# Dual Graph Partitioning Highlights a Small Group of Pseudoknot-Containing RNA Submotifs

^{1}

^{2}

^{3}

^{4}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. RNA Dual Graph Representation

- An RNA stem (or helix) with at least two canonical base pairs is considered as a vertex.
- Each loop strand between two helices is denoted as an edge. Single-residue bulges and internal loops with only one nucleotide in each strand are ignored.
- Uninterrupted hairpin loops (including helical ends) are represented as self loops.
- Unpaired bases or helical ends at the 5${}^{\prime}$ and 3${}^{\prime}$ ends of RNA molecules are not represented.

#### 2.2. Dual Graph Enumeration

#### 2.3. Dual Graph Partitioning Algorithm

#### 2.4. Representative Set of RNA Structures

#### 2.5. Defining Existing Dual Graph Topologies

## 3. Results

#### 3.1. Partitioning Dual Graphs into Subgraphs

#### 3.2. Submotifs in Ribosomal RNAs

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Crick, F. Central dogma of molecular biology. Nature
**1970**, 227, 561–563. [Google Scholar] [CrossRef] [PubMed] - Kaikkonen, M.U.; Lam, M.T.; Glass, C.K. Non-coding RNAs as regulators of gene expression and epigenetics. Cardiovasc. Res.
**2011**, 90, 430–440. [Google Scholar] [CrossRef] [PubMed][Green Version] - Patil, V.S.; Zhou, R.; Rana, T.M. Gene regulation by non-coding RNAs. Crit. Rev. Biochem. Mol. Biol.
**2014**, 49, 16–32. [Google Scholar] [CrossRef] [PubMed] - Lilley, D.M.J. Mechanisms of RNA catalysis. Philos. Trans. R. Soc. B Biol. Sci.
**2011**, 366, 2910–2917. [Google Scholar] [CrossRef] [PubMed][Green Version] - Wilson, T.J.; Liu, Y.; Lilley, D.M.J. Ribozymes and the mechanisms that underlie RNA catalysis. Front. Chem. Sci. Eng.
**2016**, 10, 178–185. [Google Scholar] [CrossRef] - Tinoco, I.; Bustamante, C. How RNA folds. J. Mol. Biol.
**1999**, 293, 271–281. [Google Scholar] [CrossRef] [PubMed] - Brion, P.; Westhof, E. Hierarchy and dynamics of RNA folding. Ann. Rev. Biophys. Biomol. Struct.
**1997**, 26, 113–137. [Google Scholar] [CrossRef] [PubMed] - Schlick, T.; Pyle, A.M. Opportunities and challenges in RNA structural modeling and design. Biophys. J.
**2017**, 113, 225–234. [Google Scholar] [CrossRef] [PubMed] - Pyle, A.M.; Schlick, T. Challenges in RNA structural modeling and design. J. Mol. Biol.
**2016**, 428, 733–735. [Google Scholar] [CrossRef] [PubMed] - Laing, C.; Schlick, T. Computational approaches to RNA structure prediction, analysis, and design. Curr. Opin. Struct. Biol.
**2011**, 21, 306–318. [Google Scholar] [CrossRef] [PubMed][Green Version] - Dawson, W.K.; Maciejczyk, M.; Jankowska, E.J.; Bujnicki, J.M. Coarse-grained modeling of RNA 3D structure. Methods
**2016**, 103, 138–156. [Google Scholar] [CrossRef] [PubMed] - Tan, R.K.; Petrov, A.S.; Harvey, S.C. YUP: A Molecular simulation program for coarse-grained and multi-scaled models. J. Chem. Theory Comput.
