## The biology and some of the mathematics of stripe patterns

### October 5, 2011

When mathematics and biology come together, interesting things can happen. For a Radboud University Nijmegen Honours Academy course entitled “Mathematics and Harmony“, given by Dr. Bernd Souvignier, I wrote a piece on the mathematical model that describes the chemical gradients which determine biological pattern formation in nature. This model is mainly based on the British mathematician Alan Turing. He was one of the founders of modern informatics, but shortly before his tragic death, also published one single paper on chemistry and biology. Published in 1952 and entitled “The Chemical Basis of Morphogenesis”, in this paper some easy to understand principles concerning chemical gradient formation that determine pattern formation are described in a relatively complex mathematical manner. In my work on this topic, I mainly concentrated on the conceptional basis behind the mathematics, since my field of study is more biochemically orientated. Nevertheless this early combination of biology, chemistry and mathematics is extremely interesting and is still keeping scientists from a number of disciplines busy.

With the help of reaction-diffusion equations, which describe the interaction and movements of chemical compounds through structures, Alan Turing postulated his hypothesis of pattern formation and morphogenesis. Figure 1 gives an easy-to-understand overview of Turing’s principle.

Figure 1: The elements and meaning of one version of a reaction-diffusion equation which was also used by Alan Turing in his 1952 publication on “The Chemical Basis of Morphogenesis”. A “thank you” to Kele’s Science blog for the “simplification of the complex“.

The above mentioned formulas helps to describe how chemical compounds, now known as transcription factors or morphogens, can interact with each other in relatively simple feed-forward or inhibition loops to create biological patterns such as stripes, spirals and much more from an original homogeneous and uniform begin situation. My small article on the topic, which can be found as a pdf document below, describes some of the implication of Turing’s model on modern-day knowledge of morphogenesis and connects this knowledge to other existing theories. In some sort of timeline research is reviewed which describes applications of Turing’s theories. Finally, an outlook in the future states that Turing’s ideas are also applicable in “real” 3D systems. A critical conclusion on the conformity of his theory with other existing morphogen theories follows as well. The article is in Dutch (all figures have journal references, so it’s also interesting for English speakers) and can be accessed here:

WiskundeHarmonie_Essay_J.Wilbertz