Emulating cellular automata in chemical reaction–diffusion networks
- 261 Downloads
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.
KeywordsReaction–diffusion Cellular automata DNA strand displacement Chemical reaction network Intelligent materials Molecular programming Programmable matter Distributed computation
The authors would like to thank Paul Rothemund, Damien Woods, Josh Fern, John Zenk, and the anonymous referees for insightful reading and comments. This work was supported by NSF-CCF-1161941 and a grant to the Turing Centenary Project by the John Templeton Foundation.
- Codon A, Kirkpatrick B, Maňuch J (2012) Reachability bounds for chemical reaction networks and strand displacement systems. DNA Computing and Molecular Programming. Springer, Heidelberg, BerlinGoogle Scholar
- Doty D (2014) Timing in chemical reaction networks. In: Proceedings of the 25th ACM-SIAM symposium on discrete algorithms, pp 772–784Google Scholar
- Du Y, Lo E, Ali S, Khademhosseini A (2008) Directed assembly of cell-laden microgels for fabrication of 3D tissue constructs. In; Proceedings of the National Academy of Sciences 105(28):9522–9527Google Scholar
- Greenfield D, McEvoy AL, Shroff H, Crooks GE, Wingreen NS, Betzig E, Liphardt J (2009) Self-organization of the Escherichia coli chemotaxis network imaged with super-resolution light microscopy. PLoS Biol. 7(6)Google Scholar
- Lukacs G, Haggie P, Seksek O, Lechardeur D, Verkman NFA (2000) Size-dependent DNA mobility in cytoplasm and nucleus. J Biol Chem 275(1625)Google Scholar
- Montagne K, Plasson R, Sakai Y, Fujii T, Rondelez Y (2011) Programming an in vitro DNA oscillator using a molecular networking strategy. Mol Sys Biol 7(1)Google Scholar
- Neary T, Woods D (2006) P-completeness of cellular automaton rule 110. LNCS 4051(132–143)Google Scholar
- von Neumann J, Burks AW (1966) The theory of self-reproducing automata. University of Illinois Press, UrbanaGoogle Scholar
- Qian L, Soloveichik D, Winfree E (2011) Efficient turing-universal computation with DNA polymers. DNA computing and molecular programming pp 123–140Google Scholar
- Soloveichik D, Seelig G, Winfree E (2010) DNA as a universal substrate for chemical kinetics. In: Proceedings of the National Academy of Sciences 107(12):5393–5398Google Scholar