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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
Article

Abstract

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.

Keywords

Boundary Component Mapping Class Group Neutral Network Oriented Edge Exponential Growth Rate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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