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Inverse folding of RNA pseudoknot structures
Algorithms for Molecular Biology volume 5, Article number: 27 (2010)
Abstract
Background
RNA exhibits a variety of structural configurations. Here we consider a structure to be tantamount to the noncrossing WatsonCrick and GUbase pairings (secondary structure) and additional crossserial base pairs. These interactions are called pseudoknots and are observed across the whole spectrum of RNA functionalities. In the context of studying natural RNA structures, searching for new ribozymes and designing artificial RNA, it is of interest to find RNA sequences folding into a specific structure and to analyze their induced neutral networks. Since the established inverse folding algorithms, RNAinverse, RNASSD as well as INFORNA are limited to RNA secondary structures, we present in this paper the inverse folding algorithm Inv which can deal with 3noncrossing, canonical pseudoknot structures.
Results
In this paper we present the inverse folding algorithm Inv. We give a detailed analysis of Inv, including pseudocodes. We show that Inv allows to design in particular 3noncrossing nonplanar RNA pseudoknot 3noncrossing RNA structuresa class which is difficult to construct via dynamic programming routines. Inv is freely available at http://www.combinatorics.cn/cbpc/inv.html.
Conclusions
The algorithm Inv extends inverse folding capabilities to RNA pseudoknot structures. In comparison with RNAinverse it uses new ideas, for instance by considering sets of competing structures. As a result, Inv is not only able to find novel sequences even for RNA secondary structures, it does so in the context of competing structures that potentially exhibit crossserial interactions.
1 Introduction
Pseudoknots are structural elements of central importance in RNA structures [1], see Figure 1. They represent crossserial base pairing interactions between RNA nucleotides that are functionally important in tRNAs, RNaseP [2], telomerase RNA [3], and ribosomal RNAs [4]. Pseudoknot structures are being observed in the mimicry of tRNA structures in plant virus RNAs as well as the binding to the HIV1 reverse transcriptase in in vitro selection experiments [5]. Furthermore basic mechanisms, like ribosomal frame shifting, involve pseudoknots [6].
Despite them playing a key role in a variety of contexts, pseudoknots are excluded from largescale computational studies. Although the problem has attracted considerable attention in the last decade, pseudoknots are considered a somewhat "exotic" structural concept. For all we know [7], the ab initio prediction of general RNA pseudoknot structures is NPcomplete and algorithmic difficulties of pseudoknot folding are confounded by the fact that the thermodynamics of pseudoknots is far from being well understood.
As for the folding of RNA secondary structures, Waterman et al[8, 9], Zuker et al[10] and Nussinov [11] established the dynamic programming (DP) folding routines. The first mfefolding algorithm for RNA secondary structures, however, dates back to the 60's [12–14]. For restricted classes of pseudoknots, several algorithms have been designed: Rivas and Eddy [15], Dirks and Pierce [16], Reeder and Giegerich [17] and Ren et al[18]. Recently, a novel ab initio folding algorithm Cross has been introduced [19]. Cross generates minimum free energy (mfe), 3noncrossing, 3canonical RNA structures, i.e. structures that do not contain three or more mutually crossing arcs and in which each stack, i.e. sequence of parallel arcs, see eq. (1), has size greater or equal than three. In particular, in a 3canonical structure there are no isolated arcs, see Figure 2.
The notion of mfestructure is based on a specific concept of pseudoknot loops and respective loopbased energy parameters. This thermodynamic model was conceived by Tinoco and refined by Freier, Turner, Ninio, and others [13, 20–24].
1.1 knoncrossing, σcanonical RNA pseudoknot structures
Let us turn back the clock: three decades ago Waterman et al.[25], Nussinov et al.[11] and Kleitman et al. in [26] analyzed RNA secondary structures. Secondary structures are coarse grained RNA contact structures, see Figure 3.
RNA secondary structures as well as RNA pseudoknot structures can be represented as diagrams, i.e. labeled graphs over the vertex set [n] = {1, ..., n} with vertex degrees ≤ 1, represented by drawing its vertices on a horizontal line and its arcs (i, j) (i < j), in the upper halfplane, see Figure 4 and Figure 1. Given an arc (i, j) we refer to (j  i) as its arclength.
