Impact Of The Energy Model On The Complexity Of RNA Folding With Pseudoknots
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1 Impact Of The Energy Model On The omplexity Of RN Folding With Pseudoknots Saad Sheikh, Rolf Backofen Yann Ponty, niversity of Florida, ainesville, S lbert Ludwigs niversity, Freiburg, ermany LIX, NRS/Ecole Polytechnique, France MIB Team-Project, INRI, Saclay, France July 5th PM 12 Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM 12 1 / 17
2 Fundamental dogma of molecular biology DN Transcription RN Translation Proteins Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM 12 2 / 17
3 Fundamental dogma of molecular biology DN Tra ns fe on r Transcription M i rat atu RN RN roles: Regulation Translation Syn thes is Messenger Translation agent Regulator Modifier atalytic... Proteins Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM 12 2 / 17
4 RN structure Primary structure Secondary structure Tertiary structure (Matching) Source: 5s rrn (PDBID: 1K73:B) Bottom-up approach to molecular biology nderstand and predict how RN folds to decypher its function(s). Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM 12 3 / 17
5 RN structure Primary structure Secondary + structure Tertiary structure (Matching) Source: 5s rrn (PDBID: 1K73:B) Bottom-up approach to molecular biology nderstand and predict how RN folds to decypher its function(s). Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM 12 3 / 17
6 rossing interactions Non-canonical base-pairs: ny base-pair other than {(-), (-), (-)} OR interacting in a non-standard way (W/W-is) [Leontis 01]. anonical base-pair (W/W-is) Non-canonical base-pair (Sugar/W-Trans) Pseudoknots: rossing sets of nested stable base-pairs roup I Ribozyme (PDBID: 1Y0Q:) Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM 12 4 / 17
7 rossing interactions Non-canonical base-pairs: ny base-pair other than {(-), (-), (-)} OR interacting in a non-standard way (W/W-is) [Leontis 01]. anonical base-pair (W/W-is) Non-canonical base-pair (Sugar/W-Trans) Pseudoknots: rossing sets of nested stable base-pairs roup I Ribozyme (PDBID: 1Y0Q:) Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM 12 4 / 17
8 rossing interactions Non-canonical base-pairs: ny base-pair other than {(-), (-), (-)} OR interacting in a non-standard way rossing (W/W-is) interactions, [Leontis 01]. once ignored, are now ubiquitous! Example: roup II Intron (PDB ID: 3II) anonical base-pair (W/W-is) Non-canonical base-pair (Sugar/W-Trans) Pseudoknots: rossing sets of nested stable base-pairs roup I Ribozyme (PDBID: 1Y0Q:) Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM 12 4 / 17
9 Problem statement RN structure S: (Partial) matching of positions in sequence w Motifs: Sequence/structure features (e.g. Base-pairs, Stacking pairs, Loops... ) Energy model: Motif Free-energy contribution ( ) R {+ } Free-Energy E w(s): Sum over (independently contributing) motifs in S Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM 12 5 / 17
10 Problem statement RN structure S: (Partial) matching of positions in sequence w Motifs: Sequence/structure features (e.g. Base-pairs, Stacking pairs, Loops... ) Energy model: Motif Free-energy contribution ( ) R {+ } Free-Energy E w(s): Sum over (independently contributing) motifs in S Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM 12 5 / 17
11 Problem statement RN structure S: (Partial) matching of positions in sequence w Motifs: Sequence/structure features (e.g. Base-pairs, Stacking pairs, Loops... ) Energy model: Motif Free-energy contribution ( ) R {+ } Free-Energy E w(s): Sum over (independently contributing) motifs in S Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM 12 5 / 17
12 Problem statement RN structure S: (Partial) matching of positions in sequence w Motifs: Sequence/structure features (e.g. Base-pairs, Stacking pairs, Loops... ) Energy model: Motif Free-energy contribution ( ) R {+ } Free-Energy E w(s): Sum over (independently contributing) motifs in S Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM 12 5 / 17
