 Research
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Fast prediction of RNARNA interaction
Algorithms for Molecular Biology volume 5, Article number: 5 (2010)
Abstract
Background
Regulatory antisense RNAs are a class of ncRNAs that regulate gene expression by prohibiting the translation of an mRNA by establishing stable interactions with a target sequence. There is great demand for efficient computational methods to predict the specific interaction between an ncRNA and its target mRNA(s). There are a number of algorithms in the literature which can predict a variety of such interactions  unfortunately at a very high computational cost. Although some existing target prediction approaches are much faster, they are specialized for interactions with a single binding site.
Methods
In this paper we present a novel algorithm to accurately predict the minimum free energy structure of RNARNA interaction under the most general type of interactions studied in the literature. Moreover, we introduce a fast heuristic method to predict the specific (multiple) binding sites of two interacting RNAs.
Results
We verify the performance of our algorithms for joint structure and binding site prediction on a set of known interacting RNA pairs. Experimental results show our algorithms are highly accurate and outperform all competitive approaches.
Background
Regulatory noncoding RNAs (ncRNAs) play an important role in gene regulation. Studies on both prokaryotic and eukaryotic cells show that such ncRNAs usually bind to their target mRNA to regulate the translation of corresponding genes. Many regulatory RNAs such as microRNAs and small interfering RNAs (miRNAs/siRNAs) are very short and have full sequence complementarity to the targets. However some of the regulatory antisense RNAs are relatively long and are not fully complementary to their target sequences. They exhibit their regulatory functions by establishing stable joint structures with target mRNA initiated by one or more looploop interactions.
In this paper we present an efficient method for the RNARNA interaction prediction (RIP) problem with multiple binding domains. Alkan et al. [1] proved that RIP, in its general form, is an NPcomplete problem and provided algorithms for predicting specific types of interactions and two relatively simple energy models  under which RIP is polynomial time solvable. We focus on the same type of interactions, which to the best of our knowledge, are the most general type of interactions considered in the literature; however the energy model we use is the joint structure energy model recently presented by Chitsaz et al. [2] which is more general than the one used by Alkan et al.
In what follows below, we first describe a combinatorial algorithm to compute the minimum free energy joint structure formed by two interacting RNAs. This algorithm has a running time of O(n^{6}) and uses O(n^{4}) space  which makes it impractical for long RNA molecules. Then we present a fast heuristic algorithm to predict the joint structure formed by interacting RNA pairs. This method provides a significant speedup over our combinatorial method, which it achieves by exploiting the observation that the independent secondary structure of an RNA molecule is mostly preserved even after it forms a joint structure with another RNA. In fact there is strong evidence [3, 4] suggesting that the probability of an ncRNA binding to an mRNA target is proportional to the probability of the binding site having an unpaired conformation. The above observation has been used by different methods for target prediction in the literature (see below for an overview). However, most of these methods focus on predicting interactions involving only a single binding site, and are not able to predict interactions involving multiple binding sites. In contrast, our heuristic approach can predict interactions involving multiple binding sites by: (1) identifying the collection of accessible regions for both input RNA sequences, (2) using a matching algorithm, computing a set of "nonconflicting" interactions between the accessible regions which have the highest overall probability of occurrence.
Note that an accessible region is a subsequence in an RNA sequence which, with "high" probability, remain unpaired in its secondary structure. Our method considers the possibility of interactions being formed between one such accessible region from an RNA sequence with more than one such region from the other RNA sequence. Thus, in step (1), it extends the algorithm by Mückstein et al. for computing the probability of a specific region being unpaired [5] to compute the joint probability of two (or more) regions remaining unpaired. Because an accessible region from an RNA typically interacts with no more than two accessible regions from the other RNA, we focus on calculating the probability of at most two regions remaining unpaired: within a given an RNA sequence of length n, our method can calculate the probability of any pair of regions of length ≤ w each, in O(n^{4}.w) time and O(n^{2}) space. In step (2), on two input RNA sequences of length n and m (n ≤ m), our method computes the most probable nonconflicting matching of accessible regions in O(n^{2}.w^{4} + n^{3}/w^{3}) time and O(w^{4} + n^{2}/w^{2}) space.