Here, vertices and arcs correspond to the nucleotides A, G, U, C and WatsonCrick (AU, GC) and (UG) base pairs, respectively.
In a diagram, two arcs (i_{1}, j_{1}) and (i_{2}, j_{2}) are called crossing if i_{1} < i_{2} < j_{1} < j_{2} holds. Accordingly, a kcrossing is a sequence of arcs (i_{1}, j_{1}), ..., (i_{ k }, j_{ k }) such that i_{1} < i_{2} < ... < i_{k} < j_{1} < j_{2} < ... < j_{ k }. We call diagrams containing at most (k  1)crossings, knoncrossing diagrams, see Figure 5.
RNA secondary structures exhibit no crossings in their diagram representation, see Figure 3 and Figure 4, and are therefore 2noncrossing diagrams satisfying some minimum arclength condition. An RNA pseudoknot structure is therefore a knoncrossing diagram for some k satisfying some minimum arclength condition.
A structure in which any stack has at least size σ is called σcanonical, where a stack of size σ is a sequence of "parallel" arcs of the form
A sequence of consecutive stacks, separated by unpaired nucleotides, i.e. where
is called a stem of length r, see Figure 6.
As a natural generalization of RNA secondary structures knoncrossing RNA structures [27–29] were introduced. A knoncrossing RNA structure of length n is knoncrossing diagram over [n] without arcs of the form (i, i + 1). In the following we assume k = 3, i.e. in the diagram representation there are at most two mutually crossing arcs, a minimum arclength of four and a minimum stacksize of three base pairs. The notion knoncrossing stipulates that the complexity of a pseudoknot is related to the maximal number of mutually crossing bonds. Indeed, most natural RNA pseudoknots are 3noncrossing [30].
1.2 Neutral networks
Before considering an inverse folding algorithm into specific RNA structures one has to have at least some rationale as to why there exists one sequence realizing a given target as mfeconfiguration. In fact this is, on the level of entire folding maps, guaranteed by the combinatorics of the target structures alone. It has been shown in [31], that the numbers of 3noncrossing RNA pseudoknot structures, satisfying the biophysical constraints grows asymptotically as c_{3}n^{5}2.03^{n}, where c_{3}> 0 is some explicitly known constant. In view of the central limit theorems of [32], this fact implies the existence of extended (exponentially large) sets of sequences that all fold into one 3noncrossing RNA pseudoknot structure, S. In other words, the combinatorics of 3noncrossing RNA structures alone implies that there are many sequences mapping (folding) into a single structure. The set of all such sequences is called the neutral network of the structure S[33, 34], see Figure 7. The term "neutral network" as opposed to "neutral set" stems from giant component results of random induced subgraphs of ncubes. That is, neutral networks are typically connected in sequence space.
By construction, all the sequences contained in such a neutral network are all compatible with S. That is, at any two positions paired in S, we find two bases capable of forming a bond (AU, UA, GC, CG, GU and UG), see Figure 8. Let s' be a sequence derived via a pointmutation of s. If s' is again compatible with S, we call this mutation "compatible".
Let C[S] denote the set of Scompatible sequences. The structure S motivates to consider a new adjacency relation within C[S]. Indeed, we may reorganize a sequence (s_{1}, ..., s_{ n } ) into the pair
where the u_{ h }denotes the unpaired nucleotides and the p_{ h }= (s_{ i }, s_{ j }) denotes base pairs, respectively, see Figure 8. We can then view and as elements of the formal cubes and implying the new adjacency relation for elements of C[S].
Accordingly, there are two types of compatible neighbors in the sequence space u and pneighbors: a uneighbor has Hamming distance one and differs exactly by a point mutation at an unpaired position. Analogously a pneighbor differs by a compensatory base pairmutation, see Figure 9.
