SLIDE 1
NP-completeness of the direct energy barrier problem without - - PowerPoint PPT Presentation
NP-completeness of the direct energy barrier problem without - - PowerPoint PPT Presentation
NP-completeness of the direct energy barrier problem without pseudoknots Jn Mauch, Anne Condon, Ladislav Stacho, Chris Thachuk Motivation: structure, pathway, barrier Motivation: structure , pathway, barrier Motivation: structure, pathway ,
SLIDE 2
SLIDE 3
Motivation: structure, pathway, barrier
SLIDE 4
Motivation: structure, pathway, barrier
Yin et al., 2008
(multiple copies
- f “fuel”)
SLIDE 5
Motivation: structure, pathway, barrier
Yin et al., 2008
SLIDE 6
Motivation: structure, pathway, barrier
Yin et al., 2008
SLIDE 7
Motivation: structure, pathway, barrier
Yin et al., 2008
SLIDE 8
Motivation: structure, pathway, barrier
Yin et al., 2008
SLIDE 9
Motivation: structure, pathway, barrier
Yin et al., 2008
SLIDE 10
Motivation: structure, pathway, barrier
unfolded
true
SLIDE 11
Motivation: structure, pathway, barrier
energy
unfolded
true
MFE barrier from true to MFE
SLIDE 12
Motivation: literature on energy barriers
SLIDE 13
Motivation: literature on energy barriers
Chen, S.J., Dill, K.A.: RNA folding energy landscapes. Proc. Nat.
- Acad. Sci. 97(2) (January 2000) 646–651
Russell, R., Zhuang, X., Babcock, H., Millett, I., Doniach, S., Chu, S., Herschlag, D.: Exploring the folding landscape of a structured
- RNA. Proc. Nat. Acad. Sci. 99 (2002) 155–160
Shcherbakova, I., Mitra, S., Laederach, A., Brenowitz, M.: Energy barriers, pathways, and dynamics during folding of large, multidomain RNAs. Curr. Opin. Chem. Biol. (2008) 655–666 Treiber, D.K., Williamson, J.R.: Beyond kinetic traps in RNA
- folding. Curr. Opin. Struc. Biol. 11 (2001) 309–314
SLIDE 14
Motivation: literature on energy barriers
Flamm, C., Fontana, W., Hofacker, I.L., Schuster, P.: RNA folding at elementary step resolution. RNA (2000) 325–338 Tang, X., Thomas, S., Tapia, L., Giedroc, D.P., Amato, N.M.: Simulating RNA folding kinetics on approximated energy
- landscapes. J. Mol. Biol. 381 (2008) 1055–1067
Wolfinger, M.T.: The energy landscape of RNA folding. Master’s thesis, University Vienna (2001) Flamm, C., Hofacker, I.L., Stadler, P.F., Wolfinger, M.T.: Barrier trees of degenerate landscapes. Zeitschrift f¨ur Physikalische
SLIDE 15
Our contribution
Q: is there a polynomial-time algorithm to calculate the min-barrier folding pathway between two structures? A: probably not
SLIDE 16
Overview
- clear statement of our result
- some details about our proof
- open questions
SLIDE 17
Notation: secondary structures
SLIDE 18
Notation: secondary structures
I F
SLIDE 19
Notation: secondary structures
I F
8 3 8 10 7
SLIDE 20
Notation: direct folding pathway
8 3 8 10 7
I F
+
- +
SLIDE 21
Notation: energy barrier
+
- +
- 3
- 8
- 18
- 13
b a r r i e r 1 5 8 3 8 10 7
I F
SLIDE 22
Notation: energy barrier
+ +
b a r r i e r 1 5 8 3 8 10 7
I F
- 3
- 8
- 18
- 13
SLIDE 23
The EB-DPKF problem
(Energy Barrier for Direct PseudoKnot Free pathways)
two (pkfree) structures I and F, and integer k is there a direct folding pathway from I to F whose energy barrier is at most k?
Given: Q:
SLIDE 24
Example
what is the min-barrier pathway?
