Journal of Mathematical Biology

, Volume 68, Issue 1–2, pp 341–375 | Cite as

Combinatorics of locally optimal RNA secondary structures

  • Éric Fusy
  • Peter Clote


It is a classical result of Stein and Waterman that the asymptotic number of RNA secondary structures is \(1.104366 \cdot n^{-3/2} \cdot 2.618034^n\). Motivated by the kinetics of RNA secondary structure formation, we are interested in determining the asymptotic number of secondary structures that are locally optimal, with respect to a particular energy model. In the Nussinov energy model, where each base pair contributes \(-1\) towards the energy of the structure, locally optimal structures are exactly the saturated structures, for which we have previously shown that asymptotically, there are \(1.07427\cdot n^{-3/2} \cdot 2.35467^n\) many saturated structures for a sequence of length \(n\). In this paper, we consider the base stacking energy model, a mild variant of the Nussinov model, where each stacked base pair contributes \(-1\) toward the energy of the structure. Locally optimal structures with respect to the base stacking energy model are exactly those secondary structures, whose stems cannot be extended. Such structures were first considered by Evers and Giegerich, who described a dynamic programming algorithm to enumerate all locally optimal structures. In this paper, we apply methods from enumerative combinatorics to compute the asymptotic number of such structures. Additionally, we consider analogous combinatorial problems for secondary structures with annotated single-stranded, stacking nucleotides (dangles).

Mathematics Subject Classification

05A16 92B05 82B30 



Figure 1 was created by W.A. Lorenz and H. Jabbari. We would like to thank the anonymous referees for their helpful comments. É. Fusy is supported by the European project ExploreMaps—-ERC StG 208471. P. Clote is supported by the National Science Foundation under grants DBI-0543506 and DMS-0817971, and by Digiteo Foundation project RNAomics. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Supplementary material

285_2012_631_MOESM1_ESM.pdf (84 kb)
ESM 1 (PDF 85KB)
285_2012_631_MOESM2_ESM.pdf (82 kb)
ESM 2 (PDF 83KB)


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Laboratoire d’Informatiques (LIX)Ecole PolytechniquePalaiseauFrance
  2. 2.Department of BiologyBoston CollegeChestnut HillUSA

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