Note, however, that a pneighbor has either Hamming distance one (GC ↦ GU) or Hamming distance two (GC ↦ CG). We call a u or a pneighbor, y, a compatible neighbor. In light of the adjacency notion for the set of compatible sequences we call the set of all sequences folding into S the neutral network of S. By construction, the neutral network of S is contained in C[S]. If y is contained in the neutral network we refer to y as a neutral neighbor. This gives rise to consider the compatible and neutral distance of the two sequences, denoted by C(s, s') and N(s, s'). These are the minimum length of a C[S]path and path in the neutral network between s and s', respectively. Note that since each neutral path is in particular a compatible path, the compatible distance is always smaller or equal than the neutral distance.
In this paper we study the inverse folding problem for RNA pseudoknot structures: for a given 3noncrossing target structure S, we search for sequences from C[S], that have S as mfe configuration.
2 Background
For RNA secondary structures, there are three different strategies for inverse folding, RNAinverse, RNASSD and INFORNA[35–37].
They all generate via a local search routine iteratively sequences, whose structures have smaller and smaller distances to a given target. Here the distance between two structures is obtained by aligning them as diagrams and counting "0", if a given position is either unpaired or incident to an arc contained in both structures and "1", otherwise, see Figure 10.
One common assumption in these inverse folding algorithms is, that the energies of specific substructures contribute additively to the energy of the entire structure. Let us proceed by analyzing the algorithms.
RNAinverse is the first inversefolding algorithm that derives sequences that realize given RNA secondary structures as mfeconfiguration. In its initialization step, a random compatible sequence s for the target T is generated. Then RNAinverse proceeds by updating the sequence s to s', s'' ... step by step, minimizing the structure distance between the mfe structure of s' and the target structure T. Based on the observation, that the energy of a substructure contributes additively to the mfe of the molecule, RNAinverse optimizes "small" substructures first, eventually extending these to the entire structure. While optimizing substructures, RNAinverse does an adaptive walk in order to decrease the structure distance. In fact, this walk is based entirely on random compatible mutations.
RNASSD inverse folds RNA secondary structures by initializing sequences using three specific subroutines. In the first a particular compatible sequence is generated, where noncomplementary nucleotides to bases adjacent to helical regions are assigned. In the second nucleotides located in unpaired positions as well as helical regions are assigned at random, using specific (nonuniform) probabilities. The third routine constitutes a mechanism for minimizing the occurrence of undesired but favourable interactions between specific sequence segments. Following these subroutines, RNASSD derives a hierarchical decomposition of the target structure. It recursively splits the structure and thereby derives a binary decomposition tree rooted in T and whose leaves correspond to Tsubstructures. Each nonleaf node of this tree represents a substructure obtained by merging the two substructures of its respective children. Given this tree, RNASSD performs a stochastic local search, starting at the leaves, subsequently working its way up to the root.
INFORNA constructs sequences folding into a given secondary structure by employing a dynamic programming method for finding a well suited initial sequence. This sequence has a lowest energy with respect to the T. Since the latter does not necessarily fold into T, (due to potentially existing competing configurations) INFORNA then utilizes an improved (relative to the local search routine used in RNAinverse) stochastic local search in order to find a sequence in the neutral network of T. In contrast to RNAinverse, INFORNA allows for increasing the distance to the target structure. At the same time, only positions that do not pair correctly and positions adjacent to these are examined.
2.1 Cross
Cross is an ab initio folding algorithm that maps RNA sequences into 3noncrossing RNA structures. It is guaranteed to search all 3noncrossing, σcanonical structures and derives some (not necessarily unique), loopbased mfeconfiguration. In the following we always assume σ ≥ 3. The input of Cross is an arbitrary RNA sequence s and an integer N. Its output is a list of N 3noncrossing, σcanonical structures, the first of which being the mfestructure for s. This list of N structures (C_{0}, C_{1}, ..., C_{N1}) is ordered by the free energy and the first listelement, the mfestructure, is denoted by Cross(s). If no N is specified, Cross assumes N = 1 as default.
