Neutral Networks of Sequence to Shape Maps

TL;DR

This paper introduces a new combinatorial framework for sequence-to-shape maps that generalizes RNA folding, demonstrating the existence of exponentially many shapes with extensive neutral networks and analyzing their structural properties.

Contribution

The paper presents a novel combinatorial model for sequence-to-shape maps, establishing the existence of many shapes with large neutral networks and a natural graph structure among shapes.

Findings

Existence of at least rac12;^{n-1} shapes with extended neutral networks

Shapes form a graph where connected shapes differ by a neutral network of distance one

Comparison with RNA folding maps highlights similarities and differences

Abstract

In this paper we present a novel framework for sequence to shape maps. These combinatorial maps realize exponentially many shapes, and have preimages which contain extended connected subgraphs of diameter n (neutral networks). We prove that all basic properties of RNA folding maps also hold for combinatorial maps. Our construction is as follows: suppose we are given a graph $H$ over the $\{1 >...,n\}$ and an alphabet of nucleotides together with a symmetric relation $\mathcal{R}$, implied by base pairing rules. Then the shape of a sequence of length n is the maximal H subgraph in which all pairs of nucleotides incident to H-edges satisfy $\mathcal{R}$. Our main result is to prove the existence of at least $\sqrt{2}^{n-1}$ shapes with extended neutral networks, i.e. shapes that have a preimage with diameter $n$ and a connected component of size at least $(\frac{1+\sqrt{5}}{2})^n+(\frac{1-\sqrt{5}}{2})^n$. Furthermore, we show that there exists a certain subset of shapes which carries a natural graph structure. In this graph any two shapes are connected by a path of shapes with respective neutral networks of distance one. We finally discuss our results and provide a comparison with RNA folding maps.