Computer simulations of mathematical Model 1.

(A) Schematic summarises the reaction regime of Model 1, a Type I Turing system. The biology underlying this model was originally proposed by Takao and colleagues (2019). Although it is not possible to precisely assign the activator and inhibitor relationships depicted in this schematic to specific terms in the reaction equations (Eq 3 and Eq 4, main text) we can approximately see that A (non-phosphorylated PLK4, red) promotes the production of more A through self-assembly, and the production of I (phosphorylated PLK4, denoted PLK4*, blue) by stimulating the self-phosphorylation of A as the concentration of A increases. I inhibits the production of A by converting it to I via phosphorylation, and I inhibits its own accumulation by promoting its own degradation/dissociation. (B–D) Graphs show how the levels of the activator species (red lines) and the inhibitor species (blue lines) change over time (arbitrary units) at the centriole surface (modelled as a continuous ring stretched out as a line between 0–2π) in computer simulations. The rate parameters for each simulation are defined in the main text and can lead to symmetry breaking to a single peak (B), or to multiple peaks when cytoplasmic PLK4 levels are increased (C), or to no symmetry breaking at all (with the accumulation of relatively high levels of activator and inhibitor species spread uniformly around the centriole surface) when PLK4 kinase activity is reduced (D). The underlying data for this figure can be found in S1 Data.