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Two-dimensional tile displacement can simulate cellular automata

Tile displacement is a newly-recognized mechanism in DNA nanotechnology that exploits principles analogous to toehold-mediated strand displacement but within the context of self-assembled DNA origami tile arrays. Here, we formulate an abstract model of tile displacement for the simplest case: individual assemblies interacting with monomer tiles in solution. We give several constructions for programmable computation by tile displacement, from circuits to cellular automata, that vary in how they use energy (or not) to drive the system forward (or not), how much space and how many tile types they require, and whether their computational power is limited to PTIME or PSPACE with respect to the size of the system. In particular, we show that tile displacement systems are Turing universal and can simulate arbitrary two-dimensional synchronous block cellular automata, where each transition rule for updating the state of a 2 by 2 neighborhood is implemented by just a single tile.