Abstract
RNA molecules play crucial roles in various biological processes. Their threedimensional configurations determine the functions and, in turn, influences the interaction with other molecules. RNAs and their interaction structures, the socalled RNA–RNA interactions, can be abstracted in terms of secondary structures, i.e., a list of the nucleotide bases paired by hydrogen bonding within its nucleotide sequence. Each secondary structure, in turn, can be abstracted into cores and shadows. Both are determined by collapsing nucleotides and arcs properly. We formalize all of these abstractions as arc diagrams, whose arcs determine loops. A secondary structure, represented by an arc diagram, is pseudoknotfree if its arc diagram does not present any crossing among arcs otherwise, it is said pseudoknotted. In this study, we face the problem of identifying a given structural pattern into secondary structures or the associated cores or shadow of both RNAs and RNA–RNA interactions, characterized by arbitrary pseudoknots. These abstractions are mapped into a matrix, whose elements represent the relations among loops. Therefore, we face the problem of taking advantage of matrices and submatrices. The algorithms, implemented in Python, work in polynomial time. We test our approach on a set of 16S ribosomal RNAs with inhibitors of Thermus thermophilus, and we quantify the structural effect of the inhibitors.
1 Introduction
Ribonucleic acid (RNA) is a linear polymer with a preferred 5–3′ direction, made of four different types of nucleotides, known as Adenine (A), Guanine (G), Cytosine (C), and Uracil (U). Each nucleotide is linked to the next one by a phosphodiester bond, referred to as a strong bond. It can at most interact with another noncontiguous one, establishing a hydrogen bond, referred to as a weak bond, and forming mainly Watson–Crick (G–C and A–U) and wobble (G–U) base pairs. Such a process, known as the folding process, induces complex threedimensional configurations (or shapes). Each of them is strictly related to the molecular functions. RNA molecules play numerous roles in cellular processes. Usually, they do not act in isolation, but they express their biological roles by interacting with other molecules [1], including other RNAs that determine the socalled RNA–RNA interactions (RRIs). Understanding the link between shape and function has been considered a challenge in biology. Disregarding the molecular spatial configuration and reducing nucleotides to dots, RNAs can be abstracted in terms of secondary structures that consist of the nucleotide bases paired by hydrogen bonding within its sequence. This abstraction represents an intermediate level between the nucleotides sequence and its shape, and it is both tractable from a computational point of view and relevant from a biological perspective. For example, under the action of inhibitors, many 16S ribosomal RNAs alter their shape by preserving the nucleotides sequence [2], and the secondary structure can capture such changes. Moreover, functional RNA families, such as tRNA, rRNA, and RNAse P, exhibit a highly conserved secondary structure but little sequence similarity [3]. Searching for sequence motifs does not work effectively with RNA, while it has been a powerful tool for DNA and protein analysis [4]. Therefore, the ability to compare and classify RNA secondary structures equipped to identify common substructures is of great interest. A way for schematically representing a secondary structure is the arc diagram, constituted by vertices that formalize the nucleotides on a straight line (backbone) and semicircular zigzag arcs in the upper halfplane that depict the weak bonds. In the following, we will call this the diagram of the structure. A secondary structure represented by a diagram is pseudoknotfree if it does not present crossings among the weak bond zigzag arcs; otherwise, it is called pseudoknotted. On the right of Figure 1A, the RNA structure is pseudoknotfree, and on the left, we have a pseudoknotted motif, which makes the whole structure pseudoknotted. An example of RNA–RNA interaction structure in terms of arc diagram is shown in Figure 1B. Each arc determines a loop. Therefore, every RNA secondary structure is composed of loops. Given two of them, we have only three situations: a loop is followed by the other one as illustrated in Figure 2A, a loop is inside another one as shown in Figure 2B, and a loop crosses with the another as illustrated in Figure 2C. We refer to such relations as concatenation, nesting, and crossing, respectively. Some approaches have been exploited to study the link between the structure and its biological functions. Maestri and Merelli studied the relationships between RNA structure and functions by using process calculi [5]. Quadrini et al. introduced algebraic languages for representing and comparing RNA secondary structures with arbitrary pseudoknots [6], [7], while Andersen et al. exploited a combinatorial approach [8]. On the other hand, Giegerich et al. introduced the concept of shape [9], while Bon et al. proposed a classification of RNA secondary structures based on a topological invariant, the genus [10]. Reidys et al. developed several algorithms for predicting pseudoknots by exploiting the concept of shadow [11], [12]. Moreover, several approaches have been proposed for searching common patterns. Algorithms based on tree data structures find the largest approximately common substructures and patterns in [13] and [14], respectively. Affix trees allow us to exact and approximate pattern matching [15]. Arslan et al. proposed a substructure search algorithm based on a binary search on a suffix array to find the largest common substructure of given RNA structures [16]. Backofen and Siebert have developed a dynamic programming approach for computing common exact sequential and structural patterns between two RNAs without pseudoknots [17]. Several proposed approaches have been based on arcannotated sequences, also called contact maps, including the longest arcannotated subsequence problem, the arc preserving subsequence problem, the maximum arcpreserving common subsequence problem, and the editdistance for arcannotated sequence problem [18]. Blin et al. introduced an approach, called maximum arcpreserving common subsequence problem to compare arcannotated sequences in [19], and Evans proposed an algorithm to find common structures excluding some classes of pseudoknots [20]. Recently, Quadrini et al. have faced the problems of identifying substructures considering both the primary and secondary structure only for RNA pseudoknotfree structure in [21].
