Emulating cellular automata in chemical reaction–diffusion networks

Abstract

Chemical reactions and diffusion can produce a wide variety of static or transient spatial patterns in the concentrations of chemical species. Little is known, however, about what dynamical patterns of concentrations can be reliably programmed into such reaction---diffusion systems. Here we show that given simple, periodic inputs, chemical reactions and diffusion can reliably emulate the dynamics of a deterministic cellular automaton, and can therefore be programmed to produce a wide range of complex, discrete dynamics. We describe a modular reaction---diffusion program that orchestrates each of the fundamental operations of a cellular automaton: storage of cell state, communication between neighboring cells, and calculation of cells' subsequent states. Starting from a pattern that encodes an automaton's initial state, the concentration of a "state" species evolves in space and time according to the automaton's specified rules. To show that the reaction---diffusion program we describe produces the target dynamics, we simulate the reaction---diffusion network for two simple one-dimensional cellular automata using coupled partial differential equations. Reaction---diffusion based cellular automata could potentially be built in vitro using networks of DNA molecules that interact via branch migration processes and could in principle perform universal computation, storing their state as a pattern of molecular concentrations, or deliver spatiotemporal instructions encoded in concentrations to direct the behavior of intelligent materials.