Program Size and Temperature in Self-Assembly

Abstract

Winfree’s abstract Tile Assembly Model (aTAM) 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 ×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, Cheng, Goel, Huang, Kempe, Moisset de Espanés, and Rothemund (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)

The first author was supported by the Molecular Programming Project under NSF grant 0832824, the second author was supported by an NSF Computing Innovation Fellowship, and the third author was supported by NSERC Discovery Grant R2824A01 and the Canada Research Chair in Biocomputing to Lila Kari and the Funding Program for Next Generation World-Leading Researchers (NEXT program) to Professor Yasushi Okuno.