Cross generates a mfestructure based on specific looptypes of 3noncrossing RNA structures. For a given structure S, let α be an arc contained in S (Sarc) and denote the set of Sarcs that cross α by . For two arcs α = (i, j) and α' = (i', j'), we next specify the partial order "≺" over the set of arcs:
All notions of minimal or maximal elements are understood to be with respect to ≺. An arc α ∈ is called a minimal, βcrossing if there exists no α' ∈ such that α' ≺ α. Note that α ∈ can be minimal βcrossing, while β is not minimal αcrossing. 3noncrossing diagrams exhibit the following four basic looptypes:
(1) A hairpinloop is a pair
where (i, j) is an arc and [i, j] is an interval, i.e. a sequence of consecutive, isolated vertices (i, i + 1, ..., j  1, j).
(2) An interiorloop, is a sequence
where (i_{2}, j_{2}) is nested in (i_{1}, j_{1}). That is we have i_{1}< i_{2}< j_{2}< j_{1}.
(3) A multiloop, see Figure 11[19], is the closed structure formed by
where denotes the substructure over the interval [ω_{ h }, τ_{ h }], subject to the condition that if all these substructures are simply stems, then there are at least two of them, see Figure 6.
A pseudoknot, see Figure 12[19], consists of the following data:
(P1) A set of arcs
where i_{1} = min{i_{ h }} and j_{ t }= max{j_{ h }}, such that

(i)
the diagram induced by the arcset P is irreducible, i.e. the dependencygraph of P (i.e. the graph having P as vertex set and in which α and α' are adjacent if and only if they cross) is connected and

(ii)
for each (i_{ h }, j_{ h }) ∈ P there exists some arc β (not necessarily contained in P) such that (i_{ h }, j_{ h }) is minimal βcrossing.
(P2) Any i_{1} < x < j_{ t }, not contained in hairpin, interior or multiloops.
Having discussed the basic looptypes, we are now in position to state
Theorem 1 Any 3noncrossing RNA pseudoknot structure has a unique loopdecomposition[19].
Figure 13 illustrates the loop decomposition of a 3noncrossing structure.
In order to discuss the organization of Cross, we introduce the basic idea behind motifs and skeleta, combinatorial structures used in the folding algorithm.
A motif is a 3noncrossing structure, having only ≺maximal stacks of size exactly σ, i.e. no stacks nested in other stacks, see Figure 14. Despite that motifs can exhibit complicated crossings, they can be inductively generated. A skeleton, S is a knoncrossing structure such that

its core, c(S) has no noncrossing arcs and

its Lgraph, L(S) is connected.
Here the core of a structure, c(S), is obtained by collapsing its stacks into single arcs (thereby reducing its length) and the graph L(S) is obtained by mapping arcs into vertices and connecting any two if they cross in the diagram representation of S, see Figure 15. A skeleton reflects all crossserial interactions of a structure.
Having introduced motifs and skeleta we can proceed by discussing the general idea of Cross. The algorithm generates 3noncrossing RNA structure "from top to bottom" via the following three subroutines:
I (SHADOW): In this routine we generate all maximal stacks of the structure. Note that a stack is maximal with respect to ≺ if it is not nested in some other stack. This is derived by "shadowing" the motifs, i.e. their σstacks are extended "from top to bottom".
II (SKELETON BRANCH): Given a shadow, the second step of Cross consists in generating, the skeletatree. The nodes of this tree are particular 3noncrossing structures, obtained by successive insertions of stacks. Intuitively, a skeleton encapsulates all crossserial arcs that cannot be recursively computed. Here the tree complexity is controlled via limiting the (total) number of pseudoknots.
III (SATURATION): In the third subroutine each skeleton is saturated via DProutines. After the saturation the mfe3noncrossing structure is derived.
Figure 16 provides an overview on how the three subroutines are combined.
3 The algorithm
The inverse folding algorithm Inv is based on the ab initio folding algorithm Cross. The input of Inv is the target structure, T. The latter is expressed as a character string of ":( )[ ]{ }", where ":" denotes unpaired base and "( )", "[ ]", "{ }" denote paired bases.
In Algorithm 7.1, we present the pseudocodes of algorithm Inv. After validation of the target structure (lines 2 to 5 in Algorithm 7.1), similar to INFORNA, Inv constructs an initial sequence and then proceeds by a stochastic local search based on the loop decomposition of the target. This sequence is derived via the routine ADJUSTSEQ. We then decompose the target structure into loops and endow these with a linear order. According to this order we use the routine LOCALSEARCH in order to find for each loop a "proper" local solution.
