Universal Computation and Optimal Construction in the Chemical Reaction Network-Controlled Tile Assembly Model

Abstract

Tile-based self-assembly and chemical reaction networks provide two well-studied models of scalable DNA-based computation. Although tile self-assembly provides a powerful framework for describing Turing-universal self-assembling systems, assembly logic in tile self-assembly is localized, so that only the nearby environment can affect the process of self-assembly. We introduce a new model of tile-based self-assembly in which a well-mixed chemical reaction network interacts with self-assembling tiles to exert non-local control on the self-assembly process. Through simulation of multi-stack machines, we demonstrate that this new model is efficiently Turing-universal, even when restricted to unbounded space in only one spatial dimension. Using a natural notion of program complexity, we also show that this new model can produce many complex shapes with programs of lower complexity. Most notably, we show that arbitrary connected shapes can be produced by a program with complexity bounded by the Kolmogorov complexity of the shape, without the large scale factor that is required for the analogous result in the abstract tile assembly model. These results suggest that controlled self-assembly provides additional algorithmic power over tile-only self-assembly, and that non-local control enhances our ability to perform computation and algorithmically self-assemble structures from small input programs.

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