Posets and spaces of $k$-noncrossing RNA Structures

TL;DR

This paper introduces a new partially ordered set (poset) of RNA diagrams called the Penner-Waterman poset, revealing its topological structure and potential applications in understanding RNA folding landscapes.

Contribution

It defines a novel poset of RNA diagrams and analyzes its topological properties using multitriangulation theory, connecting combinatorics and RNA structure modeling.

Findings

The poset is pure and has a specific rank.

Its geometric realization combines a simplicial sphere and a simplex.

Results relate to the topology of RNA structure spaces.

Abstract

RNA molecules are single-stranded analogues of DNA that can fold into various structures which influence their biological function within the cell. RNA structures can be modelled combinatorially in terms of a certain type of graph called an RNA diagram. In this paper we introduce a new poset of RNA diagrams $\mathcal{B}^r_{f,k}$, $r\ge 0$, $k \ge 1$ and $f \ge 3$, which we call the Penner-Waterman poset, and, using results from the theory of multitriangulations, we show that this is a pure poset of rank $k(2f-2k+1)+r-f-1$, whose geometric realization is the join of a simplicial sphere of dimension $k(f-2k)-1$ and an $\left((f+1)(k-1)-1\right)$-simplex in case $r=0$. As a corollary for the special case $k=1$, we obtain a result due to Penner and Waterman concerning the topology of the space of RNA secondary structures. These results could eventually lead to new ways to investigate landscapes of RNA $k$-noncrossing structures.