Skip to main content
SpringerLink
Log in

The Rainbow Spectrum of RNA Secondary Structures

  • Original Article
  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

In this paper, we analyze the length spectrum of rainbows in RNA secondary structures. A rainbow in a secondary structure is a maximal arc with respect to the partial order induced by nesting. We show that there is a significant gap in this length spectrum. We shall prove that there asymptotically almost surely exists a unique longest rainbow of length at least \(n-O(n^{1/2})\) and that with high probability any other rainbow has finite length. We show that the distribution of the length of the longest rainbow converges to a discrete limit law and that, for finite k, the distribution of rainbows of length k becomes for large n a negative binomial distribution. We then put the results of this paper into context, comparing the analytical results with those observed in RNA minimum free energy structures, biological RNA structures and relate our findings to the sparsification of folding algorithms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  • Backofen R, Tsur D, Zakov S, Ziv-Ukelson M (2011) Sparse RNA folding: time and space efficient algorithms. J Discrete Algorithms 9(1):12–31

    Article  MathSciNet  MATH  Google Scholar 

  • Barrett C, Li T, Reidys C (2016) RNA secondary structures having a compatible sequence of certain nucleotide ratios. J Comput Biol 23(11):857–873

    Article  MathSciNet  Google Scholar 

  • Barrett C, Huang F, Reidys C (2017) Sequence-structure relations of biopolymers. Bioinformatics 33(3):382–389

    Google Scholar 

  • Berman H, Westbrook J, Feng Z et al (2000) The protein data bank. Nucl Acids Res 28:235–242

    Article  Google Scholar 

  • Byun Y, Han K (2009) PseudoViewer3: generating planar drawings of large-scale RNA structures with pseudoknots. Bioinformatics 25(11):1435–1437

    Article  Google Scholar 

  • Cannone J, Subramanian S, Schnare M et al (2002) The comparative RNA Web (CRW) site: an online database of comparative sequence and structure information for ribosomal, intron, and other RNAs. BMC Bioinform 3:2

    Article  Google Scholar 

  • Clote P, Ponty Y, Steyaert J (2012) Expected distance between terminal nucleotides of RNA secondary structures. J Math Biol 65(3):581–599

    Article  MathSciNet  MATH  Google Scholar 

  • Eddy S (2001) Non-coding RNA genes and the modern RNA world. Nat Rev Genet 2(12):919–929

    Article  Google Scholar 

  • Flajolet P, Sedgewick R (2009) Analytic combinatorics. Cambridge University Press, New York

    Book  MATH  Google Scholar 

  • Graham R, Knuth D, Patashnik O (1994) Concrete mathematics: a foundation for computer science. Addison-Wesley Professional, Reading

    MATH  Google Scholar 

  • Han H, Reidys C (2012) The \(5^{\prime }\)-\(3^{\prime }\) distance of RNA secondary structures. J Comput Biol 19(7):868–878

    Article  MathSciNet  Google Scholar 

  • Hofacker I, Fontana W, Stadler P, Bonhoeffer L, Tacker M, Schuster P (1994) Fast folding and comparison of RNA secondary structures. Chem Mon 125(2):167–188

    Article  Google Scholar 

  • Hofacker I, Schuster P, Stadler P (1998) Combinatorics of RNA secondary structures. Discrete Appl Math 88(1–3):207–237

    Article  MathSciNet  MATH  Google Scholar 

  • Howell J, Smith T, Waterman M (1980) Computation of generating functions for biological molecules. SIAM J Appl Math 39(1):119–133

    Article  MathSciNet  MATH  Google Scholar 

  • Huang F, Reidys C (2012) On the combinatorics of sparsification. Algorithms Mol Biol 7(1):1–15

    Article  Google Scholar 

  • Hunter C, Sanders J (1990) The nature of \(\pi \)-\(\pi \) interactions. J Am Chem Soc 112(14):5525–5534

    Article  Google Scholar 

  • Jin E, Reidys C (2010) Irreducibility in RNA structures. Bull Math Biol 72:375–399

    Article  MathSciNet  MATH  Google Scholar 

  • Jin E, Reidys C (2010) On the decomposition of \(k\)-noncrossing RNA structures. Adv Appl Math 44:53–70

    Article  MathSciNet  MATH  Google Scholar 

  • Kim S, Sussman J, Suddath F et al (1974) The general structure of transfer RNA molecules. Proc Natl Acad Sci USA 71(12):4970–4974

    Article  Google Scholar 

  • Kruger K, Grabowski P, Zaug A, Sands J, Gottschling D, Cech T (1982) Self-splicing RNA: autoexcision and autocyclization of the ribosomal RNA intervening sequence of tetrahymena. Cell 31(1):147–157

    Article  Google Scholar 

  • Lorenz R, Bernhart S, Höner zu Siederdissen C, Tafer H, Flamm C, Stadler P, Hofacker I (2011) ViennaRNA Package 2.0. Algorithms Mol Biol 6:26

