Discretization of Chemical Reactions in a Periodic Cellular Space


We investigated spatio-temporal pattern formation in a reaction-diffusion system assuming a periodic cellular space. The reaction space is an array of cells, where diffusion is assumed to be fast inside the cells and relatively slow in the walls separating them. The simulation results showed that the spatio-temporal development of the concentration pattern on the array depends on some specific parameters such as the diffusion coefficients in the cell walls and the size of cells. In a certain parameter region, the concentration inside each cell takes either a high or low value, and moreover, these locally discretized cells generate an alternating pattern spreading into the entire space. Whole of this process can be regarded as discretization of state, time, and space of a chemical reaction system which is intrinsically continuous. This kind of discretization mechanism is expected to provide a new way of the implementation of cellular automata based on molecular reactions.

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  1. 1.

    This requirement comes from the property of DNA logic gates; if all the cells have the same set of molecules, they cannot distinguish between signals from neighboring cells and their own output signal. Therefore, the molecular species used for signaling in the cells should be alternately changed. Moreover, the logic gate molecules have to be immobilized to the cell, otherwise the state of each cell cannot be properly defined.

  2. 2.

    We adopted these reaction functions so that roles of activator (u) and inhibitor (v) would be easily understood in this simple form. Since an arbitrary chemical reaction system can be implemented using DNA reactions [10], a similar reaction system can be designed with DNA molecules.


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This work was supported by a Grant-in-Aid for Scientific Research on Innovative Areas “Molecular Robotics” (No. 24104005) of The Ministry of Education, Culture, Sports, Science, and Technology, Japan.

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Correspondence to Satoshi Murata.

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Takabatake, F., Kawamata, I., Sugawara, K. et al. Discretization of Chemical Reactions in a Periodic Cellular Space. New Gener. Comput. 35, 213–223 (2017). https://doi.org/10.1007/s00354-017-0009-z

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  • Pattern formation
  • Reaction-diffusion system
  • Periodic reaction space
  • Discretized reaction state