Alan Turing was the first scientist to use computer simulation to study the mechanisms behind the development of pattern in living organisms. When Manchester Electronic Computer, also called Ferranti Mark I, was installed in Manchester University, Turing wrote to his colleague, “I am hoping as one of the first jobs to do something about ‘chemical embryology.’ In particular I think one can account for the appearance of Fibonacci numbers in connection with fir cones.” A Fibonacci series is a series of number in which each number is the sum of the two previous number, except the first two numbers. For instance, it is 1, 3, 4 ,7, 11, 18,... Turing was inspired by zoologist Thompson’s work “On Growth and Form.” Thompson observed that the scales of a fir cone and the florets of a sunflower are grouped in the numbers in Fibonacci series. Turing formulated his reaction-diffusion model from those observation. He believed that diffusing chemicals react with each other and causes the development of forms of living organisms. Modern computer has simulated Turing’s reaction-diffusion mechanism, and it has successfully produced leopard-like, cheetah-like, and giraffe-like stripes. In his paper “Chemical basis of morphogenesis,” he called these interacting chemicals morphogens. Turing believed that genes catalyzed the production of morphogens and might influence the rate of the reaction to determine the pattern.

To be honest, I didn’t understand the maths in “Chemical basis of morphogenesis,” so I read quite a few articles that summarized Turing’s reaction-diffusion system. ∂c/∂t=f(c)+D∇2c can be used to describe such a system. f(c) represents the local chemical reaction that different chemicals are reacted and formed. D is the diffusion constants. Simply put, the equation states that the distribution of a chemical is determined by the chemical reaction that generates this chemical and the diffusion of this chemical.

Program sheet written by Alan Turing during his study in fir cone pattern formation.

The computer output. Turing wrote "How did this happen? Presumably ODE" on it.

Turing's numbering on the florets of a sunflower.

Sources:

Charvoin, J. and Sadoc, J-F. “A Phyllotactic Approach to The Structure of Collagen Fibrils” 2011. <http://arxiv.org/pdf/1102.2359v2.pdf>

Copeland, B.J.(2004) “The Essential Turing, Seminal Writings in Computing, Logic, Philosophy, Artificial Intelligence, and Artificial Life plus The Secrets of Enigma.” Oxford: Clarendon Press.

Engelhardt, R. “Modeling Pattern Formation in Reaction-Diffusion System.” 1994. <http://www.robinengelhardt.info/speciale/main.pdf>

Maini, P. K. “The impact of Turing's work on pattern formation in biology.” 1-17-2007. <people.maths.ox.ac.uk/maini/PKM%20publications/172.pdf>

Swinton, J. “Watching the Daisies Grow: Turing and Fibonacci Phyllotaxis.” 2003. <user29459.vs.easily.co.uk/wp-content/uploads/2011/05/swinton.pdf>

Turing, A. M. “The Chemical Basis of Morphogensis.” 8-14-1952. <http://links.jstor.org/sici?sici=0080-4622%2819520814%29237%3A641%3C37%3ATCBOM%3E2.0.CO%3B2-I>