Journal of Mathematical Biology

, Volume 67, Issue 5, pp 1261–1278 | Cite as

Topological classification and enumeration of RNA structures by genus

  • J.E. Andersen
  • R.C. Penner
  • C.M. Reidys
  • M.S. Waterman


To an RNA pseudoknot structure is naturally associated a topological surface, which has its associated genus, and structures can thus be classified by the genus. Based on earlier work of Harer–Zagier, we compute the generating function \(\mathbf{D}_{g,\sigma }(z)=\sum _{n}\mathbf{d}_{g,\sigma }(n)z^n\) for the number \(\mathbf{d}_{g,\sigma }(n)\) of those structures of fixed genus \(g\) and minimum stack size \(\sigma \) with \(n\) nucleotides so that no two consecutive nucleotides are basepaired and show that \(\mathbf{D}_{g,\sigma }(z)\) is algebraic. In particular, we prove that \(\mathbf{d}_{g,2}(n)\sim k_g\,n^{3(g-\frac{1}{2})} \gamma _2^n\), where \(\gamma _2\approx 1.9685\). Thus, for stack size at least two, the genus only enters through the sub-exponential factor, and the slow growth rate compared to the number of RNA molecules implies the existence of neutral networks of distinct molecules with the same structure of any genus. Certain RNA structures called shapes are shown to be in natural one-to-one correspondence with the cells in the Penner–Strebel decomposition of Riemann’s moduli space of a surface of genus \(g\) with one boundary component, thus providing a link between RNA enumerative problems and the geometry of Riemann’s moduli space.


