Simulation of the Abstract Tile Assembly Model Using Crisscross Slats

Abstract

The abstract Tile Assembly Model (aTAM) provides an excellent foundation for the mathematical study of DNA-tile-based self-assembling systems, especially those wherein logic is embedded within the designs of the tiles so that they follow prescribed algorithms. While such algorithmic self-assembling systems are theoretically powerful, being computationally universal and capable of building complex shapes using information-theoretically optimal numbers of tiles, physical DNA-based implementations of these systems still encounter formidable error rates and undesired nucleation that hinder this theoretical potential. Slat-based self-assembly is a recent development wherein DNA forms long slats that combine together in 2 layers, rather than square tiles in a plane. In this approach, the length of the slats is key; while tiles typically only bind to 2 neighboring tiles at a time, slats may bind to dozens of other slats. This increased coordination between slats means that several mismatched slats must coincidentally meet in just the right way for errors to persist, unlike tiles where only a few are required. Consequently, while still a novel technology, large slat-based DNA constructions have been successfully implemented in the lab with resilience to many tile-based construction problems. These improved error characteristics come at a cost however, as slat-based systems are often more difficult to design and simulate than tile-based ones. Moreover, it has not been clear whether slats, with their larger sizes and different geometries, have the same theoretical capabilities as tiles. In this paper, we show that slats are capable of doing anything that tiles can, at least at scale. We demonstrate that any aTAM system may be converted to and simulated by an effectively equivalent system of slats. Furthermore, we show that these simulating slat systems can be made more efficiently, using shorter slats and a smaller scale factor, if the simulated tile system avoids certain uncommon growth patterns. Specifically, we consider 5 classes of aTAM systems with increasing complexity, from zig-zag systems which grow in a rigid pattern to the full class of all aTAM systems, and show how they may be converted to equivalent slat systems. We show that the simplest class may be simulated by slats at only a 2c × 2c scale, where c is the freely chosen coordination number of the slats, and further show that the full class of aTAM systems can be simulated at only a 5c × 5c scale. These results prove that slats have the full theoretical power of aTAM tiles while also providing constructions that are compact enough for potential DNA-based implementations of slat systems that are both capable of powerful algorithmic self-assembly and possessing of the strong error resilience of slats.