Figure 1.
Level set model geometry and stress distribution.
A. The cell model assumes cylindrical symmetry. Points on the cell boundary (x∈Γ) are obtained implicitly. B. Using a viscoelastic description of the cell (Equation 3), cell boundary/membrane displacements (xm) are generated by moving the potential function (φ, not shown) according to the total stress applied, σtot (Equation 4). The spring-dashpot (K, D) elements represent the mostly elastic cortex, which moves a distance xcor. The viscous component (B) represents the cytosol, which moves a distance xcyt. Values for K, B and D were previously obtained using micropipette aspiration experiments and are given in Table 2. C. Area density maps (Dr(z) and Dz(r)), obtained by summing the cell area (in the z-r plane) one axis at a time (Equation 5). The resultant adhesion map, shown overlaid on the cell shape, is obtained by multiplying these two together. D. Protrusive stress is assume to work in the z-direction away from the furrow according to Equation 7, but only the component normal to the boundary is used. E. Geometry of contractile stress. Though myosin II acts radially, its effect is to reduce the circumference, and hence radius. This can be recreated by applying a stress (σmyo) inwards radially (shown in gray).
Figure 2.
Simulations of interphase cells under various stresses.
A. Simulation of a non-adherent cell, initialized as an ellipsoid, experiencing only passive forces. As expected, the cell rounds up relatively quickly. B. Stresses due to adhesion and protrusion were incorporated into the model to simulate traction-mediated cytofission (Video S1). The stress color scale applies for both panels A and B. Negative stress is inward-directed. C. Furrow ingression dynamics of the cell for the simulation in panel B. The point in time when the furrow diameter and length are equal is defined as the cross-over time (tX) and this distance is known as the cross-over distance. The relative furrow diameter is the ratio of furrow diameter divided by the cross-over distance.
Table 1.
Simulations considered.
Figure 3.
Simulation of myoII null cells.
Morphological changes in a model where there is a spatial difference in cortical tension for both non-adherent (A) and adherent (B) cells (Video S2). Simulation times are from the initial spherical shape. The distribution of the stresses in the adherent case is shown in Fig. S5. C. Experimental data are taken from myoII null cells dividing on a surface. Experimental times are from Video S3. Scale bar denotes 10 µm. D. Comparison of the furrow thinning trajectory. The experimental data represents mean ± SEM and are taken from reference [12]. To compare the shapes at comparable times, time is rescaled so that the cross-over points coincided (Methods). E. Curvature in a simulation of adherent cells. The curvature at the furrow initially decreases slowly but reaches a minimum before increasing. This causes the stress to increase further increasing curvature and thereby closing a positive feedback loop which leads to rapid cell ingression. The curvature of the daughter cell changes relatively little.
Figure 4.
Cell division in the presence of a contractile force.
Simulation of dividing cells in both non-adherent (A; Video S4) and adherent conditions (B; Video S5). In the latter we also considered the effect of strain-stiffening as defined by Equation 11 (C; Video S6). Simulation times are from the initial spherical shape. D. Experimental comparison is with WT cells. Experimental times are from Video S7. Scale bar denotes 10 µm. E. Comparison of furrow thinning trajectory. Experimental data represent the mean ± SEM and are taken from reference [12]. We rescaled the time axis to compare the shapes at comparable times, by shifting the time so that the cross-over times are denoted as 0 s (Methods). The elapsed time between the start of the simulation and the cross-over time for each simulation is given in the legend. F. Pole-to-pole distance as a function of time.
Figure 5.
Distribution of stresses acting on the cell.
A. Temporal and spatial profiles of different stresses in WT simulation at various time points. Negative stresses denote inward-directed forces. B. Summary of phenotypes observed in the simulations separated by the different conditions applied. Phase 1 denotes the initial breaking of spherical symmetry. Phase 2 is the progression into a dumb-bell shape.
Table 2.
Nominal Parameters.
Table 3.
Algorithm steps.