Extending the hypergraph analogy for RNA dynamic programming

Transcription

1 Extending the hypergraph analogy for RN dynamic programming Yann Ponty Balaji Raman édric Saule Polytechnique/NRS/INRI MIB France Yann Ponty, Balaji Raman, édric Saule

2 RN Folding RN = Biopolymer composed of nucleotides,,, and : denosine, : ytosine, : uanine et : racil anonical base-pairs / / / Watson/rick Wobble RN folding = Stochastic continuous process directed by (resulting) the pairing of nucleotides. nderstanding RN folding is a key step toward understanding and predicting its function(s). Yann Ponty, Balaji Raman, édric Saule

3 RN structure(s) Three levels of representation: Primary structure Secondary structure Tertiary structures Source: 5s rrn (PDB K73:B) Well, almost... Yann Ponty, Balaji Raman, édric Saule

4 RN structure(s) Three levels of representation: Primary structure Secondary + structure Tertiary structures Source: 5s rrn (PDB K73:B) Well, almost... Yann Ponty, Balaji Raman, édric Saule

5 Discarded by secondary structure Non-canonical base-pairs ny basepair other than {(-), (-), (-)} Or interacting using a non-standard edge/orientation (W/W-is) [LW0]. / canonical pair (W/W-is) non-canonical pair (Sugar/W-Trans) Pseudoknots Pseudoknots within a group I Ribozyme (PDBID: Y0Q:) More expressive model, but ab initio folding with pseudoknots: NP-omplete [LP00]... yet polynomial for restricted classes [DR + 04]. Yann Ponty, Balaji Raman, édric Saule

6 Our objective(s) uthors omplexity uthors omplexity Lyngso and Pedersen O(n 5 ) ao and hen O(n 6 ) Reeder and iegerich O(n 4 ) Dirk and Pierce O(n 5 ) kutso and emara O(n 5 ) Rivas and Eddy O(n 6 ) hen, Jabbari and ondon O(n 5 ) Many DP algorithms proposed for folding with restricted pseudoknots. Decompositions induce high complexities and/or are ambiguous, prohibiting ensemble-based approaches: Partition function, BP prob.... Is it possible to come up with highly expressive, unambiguous decompositions? Our programme To unify DP RN algorithms into a single abstract framework. To reformulate ensemble-based applications within this framework. Improve existing algorithms by working on their decomposition at an abstract level. May benet to RN-RN interaction prediction, parameterized approaches... Yann Ponty, Balaji Raman, édric Saule

7 DP Example: Nussinov/Jacobson decomposition Nearest-neighbor model Free-energy = Weighted sum over base-pairs oal: iven RN sequence ω, nd Minimum Free Energy (MFE) secondary structure compatible with base-pairing. i j = i i+ j + θ i k j N i,t = 0, t [i, i + θ] N i+,j N i,j = min j min k=i+θ+ Eω i,ω + N k i+,k + N k+,j i non apparié i apparié à k E, = E, = 3.0 kal mol, E, = E, = 2.0 kal mol, and E, = E, =.0 kal mol. Once the minimal energy is gured out, the corresponding secondary structure is obtained through a backtrack procedure. Yann Ponty, Balaji Raman, édric Saule

8 More complex/realistic example Turner model Free-energy = Weighted sum over loops E H (i, j) M E i,j = S (i, j) + M i+,j min Min i,j (E BI (i, i, j, j) + M i,j ) a + c + Min k (M i+,k + M k,j ) { M i,j = Min k min (Mi,k, b(k )) + M } k,j M { i,j = Min k b + M i,j, c + M } i,j EH(i, j): Energy of terminal loop closed by base-pair (i, j) EBI (i, j): Energy of bulge/internal loop renement closed by bp (i, j) ES (i, j): Stacking energy (i, j)/(i +, j ) a,c,b: Penalties for multiple loops, helix and unpaired base in multiloop. Yann Ponty, Balaji Raman, édric Saule

9 Beyond minimization dditional motivation: o beyond energy minimization. urrent driving hypothesis = Boltzmann ensemble of low energy Functional folding? Ill-dened folding: mrn? Bistable RN Kinetics? Decompositions will have to be unambiguous... Yann Ponty, Balaji Raman, édric Saule

