1 5 N ov 2 01 0 Parallelism , Program Size , Time , and Temperature in Self - Assembly ∗

We settle a number of questions in variants of Winfree’s abstract Tile Assembly Model (aTAM), a model of molecular algorithmic self-assembly. In the “hierarchical” aTAM, two assemblies, both consisting of multiple tiles, are allowed to aggregate together, whereas in the “seeded” aTAM, tiles attach one at a time to a growing assembly. Adleman, Cheng, Goel, and Huang (Running Time and Program Size for Self-Assembled Squares, STOC 2001) showed how to assemble an n × n square in O(n) time in the seeded aTAM using O( logn log logn ) unique tile types, showed that both of these parameters are optimal, and asked whether the hierarchical aTAM could allow a tile system to use the ability to form large assemblies in parallel before they attach to break the Ω(n) lower bound for assembly time. We show there is a tile system with the optimal O( log n log logn ) tile types that assembles an n × n square using O(log n) parallel “stages”, which is close to the optimal Ω(log n) stages, forming the final n×n from four n/2×n/2 squares, which are themselves recursively formed from n/4×n/4 squares, etc. However, despite this nearly maximal parallelism, the system requires superlinear time to assemble the square. We leave open the question of whether some hierarchical tile system can break the Ω(n) assembly time lower bound for assembling an n× n square. We extend the definition of partial order tile systems studied by Adleman et al. in a natural way to hierarchical assembly and show that no hierarchical partial order tile system can build any shape with diameter N in less than time Ω(N), demonstrating that in this case the hierarchical model affords no speedup whatsoever over the seeded model. We also strengthen the Ω(N) time lower bound for deterministic seeded systems of Adleman et al. to nondeterministic seeded systems. We then investigate the relationship between the temperature of a tile system and its size. We show that a tile system can in general require temperature that is exponentially greater than its number of tile types. On the other hand, for the special case of 2-cooperative systems, in which all binding events involve at most 2 sides of tiles, it suffices to use temperature linear in the number of tile types. We show that there is a polynomial-time algorithm that, given any tile system T specified by its desired binding behavior, finds a temperature and binding energies (at most exponential in the number of tile types of T ) that realize this behavior or reports that no such energies exist. This result is applied to show that there is a polynomial-time algorithm that, given an n×n square Sn, determines the smallest seeded system (at any temperature) that is deterministic and self-assembles Sn. This answers an open question of Adleman, Cheng, Goel, Huang, Kempe, Moisset de Espanés, and Rothemund (Combinatorial Optimization Problems in Self-Assembly, STOC 2002). The first author was supported by the Molecular Programming Project under NSF grant 0832824, the second and fourth authors were supported by NSF Computing Innovation Fellowships, and the third author was supported by NSERC Discovery Grant R2824A01 and the Canada Research Chair in Biocomputing to Lila Kari. California Institute of Technology, Pasadena, CA, USA, [email protected], [email protected] University of Western Ontario, Dept. of Computer Science, London, ON, Canada, N6A 5B7, [email protected]. University of Washington, Dept. of Computer Science, Seattle, WA, USA, [email protected].