In this paper, we face the problem of identifying a given structural pattern into the secondary structure of RNAs and or RNA–RNA interactions. We propose a method that works on patterns and structures characterized by arbitrary pseudoknots. Following the approaches proposed in [22], [23], we introduce the concept of the core. The core of a secondary structure is determined by deleting unpaired nucleotides and by collapsing arcs, appropriately. For example, the core of the structures in Figure 1 is shown in Figure 3.
In the literature, the concept of collapsing has already been introduced to reduce the complexity and classify the RNAs in equivalent classes based on topological concepts. From each diagram, it is possible to associate a shadow by removing all the noncrossing arcs and all the unpaired vertices, and then collapsing parallel zigzag arcs into an arc [24]. For example, the shadow of the structures in Figure 1 is illustrated in Figure 4. The shadow is again a diagram that is unique for the considered molecule, like to core. Based on our previous results [6], [25], we define three operators able to formalize the concatenation, nesting, and crossing between two loops. Such operators are necessary and sufficient to describe any arc diagram in terms of relations among loops. Such description allows us to uniquely associate a matrix, called relation matrix, whose elements represent the relation between the two corresponding loops for each abstraction (secondary structure, core, or shadows) of RNA and RNA–RNA interactions structures. Therefore, identifying a given structural pattern into an RNA structure corresponds to search a submatrix within a matrix. To reach the aim, we have defined four algorithms: loop determination, core determination, determination of the relation matrix, and structural relation matching. Each of them is implemented in Python and works in polynomial time. The first three algorithms take as input a secondary structure of an RNA or RNA–RNA interaction encoded as a Bpseq notation and return the list of loops of the secondary structure, of the core and the shadow, respectively. A Bpseq notation contains information about base pairs, stored in three columns: the first one represents the sequence position, the second contains information about the kind of bases (i.e., Adenine, Guanine, Cytosine, or Uracil), the third encodes the pairing base if the considered nucleotide is paired, or zero, otherwise. The other algorithm considers the output of the previous one, and it returns the relation matrix. The last one searches the relation matrix associated with the pattern into the relation matrix of the structure. The approach has been tested on structures of 16S ribosomal RNAs from the RNA Strand database [26]. In particular, we have selected 16S ribosomal RNAs with inhibitors of Thermus thermophilus to quantify the structural effect of the inhibitors.
The paper is organized as follows. In Section 2, we formally introduce the arc diagram representation within the concepts of core and shadows. Moreover, we describe the relations among loops and we define the relation matrix. In Section 3, we formally face the problem of searching a structural pattern. In Section 4, we present an application of our methodology on a set of 16S ribosomal RNAs. The paper ends with some conclusions and future perspective.
2 RNA abstractions and representation
RNA secondary structure can be represented by a diagram. Formally,
Definition 1
(Diagram). A diagram is a labeled graph over the ordered set of vertices [ℓ] = {1, …, ℓ}, in which each vertex has degree ≤3, and the edges are all the segments [i, i + 1] for i = 1, …, ℓ − 1 and some semicircular arcs (i, j) in the upper halfplane, with 1 ≤ i < j ≤ ℓ.