10 Dynamic programming Starting from an instance, or problem: Search space: May depend on instance Objective function: Dened on search space, may depend on instance Dynamic programming equation: Relates value of objective function for problem to that of its sub-problems (substructure property), relying on a decomposition of search space ompute obj. fun. for sub-problems (se backtrack to build solution). Nussinov example: Search space: Secondary structures inducing valid bps. Objective function: Free-energy DP equation/decomposition: MFE obtained by minimizing energy on subsequence(s) lternative view: DP equation generates search space ND implements a specic application. Detach two conceptually dierent entities. Yann Ponty, Balaji Raman, édric Saule

11 Existing DP abstract frameworks Parsing approaches Search space modeled by context-free grammar(s) (F). (Multitape)-attribute grammars (Lefebvre, Waldispühl et al): ompute and minimize score based on attribute algebra. Pros: aptures simple pseudoknots through synchronized multiple tapes. ons: Designed for optimization. Multi-tapes can be overkill for almost F. lgebraic Dynamic Programming (iegerich et al) Pros: dresses general applications thanks to an algebraic approach. ons: Pseudoknots require hacking the formalism. Other Hypergraphs (..., Roytberg/Finkelstein) Pros: Very exible. Mild inuence of order. ny F can be transformed into equivalent hypergraph. ons: No generic implementation (yet!). Yann Ponty, Balaji Raman, édric Saule

12 Existing DP abstract frameworks Parsing approaches Search space modeled by context-free grammar(s) (F). (Multitape)-attribute grammars (Lefebvre, Waldispühl et al): ompute and minimize score based on attribute algebra. Pros: aptures simple pseudoknots through synchronized multiple tapes. ons: Designed for optimization. Multi-tapes can be overkill for almost F. lgebraic Dynamic Programming (iegerich et al) Pros: dresses general applications thanks to an algebraic approach. ons: Pseudoknots require hacking the formalism. Other Hypergraphs (..., Roytberg/Finkelstein) Pros: Very exible. Mild inuence of order. ny F can be transformed into equivalent hypergraph. ons: No generic implementation (yet!). Yann Ponty, Balaji Raman, édric Saule

13 Existing DP abstract frameworks Parsing approaches Search space modeled by context-free grammar(s) (F). (Multitape)-attribute grammars (Lefebvre, Waldispühl et al): ompute and minimize score based on attribute algebra. Pros: aptures simple pseudoknots through synchronized multiple tapes. ons: Designed for optimization. Multi-tapes can be overkill for almost F. lgebraic Dynamic Programming (iegerich et al) Pros: dresses general applications thanks to an algebraic approach. ons: Pseudoknots require hacking the formalism. Other Hypergraphs (..., Roytberg/Finkelstein) Pros: Very exible. Mild inuence of order. ny F can be transformed into equivalent hypergraph. ons: No generic implementation (yet!). Yann Ponty, Balaji Raman, édric Saule

14 Hypergraphs Hypergaphs generalize directed graphs to arcs of arbitrary in/out degrees. Denition (Hypergraph) directed hypergaph H is a couple (V, E) such that: V is a set of vertices E is a set of hyperarcs e = (t(e) h(e)) such that t(e), h(e) E Forward hypergraphs, or F-graphs, are hypergraphs whose arcs have ingoing degree exactly. Yann Ponty, Balaji Raman, édric Saule

15 Hypergraphs Hypergaphs generalize directed graphs to arcs of arbitrary in/out degrees. Denition (Hypergraph) directed hypergaph H is a couple (V, E) such that: V is a set of vertices E is a set of hyperarcs e = (t(e) h(e)) such that t(e), h(e) E Forward hypergraphs, or F-graphs, are hypergraphs whose arcs have ingoing degree exactly. Yann Ponty, Balaji Raman, édric Saule

16 Weight: Score: 3 45 Weight: Score: 32 Weight: Score: 88 Weight: Score: 6 Weight: Score: F-graph ll F-paths starting from vertex 7 3 Denition (F-path) F-path is a tree having root s V, whose children are F-paths built from the outgoing vertices of some arc e = (s t) E. Remark: Vertices of out degree 0 (t = ) provide a terminal case to the above recursive denition. F-graph is independent i any arc is present at most once in each F-path. numerical value fonction π : E R can be assigned to each arc e E: Weight of a path is the product of its arcs' values Score of a path is the sum of its arcs' values Yann Ponty, Balaji Raman, édric Saule

