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Pi of Life

Following up on my previous post on the use of pi in science, I wanted to write about something very fascinating I found. Not only is pi a fundamental concept for geometry, it apparently has a connection to the very building blocks of life. Biological sciences appear farthest from math, yet this enigmatic number plays a role in biological pattern formation, i.e. a leopard's spots, a zebra's stripes, a leaf's feathering etc. I have to admit this concept is still not very clear, and I need to do a lot more reading to understand it fully, but here's a high-level description.

The creation of specific organs and patterns from a single group of cells was a long-standing mystery to scientists until it was unlocked by the genius that cracked Hitler's Enigma code during World War II. Regarded as the father of computer science and artificial intelligence, Alan Turing's interests went far beyond computers into the depths of the nature of life. After the end of the war, in 1951, Turing apparently became interested in mathematical biology. He was intrigued by the development of patterns and shapes in biological organisms, which he named "morphogenesis." After thinking deeply about it for months (really not that long!), he published what is considered to be his masterpiece, "The Chemical Basis of Morphogenesis" in January 1952, shortly before his death by suicide. 

Animal patterns. Image by macrovector on Freepik

His theory was that morphogenesis was caused by a system of chemicals produced by the cells that react with each other and diffuse across space. This reaction–diffusion system could create patterns if the reaction produced two chemical substances (morphogens) that diffused at different rates. The way I saw it best described was, suppose a chemical reaction produces two morphogens: catalyst A and an inhibitor B that controls the production of A. The two morphogens diffuse through tissue at different rates, ending up with some regions having A dominating and others with B, thus producing a pattern. The activator A forms local patches of spots or stripes and also keeps producing B, while the inhibitor B diffuses farther and inhibits A, thus preventing the patches from growing too close to each other. The interaction (reaction) between the two morphogens determines the pattern. 

Computer simulation of Turing's model produces an array of patterns

Although the computing power required to prove his theory did not exist during his time, Turing used systems of partial differential equations to model these catalytic chemical reactions, which he solved by hand and predicted roughly how it could create repeating patterns. It was years later that others used his model to explain spots and stripes on different cats' furs and further concluded that it can explain other patterns like feathers, hair follicles, branching pattern of lungs, and even how an embryo grows from a bunch of cells into a patterned structure consisting of a brain, lungs, and limbs with fingers. 

And finally, where pi comes in is in the size and spacing of the pattern. Any periodic process in living organisms, whether the pattern on the surface, or even heartbeats, breathing cycle, and sleep circadian rhythms are apparently controlled by pi. 


Sources:

https://ideas.ted.com/how-the-zebra-got-its-stripes-with-alan-turing/ 
https://www.biophysics.org/blog/pi-is-encoded-in-the-patterns-of-life
https://sitn.hms.harvard.edu/flash/2012/morphogenesis/

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