Membrane area gain and loss during cytokinesis

In cytokinesis of animal cells, the cell is symmetrically divided into two. Since the cell's volume is conserved, the projected area has to increase to allow for the change of shape. Here we aim to predict how membrane gain and loss adapt during cytokinesis. We work with a kinetic model in which membrane turnover depends on membrane tension and cell shape. We apply this model to a series of calculated vesicle shapes as a proxy for the shape of dividing cells. We find that the ratio of kinetic turnover parameters changes nonmonotonically with cell shape, determined by the dependence of exocytosis and endocytosis on membrane curvature. Our results imply that controlling membrane turnover will be crucial for the successful division of artificial cells.

Figure 1

Membrane shape and membrane dynamics during cytokinesis. (a) During cytokinesis, a rounded-up cell splits into two daughter cells of equal size. While the projected area $A$ of the cell increases, the cellular volume $V$ is kept constant. Several processes influence the shape and the total membrane area. (b) Since membrane area is laterally redistributed on a short timescale, the membrane relaxes rapidly toward its equilibrium shape. (c) On an intermediate timescale, the total membrane area is determined by membrane turnover. Total membrane area gain is driven through exocytosis, whereas total membrane area loss is driven through endocytosis. Both exocytosis and endocytosis act along the projected area. (d) On a large timescale, the membrane shape changes.

Figure 2

Calculated FUV shapes, mimicking the shapes of dividing cells. (a) Sequential change sequence, where first the reduced volume $\nu$ is decreased from 1 to 0.7 and then the reduced preferred curvature ${\overline{H}}_0$ is increased from 0 to 1.41. (b) Parallel change sequence, where the reduced volume $\nu$ and the preferred curvature ${\overline{H}}_0$ are changed simultaneously. While $\nu$ is decreased from 1 to 0.7, ${\overline{H}}_0$ is increased from 0 to 1. Subsequently, and similar to (a), ${\overline{H}}_0$ is increased from 1 to 1.41, while $\nu =0.7$ is kept constant.

Figure 3

Membrane gain through exocytosis and membrane loss through endocytosis. (a) In our model, membrane gain depends on the amount of internal membrane area $a_{\mathrm{i}}$ and is inversely proportional to the internal membrane tension ${\mathrm{\Gamma }}_{\mathrm{i}}$. Moreover, the tendency for exocytosis is low for a nearly flat membrane since the contact area between the vesicle (red sphere) and the membrane (red line) is small (top). The tendency for exocytosis increases when the membrane is more curved since the contact area between the vesicle and the membrane increases (bottom). (b) In our model, membrane loss is inversely proportional to the tension of the cell membrane ${\mathrm{\Gamma }}_{\mathrm{cm}}$. Moreover, the tendency for endocytosis is low for a curved membrane since the contact area between the particle (blue sphere) and the membrane (red line) is small (top). The tendency for endocytosis increases for a nearly flat membrane since the contact area between the particle and the membrane increases (bottom).

Figure 4

Membrane trafficking predicted for the shapes as displayed in Fig. 2. (a) The gain/loss factor $\gamma$ as a function of the reduced area $A/A_{\mathrm{s}}$. For the sequential change sequence (dashed blue line), $\gamma$ first decreases with increasing $A/A_{\mathrm{s}}$ before it increases. Then, $\gamma$ increases at a constant reduced area of around $A/A_{\mathrm{s}}=1.27$ toward the end of the sequence since here only the reduced preferred curvature increases. For the parallel change sequence (dashed red line), we find qualitatively similar behavior. At $A/A_{\mathrm{s}}\approx 1.26$, the fission limit is marked as a gray line. (b) The gain/loss factor $\gamma$ as a function of the reduced preferred curvature ${\overline{H}}_0$. For the sequential change sequence (blue), $\gamma$ first decreases and then increases at constant ${\overline{H}}_0$ before it increases monotonically with increasing ${\overline{H}}_0$. For the parallel change sequence (red), $\gamma$ first decreases before it increases again with increasing ${\overline{H}}_0$. At ${\overline{H}}_0\approx 1.41$, the fission limit is marked as a gray line.

Figure 5

Predicted membrane turnover for various shapes. We plot the ratio of the gain/loss factor $\gamma$ as a function of the reduced volume $\nu$ for various cases. The sequential change and parallel change sequences from Fig. 2 are shown in dashed blue and red, respectively. For both sequences, $\gamma$ first decreases with decreasing $\nu$ before it increases again with decreasing or constant $\nu$. The corresponding membrane shapes are plotted along the curves in the corresponding colors. As a limiting case, $\gamma$ is also shown for a spherocylinder of increasing length (solid orange). The solution from the expansion in spherical harmonics around the sphere (solid green) agrees with the numerical solutions when $\nu \approx 1$. In purple, the analytic values are shown for two connected spheres, a small sphere and a large sphere to show how $\gamma$ is predicted to change during the whole sequence of shapes during FUV division.

Figure 6

Membrane turnover during division for the simple loss model (a) and the simple gain model (b). We plot $\gamma \left(\nu \right)$ for the sequential change sequence (dashed blue line) and parallel change (dashed red line). The corresponding membrane shapes from Fig. 2 are plotted along the curves in the corresponding colors. In orange, we plot $\gamma \left(\nu \right)$ for a spherocylinder of increasing length. In purple, the analytical values are shown for two connected spheres, a small sphere and a large sphere.

Figure 7

Effect of the area balance equation on the projected area. The change of the non-dimensionalized sphere radius $dr/d\tau$ is plotted as a function of the sphere radius $r$. With curvature dependent kinetic turnover parameters, $dr/d\tau$ is positive for $r<1$ (blue line) and negative for $r>1$ (red line), predicting a single stable steady state at $r=1$ (gray circle). Inset: The stable solutions of $r\left(\tau \right)$ for different initial radii $r_0$ (blue and red line). Top: For initial radii $r>1$, the sphere shrinks (red). For initial radii $r<1$, the sphere grows (blue), leading to one stable sphere size (gray) at $r=1$.

Figure 8

The steady-state amplitude $f_{20}^{\mathrm{s}}$ as a function of $\gamma$ according to Eq. (D19). For $\gamma =1$, the amplitude vanishes and the spherical shape is restored. For $\gamma =1/3$, the amplitude diverges. For $1/3<\gamma <1$, the amplitude is finite, so the spherical shape is destabilized and stable shapes develop. Inset: Time evolution of $f_{20}\left(\tau \right)$ for different values of $\gamma$.