17 Weight: Score: 3 45 Weight: Score: 32 Weight: Score: 88 Weight: Score: 6 Weight: Score: F-graph ll F-paths starting from vertex 7 3 Denition (F-path) F-path is a tree having root s V, whose children are F-paths built from the outgoing vertices of some arc e = (s t) E. Remark: Vertices of out degree 0 (t = ) provide a terminal case to the above recursive denition. F-graph is independent i any arc is present at most once in each F-path. numerical value fonction π : E R can be assigned to each arc e E: Weight of a path is the product of its arcs' values Score of a path is the sum of its arcs' values Yann Ponty, Balaji Raman, édric Saule

18 Weight: Score: 3 45 Weight: Score: 32 Weight: Score: 88 Weight: Score: 6 Weight: Score: F-graph ll F-paths starting from vertex 7 3 Denition (F-path) F-path is a tree having root s V, whose children are F-paths built from the outgoing vertices of some arc e = (s t) E. Remark: Vertices of out degree 0 (t = ) provide a terminal case to the above recursive denition. F-graph is independent i any arc is present at most once in each F-path. numerical value fonction π : E R can be assigned to each arc e E: Weight of a path is the product of its arcs' values Score of a path is the sum of its arcs' values Yann Ponty, Balaji Raman, édric Saule

19 Weight: Score: 3 45 Weight: Score: 32 Weight: Score: 88 Weight: Score: 6 Weight: Score: F-graph ll F-paths starting from vertex 7 3 Denition (F-path) F-path is a tree having root s V, whose children are F-paths built from the outgoing vertices of some arc e = (s t) E. Remark: Vertices of out degree 0 (t = ) provide a terminal case to the above recursive denition. F-graph is independent i any arc is present at most once in each F-path. numerical value fonction π : E R can be assigned to each arc e E: Weight of a path is the product of its arcs' values Score of a path is the sum of its arcs' values Yann Ponty, Balaji Raman, édric Saule

20 Basic algorithms H = (s 0, V, E, π): acyclic hypergraph s 0: Initial node π: value function Some questions naturally arise: What is the (min/max)imal score m s of an F-path starting from s V? omplexities: Θ( E + V ) time/θ( V ) memory. What is the number n s of F-paths starting from s V? omplexities: Θ( E + V ) time/θ( V ) memory. What is the total weight w s of all F-paths starting from s V? omplexities: Θ( E + V ) time/θ( V ) memory. m s = min e=(s t) E ( w(e) + u t m u ) s w(e) w(e ) w(e ) t t + t 2 Yann Ponty, Balaji Raman, édric Saule

21 Basic algorithms H = (s 0, V, E, π): acyclic hypergraph s 0: Initial node π: value function Some questions naturally arise: What is the (min/max)imal score m s of an F-path starting from s V? omplexities: Θ( E + V ) time/θ( V ) memory. What is the number n s of F-paths starting from s V? omplexities: Θ( E + V ) time/θ( V ) memory. What is the total weight w s of all F-paths starting from s V? omplexities: Θ( E + V ) time/θ( V ) memory. t n s = (s t) E u t n u s t t 2 Yann Ponty, Balaji Raman, édric Saule

22 Basic algorithms H = (s 0, V, E, π): acyclic hypergraph s 0: Initial node π: value function Some questions naturally arise: What is the (min/max)imal score m s of an F-path starting from s V? omplexities: Θ( E + V ) time/θ( V ) memory. What is the number n s of F-paths starting from s V? omplexities: Θ( E + V ) time/θ( V ) memory. What is the total weight w s of all F-paths starting from s V? omplexities: Θ( E + V ) time/θ( V ) memory. t w s = w(e) e=(s h(e)) E s h(e) w s s w(e) w(e ) w(e ) t t 2 Yann Ponty, Balaji Raman, édric Saule

23 Denition (Weighted distribution) ssume a weighted, Boltzmann-like, distribution on the set T of F-Paths: e p P(p π) = π(e), p T. w s0 Ensemble related questions arise: How to generate a random F-path p from T w.r.t. weighted distribution? omplexities: Θ( E + V ) time/θ( V ) memory precomputation + O( p + e p h(e) ) time generation. What is the probability that a given arc be present in a random F-Path? Inside/outside algorithm. Distribution of additive features lgorithm: hoose e i = (s ti ) with probability p s,i such that: p s,i = w(e) s w t s w s Iterate the process over all ti 's. s w(e) w(e ) w(e ) t t t 2 Yann Ponty, Balaji Raman, édric Saule

24 Denition (Weighted distribution) ssume a weighted, Boltzmann-like, distribution on the set T of F-Paths: e p P(p π) = π(e), p T. w s0 Ensemble related questions arise: How to generate a random F-path p from T w.r.t. weighted distribution? omplexities: Θ( E + V ) time/θ( V ) memory precomputation + O( p + e p h(e) ) time generation. What is the probability that a given arc be present in a random F-Path? Inside/outside algorithm. Distribution of additive features Yann Ponty, Balaji Raman, édric Saule

