Optimal Constructions for DNA Self-Assembly of $k$-Regular Graphs

TL;DR

This paper explores optimal DNA tile constructions for assembling specific regular graphs, establishing bounds on the minimal types needed for accurate self-assembly, with applications to rook's and Kneser graphs.

Contribution

It introduces new bounds on tile and bond-edge types for DNA self-assembly of regular graphs, including methods for establishing these bounds and applying them to specific graph classes.

Findings

Lower bounds for unswappable graphs established

Upper bounds derived via vertex cover methods

New bounds proved for rook's and Kneser graphs

Abstract

Within biology, it is of interest to construct DNA complexes of a certain shape. These complexes can be represented through graph theory, using edges to model strands of DNA joined at junctions, represented by vertices. Because guided construction is inefficient, design strategies for DNA self-assembly are desirable. In the flexible tile model, branched DNA molecules are referred to as tiles, each consisting of flexible unpaired cohesive ends with the ability to form bond-edges. We thus consider the minimum number of tile and bond-edge types necessary to construct a graph $G$ (i.e. a target structure) without allowing the formation of graphs of lesser order, or nonisomorphic graphs of equal order. We emphasize the concept of (un)swappable graphs, establishing lower bounds for unswappable graphs. We also introduce a method of establishing upper bounds via vertex covers. We apply both of these methods to prove new bounds on rook's graphs and Kneser graphs.