On Times to Compute Shapes in 2D Tile Self-assembly

We study the times to grow structures within the tile self-assembly model proposed by Winfree, and the possible shapes that can be achieved. Our earlier work was confined to the growth of rectangular structures, in which the rates of attachment of border tiles and rule tiles were the same. By varying the relative rates one can engineer interesting new shapes, which have been observed in the laboratory. We show that the results from an extension of our earlier stochastic models agree remarkably closely with experimental results. This is an important further demonstration of the validity and usefulness of our stochastic models, which has also been used to study error correction in DNA self assembly. 1 The Tile Self-Assembly Model The general focus of the work here is on mathematical foundations of self assembly based on Winfrees DNA tile model [12] to be described shortly. More precisely, the emphasis is on the analysis of stochastic models. Although insightful such models and reference theories are ubiquitous in the physical sciences, they remain a fertile ground for self-assembly research in DNA-Based Computing, where stochastic analysis has only recently begun. The early work of Adleman [3] and colleagues and that of the authors [5, 4, 6] sets the stage in this area, and serves as the point of departure for the analysis here. The seminal mathematical tile model of DNA self-assembly, as developed by Winfree [12] and pursued by many others, has led to a much improved understanding of DNA self-assembly in two dimensions. At the physical layer being modeled, single-strand DNA molecules are manipulated to form DNA molecules (e.g., double-crossover molecules [7, 8]) which are designed to assemble (bond) with other such molecules in a two dimensional crystal-growth process obeying bonding rules determined by the molecular motifs [12]. These building-block molecules are modeled as tiles. There are three types of tiles involved in a self-assembly process: rule tiles, border tiles, and seed tiles. They participate in a growth process beginning at the origin of the positive lattice; the unit squares of the lattice are the potential sites occupied by tiles. • The seed tile occupies the lower-left corner of the positive lattice and is responsible for initiating the tile self-assembly process. Only border tiles, as described next, can stick to the two free (upper and right-hand) sides of the seed tiles. • A border tile can join the structure only by attaching to the seed tile or another border tile along the horizontal and vertical boundaries of the positive lattice, each such attachment extending one of the borders of the structure assembled so far.