3.1 ADJUSTSEQ
In this section we describe Steps 2 and 3 of the pseudocodes presented in Algorithm 7.1. The routine MAKESTART, see line 8, generates a random sequence, start, which is compatible to the target, with uniform probability.
We then initialize the variable seq_{min} via the sequence start and set the variable d = + ∞, where d denotes the structure distance between Cross(seq_{min}) and T.
Given the sequence start, we construct a set of potential "competitors", C, i.e. a set of structures suited as folding targets for start. In Algorithm 7.2 we show how to adjust the start sequence using the routine ADJUSTSEQ. Lines 3 to 36 of Algorithm 7.2, contain a Forloop, executed at most times. Here the looplength is heuristically determined.
For all computer experiments setting the Crossparameter N = 50, the subroutine executed in the loopbody consists of the following three steps.
Step I. Generating C^{0}(λ^{i}) via Cross. Suppose we are in the i th step of the Forloop and are given the sequence λ^{i1}where λ^{0} = start. We consider Cross(λ^{i1}, N), i.e. the list of suboptimal structures with respect to λ^{i1},
If , then Inv returns λ^{i1}. Else, in case of , we set
Otherwise we do not update seq_{min} and go directly to Step II.
Step II. The competitors. We introduce a specific procedure that "perturbs" arcs of a given RNA pseudoknot structure, S. Let a be an arc of S and let l(a), r(a) denote the start and endpoint of a. A perturbation of a is a procedure which generates a new arc a', such that
Clearly, there are nine perturbations of any given arc a (including a itself), see Figure 17.
We proceed by keeping a, replacing the arc a by a nontrivial perturbation or remove a, arriving at a set of ten structures ν(S, a).
Now we use this method in order to generate the set C^{1}(λ^{i1}) by perturbing each arc of each structure . If has A_{ h }arcs, , then
This construction may result in duplicate, inconsistent or incompatible structures. Here, a structure is inconsistent if there exists at least one position paired with more than one base, and incompatible if there exists at least one arc not compatible with λ^{i1}, see Figures 18 and 19. Here compatibility is understood with respect to the WatsonCrick and GU base pairing rules. Deleting inconsistent and incompatible structures, as well as those identical to the target, we arrive at the set of competitors,
Step III. Mutation. Here we adjust λ^{i1}with respect to T as well as the set of competitors, C(λ^{i1}) derived in the previous step. Suppose . Let p(S, w) be the position paired to the position w in the RNA structure S ∈ C(λ^{i1}), or 0 if position w is unpaired. For instance, in Figure 20, we have p(T, 1) = 4, p(T, 2) = 0 and p(T, 4) = 1. For each position w of the target T, if there exists a structure C_{ h }(λ^{i1}) ∈ C(λ^{i1}) such that p(C_{ h }(λ^{i1}, w) ≠ p(T, w) (see positions 5, 6, 9, and 11 in Figure 20) we modify λ^{i1}as follows:

1.
unpaired position: If p(T, w) = 0, we update randomly into the nucleotide , such that for each C _{ h }(λ ^{i1}) ∈ C(λ ^{i1}), either p(C _{ h }(λ ^{i1}), w) = 0 or is not compatible with where v = p(C _{ h }(λ ^{i1}), w) < 0, See position 6 in Figure 20.

2.
startpoint: If p(T, w) < w, set v = p(T, w), We randomly choose a compatible base pair () different from (, ) such that for each C _{ h }(λ ^{i1}) ∈ C(λ ^{i1}), either p(C _{ h }(λ ^{i1}), w) = 0 or is not compatible with , where u = p(C _{ h }(λ ^{i1}), w) > 0 is the endpoint paired with in C _{ h }(λ ^{i1}) (Figure 20: (5, 9). The pair GC retains the compatibility to (5, 9), but is incompatible to (5, 10)). By Figure 21 we show feasibility of this step.

