Mechanical conditions for stable symmetric cell constriction

Cell constriction is a decisive step for division in many cells. However, its physical pathway remains poorly understood, calling for a quantitative analysis of the forces required in different cytokinetic scenarios. Using a model cell composed by a flexible membrane (actin cortex and cell membrane) that encloses the cytoplasm, we study the mechanical conditions necessary for stable symmetric constriction under radial equatorial forces using analytical and numerical methods. We deduce that stable symmetric constriction requires positive effective spontaneous curvature, while spontaneous constriction requires a spontaneous curvature higher than the characteristic inverse cell size. Surface tension reduction (for example by actin cortex growth and membrane trafficking) increases the stability and spontaneity of cellular constriction. A reduction of external pressure also increases stability and spontaneity. Cells with prolate lobes (elongated cells) require lower stabilization forces than oblate-shaped cells (discocytes). We also show that the stability and spontaneity of symmetric constriction increase as constriction progresses. Our quantitative results settle the physical requirements for stable cytokinesis, defining a quantitative framework to analyze the mechanical role of the different constriction machinery and cytokinetic pathways found in real cells, so contributing to a deeper quantitative understanding of the physical mechanism of the cell division process.

Figure 1

Real cells and minimal model cell during cytokinesis. (a) HeLa cell undergoing cell constriction (cytokinesis). Upper panel: Formation of the cleavage furrow (optical microscopy image, courtesy of A. Siegel and H. C. Smith, University of Rochester). Lower panel: Scanning electron micrograph during the final stage of constriction (courtesy of A. Wilde, University of Toronto). (b) Profile of a constricted cell and its characteristic parameters. The force $F_m$ at the polar radius $R_m$ keeps the size of the lobes; the constriction force $F_c$ directed toward the cell interior causes constriction to constriction radius $R_c$; and the stabilization force $F_s$ directed to the middle of the cell $(\mathrm{\Delta }x=0)$ symmetrizes the division. External stress fields applied either transversally, as a pressure $\mathrm{\Delta }p$, or longitudinally, as a lateral membrane tension $\mathrm{\Sigma }$, complete the main systemic factors defining the shape of the cell, whose intrinsic mechanical properties are represented by an effective bending modulus $\kappa$ and an effective spontaneous curvature $C_0$.

Figure 2

Symmetrically constricted vesicle. (a) Profile $R\left(x\right)$ of a symmetrically constricted vesicle with the axis of symmetry along the $x$ axis and its characteristic parameters. Left polar cap is shaded in blue and the left half of the constriction zone is shaded in yellow. (b) Surface obtained from the revolution around the $x$ axis.

Figure 3

Constraints to the constriction pathway. Upper panel: Model shapes for prolate, spherical, and oblate cells at different constriction stages (at $s=0$: initially unconstricted vesicle; at $s=0.5$: midpoint in the constriction process; at $s=1$: maximum constriction; and at the finally fissioned state). Three different constraints to the constriction pathway are considered: constant polar radius (black shadow), constant volume constriction (red shadow), and constant area constriction (blue shadow). Lower panel: Constriction at constant polar radius requires an increase of 100% in membrane area and volume (black columns); constriction at constant volume requires a decrease in polar radius of around 20% and an increase in membrane area of around 26% (red columns); and constriction at constant area requires a decrease in both polar radius and volume of around 30% (blue columns). The cell shapes and the values in this figure were computed using the numerical solution.

