On the Loop Homology of a Certain Complex of RNA Structures

In this paper, we establish a topological framework of tau-structures to quantify the evolutionary transitions between two RNA sequence-structure pairs. tau-structures developed here consist of a pair of RNA secondary structures together with a non-crossing partial matching between the two backbones. The loop complex of a tau-structure captures the intersections of loops in both secondary structures. We compute the loop homology of tau-structures. We show that only the zeroth, first and second homology groups are free. In particular, we prove that the rank of the second homology group equals the number gamma of certain arc-components in a tau-structure and that the rank of the first homology is given by gamma-chi+1, where chi is the Euler characteristic of the loop complex.