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

TL;DR

The work studies optimal DNA self-assembly schemes for realizing $k$-regular graphs under Scenario 3 of the flexible tile model by introducing $B_3(G)$ and $T_3(G)$, the minimum numbers of bond-edge and tile types. It derives lower bounds via the concepts of unswappable graphs and vertex covers, and presents an upper-bound construction using vertex covers with neighborhood independence and 2-edge-connected induced subgraphs, yielding exact bounds for several graph families. The authors establish exact results for rook's graphs $R_{m,n}$ (large $m,n$) and provide tight bounds for Kneser graphs $Kn(n,k)$, notably $k=2$, clarifying when each bound is tight. The results advance design strategies for efficient DNA-based graph constructions and highlight computational challenges in identifying unswappability.

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.

Figures (15)

• Figure 1: Left: an example tile, or branched junction molecule, with half-edges of bond-edge types $a$, $a$, $\hat{a}$, $b$, and $\hat{x}$. Right: an illustration of how corresponding half-edges can bond to form an edge almodovar2021complexity.
• Figure 2: The pot $\{\{a^3\}, \{a, \hat{a}^2\}\}$ realizing the complete graph $K_4$ellis2019tile.
• Figure 3: Another graph realized by the pot $\{\{a^3\}, \{a, \hat{a}^2\}\}$, showing that this pot is not valid in Scenario 3.
• Figure 4: Half-edges may be swapped during self-assembly ellis2014minimal.
• Figure 5: Illustration that the Petersen graph is unswappable.

Theorems & Definitions (44)

• Lemma 2.1: Ellis-Monaghan et al.
• Definition 2.1
• Proposition 2.1
• proof
• Definition 2.2
• Corollary 2.0.1
• Theorem 2.1
• proof
• Example 1
• Definition 2.3