Related work
Early attempts to compute the joint structure of interacting RNAs started by concatenating the two interacting RNA sequences and treated them as a single sequence PairFold[6] and RNAcofold[7]. Dirks et al. present a method, as a part of NUPack, that concatenates the input sequences in some order, carefully considering symmetry and sequence multiplicities, and computes the partition function for the whole ensemble of complex species [8]. As these methods typically use secondary structure prediction methods that do not allow pseudoknots, they fail to predict joint structures formed by nontrivial interactions between a pair of RNAs.
Another set of methods ignore internal basepairing in both RNAs, and compute the minimum free energy secondary structure for their hybridization (RNAhybrid[9], UNAFold[10, 11], and RNAduplex from Vienna package [7]). These approaches work only for simple cases involving typically very short strands.
A further set of studies aim to compute the minimum free energy joint structure between two interacting RNAs. For example Pervouchine [12] devised a dynamic programming algorithm to maximize the number of base pairs among interacting strands. A follow up work by Kato et al. [13] proposed a grammar based approach to RNARNA interaction prediction. More generally Alkan et al. [1] studied the joint secondary structure prediction problem under three different models: 1) base pair counting, 2) stacked pair energy model, and 3) loop energy model. Alkan et al. proved that the general RNARNA interaction prediction under all three energy models is an NPhard problem. Therefore, they suggested some natural constraints on the topology of possible joint secondary structures which are satisfied by all examples of complex RNARNA interactions in the literature. The resulting algorithms compute the optimum structure among all possible joint secondary structures that do not contain pseudoknots, crossing interactions, and zigzags (please see [1] for the exact definition). In fact the last set of algorithms above are the only methods that have the capability to predict joint secondary structures with multiple looploop interactions. However, these algorithms all requires significant computational resources (O(n^{6}) time and O(n^{4}) spaces) and thus are impractical for sequences of even modest length.
A final group of methods are based on the observation that interaction is a multi step process [14] that involves: 1) unfolding of the two RNA structures to expose the bases needed for hybridization, 2) the hybridization at the binding site, and 3) restructuring of the complex to a new minimum free energy conformation. The main aim of these methods is to identify the potential binding sites which are going to be unfolded in order to form interactions. One such method presented by Alkan et al. [1], extends existing loop regions in independent structures to find potential binding sites. RNAup[15] presents an extension of the standard partition function approach to compute the probabilities that a sequence interval remains unpaired. IntaRNA[16] considers not only accessibility of a binding sites but also the existence of a seed to predict potential binding sites. All of these methods achieve reasonably high accuracy in predicting interactions involving single binding sites; however, their accuracy levels are not very high when dealing with interactions involving multiple binding sites.
Methods
We address the RNARNA Interaction Problem (RIP) based on the interaction energy model proposed by Chitsaz et al. [2] over the type of interaction considered by Alkan et al. [1]. Our algorithm computes the minimum free energy joint secondary structure that does not contain pseudoknots, crossing interactions, and zigzags. The zigzag constraint simply states that if two substructures from two RNAs interact, then one substructure must subsume the other.