13 Problem statement RN structure S: (Partial) matching of positions in sequence w Motifs: Sequence/structure features (e.g. Base-pairs, Stacking pairs, Loops... ) Energy model: Motif Free-energy contribution ( ) R {+ } Free-Energy E w(s): Sum over (independently contributing) motifs in S E S = Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM 12 5 / 17
14 Problem statement RN structure S: (Partial) matching of positions in sequence w Motifs: Sequence/structure features (e.g. Base-pairs, Stacking pairs, Loops... ) Energy model: Motif Free-energy contribution ( ) R {+ } Free-Energy E w(s): Sum over (independently contributing) motifs in S E S = ( ) + ( ) + ( ) + ( ) + ( ) Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM 12 5 / 17
15 Problem statement RN structure S: (Partial) matching of positions in sequence w Motifs: Sequence/structure features (e.g. Base-pairs, Stacking pairs, Loops... ) Energy model: Motif Free-energy contribution ( ) R {+ } Free-Energy E w(s): Sum over (independently contributing) motifs in S ( E S = ( + ) ( + ) ( + ) ( + ) ( + ) ( + ) ) ( + ) Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM 12 5 / 17
16 Problem statement RN structure S: (Partial) matching of positions in sequence w Motifs: Sequence/structure features (e.g. Base-pairs, Stacking pairs, Loops... ) Energy model: Motif Free-energy contribution ( ) R {+ } Free-Energy E w(s): Sum over (independently contributing) motifs in S Definition (RN-PK-FOLD(E) problem) Input: RN sequence w {,,, }. Output: Matching S, having Minimal Free-Energy E w(s ). Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM 12 5 / 17
17 Energy models Three models, based on interacting positions (i, j): Base-pair model B: Nucleotides (w i, w j ) at (i, j) B(w i, w j ) Nearest-neighbor model N : Nucl. at (i, j) and (i+1, j-1) + partners (or ) N (w i, w j, w i+1, w j 1, w mi+1, w mj 1 ) Stacking pairs model S: Nucl. at (i, j) and (i+1, j-1) only if latter paired S(w i, w j, w i+1, w j 1 ) a h i p q 5 b g j o r c f k n s d e l m t 3 a h p q g j o r c f k n d e m t Solved in O(n 3 ) [Tabaska 98] (Max-weighted matching) nrealistic! Base-pairs (B) Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM 12 6 / 17
18 Energy models Three models, based on interacting positions (i, j): Base-pair model B: Nucleotides (w i, w j ) at (i, j) B(w i, w j ) Nearest-neighbor model N : Nucl. at (i, j) and (i+1, j-1) + partners (or ) N (w i, w j, w i+1, w j 1, w mi+1, w mj 1 ) Stacking pairs model S: Nucl. at (i, j) and (i+1, j-1) only if latter paired S(w i, w j, w i+1, w j 1 ) 5 a h i p q b g j o r c f k n s d e l m t 3 a h b g j p q a h i p q g j o r c f k n s d e m t d e k n NP-hard [Lyngsø 00, kutsu 00] Too expressive? l m t Nearest neighbor(s) Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM 12 6 / 17
19 Energy models Three models, based on interacting positions (i, j): Base-pair model B: Nucleotides (w i, w j ) at (i, j) B(w i, w j ) Nearest-neighbor model N : Nucl. at (i, j) and (i+1, j-1) + partners (or ) N (w i, w j, w i+1, w j 1, w mi+1, w mj 1 ) Stacking pairs model S: Nucl. at (i, j) and (i+1, j-1) only if latter paired S(w i, w j, w i+1, w j 1 ) a h i p q 5 b g j o r p q o r c f k n s d e l m t 3 c f d e Stacking pairs (S) aptures stablest motifs Still NP-hard [Lyngsø 04]... but PTS [Lyngsø 04] Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM 12 6 / 17
