# Alan Turing and fibonacci phyllotaxis

- 14:00 11th March 2016 ( week 8, Hilary Term 2016 )Mathematical Institute, Room L3

Turing's seminal 1952 paper on morphogenesis is widely known. Less well known is that he continued to work on his morphogenetic theory over the last years of his life, using the new Manchester computer to generate solutions to reaction-diffusion systems. Among other things, he claimed at one point to be able to explain the phenomenon of 'Fibonacci phyllotaxis': the appearance of Fibonacci numbers in the structures of plants. This work studied the properties of lattices arising from repeated placements of mutually repelling organs, and aimed to show that as the patterns deform under growth the stable branch of the resulting bifurcation pattern maintains Fibonacci structure. This work was incomplete at his death and it was left to later mathematicians and physicists, unaware of it, to independently establish the generic nature of Fibonacci phyllotaxis in exactly this mathematical sense.

Despite Turing's original ambition, it turned out that much developmental patterning could be explained not as a reaction-diffusion outcome, but as a digital hierarchy of binary genetic switches, and it has been argued by prominent evolutionary developmental biologists (eg Carroll 2006) that Turing's theories were beautiful but wrong. In the mean time, the molecular developmental biology of plant organ placement has been increasingly well characterised, with little input from or understanding generated by the known mathematical theories. However sunflowers really do commonly have 89 spirals in their seedheads and it seems implausible that this fact will be explicable on a molecular basis without invoking some version of this mathematics, and in which Turing's insight (if the credit is indeed his) will be essential.

Given this mathematical heritage it is simple to sketch out a mathematical theory for the development of the sunflower but this very simplicity makes testing in a biological context difficult: asserting that a model and a sunflower both have Fibonacci structure is unhelpful. One key approach will be to examine how plant developmental structure responds to perturbation and defect. Almost all of the sunflower heads previously described have, not unnaturally, been perfect specimens with clear Fibonacci structure, but these are not universal and the ways in which they depart from order should be very useful for model testing. Thanks to a novel set of sunflower head structures analysed as part of a citizen science experiment run by the Museum of Science and Industry, Manchester we are able to articulate some of the challenges these models face. Fibonacci structure in plants should provide a fertile ground for a new generation of mathematical biology.