  • McCarthy B, Holland J (1965) Denatured DNA as a direct template for in vitro protein synthesis. Pro Natl Acad Sci USA 54(3):880–886

    Article  Google Scholar 

  • Penner R, Waterman M (1993) Spaces of RNA secondary structures. Adv Math 217:31–49

    Article  MathSciNet  MATH  Google Scholar 

  • Robart A, Chan R, Peters J, Rajashankar K, Toor N (2014) Crystal structure of a eukaryotic group II intron lariat. Nature 514(7521):193–197

    Article  Google Scholar 

  • Robertus J, Ladner J, Finch J, Rhodes D, Brown R, Clark B, Klug A (1974) Structure of yeast phenylalanine tRNA at 3 \(\mathring{A}\) resolution. Nature 250(5467):546–551

    Article  Google Scholar 

  • Salari R, Möhl M, Will S, Sahinalp S, Backofen R (2010) Time and space efficient RNA–RNA interaction prediction via sparse folding. In: Berger B (ed) Research in computational molecular biology, No. 6044 in Lecture Notes in Computer Science. Springer, Berlin, Heidelberg, pp 473–490

    Google Scholar 

  • Schmitt W, Waterman M (1994) Linear trees and RNA secondary structure. Discrete Appl Math 51:317–323

    Article  MathSciNet  MATH  Google Scholar 

  • Smith T, Waterman M (1978) RNA secondary structure. Math Biol 42:31–49

    MATH  Google Scholar 

  • Šponer J, Leszczynski J, Hobza P (2001) Electronic properties, hydrogen bonding, stacking, and cation binding of DNA and RNA bases. Biopolymers 61(1):3–31

    Article  Google Scholar 

  • Šponer J, Sponer J, Mládek A, Jurečka P, Banáš P, Otyepka M (2013) Nature and magnitude of aromatic base stacking in DNA and RNA: quantum chemistry, molecular mechanics, and experiment. Biopolymers 99(12):978–988

    Google Scholar 

  • Stein P, Waterman M (1979) On some new sequences generalizing the Catalan and Motzkin numbers. Discrete Math 26(3):261–272

    Article  MathSciNet  MATH  Google Scholar 

  • Waterman M (1978) Secondary structure of single-stranded nucleic acids. In: Rota GC (ed) Studies on foundations and combinatorics, Advances in Mathematics Supplementary Studies, vol 1. Academic Press, New York, pp 167–212

    Google Scholar 

  • Waterman M (1979) Combinatorics of RNA Hairpins and Cloverleaves. Stud Appl Math 60(2):91–98

    Article  MathSciNet  MATH  Google Scholar 

  • Waterman M, Smith T (1986) Rapid dynamic programming algorithms for RNA secondary structure. Adv Appl Math 7(4):455–464

    Article  MathSciNet  MATH  Google Scholar 

  • Wexler Y, Zilberstein C, Ziv-Ukelson M (2007) A study of accessible motifs and RNA folding complexity. J Comput Biol 14(6):856–872

    Article  MathSciNet  MATH  Google Scholar 

  • Woese C, Magrum L, Gupta R, Siegel R, Stahl D, Kop J, Crawford N, Brosius J, Gutell R, Hogan J, Noller H (1980) Secondary structure model for bacterial 16S ribosomal RNA: phylogenetic, enzymatic and chemical evidence. Nucl Acids Res 8(10):2275–2293

    Article  Google Scholar 

  • Yoffe A, Prinsen P, Gelbart W, Ben-Shaul A (2011) The ends of a large RNA molecule are necessarily close. Nucl Acids Res 39(1):292–299

    Article  Google Scholar 

  • Zuker M (1989) On finding all suboptimal foldings of an RNA molecule. Science 244(4900):48–52

    Article  MathSciNet  MATH  Google Scholar 

  • Zuker M, Sankoff D (1984) RNA secondary structures and their prediction. Bull Math Biol 46(4):591–621

    Article  MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank the reviewers for their comments and suggestions and specifically for pointing out a gap in the proof of Lemma 1. We gratefully acknowledge the help of Kevin Shinpaugh and the computational support team at BI. Many thanks to Christopher L. Barrett and Henning Mortveit for discussions. The second author is a Thermo Fisher Scientific Fellow in Advanced Systems for Information Biology and acknowledges their support of this work.

Author information

Authors and Affiliations

  1. Biocomplexity Institute of Virginia Tech, Blacksburg, VA, 24061, USA

    Thomas J. X. Li & Christian M. Reidys

Authors
  1. Thomas J. X. Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christian M. Reidys
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christian M. Reidys.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, T.J.X., Reidys, C.M. The Rainbow Spectrum of RNA Secondary Structures. Bull Math Biol 80, 1514–1538 (2018). https://doi.org/10.1007/s11538-018-0411-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-018-0411-9

Keywords

Mathematics Subject Classification

Search

Navigation