Valid Plane Trees: Combinatorial Models for RNA Secondary Structures with Watson-Crick Base Pairs

The combinatorics of RNA plays a central role in biology. Mathematical biologists have several commonly-used models for RNA: words in a fixed alphabet (representing the primary sequence of nucleotides) and plane trees (representing the secondary structure, or folding of the RNA sequence). This paper considers an augmented version of the standard model of plane trees, one that incorporates some observed constraints on how the folding can occur. In particular we assume the alphabet consists of complementary pairs, for instance the Watson-Crick pairs A-U and C-G of RNA. Given a word in the alphabet, a valid plane tree is a tree for which, when the word is folded around the tree, each edge matchs two complementary letters. Consider the graph whose vertices are valid plane trees for a fixed word and whose edges are given by Condon, Heitsch, and Hoos’s local moves. We prove this graph is connected. We give an explicit algorithm to construct a valid plane tree from a primary sequence, assuming that at least one valid plane tree exists. The tree produced by our algorithm has other useful characterizations, including a uniqueness condition defined by local moves. We also study enumerative properties of valid plane trees, analyzing how the number of valid plane trees depends on the choice of sequence length and alphabet size. Finally we show that words with at least one valid plane tree are unusual, in the sense that the proportion of words with at least one valid plane tree goes to zero as the word size increases. Open questions and conjectures are given throughout the document.