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
Stefan Janssen (Bielefeld University) Longer, deeper and affordable kisses. LIX 2010, Paris 1 / 28
Folding nested structures
Stefan Janssen (Bielefeld University) Longer, deeper and affordable kisses. LIX 2010, Paris 2 / 28
maximizing base-pairs
minimizing free energy
O(n3) time
O(n2) space
Folding crossed structures
Stefan Janssen (Bielefeld University) Longer, deeper and affordable kisses. LIX 2010, Paris 3 / 28
general pseudoknotfolding is NP-hard[Lyngsø 2000, Akutsu 2000]
polynomial algorithmsfor special classes
Folding simple pseudoknots
Stefan Janssen (Bielefeld University) Longer, deeper and affordable kisses. LIX 2010, Paris 4 / 28
Folding simple recursive pseudoknots
Stefan Janssen (Bielefeld University) Longer, deeper and affordable kisses. LIX 2010, Paris 5 / 28
loop regions may contain substructures ...
... even simple recursive pseudoknots
already computed
Folding simple recursive pseudoknots
Stefan Janssen (Bielefeld University) Longer, deeper and affordable kisses. LIX 2010, Paris 6 / 28
8 moving boundaries
→ O(n8) time→ O(n2) space
too much computing time
→ canonization= pknotsRG
[Reeder & Giegerich 2004]
Folding canonical simple recursive pseudoknots
Stefan Janssen (Bielefeld University) Longer, deeper and affordable kisses. LIX 2010, Paris 7 / 28
Three rules of canonization1 perfect helices: no loops, no bulges
Folding canonical simple recursive pseudoknots
Stefan Janssen (Bielefeld University) Longer, deeper and affordable kisses. LIX 2010, Paris 8 / 28
Three rules of canonization1 perfect helices: no loops, no bulges
2 maximal helices
Folding canonical simple recursive pseudoknots
Stefan Janssen (Bielefeld University) Longer, deeper and affordable kisses. LIX 2010, Paris 9 / 28
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
What canonization might miss
Stefan Janssen (Bielefeld University) Longer, deeper and affordable kisses. LIX 2010, Paris 10 / 28
Folding kissing hairpins
Stefan Janssen (Bielefeld University) Longer, deeper and affordable kisses. LIX 2010, Paris 11 / 28
Folding kissing hairpins
Stefan Janssen (Bielefeld University) Longer, deeper and affordable kisses. LIX 2010, Paris 12 / 28
12 moving boundaries
→ O(n12) time→ O(n2) space
way too high compute time
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
Kissing hairpin as an overlay of two pseudoknots
Stefan Janssen (Bielefeld University) Longer, deeper and affordable kisses. LIX 2010, Paris 14 / 28
Problem 1: Violating Bellman’s Principle
Stefan Janssen (Bielefeld University) Longer, deeper and affordable kisses. LIX 2010, Paris 15 / 28
Bellman’s Principle: Monotonicity of scoring wrto optimizationE (Kiss) = E (left PK ) + E (right PK )− E (overlap)
Problem 2: Incompatible pseudoknots
Stefan Janssen (Bielefeld University) Longer, deeper and affordable kisses. LIX 2010, Paris 16 / 28
Strategy A
Stefan Janssen (Bielefeld University) Longer, deeper and affordable kisses. LIX 2010, Paris 17 / 28
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
Stefan Janssen (Bielefeld University) Longer, deeper and affordable kisses. LIX 2010, Paris 18 / 28
Strategy B
Stefan Janssen (Bielefeld University) Longer, deeper and affordable kisses. LIX 2010, Paris 19 / 28
Strategy C
Stefan Janssen (Bielefeld University) Longer, deeper and affordable kisses. LIX 2010, Paris 20 / 28
l = boundaryleft (rpk (k))rpk(k) = min (E (PKh,k+2,m,j) , rpk(k + 1))
Relating strategies
Strategies A,B,C are heuristic, but increasingly general.For any sequenxe x :
SearchspaceA(x) ⊆ SearchspaceB(x) ⊆ SearchspaceC (x) ⊂ SearchspacecsrKH(x)
Stefan Janssen (Bielefeld University) Longer, deeper and affordable kisses. LIX 2010, Paris 21 / 28
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
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
Stefan Janssen (Bielefeld University) Longer, deeper and affordable kisses. LIX 2010, Paris 22 / 28
Co-transcriptional folding
Stefan Janssen (Bielefeld University) Longer, deeper and affordable kisses. LIX 2010, Paris 23 / 28
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)?
Co-transcriptional folding
Stefan Janssen (Bielefeld University) Longer, deeper and affordable kisses. LIX 2010, Paris 24 / 28
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)?
Stefan Janssen (Bielefeld University) Longer, deeper and affordable kisses. LIX 2010, Paris 25 / 28
Stefan Janssen (Bielefeld University) Longer, deeper and affordable kisses. LIX 2010, Paris 26 / 28
Thanks for your attention
Problem with Strategy B
Stefan Janssen (Bielefeld University) Longer, deeper and affordable kisses. LIX 2010, Paris 27 / 28
Extra Heuristic for Strategy B
Stefan Janssen (Bielefeld University) Longer, deeper and affordable kisses. LIX 2010, Paris 28 / 28