Topological language for RNA

Abstract: In this paper we introduce a novel, context-free grammar, {\it 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. {\it RNAFeatures$^*$} acts on formal, arc-labeled RNA secondary structures, called $\lambda$-structures. $\lambda$-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. {\it 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 {\it 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 $O(n log(n))$ runtime algorithm which samples pk-structures based on ninety tRNA sequences and structures from the Nucleic Acid Database (NDB).