Bulletin of Mathematical Biology

, Volume 80, Issue 6, pp 1514–1538 | Cite as

The Rainbow Spectrum of RNA Secondary Structures

  • Thomas J. X. Li
  • Christian M. Reidys
Original Article


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.


Secondary structure Rainbow Length spectrum Gap Arc Generating function Singularity analysis 

Mathematics Subject Classification

05A16 92E10 92B05 



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.


  1. Backofen R, Tsur D, Zakov S, Ziv-Ukelson M (2011) Sparse RNA folding: time and space efficient algorithms. J Discrete Algorithms 9(1):12–31MathSciNetCrossRefMATHGoogle Scholar
  2. Barrett C, Li T, Reidys C (2016) RNA secondary structures having a compatible sequence of certain nucleotide ratios. J Comput Biol 23(11):857–873MathSciNetCrossRefGoogle Scholar
  3. Barrett C, Huang F, Reidys C (2017) Sequence-structure relations of biopolymers. Bioinformatics 33(3):382–389Google Scholar
  4. Berman H, Westbrook J, Feng Z et al (2000) The protein data bank. Nucl Acids Res 28:235–242CrossRefGoogle Scholar
  5. Byun Y, Han K (2009) PseudoViewer3: generating planar drawings of large-scale RNA structures with pseudoknots. Bioinformatics 25(11):1435–1437CrossRefGoogle Scholar
  6. 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 3:2CrossRefGoogle Scholar
  7. Clote P, Ponty Y, Steyaert J (2012) Expected distance between terminal nucleotides of RNA secondary structures. J Math Biol 65(3):581–599MathSciNetCrossRefMATHGoogle Scholar
  8. Eddy S (2001) Non-coding RNA genes and the modern RNA world. Nat Rev Genet 2(12):919–929CrossRefGoogle Scholar
  9. Flajolet P, Sedgewick R (2009) Analytic combinatorics. Cambridge University Press, New YorkCrossRefMATHGoogle Scholar
  10. Graham R, Knuth D, Patashnik O (1994) Concrete mathematics: a foundation for computer science. Addison-Wesley Professional, ReadingMATHGoogle Scholar
  11. Han H, Reidys C (2012) The \(5^{\prime }\)-\(3^{\prime }\) distance of RNA secondary structures. J Comput Biol 19(7):868–878MathSciNetCrossRefGoogle Scholar
  12. 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):167–188CrossRefGoogle Scholar
  13. Hofacker I, Schuster P, Stadler P (1998) Combinatorics of RNA secondary structures. Discrete Appl Math 88(1–3):207–237MathSciNetCrossRefMATHGoogle Scholar
  14. Howell J, Smith T, Waterman M (1980) Computation of generating functions for biological molecules. SIAM J Appl Math 39(1):119–133MathSciNetCrossRefMATHGoogle Scholar
  15. Huang F, Reidys C (2012) On the combinatorics of sparsification. Algorithms Mol Biol 7(1):1–15CrossRefGoogle Scholar
  16. Hunter C, Sanders J (1990) The nature of \(\pi \)-\(\pi \) interactions. J Am Chem Soc 112(14):5525–5534CrossRefGoogle Scholar
  17. Jin E, Reidys C (2010) Irreducibility in RNA structures. Bull Math Biol 72:375–399MathSciNetCrossRefMATHGoogle Scholar
  18. Jin E, Reidys C (2010) On the decomposition of \(k\)-noncrossing RNA structures. Adv Appl Math 44:53–70MathSciNetCrossRefMATHGoogle Scholar
  19. Kim S, Sussman J, Suddath F et al (1974) The general structure of transfer RNA molecules. Proc Natl Acad Sci USA 71(12):4970–4974CrossRefGoogle Scholar
  20. 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):147–157CrossRefGoogle Scholar
  21. Lorenz R, Bernhart S, Höner zu Siederdissen C, Tafer H, Flamm C, Stadler P, Hofacker I (2011) ViennaRNA Package 2.0. Algorithms Mol Biol 6:26Google Scholar
  22. McCarthy B, Holland J (1965) Denatured DNA as a direct template for in vitro protein synthesis. Pro Natl Acad Sci USA 54(3):880–886CrossRefGoogle Scholar
  23. Penner R, Waterman M (1993) Spaces of RNA secondary structures. Adv Math 217:31–49MathSciNetCrossRefMATHGoogle Scholar
  24. Robart A, Chan R, Peters J, Rajashankar K, Toor N (2014) Crystal structure of a eukaryotic group II intron lariat. Nature 514(7521):193–197CrossRefGoogle Scholar
  25. 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):546–551CrossRefGoogle Scholar
  26. 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 473–490Google Scholar
  27. Schmitt W, Waterman M (1994) Linear trees and RNA secondary structure. Discrete Appl Math 51:317–323MathSciNetCrossRefMATHGoogle Scholar
  28. Smith T, Waterman M (1978) RNA secondary structure. Math Biol 42:31–49MATHGoogle Scholar
  29. Šponer J, Leszczynski J, Hobza P (2001) Electronic properties, hydrogen bonding, stacking, and cation binding of DNA and RNA bases. Biopolymers 61(1):3–31CrossRefGoogle Scholar
  30. 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):978–988Google Scholar
  31. Stein P, Waterman M (1979) On some new sequences generalizing the Catalan and Motzkin numbers. Discrete Math 26(3):261–272MathSciNetCrossRefMATHGoogle Scholar
  32. 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 167–212Google Scholar
  33. Waterman M (1979) Combinatorics of RNA Hairpins and Cloverleaves. Stud Appl Math 60(2):91–98MathSciNetCrossRefMATHGoogle Scholar
  34. Waterman M, Smith T (1986) Rapid dynamic programming algorithms for RNA secondary structure. Adv Appl Math 7(4):455–464MathSciNetCrossRefMATHGoogle Scholar
  35. Wexler Y, Zilberstein C, Ziv-Ukelson M (2007) A study of accessible motifs and RNA folding complexity. J Comput Biol 14(6):856–872MathSciNetCrossRefMATHGoogle Scholar
  36. 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):2275–2293CrossRefGoogle Scholar
  37. 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):292–299CrossRefGoogle Scholar
  38. Zuker M (1989) On finding all suboptimal foldings of an RNA molecule. Science 244(4900):48–52MathSciNetCrossRefMATHGoogle Scholar
  39. Zuker M, Sankoff D (1984) RNA secondary structures and their prediction. Bull Math Biol 46(4):591–621CrossRefMATHGoogle Scholar

Copyright information

© Society for Mathematical Biology 2018

Authors and Affiliations

  1. 1.Biocomplexity Institute of Virginia TechBlacksburgUSA

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