Bulletin of Mathematical Biology

, Volume 70, Issue 4, pp 951–970 | Cite as

Asymptotic Enumeration of RNA Structures with Pseudoknots

Original Article


In this paper, we present the asymptotic enumeration of RNA structures with pseudoknots. We develop a general framework for the computation of exponential growth rate and the asymptotic expansion for the numbers of k-noncrossing RNA structures. Our results are based on the generating function for the number of k-noncrossing RNA pseudoknot structures, \({\mathsf{S}}_{k}(n)\) , derived in Bull. Math. Biol. (2008), where k−1 denotes the maximal size of sets of mutually intersecting bonds. We prove a functional equation for the generating function \(\sum_{n\ge 0}{\mathsf{S}}_{k}(n)z^{n}\) and obtain for k=2 and k=3, the analytic continuation and singular expansions, respectively. It is implicit in our results that for arbitrary k singular expansions exist and via transfer theorems of analytic combinatorics, we obtain asymptotic expression for the coefficients. We explicitly derive the asymptotic expressions for 2- and 3-noncrossing RNA structures. Our main result is the derivation of the formula \({\mathsf{S}}_{3}(n)\sim \frac{10.4724\cdot4!}{n(n-1)\cdots(n-4)}(\frac{5+\sqrt{21}}{2})^{n}\) .