RNARNA joint structure prediction
Recently Chitsaz et al. [2] present an energy model for joint structure of two nucleic acid strands over the type of interaction introduced by Alkan et al. [1]. Based on the presented energy model they propose an algorithm that consider all possible joint secondary structures to compute the partition function for two interacting nucleic acid strands. The specified algorithm with some minor changes can be used to compute the minimum free energy joint structure of two interacting nucleic acid strands. Following we shortly describe the dynamic programming algorithm to predict the minimum free energy RNARNA interaction. We are given two RNA sequences R and S of lengths n and m. Strand R is indexed from 1 to n in 5' to 3' direction and S is indexed from 1 to m in 3' to 5' direction. Note that the two strands interact in opposite directions, i.e. R in 5' → 3' with S in 3' ← 5' direction. Each nucleotide is paired with at most one nucleotide in the same or the other strand. We refer to the i^{th}nucleotide in R and S by i_{ R }and i_{ S }respectively. The subsequence from the i^{th}nucleotide to the j^{th}nucleotide in one strand is denoted by [i, j]. We denote a base pair between the nucleotides i and j by i·j. MFE(i, j) denotes the minimum free energy structure of [i, j], and MFE(i_{ R }, j_{ R }, i_{ S }, j_{ S }) denotes the minimum free energy joint structure of [i_{ R }, j_{ R }] and [i_{ S }, j_{ S }].
Figure 1 shows the recursion diagram of the MFE joint structure of [i_{ R }, j_{ R }] and [i_{ S }, j_{ S }]. In this figure a horizontal line indicates the phosphate backbone, a dashed curved line encloses a subsequence and denotes its two terminal bases which may be paired or unpaired. A solid vertical line indicates an interaction base pair, a dashed vertical line denotes two terminal bases which may be base paired or unpaired, and a dotted vertical line denotes two terminal bases which are assumed to be unpaired. Grey regions indicate a reference to the substructure of single sequences.
The joint structure of two subsequences derived from one of the following cases. The first possibility is when there is no interaction between the two subsequences. If there are some interaction bonds, the structure has two cases: either the leftmost bond is closed by base pair in at least one of the subsequences or not. If the joint structure starts with a bond which is not closed by any base pair we denote the case by Ib, otherwise the structure starts with a bond which is closed by base pair in at least one subsequence and the case is denoted by Ia. Therefore, MFE(i_{ R }, j_{ R }, i_{ S }, j_{ S }) is calculated by the following dynamic programming:
in which MFE^{Ib}(k_{1}, j_{ R }, k_{2}, j_{ S }) is the minimum free energy for the joint structure of [k_{1}, j_{ R }] and [k_{2}, j_{ S }] assuming k_{1}·k_{2} is an interaction bond, and MFE^{Ia}(k_{1}, j_{ R }, k_{2}, j_{ S }) is the minimum free energy for the joint structure of [k_{1}, j_{ R }] and [k_{2}, j_{ S }] assuming the leftmost interaction bond is covered by a base pair in at least one subsequence. The corresponding dynamic programing for computing the MFE^{Ib}and MFE^{Ia}can be derived from the cases explained in [2] in a similar way.
Similar to the partition function algorithm, the minimum free energy joint structure prediction algorithm has O(n^{6}) running time and O(n^{4}) space requirements. However the algorithm is highly accurate (see experimental results), but it requires substantial computational resources. Thus it could be prohibitive for predicting the joint secondary structures of long RNA molecules. In next section we present a fast heuristic algorithm to predict RNARNA interaction without applying any restriction on type of interaction and energy model.
RNARNA binding sites prediction
Our heuristic algorithm for prediction of RNARNA interactions involving multiple binding sites is based on the idea that the external interactions mostly occur between unpaired regions of two RNA structures. The heuristic algorithm contains the following steps:

Predict highly accessible regions in each strands. These regions include the loop regions in native structure of RNA strand. In order to predict accessible regions we chose all the regions which remain unpaired with high probability.

Predict the optimal nonconflicting interactions between the accessible regions. For every pair of accessible regions of two interacting RNAs a cost of interaction is calculated. Then a matching algorithm runs to find the minimum cost nonconflicting subset of interactions.
Accessible regions
For a single RNA sequence an accessible region is a subsequence that remains unpaired in equilibrium with high probability. The probability of an unpaired region can be calculated based on the algorithm presented in RNAup [5]. Since we are interested in multiple unpaired regions, we need to consider the joint probabilities for all possible subsets of intervals. However, computation of all joint probabilities requires substantial time and space and thus in this paper we only consider the joint probability of two unpaired subsequences as well as the probability of an unpaired subsequence.
