Analytic Combinatorics Applied to RNA Structures
Abstract
In recent years it has been shown that the folding pattern of an RNA molecule plays an important role in its function, likened to a lock and key system. γ-structures are a subset of RNA pseudoknot structures filtered by topological genus that lend themselves nicely to combinatorial analysis. Namely, the coefficients of their generating function can be approximated for large n. This paper is an investigation into the length-spectrum of the longest block in random γ-structures. We prove that the expected length of the longest block is on the order n - O(n^1/2). We further compare these results with a similar analysis of the length-spectrum of rainbows in RNA secondary structures, found in Li and Reidys (2018). It turns out that the expected length of the longest block for γ-structures is on the order the same as the expected length of rainbows in secondary structures.
General Audience Abstract
Ribonucleic acid (RNA), similar in composition to well-known DNA, plays a myriad of roles within the cell. The major distinction between DNA and RNA is the nature of the nucleotide pairings. RNA is single stranded, to mean that its nucleotides are paired with one another (as opposed to a unique complementary strand). Consequently, RNA exhibits a knotted 3D structure. These diverse structures (folding patterns) have been shown to play important roles in RNA function, likened to a lock and key system. Given the cost of gathering data on folding patterns, little is known about exactly how structure and function are related. The work presented centers around building the mathematical framework of RNA structures in an effort to guide technology and further scientific discovery. We provide insight into the prevalence of certain important folding patterns.
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