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Can Math Explain How Animals Get Their Patterns?


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How Alan Turing's Reaction-Diffusion Model Simulates Patterns in Nature
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Reaction Diffusion Simulation (Gray-Scott model)


Credits (and Twitter handles):
Script Writer: Rachel Becker (@RA_Becks)
Script Editor: Emily Elert (@eelert)
Video Illustrator: Ever Salazar (@eversalazar)
Video Director: Emily Elert (@eelert)
Video Narrator: Emily Elert (@eelert)
With Contributions From: Henry Reich, Alex Reich, Kate Yoshida, Omkar Bhagat, Peter Reich, David Goldenberg
Music by: Nathaniel Schroeder:

Also, special thanks to the following scientists:
Greg Barsh: Investigator, HudsonAlpha Institute for Biotechnology (
Jeremy Green: Professor of developmental biology, King’s College London (
Thomas Hiscock: Graduate student in systems biology, Harvard University (
Shigeru Kondo: Professor, Osaka University (
James Sharpe: Coordinator of EMBL-CRG Systems Biology Unit and ICREA research professor (
Ian Stewart: Emeritus professor of mathematics, University of Warwick and author of The Mathematics of Life (
Thomas Woolley: Postdoctoral scientist, St John's College Oxford (

Image Credits:
- Mouse palate images provided courtesy of Jeremy Green, King’s College London.
- Digit patterns image provided courtesy of Luciano Marcon and Jelena Raspopovic.
- Angelfish and zebrafish images provided courtesy of Shigeru Kondo.

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Here are some handy keywords to get your googling started:

Reaction-diffusion system: A hypothetical system in which multiple chemical substances diffuse through a defined space at different rates and react with one another, thereby generating a pattern.

Turing pattern: A periodic pattern that forms in a space where the initial distribution of ‘activator’ and ‘inhibitor’ is the same.

Morphogenesis: The processes during development that give rise to the form or shape of the organism or a structure

Alan Turing: Alan Turing was a British mathematician and the father of modern computer science. During World War II, he broke Germany’s Enigma code used to encrypt communications.



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Economou, A. D., & Green, J. B. (2014). Modelling from the experimental developmental biologists viewpoint. Seminars in Cell & Developmental Biology, 35, 58-65. doi:10.1016/j.semcdb.2014.07.006

Green, J. B., & Sharpe, J. (2015). Positional information and reaction-diffusion: Two big ideas in developmental biology combine.Development, 142(7), 1203-1211. doi:10.1242/dev.114991

Kimura, Y. T. (2016, May 24). The mathematics of patterns. Retrieved from

Kimura, Y. T. (2014). The Mathematics of Patterns: The modeling and analysis of reaction-diffusion equations (Thesis, Princeton University). Http://

Kondo, S., & Asai, R. (1995). A reaction-diffusion wave on the skin of the marine angelfish Pomacanthus. Nature, 376(6543), 765-768. doi:10.1038/376765a0

Kondo, S., & Miura, T. (2010). Reaction-Diffusion Model as a Framework for Understanding Biological Pattern Formation. Science, 329(5999), 1616-1620. doi:10.1126/science.1179047

Marcon, L., & Sharpe, J. (2012). Turing patterns in development: What about the horse part? Current Opinion in Genetics & Development, 22(6), 578-584. doi:10.1016/j.gde.2012.11.013

Raspopovic, J., Marcon, L., Russo, L., & Sharpe, J. (2014). Digit patterning is controlled by a Bmp-Sox9-Wnt Turing network modulated by morphogen gradients. Science, 345(6196), 566-570. doi:10.1126/science.1252960

Stewart, I. (2012). The mathematics of life. Philadelphia, PA: Basic Books. (

Turing, A. M. (1952). The Chemical Basis of Morphogenesis. Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 237(641), 37-72. Retrieved from

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