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 o