25 Inside/outside algorithm F-path using a given arc e (s t ) Inside: Some F-path built from t Distinguished arc e Outside: ontext where e is used: Path p = (s 0, s,..., s ) using F-arcs (e,..., e k ) For each vertex in some h(e i )/p complete F-path by adding siblings to the vertices in p. s 0 s t t 2 Yann Ponty, Balaji Raman, édric Saule

26 Inside/outside algorithm F-path using a given arc e (s t ) Inside: Some F-path built from t Distinguished arc e Outside: ontext where e is used: Path p = (s 0, s,..., s ) using F-arcs (e,..., e k ) For each vertex in some h(e i )/p complete F-path by adding siblings to the vertices in p. s 0 s t t 2 Yann Ponty, Balaji Raman, édric Saule

27 Inside/outside algorithm F-path using a given arc e (s t ) Inside: Some F-path built from t Distinguished arc e Outside: ontext where e is used: Path p = (s 0, s,..., s ) using F-arcs (e,..., e k ) For each vertex in some h(e i )/p complete F-path by adding siblings to the vertices in p. s 0 s t t 2 Yann Ponty, Balaji Raman, édric Saule

28 Inside/outside algorithm F-path using a given arc e (s t ) Inside: Some F-path built from t Distinguished arc e Outside: ontext where e is used: Path p = (s 0, s,..., s ) using F-arcs (e,..., e k ) For each vertex in some h(e i )/p complete F-path by adding siblings to the vertices in p. s 0 s t t 2 Yann Ponty, Balaji Raman, édric Saule

29 Inside/outside algorithm F-path using a given arc e (s t ) Inside: Some F-path built from t Distinguished arc e Outside: ontext where e is used: Path p = (s 0, s,..., s ) using F-arcs (e,..., e k ) For each vertex in some h(e i )/p complete F-path by adding siblings to the vertices in p. s 0 s t t 2 Yann Ponty, Balaji Raman, édric Saule

30 Inside/outside algorithm F-path using a given arc e (s t ) Inside: Some F-path built from t Distinguished arc e Outside: ontext where e is used: Path p = (s 0, s,..., s ) using F-arcs (e,..., e k ) For each vertex in some h(e i )/p complete F-path by adding siblings to the vertices in p. s s t t 2 Yann Ponty, Balaji Raman, édric Saule

31 Inside/outside algorithm F-path using a given arc e (s t ) Inside: Some F-path built from t Distinguished arc e Outside: ontext where e is used: Path p = (s 0, s,..., s ) using F-arcs (e,..., e k ) For each vertex in some h(e i )/p complete F-path by adding siblings to the vertices in p. s 0 s t t 2 If F-graph is acyclic and independent, this decomposition is complete and unambiguous, and induces the following DP equations for the cumulated probability p e of all F-paths featuring e = (s t ): bs π(e) s p e = t w s b s = s=s0 + w s0 e =(s t ) E s. t. s t π(e ) b s s t s s w s, s V Yann Ponty, Balaji Raman, édric Saule

32 Ensemble applications Denition (Weighted distribution) ssume a weighted, Boltzmann-like, distribution on the set T of F-Paths: e p P(p π) = π(e), p T. w s0 Ensemble related questions arise: How to generate a random F-path p from T w.r.t. weighted distribution? omplexities: Θ( E + V ) time/θ( V ) memory precomputation + O( p + e p h(e) ) time generation. What is the probability that a given arc be present in a random F-Path? Inside/outside algorithm omplexities: O( V + E + e E h(e) 2 )time/θ( V ) memory How to characterize the distribution of some additive features? Extraction of generalized moments Yann Ponty, Balaji Raman, édric Saule

33 Moments of a feature distribution Let T be the set of F-paths associated with a given hypergraph. Denition (Feature) feature is a function α : E R inherited additively by F-paths through α(p) = α(e) e p Example: Let us consider additive price α function over F-arcs and its associated random variable X α. Within the weighted distribution, what is the expected price of an F-path? B... the variance of the price X α of an F-path? How does the price X α correlates with the weight X π? = p T π(p) α(p) ws0 = E(X α) B = = p T π(p) α(p)2 ws0 E(X α X π) E(X α)e(x π) (E(X 2 α ) E(X α) 2 )(E(Xπ 2 ) E(Xπ)2 ) E(X α) 2 = E(X 2 α) E(X α) 2 an be expressed in term of the moments of the feature(s) distribution. How to extract them? Yann Ponty, Balaji Raman, édric Saule

