Pattern Formation

Artist's impression

The picture above, by artist Ricardo Solís, gives an idea for how a tropical fish came to have its distinctive patterns. The real-life process is just as exciting and mysterious! Can we use mathematical models to understand how these patterns are formed? Can the biological processes that we know and trust be combined to produce the stripes on a fish?

We journey back to the year 1952 and to the University of Manchester, where a famous mathematician was also interested in these questions. Alan Turing is better known for breaking the Enigma codes, especially with the recent movie “The Imitation Game”. But in 1952 he published an important piece of work into the formation of patterns: “The Chemical Basis of Morphogenesis”. In this, he describes a mathematical model for how patterns can arise.

First, what exactly do we  mean by a “pattern”? There are some images that come to mind when we think of what a pattern is, but how can we make this mathematically precise? It’s important to think about why and how patterns form, and to predict their structure.

Caption
Pattern design from White Stuff clothing, see below for info on the maths equations.

We define a pattern to be a temporally stable but spatially heterogeneous mixture of substances. Let’s pick apart what this means in two stages:

  1. Spatially heterogeneous means “different colours in different places” – it’s not much of a pattern if it’s the same colour everywhere
  2. Temporally stable means it stays that way – we wouldn’t call something a pattern if it started out patterned but the colours rapidly mixed into one

To make this definition mathematically precise we have to quantify these factors. For example, we might say that the colours have to be distinguishable by the average human eye and the pattern has to have no visible changes over the course of one day.

Turing showed that patterns can emerge from an initially homogeneous mix of substances by a reaction-diffusion system with one activator and one inhibitor. Equations for this process were given in the later paper from 1972 “A Theory of Biological Pattern Formation” by Gierer and Meinhardt. The actual equations are in the picture above.

The equations are pretty simple. We have a stable system, and then two diffusion terms D_a and D_h added on, and somehow we create patterns. Turing’s insight is summarized in this article by Philip Maini:

One can take a system which has stabilizing reaction kinetics, add to it diffusion (which we also think of as stabilizing) and the resultant system is unstable!

The biological relevance of the model today may be disputed. But, from a mathematical perspective, it is fascinating to see how such a simple system of two equations can produce some of the patterns we observe in nature:

With slight modifications to the model, patterns can be produced that look like those seen in nature. This collection of comparisons is from the paper
With slight modifications to the model, we can make patterns that look like those in nature. This collection of comparisons is from the paper “Reaction-Diffusion Model as a Framework for Understanding Biological Pattern Formation” (see the paper to find out where each individual picture came from).

I first learnt about the Turing “Reaction-Diffusion” model for pattern formation at a summer school hosted by the Altschuler-Wu Lab – thanks to them for this opportunity!

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