20 State of the art omp. Base-pairs Stacking-Pairs Nearest-Neighbor P [Nussinov 80] P [Ieong 03] P [Zuker 81] Non-crossing pprox. Planar omp.??? pprox. omp. 2-approx. [Ieong 03] P [Tabaska 98] NP-Hard [Ieong 03] 2-approx. [Ieong 03] NP-Hard [Lyngsø 04] eneral pprox. ε-approx. O(n 41/ε ) [Lyngsø 04] NP-Hard [Ieong 03]??? NP-Hard [Lyngsø 00, kutsu 00] Missing: Qualitative difference between Stacking-pairs and Nearest-Neighbor models? Influence of M on hardness/approx. ratio (only unit-valued studied)??? Biologists demand (Biology deserves) honest hardness results: Energy model as input: Pandora s box (e.g. RN folding on infinite alphabet!) Model as parameter: Is problem hard... Sometimes ( M)? Dishonest lways ( M)? lmost surely (w. p. 1)? Honest nder reasonable assumptions + parameterization? lmost honest Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM 12 7 / 17
21 State of the art omp. Base-pairs Stacking-Pairs Nearest-Neighbor P [Nussinov 80] P [Ieong 03] P [Zuker 81] Non-crossing pprox. Planar omp.??? pprox. omp. 2-approx. [Ieong 03] P [Tabaska 98] NP-Hard [Ieong 03] 2-approx. [Ieong 03] NP-Hard [Lyngsø 04] eneral pprox. ε-approx. O(n 41/ε ) [Lyngsø 04] NP-Hard [Ieong 03]??? NP-Hard [Lyngsø 00, kutsu 00] Missing: Qualitative difference between Stacking-pairs and Nearest-Neighbor models? Influence of M on hardness/approx. ratio (only unit-valued studied)??? Biologists demand (Biology deserves) honest hardness results: Energy model as input: Pandora s box (e.g. RN folding on infinite alphabet!) Model as parameter: Is problem hard... Sometimes ( M)? Dishonest lways ( M)? lmost surely (w. p. 1)? Honest nder reasonable assumptions + parameterization? lmost honest Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM 12 7 / 17
22 State of the art omp. Base-pairs Stacking-Pairs Nearest-Neighbor P [Nussinov 80] P [Ieong 03] P [Zuker 81] Non-crossing pprox. Planar omp.??? pprox. omp. 2-approx. [Ieong 03] P [Tabaska 98] NP-Hard [Ieong 03] 2-approx. [Ieong 03] NP-Hard [Lyngsø 04] eneral pprox. ε-approx. O(n 41/ε ) [Lyngsø 04] NP-Hard [Ieong 03]??? NP-Hard [Lyngsø 00, kutsu 00] Missing: Qualitative difference between Stacking-pairs and Nearest-Neighbor models? Influence of M on hardness/approx. ratio (only unit-valued studied)??? Biologists demand (Biology deserves) honest hardness results: Energy model as input: Pandora s box (e.g. RN folding on infinite alphabet!) Model as parameter: Is problem hard... Sometimes ( M)? Dishonest lways ( M)? lmost surely (w. p. 1)? Honest nder reasonable assumptions + parameterization? lmost honest Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM 12 7 / 17
23 State of the art omp. Base-pairs Stacking-Pairs Nearest-Neighbor P [Nussinov 80] P [Ieong 03] P [Zuker 81] Non-crossing pprox. Planar omp.??? pprox. omp. 2-approx. [Ieong 03] P [Tabaska 98] NP-Hard [Ieong 03] 2-approx. [Ieong 03] NP-Hard [Lyngsø 04] eneral pprox. ε-approx. O(n 41/ε ) [Lyngsø 04] NP-Hard [Ieong 03]??? NP-Hard [Lyngsø 00, kutsu 00] Missing: Qualitative difference between Stacking-pairs and Nearest-Neighbor models? Influence of M on hardness/approx. ratio (only unit-valued studied)??? Biologists demand (Biology deserves) honest hardness results: Energy model as input: Pandora s box (e.g. RN folding on infinite alphabet!) Model as parameter: Is problem hard... Sometimes ( M)? Dishonest lways ( M)? lmost surely (w. p. 1)? Honest nder reasonable assumptions + parameterization? lmost honest Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM 12 7 / 17
24 State of the art omp. Base-pairs Stacking-Pairs Nearest-Neighbor P [Nussinov 80] P [Ieong 03] P [Zuker 81] Non-crossing pprox. Planar omp.??? pprox. omp. 2-approx. [Ieong 03] P [Tabaska 98] NP-Hard [Ieong 03] 2-approx. [Ieong 03] NP-Hard [Lyngsø 04] eneral pprox. ε-approx. O(n 41/ε ) [Lyngsø 04] NP-Hard [Ieong 03]??? NP-Hard [Lyngsø 00, kutsu 00] Missing: Qualitative difference between Stacking-pairs and Nearest-Neighbor models? Influence of M on hardness/approx. ratio (only unit-valued studied)??? Biologists demand (Biology deserves) honest hardness results: Energy model as input: Pandora s box (e.g. RN folding on infinite alphabet!) Model as parameter: Is problem hard... Sometimes ( M)? Dishonest lways ( M)? lmost surely (w. p. 1)? Honest nder reasonable assumptions + parameterization? lmost honest Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM 12 7 / 17
