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We demonstrate a computing method in which a DNA nano-object representing the solution of a problem emerges as a result of self-assembly. We report an experiment in which three-vertex colorability for a 6-vertex graph with 9 edges is solved by constructing a DNA molecule representing the colored graph itself. Our findings show that computation based on "shape processing" is a viable alternative to symbol processing when computing by molecular self-assembly.Natural processes from which three dimensional molecular structures emerge can be seen as "structural" computation in nature [1]; the folding of protein chains into their tertiary structures is a familiar example. By contrast, classical models of computing and information processing are based not on direct physico-chemcal processes, but rather on operations on sequences of symbols [2]; we are all familiar with the representation of variables by 0s and 1s in electronic computers. In traditional DNA-based computing, DNA sequences have been used to represent binary symbols and linear DNA sequences encoding vertices, edges and paths have been used to represent graphs to solve problems involving graphs; by performing parallel molecular operations with these molecules, answers have been extracted by employing protocols that entail a series of sequential screening operations [3][4][5][6][7]. Similarly, computational methods involving the self-assembly of DNA tiles into one-dimensional [8] or two-dimensional arrays [9] use the cohesive ends of the tiles to encode binary symbols. By contrast, stable branched DNA molecules [10] offer the opportunity to produce a molecular version of the graph structure, one that acts both as an information processing tool and as a solution to a computation. As abstract mathematical objects in electronic computers graphs are usually represented by their adjacency matrix, which indicates the connectivity of the vertices by the edges.However, graphs are often drawn as diagrams, by indicating vertices as points in space and edges as arcs or curves connecting the vertices. It is well known that such graph representations can be embedded in 3-space, so that the curves representing the edges do not intersect except at the vertices. Our approach uses precisely such a spatial representation of the graph, thereby avoiding the encryption of its structure into a set of symbols that in turn correspond to DNA

Through self-assembly of branched junction molecules many different DNA structures (graphs) can be assembled. We show that every multigraph can be assembled by DNA such that there is a single strand that traces each edge in the graph at least once. This strand corresponds to a boundary component of a two-dimensional orientable surface that has the given graph as a deformation retract. This boundary component traverses every edge at least once, and it defines a circular path in the graph that “preserves the graph structure” and traverses each edge.

No abstract

A functional machine is not only an assembly of parts, but also an assembly of processes. The processing of each part must obey laws that respect to the property of this part. For example, building any kind of computer entails selecting appropriate components and assembling their properties to function in computation. Here, we describe computation using a DNA strand as the basic unit and we have used this unit to achieve the function of multiplication. We exploit the phenomenon of DNA hybridization, in which each strand can represent two individual units that can pair to form a single unit. We represent the numbers we multiply in binary, with different lengths representing each digit present in the number. In principle, all combinations of the numbers will be present in solution. Following hybridization, there is present a collection of duplex molecules that are tailed by single-stranded ends. These intermediates are converted to fully duplex molecules by filling in the ends with DNA polymerase. The lengths that are present represent the digits that are present, and they may be separated by denaturing PAGE. The results give a series of bands for each power of two. The number of bands in the size domain for a particular power of two is converted to binary and the sum of all present bands is then added together. Experimentally, the result of this process always yields the correct answer.