Figure 4

Stabilization and constriction forces for different scenarios. Stabilization and constriction forces, $F_s$ and $F_c$, at the beginning of constriction $(s=0.2)$ as a function of the adimensionalized tension $\mathrm{\Sigma }{R}_m^2/\kappa$ ($y$ axis) and the adimensionalized pressure $\mathrm{\Delta }p{R}_m^3/\kappa$ ($x$ axis) for different values of the adimensionalized spontaneous curvature: $C_0{R}_m=-0.8$ (a), $C_0{R}_m=-0.3$ (b), $C_0{R}_m=0.3$ (c), $C_0{R}_m=0.8$ (d), $C_0{R}_m=1.3$ (e), and $C_0{R}_m=1.8$ (f). Conditions predicting unstable and nonspontaneous constrictions are shaded in red; conditions predicting stable but nonspontaneous constrictions are shaded in yellow; conditions predicting stable and spontaneous constrictions are shaded in green; and conditions under which equatorial constriction is impossible (imaginary analytical results and no numerical solution) are shaded in black. Stable symmetric constriction requires positive spontaneous curvature [(c)–(f)] while spontaneous symmetric constriction requires that the spontaneous curvature be higher than the characteristic radius of the cell [(e)–(f)]. Panel (g) graphically summarizes the results showing the different regimes, their locations, and separation conditions; cell shapes of three illustrative cases are represented. High surface tensions $\mathrm{\Sigma }$ and high external pressures $\mathrm{\Delta }p$ give rise to oblate vesicles, which require large constriction and stabilization forces. In contrast, if $\mathrm{\Sigma }$ and $\mathrm{\Delta }p$ are low, our model predicts prolate shapes requiring smaller forces. Dashed lines correspond to spherical vesicles [curves with $\mathrm{\Lambda }=1$, Eq. (6)], the case that separates prolates $(0<\mathrm{\Lambda }<1)$ and oblates ($\mathrm{\Lambda }>1$) (note that the curve with $\mathrm{\Lambda }=1$ for $C_0{R}_m=-0.8$ is not in the region shown). The stabilization force $F_s$ and the constriction force $F_c$ were determined assuming constant volume constriction, but the same conclusions apply for constant area constriction. This figure represents the results obtained with the analytical solutions, verified by comparison with the numerical solutions in a representative grid in the plotted ranges.

Figure 5

Energy landscape during cell constriction for different scenarios. Energy of the asymmetric configuration as a function of the constriction stage $s$ and the displacement of the constriction ring from the middle point $\mathrm{\Delta }x/R_0$ for $\mathrm{\Sigma }{R}_0^2/\kappa =\mathrm{\Delta }p{R}_0^3/\kappa =0.25$ and for $C_0{R}_0=-0.3$ (a), 0.3 (b), and 1.8 (c), with $R_0$ the initial maximum radius. Dotted line represents the symmetric constriction pathway. Unstable, stable, and spontaneous regions are shaded, respectively, in red, yellow, and green. Membranes characterized by negatives values of $C_0$ (a) present unstable (convex profile) and nonspontaneous constriction (uphill energy) during the whole process. If, instead, $C_0$ is positive but lower than the characteristic inverse cell size (b), the constriction is nonspontaneous (uphill energy) but reaches stability at some point of the division (concave profile). Finally, when $C_0$ is higher than the characteristic inverse cell size (c), constriction starts as a nonspontaneous and unstable process but becomes stable (concave profile) and spontaneous (downhill energy) at final stages. These results were determined assuming constant volume constriction, but the same conclusions apply for constant area constriction. This figure represents the results obtained with the analytical solutions, verified by comparison with the numerical solutions in a representative grid in the plotted ranges.

Figure 6

Constant volume constriction. Decrease of the polar radius (left-hand column) and increase of membrane area (right-hand column) along the constriction pathway maintaining a constant volume. Comparison between analytical and numerical results. (a) Influence of the surface tension $\mathrm{\Sigma }$. (b) Influence of the osmotic pressure $\mathrm{\Delta }p$. (c) Influence of the spontaneous curvature of the membrane $C_0$. Constriction process at constant volume requires a final decrease in polar radius of around 20% and a final increase in the membrane area of around 26%, independently of $C_0$, $\mathrm{\Sigma }$, and $\mathrm{\Delta }p$.

Figure 7

Constant area constriction. Decrease of both polar radius (left-hand column) and the volume enclosed in the vesicle (right-hand column) during constant area constriction. Comparison between analytical and numerical results. (a) Influence of the surface tension $\mathrm{\Sigma }$. (b) Influence of the osmotic pressure $\mathrm{\Delta }p$. (c) Influence of the spontaneous curvature of the membrane $C_0$. Constriction process at constant area requires a final decrease in both polar radius and volume enclosed on the vesicle of around 30%, independently of $C_0$, $\mathrm{\Sigma }$, and $\mathrm{\Delta }p$.

Figure 8

Asymmetrically constricted vesicle. (a) Asymmetric profile after displacing the constriction ring a length $\mathrm{\Delta }x$ from the middle point. (b) Surface obtained from the revolution around the $x$ axis.

Figure 9

Stability coefficients for constant volume and constant area constrictions. Stability coefficients maintaining volume constant $k_V/\kappa$ (above) and maintaining membrane area constant $k_A/\kappa$ (below) at all stages of constriction. Comparison of the exact numerical results with the analytical approximate solutions. (a) Influence of the surface tension $\mathrm{\Sigma }$. (b) Influence of the osmotic pressure $\mathrm{\Delta }p$. (c) Influence of the spontaneous curvature of the membrane $C_0$. Regions with stable configurations $(k_{V,A}>0)$ are shaded in yellow.