Denoting the set of secondary structures in which the sequence interval [k, l] remains unpaired by S^{u [k, l]}, the corresponding partition function is
where R is the universal gas constant and T is the temperature. In order to compute the Q^{u [k, l]}, the standard recursion for the partition function folding algorithm [17] can be extended based on the recursion cases in Figure 2. Therefore,
where i ≤ k ≤ l ≤ j and k_{1}·k_{2} is the leftmost base pair. Note that without loss of generality we assumed i ≤ k ≤ l ≤ j. Clearly if [k, l] is not a subsequence of [i, j], we have . In fact for any arbitrary interval [k, l] is equivalent to such that [k', l'] is the common subsequence between [i, j] and [k, l].
Partition functions (where i·j is a base pair) and (where [i, j] is inside a multiloop and constitutes at least one base pair) while the interval [k, l] remains unpaired are derived from the standard algorithm in a similar way. Furthermore, probability of a base pair p·q while [k, l] remains unpaired, ℙ(p·qu [k, l]), can be calculated by applying the McCaskill algorithm [17] for computing the base pair probability on Q^{u [k, l]}. It is easy to see that the desired partition function Q^{u [k, l]}and base pair probability ℙ(p·qu [k, l]) are computed in same time and space complexity as the standard algorithm by McCaskill  it has O(n^{3}) time and O(n^{2}) space complexity.
Mückstein et al. [5] introduce an algorithm to compute the probability of unpaired region ℙ(u [i, j]) for a given sequence interval [i, j]. Here, we extend the specified algorithm to compute ℙ(u [i, j]u [k, l]) which is the probability of unpaired sequence interval [i, j] while interval [k, l] remains unpaired. Clearly if some part of [i, j] is within the interval [k, l], the corresponding probability for that part is equal to one. Hence, for computing the probability only those parts of [i, j] which are exterior to [k, l] should be considered. Here, without loss of generality we assume k ≤ l ≤ i ≤ j.
For an unpaired interval [i, j] there are two general cases: either it is not closed by any base pair, or it is part of a loop. Figure 3 summarizes the cases of unpaired interval [i, j] as a part of the loop enclosed by base pair p·q while interval [k, l] remains unpaired. In case x interval [p, q] does not contain interval [k, l], and in the other cases (a  e) interval [k, l] lies in interval [p, q]. Probability ℙ(u [i, j]u [k, l]) can be calculated as follows:
The partition function Q^{pq}[i, j] which is introduced by Mückstein et al. considers all structures on [p, q] while [i, j] is part of the loop closed by base pair p·q. The quantity Q^{pq, u [k, l]}[i, j] is a variant of Q^{pq}[i, j] while [k, l] lies in [p, q]. Recursion of Q^{pq, u [k, l]}[i, j] on cases (a  e) displayed in Figure 3, is based on different types of loop and position of [k, l]. Therefore, we have
where Q^{m 2}is the partition function of a subsequence inside a multiloop that constitutes at least two base pairs. Q^{m 2}which is introduced in Mückstein et al. algorithm can be extended to calculate Q^{m 2, u [k, l]}:
where is the partition function of a subsequence inside a multiloop that constitutes exactly one base pair such that k_{1} is one terminal of that base pair. Recursion of can be simply derived from. recursion of . Therefore, the joint probability of two unpaired regions is obtained using
The Mückstein et al. algorithm requires O(n^{3}) running time and O(n^{2}) space complexity to compute the probability of unpaired region ℙ(u [i, j]) for every possible interval [i, j] assuming the interval length is limited to size w. Using the extended algorithm, given sequence interval [k, l] computing ℙ(u [i, j], u [k, l]) for every possible interval [i, j] requires the same time and space complexity. Note that for each interval [k, l], Q^{u [k, l]}should be computed separately. Since there are O(n.w) different intervals for a limited interval length w, with O(n^{4}.w) running time and O(n^{2}) space complexity we are able to compute the joint probabilities for all pairs of unpaired regions. The same idea can be used to compute the joint probability of multiple unpaired regions. However, considering each extra interval increases the running time by a factor of O(n.w).
