Recent research using real RNA sequences has repeatedly revealed the same phenomenon: after RNA folds into a three-dimensional structure, the ends of the strand are close to each other. We place these analyses into a combinatorial framework, and conclude that for several different models of branching structures the ends of the structures are almost certainly proximal. Ultimately, this implies that end proximity is a property of branching structures in general, as opposed to a special property of RNA. Using analytic combinatorics, our results are framed in terms of Dyck paths, Motzkin paths, and context-free grammars.

This is joint work with Christine Heitsch. C.H. was partially supported by NIH R01GM126554, and by the NSF-Simons Southeast Center for Mathematics and Biology (SCMB) through the grants NSF DMS #1764406 and Simons Foundation/SFARI 594594.