3.
endpoint: If 0 < p(T, w) < w, then by construction the nucleotide has already been considered in the previous step.
Therefore, updating all the nucleotides of λ^{i1}, we arrive at the new sequence .
Note that the above mutation steps heuristically decrease the structure distance. However, the resulting sequence is not necessarily incompatible to all competitors. For instance, consider a competitor C_{ h }whose arcs are all contained T. Since λ^{i}is compatible with T, λ^{i}is compatible with C_{ h }. Since competitors are obtained from suboptimal folds such a scenario may arise.
In practice, this situation represents not a problem, since these type of competitors are likely to be ruled out by virtue of the fact that they have a mfe larger than that of the target structure.
Accordingly we have the following situation, competitors are eliminated due to two, equally important criteria: incompatibility as well as minimum free energy considerations.
If the distance of Cross(λ^{i}) to T is less than or equal to d_{min} + 5, we return to Step I (with λ^{i}). Otherwise, we repeat Step III (for at most 5 times) thereby generating and set where d(Cross(), T) is minimal.
The procedure ADJUSTSEQ employs the negative paradigm [16] in order to exclude energetically close conformations. It returns the sequence seq_{middle} which is tailored to realize the target structure as mfefold.
3.2 DECOMPOSE and LOCALSEARCH
In this section we introduce two the routines, DECOMPOSE and LOCALSEARCH. The routine DECOMPOSE partitions T into linearly ordered energy independent components, see Figure 13 and Section 2.1. LOCALSEARCH constructs iteratively an optimal sequence for T via local solutions, that are optimal to certain substructures of T.
DECOMPOSE: Suppose T is decomposed as follows,
where the T_{ w }are the loops together with all arcs in the associated stems of the target.
We define a linear order over B as follows: T_{ w }< T_{ h }if either

1.
T _{ w }is nested in T _{ h }, or

2.
the startpoint of T _{ w }precedes that of T _{ h }.
In Figure 22 we display the linear order of the loops of the structure shown in Figure 13.
Next we define the interval
projecting the loop T_{ w }onto the interval [l(T_{ w }), r(T_{ w })] and b_{ w }= [l', r'] ⊃ a_{ w }, being the maximal interval consisting of a_{ w }and its adjacent unpaired consecutive nucleotides, see Figure 13. Given two consecutive loops T_{ w }< T_{w + 1}, we have two scenarios:

either b_{ w }and b_{w+1}are adjacent, see b_{5} and b_{6} in Figure 22,

or b_{ w }⊆ b_{w + 1}, see b_{1} and b_{2} in Figure 22.
Let , then we have the sequence of intervals a_{1}, b_{1}, c_{1}, ..., a_{ m' }, b_{ m' }, c_{ m' }. If there are no unpaired nucleotides adjacent to a_{ w }, then a_{ w }= b_{ w }and we simply delete all such b_{ w }. Thereby we derive the sequence of intervals I_{1}, I_{2}, ..., I_{ m }. In Figure 23 we illustrate how to obtain this interval sequence: here the target decomposes into the loops T_{1}, T_{2} and we have I_{1} = [3, 5], I_{2} = [3, 6], I_{3} = [2, 9], and I_{4} = [1, 10].
LOCALSEARCH: Given the sequence of intervals I_{1}, I_{2}, ..., I_{ m }. We proceed by performing a local stochastic search on the subsequences (initialized via seq = seq_{middle} and where s_{[x, y]}= s_{ x }s_{x + 1}... s_{ y }). When we perform the local search on , only positions that contribute to the distance to the target, see Figure 10, or positions adjacent to the latter, will be altered. We use the arrays U_{1}, U_{2} to store the unpaired and paired positions of T. In this process, we allow for mutations that increase the structure distance by five with probability 0.1. The latter parameter is heuristically determined. We iterate this routine until the distance is either zero or some halting criterion is met.
4 Discussion
The main result of this paper is the presentation of the algorithm Inv, freely available at http://www.combinatorics.cn/cbpc/inv.html
Its input is a 3noncrossing RNA structure T, given in terms of its base pairs (i_{1}, i_{2}) (where i_{1} < i_{2}). The output of Inv is an RNA sequences s = (s_{1}s_{2}...s_{ n }), where s_{ h }∈ {A, C, G, G} with the property Cross(s) = T, see Figure 24.
