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Combinatorics of locally optimal RNA secondary structures

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Abstract

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).

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Notes

  1. If the energy \(E(S)=0\) or if the temperature \(T=+\infty \), then the partition function is exactly equal to the number of secondary structures.

  2. Jean Gaston Darboux (1842–1917).

  3. The shape of a secondary structure was defined by Voss et al. (2006) to represent its branching topology; for instance, the shape of the well-known clover-leaf structure of tRNA is \(\,\mathtt [ \,\,\mathtt [ \,\,\mathtt ] \,\,\mathtt [ \,\,\mathtt ] \,\,\mathtt [ \,\,\mathtt ] \,\,\mathtt ] \,\). The asymptotic number of shapes for a length \(n\) sequence yields the run time for the Giegerich Lab software RNAshapes on length \(n\) sequences, since Steffen et al. (2006) report that RNAshapes runs in time \(O(n^3 k s)\) for \(s\) sequences, each of length at most \(n\) and \(k\) shapes.

  4. To the best of our knowledge, UNAFOLD is currently the only software that computes the partition function over all secondary structures in a mathematically rigorous manner.

  5. This is done, for instance, in grammar \(G_4\) by replacing the rule \(\bullet _{\ge \theta } \rightarrow \bullet \) by \(\bullet _{\ge \theta } \rightarrow \bullet ^{\theta }\), where \(\bullet ^{\theta }\) consists of \(\theta \) occurrences of \(\bullet \).

  6. Our grammar \(G_1\) is equivalent to the “tree grammar nussinov78” from Steffen and Giegerich (2005).

  7. It is clear that the number of structures equals the partition function \(\sum _{S} \exp (-E(S)/RT)\) provided that \(E(S)=0\).

  8. Alternatively, and more simply, we could have produced this curve from the Taylor coefficients of the expressions to the right of the limit in equations (3) and (6), after first solving for \(S(z,u)\) [resp. \(S(z,u,v)\)] in equation (1) [resp. (4)].

    Fig. 2
    figure 2

    Theoretical melting curve for two simple energy models of RNA secondary structure. Temperature in Celsius is given on the \(x\)-axis, while expected number of base pairs is given on the \(y\)-axis. We implemented an algorithm, using dynamic programming, with run time \(O(n^5)\) and space \(O(n^3)\), to compute the partition function \(Z_k = \sum _{S \in \mathbb S _k} \exp (-E(S)/RT)\), where \(\mathbb ( S)_k\) denotes the set of all secondary structures for a homopolymer of length 100 nt, having exactly \(k\) base pairs. The expected number of base pairs is thus \(\sum _k k \cdot p_k\), where \(p_k = \frac{Z_k}{Z}\) denotes the probability that a secondary structure has \(k\) base pairs, and \(Z\) denotes the full partition function \(Z = \sum _{S} \exp (-E(S)/RT) = \sum _k Z_k\). (Alternatively, and more simply, we could have produced this curve from the Taylor coefficients of the expressions to the right of the limit in equations (3) and (6), after first solving for \(S(z,u)\) [resp. \(S(z,u,v)\)] in equation (1) [resp. (4)].) In the Nussinov–Jacobson energy model (Nussinov and Jacobson 1980), \(E(S)\) is defined to be \(-1\cdot |S|\); i.e. \(-1\) times the number of base pairs of \(S\). In the base stacking energy model, \(E(S)\) is defined to be \(-1\) times the number of stacked base pairs of \(S\). Although both models are quite similar, we see that the melting curves are indeed different, where the base stacking model entails more cooperative folding (see Dill and Bromberg 2002 for discussion of cooperative folding)

    Full size image

  9. We use the subscript notation for partial derivatives.

  10. Exact base stacking parameters are ignored as is entropy; however, the context-free grammar allows the separate marking of distinct features, such as stacked base pairs, hairpins, bulges, internal loops, multiloops.

  11. Sheikh et al. (2012) show that minimum energy pseudoknotted structure prediction is NP-complete, in contrast with the existence of a cubic time algorithm for the Nussinov energy model (Tabaska et al. 1998).

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Acknowledgments

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.

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  1. Laboratoire d’Informatiques (LIX), Ecole Polytechnique, 91128 , Palaiseau, France

    Éric Fusy

  2. Department of Biology, Boston College, Chestnut Hill, MA, 02467, USA

    Peter Clote

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  1. Éric Fusy
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Correspondence to Éric Fusy or Peter Clote.

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Fusy, É., Clote, P. Combinatorics of locally optimal RNA secondary structures. J. Math. Biol. 68, 341–375 (2014). https://doi.org/10.1007/s00285-012-0631-9

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