Reaction-Diffusion Systems – The physics and chemistry of spots, stripes, and swirls

Butterfly wing (photo from Pixabay), Blue discus stripes (photographer Patrick Farrelly), giraffe hide (photo from Unsplash), Honeycomb toby spots (photographer Gary McKinney), leopard skin (photo from Pixabay), and zebra coat (photo from Pixabay), all exhibit classic Turing Patterns

Have you ever stopped to consider the stripes on a zebra’s hide, the beautiful, concentric spots on a butterfly’s wing, or the complex, undulating patterns of certain fish? What if all these phenomena were governed by the same law? What exactly is going on here?

In 1952 Alan Turning – yes that Alan Turing – published a paper entitled The Chemical Basis of Morphogenesis (Turing, 1952). Morphogenesis is a biological term for how cells, tissues and organisms develop their shape. Turning proposed that these patterns, now called Turing Patterns, can arise naturally from an interplay between the differential diffusion of two chemical species and a chemical reaction between those species. Let’s break down those concepts:

Diffusion refers to the spontaneous movement of particles from a region of high to low concentration. It is governed by Fick’s Law:

If these symbols don’t mean anything to you, that’s okay! The term on the left is the partial derivative with respect to time of the concentration – it measures how much the concentration changes over time at certain point. The triangle on the right is called the Laplacian – it indicates how much the concentration at that point deviates from its neighbors. D is the Diffusion Coefficient; it measures how quickly this diffusion process happens. Fick’s law, in essence, says that if concentration varies across space, particles move to smooth out these inhomogeneities. Once the concentration is uniform, the system reaches equilibrium and stops changing.

A chemical reaction is a process in which one or more substance (reactant) is converted into a different substance (product). For example, take two chemicals A and B. they combine in a certain way, in a certain ratio, under a certain set of conditions, to produce a new species, C. We can write this in equation form as 2A+3B→5C

Great – now we’re all set to understand Turing’s work.  Turing proposed that when morphogens (chemical species) with different diffusion coefficients react and diffuse in embryonic tissue, patterns can spontaneously emerge. This is rather surprising – we imagine diffusion as having a stabilizing influence on our system – yet Turing’s model now shows that, depending on conditions, this complex interplay of diffusion and reaction can produce spots, stripes, swirls and many other of the patterns we see in nature.

These so-called reaction-diffusion systems can be represented mathematically as

The two terms on the right are from our familiar Fick’s equation, which have been modified by R(A) to account for local reactions. Several Turing patterns are simulated below using the Gray-Scott Model, a simple implementation of Turning’s morphogen theory, for the following system:

Our system contains two chemical morphogens, A and B, (concentrations denoted as a and b respectively), with a simple reaction that can happen between them:

In order to keep the reaction going, A is added at a feed rate f and B is removed at a kill rate k.

Each chemical has a diffusion coefficient Da and Db. The Gray-Scott Model uses the following two equations

The first additional term describes change in concentrations as a result of the chemical reaction between A and B. The second describes the change in contrations due to the feed and kill rates f and k. For most combinations of f and f, nothing interesting happens.  The system fills completely with A or B. However, for certain combinations, familiar patterns emerge… (I highly recommend checking out
Karl Sim’s Interactive Reaction Diffusion Model, tweaking f and k, and seeing what cool patterns you can come up with!)

In these images, the colour gradient corresponds to the concetration of morphogen B, who’s diffusion coefficient is smaller than that of morphogen A.

In his 2012 paper, mathematical biologist James D. Murray used the analogy of a sweaty grasshopper to explain the Turing mechanism (Murray, 2003): Imagine a dry field evenly distributed with grasshoppers. Fire catches in a few small parts and quickly spreads. The heat causes the grasshoppers to persperate, damening the grass around them. After the fire has burnt out, instead of an uniformly cremated field, there are spots of unburnt grass which have restricted and channelled the charred areas into finite domains.

While the sweaty grasshopper analogy is a rather fanciful one, real, recent research has been done to try verify Turing’s claims. A paper was published in 2019 above the potential for the VEGFC protein to form Turing patterns in Zebrafish embryos (Wartlick et al., 2019). Is still debated whether this model can account for all these striking and unusual patterns in nature  – Turing himself acknowledged the simplicity of the model in his paper, cautiously extending his theory to the tentacle rings of microscopic Hydra. That being said, the reaction-diffusion model has been used accurately model many phenomena across a wide range of fields, from ecological invasions, crystal growth, fission waves in nuclear materials, non-linear optics, wound healing, and even psychedelic hallucinations! While the physical/chemical/ecological principles underlying these phenomena may differ, the math describing these systems is the same!

Several other examples of Turing patterns. On the left, winds create long zebra-like stripes on the desert’s surface (photo from Adobe stock), in the middle, the dappled vegetation growth of the Tiger Bush (photographer Nicolas Barbier) , in the center, a ferrofluid under the influence magnetic field (image from Chemical bouillon), all exhibiting classic Turing patterns.

References

• 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.

• Murray, J. D. (2003). Mathematical Biology II: Spatial Models and Biomedical Applications. Springer.

• Pearson, J. E. (1993). Complex patterns in a simple system. Science, 261(5118), 189–192.

• Wartlick, O., et al. (2019). VEGFC forms a Turing pattern in the zebrafish tail fin. Nature Cell Biology.

• Ermentrout, G. B., & Cowan, J. D. (1979). A mathematical theory of visual hallucination patterns. Biological Cybernetics, 34(3), 137–150.

• Nervous System. (n.d.). Reaction Diffusion Simulation Workshop. Retrieved from https://n-e-r-v-o-u-s.com/education/simulation/ethworkshop.php

• Navarro, P. (n.d.). The Gray-Scott model in Python and Fortran. Retrieved from https://pnavaro.github.io/python-fortran/06.gray-scott-model.html

• Holmes, E. E.; Lewis, M. A.; Banks, J. E.; Veit, R. R. (1994). “Partial Differential Equations in Ecology: Spatial Interactions and Population Dynamics” Wiley: 17–29.
• Osborne, Andrew G.; Deinert, Mark R. (October 2021). “Stability instability and Hopf bifurcation in fission waves”. Cell Reports Physical Science.
•  Sherratt, J. A.; Murray, J. D. (July 23, 1990). “Models of epidermal wound healing”. Proceedings of the Royal Society B: Biological Sciences. The Royal Society: 29–36
• Gray-Scott Model of a Reaction-Diffusion System, Pattern Formation, by Katharina Käfer and Mirjam Schulz: Retrieved from https://itp.uni-frankfurt.de/~gros/StudentProjects/Projects_2020/projekt_schulz_kaefer/#john-pearson
• Turing’s Cake (and other wrinkly math),\T by Physics for the Birds: Retrieved from https://www.youtube.com/watch?v=icQ_BTtNGEo