Two-Handed Assembly Model (2HAM)

Informal model description

The 2HAM is a generalization of the aTAM meant to model systems where self-assembly of multiple sub-assemblies can occur separately and in parallel, and then those sub-assemblies can combine with each other. The "2-handed" portion of the name comes from the fact that each combination is of exactly two assemblies at a time. Note that variations of this model have appeared in several papers and by several different names (e.g. hierarchical self-assembly, polyominoes, etc.) We now give a brief, informal, sketch of the 2HAM.

The 2HAM is formulated without a seed structure, so that all individual tiles have equal status in the initial solution, and assembly begins as separate assemblies nucleate in parallel. Each step of assembly occurs as any two existing assemblies (which at first are just the singleton tiles) which are able to bind to each other, with strength at least equal to the temperature parameter and without any overlaps, combine to form a new assembly. Since it is experimentally challenging to enforce the seeded nature of growth in the aTAM (see ktam), the 2HAM provides a perhaps more experimentally feasible model in that respect, by removing the seed constraint. However, since the 2HAM allows for pairs of arbitrarily large assemblies to combine with each other as long as there are no overlaps of any portions of those assemblies in the final configuration, two new difficulties arise in terms of experimental viability. First, the rate of diffusion of assemblies will decrease as their sizes increase, making it less and less likely for combinations of larger assemblies to occur. Second, in order to enforce the requirement that pairs of assemblies can only join in configurations in which they don't contain overlaps, it would need to be the case that assemblies are completely rigid (which is certainly not the case with DNA implementations of tiles) so that portions of the assemblies couldn't bend to avoid the overlaps. The fact that the 2HAM allows for the combination of arbitrarily large assemblies gives rise to the phenomenon that, although all interactions are local in the context of being between exactly two assemblies which are immediately adjacent to each other, there is also a notion of instantaneous long range interactions on the scale of individual tiles. This is because the existence of a tile at a location arbitrarily far from another can dictate whether or not that tile will be able to bind to a tile in another assembly by perhaps providing enough cooperative binding, or instead perhaps by blocking the assemblies from achieving a binding configuration. This long range interaction provides for a great amount of difference in the power of the 2HAM versus the aTAM, and is also the reason that the 2HAM isn't immediately similar to ACA systems (see wang-vs-atam).