Combinatorial Optimization and Verification in Self-Assembly

Two of the most studied tile self-assembly models in the literature are the abstract Tile Assembly Model (aTAM) [7] and the Two-Handed Tile Assembly Model (2HAM) [4]. Both models constitute a mathematical model of self-assembly in which system components are four-sided Wang tiles with glue types assigned to each tile edge. Any pair of glue types are assigned some nonnegative interaction strength denoting how strongly the pair of glues bind. The models differ in their rules for growth in that the aTAM allows singleton tiles to attach one at a time to a growing seed, whereas the 2HAM permits any two previously built assemblies to combine given enough affinity for attachment. In more detail, an aTAM system is an ordered triplet .T; ; / consisting of a set of tiles T , a positive integer threshold parameter called the system’s temperature, and a special tile 2 T denoted as the seed tile. Assembly proceeds by attaching copies of tiles from T to a growing seed assembly whenever the placement of a tile on the 2D grid achieves a total strength of attachment from abutting edges, determined by the sum of pairwise glue interactions, that meets or exceeds the temperature parameter . An additional twist that is often considered is the ability to specify a relative concentration distribution on the tiles in T . The growth from the initial seed then proceeds randomly with higher concentrated tile types attaching more quickly than lower concentrated types. Even when the final assembly is deterministic, adjusting concentration profiles may substantially alter the expected time to reach the unique terminal state. The Two-Handed Tile Assembly Model (2HAM) [4] is similar to the aTAM, but removes the concept of a seed tile. Instead, a 2HAM system .T; / produces a new assembly whenever any two previously built (and potentially large) assemblies may translate together into a new stable assembly based on glue interactions and temperature. The distinction between the 2HAM and the aTAM is that the 2HAM allows large assemblies to grow independently and attach as large, pre-