The diagram is denoted by
Taking advantage of such an enumeration, we impose an order on the loops. Each nucleotide can interact at most with another one. As a consequence, each nucleotide can be involved at most in a pair. This means that the choice of a loop is unique. Moreover, the three relations, concatenation, nesting, and crossing, are necessary and sufficient to describe any RNA secondary structure with arbitrary pseudoknots. In fact, given two loops, L _{ s } and L _{ t } with s < t, it is equivalent to consider the two pairs of natural number (i _{ s }, j _{ s }) and (i _{ t }, j _{ t }) such that i _{ s } < j _{ s }, i _{ t } < j _{ t } and j _{ s } < j _{ t }. It follows that j _{ s } is the greatest number. From the theory of combinations, we have 6 different order relations over i _{ s }, j _{ s }, and i _{ t }, which became 3 considering the constraints i _{ s } < j _{ s } and i _{ t } < j _{ t }. For each structure, we can uniquely associate a relation matrix. Each element a _{ ij } of the matrix is the relation between the loops L _{ i } and L _{ j }. The relation matrix of the structure in Figure 5 is given in Table 1. By appropriately exchanging some rows and columns, we obtain the following matrix that allows distinguishing the relations between the two molecules and the relations due to the interaction (Table 2). In this way, for each RNA–RNA interaction structure, we can identify structural patterns of the two single molecules or interactions between them. The ability to distinguish if a set of loop relations characterize the interactions between molecules or a single molecule is useful for studying how the molecules interact with each other us allowing to study how such interactions change according to the environment.
L _{1}  L _{2}  L _{3}  L _{4}  L _{5}  L _{6}  L _{7}  L _{8}  L _{9}  
L _{1}  –  ⊙  ⊙  ⊙  ⊙  ⊙  ⊙  ⊙  
L _{2}  –  ⊙  ⊙  ⊙  ⊙  
L _{3}  –  ⋒  ⋒  ⋒  ⋒  
L _{4}  –  ⋒  ⋒  ⋒  
L _{5}  –  ⊙  ⊙  
L _{6}  –  ⋒  ⊙  ⊙  
L _{7}  –  ⊙  ⊙  
L _{8}  –  ⋒  
L _{9}  – 
L _{1}  L _{2}  L _{3}  L _{4}  L _{5}  L _{6}  L _{7}  L _{8}  L _{9}  
L _{1}  –  ⊙  ⊙  ⊙  ⊙  ⊙  ⊙  ⊙  
L _{2}  –  ⊙  ⊙  ⊙  ⊙  
L _{4}  –  ⋒  ⋒  ⋒  
L _{5}  –  ⊙  ⊙  
L _{3}  –  ⋒  ⋒  ⋒  ⋒  
L _{6}  –  ⋒  ⊙  ⊙  
L _{7}  –  ⊙  ⊙  
L _{8}  –  ⋒  
L _{9}  – 
3 Structural matching
We face the problem of searching a given structural pattern into a secondary structure or its abstraction, i.e., core and shape, of RNAs and RNA–RNA interactions with arbitrary pseudoknots. Formally, we address the problem of the arcpreserving subsequence (APS) problem with a particular restriction. Let
For each RNA secondary structure represented as an arc diagram, we uniquely determine its relation matrix using Algorithm 3, determination of the relation matrix. The algorithm, whose pseudocode is reported in Appendix A, takes as input the set B of the pairs (i
_{
s
}, j
_{
s
}) and returns a matrix, whose element a
_{
k,t
} represents the relation between the loops L
_{
k
} and L
_{
t
}. It is computed with time complexity of
L _{1}  L _{2}  L _{3}  L _{4}  L _{5}  L _{6}  
L _{1}  –  ⊙  ⊙  ⊙  
L _{2}  –  ⋒  ⊙  ⊙  
L _{3}  –  ⊙  ⊙  
L _{4}  –  
L _{5}  –  ⋒  
L _{6}  –  
L _{1}  L _{2}  L _{3}  
L _{1}  –  
L _{2}  –  ⋒  
L _{3}  – 
In the literature, as mentioned in the Introduction, different approaches have been introduced to extract patterns of RNA molecules using several data structures, including arcannotated sequences, affix trees, and suffix arrays. In Table 4, we report some approaches underlying the data structure used, the performance in terms of computational cost, supporting pseudoknots.
