RNA
consists of following rules (here,
balanced parentheses correspond to
Watson-Crick or GU base pairs).
S λ | AS | CS | GS | US | ASU |
USA | CSG | GSC | GSU | USG |
SS
Peter Clote
July 2007

• Branching given by rule S SS
• Watson-Crick base pairings given by rules
S ASU | GSC | etc.
• Unpaired bases (e.g. in hairpin loop) given
by rules
S AS | GS | etc.
• Stochastic grammar has probabilities
associated with rule applications
Peter Clote
July 2007

http://www.genetics.wustl.edu/eddy/tRNAscan-SE/
Peter Clote
July 2007

Secondary structure
ACGUACGUACGU
((((....))))
RNA secondary structure is an
outerplanar graph, or equivalently, a
balanced parenthesis expression (only
one kind of parenthesis).
Peter Clote
July 2007

Secondary structure definition
Peter Clote
July 2007

Peter Clote
July 2007

Ding and Lawrence, A statistical sampling algorithm for {RNA} secondary structure
prediction, Nucleic Acids Res., 31(24):7280--7301 (2003)
Peter Clote
July 2007

Type H pseudoknot, taken from Dirks, Pierce, “A partition function
algorithm for nucleic acid secondary structure including pseudoknots
J Comput Chem, 24(13):1664-1677, 2003
Peter Clote
July 2007

PSEUDOBASE collection of classified pseudoknots
http://wwwbio.leidenuniv.nl/~batenburg/pkb.html
Peter Clote
July 2007

Example of pseudoknot causing frameshift, taken from
Von Batenburg’s PSEUDOBASE
http://wwwbio.leidenuniv.nl/~batenburg/PKBase/PKB00240.html
Peter Clote
July 2007

Counting number of
secondary structures
Peter Clote
July 2007

Counting number of secondary
structures
• Secondary structures are balanced parenthesis
expressions with dots
• Catalan numbers give the number of balanced
parenthesis expressions (without dots)
Peter Clote
July 2007

Catalan number c(n-1) is the number of
balanced parenthesis expressions of length n.
Initial Goal: Count number of secondary
structures (Motzkin numbers)
Peter Clote
July 2007

Case 1: n+1 not base paired
Case 2: n+1 base paired with intermediate j
Peter Clote
July 2007

Peter Clote
July 2007

General treatment of cases
Case 1: n+1 not base paired
Case 2: n+1 base paired with i
Case 3: n+1 base paired with intermediate j
Peter Clote
July 2007

Peter Clote
July 2007

Peter Clote
July 2007

Peter Clote
July 2007

Peter Clote
July 2007

Peter Clote
July 2007

Asymptotics for Motzkin numbers
• Is there a closed formula for the asymptotic
number S(n) of secondary structures, where any
base can pair with any base, and where the
threshold for hairpin loops is 1?
• Two methods of solution:
– Viennot, Vauchaussade de Chaumont, “Enumeration
of RNA secondary structures by complexity”, Lecture
Notes in Biomaths. 57 (1985) 360-365. Uses generating
functions and influenced Nebel’s later work.
– Stein, Waterman: applied Bender’s theorem
Peter Clote
July 2007

Peter Clote
July 2007

Peter Clote
July 2007

Peter Clote
July 2007

Peter Clote
July 2007

Bender’s theorem for asymptotics
Peter Clote
July 2007

Peter Clote
July 2007

Peter Clote
July 2007

Peter Clote
July 2007

Peter Clote
July 2007

Peter Clote
July 2007

Peter Clote
July 2007

• Asymptotic (exponential) formula for the number
S(n) of secondary structures is STARTING
POINT for NEUTRAL NETWORKS (Schuster,
Fontana, Stadler, Hofacker, et al.)
Sequence Space {A,C,G,U} n
(Genotype)
Shape Space {‘(‘,’)’,’.’} n
(Phenotype)
Peter Clote
July 2007

• What is the number of secondary structures for a
given RNA nucleotide sequence, where base pairs
are either Watson-Crick or GU wobble pairs, and
hairpin loops have at least 3 (or more generally θ)
unpaired bases?
Peter Clote
July 2007

From combinatorics to
secondary structure energy
minimization algorithms
Peter Clote
July 2007

Key idea
• Absence of pseudoknots in secondary structures
allows determination of optimal secondary
structure for s i ,…,s j to be made independently of
s k ,…,s l whenever i

• Pascal’s triangle is a dynamic programming
computation of binomial coefficients:
C(n,k) = C(n-1,k)+C(n-1,k-1)
1
11
121
1331
14641
Peter Clote
July 2007

Idea of Nussinov-Jacobson: add the contribution
due to each base pair, ignoring stabilizing energies for
stacked base pairs or destabilizing energies of loops
(unpaired) regions.
Peter Clote
July 2007

• M i,j equals the maximum number of base pairs in
s i ,…,s j
• M j,i equals index r such that if r,j base pair then
maximum number of base pairs is realized in
s i ,…,s j
• M j,i used for linear time traceback
Peter Clote
July 2007

Peter Clote
July 2007

Peter Clote
July 2007

Peter Clote
July 2007

Dynamic programming order of filling in matrix:
left to right along principal diagonal, then successively
fill off-diagonals proceeding outwards.
Peter Clote
July 2007

Compute energy matrix M
Cubic time FILL algorithm
Upper triangular part of M contains energies
Lower triangular part of M contains crumbs
for traceback

acktrack(i,j)
Linear time backtracking algorithm
after the energy matrix is filled