25 (lmost!)-honest hardness of RN-PK-FOLD(S) For any stacking energy model S, such that: Only /, / and / pairs are allowed ny other X/Y pair forbidden S(X, Y,, ) = + (Such BPs are rarely observed [Stombaugh 09] nstable) rbitrary energies associated with valid stackings S(X, Y, X, Y ) < 0 Theorem RN-PK-FOLD(S) is NP-hard. Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM 12 8 / 17
26 Proof Definition (3-PRTITION problem) Input: Sequence of integers X = {x i } n i=1, summing to n/3 K, K N. Output: True iff X can be split into m := n/3 triplets {(x aj, x bj, x cj )} m j=1 s. t. x aj + x bj + x cj = K, j [1, m]. Proof. Reduction from 3-PRTITION: Let w X := x 1 x 2 x 3 xn } K K {{ K } and δ := S(,,, ) m times Best matching S for w X has free-energy E(S ) wx E := δ (K 3) m. If X 3-partitionable, then matching induced by partition gives E(S ) wx = E. If E(S ) wx = E, then S saturates each K block, using three blocks ( a, b, c ). Since w X O(n P(n)), then RN-PK-FOLD(S) P 3-PRTITION P. Reminder: 3-PRTITION is strongly NP-Hard [arey 75], i.e. still hard if x i < P(n). Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM 12 9 / 17
27 Proof Definition (3-PRTITION problem) Input: Sequence of integers X = {x i } n i=1, summing to n/3 K, K N. Output: True iff X can be split into m := n/3 triplets {(x aj, x bj, x cj )} m j=1 s. t. x aj + x bj + x cj = K, j [1, m]. Proof. Reduction from 3-PRTITION: Let w X := x 1 x 2 x 3 xn } K K {{ K } and δ := S(,,, ) m times Best matching S for w X has free-energy E(S ) wx E := δ (K 3) m. If X 3-partitionable, then matching induced by partition gives E(S ) wx = E. If E(S ) wx = E, then S saturates each K block, using three blocks ( a, b, c ). Since w X O(n P(n)), then RN-PK-FOLD(S) P 3-PRTITION P. Reminder: 3-PRTITION is strongly NP-Hard [arey 75], i.e. still hard if x i < P(n). Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM 12 9 / 17
28 Proof Definition (3-PRTITION problem) Input: Sequence of integers X = {x i } n i=1, summing to n/3 K, K N. Output: True iff X can be split into m := n/3 triplets {(x aj, x bj, x cj )} m j=1 s. t. x aj + x bj + x cj = K, j [1, m]. Proof. Reduction from 3-PRTITION: Let w X := x 1 x 2 x 3 xn } K K {{ K } and δ := S(,,, ) m times Best matching S for w X has free-energy E(S ) wx E := δ (K 3) m. If X 3-partitionable, then matching induced by partition gives E(S ) wx = E. If E(S ) wx = E, then S saturates each K block, using three blocks ( a, b, c ). Since w X O(n P(n)), then RN-PK-FOLD(S) P 3-PRTITION P. Reminder: 3-PRTITION is strongly NP-Hard [arey 75], i.e. still hard if x i < P(n). Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM 12 9 / 17
29 Proof Definition (3-PRTITION problem) Input: Sequence of integers X = {x i } n i=1, summing to n/3 K, K N. Output: True iff X can be split into m := n/3 triplets {(x aj, x bj, x cj )} m j=1 s. t. x aj + x bj + x cj = K, j [1, m]. Proof. Reduction from 3-PRTITION: Let w X := x 1 x 2 x 3 xn } K K {{ K } and δ := S(,,, ) m times Best matching S for w X has free-energy E(S ) wx E := δ (K 3) m. If X 3-partitionable, then matching induced by partition gives E(S ) wx = E. If E(S ) wx = E, then S saturates each K block, using three blocks ( a, b, c ). Since w X O(n P(n)), then RN-PK-FOLD(S) P 3-PRTITION P. Reminder: 3-PRTITION is strongly NP-Hard [arey 75], i.e. still hard if x i < P(n). Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM 12 9 / 17
30 Proof Definition (3-PRTITION problem) Input: Sequence of integers X = {x i } n i=1, summing to n/3 K, K N. Output: True iff X can be split into m := n/3 triplets {(x aj, x bj, x cj )} m j=1 s. t. x aj + x bj + x cj = K, j [1, m]. Proof. Reduction from 3-PRTITION: Let w X := x 1 x 2 x 3 xn } K K {{ K } and δ := S(,,, ) m times Best matching S for w X has free-energy E(S ) wx E := δ (K 3) m. If X 3-partitionable, then matching induced by partition gives E(S ) wx = E. If E(S ) wx = E, then S saturates each K block, using three blocks ( a, b, c ). Since w X O(n P(n)), then RN-PK-FOLD(S) P 3-PRTITION P. Reminder: 3-PRTITION is strongly NP-Hard [arey 75], i.e. still hard if x i < P(n). Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM 12 9 / 17
