This simulation shows the spatial pattern formation in three component reaction diffusion systems using Schnakenberg model exhibiting the Turing type instability. 

Expected Result: The simulation should exhibit spatially inhomogeneous, temporally stable patterns under the proper selection of system parameters.

Schnakenberg kinetics, also known as activator-depleted kinetics when modelled with diffusive systems, is a well-known reaction kinetics exhibiting the spatial dynamics of patterns [1]. The evolution of self-organizing pattern formation was introduced by Alan Turing [2] providing a detailed explanation of how reaction-diffusion systems could be responsible for the emergence of pattern formation in nature. For such patterns to evolve, a short-range activation and long-range inhibition mechanism is a requirement. This means that the diffusion coefficient of the inhibitor is significantly larger than the diffusion coefficient of the activator [3,4]. Specifically, our model includes one activator and two inhibitors coupled with three components. For the model setting, we refer readers to the works in [5,6]. One can see how diffusion coefficients are selected as well as how reaction functions emerge as an additional term in the XML file. Through CompuCell3D, the simulation effectively illustrates the emergence of elegant spots and circular stripes.

References:

[1] Schnakenberg, J. (1979). Simple chemical reaction systems with limit cycle behaviour,
Journal of theoretical biology, 81(3), 389–400.
[2] Turing, A. M. (1990). The chemical basis of morphogenesis. Bulletin of mathematical biology, 52, 153-197.
[3] Murray J.D. (2001). Mathematical biology II: Spatial models and biomedical applications, Vol. 3, Springer New York.
[4] Gierer A., Meinhardt H. (1972) A theory of biological pattern formation, Kybernetik 12: 30–39.
[5] Xie, S., Kolokolnikov, T., Nishiura, Y. (2021). Complex oscillatory motion of multiple spikes in a three-component Schnakenberg system. Nonlinearity, 34(8), 5708.
[6] Tzou, J. C., Xie, S. (2023). Oscillatory translational instabilities of spot patterns in the Schnakenberg system on general 2D domains. Nonlinearity, 36(5), 2473.


Author: Gülsemay YİĞİT, 2024