The Expected Order of Saturated RNA Secondary Structures

Abstract Over the last 30 years the development of RNA secondary structure prediction algorithms have been guided and inspired by corresponding combinatorial studies where the RNA molecules are modeled as certain kind of planar graphs. The other way round, new algorithmic ideas gave rise to interesting combinatorial problems asking for a deeper understanding of the structures processed. One such example is the notion order of a secondary structure as introduced by Waterman (as a parameter on graphs) in 1978, which reflects a structure’s overall complexity: Regarding so-called hairpinloops as the building blocks of a secondary structure, the order provides information on the (balanced) nesting-depth of hairpin-loops and thus on the overall complexity of the structure. In related prediction algorithms, one first searches for order 1 structures, increasing the allowed order step by step and thus considering an improved structural complexity in every iteration. Subsequently, Zucker et al. and Clote introduced a more realistic combinatorial model for RNA secondary structures, the so-called saturated secondary structures. Compared to the traditional model of Waterman, unpaired nucleotides (vertices) which are in favorable position for a pairing do not exist, i.e. no base pair (edge) can be added without violating at least one restriction for the graphs. That way, one major shortcoming of the traditional model has been cleared. However, the resulting model gets much more challenging from a mathematical point of view. As a consequence, so far only little is known about the combinatorics of RNA saturated structures. In this paper we show how it is possible to attack saturated structures and especially how to analyze their order. This is of special interest since in the past it has been proven to be one of the most demanding parameters to address (for the traditional model it has been an open problem for more than 20 years to find asymptotic results for the number of structures of given order and similar). We show the expected order of RNA saturated secondary structures of size n is log4 n (