31 Example X = x 1 x 2 x 3 x 4 x 5 x 6 (K = 7) B w X = x 1 x 2 x 3 x 4 x 5 x 6 K K D K x 1 x 3 x x 2 x 5 x 6 Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM / 17
32 Example X = x 1 x 2 x 3 x 4 x 5 x 6 (K = 7) B w X = x 1 x 2 x 3 x 4 x 5 x 6 K K D K x 1 x 3 x x 2 x 5 x 6 Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM / 17
33 Example X = x 1 x 2 x 3 x 4 x 5 x 6 (K = 7) B w X = x 1 x 2 x 3 x 4 x 5 x 6 K K D K x 1 x 3 x x 2 x 5 x 6 Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM / 17
34 Honest O(n 3 ) 5-approximation for RN-PK-FOLD(S) Existence of polynomial time approximation scheme (in O(n 41/ε )) [Lyngsø 04] Base-pair maximization (unit cost) rbitrary energies??? lgorithm: 1 Build weighted adjacency graph = (V, E) Vertices: Pairs of consecutive pos. (i, i + 1) Edges: (i, i + 1) (j 1, j) with weight S (w i, w j, w i+1, w j 1 ) 2 ompute maximal-weighted matching m. 3 Loop over p = (i, i + 1), (j, j 1) m, ordered by decreasing weight: dd result to output m, remove any p m conflicting with p 4 Return m Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM / 17
35 Honest O(n 3 ) 5-approximation for RN-PK-FOLD(S) Existence of polynomial time approximation scheme (in O(n 41/ε )) [Lyngsø 04] Base-pair maximization (unit cost) rbitrary energies??? lgorithm: 1 Build weighted adjacency graph = (V, E) Vertices: Pairs of consecutive pos. (i, i + 1) Edges: (i, i + 1) (j 1, j) with weight S (w i, w j, w i+1, w j 1 ) 2 ompute maximal-weighted matching m. 3 Loop over p = (i, i + 1), (j, j 1) m, ordered by decreasing weight: dd result to output m, remove any p m conflicting with p 4 Return m Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM / 17
36 Honest O(n 3 ) 5-approximation for RN-PK-FOLD(S) Existence of polynomial time approximation scheme (in O(n 41/ε )) [Lyngsø 04] Base-pair maximization (unit cost) rbitrary energies??? lgorithm: 1 Build weighted adjacency graph = (V, E) Vertices: Pairs of consecutive pos. (i, i + 1) Edges: (i, i + 1) (j 1, j) with weight S (w i, w j, w i+1, w j 1 ) 2 ompute maximal-weighted matching m. 3 Loop over p = (i, i + 1), (j, j 1) m, ordered by decreasing weight: dd result to output m, remove any p m conflicting with p 4 Return m Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM / 17
37 Honest O(n 3 ) 5-approximation for RN-PK-FOLD(S) Existence of polynomial time approximation scheme (in O(n 41/ε )) [Lyngsø 04] Base-pair maximization (unit cost) rbitrary energies??? lgorithm: 1 Build weighted adjacency graph = (V, E) Vertices: Pairs of consecutive pos. (i, i + 1) Edges: (i, i + 1) (j 1, j) with weight S (w i, w j, w i+1, w j 1 ) 2 ompute maximal-weighted matching m. 3 Loop over p = (i, i + 1), (j, j 1) m, ordered by decreasing weight: dd result to output m, remove any p m conflicting with p 4 Return m Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM / 17
38 Honest O(n 3 ) 5-approximation for RN-PK-FOLD(S) Existence of polynomial time approximation scheme (in O(n 41/ε )) [Lyngsø 04] Base-pair maximization (unit cost) rbitrary energies??? lgorithm: 1 Build weighted adjacency graph = (V, E) Vertices: Pairs of consecutive pos. (i, i + 1) Edges: (i, i + 1) (j 1, j) with weight S (w i, w j, w i+1, w j 1 ) 2 ompute maximal-weighted matching m. 3 Loop over p = (i, i + 1), (j, j 1) m, ordered by decreasing weight: dd result to output m, remove any p m conflicting with p 4 Return m Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM / 17