Asymptotic enumeration RNA secondary structure k-noncrossing RNA structure Pseudoknot Generating function Transfer theorem Hankel contour Singular expansion 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Akutsu, T., 2000. Dynamic programming algorithms for RNA secondary structure prediction with pseudoknots. Discret. Appl. Math. 104, 45–62. MATHCrossRefMathSciNetGoogle Scholar
  2. Chamorro, M., Parkin, N., Varmus, H.E., 1991. An RNA pseudoknot and an optimal heptameric shift site are required for highly efficient ribosomal frameshifting on a retroviral messenger RNA. Proc. Natl. Acad. Sci. USA 89, 713–717. CrossRefGoogle Scholar
  3. Chen, W.Y.C., Deng, E.Y.P., Du, R.R.X., Stanley, R.P., Yan, C.H., 2007. Crossings and nestings of matchings and partitions. Trans. Am. Math. Soc. 359, 1555–1575. MATHCrossRefMathSciNetGoogle Scholar
  4. Flajolet, P., 1999. Singularity analysis and asymptotics of Bernoulli sums. Theor. Comput. Sci. 215(1–2), 371–381. MATHCrossRefMathSciNetGoogle Scholar
  5. Flajolet, P., Fill, J.A., Kapur, N., 2005. Singularity analysis, Hadamard products, and tree recurrences. J. Comput. Appl. Math. 174, 271–313. MATHCrossRefMathSciNetGoogle Scholar
  6. Flajolet, P., Grabiner, P., Kirschenhofer, P., Prodinger, H., Tichy, R.F., 1994. Mellin transforms and asymptotics: digital sums. Theor. Comput. Sci. 123, 291–314. MATHCrossRefGoogle Scholar
  7. Gao, Z., Richmond, L.B., 1992. Central and local limit theorems applied to asymptotic enumeration. J. Appl. Comput. Anal. 41, 177–186. MATHMathSciNetGoogle Scholar
  8. Gessel, I.M., Zeilberger, D., 1992. Random walk in a Weyl chamber. Proc. Am. Math. Soc. 115, 27–31. MATHCrossRefMathSciNetGoogle Scholar
  9. Haslinger, C., Stadler, P.F., 1999. RNA Structures with pseudo-knots. Bull. Math. Biol. 61, 437–467. CrossRefGoogle Scholar
  10. Hofacker, I.L., Schuster, P., Stadler, P.F., 1998. Combinatorics of RNA secondary structures. Discret. Appl. Math. 88, 207–237. MATHCrossRefMathSciNetGoogle Scholar
  11. Howell, J.A., Smith, T.F., Waterman, M.S., 1980. Computation of generating functions for biological molecules. SIAM J. Appl. Math. 39, 119–133. MATHCrossRefMathSciNetGoogle Scholar
  12. Jin, E.Y., Qin, J., Reidys, C.M., 2008. Combinatorics of RNA structures with pseudoknots. Bull. Math. Biol. 70, 45–67. CrossRefGoogle Scholar
  13. Konings, D.A.M., Gutell, R.R., 1995. A comparison of thermodynamic foldings with comparatively derived structures of 16s and 16s-like rRNAs. RNA 1, 559–574. Google Scholar
  14. Lindstroem, B., 1973. On the vector representation of induced matroids. Bull. Lond. Math. Soc. 5, 85–90. MATHCrossRefGoogle Scholar
  15. Loria, A., Pan, T., 1996. Domain structure of the ribozyme from eubacterial ribonuclease p. RNA 2, 551–563. Google Scholar
  16. Lyngso, R., Pedersen, C., 1996. Pseudoknots in RNA secondary structures. In: H. Flyvbjerg, J. Hertz, M.H. Jensen, O.G. Mouritsen, K. Sneppen (Eds.), Physics of Biological Systems: From Molecules to Species. Springer, Berlin. Google Scholar
  17. Mapping RNA Form and Function, 2005. Science 2, September 2005. Google Scholar
  18. McCaskill, J.S., 1990. The equilibrium partition function and base pair binding probabilities for RNA secondary structure. Biopolymers 29, 1105–1119. CrossRefGoogle Scholar
  19. Odlyzko, A.M., 1992. Explicit tauberian estimates for functions with positive coefficients. J. Comput. Appl. Math. 41, 187–197. MATHCrossRefMathSciNetGoogle Scholar
  20. Odlyzko, A.M., 1995. Handbook of Combinatorics. Elsevier, Amsterdam. Chapter 22. Google Scholar
  21. Penner, R.C., Waterman, M.S., 1993. Spaces of RNA secondary structures. Adv. Math. 101, 31–49. MATHCrossRefMathSciNetGoogle Scholar
  22. Popken, A., 1953. Asymptotic expansions from an algebraic standpoint. Indag. Math. 15, 131–143. MathSciNetGoogle Scholar
  23. Rivas, E., Eddy, S., 1999. A dynamic programming algorithm for RNA structure prediction inclusing pseudoknots. J. Mol. Biol. 285, 2053–2068. CrossRefGoogle Scholar
  24. Schmitt, W.R., Waterman, M.S., 1994. Linear trees and RNA secondary structure. Discret. Appl. Math. 51, 317–323. MATHCrossRefMathSciNetGoogle Scholar
  25. Tacker, M., Fontana, W., Stadler, P.F., Schuster, P., 1994. Statistics of RNA melting kinetics. Eur. Biophys. J. 23, 29–38. Google Scholar
  26. Tacker, M., Stadler, P.F., Bauer, E.G., Hofacker, I.L., Schuster, P., 1996. Algorithm independent properties of RNA secondary structure predictions. Eur. Biophys. J. 25, 115–130. CrossRefGoogle Scholar
  27. Titchmarsh, E.C., 1939. The Theory of Functions. Oxford University Press, London. MATHGoogle Scholar
  28. Tuerk, C., MacDougal, S., Gold, L., 1992. RNA pseudoknots that inhibit human immunodeficiency virus type 1 reverse transcriptase. Proc. Natl. Acad. Sci. USA 89, 6988–6992. CrossRefGoogle Scholar
  29. Uemura, Y., Hasegawa, A., Kobayashi, S., Yokomori, T., 1999. Tree adjoining grammars for RNA structure prediction. Theor. Comput. Sci. 210, 277–303. MATHCrossRefMathSciNetGoogle Scholar
  30. Waterman, M.S., 1978. Secondary structure of single-stranded nucleic acids. Adv. Math. I (suppl.) 1, 167–212. MathSciNetGoogle Scholar
  31. Waterman, M.S., 1979. Combinatorics of RNA hairpins and cloverleafs. Stud. Appl. Math. 60, 91–96. MathSciNetGoogle Scholar
  32. Waterman, M.S., Smith, T.F., 1986. Rapid dynamic programming algorithms for RNA secondary structure. Adv. Appl. Math. 7, 455–464. MATHCrossRefMathSciNetGoogle Scholar
  33. Westhof, E., Jaeger, L., 1992. RNA pseudoknots. Curr. Opin. Struct. Biol. 2, 327–333. CrossRefGoogle Scholar
  34. Wong, R., Wyman, M., 1974. The method of Darboux. J. Approx. Theory 10, 159–171. MATHCrossRefMathSciNetGoogle Scholar
  35. Zuker, M., Sankoff, D., 1984. RNA secondary structures and their prediction. Bull. Math. Biol. 46(4), 591–621. MATHGoogle Scholar

Copyright information

© Society for Mathematical Biology 2008

Authors and Affiliations

  1. 1.Center for Combinatorics, LPMC-TJKLCNankai UniversityTianjinPeople’s Republic of China

Personalised recommendations