RNA Secondary Structures Having a Compatible Sequence of Certain Nucleotide Ratios

RNA substructure as a random matrix ensemble

Combinatorial analysis of a certain abstraction of RNA structures has been studied to investigate their statistics. Our approach regards the backbone of secondary structures as an alternate sequence of paired and unpaired sets of nucleotides, which can be described by a random matrix model. We obtain the generating function of the structures using the Hermitian matrix model with the Chebyshev polynomial of the second kind and analyze the statistics with respect to the number of stems. To match the experimental findings of the statistical behavior, we consider the structures in a grand canonical ensemble and find a fugacity value corresponding to an appropriate number of stems.

The Rainbow Spectrum of RNA Secondary Structures

In this paper, we analyze the length spectrum of rainbows in RNA secondary structures. A rainbow in a secondary structure is a maximal arc with respect to the partial order induced by nesting. We show that there is a significant gap in this length spectrum. We shall prove that there asymptotically almost surely exists a unique longest rainbow of length at least [Formula: see text] and that with high probability any other rainbow has finite length. We show that the distribution of the length of the longest rainbow converges to a discrete limit law and that, for finite k, the distribution of rainbows of length k becomes for large n a negative binomial distribution. We then put the results of this paper into context, comparing the analytical results with those observed in RNA minimum free energy structures, biological RNA structures and relate our findings to the sparsification of folding algorithms.