Strong Fault-Tolerance for Self-Assembly with Fuzzy Temperature

Abstract

We consider the problem of fault-tolerance in nanoscale algorithmic self-assembly. We employ a standard variant of Winfree's abstract Tile Assembly Model (aTAM), the two-handed aTAM, in which square "tiles" -a model of molecules constructed from DNA for the purpose of engineering self-assembled nanostructuresaggregate according to specific binding sites of varying strengths, and in which large aggregations of tiles may attach to each other, in contrast to the seeded aTAM, in which tiles aggregate one at a time to a single speciallydesignated "seed" assembly. We focus on a major cause of errors in tile-based self-assembly: that of unintended growth due to "weak" strength-1 bonds, which if allowed to persist, may be stabilized by subsequent attachment of neighboring tiles in the sense that at least energy 2 is now required to break apart the resulting assembly; i.e., the errant assembly is stable at temperature 2.We study a common self-assembly benchmark problem, that of assembling an n × n square using O(log n) unique tile types, under the two-handed model of self-assembly. Our main result achieves a much stronger notion of fault-tolerance than those achieved previously. Arbitrary strength-1 growth is allowed; however, any assembly that grows sufficiently to become stable at temperature 2 is guaranteed to assemble into the correct final assembly of an n × n square. In other words, errors due to insufficient attachment, which is the cause of errors studied in earlier papers on fault-tolerance, are prevented absolutely in our main construction, rather than only with high probability and for sufficiently small structures, as in previous faulttolerance studies.

Parallelism and Time in Hierarchical Self-Assembly

“…This is because some sub-square has concentration at most O(1/n 2 ), so the time for even a single step of hierarchical assembly is at least Ω(n 2 ) by standard models of chemical kinetics. We note, however, that there are other theoretical advantages to the hierarchical model, for instance, the use of steric hindrance to enable algorithmic fault-tolerance [22]. For this reason, our highly parallel square construction may be of independent interest despite the fact that the parallelism does not confer a speedup.…”

“…two-handed, recursive, multiple tile, q-tile, aggregation, polyomino) aTAM allows non-seed tiles to aggregate together into an assembly, allows this assembly to then aggregate to other assemblies, and possibly (depending on the model) dispenses completely with the idea of a seed. Variants of the hierarchical aTAM have recently received extensive theoretical study [1,2,4,6,[16][17][18]22,25,36,38,40,57]. It is intuitively conceivable that by allowing two large assemblies to form in parallel and combine in one step, it may be possible to recursively build an n × n square in o(n) time, perhaps even O(log n) or O(polylog(n)) time.…”

Efficient 3-SAT algorithms in the tile assembly model

“…Some of the ideas used in the definitions of nondeterministic computation originate from Winfree et al (Winfree et al 1998). I will first define the basics of the tile assembly model in section 2.1 and then the concept of computation within the model in section 2.2 While there exist a number of variations of the tile assembly model (Aggarwal et al 2005;Demaine et al 2008;Doty et al 2010;Kao and Schweller 2006), some of which are more biologically accurate, I do not focus on them in this paper.…”

Size-separable tile self-assembly: a tight bound for temperature-1 mismatch-free systems

“…An alternative model, called the two-handed assembly model (2HAM) [1,2,4,6], hierarchical tile assembly model [3,14], or polyomino tile assembly model [9,10], allows "seedless" assembly, where tiles can attach spontaneously to form large assemblies that may attach to each other. This seedless assembly was proved by Cannon et al [2] to be capable of simulating any seeded assembly process, while also achieving more efficient assembly of some classes of shapes.…”