longer, deeper and affordable kisses. · folding kissing hairpins stefan janssen (bielefeld...
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Longer, deeper and affordable kisses.
Corinna Theis, Stefan Janssen and Robert Giegerich
Faculty of Technology & Center for BiotechnologyBielefeld University, Germany
LIX Bioinformatics Colloquium,8-10th November 2010,
Ecole Polytechnique, Paris
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Folding nested structures
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maximizing base-pairs
minimizing free energy
O(n3) time
O(n2) space
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Folding crossed structures
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general pseudoknotfolding is NP-hard[Lyngsø 2000, Akutsu 2000]
polynomial algorithmsfor special classes
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Folding simple pseudoknots
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Folding simple recursive pseudoknots
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loop regions may contain substructures ...
... even simple recursive pseudoknots
already computed
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Folding simple recursive pseudoknots
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8 moving boundaries
→ O(n8) time→ O(n2) space
too much computing time
→ canonization= pknotsRG
[Reeder & Giegerich 2004]
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Folding canonical simple recursive pseudoknots
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Three rules of canonization1 perfect helices: no loops, no bulges
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Folding canonical simple recursive pseudoknots
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Three rules of canonization1 perfect helices: no loops, no bulges
2 maximal helices
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Folding canonical simple recursive pseudoknots
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Three rules of canonization1 perfect helices: no loops, no bulges
2 maximal helices
3 helices must not overlap
The clue
optimal helices can beprecomputed in O(n2) time andspace
pknotsRG: 4 moving boundaries
→ O(n4) time→ O(n2) space
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What canonization might miss
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Folding kissing hairpins
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Folding kissing hairpins
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12 moving boundaries
→ O(n12) time→ O(n2) space
way too high compute time
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Folding kissing hairpins
Stefan Janssen (Bielefeld University) Longer, deeper and affordable kisses. LIX 2010, Paris 13 / 28
12 moving boundaries
→ O(n12) time→ O(n2) space
way too high compute time
with canonization
→ O(n6) time→ O(n2) space
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Kissing hairpin as an overlay of two pseudoknots
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Problem 1: Violating Bellman’s Principle
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Bellman’s Principle: Monotonicity of scoring wrto optimizationE (Kiss) = E (left PK ) + E (right PK )− E (overlap)
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Problem 2: Incompatible pseudoknots
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Strategy A
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Summary Strategy A
Strategy A is correct, symmetric and heuristic:
1 Assume left PK optimal and find best consistent right PK
2 Assume right PK optimal and find best consistent left PK
3 Return better overlay structure
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Strategy B
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Strategy C
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l = boundaryleft (rpk (k))rpk(k) = min (E (PKh,k+2,m,j) , rpk(k + 1))
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Relating strategies
Strategies A,B,C are heuristic, but increasingly general.For any sequenxe x :
SearchspaceA(x) ⊆ SearchspaceB(x) ⊆ SearchspaceC (x) ⊂ SearchspacecsrKH(x)
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O(n4) O(n4) O(n5) O(n6) timeO(n2) O(n3) O(n2) O(n2) space
2.4% 5.5% 2.4% 0% missed opt.csrKH
1 1.2 11.2 508.0 time1 9.2 1.3 4.5 space
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Evaluation
Evaluation in terms of Pseudobase:61 pseudoknotted structures, including (only) 6 kissing hairpin motifsAgreement of strategies:
C 6= B 6= A C = B 6= A C = B = A
2 2 57
Correctness of predictions:
A,B,C agree and correctly predict the 6 true kissing hairpins
for 7 simpler pseudoknots, a kissing hairpin is predicted
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Co-transcriptional folding
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Strategy A: in all cases, it is the left PK which is locally optimal
→ double speed of Strategy A, by dropping symmetric case?
→ more complex motifs as successive overlays in O(n4) / O(n2)?
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Co-transcriptional folding
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Strategy A: in all cases, it is the left PK which is locally optimal
→ double speed of Strategy A, by dropping symmetric case?
→ more complex motifs as successive overlays in O(n4) / O(n2)?
![Page 25: Longer, deeper and affordable kisses. · Folding kissing hairpins Stefan Janssen (Bielefeld University) Longer, deeper and a ordable kisses. LIX 2010, Paris 13 / 28 12 moving boundaries!O(n12)](https://reader034.vdocument.in/reader034/viewer/2022042309/5ed6d3f0126754677f63015c/html5/thumbnails/25.jpg)
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Thanks for your attention
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Problem with Strategy B
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Extra Heuristic for Strategy B
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