Fast prediction of RNARNA interaction
 Raheleh Salari^{1},
 Rolf Backofen^{2} and
 S Cenk Sahinalp^{1}Email author
DOI: 10.1186/1748718855
© Salari et al; licensee BioMed Central Ltd. 2010
Received: 16 July 2009
Accepted: 4 January 2010
Published: 4 January 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 }].
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.
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.
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.
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.
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.
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.
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).
Prediction accuracy of competitive RNARNA joint secondary structure prediction methods.
Sensitivity  PPV  Fmeasure  

RNARNA interaction pairs  inRNAs  EBM  SPM  LM  inRNAs  EBM  SPM  LM  inRNAs  EBM  SPM  LM 
CopACopT  1.000  0.909  0.955  0.864  0.846  0.800  0.778  0.760  0.917  0.851  0.857  0.809 
DISDIS  1.000  0.786  0.786  0.786  1.000  0.786  0.786  0.786  1.000  0.786  0.786  0.786 
IncRNA_{54}RepZ  0.875  0.917  0.875  0.875  0.792  0.830  0.778  0.778  0.831  0.871  0.824  0.824 
R1invR2inv  0.900  0.900  1.000  1.000  0.900  0.947  1.000  1.000  0.900  0.923  1.000  1.000 
TarTar*  1.000  1.000  1.000  1.000  0.875  0.933  0.875  0.875  0.933  0.965  0.933  0.933 
Average  0.955  0.902  0.923  0.905  0.883  0.859  0.843  0.840  0.916  0.879  0.880  0.870 
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.
Prediction accuracy of competitive RNARNA binding sites prediction methods.
Sensitivity  PPV  Fmeasure  

RNARNA interaction pairs  inRNAs  IntaRNA  RNAup  inRNAs  IntaRNA  RNAup  inRNAs  IntaRNA  RNAup 
CopACopT  0.889  1.000  0.556  0.828  0.391  0.652  0.857  0.562  0.600 
DISDIS  1.000  1.000  1.000  1.000  1.000  1.000  1.000  1.000  1.000 
IncRNA_{54}RepZ  1.000  0.738  0.750  0.889  0.850  0.857  0.941  0.790  0.800 
R1invR2inv  1.000  1.000  1.000  0.778  1.000  0.778  0.875  1.000  0.875 
TarTar*  1.000  1.000  1.000  0.833  0.833  0.833  0.909  0.909  0.909 
DsrARpoS  0.808  0.808  0.808  0.778  0.778  0.778  0.793  0.793  0.793 
GcvBargT  0.950  0.950  0.900  0.864  0.950  0.947  0.905  0.950  0.923 
GcvBdppA  1.000  1.000  1.000  0.850  0.586  0.459  0.919  0.739  0.629 
GcvBgltI  0.750  0.000  0.000  0.500  0.000  0.000  0.600  0.000  0.000 
GcvBlivJ  0.634  0.955  0.955  0.824  0.955  0.955  0.717  0.955  0.955 
GcvBlivK  0.540  0.542  0.542  0.570  0.565  0.565  0.555  0.553  0.553 
GcvBoppA  1.000  1.000  1.000  0.733  0.957  0.957  0.846  0.978  0.978 
GcvBSTM4351  0.760  0.760  0.880  1.000  0.905  0.957  0.864  0.826  0.917 
IstRtisAB  0.722  0.879  0.667  1.000  0.960  1.000  0.839  0.918  0.800 
MicAompA  1.000  1.000  1.000  1.000  1.000  1.000  1.000  1.000  1.000 
MicAlamB  1.000  1.000  0.826  1.000  0.821  0.704  1.000  0.902  0.760 
MicCompC  1.000  1.000  0.727  1.000  0.537  0.410  1.000  0.699  0.524 
MicFompF  0.960  0.960  0.800  0.960  0.960  0.952  0.960  0.960  0.869 
OxySfhlA  0.813  0.500  0.375  1.000  1.000  1.000  0.897  0.667  0.545 
RyhBsdhD  0.618  0.588  0.794  0.955  1.000  0.794  0.750  0.741  0.794 
RyhBsodB  1.000  1.000  1.000  1.000  0.818  0.900  1.000  0.900  0.947 
SgrSptsG  0.566  0.739  0.739  0.765  1.000  1.000  0.651  0.850  0.850 
Spot42galK  0.432  0.409  0.523  0.760  0.643  0.523  0.551  0.500  0.523 
Average  0.845  0.819  0.776  0.865  0.805  0.784  0.845  0.791  0.763 
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.
Declarations
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.