34 oal: To extract the (combined) moment of an additive feature : E(X k α X α km m ) = π(p) w s0 p T m α i (p) km Diculty: Mixing additive (features) and multiplicative algebraic aspects (weighted distribution). Solution: Modify hypergraph to introduce controlled ambiguity. Denition (Feature-pointed F-graph) Let H = (s 0, V, E, π) be a weighted F-graph and α : E R some feature. The α-pointing of H is an F-graph H α = (s α 0, V α, E α, π α ) such that: Pointed versions of vertices are introduced V α := V {s α s V } New arcs are introduced to propagate a point or erase it. Propagation P α := { (s t) E s α (t,..., t α,..., t i k ) i [, k] } i= t Point erasure M α := (s t) E {(s α t)} E α := E P α M α Weight function { is extended to partially incorporate feature function π α (e π(e α ) If e P E ) := α(e ) π(e α ) Otherwise (e, M α e E α ) Yann Ponty, Balaji Raman, édric Saule

35 Example: Simple 0 (dashed) or (plain) feature function α Initial F-graph 4 Pointed F-graph π = π = π = π = 3 6 π = 4 6 π = π = 4 π = Initial F-Paths 6 nalogs in H of boxed F-path Rationale: Through pointing, each F-path p in H is duplicated p times in H ( analogs of p). Each copy features a point erasure over a dierent F-arc. The total weight, under π α, of all analogs of p is now π(p) α(p). Yann Ponty, Balaji Raman, édric Saule

36 Extracting moments Input: F-graph H = (s 0, V, E, π) oal: To extract the (higher order) moment of an additive feature : lgorithm E(X k α X α km m ) = π(p) w s0 p T pply weighted count algorithm to H w s0. m α i (p) km i= Point H repeatedly (commutative transform) H. pply weighted count algorithm to H Return w /w s0. w = p T π(p) m α i (p) km. omplexities: O ( 2 k V + e E ( h(e) + ) ( h(e) + 2)k) time O(2 k V ) memory, with k = m i= k i. Remarks: an be improved by pointing k i times in one go instead of pointing repeatedly. Without loss of generality, h(e) = 2 can be assumed (NF). i= Yann Ponty, Balaji Raman, édric Saule

37 Half time summary Message # Specic applications of Dynamic Programming could (and should) be detached from the equation, and be expressed at an abstract level. Sec. str. Nussinov Sec. str. Turner PK Dirks/Pierce Moments extraction Random generation ount Min/max score Weighted count rc probability... Message #2 Thanks to a pointing operator (formal derivative), one can extract arbitrary moments of features distribution under a weighted/boltzmann distribution. redits: Roytberg and Finkelstein for Hypergraph DP in Bioinformatics, L. Hwang for algebraic hypergraph DP, Flajolet et al for formal derivative through pointing... Let us now address the construction of suitable F-graphs... Yann Ponty, Balaji Raman, édric Saule

38 Half time summary Message # Specic applications of Dynamic Programming could (and should) be detached from the equation, and be expressed at an abstract level. Sec. str. Nussinov Sec. str. Turner PK Dirks/Pierce Hypergraph Moments extraction Random generation ount Min/max score Weighted count rc probability... Message #2 Thanks to a pointing operator (formal derivative), one can extract arbitrary moments of features distribution under a weighted/boltzmann distribution. redits: Roytberg and Finkelstein for Hypergraph DP in Bioinformatics, L. Hwang for algebraic hypergraph DP, Flajolet et al for formal derivative through pointing... Let us now address the construction of suitable F-graphs... Yann Ponty, Balaji Raman, édric Saule

39 Half time summary Message # Specic applications of Dynamic Programming could (and should) be detached from the equation, and be expressed at an abstract level. Sec. str. Nussinov Sec. str. Turner PK Dirks/Pierce Hypergraph Moments extraction Random generation ount Min/max score Weighted count rc probability... Message #2 Thanks to a pointing operator (formal derivative), one can extract arbitrary moments of features distribution under a weighted/boltzmann distribution. redits: Roytberg and Finkelstein for Hypergraph DP in Bioinformatics, L. Hwang for algebraic hypergraph DP, Flajolet et al for formal derivative through pointing... Let us now address the construction of suitable F-graphs... Yann Ponty, Balaji Raman, édric Saule