**2006**, 2, 529–540. [Google Scholar] [CrossRef] [PubMed] - Jonikas, M.A.; Radmer, R.J.; Laederach, A.; Das, R.; Pearlman, S.; Herschlag, D.; Altman, R.B. Coarse-grained modeling of large RNA molecules with knowledge-based potentials and structural filters. RNA
**2009**, 15, 189–199. [Google Scholar] [CrossRef] [PubMed][Green Version] - Krokhotin, A.; Houlihan, K.; Dokholyan, N.V. iFoldRNA v2: Folding RNA with constraints. Bioinformatics
**2015**, 31, 2891–2893. [Google Scholar] [CrossRef] [PubMed] - Mustoe, A.M.; Al-Hashimi, H.M.; Brooks, C.L. Coarse grained models reveal essential contributions of topological constraints to the conformational free energy of RNA Bulges. J. Phys. Chem. B
**2014**, 118, 2615–2627. [Google Scholar] [CrossRef] [PubMed] - Xu, X.; Chen, S.J. Physics-based RNA structure prediction. Biophys. Rep.
**2015**, 1, 2–13. [Google Scholar] [CrossRef] [PubMed][Green Version] - Boniecki, M.J.; Lach, G.; Dawson, W.K.; Tomala, K.; Lukasz, P.; Soltysinski, T.; Rother, K.M.; Bujnicki, J.M. SimRNA: A coarse-grained method for RNA folding simulations and 3D structure prediction. Nucl. Acid. Res.
**2015**, 44, e63. [Google Scholar] [CrossRef] [PubMed] - Xia, Z.; Bell, D.R.; Shi, Y.; Ren, P. RNA 3D Structure prediction by using a coarse-grained model and experimental data. J. Phys. Chem. B
**2013**, 117, 3135–3144. [Google Scholar] [CrossRef] [PubMed] - Cragnolini, T.; Laurin, Y.; Derreumaux, P.; Pasquali, S. Coarse-grained HiRE-RNA model for ab Initio RNA folding beyond simple molecules, including noncanonical and multiple base pairings. J. Chem. Theory Comput.
**2015**, 11, 3510–3522. [Google Scholar] [CrossRef] [PubMed] - Kim, N.; Fuhr, K.N.; Schlick, T. Graph applications to RNA structure and function. In Biophysics of RNA Folding; Russell, R., Ed.; Springer: New York, NY, USA, 2013; pp. 23–51. [Google Scholar]
- Kim, N.; Petingi, L.; Schlick, T. Network theory tools for RNA modeling. WSEAS Trans. Math.
**2013**, 9, 941–955. [Google Scholar] [PubMed] - Schlick, T. Adventures with RNA Graphs. Methods
**2018**. [Google Scholar] [CrossRef] [PubMed] - Waterman, M. Secondary structure of single-stranded nucleic acids. Adv. Math. Suppl. Stud.
**1978**, 1, 167–212. [Google Scholar] - Nussinov, R.; Jacobson, A.B. Fast algorithm for predicting the secondary structure of single-stranded RNA. Proc. Natl. Acad. Sci. USA
**1980**, 77, 6309–6313. [Google Scholar] [CrossRef] [PubMed] - Le, S.; Nussinov, R.; Maizel, J. Tree graphs of RNA secondary structures and their comparisons. Comput. Biomed. Res.
**1989**, 22, 461–473. [Google Scholar] [CrossRef] - Shapiro, B.A.; Zhang, K. Comparing multiple RNA secondary structures using tree comparisons. Bioinformatics
**1990**, 6, 309–318. [Google Scholar] [CrossRef] - Reinharz, V.; Soulé, A.; Westhof, E.; Waldispühl, J.; Denise, A. Mining for recurrent long-range interactions in RNA structures reveals embedded hierarchies in network families. Nucl. Acids Res.
**2018**, 46, 3841–3851. [Google Scholar] [CrossRef] [PubMed][Green Version] - Gan, H.H.; Pasquali, S.; Schlick, T. Exploring the repertoire of RNA secondary motifs using graph theory; implications for RNA design. Nucl. Acids Res.
**2003**, 31, 2926–2943. [Google Scholar] [CrossRef] [PubMed][Green Version] - Gan, H.H.; Fera, D.; Zorn, J.; Shiffeldrim, N.; Tang, M.; Laserson, U.; Kim, N.; Schlick, T. RAG: RNA-As-Graphs database—Concepts, analysis, and features. Bioinformatics
**2004**, 20, 1285–1291. [Google Scholar] [CrossRef] [PubMed] - Pasquali, S.; Gan, H.H.; Schlick, T. Modular RNA architecture revealed by computational analysis of existing pseudoknots and ribosomal RNAs. Nucl. Acids Res.
