Alan Turing’s legacy and an introduction to the mathematics of pattern formation
Patterns and symmetry are woven into the fabric of nature, from the golden spiral of a nautilus shell and the fractal nature of snowflakes to the speckles on trout. While these designs may seem like random accidents of biology, they follow underlying mathematical principles.
The study of these principles falls within the growing field of mathematical biology, where mathematics is used to uncover the rules governing the complex nature of life. By translating biological phenomena into models and equations, scientists can simulate population growth, the spread of disease, developmental patterns and gene regulation. ¹
But how exactly does maths explain such patterns? Why do these patterns exist? What laws are controlling phyllotaxis or the spots on a leopard?
These questions plagued Alan Turing – not only the brilliant mathematician behind breaking the Enigma code, but also a pioneer in mathematical biology.
Turing’s Ground-breaking Idea
In his 1952 paper The Chemical Basis of Morphogenesis, ² Turing proposed a revolutionary idea that laid the foundation for mathematical biology: the intricate patterns seen in nature could be modelled by simple mathematical principles.
Morphogenesis – derived from the Greek morphe (form) and genesis (origin) ³ – is the process by which an organism develops its shape. Earlier scientists, like Johann Wolfgang von Goethe and D’Arcy Thompson had studied the relationship between form and growth, ³ the latter of whom heavily influenced Turing’s mathematical framework to explain pattern formation.
Turing proposed that patterns emerged from the interaction of two chemicals, called morphogens. He explains that small random disturbances in an initially homogenous system could lead to instability (diffusion-driven instability) ², also known as Turing Instability. This instability leads to the formation of spatially organised structures, i.e. Turing Patterns. By modelling the behaviour of morphogens using differential equations, Turing demonstrated how the reaction and diffusion of these chemicals across a developing organism formed patterns, such as the “dappling” on a cow’s coat. Thus, creating a reaction-diffusion system.
Figure 1 ⁵ below gives an example of the differential equations governing the system, showing the different factors that might affect the concentration of the morphogens.
However, like any scientific model, Turing’s equations came with simplifications. He acknowledged that certain assumptions must be made, noting that the model was “a simplification and an idealization, and consequently a falsification”. ² To make his model as accurate as possible, he assumed a uniform medium, constant rates of reaction and diffusion which caused the disturbances in the homogenous system, creating different patterns.
Explaining the Reaction-Diffusion System
In 1972, Alfred Gierer and Hans Meinhardt refined Turing’s reaction-diffusion system, introducing an activator-inhibitor system. Their version proposed the interaction of two morphogens: a short-range activator (autocatalyzing substance), and a long-range inhibitor (suppresses activator). ⁶ (See Fig. 2.) ⁷
The activator diffuses slowly, remaining concentrated in localised areas, and promotes the production of both itself and the inhibitor. In contrast, the inhibitor diffuses much faster, spreading over a broader area and preventing the activator from clustering. The difference in diffusion rates creates different patterns, such as spots or stripes.
This mechanism is more easily visualised in Fig. 3. ⁵ Molecule P acts as an activator, and molecule S as the inhibitor. Initially (Time 1), no pattern forms as both substances are randomly distributed. Time 2 and 3 shows autocatalysis of P and production of S. The faster diffusion of S and inhibition of P, causes localised and isolated peaks of P.
Experimental Proof: the CIMA reaction
Turing’s work lacked experimental confirmation and was hidden by the discovery of the structure of DNA in 1953, a year after he published his paper. The lack of computational power to prove Turing’s work served as its main challenge. Nearly 40 years later, in 1990, a research group at the University of Bordeaux conducted an experiment, providing the first experimental evidence for Turing patterns using the chlorite-iodide-malonic acid (CIMA) reaction system. ⁸
In this reaction, iodide ions (activator) and chlorite ions (inhibitor) are fed into a gel from opposite ends (see Fig. 4). ⁹ Starch was used as an indicator, forming a blue-black complex upon binding to iodide ions, making the reaction visible. The difference in concentrations of these two morphogens creates spots (localized areas of high concentration of iodide ions) and stripes (alternating high and low areas of concentration of iodide).
Shown in Figure 5 are the results from the CIMA reaction – various patterns, from hexagonal spots to labyrinth-like striped patterns. ¹⁰
Further Developments: A Wider Impact
Future research into Turing’s reaction-diffusion systems continues to be applied to various applications in biology. Studies have shown the role of these mechanisms in feather ¹¹ and hair follicle ¹² formation, digit formation ¹³ in limb development (and regulation of Hox genes ¹⁴ – transcription factors controlling patterns of development in the early embryonic stage in vertebrates) and many other aspects of developmental biology. Beyond biology, reaction-diffusion principles have been applied in development of water filtration devices ¹⁵ and in bismuth monolayers on niobium diselenide ¹⁶, a lubricant and superconductor, showcasing Turing patterns at the atomic scale.
