Fig. 1. Biological background and model network.

a Cell polarisation in the C. elegans embryo during the establishment (top) and maintenance (bottom) phases; sketch adapted from ref. ⁵. b Illustration of protein flux between the cytosol and membrane. As proteins detach from the membrane when phosphorylated, they cannot immediately rebind to the membrane. There is therefore an intrinsic delay before dephosphorylation permits rebinding. c The biochemical reaction network is comprised of two mutually antagonistic sets of proteins, aPARs and pPARs. Dephosphorylated (active) A₁ and P attach to the membrane with rates $k_{\mathrm{a}}^{\mathrm{on}}$ and $k_{\mathrm{p}}^{\mathrm{on}}$, respectively. Both active proteins may also detach spontaneously from the membrane with rates $k_{\mathrm{a}}^{\mathrm{off}}$ and $k_{\mathrm{p}}^{\mathrm{off}}$, respectively. A₁ acts as a scaffold protein: once bound to the membrane, it recruits A₂ with rate k_d and forms a membrane-bound hetero-dimeric aPAR complex A₁₂. The heterodimer A₁₂ may itself spontaneously detach from the membrane with rate $k_{\mathrm{a}}^{\mathrm{off}}$ and dissociate into A₂ and active A₁. Membrane-bound A₁ and A₁₂ can also be phosphorylated by P with rate k_Ap[P], thereby initiating dissociation of the aPAR complex and release of aPAR proteins into the cytosol. While reattachment of the scaffold protein A₁ is delayed by the requirement for dephosphorylation (reactivation), detached A₂ can be recruited to the membrane by membrane-bound A₁ immediately. Similarly, P is phosphorylated by the heterodimer A₁₂ at rate k_Pa[A₁₂], and is consequently released as inactive P into the cytosol. In the same way as A₁, also P must be dephosphorylated before it can bind again to the membrane. For simplicity, we take identical dephosphorylation (reactivation) rates λ for inactive A₁ and P. The ensuing reaction–diffusion equations are provided in the Methods section Model, and a table listing the values of the rate constants can be found in ref. ¹. d Sketch of the cell's geometry: prolate spheroid with long axis a and short axis b, and with short- (left) and long-axis (right) polarisation.