Program Size and Temperature in Self-Assembly

Winfree’s abstract Tile Assembly Model is a model of molecular self‐assembly of DNA complexes known as tiles, which float freely in solution and attach one at a time to a growing ‘seed’ assembly based on specific binding sites on their four sides. We show that there is a polynomial‐time algorithm that, given an $n\times n$ square, finds the minimal tile system (i.e., the system with the smallest number of distinct tile types) that uniquely self‐assembles the square, answering an open question of Adleman et al. (Combinatorial optimization problems in self‐assembly, STOC 2002). Our investigation leading to this algorithm reveals other positive and negative results about the relationship between the size of a tile system and its ‘temperature’ (the binding strength threshold required for a tile to attach).