40 Mfold/nafold decomposition This decomposition is provably unambiguous and complete, and yields an F-graph that is acyclic, independent, and has Θ(n 2 ) vertices and O(n 3 ) arcs. Yann Ponty, Balaji Raman, édric Saule

41 Mfold/nafold decomposition This decomposition is provably unambiguous and complete, and yields an F-graph that is acyclic, independent, and has Θ(n 2 ) vertices and O(n 3 ) arcs. pplication lgorithm Time Memory Reference Energy minimization Min. score + π T O(n 3 ) O(n 2 ) [ZS8] Partition function Base-pairing probabilities Statistical sampling (k-samples) Moments of energy (Mean, Var.) k-th moment of additive features orrelations of additive features eneralized moments π count + e π T RT O(n 3 ) O(n 2 ) [Mc90] rc prob. + e π T RT O(n 3 ) O(n 2 ) [Mc90] Rand. gen. + e π T RT O(n 3 + k n log n) O(n 2 ) [DL03, Pon08] Moments + e π T RT O(n 3 ) O(n 2 ) [MMN05] Moments + e π T RT O(k 3.n 3 ) O(k.n 2 ) Moments + e π T RT O(n 3 ) O(n 2 ) Moments + e π T RT O(4 k.n 3 ) O(2 k.n 2 ) Yann Ponty, Balaji Raman, édric Saule

42 Basic pseudoknots (kutsu) Yann Ponty, Balaji Raman, édric Saule

43 Basic pseudoknots (kutsu) nambiguous decomposition capturing simple pseudoknots kutsu et al. yields O(n 4 )/O(n 4 ) time/memory algorithms for nearest neighbor energy model, and O(n 5 ) time/o(n 4 ) memory for Turner model. We can now answer new questions: What probability for the MFE within kutsu's class of conformations? What is the expected number of pseudoknots within a given sequence?... Yann Ponty, Balaji Raman, édric Saule

44 onclusion sing Hypergraphs, we were able to successfully detach conformation space from application. Implementation? We still have to mess with indices :( Start from F? Limited... (and already done by DP) but maybe worthy se Mathias Möhl's split types? ccount for additional parameters (RNMutants, RNBor... ) Pseudoknots gene scanning (in progress). Non-canonical motifs? Rebuild distributions from truncated moments ( Economics??) Yann Ponty, Balaji Raman, édric Saule

45 References I. ondon, B. Davy, B. Rastegari, S. Zhao, and F. Tarrant. lassifying RN pseudoknotted structures. Theoretical omputer Science, 320():3550, Y. Ding and E. Lawrence. statistical sampling algorithm for RN secondary structure prediction. Nucleic cids Res, 3(24): , R. B. Lyngsø and. N. S. Pedersen. RN pseudoknot prediction in energy-based models. Journal of omputational Biology, 7(3-4):409427, N. Leontis and E. Westhof. eometric nomenclature and classication of RN base pairs. RN, 7:49952, 200. J.S. Mcaskill. The equilibrium partition function and base pair binding probabilities for RN secondary structure. Biopolymers, 29:059, 990. István Miklós, Irmtraud M Meyer, and Borbála Nagy. Moments of the boltzmann distribution for rna secondary structures. Bull Math Biol, 67(5):03047, Sep Y. Ponty. Ecient sampling of RN secondary structures from the boltzmann ensemble of low-energy: The boustrophedon method. J Math Biol, 56(-2):0727, Jan M. Zuker and P. Stiegler. Optimal computer folding of large RN sequences using thermodynamics and auxiliary information. Nucleic cids Res, 9:3348, 98. Yann Ponty, Balaji Raman, édric Saule

46 omplexity of secondary structure folding Figure: lternative strategy for interior loops, creating the symmetric part rst and the asymmetric part afterward. Due to interior loops, the set F-arcs generated for the Q case have apparent cardinality in O(n 4 ), while we claims O(n 3 ) complexities. pragmatic answer may point out that it is common practice to upper bound the interior loop size (j j) + (i i) to a predened constant K = 30, bringing back the overall complexity to O(n 3 ). more theoretically-sound approach achieves a similar complexity while remaining exact by further decomposing internal loops into a symmetric loop followed by a fully asymmetric one, as illustrated in Figure. Such a modication introduces a new case in the decomposition and captures with l the asymmetry of the loop, but the log nature of the entropy term in the Turner model then requires a small approximation. Yann Ponty, Balaji Raman, édric Saule