The block spectrum of RNA pseudoknot structures.

In this paper we analyze the length-spectrum of blocks in [Formula: see text]-structures. [Formula: see text]-structures are a class of RNA pseudoknot structures that play a key role in the context of polynomial time RNA folding. A [Formula: see text]-structure is constructed by nesting and concatenating specific building components having topological genus at most [Formula: see text]. A block is a substructure enclosed by crossing maximal arcs with respect to the partial order induced by nesting. We show that, in uniformly generated [Formula: see text]-structures, there is a significant gap in this length-spectrum, i.e., there asymptotically almost surely exists a unique longest block of length at least [Formula: see text] and that with high probability any other block has finite length. For fixed [Formula: see text], we prove that the length of the complement of the longest block converges to a discrete limit law, and that the distribution of short blocks of given length tends to a negative binomial distribution in the limit of long sequences. We refine this analysis to the length spectrum of blocks of specific pseudoknot types, such as H-type and kissing hairpins. Our results generalize the rainbow spectrum on secondary structures by the first and third authors and are being put into context with the structural prediction of long non-coding RNAs.

Statistics of topological RNA structures.

In this paper we study properties of topological RNA structures, i.e. RNA contact structures with cross-serial interactions that are filtered by their topological genus. RNA secondary structures within this framework are topological structures having genus zero. We derive a new bivariate generating function whose singular expansion allows us to analyze the distributions of arcs, stacks, hairpin- , interior- and multi-loops. We then extend this analysis to H-type pseudoknots, kissing hairpins as well as 3-knots and compute their respective expectation values. Finally we discuss our results and put them into context with data obtained by uniform sampling structures of fixed genus.

Topological language for RNA.

In this paper we introduce a novel, context-free grammar, RNAFeatures*, capable of generating any RNA structure including pseudoknot structures (pk-structure). We represent pk-structures as orientable fatgraphs, which naturally leads to a filtration by their topological genus. Within this framework, RNA secondary structures correspond to pk-structures of genus zero. RNAFeatures* acts on formal, arc-labeled RNA secondary structures, called λ-structures. λ-structures correspond one-to-one to pk-structures together with some additional information. This information consists of the specific rearrangement of the backbone, by which a pk-structure can be made cross-free. RNAFeatures* is an extension of the grammar for secondary structures and employs an enhancement by labelings of the symbols as well as the production rules. We discuss how to use RNAFeatures* to obtain a stochastic context-free grammar for pk-structures, using data of RNA sequences and structures. The induced grammar facilitates fast Boltzmann sampling and statistical analysis. As a first application, we present an O(nlog (n)) runtime algorithm which samples pk-structures based on ninety tRNA sequences and structures from the Nucleic Acid Database (NDB). AVAILABILITY: the source code for simulation results is available at http://staff.vbi.vt.edu/fenixh/TPstructure.zip. The code is written in C and compiled by Xcode.