Boundary Component Mapping Class Group Neutral Network Oriented Edge Exponential Growth Rate 
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  1. Andersen JE, Bene AJ, Meilhan J-B, Penner RC (2010) Finite type invariants and fatgraphs. Adv Math 225:2117–2161MathSciNetCrossRefzbMATHGoogle Scholar
  2. Andersen JE, Mattes J, Reshetikhin N (1996) The poisson structure on the moduli space of flat connections and chord diagrams. Topology 35:1069–1083MathSciNetCrossRefzbMATHGoogle Scholar
  3. Andersen JE, Mattes J, Reshetikhin N (1998) Quantization of the algebra of chord diagrams. Math Proc Camb Phil Soc 124:451–467MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bar-Natan D (1995) On the Vassiliev knot invariants. Topology 34:423–475MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bar-Natan D (1997) Lie algebras and the four colour problem. Combinatorica 17:43–52MathSciNetCrossRefzbMATHGoogle Scholar
  6. Bender EA, Rodney Canfield E (1988) The asymptotic number of tree-rooted maps on a surface. J Comb Theory Ser A 48(2):156–164CrossRefzbMATHGoogle Scholar
  7. Bon M, Vernizzi G, Orland H, Zee A (2008) Topological classification of RNA structures. J Mol Biol 379:900–911CrossRefGoogle Scholar
  8. Campoamor-Stursberg R, Manturov VO (2004) Invariant tensor formulas via chord diagrams. J Math Sci 108:3018–3029MathSciNetGoogle Scholar
  9. dell’Erba MG, Zemba GR (2009) Thermodynamics of a model for RNA folding. Phys Rev E 79:011913CrossRefGoogle Scholar
  10. Euler L (1752) Elementa doctrinae solidorum. Novi Comm Acad Sci Imp Petropol 4:109–140Google Scholar
  11. Flajolet P (1980) Combinatorial aspects of continued fractions. Discret Math 32:125–161MathSciNetzbMATHGoogle Scholar
  12. Flajolet P, Francon J, Vuillemin J (1980) Sequence of operations analysis for dynamic data structures. J Algorithms 1:111–141MathSciNetCrossRefzbMATHGoogle Scholar
  13. Flajolet P, Sedgewick R (2009) Analytical combinatorics. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  14. Gao JZM, Li LYM, Reidys CM (2010) Inverse folding of RNA pseudoknot structures. Algorithms Mol Biol 5:R27CrossRefGoogle Scholar
  15. Garg I, Deo N (2009) RNA matrix models with external interactions and their asymptotic behavior. Phys Rev E 79:061903CrossRefGoogle Scholar
  16. Goulden P, Nica A (2005) A direct bijection for the Harer–Zagier formula. J Comb Theory (A) 111:224–238MathSciNetCrossRefzbMATHGoogle Scholar
  17. Goupil A, Schaeffer G (1998) Factoring n-cycles and counting maps of given genus. Eur J Comb 19(7): 819–834Google Scholar
  18. Grüner WG, Strothmann D, Reidys CM, Weber J, Hofacker IL, Stadler PF, Schuster P (1996) Analysis of RNA sequence structure maps by exhaustive enumeration II. Neutral Netw Chem Mon 127:375–389CrossRefGoogle Scholar
  19. Grüner WG, Strothmann D, Reidys CM, Weber J, Hofacker IL, Stadler PF, Schuster P (1996) Analysis of RNA sequence structure maps by exhaustive enumeration I. Neutral Netw Chem Mon 127:355–374CrossRefGoogle Scholar
  20. Harer J, Zagier D (1986) The Euler characteristic of the moduli space of curves. Invent Math 85:457–485MathSciNetCrossRefzbMATHGoogle Scholar
  21. Haslinger C, Stadler PF (1999) RNA structures with pseudo-knots. Bull Math Biol 61:437–467CrossRefGoogle Scholar
  22. Jin EY, Reidys CM (2011) Random induced subgraphs of Cayley graphs induced by transpositions. Discret Math 21(311):2496–2511MathSciNetCrossRefGoogle Scholar
  23. Kimura M (1983) The neutral theory of molecular evolution. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  24. Konings DAM, Gutell RR (1995) A comparison of thermodynamic foldings with comparatively derived structures of 16s and 16s-like r RNAs. RNA 1:559–574Google Scholar
  25. Kontsevich M (1993) Vassiliev’s knot invariants. Adv Sov Math 16:137–150MathSciNetGoogle Scholar
  26. Lando SK, Zvonkin AK (2004) Graphs on surfaces and their applications: with an appendix by Don B. Zagier. Encyclopaedia of Mathematical Sciences, 141. Low-Dimensional Topology, II. Springer-Verlag, BerlinGoogle Scholar
  27. Li TJX, Reidys CM (2012) The genus filtration of \(\gamma \)-structures. Math Biosci (submitted)Google Scholar
  28. Loria A, Pan T (1996) Domain structure of the ribozyme from eubacterial ribonuclease. RNA 2:551–563Google Scholar
  29. Milgram RJ, Penner RC (1993) Riemann’s moduli space and the symmetric groups. In: Bödigheimer C-F, Hain RM (eds) Mapping class groups and moduli spaces of Riemann surfaces. AMS contemporary math, vol 150. pp 247–290Google Scholar
