Turing's Decipherment of Nature Codes



Link: Online Art Show for Alan Turing's Centenary

From spiral galaxies and wind-combed dunes to coral reefs and peacock feathers - patterns in nature offer us the greatest visual experience and aesthetic pleasure. Self-organized forms and structures in natural systems had inspired the legendary artist-scientists like Leonardo di Vinci and Ernst Haeckel. The existence of Fibonacci numbers in phyllotactic architecture allows plants to optimize access to resources like moisture and sunlight, producing stunning fractal structures in many plants. The last, unpublished work of Alan Turing, the father of computer science and artificial intelligence, was to explain the Fibonacci phyllotaxis. Turing has been well-known of his remarkable achievements of cracking the German Enigma Machine during World War II and the Turing Test; however, his pioneering work on mathematical biology has received relatively less recognition. During his last few years, Turing developed the Reaction-Diffusion Theory of Morphogenesis (or the Turing Instability) to explain the mechanism of pattern formation in biological systems, such as animal skin pigmentations [1]. More than fifty years later, biologists provided the first biological evidence to show that Turing's model can experimentally control hair follicle spacing of mice [2]. Recently tissue engineering scientists have made a significant discovery of the left-right symmetry breaking of vascular stem cells due to mechanical stimuli, and successfully created targeted cellular patterns based on Turing's morphogenetic model [3]. Turing's Reaction-Diffusion Theory not only explains millions of amazing patterns in nature, but also has significant implications for developmental biology, regenerative medicine and complex systems.

My art project "Turing's Decipherment of Nature Codes" is a tribute to Alan Turing for his groundbreaking and counterintuitive idea of using Reaction-Diffusion systems to explain the complex mechanism of patterning in nature. I have developed a set of computer programs to mimic these Reaction-Diffusion systems. I have also extended Turing's two-component Reaction-Diffusion model to three-component, using the three additive primary colors, red, green, and blue, to represent three types of morphogens that react and diffuse in the system. Similar to oscillating chemical reactions, the density of morphogens in the domain will change in an iterative fashion due to reaction and diffusion, finally reaching a stable equilibrium state. Such stable equilibrium does not always occur; it requires careful selection of initial conditions, such as the diffusion and reaction rates. These initial conditions, and the initial placement of morphogens, also affect the periodicity of the resultant patterns. Based on the anisotropic migration model described in [3], I modified the system to guide the diffusion direction of morphogens which transforms resultant patterns from labyrinth to unidirectional.

[1] Turing, A. M. (1952). "The Chemical Basis of Morphogenesis". Philosophical Transactions of the Royal Society of London, series B 237 (641): 37-72.
[2] Sick, S., Reinker, S., Timmer, J., Schlake, T. (2006). "WNT and DKK determine hair follicle spacing through a reaction-diffusion mechanism". Science 314, 1447-1450.
[3] Chen, T.-H., Hsu, J. J., Zhao, X., Guo, C., Wong, M., Huang, Y., Li, Z., Garfinkel, A., Ho, C.-M., Tintut, Y., Demer, L. L. (2012). "Left-Right Symmetry Breaking in Tissue Morphogenesis via Cytoskeletal Mechanics". Circulation Research, DOI: 10.1161/CIRCRESAHA.111.25592.