Asymptotics of Canonical RNA Secondary Structures

It is a classical result of Stein and Waterman that the asymptotic number S(n) of RNA secondary structures is 1.104366 ldr n^-3/2 ldr 2.618034ⁿ, where the combinatorial model of RNA concerns a length n homopolymer, such that any base can pair with any other base, subject to the usual convention that hairpin loops must contain at least thetas = 1 unpaired bases. The result of Stein and Waterman is proved by developing recursions,using generating functions and applying Bender´s theorem. These recursions form the basis to compute the minimum free energy secondary structure for a given RNA sequence, with respect to the Nussinov energy model, later extended by Zuker to substantially more complicated resursions for the Turner nearest neighbor energy model. In this paper, we study combinatorial asymptotic for two special subclasses of RNA secondary structures - canonical and saturated structures. Canonical secondary structures are defined to have no lonely (isolated) base pairs. This class of secondary structures was introduced b y Bompfuenewerer et al., who noted that the runtime of Vienna RNA Package is substantially decreased when restricting computations to canonical structures. Here we provide an explanation for the speed-up, by proving that the asymptotic number of canonical RNA secondary structures is 2.1614 ldr n^-3/2 ldr 1.96798ⁿ. Saturated secondary structures have the property that no base pairs can be added without violating the definition of secondary structure (i.e. introducing a pseudoknotor base triple). In the Nussinov energy model,where the energy for a base pair is -1, saturated structures correspond to kinetic traps.n prior work, we showed that the asymptotic number of saturated structures of a length n homopolymer is 1.07427 ldr n^-3/2 ldr 2.35467ⁿ. In this paper, we show that the expected number of base pairs of random saturated structures, generated by a natural stochastic procedure, is - (z^thetas+1)/((1-z)²) (-z-Sigma_i=0 ^thetas (z²)/(i+1)) (int e (z+Sigma_i=0 ^thetas (z²)/(i+1))dz).