21st Century Turing

In this post Dr Thomas Woolley describes one of the original, biggest and most outrageous ideas of mathematical biology. Thomas achieved his doctorate working on mathematical pattern formation within the WCMB in 2012, under the supervision of Prof. Philip Maini, Dr Ruth Baker and Dr Eamonn Gaffney. He is currently a Junior Research Fellow for St John’s College, Oxford and works on the solid mechanics behind cellular motion. However, he still likes to keep an eye on pattern formation. He runs his own mathematical outreach website as well as working as a mathematical consultant on the TV show “Dara O’Briain’s School of Hard Sums”. 

On the 7th June 1954, just a couple of weeks before his 42nd birthday, the world lost one of its greatest mathematicians. Betrayed by the very government that he had fought so hard to protect during World War II, Alan Mathison Turing (Figure 1) took his own life by eating an apple laced with cyanide. Although his tale is an extremely sad one, this post is designed to focus on his achievements and genius, rather than his depression and death.

Figure 1. A black and white vector portrait of Alan Turing. © Thomas Woolley.

Although Turing is well known as an eminent figure in computation, logic and cryptography, he is my hero because of the work he did just two years before his untimely death. Specifically, he was interested in such questions as:

• Why is there something rather than nothing?
• How are complex structures formed from simple components?
• Why doesn't everything tend to a state of uniformity?

In 1952 he published a paper, The Chemical Basis of Morphogenesis, trying to answer these questions. His ideas were revolutionary and not fully appreciated at the time. However, his paper was the start of a whole new field of mathematics, which 60 years later still uses his ideas as their foundation.

In order to produce patterns, Turing coupled two components together that individually would not lead to patterning. First, he considered a stable system of two chemicals. The chemicals reacted in such a way that if they were put in a small container the system eventually produced a uniform density of products, thus, no patterns. Second, he added the mechanism of diffusion. This meant that the chemicals could move. Incredibly, he showed that if the chemicals are put in a larger container, then diffusion could make the equilibrium state unstable, leading to a spatial pattern. This process is now known as a diffusion-driven instability.

To see how remarkable this is, consider putting a spot of ink in water, without stirring (Figure 2). The ink will diffuse throughout the water and eventually the solution will be one shade of colour. There will be no patches that are darker or lighter than the rest. However, in Turing’s mechanism diffusion does create patterns. When this happens the system is said to have self-organised and the resultant pattern is an emergent property. In this respect, Turing was many years ahead of his time: he showed that understanding the integration of the parts of a system plays a role as important (if not more so) as the identification of the parts themselves. 

Figure 2. A 3D rendering of diffusion of purple dye in water. Image used under the Creative Commons License. Original from here.

Turing termed the chemicals in his framework morphogens and hypothesised that cells would adopt a certain fate if the individual morphogen concentrations breached a certain threshold value. In this way, the spatial pattern in morphogen concentration would serve as a pre-pattern to which cells would respond by differentiating accordingly.

Of course, it is possible to produce patterns by other means. One such mechanism is known as the French flag model. Suppose that a row of cells has a constant source of morphogen at its left-hand boundary. If this morphogen is allowed to diffuse outwards from the source it produces a concentration profile, or gradient, that is higher on the left than the right (Figure 3). The cells on the left sense a higher concentration of morphogen and respond in some way, e.g. they turn red. The central cells sense an intermediate concentration and the right-hand cells sense a low concentration, and so they produce different responses, e.g. they turn white and blue, respectively. Hence, through the diffusion of a single morphogen we are able to produce the so-called French flag pattern. Importantly, we do not need the morphogens to react.

Figure 3. French flag model pattern producible by a source of diffusible chemical assumed to lie along the left-hand boundary. The source produces an exponentially decreasing profile similar to that shown in the top image. The different levels of concentration sensed by the polygons (denoting cells) are visualised by their given colour in the bottom image. The cells are red if they can sense a high concentration, white for medium concentrations and blue for low concentrations. © Thomas Woolley.

However, the French flag pattern fundamentally assumes the heterogeneous localisation of a source. Namely, if the source was not only along one side, but instead uniform everywhere then no pattern would emerge. The Turing mechanism of pattern formation needs no such assumption. The reactions and diffusion are defined to happen equally everywhere across the domain. Yet heterogeneity can be produced (Figure 4).

Figure 4. An evolving morphogen concentration producing a Turing pattern. The x-axis shows the space, whilst the y-axis shows the concentration of one of the interacting morphogens, u. The simulation starts from a random initial condition and tends to a three-peak Turing pattern. © Thomas Woolley.

Hopefully, my next post on this blog will detail how mathematical biologists use these patterns to understand animal skin pigmentation, limb formation and maybe even heart attacks.