**2005**, 33, 1384–1398. [Google Scholar] [CrossRef] [PubMed] - Kim, N.; Zheng, Z.; Elmetwaly, S.; Schlick, T. RNA Graph Partitioning for the discovery of RNA modularity: A novel application of graph partition algorithm to biology. PLoS ONE
**2014**, 9, e106074. [Google Scholar] [CrossRef] [PubMed] - Shu, W.; Bo, X.; Zheng, Z.; Wang, S. A novel representation of RNA secondary structure based on element-contact graphs. BMC Bioinform.
**2008**, 9, 188. [Google Scholar] [CrossRef] [PubMed] - Huang, J.; Li, K.; Gribskov, M. Accurate classification of RNA structures using topological fingerprints. PLoS ONE
**2016**, 11, 1–19. [Google Scholar] [CrossRef] [PubMed] - Kim, N.; Shiffeldrim, N.; Gan, H.H.; Schlick, T. Candidates for novel RNA topologies. J. Mol. Biol.
**2004**, 341, 1129–1144. [Google Scholar] [CrossRef] [PubMed] - Jain, S.; Laederach, A.; Ramos, S.B.V.; Schlick, T. A pipeline for computational design of novel RNA-like topologies. Nucl. Acid. Res.
**2018**. [Google Scholar] [CrossRef] - Baba, N.; Elmetwaly, S.; Kim, N.; Schlick, T. Predicting large RNA-Like topologies by a knowledge-based clustering approach. J. Mol. Biol.
**2016**, 428, 811–821. [Google Scholar] [CrossRef] [PubMed] - Fera, D.; Kim, N.; Shiffeldrim, N.; Zorn, J.; Laserson, U.; Gan, H.H.; Schlick, T. RAG: RNA-As-Graphs web resource. BMC Bioinform.
**2004**, 5, 88. [Google Scholar] [CrossRef] [PubMed][Green Version] - Kim, N.; Laing, C.; Elmetwaly, S.; Jung, S.; Curuksu, J.; Schlick, T. Graph-based sampling for approximating global helical topologies of RNA. Proc. Natl. Acad. Sci. USA
**2014**, 111, 4079–4084. [Google Scholar] [CrossRef] [PubMed] - Kim, N.; Zahran, M.; Schlick, T. Chapter five—Computational Prediction of Riboswitch Tertiary Structures Including Pseudoknots by RAGTOP: A Hierarchical Graph Sampling Approach. In Computational Methods for Understanding Riboswitches; Chen, S.J., Burke-Aguero, D.H., Eds.; Methods in Enzymology; Academic Press: Waltham, MA, USA, 2015; Volume 553, pp. 115–135. [Google Scholar] [CrossRef]
- Bayrak, C.S.; Kim, N.; Schlick, T. Using sequence signatures and kink-turn motifs in knowledge-based statistical potentials for RNA structure prediction. Nucl. Acids Res.
**2017**, 45, 5414–5422. [Google Scholar] [CrossRef] [PubMed] - Jain, S.; Schlick, T. F-RAG: Generating atomic models from RNA graphs using fragment assembly. J. Mol. Biol.
**2017**, 429, 3587–3605. [Google Scholar] [CrossRef] [PubMed] - Izzo, J.A.; Kim, N.; Elmetwaly, S.; Schlick, T. RAG: An update to the RNA-As-Graphs resource. BMC Bioinform.
**2011**, 12, 219. [Google Scholar] [CrossRef] [PubMed] - Staple, D.W.; Butcher, S.E. Pseudoknots: RNA structures with diverse functions. PLoS Biol.
**2005**, 3, e213. [Google Scholar] [CrossRef] [PubMed][Green Version] - Brierley, I.; Gilbert, R.J.; Pennell, S. RNA pseudoknots and the regulation of protein synthesis. Biochem. Soc. Trans.
**2008**, 36, 684–689. [Google Scholar] [CrossRef] [PubMed][Green Version] - Gultyaev, A.P.; Olsthoorn, R.C.; Pleij, C.W.; Westhof, E. RNA Structure: Pseudoknots. eLS
**2012**. [Google Scholar] [CrossRef] - Zahran, M.; Bayrak, C.S.; Elmetwaly, S.; Schlick, T. RAG-3D: A search tool for RNA 3D substructures. Nucl. Acids Res.
**2015**, 43, 9474–9488. [Google Scholar] [CrossRef] [PubMed] - Petingi, L.; Schlick, T. Partitioning RNAs into pseudonotted and pseudoknot-free regions modeled as Dual Graphs. arXiv, 2016; arXiv:1601.04259. [Google Scholar]
- Petingi, L.; Schlick, T. Partitioning and classification of RNA secondary structures into pseudonotted and pseudoknot-free regions using a graph-theoretical approach. IAENG Int. J. Comput. Sci.
