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Topological classification and enumeration of RNA structures by genus

Andersen, J. E. and Penner, R. C. and Reidys, C. M. and Waterman, M. S. (2013) Topological classification and enumeration of RNA structures by genus. Journal of Mathematical Biology, 67 (5). pp. 1261-1278. ISSN 0303-6812.

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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 D_(g,σ)(z)=∑_n d_(g,σ)(n)z^n for the number d_(g,σ)(n) of those structures of fixed genus g and minimum stack size σ with n nucleotides so that no two consecutive nucleotides are basepaired and show that D_(g,σ)(z) is algebraic. In particular, we prove that d_(g,2)(n)∼k_g n^(3(g−1/2))γ^n_2, where γ_2 ≈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.

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Additional Information:© 2012 Springer-Verlag. Received: 21 May 2011; Revised: 27 June 2012; Published online: 2 October 2012. J.E. Andersen and R.C. Penner are supported by QGM (Centre for Quantum Geometry of Moduli Spaces, funded by the Danish National Research Foundation).
Funding AgencyGrant Number
Centre for Quantum Geometry of Moduli Spaces (QGM)UNSPECIFIED
Danish National Research FoundationUNSPECIFIED
Issue or Number:5
Record Number:CaltechAUTHORS:20131112-112051499
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:42381
Deposited By: Bianca Rios
Deposited On:12 Nov 2013 21:43
Last Modified:03 Oct 2019 05:57

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