On quantitative effects of RNA shape abstraction

Abstract

Over the last few decades, much effort has been taken to develop approaches for identifying good predictions of RNA secondary structure. This is due to the fact that most computational prediction methods based on free energy minimization compute a number of suboptimal foldings and we have to identify the native folding among all these possible secondary structures. Using the abstract shapes approach as introduced by Giegerich et al. (Nucleic Acids Res 32(16):4843–4851, 2004), each class of similar secondary structures is represented by one shape and the native structures can be found among the top shape representatives. In this article, we derive some interesting results answering enumeration problems for abstract shapes and secondary structures of RNA. We compute precise asymptotics for the number of different shape representations of size n and for the number of different shapes showing up when abstracting from secondary structures of size n under a combinatorial point of view. A more realistic model taking primary structures into account remains an open challenge. We give some arguments why the present techniques cannot be applied in this case.

Appendix

During our investigations, we computed precise asymptotics for the size of the folding space F(s) for different models of secondary structures. The models differ with respect to structural restrictions (minimal length of hairpin loops, isolated base pairs) and the complementary assumed (Watson–Crick pairings only, wobble GU pairs allowed), expecting a uniform distribution for the bases or a skewed one (p _A = p _U = 2/10, p _C  = p _G = 3/10), according to the experiments performed in Giegerich et al. (2004) and Voß et al. (2006).

Even if they were of no use to our investigations related to abstract shapes due to the problems reported when analyzing shape spaces, we expect those results to be of use for the future and, therefore, decided to present them in this appendix without proof. Few of those results may already be found in literature (see e.g., Nebel 2004), but such a complete presentation does not exist.

Theorem 6.1

Considering a uniform distribution of the bases A, C, G and U resp. the skewed distribution p _A  = p _U = 2/10, p _C = p _G = 3/10, regarding Watson–Crick pairings only or allowing wobble GU pairs and under the assumption of each possible combination of a minimum hairpin loop length minL_hairpin ∈ {1, 3}, and a minimum helix length minL_ladder ∈ {1, 2}, the asymptotic expected folding space sizes card(F(s)) for a random primary structure s of size $n, n \rightarrow \infty,$ are those given in Table 3 shown in roman resp. italics.

Table 3 Asymptotics for the expected sizes of the folding space for a random primary structure s of size n assuming a uniform distribution of the bases A, C, G, U (results in roman) or the skewed distribution p _A  = p _U  = 2/10, p _C  = p _G  = 3/10 (results in italics), a minimum hairpin length minL_hairpin ∈ {1, 3} and a minimum ladder length minL_ladder ∈ {1, 2}