NP-completeness of the direct energy barrier problem without - PowerPoint PPT Presentation

NP-completeness of the direct energy barrier problem without pseudoknots Ján Mañuch, Anne Condon, Ladislav Stacho, Chris Thachuk

Motivation: structure, pathway, barrier

Motivation: structure , pathway, barrier

Motivation: structure, pathway , barrier (multiple copies of “fuel”) Yin et al., 2008

Motivation: structure, pathway , barrier Yin et al., 2008

Motivation: structure, pathway , barrier Yin et al., 2008

Motivation: structure, pathway , barrier Yin et al., 2008

Motivation: structure, pathway , barrier Yin et al., 2008

Motivation: structure, pathway, barrier Yin et al., 2008

Motivation: structure, pathway, barrier unfolded true

Motivation: structure, pathway, barrier unfolded barrier from true to MFE energy true MFE

Motivation: literature on energy barriers

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

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

Our contribution Q: is there a polynomial-time algorithm to calculate the min-barrier folding pathway between two structures? A: probably not

Overview • clear statement of our result • some details about our proof • open questions

Notation: secondary structures

Notation: secondary structures I F

Notation: secondary structures 7 I 8 3 F 10 8

Notation: direct folding pathway 7 - - - I + + 8 3 F 10 8

Notation: energy barrier 7 - - - I + + 8 3 F -3 10 8 5 -8 1 r e i r -13 r a b -18

Notation: energy barrier 7 I + + 8 3 F -3 10 8 5 -8 1 r e i r -13 r a b -18

The EB-DPKF problem (Energy Barrier for Direct PseudoKnot Free pathways) two (pkfree) structures I and F, Given: and integer k is there a direct folding pathway Q: from I to F whose energy barrier is at most k?

Example what is the min-barrier pathway? 3 7 4 5 8 10 8 10

Example what is the min-barrier pathway? 3 7 - - - - - + + + 4 5 8 10 8 10

Example what is the min-barrier pathway? 3 7 - - - - - + + + 4 6 8 15 10 2 1 10 8 r 10 e 5 i r r a b 0

Main result The EB-DPKF problem is NP-complete

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

3-Partition problem: example Given: integers 10, 9, 8, 7, 7, 7 Q: can we partition the integers into triples, with the integers in each triple summing to 24?

3-Partition problem: example Given: integers 10, 9, 8, 7, 7, 7 Q: can we partition the integers into triples, with the integers in each triple summing to 24? A: yes! {9,8,7} and {10,7,7}

3-Partition problem Given: integers a 1 , . . . ,a 3m and A (in unary), where A/4 < a i < A/2 and the sum of the a i ’s is mA Q: can we partition the integers into m triples, with the integers in each triple summing to A?

Reduction: 3-partition → EB-DPKF

Reduction: 3-partition → EB-DPKF → 10, 9, 8, 7, 7, 7 k ➓➒➑➐➐➐ ➉➈➇➆➆➆ triple1 triple2

Reduction: 3-partition → EB-DPKF → 10, 9, 8, 7, 7, 7 k ➓➒➑➐➐➐ ➉➈➇➆➆➆ triple1 triple2 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.

Reduction: 3-partition → EB-DPKF → 10, 9, 8, 7, 7, 7 k ➓➒➑➐➐➐ ➉➈➇➆➆➆ triple1 triple2 Correctness (yes → yes): consider folding pathway: triple2 ➒➑➐➉ ➆➆ ➓ ➈ ➇ ➆ ➐ ➐ triple1

Reduction: 3-partition → EB-DPKF → 10, 9, 8, 7, 7, 7 k ➓➒➑➐➐➐ ➉➈➇➆➆➆ triple1 triple2 Correctness (yes → yes): consider folding pathway: triple2 ➒➑➐➉ ➆➆ ➓ ➈ ➇ ➆ ➐ ➐ triple1 { { { triple-choosing triple-validating clean-up

Reduction: 3-partition → EB-DPKF → 10, 9, 8, 7, 7, 7 k ➓➒➑➐➐➐ ➉➈➇➆➆➆ triple1 triple2 Correctness (yes → yes): consider folding pathway: triple2 ➒➑➐➉ ➆➆ ➓ ➈ ➇ ➆ ➐ ➐ triple1 k

Reduction: 3-partition → EB-DPKF → 9, 9, 9, 7, 7, 7 k ➒ ➒ ➒ ➐➐➐ ➈ ➈ ➈ ➆➆➆ triple1 triple2 Correctness (no → no): all folding pathways exceed barrier: triple2 ➒ ➒ ➈ ➈ ➆ ➐ ➐ triple1 ➒ ➐ ➈ ➆➆ k

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

Open problems • is the problem of determining the min- barrier pathway from I to F hard, if repeat and/or temporary arcs are allowed?

Example 4 4 4 8 5 4 8 2 6 7 10 8 10 8 12 4 weights enforce “choose-and-rank” strategy, to stay within barrier of 12