New Generation Computing

, Volume 35, Issue 3, pp 213–223 | Cite as

Discretization of Chemical Reactions in a Periodic Cellular Space

  • Fumi Takabatake
  • Ibuki Kawamata
  • Ken Sugawara
  • Satoshi Murata
Research Paper
  • 215 Downloads

Abstract

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.

Keywords

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

References

  1. 1.
    Turing, A.M.: The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B 237, 37–72 (1952)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Kondo, S., Miura, T.: Reaction-diffusion model as a framework for understanding biological pattern formation. Science 329, 1616–1620 (2010)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Murray, J.D.: Mathematical Biology. Springer, Berlin (1989)CrossRefMATHGoogle Scholar
  4. 4.
    Tyson, J.J., Fife, P.C.: Target patterns in a realistic model of the Belousov–Zhabotinskii reaction. J. Chem. Phys. 73, 2224–2237 (1980)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Miura, T., Maini, P.K.: Periodic pattern formation in reaction-diffusion systems: an introduction for numerical simulation. Anat. Sci. Int. 79, 112–123 (2004)CrossRefGoogle Scholar
  6. 6.
    Lyons, M.J., Harrison, L.G.: Stripe selection: an intrinsic property of some pattern-forming models with nonlinear dynamics. Dev. Dyn. 195, 201–215 (1992)CrossRefGoogle Scholar
  7. 7.
    Ermentrout, B.: Stripes or spots? Nonlinear effects in bifurcation of reaction-diffusion equations on the square. Proc. R. Soc. Lond. A 434, 413–417 (1991)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Kashima, K., Ogawa, T., Sakurai, T.: Selective pattern formation control: spatial spectrum consensus and Turing instability approach. Automatica 56, 25–35 (2015)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Chen, Y., Dalchau, N., Srinivas, N., Phillips, A., Cardelli, L., Soloveichik, D., Seelig, G.: Programmable chemical controllers made from DNA. Nat. Nanotechnol. 8, 755–762 (2013)CrossRefGoogle Scholar
  10. 10.
    Soloveichika, D., Seeliga, G., Winfree, E.: DNA as a universal substrate for chemical kinetics. Proc. Natl. Acad. Sci. USA 107, 5393–5398 (2010)CrossRefGoogle Scholar
  11. 11.
    Padirac, A., Fujii, T., Estévez-Torres, A., Rondelez, Y.: Spatial waves in synthetic biochemical networks. J. Am. Chem. Soc. 135, 14586–14592 (2013)CrossRefGoogle Scholar
  12. 12.
    Scalise, D., Schulman, R.: Designing modular reaction-diffusion programs for complex pattern formation. Technology 02, 55–66 (2014)CrossRefGoogle Scholar
  13. 13.
    Wolfram, S.: A New Kind of Science. Wolfram Media, Champaign (2002)MATHGoogle Scholar
  14. 14.
    Wolfram, S.: Statistical mechanics of cellular automata. Rev. Mod. Phys. 55, 601–644 (1983)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Wolfram, S.: Cellular automata as models of complexity. Nature 311, 419–424 (1984)CrossRefGoogle Scholar
  16. 16.
    Rendell, P.: Turing universality of the game of life. In: Collision-Based Computing, pp. 513–539 (2002)Google Scholar
  17. 17.
    Cook, M.: Universality in elementary cellular automata. Complex Syst. 15, 1–40 (2004)MathSciNetMATHGoogle Scholar
  18. 18.
    Scalise, D., Schulman, R.: Emulating cellular automata in chemical reaction–diffusion networks. Nat. Comput. 15, 197–214 (2015)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Mimura, M. (Ed.): Pattern Formation and its Dynamics. University of Tokyo Press, Tokyo (2006) (in Japanese)Google Scholar
  20. 20.
    Murata, S., Konagaya, A., Kobayashi, S., Saito, H., Hagiya, M.: Molecular robotics: a new paradigm for artifacts. New Gener. Comput. 31, 27–45 (2013)CrossRefGoogle Scholar
  21. 21.
    Hagiya, M., Konagaya, A., Kobayashi, S., Saito, H., Murata, S.: Molecular robots with sensors and intelligence. Acc. Chem. Res. 47, 1681–1690 (2014)CrossRefGoogle Scholar
  22. 22.
    Hagiya, M., Wang, S., Kawamata, I., Murata, S., Isokawa, T., Peper, F., Imai, K.: On DNA-based gellular automata. UCNC 2014 LNCS, vol. 8553, pp. 177–189 (2014)Google Scholar

Copyright information

© Ohmsha, Ltd. and Springer Japan 2017

Authors and Affiliations

  • Fumi Takabatake
    • 1
  • Ibuki Kawamata
    • 1
  • Ken Sugawara
    • 2
  • Satoshi Murata
    • 1
  1. 1.Tohoku UniversityAoba-ku SendaiJapan
  2. 2.Tohoku Gakuin UniversityIzumi-ku SendaiJapan

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