39 Honest O(n 3 ) 5-approximation for RN-PK-FOLD(S) Existence of polynomial time approximation scheme (in O(n 41/ε )) [Lyngsø 04] Base-pair maximization (unit cost) rbitrary energies??? lgorithm: 1 Build weighted adjacency graph = (V, E) Vertices: Pairs of consecutive pos. (i, i + 1) Edges: (i, i + 1) (j 1, j) with weight S (w i, w j, w i+1, w j 1 ) 2 ompute maximal-weighted matching m. 3 Loop over p = (i, i + 1), (j, j 1) m, ordered by decreasing weight: dd result to output m, remove any p m conflicting with p 4 Return m Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM / 17
40 Honest O(n 3 ) 5-approximation for RN-PK-FOLD(S) Existence of polynomial time approximation scheme (in O(n 41/ε )) [Lyngsø 04] Base-pair maximization (unit cost) rbitrary energies??? lgorithm: 1 Build weighted adjacency graph = (V, E) Vertices: Pairs of consecutive pos. (i, i + 1) Edges: (i, i + 1) (j 1, j) with weight S (w i, w j, w i+1, w j 1 ) 2 ompute maximal-weighted matching m. 3 Loop over p = (i, i + 1), (j, j 1) m, ordered by decreasing weight: dd result to output m, remove any p m conflicting with p 4 Return m Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM / 17
41 Honest O(n 3 ) 5-approximation for RN-PK-FOLD(S) Existence of polynomial time approximation scheme (in O(n 41/ε )) [Lyngsø 04] Base-pair maximization (unit cost) rbitrary energies??? lgorithm: 1 Build weighted adjacency graph = (V, E) Vertices: Pairs of consecutive pos. (i, i + 1) Edges: (i, i + 1) (j 1, j) with weight S (w i, w j, w i+1, w j 1 ) 2 ompute maximal-weighted matching m. 3 Loop over p = (i, i + 1), (j, j 1) m, ordered by decreasing weight: dd result to output m, remove any p m conflicting with p 4 Return m omplexity: t most O(n 3 ) (Max-weighted matching) pprox. ratio: Initial matching m has total energy smaller than OPT. Loop 3: Each stacking pair p conflicts with 4 pairs in m, having greater energy. Returned matching has free-energy 1/5 of OPT ( S Honest) Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM / 17
42 Half-time summary omp. Base-pairs Stacking-Pairs Nearest-Neighbor P [Nussinov 80] P [Ieong 03] P [Zuker 81] Non-crossing pprox. Planar omp.??? pprox. omp. 2-approx. [Ieong 03] P [Tabaska 98] eneral pprox. NP-Hard [Ieong 03] 2-approx. [Ieong 03] NP-Hard [Lyngsø 04] (any model) ε-approx. O(n 41/ε ) [Lyngsø 04] 1/5 (any model) NP-Hard [Ieong 03]??? NP-Hard [Lyngsø 00, kutsu 00]??? How hard is it to approximate the nearest neighbor model? Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM / 17
43 (Dishonest!) Inapproximability of Nearest-Neighbor model Theorem For some nearest-neighbor model N, one has RN-PK-FOLD(N ) / PX. Proof. onsider the RN seq. built from some 3-PRTITION instance X: w X = x 1 x 2 x 3m } K K {{ K } m times and the energy model: 2m N : () 1, i < j, i i+1 j-1 j (B) X Y 1, i < j, X, Y, i i+1 j-1 j (i +1 and j 1 must both base-pair somewhere, possibly together) () X Y 1, i < j, (X, Y ), i i+1 j-1 j (i +1 and j 1 must both base-pair somewhere, possibly together) (D) ny other motif +, i < j, Lemma: The energy of any matching of w X is either 0 (no base-pair), w X < 0 ( X is 3-partitionable) or + (any other case). Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM / 17
44 (Dishonest!) Inapproximability of Nearest-Neighbor model Theorem For some nearest-neighbor model N, one has RN-PK-FOLD(N ) / PX. Proof. onsider the RN seq. built from some 3-PRTITION instance X: w X = x 1 x 2 x 3m } K K {{ K } m times and the energy model: 2m N : () 1, i < j, i i+1 j-1 j (B) X Y 1, i < j, X, Y, i i+1 j-1 j (i +1 and j 1 must both base-pair somewhere, possibly together) () X Y 1, i < j, (X, Y ), i i+1 j-1 j (i +1 and j 1 must both base-pair somewhere, possibly together) (D) ny other motif +, i < j, Lemma: The energy of any matching of w X is either 0 (no base-pair), w X < 0 ( X is 3-partitionable) or + (any other case). Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM / 17