All the regions that have probability of being unpaired more than some fixed threshold are selected as accessible regions r_{ i }from sequecen R (as well as s_{ j }from sequecen S). For two consecutive intervals, r_{ i }= [k_{ i }, l_{ i }] and r_{ i }+1 = [k_{i+1}, l_{i+1}], in order to decide whether the concatenated region should be considered the joint probability ℙ(u [r_{ i }], u [r_{i+1}]) and single probability ℙ(u [k_{ i }...l_{i+1}]) are compared. The selected intervals are extended by some limited number of nucleotides (< 5) in each side.
Interaction matching algorithm
Given two lists of nonoverlapping accessible regions T_{ R }= {r_{1}, r_{2}, ..., r_{ n' }} and T_{ S }= {s_{1}, s_{2}, ..., s_{ m' }}sorted according to their orders in interacting sequences R and S, we aim to calculate the optimal set of interactions between the accessible regions under the following constraints:

Each accessible region can interact with at most two accessible regions from the other sequence.

There is no crossing interaction.
For computing the interaction between accessible regions, IntaRNA minimizes the free energy of interaction and RNAup maximizes the probability of interaction while no internal base pair is allowed. Both approaches use RNAhybrid energy model for interaction. As mentioned before, we select a set of high probable unpaired intervals and extend them by some limited number of nucleotides. This extension is motivated by the observation that suggests usually the hybridization initiated at the accessible regions, and then some adjacent internal base pairs open up to form new interactions and make the complex more stable [14]. In order to not always prefer interaction rather than internal base pair in accessible regions, our method allows internal base pairs as well as interactions between accessible regions. We consider both options of minimizing the free energy of interaction and maximizing the probability of interaction while the interaction energy model introduced by [2] has been used.
Let be the partition function over all possible joint structures of two subsequences r_{ i }and s_{ j }, which can be calculated by interaction between accessible piRNA[2]. Define as the partition function for the set of joint structures that contain some interactions. We denote two interacting subsequences r_{ i }and s_{ j }by r_{ i }∘ s_{ j }. Therefore, probability of interaction for two accessible regions ri and s_{ j }is considered as . The interaction between two accessible regions r_{ i }and s_{ j }is considered if and only if ℙ(r_{ i }∘ s_{ j }) > 1/2, i.e. the probability of interaction for two accessible regions is higher than the probability of forming independent single structures. In this case the ensemble free energy of interacting joint structure for the two accessible regions is
Also the minimum free energy of interaction for two accessible regions r_{ i }and s_{ j }, MFE(r_{ i }, s_{ j }), can be calculated by using the dynamic programming algorithm explained in previous section. If our goal is to minimize the free energy of interaction, accessible regions r_{ i }and s_{ j }are considered to be able to interact if and only if MFE(r_{ i }, s_{ j }) <MFE(r_{ i }) + MFE(s_{ j }), i.e. there are some interaction bonds in the minimum free energy joint structure.