4 8 7 5 10 8 3 10
SLIDE 25
Example
what is the min-barrier pathway?
4 8 7 5 10 8 3 10
+
- +
- +
SLIDE 26
Example
what is the min-barrier pathway?
4 8 7 6 10 8 3 10
+
- +
- +
15 10 5 b a r r i e r 1 2
SLIDE 27
Main result
The EB-DPKF problem is NP-complete
SLIDE 28
Main result
The EB-DPKF problem is NP-complete Proof: we show a polynomial time reduction from a known NP-complete problem, namely 3-Partition, to EB-DPKF
SLIDE 29
3-Partition problem: example
integers 10, 9, 8, 7, 7, 7 can we partition the integers into triples, with the integers in each triple summing to 24? Given: Q:
SLIDE 30
3-Partition problem: example
integers 10, 9, 8, 7, 7, 7 can we partition the integers into triples, with the integers in each triple summing to 24? yes! {9,8,7} and {10,7,7} Given: Q: A:
SLIDE 31
3-Partition problem
integers a1, . . . ,a3m and A (in unary), where A/4 < ai < A/2 and the sum of the ai’s is mA can we partition the integers into m triples, with the integers in each triple summing to A? Given: Q:
SLIDE 32
Reduction: 3-partition → EB-DPKF
SLIDE 33
Reduction: 3-partition → EB-DPKF
➓➒➑➐➐➐ ➉➈➇➆➆➆
triple1 triple2
→
10, 9, 8, 7, 7, 7
k
SLIDE 34
Reduction: 3-partition → EB-DPKF
➓➒➑➐➐➐ ➉➈➇➆➆➆
triple2
→
10, 9, 8, 7, 7, 7
k
Correctness: Suppose that the reduction maps instance x of 3-Partition to instance y of EB-DPKF. Then x is a “yes”-instance → y is a “yes”-instance, and x is a “no”-instance → y is a “no”-instance. triple1
SLIDE 35
Reduction: 3-partition → EB-DPKF
➓ ➈ ➇ ➆ ➐ ➐ triple1
triple2 ➒➑➐➉ ➆➆
➓➒➑➐➐➐ ➉➈➇➆➆➆
triple2
→
10, 9, 8, 7, 7, 7
Correctness (yes → yes): consider folding pathway:
k
triple1
SLIDE 36
Reduction: 3-partition → EB-DPKF
➓ ➈ ➇ ➆ ➐ ➐ triple1
triple2 ➒➑➐➉ ➆➆
{
triple-choosing
{
triple-validating
{
clean-up
➓➒➑➐➐➐ ➉➈➇➆➆➆
triple2
→
10, 9, 8, 7, 7, 7
Correctness (yes → yes): consider folding pathway:
k
triple1
SLIDE 37
Reduction: 3-partition → EB-DPKF
➓ ➈ ➇ ➆ ➐ ➐ triple1
triple2 ➒➑➐➉ ➆➆
➓➒➑➐➐➐ ➉➈➇➆➆➆
triple2
→
10, 9, 8, 7, 7, 7
Correctness (yes → yes): consider folding pathway:
k k
triple1
SLIDE 38
Reduction: 3-partition → EB-DPKF
➈ ➆ ➐ ➐ triple1
triple2 ➒
➐ ➆➆ ➒ ➐➐➐ ➈ ➆➆➆
triple2
→
9, 9, 9, 7, 7, 7
Correctness (no → no): all folding pathways exceed barrier:
➒ ➒ ➈ ➈ ➒ ➈ ➒ ➈
k k
triple1
SLIDE 39
Observations
- the EB-DPKF instance obtained from a
“yes”-instance of 3-Partition has an energy landscape such that
- there are exponentially many “initial”
paths from I that stay within barrier
- few of these continue to F within
barrier
SLIDE 40
Open problems
- is the problem of determining the min-
barrier pathway from I to F hard, if repeat and/or temporary arcs are allowed?
SLIDE 41
SLIDE 42