Approach  Data structure  Computational cost  Pseudoknots  Sequence/structure 

Exact pattern matching  Affix tree 

No  Both 
Exact the largest common substructure  Affix array 

No  Both 
Maximum arcpreserving common  Arcannotated sequence  NP  Yes  Both 
subsequence  
Arc preserving subsequence problem  Arcannotated sequence  NP  Yes  Both 
The longest arcannotated  Arcannotated sequence  NP  Yes  Both 
subsequence problem  
Structural matching  Relation matrix 

Yes  Structure 

n and m are the length of the pattern and the structure.
Our approach allows us to search a given structural pattern into an RNA secondary structure with arbitrary pseudoknots in O(n ⋅ m) by considering pairwise relations between loops. It corresponds to the arcpreserving subsequence problem with a particular restriction. In particular, we require that no arcs will be deleted if the last paired nucleotide is inside the considered substructure. This constraint follows the folding process: each nucleotide performs a hydrogen bond with another already synthesized one. In our formalism, the first nucleotide of a loop is synthesized before the last one. This restriction allows us to extract patterns, which correspond to the local substructure taking into account the folding process. This approach has a strong impact on the RNA structures analysis because it can consider the structural formations of the molecule.
4 Applications
We test our approach on a set of 16S ribosomal RNAs of T. thermophilus. The aim is to study the effect of some inhibitors, including antibiotics, tetracycline, hygromycin B. Accordingly, we take into account the RNA molecules with ID PDB_00478, PDB_00408, PDB_00436, PDB_00438, PDB_00589 from the RNA strand database [26]. RNA Strand is a database containing known secondary structures of any type and organism drawn from public databases, searchable and downloadable in several formats. It is an easy online tool for searching, analyzing, and downloading userselected entries, and is publicly available at http://www.rnasoft.ca/strand. In our experiment, we first consider the shadows of the selected molecules (that we compute applying the algorithm shadow determination). We observe that the same shadow, shown in Figure 8A, characterizes each of the molecules. Therefore, the shadow is not able to capture the effects of the inhibitors over these molecules by confirming that the inhibitors act locally. The antibiotics bind to discrete sites on the 16S submit to effect on ribosome function [29]. To check our algorithm, we compute the relation matrix, and we trivially observe that the pattern, illustrated in Figure 8B, is contained in each shape twice.
Moreover, we consider the core of these molecules between the 800th and 900th nucleotides. The core of substructure molecules with ID PDB_00478, PDB_00408, PDB_00436, PDB_00438 is shown in Figure 9A, while the core illustration of substructure PDB_00589 is in Figure 9B. Finally, the patterns that we identify in the cores are shown in Figure 9C and D. In this case, we observe that the one in Figure 9C is not present in the molecules PDB_00589, while it involves loops L _{4}, L _{5}, L _{6}, L _{7}, L _{8} of the other substructures. However, the pattern in Figure 9D occurs once involving loops L _{1}, L _{2}, L _{3}. In Appendix B, we report the relation matrices and the pattern occurrences in RNA structures of the considered molecules. This result shows that the inhibitors do not influence the molecules between 800th to 900th nucleotides. However, we are able to capture and quantifie the structural changes due to the codon and nearcognate tRNA anticodon stemloop presence (PDB_00589 molecule).
5 Conclusion and future works
RNA functions depend on their threedimensional configuration. Understanding the relationship between structure and biological function has been considered one of the challenges in biology. In this work, we have faced the problem of identifying a given structural pattern into secondary structures of RNA and RNA–RNA interactions or their abstractions (cores and shadows) with arbitrary pseudoknots. We have used algebraic operators to formalize such RNA secondary structures and their abstractions as a combination of loops. Moreover, we have defined procedures to represent the secondary structure in terms of loops to determine the core and shadows. Finally, we have defined two procedures: determination of the relation matrix and structural relation matching. The former maps each RNA secondary structure into a matrix and the latter identifies each pattern of the RNA structure by searching for a submatrix. We have implemented the proposed methodology in Python, and we have tested our approach on a set of 16S ribosomal RNAs of T. thermophilus to understand the effects of some inhibitors. The Python code is available on https://github.com/michelaquadrini/RNARelationPattern. The results show that the approach can capture the local, intermediate and global structural changes by extracting patterns from the molecule, its core, and its shadow, respectively, and taking into account the folding of RNA molecules. The approach can be applied in different scenarios with different aims.
