1. CISC 467/667 Intro to Bioinformatics (Spring 2007) RNA secondary structure
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2. Mainly as “information carrier” in protein synthesis
Facts about RNAs
Mainly as “information carrier” in protein synthesis
mRNA
tRNA
Also as catalysts
Ribozymes
Also as regulator
RNAi
Single strand
Intra-molecular pairing
Secondary structure
Essential for sequence stability and function
Similarity in secondary structure plays a more important role than primary sequence in determining homology for RNAs
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6. hybridization pairing: A-U, and C-G
RNA stem loop
hybridization pairing: A-U, and C-G
where “-” stands for a pairing, and “x” for no pairing.
In the above example, seq1 and seq2 fold into a similar structure, whereas seq3 does not.
Pairwise alignments disregarding such structural restrictions may be misleading; seq2 and seq3 have 70% sequence identity, seq1 and seq3 have 60%, whereas seq1 and seq2 have only 30%. G C A U U A C G C A G A U x C C x U G x G
seq1
seq2
seq3
seq1
C A G G A A A C U G
seq2
G C U G C A A A G C
seq3
G C U G C A A C U G
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7. Representations for palindromic structures 1)2) 3) G C A U
C A G G A A A C U G
C A G G A A A C U G
( ( ( ) ) )
There is a 1-1 correspondence between RNA secondary structures and well-balanced parenthesis expressions, where the balancing parentheses correspond to base pairings via hydrogen bonds.
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8. Secondary structure prediction
Base pair maximization algorithm [Nussinov]
mi,j: maximal # of base pairs that can be formed for sequence si…sj.
di,j = 1, if si and sj are paired
= 0, otherwise
Initialization
mi,i-1 = for i= [2, L]
mi,i = for i = [1,L]
Recursion
mi,j= max {mi+1,j,
mi,j-1,
mi+1,j-1+ di,j,
maxi<k<j {mi,k+ mk+1,j} // bifurcation }
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9. Dynamic programming table1 2 3 4 5 6 7 8 9 G G G A A A U C C 1 G 1 2 3 2 G 1 2 3 3 G 1 2 2 4 A 1 1 1 A A 5 A 1 1 1 6 A 1 1 1 A U 7 U G C 8 C G C 9 C G
Time Complexity: O(L3), where L is sequence length
Space complexity: O(L2)
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10. Secondary structure prediction
Minimum energry algorithm [Zuker]
Ei,j: minimum energy for an optimal fold formed by sequence si…sj.
ei,j = -5, if si , sj is CG or GC
= -4, if si , sj is AU or UA
= -1, if si , sj is GU or UG
Initialization
ei,i-1 = for i= [2, L]
ei,i = for i = [1,L]
Recursion
Ei,j= min {Ei+1,j,
Ei,j-1,
Ei+1,j-1+ ei,j,
mini<k<j {Ei,k+ Ek+1,j} // bifurcation }
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11. We know the “language” is not random
Motivation:
Both DNA and protein sequences can be considered as sentences in a “language”.
We know the “language” is not random
What’s the grammar of the language?
“The linguistics of DNA” – David Searls, American Scientist, 80(
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12. Chomsky hierarchy of transformational grammars (1956)
Regular expression (no recursive structure, cannot handle palindromes)
W  aW, or W  a
Context-free
W  
Context-sensitive
1W 2  1  2
Unrestricted grammars
1W 2  
Notations:
a: any terminals;
, : any string of non-terminals and/or terminals, including the null string;
: any string of non-terminals and/or terminals, not including the null string.
Note: Chomsky normal form: any CFG can be written such that all productions have the following forms:
Wv  Wy Wz
Wv a
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13. To capture the palindromic structure, a context-free grammar can be given as follows.
S  aW1u | cW1g | gW1c | uW1a
W1 aW2u| cW2g| gW2c| uW2a
W2 aW3u| cW3g| gW3c| uW3a
W3  gaaa | gcaa
seq1
C A G G A A A C U G
seq2
G C U G C A A A G C
c a g g a a a c u g w3 w2 w1 S
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14. Stochastic grammars Motivations:
Irregularities (or exceptions to the grammar) in languages
Such irregularities can be taken care of by “growing” your grammar; introducing new non-terminals and new production rules.
It is useful to differentiate productions that account for a large portion of the language from those that are rare exceptions.
This can be achieved by assigning probabilities to various productions.
For any non-terminals, the probabilities of all its possible productions must add to 1.
Given a sentence x, a stochastic grammar  parsing the sentence will assign a probability P(x|) that the sentence belongs to the language specified by the grammar, whereas, a conventional grammar will just give a yes-or-no answer.
x language P(x| ) =1
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15. Stochastic CFG for sequence modeling
Calculate optimal alignment of a sequence to a parameterized SCFG. [Cocke-Younger-Kasami algorithm]
Calculate probability for a given sequence to belong to the language specified by SCFG. [inside algorithm, and inside-outside algorithm]
Given a set of example sequences/structures, estimate optimal probability parameters for a SCFG.
Note:
The issues addressed above by SCFGs are quite similar to those for HMMs; actually it can be proved that HMMs are equivalent to stochastic regular grammars.
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16. Resources
Lecture notes:
RNA informatics:
Software:
-Vienna RNA fold
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