Self-Assembly of Irregular Graphs Whose Edges Are DNA Helix Axes Phiset Sa-Ardyen, ²,§ Natasha Jonoska,* ,‡ and Nadrian C. Seeman* ,² Contribution from the Department of Chemistry, New York UniVersity, New York, New York 10003, and Department of Mathematics, UniVersity of South Florida, Tampa, Florida 33620 Received January 4, 2004; E-mail: ned.seeman@nyu.edu; jonoska@math.usf.edu Abstract: A variety of computational models have been introduced recently that are based on the properties of DNA. In particular, branched junction molecules and graphlike DNA structures have been proposed as computational devices, although such models have yet to be confirmed experimentally. DNA branched junction molecules have been used previously to form graph-like three-dimensional DNA structures, such as a cube and a truncated octahedron, but these DNA constructs represent regular graphs, where the connectivities of all of the vertexes are the same. Here, we demonstrate the construction of an irregular DNA graph structure by a single step of self-assembly. A graph made of five vertexes and eight edges was chosen for this experiment. DNA branched junction molecules represent the vertexes, and duplex molecules represent the edges; in contrast to previous work, specific edge molecules are included as components. We demonstrate that the product is a closed cyclic single-stranded molecule that corresponds to a double cover of the graph and that the DNA double helix axes represent the designed graph. The correct assembly of the target molecule has been demonstrated unambiguously by restriction analysis. Introduction Structural DNA nanotechnology uses reciprocal exchange between DNA double helices or hairpins to produce branched DNA motifs. These branched motifs can be combined via sticky- ended cohesion to produce specific graphlike structures. 1,2 The power of sticky-ended cohesion is that it leads to predictable adhesion between components whose product has a known structure. 3 DNA stick-polyhedra, such as a cube 4 and a truncated octahedron, 5 have been constructed from simple branched junctions; the vertexes correspond to the branch points of branched junctions, and the edges are DNA double helices that connect these vertexes. The cube was built by combining squares in solution, and the truncated octahedron entailed the use of a solid support methodology; 6 both syntheses contained a number of steps. These Platonic and Archimedean solids are regular graphs, all of whose vertexes have the same degree of connectivity, three. From the computational point of view, it is important to be able to construct irregular graphs, molecules whose vertexes have varying degrees of connectivity. From the perspective of computation, it is most desirable to construct these graphs by a single step of self-assembly, because, in this way, the power of parallel computation using DNA can be maximized and the number of computational steps will not depend on the size of the graph. It has been proposed that through the use of three-dimensional graph structures achieved by DNA self-assembly, it is possible to solve a number of NP-complete problems with a constant number of steps. Algorithms for solving the Hamiltonian Path problem, 7 the three-vertex colorability problem, 8 and the three- SAT problem 9 have been described. In contrast to many algorithms proposed for a large class of NP-complete problems where DNA molecules are treated as linear strings, the general idea here is to encode a problem in branched DNA molecules that represent the vertexes of a graph encoding a solution. A graph corresponding to a solution to the problem may then be obtained, through the assembly of a set of vertex and edge building blocks, which represent the variable inputs and the rules they must obey to generate the output. It can be shown that the whole graph can be constructed from DNA, if and only if a solution to the problem exists. Consequently, to establish that a solution exists for a particular problem, one need only verify that DNA graphs corresponding to the solution have actually resulted from the assembly. The details of the solution can be analyzed by conventional DNA analytical techniques, such as sequencing or restriction analysis. Encoding rules and actual graphs vary according to the nature of the problem. Nevertheless, ² New York University. ‡ University of South Florida. § Current Address: Department of Biochemistry, Faculty of Science, Chulalongkorn University, Phyathai Road, Patumwan, Bangkok Thailand 10330. (1) Seeman, N. C. J. Theor. Biol. 1982, 99, 237-247. (2) Seeman, N. C. Nature 2003, 421, 427-433. (3) Qiu, H.; Dewan, J. C.; Seeman, N. C. J. Mol. Biol. 1997, 267, 881-898. (4) Chen, J.; Seeman, N. C. Nature 1991, 350, 631-633. (5) Zhang, Y.; Seeman, N. C. J. Am. Chem. Soc. 1994, 116, 1661-1669. (6) Zhang, Y.; Seeman, N. C. J. Am. Chem. Soc. 1992, 114, 2656-2663. (7) Jonoska, N.; Karl, S. A.; Saito, M. In DNA Computers III; Rubin, H., Wood, D., Eds.; American Mathematical Society: Providence, RI, 1999; pp 123- 135. (8) Jonoska, N.; Karl, S. A.; Saito, M. Biosystems 1999, 52, 143-153. (9) Jonoska, N.; Sa-Ardyen, P.; Seeman, N. C. J. Genetic Programming And EVolVable Machines 2003, 4, 123-137. Published on Web 05/07/2004 6648 9 J. AM. CHEM. SOC. 2004, 126, 6648-6657 10.1021/ja049953d CCC: $27.50 © 2004 American Chemical Society