Figure 2. Increasing order leads to a phase transition from symmetric to asymmetric elongation in silico.

We use an order parameter that quantifies the bilaterally symmetric/asymmetric distribution of cells into the left and right PSM. (A) The average order-parameter, left-right population difference ⟨N_diff⟩ plotted versus noise (η) at fixed repulsion and adhesion parameters. (B) ⟨N_diff⟩ plotted versus adhesion F_adh, at fixed noise and repulsion. (C) ⟨N_diff⟩ plotted versus repulsion F_rep, at fixed noise and adhesion. Increasing the model parameters “angular noise” and “repulsion” increases disorder while increasing the model parameter “adhesion” increases order. The simulations show that regardless of the specific parameter that is varied, disordered motion correlates with symmetry and ordered motion correlates with asymmetry. (D) The phase boundaries between the symmetric and asymmetric phases of elongation, which are tuned by three dimensionless parameters (noise, adhesion and repulsion), are organized into a 3D phase diagram. Red lines on the surface correspond to 'constant-noise' contours, the blue lines correspond to 'constant-repulsion' contours, and the green line corresponds to a 'constant-adhesion' contour. (E) The variance of order-parameter versus noise. (F) The Binder cumulant G versus noise. Panels E and F correspond to the transition cure in panel A. The sharp minimum in the Binder cumulant near the transition point implies a 1st order transition. (G) The distribution of the order-parameter across the transition. Left: Below the threshold noise; Middle: around the threshold noise; Right: above the threshold noise. Note that near the transition point the distribution is bimodal showing coexistence of phases, which is a signature of a 1st order transition. (H) The normalized angular momentum per particle m quantifies the leftward (>0) or rightward (<0) bias in cell flow in the posterior tailbud. Time traces of m show that with disordered motion the cell flow switches between leftward and rightward flow (red), but with ordered motion anisotropic flow is persistent (blue). (I) The average angular momentum per particle⟨|m|⟩ decreases with increasing noise. These data indicate that flow approaches symmetry as disorder increases. (J) The probability distribution of angular momentum p(|m|). These data suggest that the system exists in two phases: asymmetric flow (caused by ordered motion) or symmetric flow (caused by disordered motion). In panels H–J the adhesion and repulsion parameters are F_adh = 0.75, F_rep = 30 . Other parameters are specified in Methods.