Now, we are working on the tool development by improving its computational performance and making it userfriendly for biologists. Moreover, we want to add other molecular encodings as accepted input, i.e., including dotbracket and CT files. Dotbracket is a notation used to encode RNA secondary structure topology, and CT is a format that describes molecules and chemical reactions. Both represent the secondary structures of RNAs and RNA–RNAs interactions as well as the Bpseq notation. However, adding all these input types makes the tool more userfriendly by avoiding format change problems due to the nonexistence of a universal notation for encoding secondary RNA structures.
Moreover, we are analyzing RNAs of 16S ribosomal of T. thermophilus and Escherichia coli to evaluate the effects of inhibitors as a function of thermal differences. This evaluation will be carried out in collaboration with experts of the biological domain to test the impact of our approach on biology. In future work, we want to generalize the approach considering the sequences of nucleotides. In other words, we want to face the problem of finding a given structural pattern into an RNA with arbitrary pseudoknots taking into both the primary and secondary structure of molecules. Although functional RNAs exhibit a highly conserved secondary structure with little sequence similarity, the sequence influences the molecular interactions. In other words, the nucleotide sequence plays a role in the study and prediction of the RNA–RNA interaction structures.

Author contribution: The author has accepted responsibility for the entire content of this submitted manuscript and approved submission.

Research funding: This work was supported by the ”GNCS  INdAM”.

Conflict of interest statement: The author declares no conflicts of interest regarding this article.
In this appendix, we define the pseudocode of four algorithms mentioned in the paper, loop determination, core determination, determination of the relation matrix, structural relation matching.
To test out approach, we selected the molecules with the following ID from RNA strand database [26]
PDB_00408
PDB_00436
PDB_00438
PDB_00478
PDB_00589
Each of them is characterized by the same shadow, whose relation matrix is in Table 5.
We extracted substructure from the 800 to 900 nucleotides of the selected molecules and we determined the core. The relation matrix of the core of the substructure (from 800th to 900th nucleotides) PDB_00408, PDB_00436, PDB_00438, PDB_00478 is shown in Table 6, while the relation matrix of core associate to the substructure with ID PDB_00589 is presented in Table 7.
L _{1}  L _{2}  L _{3}  L _{4}  L _{5}  L _{6}  L _{7}  L _{8}  L _{9}  
L _{1}  –  ⊙  ⊙  ⊙  ⊙  ⊙  ⊙  ⊙  
L _{2}  –  ⋒  ⋒  ⋒  ⋒  ⋒  ⋒  ⋒  
L _{3}  –  ⋒  ⋒  ⋒  ⊙  ⊙  
L _{4}  –  ⊙  ⊙  ⊙  ⊙  
L _{5}  –  ⋒  ⋒  ⊙  ⊙  
L _{6}  –  ⊙  ⊙  
L _{7}  –  ⊙  ⊙  
L _{8}  –  
L _{9}  – 
L _{1}  L _{2}  L _{3}  L _{4}  L _{5}  L _{6}  L _{7}  L _{8}  
L _{1}  –  ⋒  ⋒  ⊙  ⊙  ⊙  ⊙  ⊙ 
L _{2}  –  ⊙  ⊙  ⊙  ⊙  ⊙  ⊙  
L _{3}  –  ⊙  ⊙  ⊙  ⊙  ⊙  
L _{4}  –  ⋒  ⋒  ⊙  ⊙  
L _{5}  ⋒  ⊙  ⊙  
L _{6}  –  ⊙  ⊙  
L _{7}  –  ⋒  
L _{8}  – 
L _{1}  L _{2}  L _{3}  L _{4}  L _{5}  L _{6}  L _{7}  
L _{1}  –  ⋒  ⋒  ⊙  ⊙  ⊙  ⊙ 
L _{2}  –  ⊙  ⊙  ⊙  ⊙  ⊙  
L _{3}  –  ⊙  ⊙  ⊙  ⊙  
L _{4}  –  ⋒  ⊙  ⊙  
L _{5}  –  ⊙  ⊙  
L _{6}  ⋒  
L _{7}  – 
The relation matrixes of the two molecules (or patterns), illustrated in Figure 9C and Figure 9D, are shown in Tables 8 and 9.