45 (Dishonest!) Inapproximability of Nearest-Neighbor model Theorem For some nearest-neighbor model N, one has RN-PK-FOLD(N ) / PX. Proof (continued). The energy of any matching of w X under N is either 0 (no base-pair), w X < 0 ( X is 3-partitionable) or + (any other case). Reminder: Polynomial-time 1/f (n)-approximation algorithm bound to produce solution having free-energy f (n) OPT. ny 1/f (n)-approx. algorithm, f (n) > 0, produces a matching of negative free-energy f (n) E < 0 iff a matching of energy E < 0 exists i.e. iff X is 3-partitionable! nless P = NP, there is no polynomial-time approximation algorithm of (non-necessarily constant) positive ratio for RN-PK-FOLD(N ). Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM / 17
46 (Dishonest!) Inapproximability of Nearest-Neighbor model Theorem For some nearest-neighbor model N, one has RN-PK-FOLD(N ) / PX. Proof (continued). The energy of any matching of w X under N is either 0 (no base-pair), w X < 0 ( X is 3-partitionable) or + (any other case). Reminder: Polynomial-time 1/f (n)-approximation algorithm bound to produce solution having free-energy f (n) OPT. ny 1/f (n)-approx. algorithm, f (n) > 0, produces a matching of negative free-energy f (n) E < 0 iff a matching of energy E < 0 exists i.e. iff X is 3-partitionable! nless P = NP, there is no polynomial-time approximation algorithm of (non-necessarily constant) positive ratio for RN-PK-FOLD(N ). Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM / 17
47 (Dishonest!) Inapproximability of Nearest-Neighbor model Theorem For some nearest-neighbor model N, one has RN-PK-FOLD(N ) / PX. Proof (continued). The energy of any matching of w X under N is either 0 (no base-pair), w X < 0 ( X is 3-partitionable) or + (any other case). Reminder: Polynomial-time 1/f (n)-approximation algorithm bound to produce solution having free-energy f (n) OPT. ny 1/f (n)-approx. algorithm, f (n) > 0, produces a matching of negative free-energy f (n) E < 0 iff a matching of energy E < 0 exists i.e. iff X is 3-partitionable! nless P = NP, there is no polynomial-time approximation algorithm of (non-necessarily constant) positive ratio for RN-PK-FOLD(N ). Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM / 17
48 (Dishonest!) Inapproximability of Nearest-Neighbor model Theorem For some nearest-neighbor model N, one has RN-PK-FOLD(N ) / PX. Proof (continued). The energy of any matching of w X under N is either 0 (no base-pair), w X < 0 ( X is 3-partitionable) or + (any other case). Reminder: Polynomial-time 1/f (n)-approximation algorithm bound to produce solution having free-energy f (n) OPT. ny 1/f (n)-approx. algorithm, f (n) > 0, produces a matching of negative free-energy f (n) E < 0 iff a matching of energy E < 0 exists i.e. iff X is 3-partitionable! nless P = NP, there is no polynomial-time approximation algorithm of (non-necessarily constant) positive ratio for RN-PK-FOLD(N ). Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM / 17
49 (Dishonest!) Inapproximability of Nearest-Neighbor model Theorem For some nearest-neighbor model N, one has RN-PK-FOLD(N ) / PX. Proof (continued). The energy of any matching of w X under N is either 0 (no base-pair), w X < 0 ( X is 3-partitionable) or + (any other case). Reminder: Polynomial-time 1/f (n)-approximation algorithm bound to produce solution having free-energy f (n) OPT. ny 1/f (n)-approx. algorithm, f (n) > 0, produces a matching of negative free-energy f (n) E < 0 iff a matching of energy E < 0 exists i.e. iff X is 3-partitionable! nless P = NP, there is no polynomial-time approximation algorithm of (non-necessarily constant) positive ratio for RN-PK-FOLD(N ). Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM / 17