The core of Inv is a stochastic local search routine which is based on the fact that each 3noncrossing RNA structure has a unique loopdecomposition, see Theorem 1 in Section 2.1. Inv generates "optimal" subsequences and eventually arrives at a global solution for T itself. Inv generalizes the existing inverse folding algorithm by considering arbitrary 3noncrossing canonical pseudoknot structures. Conceptually, Inv differs from INFORNA in how the start sequence is being generated and the particulars of the local search itself.
As discussed in the introduction it has to be given an argument as to why the inverse folding of pseudoknot RNA structures works. While folding maps into RNA secondary structures are well understood, the generalization to 3noncrossing RNA structures is nontrivial. However the combinatorics of RNA pseudoknot structures [27, 28, 38] implies the existence of large neutral networks, i.e. networks composed by sequences that all fold into a specific pseudoknot structure. Therefore, the fact that it is indeed possible to generate via Inv sequences contained in the neutral networks of targets against competing pseudoknot configurations, see Figure 24 and Figure 25 confirms the predictions of [31].
An interesting class are the 3noncrossing nonplanar pseudoknot structures. A nonplanar pseudoknot structure is a 3noncrossing structure which is not a bisecondary structure in the sense of Stadler [30]. That is, it cannot be represented by noncrossing arcs using the upper and lower half planes. Since DPfolding paradigms of pseudoknots folding are based on gapmatrices [15], the minimal class of "missed" structures (given the implemented truncations) are exactly these, nonplanar, 3noncrossing structures. In Figure 26 we showcase a nonplanar RNA pseudoknot structure and 3 sequences of its neutral network, generated by Inv.
As for the complexity of Inv, the determining factor is the subroutine LOCALSEARCH. Suppose that the target is decomposed into m intervals with the length ℓ_{1}, ...., ℓ_{ m }. For each interval, we may assume that line 2 of LOCALSEARCH runs for f_{ h }times, and that line 14 is executed for g_{ h }times. Since LOCALSEARCH will stop (line 4) if T_{ start }= T (line 3), the remainder of LOCALSEARCH, i.e. lines 7 to 41 run for (f_{ h } 1) times, each such execution having complexity O(ℓ_{ h }). Therefore we arrive at the complexity
where c(ℓ) denotes the complexity of the Cross. The multiplicities f_{ h }and g_{ h }depend on various factors, such as start, the random order of the elements of U_{1}, U_{2} (see Algorithm 7.3) and the probability p. According to [32] the complexity of c(ℓ_{ h }) is and accordingly the complexity of Inv is given by
In Figure 27 we present the average inverse folding time of several natural RNA structures taken from the PKdatabase [39]. These averages are computed via generating 200 sequences of the target's neutral networks. In addition we present in Table 1 the total time for 100 executions of Inv for an additional set of RNA pseudoknot structures.
7 Appendix
7.1 Algorithm 7.1  INVERSEFOLD
Input: knoncrossing target structure T
Output: an RNA sequence seq
Require: k ≤ 3 and T is composed of ":( ) [ ] { }"
Ensure: Cross(seq) = T
1. ▻ Step 1: Validate structure
2. if false = CHECKSTRU(T) then
3. print incorrect structure
4. return NIL
5. end if
6.
7. ▻ Step 2: Generate the start sequence
8. start ← MAKESTART(T)
9.
10. ▻ Step 3: Adjust the start sequence
11. seq _{middle} ← ADJUSTSEQ(start, T)
12.
13. ▻ Step 4: Decompose T and derive the ordered intervals.
14. Interval array I
15. m ← I ▻ I satisfies I _{ m }= T
16.
17. ▻ Step 5: Stochastic Local Search
18. seq ← seq _{middle}
19. for all intervals in the array I _{ w } do
20. l ← startpoint(I _{ w })
21. r ← endpoint(I _{ w })
22. s' ← seq_{[l, r]}▻ get subsequence
23. seq_{[l, r]}LOCALSEARCH(s', I _{ w })
24. end for
25.
