Spaces of RNA Secondary Structures

We prove two topological theorems in physical chemistry. Namely, we introduce a hybrid of transverse and tangential measures on train tracks to prove sphericity of various simplicial complexes which arise from certain idealized models of physical chemistry. These complexes are at once identified with Thurston’s space of projective geodesic laminations on an ideal polygon and with the analogue of a compactification (described elsewhere) of the moduli space of a punctured Riemann surface. The physical structures we study are various sub-collections of the set of all possible planar chemical bonds among the sites of a linear macromolecule. Each such collection we consider has a natural partial ordering, and the geometric realizations of appropriate posets are shown to be topological spheres. Such a topological statement encodes a wealth of combinatorial data, as we briefly discuss. In fact, our primary motivation here is to study secondary structures on RNA. This imposes the further restriction that there can be at most one base-pair supported at a given site of underlying linear macromolecule, and imposing this restriction leads to the class of “binary macromolecules.” Our main results here assert the sphericity of certain topological spaces of both arbitrary and binary macromolecules, and it is the latter which we hope may have applications to RNA. Our techniques are largely elaborations of elementary topological techniques from Techmiiller theory and the theory of train tracks.