50 onclusion Dishonest inapproximability result for nearest-neighbor model lmost honest general hardness result for stacking model Honest 5-approximation for stacking model Nearest Neighbor model: Dishonest unapproximability Hardness of approximating within ratio f (r)? where r is largest ratio between contributions of motifs. Stacking model: Honest + efficient polynomial-time approximation scheme pproximations do not guarantee any overlap with best solution. Polynomial k-overlap algorithm? (Seems unlikely... ) Thanks to Thanks for listening Questions? Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM / 17
51 onclusion Dishonest inapproximability result for nearest-neighbor model lmost honest general hardness result for stacking model Honest 5-approximation for stacking model Nearest Neighbor model: Dishonest unapproximability Hardness of approximating within ratio f (r)? where r is largest ratio between contributions of motifs. Stacking model: Honest + efficient polynomial-time approximation scheme pproximations do not guarantee any overlap with best solution. Polynomial k-overlap algorithm? (Seems unlikely... ) Thanks to Thanks for listening Questions? Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM / 17
52 onclusion Dishonest inapproximability result for nearest-neighbor model lmost honest general hardness result for stacking model Honest 5-approximation for stacking model Nearest Neighbor model: Dishonest unapproximability Hardness of approximating within ratio f (r)? where r is largest ratio between contributions of motifs. Stacking model: Honest + efficient polynomial-time approximation scheme pproximations do not guarantee any overlap with best solution. Polynomial k-overlap algorithm? (Seems unlikely... ) Thanks to Thanks for listening Questions? Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM / 17
53 onclusion Dishonest inapproximability result for nearest-neighbor model lmost honest general hardness result for stacking model Honest 5-approximation for stacking model Nearest Neighbor model: Dishonest unapproximability Hardness of approximating within ratio f (r )? where r is largest ratio between contributions of motifs. Stacking model: Honest + efficient polynomial-time approximation scheme pproximations do not guarantee any overlap with best solution. Polynomial k -overlap algorithm? (Seems unlikely... ) Thanks for listening Questions? Thanks to Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM / 17
54 References I Tatsuya kutsu. Dynamic programming algorithms for RN secondary structure prediction with pseudoknots. Discrete ppl. Math., vol. 104, no. 1-3, pages 45 62, M. R. arey & D. S. Johnson. omplexity Results for Multiprocessor Scheduling under Resource onstraints. SIM Journal on omputing, vol. 4, no. 4, pages , Samuel Ieong, Ming yang Kao, Tak wah Lam, Wing kin Sung & Siu ming Yiu. Predicting RN Secondary Structures with rbitrary Pseudoknots by Maximizing the Number of Stacking Pairs. Journal Of omputational Biology, vol. 10, no. 6, pages , N. Leontis & E. Westhof. eometric nomenclature and classification of RN base pairs. RN, vol. 7, pages , R. B. Lyngsø&. N. S. Pedersen. RN Pseudoknot Prediction in Energy-Based Models. Journal of omputational Biology, vol. 7, no. 3-4, pages , Rune Lyngsø. omplexity of Pseudoknot Prediction in Simple Models. In Proceedings of ILP, R. Nussinov &.B. Jacobson. Fast algorithm for predicting the secondary structure of single-stranded RN. Proc Natl cad Sci S, vol. 77, pages , Jesse Stombaugh, raig L. Zirbel, Eric Westhof & Neocles B. Leontis. Frequency and isostericity of RN base pairs. Nucleic cids Research, vol. 37, no. 7, pages , Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM / 17
55 References II J. E. Tabaska, R. B. ary, H. N. abow &. D. Stormo. n RN folding method capable of identifying pseudoknots and base triples. Bioinformatics, vol. 14, no. 8, pages , M. Zuker & P. Stiegler. Optimal computer folding of large RN sequencesusing thermodynamics and auxiliary information. Nucleic cids Res., vol. 9, pages , Yann Ponty (NRS/Polytechnique, France) Energy models and RN pseudoknotted folding July 5th PM / 17
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