  30. Orland H, Zee A (2002) RNA folding and large N matrix theory. Nucl Phys B 620:456–476MathSciNetCrossRefzbMATHGoogle Scholar
  31. Penner RC (1987) The Teichmuller space of a punctured surface. Commun Math PhysGoogle Scholar
  32. Penner RC (1988) Perturbative series and the moduli space of Riemann surfaces. J Diff Geom 27:35–53MathSciNetzbMATHGoogle Scholar
  33. Penner RC (1992) Weil–Petersson volumes. J Diff Geom 35:559–608MathSciNetzbMATHGoogle Scholar
  34. Penner RC (2004) Cell decomposition and compactification of Riemann’s moduli space in decorated Teichmüller theory. In: Tongring N, Penner RC (eds) Woods hole mathematics-perspectives in math and physics. World Scientific, Singapore, pp 263–301 (arXiv)Google Scholar
  35. Penner RC, Knudsen M, Wiuf C, Andersen J (2010) Fatgraph model of proteins. Comm Pure Appl Math 63:1249–1297MathSciNetCrossRefzbMATHGoogle Scholar
  36. Penner RC, Waterman MS (1993) Spaces of RNA secondary structures. Adv Math 101:31–49MathSciNetCrossRefzbMATHGoogle Scholar
  37. Pillsbury M, Orland H, Zee A (2005) Steepest descent calculation of RNA pseudoknots. Phys Rev E 72:011911CrossRefGoogle Scholar
  38. Pillsbury M, Taylor JA, Orland H, Zee A (2005) An algorithm for RNA pseudoknots. arXiv: cond-mat/0310505v2Google Scholar
  39. Reidys CM, Huang FWD, Andersen JE, Penner RC, Stadler PF, Nebel ME (2011) Topology and prediction of RNA pseudoknots. Bioinformatics. doi: 10.1093/bioinformatics/btr090
  40. Reidys CM (2011) Combinatorial computational biology of RNA. Springer, New YorkCrossRefzbMATHGoogle Scholar
  41. Reidys CM, Wang RR, Zhao AYY (2010) Modular, \(k\)-noncrossing diagrams. Electr J Comb 1(17):R76MathSciNetGoogle Scholar
  42. Reidys CM, Stadler PF, Schuster PK (1997) Generic properties of combinatory maps and neutral networks of RNA secondary structrures. Bull Math Biol 59:339–397CrossRefzbMATHGoogle Scholar
  43. Reidys CM, Stadler PF (2002) Combinatorial landscapes. SIAM Rev 44:3–54MathSciNetCrossRefzbMATHGoogle Scholar
  44. Rivas E, Eddy SR (1999) A dynamic programming algorithm for RNA structure prediction including pseudoknots. J Mol Biol 285:2053–2068CrossRefGoogle Scholar
  45. Reidys CM, Forst CV, Schuster P (2001) Replication and mutation on neutral networks. Bull Math Biol 63:57–94CrossRefGoogle Scholar
  46. Reidys CM (2009) Large Components of Random induced subgraphs of n-cubes. Discret Math 309:3113–3124MathSciNetCrossRefzbMATHGoogle Scholar
  47. Stanley RP (1997) Enumerative combinatorics. Cambridge studies in advanced mathematics, vol 49. Cambridge University Press, CambridgeGoogle Scholar
  48. Strebel K (1984) Quadratic differentials. Springer, BerlinCrossRefzbMATHGoogle Scholar
  49. Staple DW, Butcher SE (2005) Pseudoknots: RNA structures with diverse functions. PLoS Biol 3(6): 956–959Google Scholar
  50. Vernizzi G, Orland H, Zee A (2005) Enumeration of RNA structures by matrix models. Phys Rev Lett 94:168103CrossRefGoogle Scholar
  51. Vernizzi G, Ribecca P, Orland H, Zee A (2006) Topology of pseudoknotted homopolymers. Phys Rev E 73:031902CrossRefGoogle Scholar
  52. Waterman M (1979) Combinatorics of RNA hairpins and cloverleafs. Stud Appl Math 60:91–96MathSciNetzbMATHGoogle Scholar
  53. Waterman M (1978) Secondary structure of single-stranded nucleic acids. Adv Math (Suppl Stud) 1:167–212Google Scholar
  54. Howell J, Smith T, Waterman M (1980) Computation of generating functions for biological molecules. SIAM J Appl Math 39:119–133MathSciNetCrossRefzbMATHGoogle Scholar
  55. Waterman M, Schmitt W (1994) Linear trees and RNA secondary structure. Discret Appl Math 51:317–323MathSciNetCrossRefzbMATHGoogle Scholar
  56. Waterman MS (1995) An introduction computational biology. Chapman and Hall, New YorkCrossRefGoogle Scholar
  57. Westhof E, Jaeger L (1992) RNA pseudoknots. Curr Opin Chem Biol 2:327–333Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • J.E. Andersen
    • 1
  • R.C. Penner
    • 1
    • 2
  • C.M. Reidys
    • 3
  • M.S. Waterman
    • 4
  1. 1.Center for the Quantum Geometry of Moduli SpacesAarhus UniversityAarhus CDenmark
  2. 2.Departments of Math and PhysicsCaltechPasadenaUSA
  3. 3.Institute for Mathematics and Computer ScienceUniversity of Southern DenmarkOdenseDenmark
  4. 4.Departments of Biological Sciences, Mathematics, Computer ScienceUniversity of Southern CaliforniaLos AngelesUSA

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