Let E_{ u }(ri) as the energy difference between the complete ensemble and the ensemble in which the interacting subsequences are left unpaired for accessible region r_{ i }. We have
The cost of interaction between two accessible regions r_{ i }and s_{ j }, C(r_{ i }, s_{ j }), is the sum of the following terms: (i) E_{ u }(r_{ i }), (ii) E_{ u }(s_{ j }), and (iii) E_{ I }(r_{ i }, s_{ j }) or MFE(r_{ i }, s_{ j }). Cost of interaction between an accessible region r_{ i }and two other accessible regions s_{ k }and s_{ j }is defined as
where s_{ k }s_{ j }is the concatenation of two subsequences, and E_{ u }(s_{ k }, s_{ j }) = (RT) ln(ℙ(u [s_{ k }], u [s_{ j }])). Similarly the cost of interaction between two accessible regions from R and one accessible region from S is defined. Also the cost of interaction where minimum free energy MFE(r_{ i }, s_{ k }s_{ j }) is used instead of ensemble energy E_{ I }(r_{ i }, s_{ k }s_{ j }) can be defined in a similar way.
With H(i, j), we denote the minimum cost nonconflicting set of interactions between the accessible regions {r_{1}, ..., r_{ i }} and {s_{1}, ..., s_{ j }}. The following dynamic programming computes H(i, j):
where 1 ≤ i ≤ n' and 1 ≤ j ≤ m'. The algorithm starts by calculating H(1, 1) and explores all H(i, j) by increasing i and j until i = n' and j = m'. The DP algorithm has O(n'^{2}.m' + n'.m'^{2}) time and O(n'.m') space requirements. Also we need O(n'.m'.w^{6}) time and O(w^{4}) space to compute the cost of interaction for every pair of accessible regions. Assuming n' ≥ m' and n' ≤ n/w, we can conclude that this step of the algorithm requires O(n^{2}.w^{4} + n^{3}/w^{3}) time and O(w^{4} + n^{2}/w^{2}) space.
CopACopT is a well known antisense RNAtarget complex observed in E. coli [18]. The joint structure of CopACopT contains two disjoint binding sites. Figure 4 shows the identified accessible regions in CopA and CopT. Two regions connected by an edge are able to interact. Figure 5 shows the known and predicted interaction bonds between CopA and CopT. Note that internal bonds of both RNAs are not displayed in this figure.
Results and Discussion
Dataset
In our experiments we use a dataset of 23 known RNARNA interactions which contains two recently compiled test sets. The first set includes 5 pairs of RNAs which are known to have looploop interactions and have been used by Kato et al. [13] to evaluate the proposed grammatical parsing approach for RNARNA joint structure prediction. The next 18 sRNAtarget pairs are compiled and used as test set by Busch et al. in IntaRNA[16]. In our dataset OxySfhlA and CopACopT are the only ones that have two disjoint binding sites.
Joint secondary structure prediction
In our first experiment, we assess the performance of our prediction algorithm for minimum free energy joint structure. For this purpose we use the 5 RNARNA complexes from Kato et al. [13] test set. We compare our results with two other stateoftheart methods for joint structure prediction: (1) the grammatical approach by Kato et al. [13] (denoted by EBM as energybased model), and (2) the DP algorithms for two energy models presented by Alkan et al. [1] (denoted by SPM as stackedpair model and LM as loop model).
In order to estimate the accuracy of prediction, we measure the sensitivity and PPV defined as follows:
As another measure of accuracy we calculate Fmeasure which considers both sensitivity and PPV. Fmeasure is the harmonic mean of sensitivity and PPV, and its formula is as follows:
Table 1 shows the accuracy results of our method and the other competitors for joint structure prediction. We refer to our method by inRNAs as an algorithm for prediction the interactions between RNAs. As it can be seen in Table 1, our method based on the three accuracy measures outperforms the competitors. For TarTar* and R1invR2inv pairs that both RNAs are relatively short (~20 nt), all methods are accurate enough. However, for DISDIS which is not still long (35 nt), only our method is able to predict the interaction while the other approaches return no interaction. CopACopT and IncRNA_{54}RepZ are a bit longer (~60 nt); CopACopT has two disjoint binding sites and IncRNA_{54}RepZ has a continuous binding site. Our method outperforms the others in predicting the joint structure of CopACopT, while IncRNA_{54}RepZ is predicted more accurately by EBM. We do not compare the running time between these methods due to the fact that each one uses different platform and hardware. Our method on one Sun Fire processor X4600 2.6 GHz with 64 GB RAM runs for ~4000(sec) to predict the joint structures of CopACopT and IncRNA_{54}RepZ.