While Turing faced limitations due to lack of computational and graphical power, modern advancements are revolutionising the scientific field. Artificial Intelligence (AI) has emerged as strong tool in recent years (Turing wrote Computing Machinery and Intelligence ¹⁷ on the topic of AI), as demonstrated with innovations such as AlphaFold. AI could be used to solve complex differential equations and visualise 2D and 3D patterns. With significant advancement in the development of technological tools, researchers can gain deeper insights into dynamic systems in many aspects of biology, such as developmental and systems biology, enhancing our understanding of morphogenesis and self-organisation during development. ¹⁸
Turing’s final contribution to the field was his collaboration with then research student, Bernard Richards, on computational simulation of surface structures of spherical organisms, using the Manchester Mark I computer prototype (i.e. The Baby). Richards’ computationally solved equations produced numerical patterns resembling the structure of Radiolaria, single-celled spherical marine organisms. ¹⁹
Turing’s theory was validated by Richards’ work in late June of 1954. Alan Turing died on 7th June 1954.
Bibliography
1. Ay, A. and Arnosti, D.N. (2011) ‘Mathematical modeling of gene expression: a guide for the perplexed biologist’, Critical Reviews in Biochemistry and Molecular Biology, 46(2), pp. 137–151. Available at: https://doi.org/10.3109/10409238.2011.556597
2. Turing, A.M. (1990) ‘The Chemical Basis of Morphogenesis’, Bulletin of Mathematical Biology, 52(1-2), pp. 153–197. Available at: https://doi.org/10.1007/bf02459572.
3. On Morphogenesis | Research groups | Imperial College London (2023) Imperial.ac.uk. Available at: https://www.imperial.ac.uk/plant-morphogenesis/onmorphogenesis/
4. Woolley, T.E., Baker, R.E. and Maini, P.K. (2017). “Turing’s Theory of Morphogenesis: Where We Started, Where We Are and Where We Want to Go”. The Incomputable, pp. 219–235. Available at:
5. Saeed, R.K. and Mustafa, A.A. (2015) ‘Homotopy Analysis Method to solve Glycolysis system in one dimension’, ResearchGate, 27(2), pp. 61–70. Available at: https://www.researchgate.net/publication/277142650_Homotopy_Analysis_Method_to_solve_Glycolysis_system_in_one_dimension (Fig. 1.)
6. Gierer, A. and Meinhardt, H. (1972) ‘A theory of biological pattern formation’, Kybernetik, 12(1), pp. 30–39. Available at: https://doi.org/10.1007/bf00289234
7. Zhu, M. (2018) Activator-Inhibitor Model for Seashell Pattern Formation. Available at: https://www.semanticscholar.org/paper/Activator-Inhibitor-Model-for-SeashellPattern-Zhu/c145f93b695b053dd13c8759422f9c403da4dd6e (Fig. 2.)
8. V. Castets et al. (1990) ‘Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern’, 64(24), pp. 2953–2956. Available at:
9. Konow, C., M Dolnik and Epstein, I.R. (2021) ‘Insights from chemical systems into Turing-type morphogenesis’, Philosophical Transactions of the Royal Society A, 379(2213). Available at: https://doi.org/10.1098/rsta.2020.0269.
10. Rudovics, B., Dulos, E. and Kepper, P. de (1996) ‘Standard and nonstandard Turing patterns and waves in the CIMA reaction’, Physica Scripta, T67, pp. 43–50. Available at: https://doi.org/10.1088/0031-8949/1996/t67/009. (Fig. 5.)
11. Jung, H.-S. et al. (1998) ‘Local Inhibitory Action of BMPs and Their Relationships with Activators in Feather Formation: Implications for Periodic Patterning’, Developmental Biology, 196(1), pp. 11–23. Available at:
12. Sick, S. et al. (2006) ‘WNT and DKK Determine Hair Follicle Spacing Through a Reaction-Diffusion Mechanism’, Science, 314(5804), pp. 1447–1450. Available at: https://doi.org/10.1126/science.1130088.
13. Raspopovic, J. et al. (2014) ‘Digit patterning is controlled by a Bmp-Sox9-Wnt Turing network modulated by morphogen gradients’, Science, 345(6196), pp. 566–570. Available at: https://doi.org/10.1126/science.1252960.
14. Sheth, R. et al. (2012) ‘Hox Genes Regulate Digit Patterning by Controlling the Wavelength of a Turing-Type Mechanism’, Science, 338(6113), pp. 1476–1480. Available at: https://doi.org/10.1126/science.1226804.
15. Tan, Z. et al. (2018) ‘Polyamide membranes with nanoscale Turing structures for water purification’, 360(6388), pp. 518–521. Available at:
16. Fuseya, Y. et al. (2021) ‘Nanoscale Turing patterns in a bismuth monolayer’, Nature Physics, 17(9), pp. 1031–1036. Available at: https://doi.org/10.1038/s41567-02101288-y.
17. Turing, A. (1950). Computing Machinery and Intelligence. Mind, [online] 59(236), pp.433–460. Available at: https://www.cs.ox.ac.uk/activities/ieg/elibrary/sources/t_article.pdf.
18. Kondo, S. (2022) ‘The present and future of Turing models in developmental biology’, Development, 149(24). Available at: https://doi.org/10.1242/dev.200974.
19. Richards, B. (2006). The Rutherford Journal - The New Zealand Journal for the History and Philosophy of Science and Technology. [online] Rutherfordjournal.org. Available at: https://www.rutherfordjournal.org/article010109.html.
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