A Phase Transition in Energy-Filtered RNA Secondary Structures

In this article we study the effect of energy parameters on minimum free energy (mfe) RNA secondary structures. Employing a simplified combinatorial energy model that is only dependent on the diagram representation and is not sequence-specific, we prove the following dichotomy result. Mfe structures derived via the Turner energy parameters contain only finitely many complex irreducible substructures, and just minor parameter changes produce a class of mfe structures that contain a large number of small irreducibles. We localize the exact point at which the distribution of irreducibles experiences this phase transition from a discrete limit to a central limit distribution and, subsequently, put our result into the context of quantifying the effect of sparsification of the folding of these respective mfe structures. We show that the sparsification of realistic mfe structures leads to a constant time and space reduction, and that the sparsification of the folding of structures with modified parameters leads to a linear time and space reduction. We, furthermore, identify the limit distribution at the phase transition as a Rayleigh distribution.