Zuker-Stiegler modification of
Nussinov-Jacobson
1) An isolated base pair contributes no energy by itself;
instead consider experimentally measured stacking
energies
5’-XA-3’
3’-YB-5’
2) Consider experimentally measured loop energies for
hairpin,bulge and internal loops (not multiloops).
3) Affine approximation for multiloop energy
a+bI+cU
for I internal base pairs and U unpaired bases in multiloop
having external pair (i,j), i.e. within i+1,…,j-1.
Peter Clote
July 2007

mfold base stacking energies at 37
degrees Celsius
Peter Clote
July 2007

Nearest neighbor model (stacked
base pairs)
See “Thermodynamic parameters for an expanded nearest-neighbor model for
Formation of RNA duplexes with Watson-Crick base pairs”, Xia et al.
Biochemistry 37:14719-14735 (1998)
Peter Clote
July 2007

See “Thermodynamic parameters for an expanded nearest-neighbor model for
Formation of RNA duplexes with Watson-Crick base pairs”, Xia et al.
Biochemistry 37:14719-14735 (1998)
Peter Clote
July 2007

See “Thermodynamic parameters for an expanded nearest-neighbor model for
Formation of RNA duplexes with Watson-Crick base pairs”, Xia et al.
Biochemistry 37:14719-14735 (1998)
Peter Clote
July 2007

Loop energies from Vienna RNA
package v1.4
PUBLIC int hairpin37[31] = {
INF, INF, INF, 570, 560, 560, 540, 590, 560, 640, 650,
660, 670, 678, 686, 694, 701, 707, 713, 719, 725,
730, 735, 740, 744, 749, 753, 757, 761, 765, 769};
PUBLIC int bulge37[31] = {
INF, 380, 280, 320, 360, 400, 440, 459, 470, 480, 490,
500, 510, 519, 527, 534, 541, 548, 554, 560, 565,
571, 576, 580, 585, 589, 594, 598, 602, 605, 609};
PUBLIC int internal_loop37[31] = {
INF, INF, 410, 510, 170, 180, 200, 220, 230, 240, 250,
260, 270, 278, 286, 294, 301, 307, 313, 319, 325,
330, 335, 340, 345, 349, 353, 357, 361, 365, 369};
Peter Clote
July 2007

Classify loops by number of base pairs visible from
Closing pair (i,j).
Hairpin loop: 0-loop
Peter Clote
July 2007

Helix (stacked base pairs),
Bulge, interior loop: 1-loops
Peter Clote
July 2007

Ding and Lawrence, A statistical sampling algorithm for {RNA} secondary structure
prediction, Nucleic Acids Res., 31(24):7280--7301 (2003)
Peter Clote
July 2007

Peter Clote
July 2007

• Waterman introduced the notion of order of
multiloop, which yields an inefficient algorithm
• Nevertheless, this classification is interesting.
Peter Clote
July 2007

Lyngsø and Pederson, RNA pseudoknot prediction in energybased models,
J. Comput. Biol., 7:409-427 (2000)
Peter Clote
July 2007

Suboptimal secondary
structures
Zuker
“On finding all suboptimal foldings of an RNA
molecule”, Zuker, Science 244(4900):48-52
(1989)

Lyngsø and Pederson, RNA pseudoknot prediction in energybased models,
J. Comput. Biol., 7:409-427 (2000)
Peter Clote
July 2007

match. i-match
• “An RNA folding method capable of identifying
pseudoknots and base triples”, by Tabaska, Cary,
Gabow and Stormo Bioinformatics 14(8) 1998.
• BASIC IDEA: Given RNA sequence of length n,
apply maximum weight matching to complete,
weighted (undirected) graph G = (V,E)
– V = {1,2,…,n}
– E = all possible Watson-Crick and GU pair positions
{i,j}
– weight edges by mutual information using reliable
sequence-structure alignment of many orthologous
RNAs.
Peter Clote
July 2007

M
i
= ! , j fxi
x log j 2(
fxi,
x / f j xf
i x )
,
j
x
i
, x
j
Cary-Stormo in ISMB’95 weighted edges
{i,j} by mutual information, and applied Ed
Rothberg’s implementation wmatch of
Gabow’s O(n 3 ) maximum weight matching
algorithm.
Peter Clote
July 2007

i-match algorithm
Pseudoknots can be detected using Cary-Stormo algorithm.
“An RNA folding method capable of identifying pseudoknots
and base triples”, Tabaska et al., Bioinformatics 14 (8) 1998
Peter Clote
July 2007

i-match algorithm contd.
“An RNA folding method capable of identifying pseudoknots
and base triples”, Tabaska et al., Bioinformatics 14 (8) 1998
Peter Clote
July 2007

Handling base triples with bmatch
“An RNA folding method capable of identifying pseudoknots
and base triples”, Tabaska et al., Bioinformatics 14 (8) 1998
Peter Clote
July 2007

Handling base triples, contd.
“An RNA folding method capable of identifying pseudoknots
and base triples”, Tabaska et al., Bioinformatics 14 (8) 1998
Peter Clote
July 2007

Handling base triples, contd.
“An RNA folding method capable of identifying pseudoknots
and base triples”, Tabaska et al., Bioinformatics 14 (8) 1998
Peter Clote
July 2007

Raw output of i-match on tRNA
“An RNA folding method capable of identifying pseudoknots
and base triples”, Tabaska et al., Bioinformatics 14 (8) 1998
Peter Clote
July 2007

tRMA after i-match output
filtration
“An RNA folding method capable of identifying pseudoknots
and base triples”, Tabaska et al., Bioinformatics 14 (8) 1998
Peter Clote
July 2007

match
“An RNA folding method capable of identifying pseudoknots
and base triples”, Tabaska et al., Bioinformatics 14 (8) 1998
Peter Clote
July 2007