Dynamic Force Balances and Cell Shape Changes during Cytokinesis

During the division of animal cells, an actomyosin ring is formed in the cell cortex. The contraction of this ring induces shape changes of the cell and the formation of a cytokinesis furrow. In many cases, a cell-cell interface forms that separates the two new cells. Here we present a simple physical description of the cell shape changes and the dynamics of the interface closure, based on force balances involving active stresses and viscous friction. We discuss conditions in which the interface closure is either axially symmetric or asymmetric. We show that our model can account for the observed dynamics of ring contraction and interface closure in the C. elegans embryo.

Figure 1

Schematic representation of a dividing cell with axisymmetric geometry of the contractile process. The initial spherical cell (a) is deformed by the constriction of a contractile ring (b), shown as green dots in all the vertical sections. A septum forms (c). It shrinks to divide the cell (d). Division leading to vanishing contact area between the daughters (e). The geometry of the cell is characterized by two intersecting spheres of radius $R$ which meet at an angle $\psi$ (f). The septum has the shape of an annulus with inner and outer radii $r$ and $r+\delta$, respectively (g).

Figure 2

Normalized energy $\overline{F}$ as a function of normalized line tension $\overline{\mathrm{\Sigma }}$, shown for three different values of normalized membrane tension $\overline{\sigma }$ as indicated (colored lines). For $\overline{\mathrm{\Sigma }}=0$, the lowest energy shape is a sphere. Constricted shapes without formation of a septum correspond to the line with $\overline{\delta }=0$ (black). These shapes exhibit a furrow with increasingly strong indentation as $\overline{\mathrm{\Sigma }}$ increases. At the spinodal indicated “$+$”, the line folds back along a branch corresponding to locally unstable shapes (broken black line) until the radius $\overline{r}$ vanishes at $\overline{\mathrm{\Sigma }}=0$. Lines corresponding to shapes with septum with $\overline{\delta }\ne 0$ branch off (colored lines). The star denotes the point where the line $\overline{\sigma }=0.5$ (green) branches off the black line via the formation of a septum. The horizontal green line corresponds to the fully constricted state with $r=0$ for which the energy does not depend on $\overline{\mathrm{\Sigma }}$. Similar curves are also shown for $\overline{\delta }=0.25$ (blue) and 0.75 (red). The intersection of the colored solid lines corresponding to fully contracted states with the black solid $\overline{\delta }=0$ line marks a first order transition point where the fully constricted state becomes energetically favored. The inset shows the normalized line tension $\overline{\mathrm{\Sigma }}$ as a function of normalized ring radius $r$ for the values of $\overline{\sigma }$ indicated.

Figure 3

Time evolution of the radius $r$ of the contractile ring during cytokinesis. The circles are the experimental data of Ref. [5] obtained for C. elegans embryos, averaged over five embryos. The solid line is the best fit solution of Eq. (3) to the experimental data with $r_0=14\mu \mathrm{m}$ and dimensionless parameters $\tilde{\sigma }=0.96$, ${\tau }_0\approx 38\mathrm{s}$, and ${\tilde{\zeta }}_L^{-1}\approx 0$.

Figure 4

Formation of an asymmetric septum during cleavage (a). Cross-sectional view of the furrow in the $x$-$y$ plane (b). Parametrization of the septum shape by angles $\varphi$, $\theta$, and height $h$ (c).

Figure 5

Ring contraction during cytokinesis with an asymmetric septum. Fluorescence microscopy images of cytokinesis at different times (shown in seconds) in a C. elegans embryo in the four-cell stage [3] (a); scale bar $5\mu \mathrm{m}$. The contractile ring (see arrow) is stained by septin labeled green fluorescent protein. The septum emerges at the side of the cell which is not adhering to a neighboring cell. Quantification of the normalized perimeter $L_p(t)$ of the ring as a function of time (b). The symbols distinguish different cell generations $n$. The inset shows original data from Ref. [3] for generations $n=1$ to 5 (top to bottom). The solid line is a fit of the solution to the equations for $\dot{h},\dot{\varphi }$ to the data for $n=1,2,3$, with $r_0=14\mu \mathrm{m}$, ${\zeta }_L^{\prime }={\zeta }_L$ and fit parameters $\tilde{\sigma }=1.43$, ${\tau }_0=62.5\mathrm{s}$, ${\mathrm{\Sigma }}_0^{\prime }/{\mathrm{\Sigma }}_0=2.5$, ${\tilde{\zeta }}_L^{-1}\approx 0$, and ${\tilde{\zeta }}_{\varphi }/{\tilde{\zeta }}_L\approx 0$. Calculated septum shapes corresponding to the solid line in (b) [(c)]. Calculated septum shapes for smaller values of $\tilde{\sigma }=0.5$ and ${\mathrm{\Sigma }}_0^{\prime }/{\mathrm{\Sigma }}_0=1.5$ (d). Panel (a) and the inset of (b) are adapted from Ref. [3], with permission.