**2017**, 44, 241–246. [Google Scholar] - Database of RNA Dual Graphs. Available online: http://www.biomath.nyu.edu/?q=rag/dual_vertices.php (accesed on 19 July 2018).
- Fiedler, M. Algebraic connectivity of graphs. Czechoslov. Math. J.
**1973**, 23, 298–305. [Google Scholar] - Hopcroft, J.; Tarjan, R. Algorithm 447: Efficient algorithms for graph manipulation. Commun. ACM
**1973**, 16, 372–378. [Google Scholar] [CrossRef] - Petingi, L. Dual Graph Partitioning Code. Available online: https://github.com/Louis-Petingi/Partition-Algorithm-2/ (accesed on 19 July 2018).
- Representative Set of RNA 3D Structures. Available online: http://rna.bgsu.edu/rna3dhub/nrlist/ (accesed on 19 July 2018).
- Leontis, N.B.; Zirbel, C.L. Nonredundant 3D structure datasets for RNA knowledge extraction and benchmarking. In RNA 3D Structure Analysis and Prediction; Leontis, N., Westhof, E., Eds.; Springer: Berlin/Heidelberg, Germany, 2012; pp. 281–298. [Google Scholar]
- Yang, H.; Jossinet, F.; Leontis, N.; Chen, L.; Westbrook, J.; Berman, H.; Westhof, E. Tools for the automatic identification and classification of RNA base pairs. Nucl. Acids Res.
**2003**, 31, 3450–3460. [Google Scholar] [CrossRef] [PubMed][Green Version] - Butcher, S.E.; Pyle, A.M. The molecular interactions that stabilize RNA tertiary structure: RNA motifs, patterns, and networks. Acc. Chem. Res.
**2011**, 44, 1302–1311. [Google Scholar] [CrossRef] [PubMed] - Hsiao, C.; Mohan, S.; Kalahar, B.K.; Williams, L.D. Peeling the onion: Ribosomes are ancient molecular fossils. Mol. Biol. Evol.
**2009**, 26, 2415–2425. [Google Scholar] [CrossRef] [PubMed] - Petrov, A.S.; Bernier, C.R.; Hsiao, C.; Norris, A.M.; Kovacs, N.A.; Waterbury, C.C.; Stepanov, V.G.; Harvey, S.C.; Fox, G.E.; Wartell, R.M.; et al. Evolution of the ribosome at atomic resolution. Proc. Natl. Acad. Sci. USA
**2014**, 111, 10251–10256. [Google Scholar] [CrossRef] [PubMed][Green Version] - Wong, T.K.; Lam, T.; Sung, W.K.; Cheung, B.W.; Yiu, S. Structural alignment of RNA with complex pseudoknot structure. J. Comput. Biol.
**2011**, 18, 97–108. [Google Scholar] [CrossRef] [PubMed][Green Version] - Han, K.; Byun, Y. PseudoViewer2: Visualization of RNA pseudoknots of any type. Nucl. Acids Res.
**2003**, 31, 3432–3440. [Google Scholar] [CrossRef] [PubMed] - Kucharík, M.; Hofacker, I.L.; Stadler, P.F.; Qin, J. Pseudoknots in RNA folding landscapes. Bioinformatics
**2016**, 32, 187–194. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Dual graph representations of common RNA 2D structure building blocks. In dual graphs, stems with at least two base pairs are denoted by vertices. All loop strands (with or without residues) are represented as edges, except for single-residue bulges and internal loops with only one residue in each strand, which are ignored. The tree graph representations are also shown for comparison. Note that pseudoknots cannot be represented by tree graphs (indicated by Not Applicable (NA)) because they contain intertwined and non-nested base pairs.