Binding sites prediction
In another experiment, we test the performance of our heuristic algorithm for interaction prediction. In order to identify the set of accessible regions in each sequence we set w = 25 and use E_{ u }< min{E_{ u }} + 2(kcal/mol) as cutoff. For assessing the predictive power of our algorithm, we compare our algorithm with IntaRNA[16] and RNAup[15]. Based on the experimental results presented by IntaRNA, both IntaRNA and RNAup which incorporate accessibility of target regions, perform better than the other competitive programs (TargetRNA[19], RNAhybrid[9], and RNAplex[20]).
The results of these two programs for the first 18 RNA pairs are as presented in [16]. For the next 5 RNA pairs, we run IntaRNA with its default settings and RNAup with the same setting that has been used by the experiment in [16]  RNAup has been run using parameter b which considers the probability of unpaired regions in both RNAs and the maximal length of interaction to 80. In order to estimate accuracy of the programs, we measure the sensitivity, PPV and Fmeasure such that only interacting base pairs are considered.
Table 2 shows the results of our programs as well as IntaRNA and RNAup. In this dataset OxySfhlA and CopACopT are the only ones that have two disjoint binding sites, and our method clearly outperforms IntaRNA and RNAup by up to 30% improvement in Fmeasure. For the OxySfhlA complex with two looploop interactions, our method is able to find both binding sites. However, the other methods find only one of the binding sites. For CopACopT complex which contains one looploop interaction and one uncovered interaction site, again our method finds both binding sites. IntaRNA predicts one continues long binding site and RNAup predicted only the binding site within the looploop interaction. Another interesting case is GcvBgltI complex. Both RNAup and IntaRNA can not predict any correct bond for GcvBgltI, since they missed the binding site. However, IntaRNA can get 80% accuracy by considering the first suboptimal prediction which is close to the accuracy that we have achieved. In overall, the results demonstrate that our method predicts RNARNA interactions more accurately in compare to the competitive methods.
Conclusions
This paper introduce a fast algorithm for RNARNA interaction prediction. Our heuristic algorithm for the RNARNA interaction prediction problem incorporates the accessibility of multiple unpaired regions, and a matching algorithm to compute the optimal set of interactions involving multiple binding sites. The algorithm requires O(n^{4}.w) running time and O(n^{2}) space complexity. Note that the simplified version that allows each accessible region interact with at most one accessible region from the other sequence can be done in O(n^{3}) running time. The main advantage of our method is its ability to predict multiple binding sites which have been predictable only by expensive algorithms [1, 13] so far. On a set of several known RNARNA complexes, our proposed algorithm shows a reliable accuracy. Especially, for complexes with multiple binding sites our approach is able to outperform the competitive methods.
It would be interesting to design a method to efficiently compute the joint probability of multiple unpaired regions. Furthermore, the improvement of IntaRNA which get some benefit by considering seed features in comparison to RNAup, encourages us to take into account the existence of seed in the follow up work.
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Acknowledgements
RS was supported by Mitacs Research Grant. R. Backofen received funding from the German Research Foundation (DFG grant BA 2168/21 SPP 1258), and from the German Federal Ministry of Education and Research (BMBF grant 0313921 FRISYS). SCS was supported by Michael Smith Foundation for Health Research Career Award.
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The authors declare that they have no competing interests.
Authors' contributions
RS participated in the design of the algorithm, performed the experiments, and drafted the manuscript. RB contributed to the design of the algorithm. SCS conceived of the study, contributed to the algorithm design, and supervised the project. All authors contributed to the writing of the manuscript.
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Keywords
 Partition Function
 Energy Model
 Minimum Free Energy
 Joint Structure
 Accessible Region