L _{1}  L _{2}  L _{3}  L _{4}  L _{5}  
L _{1}  –  ⋒  ⋒  ⊙  ⊙ 
L _{2}  –  ⋒  ⊙  ⊙  
L _{3}  –  ⊙  ⊙  
L _{4}  –  ⋒  
L _{5}  – 
L _{1}  L _{2}  L _{3}  
L _{1}  –  ⋒  ⋒ 
L _{2}  –  ⊙  
L _{3}  – 
The occurrence of pattern shown in Figure 9C in the structures illustrated in Figure 9A is (L _{7}; L _{8}); (L _{6}; L _{7}); (L _{6}; L _{8}); (L _{5}; L _{6}); (L _{5}; L _{7}); (L _{5}; L _{8}); (L _{4}; L _{5}); (L _{4}; L _{6}); (L _{4}; L _{7}); (L _{4}; L _{8}), while there is no occurrence of the patterns for the structures illustrated in Figure 9B. The pattern Figure 9D occurs in all considered structures once. The occurrence is (L _{2}; L _{3}); (L _{1}; L _{2}); (L _{1}; L _{3}).
References
1. Alberts, B, Bray, D, Hopkin, K, Johnson, AD, Lewis, J, Raff, M, et al.. Essential cell biology. New York: Garland Science; 2013.10.1201/9781315815015Search in Google Scholar
2. Carter, AP, Clemons, WM, Brodersen, DE, MorganWarren, RJ, Wimberly, BT, Ramakrishnan, V. Functional insights from the structure of the 30S ribosomal subunit and its interactions with antibiotics. Nature 2000;407:340–8. https://doi.org/10.1038/35030019.Search in Google Scholar PubMed
3. Höchsmann, M, Voss, B, Giegerich, R. Pure multiple RNA secondary structure alignments: a progressive profile approach. IEEE/ACM Trans Comput Biol Bioinf 2004;1:53–62. https://doi.org/10.1109/tcbb.2004.11.Search in Google Scholar
4. Li, K, Rahman, R, Gupta, A, Siddavatam, P, Gribskov, M. Pattern matching in RNA structures. In: Proceedings of the 4th international conference on bioinformatics research and applications. ISBRA’08. SpringerVerlag; 2008:317–30 pp.10.1007/9783540794509_30Search in Google Scholar
5. Maestri, S, Merelli, E. Process calculi may reveal the equivalence lying at the heart of RNA and proteins. Sci Rep 2019;9:1–9. https://doi.org/10.1038/s41598018369651.Search in Google Scholar PubMed PubMed Central
6. Quadrini, M, Tesei, L, Merelli, E. An algebraic language for RNA pseudoknots comparison. BMC Bioinf 2019;20:161. https://doi.org/10.1186/s1285901926895.Search in Google Scholar PubMed PubMed Central
7. Quadrini, M, Tesei, L, Merelli, E. ASPRAlign: a tool for the alignment of RNA secondary structures with arbitrary pseudoknots. Bioinformatics 2020;36:3578–9. https://doi.org/10.1093/bioinformatics/btaa147.Search in Google Scholar PubMed
8. Andersen, JE, Huang, FW, Penner, RC, Reidys, CM. Topology of RNARNA interaction structures. J Comput Biol 2012;19:928–43. https://doi.org/10.1089/cmb.2011.0308.Search in Google Scholar PubMed
9. Giegerich, R, Voß, B, Rehmsmeier, M. Abstract shapes of RNA. Nucleic Acids Res 2004;32:4843–51. https://doi.org/10.1093/nar/gkh779.Search in Google Scholar PubMed PubMed Central
10. Bon, M, Vernizzi, G, Orland, H, Zee, A. Topological classification of RNA structures. J Mol Biol 2008;379:900–11. https://doi.org/10.1016/j.jmb.2008.04.033.Search in Google Scholar PubMed
11. Reidys, CM, Huang, FW, Andersen, JE, Penner, RC, Stadler, PF, Nebel, ME. Topology and prediction of RNA pseudoknots. Bioinformatics 2011;27:1076–85. https://doi.org/10.1093/bioinformatics/btr090.Search in Google Scholar PubMed
12. Huang, FW, Reidys, CM. Topological language for RNA. Math Biosci 2016;282:109–20. https://doi.org/10.1016/j.mbs.2016.10.006.Search in Google Scholar PubMed
13. Wang, JTL, Shapiro, BA, Shasha, D, Zhang, K, Currey, KM. An algorithm for finding the largest approximately common substructures of two trees. IEEE Trans Pattern Anal Mach Intell 1998;20:889–95. https://doi.org/10.1109/34.709622.Search in Google Scholar
14. Hochsmann, M, Toller, T, Giegerich, R, Kurtz, S. Local similarity in RNA secondary structures. In: Computational systems bioinformatics. Proceedings of the 2003 IEEE bioinformatics conference. CSB2003. IEEE; 2003:159–68 pp.10.1109/CSB.2003.1227315Search in Google Scholar
