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BulletinofMathematicalBiology (2026) 88:160https://doi.org/10.1007/s11538-026-01718-zORIGINAL RESEARCHMakingEndsMeetorJustMeetingattheEnds?AssessingEnd-to-EndDistanceinFoldedRNASequencesandOtherBranchedStructuresTorin Greenwood 1 · Christine Heitsch 2Received:21January2026/Accepted:18July2026©TheAuthor(s)2026AbstractResearchers have repeatedly found thattheends ofan RNAsequence aresignif i cantlycloser than expected for a random linear chain. However, we prove that the ends of abranched structure are almost certainly close. Our results are obtained via combinato-rial branching models of increasing complexity using tools from multivariate analyticcombinatorics. We completely characterize parameters tracking end-to-end distance,including means and variances. Then, we compare to existing datasets of known rnastructures, as well as the minimum free-energy structures of randomized shuff l es. Wef i nd that the shuff l ed structures resemble our theoretical distributions while the knownrna structures have similar parameter values but are more concentrated. The heliceson the exterior loop of a structure draw the ends of the sequence together, and wef i nd evidence that these helices are more stable than expected when compared to theircounterparts in uniformly random branched structures.Keywords Dyck · Motzkin · Pfold · Stochastic context-free grammar · Analyticcombinatorics · RNA · Secondary structuresRna molecules are not merely passive carriers of genetic information but playdirect roles in many of the cell’s essential processes. The function of an rna moleculedepends critically on the three-dimensional conformation induced by intramolecularbase pairing. These interactions generate characteristic structural motifs that governstability, specif i city, and interaction with proteins and other nucleic acids. Identify-ing the three-dimensional structure can be technically challenging, time-consuming,and often infeasible. Instead, it is more tractable to identify a two-dimensionalapproximationcalledthesecondarystructure,whichstillcapturesessentialfunctionalB Torin Greenwoodtorin.greenwood@ndsu.eduB Christine Heitschheitsch@math.gatech.edu1Department of Mathematics, North Dakota State University, 1210 Albrecht Boulevard, Fargo,ND 58102, USA2School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332, USA0123456789().: V,-vol 123