Computation of Minimum Numbers of Tile and Bond-Edge Types for DNA Self-Assembly of Select Archimedean Graphs

TL;DR

This work develops a mathematical framework for DNA self-assembly of Archimedean graphs under the strict Scenario 3 of the flexible tile model, focusing on minimizing tile types $T_3(G)$ and bond-edge types $B_3(G)$. It formalizes the Substructure Realization Problem via the construction matrix $M(P)$ and spectrum $\,S(P)$ and leverages Python tools SRPS and cyclecode to compute exact or bounded values for six order-12/24 graphs. The authors show cuboctahedron, small rhombicuboctahedron, and snub cube are unswappable, yielding precise $B_3(G)$ bounds and $T_3(G)=|V(G)|$, while swappable graphs like the truncated tetrahedron and truncated cube do not lower $B_3$ or $T_3$, with the truncated octahedron presenting a notable case of $T_3(G)=18$. These results inform cost-effective DNA nanostructure design and point to future work on tighter bounds and larger Archimedean graphs.

Abstract

This project mathematically models the self-assembly of DNA nanostructures in the shape of select Archimedean graphs using the flexible tile model. Under three different sets of restrictions called scenarios, we employ principles of linear algebra and graph theory to determine the minimum number of different DNA branched molecules and bond types needed to construct the desired shapes, theoretically reducing laboratory costs and the waste of biomaterials. We determine exact values for $T_3(G)$, the minimum number of molecule (or ``tile") types needed for all six order 12 and 24 Archimedean graphs. We also determine exact values for $B_3(G)$, the minimum number of strand (or ``bond-edge") types, for three of the six graphs and establish bounds for the remaining three. Two algorithms, implemented as Python scripts, are used to analyze proposed design strategies for the graphs.

Figures (9)

• Figure 1: $k$-armed branched junction molecule
• Figure 2: A pot of four tile types; $P = \{\{a,b,x\},\{\hat{a},\hat{b},\hat{x}\},\{a,\hat{b},x\},\{\hat{a},b,\hat{x}\}\}$.
• Figure 3: $K_4$ (left), smaller order graph (center), and nonisomorphic graph of equal order (right) as realized by $P = \{\{a^3\},\{\hat{a}^3\},\{a^2,\hat{a}\},\{\hat{a}^2,a\}\}$.
• Figure 4: Nonisomorphic graph realized by pot when $\lambda(v_1,v_2)=\lambda(v_6,v_{10})$ (left); Scenario 3 labeling of cuboctahedron (right).
• Figure 5: Scenario 3 labeling of small rhombicuboctahedron

Theorems & Definitions (49)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Definition 1.5
• Definition 1.6
• Example 1.7
• Definition 2.1
• Definition 2.2
• Example 2.3