Topological Classification of RNA Structures via Intersection Graph

We introduce a new algebraic representation of RNA secondary structures as composition of hairpins and we define an appropriate abstract algebraic representation. Moreover, we propose a novel method to classify the RNA structures based on two topological invariants, the genus and the number of crossing. Starting from the classic arc representation of RNA secondary structures, the proposed method takes advantage of the algebraic representation to easy obtain an interaction graph of RNA molecule through an appropriate procedure. Each vertex of the graph is a loop and each edge represents the interaction between the two loops, thus the cardinality of edges is the number of crossing of the RNA molecule. Through the definition and application of a new procedure, the intersection graph of the RNA shape is obtained. The cardinality of the resulting graph corresponds to the crossing number of the shape associated to the RNA molecule. The aforementioned crossing number is a topological invariant, as well as the genus. Both do not uniquely identify an RNA graph, but the crossing number permits to add a term which is proportional to the standard free energy of the RNA molecule. Thus, a more precise free-energy parametrization can be obtained. Finally, our method is validated over a subset of real RNA structures from Pseudobase++ databases, and we classify the RNA structures according to their topological genus and crossing number.