15. Mauri, G, Pavesi, G. Algorithms for pattern matching and discovery in RNA secondary structure. Theor Comput Sci 2005;335:29–51. https://doi.org/10.1016/j.tcs.2004.12.015.Search in Google Scholar
16. Arslan, AN, Anandan, J, Fry, E, Monschke, K, Ganneboina, N, Bowerman, J. Efficient RNA structure comparison algorithms. J Bioinf Comput Biol 2017;15:1740009. https://doi.org/10.1142/s0219720017400091.Search in Google Scholar PubMed
17. Backofen, R, Siebert, S. Fast detection of common sequence structure patterns in RNAs. J Discrete Algorithm 2007;5:212–28. https://doi.org/10.1016/j.jda.2006.03.015.Search in Google Scholar
18. Blin, G, Crochemore, M, Vialette, S. Algorithmic aspects of arcannotated sequences. In: Algorithms in molecular biology: techniques, approaches, and applications. Wiley; 2011.10.1002/9780470892107.ch6Search in Google Scholar
19. Blin, G, Fertin, G, Herry, G, Vialette, S. Comparing RNA structures: towards an intermediate model between the edit and the lapcs problems. In: Brazilian symposium on bioinformatics. Springer; 2007:101–12 pp.10.1007/9783540737315_10Search in Google Scholar
20. Evans, PA. Finding common subsequences with arcs and pseudoknots. In: Annual symposium on combinatorial pattern matching. Springer; 1999:270–80 pp.10.1007/3540484523_20Search in Google Scholar
21. Quadrini, M, Merelli, E, Piergallini, R. Loop grammars to identify RNA structural patterns. In: Proceedings of the 12th intenational joint conference on biomedical engineering systems and technologies  volume 3: Bioinformatics. SciTePress; 2019:302–9 pp.10.5220/0007576603020309Search in Google Scholar
22. Quadrini, M, Piergallini, R, Merelli, E. Label core for understanding RNA structures. In: Proceedings of the 16th international conference on computational intelligence methods for bioinformatics and biostatistics; 2020. Accepted for publication.10.1007/9783030630614_16Search in Google Scholar
23. Quadrini, M, Culmone, R, Merelli, E. Topological classification of RNA structures via intersection graph. In: International conference on theory and practice of natural computing. Springer; 2017:203–15 pp.10.1007/9783319710693_16Search in Google Scholar
24. Reidys, CM, Wang, RR. Shapes of RNA pseudoknot structures. J Comput Biol 2010;17:1575–90. https://doi.org/10.1089/cmb.2010.0006.Search in Google Scholar
25. Quadrini, M. Searching RNA substructures with arbitrary pseudoknots. In: International conference on practical applications of computational biology & bioinformatics. Springer; 2020:123–33 pp.10.1007/9783030545680_13Search in Google Scholar
26. Andronescu, M, Bereg, V, Hoos, HH, Condon, A. RNA strand: the secondary structure and statistical analysis database. BMC Bioinf 2008;9:340. https://doi.org/10.1186/147121059340.Search in Google Scholar
27. Gramm, J, Guo, J, Niedermeier, R. Pattern matching for arcannotated sequences. In: International conference on foundations of software technology and theoretical computer science. Springer; 2002:182–93 pp.10.1007/3540362061_17Search in Google Scholar
28. Blin, G, Fertin, G, Rizzi, R, Vialette, S. What makes the arcpreserving subsequence problem hard? In: Transactions on computational systems biology II. Berlin: Springer; 2005:1–36 pp.10.1007/11567752_1Search in Google Scholar
29. Brodersen, DE, Clemons, WMJr, Carter, AP, MorganWarren, RJ, Wimberly, BT, Ramakrishnan, V. The structural basis for the action of the antibiotics tetracycline, pactamycin, and hygromycin B on the 30S ribosomal subunit. Cell 2000;103:1143–54. https://doi.org/10.1016/s00928674(00)002166.Search in Google Scholar
© 2021 Michela Quadrini, published by De Gruyter, Berlin/Boston
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