**Figure 2.**Dual graph partitioning. Illustration for (

**a**) structure with pseudoknots; and (

**b**) structure with junctions. Dashed black lines show the articulation points, where the graph will be cut.

**Figure 3.**Common existing dual graph topologies. Existing dual graph IDs with five or more structures in the representative dataset of RNA structures are shown. See Table S1 in the Supplementary data for complete details of all 94 existing dual graph topologies.

**Figure 4.**Subgraph block topologies with 2–9 vertices in decreasing order of occurrence frequency. The 56 subgraph block topologies with 2–9 vertices in the representative dataset of RNA structures are shown, along with graph IDs and number of occurrences. Red graphs are part of existing dual graph topologies corresponding to the representative RNA 3D structure dataset (Figure 3 and Supplementary Table S1). Black graphs are those that only emerge as subgraphs. The IDs of subgraphs with pseudoknots are highlighted in magenta. Subgraphs with 10 or more occurrences are shown in Figure 5 (pseudoknot-free) and Figure 6 (with pseudoknots). Atomic fragments corresponding to these 56 subgraph blocks are cataloged in the RAG-3Dual dataset of RNA 3D substructures.

**Figure 5.**Common subgraph blocks without pseudoknots. Shown for each dual graph block topology with 10 or more occurrences (found by partitioning dual graphs in the representative RNA dataset) are the 2D and 3D structure fragments of one representative example. The same colors in the 2D and 3D structures correspond to similar regions. Helices are marked as Hi, where i indicates the helix number.

**Figure 6.**Common subgraph blocks with pseudoknots. Shown for each dual graph block topology with 10 or more occurrences (found by partitioning dual graphs in the representative RNA dataset) are the 2D and 3D structure fragments of one representative example. The same colors in the 2D and 3D structures correspond to similar regions. Pseudoknots are marked as PK and are shown in magenta. Standard helices are marked as Hi, where i indicates the helix number.

**Figure 7.**Subgraphs in ribosomes. Dual graph topologies that emerge as subgraph blocks in the (

**a**) small (16S and 18S ribosomal RNAs (rRNAs)) and (

**b**) large (23S and 25–28S/5.8S rRNAs) ribosomal subunits of various prokaryotic and eukaryotic species (see Tables S5–S6 in the Supplementary data for PDB files used). Subgraph IDs highlighted in red are also common subgraphs shown in Figure 5 and Figure 6. Subgraphs with a black box are unique to rRNA structures in the entire representative RNA structure dataset.

**Figure 8.**Dual graphs for large subunit rRNAs. Dual graphs corresponding to the 23S (prokaryotes) and 25–28S/5.8S (eukaryotes) rRNAs of large ribosomal subunits for a few representative species. Different colors highlight different subgraph blocks. The subgraph blocks with > 40 vertices are shown in brown. The 21-vertex subgraph in Homo sapiens in shown in light brown. Smaller subgraphs that occur in large numbers (2_2, 3_5, 4_19, 5_2, and 2_1) are all colored red and not highlighted separately.

**Figure 9.**Dual graphs for small subunit rRNAs. Dual graphs corresponding to the 16S (prokaryotes) and 18S (eukaryotes) rRNAs of small ribosomal subunits for a few representative species. Different colors highlight different subgraph blocks. The subgraph block with 41 vertices in the dual graph of Saccharomyces cerevisiae is highlighted in brown. Smaller subgraphs that occur in large numbers (2_2, 3_5, 2_1, and 4_19) are all colored red and not highlighted separately.

**Figure 10.**Proposed design pipeline for dual graphs. Fragment assembly for design of novel RNA topologies for tree and dual graphs using the RNA-As-Graph (RAG) subgraph and atomic fragment libraries. The tree graph design results are taken from [35], and a similar pipeline is proposed for dual graphs.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Jain, S.; Bayrak, C.S.; Petingi, L.; Schlick, T. Dual Graph Partitioning Highlights a Small Group of Pseudoknot-Containing RNA Submotifs. *Genes* **2018**, *9*, 371.
https://doi.org/10.3390/genes9080371

**AMA Style**

Jain S, Bayrak CS, Petingi L, Schlick T. Dual Graph Partitioning Highlights a Small Group of Pseudoknot-Containing RNA Submotifs. *Genes*. 2018; 9(8):371.
https://doi.org/10.3390/genes9080371

**Chicago/Turabian Style**

Jain, Swati, Cigdem S. Bayrak, Louis Petingi, and Tamar Schlick. 2018. "Dual Graph Partitioning Highlights a Small Group of Pseudoknot-Containing RNA Submotifs" *Genes* 9, no. 8: 371.
https://doi.org/10.3390/genes9080371