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Alan Turing's reaction-diffusion equations

Alan Turing is arguably one of the greatest mathematicians and computer scientists of all time, best known for his code breaking work during the Second World War. However, what is less known about him is that he also played a significant role in unlocking the key to biological patterns, from how spots on cheetahs form, to the unique nature of everyone’s fingerprints. His last ever published paper was titled “The Chemical Basis of Morphogenesis”. Morphogenesis is the development of shape and form in an organism which takes place during embryonic development. Morphogens are a type of secreted molecule which diffuses through tissues and controls the development of tissue and patterns.

Deriving the diffusion equation and linking it to Turing’s RD equations

Turing’s reaction-diffusion equations are used to model the change in concentration of such morphogens against time and space. This specific set of equations is a form of two-component RD equation, meaning that it models the interactions between two morphogens as they diffuse out. We can take ‘a’ and ‘b’ as the concentrations of an activator and inhibitor respectively. For example, this could be a chemical that changes the skin pigment of an animal, with the other stopping it. As time goes on and both chemicals react while diffusing, they eventually settle into clusters or ‘pools’ of chemicals. Where there are clusters of activators, a change is triggered- in our case, a change in pigment takes place. This is how giraffes get their mosaic-like patterns and how fish get their stripes.

By changing variables in Turing’s RD equations, such beautiful patterns can drastically differ. If the activator has a greater diffusion constant (i.e. it diffuses quicker), that means different clusters of activator are more prone to merging together, forming a type of labyrinth pattern which is commonly seen in corals. If morphogenesis occurs when the embryo is still smaller, this means that fewer pools of activator and inhibitor will form as there is not a lot of space, so the pattern that emerges has less spots. For example, morphogenesis occurs in a cow at a much earlier stage than for animals like cheetahs. This is why cows have much larger, but less complex spots, while cheetahs are decorated with much more spots. Furthermore, morphogenesis also occurs on a less apparent level including controlling how tendons are attached to bones and how cells change shape.

The reaction-diffusion theory of morphogenesis has been integral to the field of theoretical biology and it is simply astonishing how mathematics and such simple equations are able to describe the vast majority of nature.

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