The Expected Order of Saturated RNA Secondary Structures

Regarding so-called hairpin-loops as the building blocks of a RNA secondary structure, the order (as introduced by Waterman as a parameter on graphs in 1978) provides information on the (balanced) nesting-depth of hairpin-loops and thus on the overall complexity of the structure. Subsequent to Waterman's seminal work, 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 ofWaterman, 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, the current state of knowledge concerning saturated structures is limited to (1) their asymptotic number, (2) the expected number of base pairs, (3) the asymptotic normal density of states [4]. Nothing is known about the nature of the branching topology of saturated structures -- a topic that the current paper completely solves. 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 (1 + O (1/log2n)), if we select the saturated secondary structure uniformly at random. Furthermore, the order of saturated secondary structures is sharply concentrated around its mean. As a consequence saturated structures and structures in the traditional model behave the same with respect to the expected order. Thus we may conclude that the traditional model has already drawn the right picture and conclusions inferred from it with respect to the order (the overall shape) of a structure remain valid even if enforcing saturation (at least in expectation).