285 research outputs found
Asymptotics of Canonical and Saturated RNA Secondary Structures
It is a classical result of Stein and Waterman that the asymptotic number of
RNA secondary structures is . In this paper, we
study combinatorial asymptotics for two special subclasses of RNA secondary
structures - canonical and saturated structures. Canonical secondary structures
were introduced by Bompf\"unewerer et al., who noted that the run time of
Vienna RNA Package is substantially reduced when restricting computations to
canonical structures. Here we provide an explanation for the speed-up.
Saturated secondary structures have the property that no base pairs can be
added without violating the definition of secondary structure (i.e. introducing
a pseudoknot or base triple). Here we compute the asymptotic number of
saturated structures, we show that the asymptotic expected number of base pairs
is , and the asymptotic number of saturated stem-loop structures is
, in contrast to the number of (arbitrary)
stem-loop structures as classically computed by Stein and Waterman. Finally, we
show that the density of states for [all resp. canonical resp. saturated]
secondary structures is asymptotically Gaussian. We introduce a stochastic
greedy method to sample random saturated structures, called quasi-random
saturated structures, and show that the expected number of base pairs of is
.Comment: accepted: Journal of Bioinformatics and Computational Biology (2009)
22 page
Combinatorial analysis of interacting RNA molecules
Recently several minimum free energy (MFE) folding algorithms for predicting
the joint structure of two interacting RNA molecules have been proposed. Their
folding targets are interaction structures, that can be represented as diagrams
with two backbones drawn horizontally on top of each other such that (1)
intramolecular and intermolecular bonds are noncrossing and (2) there is no
"zig-zag" configuration. This paper studies joint structures with arc-length at
least four in which both, interior and exterior stack-lengths are at least two
(no isolated arcs). The key idea in this paper is to consider a new type of
shape, based on which joint structures can be derived via symbolic enumeration.
Our results imply simple asymptotic formulas for the number of joint structures
with surprisingly small exponential growth rates. They are of interest in the
context of designing prediction algorithms for RNA-RNA interactions.Comment: 22 pages, 15 figure
- …