Authors’ Affiliations
References
 Alkan C, Karakoc E, Nadeau J, Sahinalp S, Zhang K: RNARNA Interaction Prediction and Antisense RNA Target Search. Journal of Computational Biology. 2006, 13 (2): 267282. 10.1089/cmb.2006.13.267PubMedView ArticleGoogle Scholar
 Chitsaz H, Salari R, Sahinalp SC, Backofen R: A partition function algorithm for interacting nucleic acid strands. Bioinformatics. 2009, 25: i365373. 10.1093/bioinformatics/btp212PubMedPubMed CentralView ArticleGoogle Scholar
 Meisner N, Hackermüller J, Uhl V, Aszódi A, Jaritz M, Auer M: mRNA openers and closers: modulating AUrich elementcontrolled mRNA stability by a molecular switch in mRNA secondary structure. Chembiochem. 2004, 5: 14321447. 10.1002/cbic.200400219PubMedView ArticleGoogle Scholar
 Hackermüller J, Meisner N, Auer M, Jaritz M, Stadler P: The effect of RNA secondary structures on RNAligand binding and the modifier RNA mechanism: a quantitative model. Gene. 2005, 345: 312. 10.1016/j.gene.2004.11.043PubMedView ArticleGoogle Scholar
 Mückstein U, Tafer H, Hackermüller J, Bernhart S, HernandezRosales M, Vogel J, Stadler P, Hofacker I: Translational control by RNARNA interaction: Improved computation of RNARNA binding thermodynamics. Bioinformatics Research and Development. 2008, 13: 114127. full_text.Google Scholar
 Andronescu M, Zhang Z, Condon A: Secondary structure prediction of interacting RNA molecules. J Mol Biol. 2005, 345: 9871001. 10.1016/j.jmb.2004.10.082PubMedView ArticleGoogle Scholar
 Bernhart S, Tafer H, Mückstein U, Flamm C, Stadler P, Hofacker I: Partition function and base pairing probabilities of RNA heterodimers. Algorithms Mol Biol. 2006, 1: 310.1186/1748718813PubMedPubMed CentralView ArticleGoogle Scholar
 Dirks R, Bois J, Schaeffer J, Winfree E, Pierce N: Thermodynamic Analysis of Interacting Nucleic Acid Strands. SIAM Review. 2007, 49: 6588. 10.1137/060651100. 10.1137/060651100View ArticleGoogle Scholar
 Rehmsmeier M, Steffen P, Hochsmann M, Giegerich R: Fast and effective prediction of microRNA/target duplexes. RNA. 2004, 10: 15071517. 10.1261/rna.5248604PubMedPubMed CentralView ArticleGoogle Scholar
 Dimitrov R, Zuker M: Prediction of Hybridization and Melting for DoubleStranded Nucleic Acids. Biophysical Journal. 2004, 87: 215226. 10.1529/biophysj.103.020743PubMedPubMed CentralView ArticleGoogle Scholar
 Markham N, Zuker M: UNAFold: software for nucleic acid folding and hybridization. Methods Mol Biol. 2008, 453: 331.PubMedView ArticleGoogle Scholar
 Pervouchine D: IRIS: intermolecular RNA interaction search. Genome Inform. 2004, 15: 92101.PubMedGoogle Scholar
 Kato Y, Akutsu T, Seki H: A grammatical approach to RNARNA interaction prediction. Pattern Recogn. 2009, 42 (4): 531538. 10.1016/j.patcog.2008.08.004. 10.1016/j.patcog.2008.08.004View ArticleGoogle Scholar
 Brunel C, Marquet R, Romby P, Ehresmann C: RNA looploop interactions as dynamic functional motifs. Biochimie. 2002, 84: 925944. 10.1016/S03009084(02)014013PubMedView ArticleGoogle Scholar
 Mückstein U, Tafer H, Hackermüller J, Bernhart S, Stadler P, Hofacker I: Thermodynamics of RNARNA binding. Bioinformatics. 2006, 22: 11771182. 10.1093/bioinformatics/btl024PubMedView ArticleGoogle Scholar
 Busch A, Richter AS, Backofen R: IntaRNA: efficient prediction of bacterial sRNA targets incorporating target site accessibility and seed regions. Bioinformatics. 2008, 24 (24): 284956. 10.1093/bioinformatics/btn544PubMedPubMed CentralView ArticleGoogle Scholar
 McCaskill J: The equilibrium partition function and base pair binding probabilities for RNA secondary structure. Biopolymers. 1990, 29: 11051119. 10.1002/bip.360290621PubMedView ArticleGoogle Scholar
 Wagner E, Flärdh K: Antisense RNAs everywhere?. Trends Genet. 2002, 18: 223226. 10.1016/S01689525(02)026586PubMedView ArticleGoogle Scholar
 Tjaden B, Goodwin S, Opdyke J, Guillier M, Fu D, Gottesman S, Storz G: Target prediction for small, noncoding RNAs in bacteria. Nucleic Acids Res. 2006, 34: 27912802. 10.1093/nar/gkl356PubMedPubMed CentralView ArticleGoogle Scholar
 Tafer H, Hofacker IL: RNAplex: a fast tool for RNARNA interaction search. Bioinformatics. 2008, 24: 26572663. 10.1093/bioinformatics/btn193PubMedView ArticleGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.