PMC

Cytokinesis duration is independent of initial cell size. Furrow radius r_f as a function of time t = T_a for four initial cell radii R₀ = 0.5, 1, 2, and 4. (Inset) Corresponding Gaussian activity signals of width w proportional to R₀, plotted as a function of the membrane midline contour length s. To see this figure in color, go online.

Influence of turnover on constriction. Furrow radius r_f = R₀ as a function of normalized time t = T_a for three turnover rates: k_dT_a = 30 (①), k_dT_a = 40 (②), and k_dT_a = 80 (③). (Inset) Corresponding steady-state membrane thickness e₁ = e₀ along the rescaled contour length s = L from equator (L is the total membrane midline length). To see this figure in color, go online.

Furrow constriction dynamics. Time evolution of the furrow radius (r_f = R₀), and pole-to-pole half distance (L_p = R₀). Numerical results (line) are compared to experimental measurements of a sand-dollar zygote (points, data from the same DIC microscopy images of ). (Vertical dashed lines) Delimiters of the four phases of constriction described in the text and numbered from 0 to 3. (Inset) Equatorial signal I(t) applied as a function of time. To see this figure in color, go online.

Constriction completion and failure. (A) Bifurcation diagram representing the final furrow radius r_f^∞/R₀ as a function of the amplitude of equatorial overactivity δζ^∞/ζ₀. The diagram displays a jump from constriction failure to completion for a critical amplitude δζ^∞/ζ₀ ≈ 40. Final cell shapes are plotted for the six activity signals ζ, of amplitudes δζ^∞/ζ₀ = 10 (❶), 25 (❷), 40 (❸), 50 (❹), 75 (❺), and 100 (❻) as represented (inset) as a function of the contour length from equator s/L along the membrane (of length L). (Arrows) Hysteresis loop. Starting from a divided state above the threshold (❹, for example), we decrease the equatorial signal: the cell remains divided, unless the signal is dropped down to 0—the point at which it goes back to the spherical state. (B) Furrow radius evolution r_f/R₀ as a function of time t/T_a for the six signals ζ (represented in panel A, inset). To see this figure in color, go online.

Constriction dynamics in scaling. Normalized furrow radius evolution r_f= R₀ with time t = T_a. (A) For κ = 0.1, 0.25, 0.4, 0.5, 0.75, and 1, with constant λ = 0.1. For κ ≲ 0.4, the furrow radius reaches a plateau indicating constriction failure, whereas for κ ≳ 0.4 constriction is complete and its speed increases with κ. (B) For e_f = e₀ between 1 and 4, keeping ζ_f/ζ₀ = 8 and w = R₀ = 0.1 constant. Constriction slows down when e_f = e₀ decreases from 4 to 1.5, and can even fail when it drops to 1. (C) For four initial cell radii R₀ = 0.5, 1, 2, and 4, where the ring width w is increased proportionally, w = 0.05, 0.1, 0.2, and 0.4. The values e_f = e₀ = 2 and ζ_f/ζ₀ = 8 are maintained constant. The rate of constriction increases proportionally to the ring width, leading to the same constriction duration for the four cell sizes. (D) For a cell with dissipation due to the ring constriction only and with dissipation due to poles stretching and ring constriction (ζ_f/ζ₀ = 8, e_f = e₀ = 2, w = R₀ = 0.1). (Dashed line) Constriction of an isolated ring (no poles) fitted with the exponential function e^−t/τ with τ = 2η/ζ_f Δμ. To see this figure in color, go online.

Scaling model. (A) Sketch of the minimal geometry proposed by Yoneda and Dan (): Two portions of sphere of surface A_p and surface tension N^a₀ are pinched by an equatorial ring of radius r_f, of width w, and of line tension γ. The opening angle θ characterizes the constriction state of the cell and the cytoplasmic volume-enclosed V₀ is conserved. (B) Mechanical energy profile E = E₀ as a function of the constriction state θ for four values of κ = γ/2R₀N^a₀. Local minima of the energy correspond to equilibrium states (darker points), above which are plotted the corresponding cell shapes. (C) Bifurcation diagram representing the final furrow radius r^∞_f = R₀ as a function of the control parameter κ. The upper branch and the branch r_f = 0 are stable branches, but one branch (dot-dashed) is unstable. The critical point is a saddle-node, and the bifurcation classically exhibits an hysteresis (see arrows). Final cell shapes, starting from a spherical cell, are plotted for the six following values of the control parameter: κ = 0, 0.1, 0.25, 0.4, 0.5, and 0.75. To see this figure in color, go online.

(Left) Numerical cell shape and cortex thickness evolution. (t₀) Initial spherical cortex of radius R₀ and main ingredients of the model. The membrane is axisymmetric around the axis e_z and is subjected to internal tensions N_s and N_φ in its axial t and azimuthal e_φ principal directions, and to the cytoplasmic pressure P along its normal n. The acto-myosin layer of initial thickness e₀ undergoes permanent turnover. Approximately 100 Lagrangian nodes are represented to follow the tangential membrane deformation over time (not all simulations nodes are shown). (t₁, t₂, and t₃) Cell cortex snapshots at successive times of constriction, in response to the rescaled myosin activity signal ζ/ζ_max illustrated by the color shading. The value r_f is the furrow radius and L_p is the half pole-to-pole distance. Cortical flows along the membrane are represented by arrows of size proportional to the local tangent velocity. (Right) DIC microscopy images of a sand-dollar zygote (Dendraster) deprived of its hyaline layer and jelly coat at four equivalent times of furrow constriction. The cell is not flattened and scale bar is 20 μm. (Credits: G. Von Dassow.) To see this figure in color, go online.