Asynchronous Signal Passing for Tile Self-assembly: Fuel Efficient Computation and Efficient Assembly of Shapes

In this paper we study the power of a model of tile self-assembly in which individual tiles of the system have the ability to turn on or off glue types based on the bonding of other glues on the given tile. This work is motivated by the desire for a system that can effect recursive self assembly, and is guided by the consideration of a DNA origami implementation of four-sided tiles which have a small set of queued up reactions based on DNA strand replacement. We formulate the Signal passing Tile Assembly Model which is based on the model of Padilla, et al. (2011) [21], but is asynchronous in that any action of turning a glue on or off, or attaching a new tile, or the breaking apart of an assembly, may happen in any order. Within this most general case we study tile self-assembly problems that have been addressed within the abstract Tile Assembly Model, as well as other variants of that model, and show that signal passing tiles allow for substantial improvement across multiple metrics. Our first results focus on the tile type efficient assembly of linear structures, which achieves assembly using provably fewer tile types than what is possible in standard tile assembly models by utilizing a recursive assembly process. We then present a system of signal passing tiles that simulate an arbitrary Turing machine which is fuel efficient in that only a constant number of tiles are used up per computation step. To the best of our knowledge, this is the first tile self-assembly model that is able to achieve this. Finally, we consider the assembly of the discrete Sierpinski triangle and show that this pattern can be strictly self-assembled within the signal tile passing model. This result is of particular interest in that it is known that this pattern cannot self-assemble within a number of well studied tile self-assembly models.