26. ▻ Step 6: output
27. if seq _{min} = Cross(seq) then
28. return seq
29. else
30. print Failed!
31. return NIL
32. end if
7.2 Algorithm 7.2  ADJUSTSEQ
Input: the original start sequence start
Input: the target structure T
Output: an initialized sequence seq_{middle}
1. n ← length of T
2. d _{min} ← + ∞, seq _{min} ← start
3. for i = 1 to do
4. ▻ Step I: generate the set C ^{0}(λ ^{i  1}) via Cross
5. C ^{0}(λ ^{i  1}) ← Cross(λ ^{i  1}, N)
6.
7. if d = 0 then
8. return λ ^{i  1}
9. else if d < d _{min} then
10. d _{min} ← d, seq _{min} ← λ ^{i  1}
11. end if
12.
13. ▻ Step II: generate the competitor set C(λ ^{i  1})
14. C ^{1}(λ ^{i1}) ← ϕ
15. for all ∈ C ^{1}(λ ^{i1}) do
16. for all arc of do
17.
18. end for
19. end for
20. C(λ ^{i  1}) =
21. { is valid}
22.
23. ▻ Step III: mutation
24. seq ← λ ^{i  1}
25. for w = 1 to n do
26. if ∃ C _{ h }(λ ^{i1}) ∈ C(λ ^{i1}) s.t. p(C _{ h }, w) ≠ p(T, w) then
27. seq[w] ← random nucleotide or pair, s.t. seq ∈ C[T] and seq ∉ C[C _{ h }(λ ^{i1})]
28. end if
29. end for
30. T _{ seq }← Cross(seq)
31. if d(T _{ seq }, T) < d _{min} + 5 then
32. seq _{middle} ← seq
33. else if Step III run less than 5 times then
34. goto Step III
35. end if
36. end for ▻ loop to line 3
37.
38. return seq _{middle}
7.3 Algorithm 7.3  LOCALSEARCH
Input: seq _{middle}
Input: the target T
Output: seq
Ensure: Cross(seq) = T
1. seq ← seq _{middle}
2. if Cross(seq) = T then
3. return seq
4. end if
5. decompose T and derive the ordered intervals
6. I ← [I _{1}, I _{2}, ..., I _{ m }]
7. for all I_{ w } in I do
8. ▻ Phase I: Identify positions
9. ▻ initialize d _{min}
10.
11. derive U _{1} via
12. derive U _{2} via
13.
14. ▻ Phase II: Test and Update
15. for all p in U _{1} do
16. random T compatible mutate seq _{ p }
17. end for
18. for all [p, q] in U _{2} do
19. random T compatible mutate seq _{ p }
20. end for
21.
22. E ← ϕ
23. for all p ∈ U _{1}, U _{2} do
24. d ← d(T, Cross(seq _{ p }))
25. if d < d _{ min } then
26. d _{min} ← d, seq ← seq _{ p }
27. goto Phase I
28. else if d _{ min }< d < d _{ min }+ 5 then
29. goto Phase I with the probability 0.1
30. end if
31. if d = d _{ min } then
32. E ← E ∪ {seq}
33. end if
34. end for
35. seq ← e _{0} ∈ E, where e _{0} has the lowest mfe in E
36. if Phase I run less than 10 n times then
37. goto Phase I
38. end if
39. end for
40. return seq
8 Acknowledgements
We are grateful to Fenix W.D. Huang for discussions. Special thanks belongs to the two anonymous referee's whose thoughtful comments have greatly helped in deriving an improved version of the paper. This work was supported by the 973 Project, the PCSIRT of the Ministry of Education, the Ministry of Science and Technology, and the National Science Foundation of China.
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Gao, J.Z., Li, L.Y. & Reidys, C.M. Inverse folding of RNA pseudoknot structures. Algorithms Mol Biol 5, 27 (2010). https://doi.org/10.1186/17487188527
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Keywords
 Target Structure
 Neutral Network
 Structure Distance
 Stochastic Local Search
 Pseudoknot Structure