Engineered Patterns in Biology I: Intro to Turing Pattern Formation

What is this post about?

This post will cover an overview of pattern formation, specifically Turing pattern formation in a non-mathematical way. The whole point here is to appreciate the beauty of what can Turing's reaction-diffusion model describe. If you are excited enough about the topic and want to cover some technicalities, a subsequent post will cover my Master's thesis work on the design of genetic circuits from first principles that makes use of Turing's reaction-diffusion model.

Pattern formation in Biology

It is best to start by explaining pattern formation in biology. The easiest and most obvious way to see patterns is within animals. Zebras; fish; cheetahs are some of the few animal examples that exhibit pigment patterns on their skin [1]. If you accept the fact that pigments are no more than proteins expressed within a group of cells [2], one can reasonably ask the question: How can cells determine their relative position to other cells so they end up expressing different pigments uniformly? To understand the previous question, one more question needs to be asked; how do cells communicate with each other? By answering these questions, a fuller idea of Turing's pattern formation (from a non-mathematical perspective) will hopefully be achieved.

Figure 1: Pattern formation in fish [1].

In 1952, Alan Turing published groundbreaking work in biology and genetics. The paper's title is The Chemical Basis of Morphogenesis [3]. He proposed that cells communicate with each other by secreting signalling molecules, we shall use the name morphogens to describe these signalling molecules, as proposed by Turing. This theory assumes two types of morphogens, an activator morphogen and an inhibitor one. A higher concentration of the former leads to activation or expression of proteins (see this article for a better understanding of gene expression). By contrast, a high concentration of the latter results in inhibiting gene expression. It does not matter which proteins get expressed or inhibited, the point of this theory is to propose a communication mechanism consisting of two morphogens that control gene expression. The activator morphogen activates itself and the inhibiting morphogen (as in increase in concentration), while the inhibitor inhibits the reduces the activation (figure 2).

Figure 2: Two interacting morphogens are shown on the left, the activator is the red circle, while the inhibitor is the blue one.
The Turing reaction gives rise to a non-homogenous distribution of morphogens, resulting in complex patterns [1].

At the time when Turing published his work, the theory was a pure mathematical hypothesis with no experimental proof of the existence of morphogens. Recently, however, experimental biologists have been supporting the reaction-diffusion model. One such study experimented with two morphogens responsible for the formation of patterns in mice's palates [4]; changing the ratio between the activator and the inhibitor changes the patterns formed (figure 3)

Figure 3: Part (a): a wild-type mouse’s palate shows regularly inter-spaced rugae patterns. Part (b) shows the pattern when FGF (fibroblast growth factor) proteins are increased in Spry1 and Spry2 genes [4].

The dynamics of these morphogens vary in both time and space – think about these morphogens as being secreted from one cell and received by another– which can be described mathematically in the form:

 A more thorough explanation of the equations will be given in following posts. Meanwhile, it is worth keeping in mind that in order for Turing patterns to occur, it is important that the inhibitor diffuses faster than the activator (e.g. Dv > Du )

Forward engineering

Since genetic engineering and synthetic biology are becoming accessible lately, one can imagine the wonderful engineering opportunities using the reaction-diffusion model as a first principle rule for engineering living organism, or any programmable machine for that matter. As an example of the latter, it is quite possible to use this rather abstract mathematical formula of pattern formation to program robots to self-form without giving specific instructions on which patterns to form [5]. That means even if the pattern is broken between the robots, they communicate with each other to "repair" and reform the pattern again, as seen in this video:

In the next post, I will give a deeper insight into the maths behind the reaction-diffusion theory. Once the mathematics is understood, I will dive into the design principles of genetic circuits and explore designing a gene regulatory network to produce patterns in bacteria.








[1] Kondo, S. and Miura, T., 2010. Reaction-diffusion model as a framework for understanding biological pattern formation. science, 329(5999), pp.1616-1620.

[2] Encyclopedia Britannica. (2020). Chromatophore | biological pigment. [online] Available at: https://www.britannica.com/science/chromatophore [Accessed 9 Feb. 2020].

[3] Turing, A.M., 1990. The chemical basis of morphogenesis. Bulletin of mathematical biology, 52(1-2), pp.153-197.

[4] Economou, A.D., Ohazama, A., Porntaveetus, T., Sharpe, P.T., Kondo, S., Basson, M.A., Gritli-Linde, A., Cobourne, M.T. and Green, J.B., 2012. Periodic stripe formation by a Turing mechanism operating at growth zones in the mammalian palate. Nature genetics, 44(3), p.348.

[5] Slavkov, I., Carrillo-Zapata, D., Carranza, N., Diego, X., Jansson, F., Kaandorp, J., Hauert, S. and Sharpe, J., 2018. Morphogenesis in robot swarms. Science Robotics, 3(25), p.eaau9178.