Figure 4. Physical description of spontaneous symmetry breaking.

(A) Principal tenets of physical description of spontaneous symmetry breaking mechanism; see Appendix for details. (B, D, F, H) Phase diagrams showing the development of 0, 1 or 2 PAR-2 domains, as indicated, as a function of the preferential PAR-2 attachment rate $\alpha /l$ and the induced decrease of cortical contractility $\beta \tau /{\rho }_P^0$ in control (B) or air-1(RNAi) (D) embryos, as well as in control or air-1(RNAi) embryos in triangular chambers (F and H, respectively). (C, E, G, I) Simulation kymographs of PAR-2 distribution corresponding to the juxtaposed phase diagrams. Moderate values for $\alpha$ and $\beta$, away from the extremes, were chosen for the numerical simulation: $\alpha /l$=1.2 $\beta \tau /{\rho }_P^0$ 100.

Figure 4—figure supplement 1. Simulations of symmetry breaking and relaxation to the poles.

(A) Numerical simulation of cortical distributions of anterior (red) and posterior (green) PAR proteins distribution dynamics in control embryos. (B) Simulation kymograph of PAR-2 distribution in control embryos, mimicking an experiment in which all densities are shifted 55°C away from the posterior pole after polarity establishment (Mittasch et al., 2018). Note correction of the PAR-2 domain over time towards the posterior pole. Parameters: α/l = 1.2 and β τ⁄(ρP0)=100. (C) Control embryo expressing RFP::NMY-2 (red), as well as GFP::PAR-2 and GFP::SAS-7 (both green) exemplifying a symmetry breaking event away from the posterior pole (t = 00:00), as evidenced by NMY-2 clearing, followed by correction of the PAR-2 domain to the posterior pole (t = 05:00, arrow indicates centrosome trajectory). (D–F) Standard deviation of the distance from the pole (0 on the X axis) to the center of the PAR-2 domain (posterior or anterior, as indicated) over time (t = 0 corresponds to the initiation of the PAR-2 domain) in randomly selected control and air-1(RNAi) embryos (N = 15 each). Note that the offset of the PAR-2 initiation site did not differ in control and air-1(RNAi) embryos on the posterior side, whereas it arose with a larger offset on the anterior side, possibly because of invaginations resulting from the polar body extrusion that give rise to pronounced membrane curvatures. (G) Numerical simulation of cortical distributions of anterior (red) and posterior (green) PAR proteins distribution dynamics in air-1(RNAi) embryos.