Construction of Nuclear Envelope Shape by a High-Genus Vesicle with Pore-Size Constraint

Abstract

Nuclear pores have an approximately uniform distribution in the nuclear envelope of most living cells. Hence, the morphology of the nuclear envelope is a spherical stomatocyte with a high genus. We have investigated the morphology of high-genus vesicles under pore-size constraint using dynamically triangulated membrane simulations. Bending-energy minimization without volume or other constraints produces a circular-cage stomatocyte, where the pores are aligned in a circular line on an oblate bud. As the pore radius is reduced, the circular-pore alignment is more stabilized than a random pore distribution on a spherical bud. However, we have clarified the conditions for the formation of a spherical stomatocyte: a small perinuclear volume, osmotic pressure within nucleoplasm, and repulsion between the pores. When area-difference elasticity is taken into account, the formation of cylindrical or budded tubules from the stomatocyte and discoidal stomatocyte is found.

Figure 1

Ring to constrain the pore size. The membrane and ring are displayed in light gray (green) and in dark gray (red), respectively. The front half of the membrane is removed for clarity. To see this figure in color, go online.

Figure 2

Comparison of genus-5 vesicles without pore-size constraint (no ring) and with six rings of $r_{\text{po}}/R_{\text{ves}}=0.14$ and 0.2. (a and b) Snapshots at (a) $V^{\ast }=0.6$ and (b) $V^{\ast }=0.45$ with no ring. Front and bird’s eye views are shown in (a). (c–e) Dependence on the reduced volume $V^{\ast }$. (c) Free-energy profile F. (d) Mean aplanarity $\langle {\alpha }_{\text{pl}}\rangle$. (e) Mean area difference $\langle \Delta a\rangle$. Error bars are shown at several data points. To see this figure in color, go online.

Figure 3

Dependence on the maximum pore radius $r_{\text{po}}$ for genus-5 vesicles with or without the volume constraint. (a–d) Snapshots at (a) and (c) $r_{\text{po}}/R_{\text{ves}}=0.077$ and (b) and (d) $r_{\text{po}}/R_{\text{ves}}=0.2$. The rings are depicted with a half of the real minor radius for clarity. (e–g) Dependence of the averages of (e) aplanarity $\langle {\alpha }_{\text{pl}}\rangle$, (f) distance $\langle d_{\text{po}}\rangle$ between centers of the rings and vesicle, (g) reduced volume $\langle V^{\ast }\rangle$, and area difference $\langle \Delta a\rangle$. The solid and dashed lines in (g) represent $\langle V^{\ast }\rangle$ and $\langle \Delta a\rangle$, respectively. To see this figure in color, go online.

Figure 4

Genus-5 vesicles with $n_{\text{in}}$ particles in the nucleoplasm for $r_{\text{po}}/R_{\text{ves}}=0.2$. (a and b) Snapshots at (a) $n_{\text{in}}=6$ and (b) $n_{\text{in}}=10$ without the volume constraint. (c–e) Dependence on $V^{\ast }$ at $n_{\text{in}}=0$, 6, and 10. (c) Free-energy profile F. (d) Mean aplanarity $\langle {\alpha }_{\text{pl}}\rangle$. (e) Mean area difference $\langle \Delta a\rangle$. Error bars are shown at several data points. To see this figure in color, go online.

Figure 5

Genus-5 vesicles with repulsion between the ring centers. (a and b) Snapshots at (a) $r_{\text{ex}}/R_{\text{ves}}=0.22$ and (b) $r_{\text{ex}}/R_{\text{ves}}=0.37$ for $r_{\text{po}}/R_{\text{ves}}=0.077$. (c–e) Dependence of the averages of (c) aplanarity $\langle {\alpha }_{\text{pl}}\rangle$, (d) distance $\langle d_{\text{po}}\rangle$ between the centers of the rings and the vesicle, (e) reduced volume $\langle V^{\ast }\rangle$, and area difference $\langle \Delta a\rangle$. The solid and dashed lines in (e) represent $\langle V^{\ast }\rangle$ and $\langle \Delta a\rangle$, respectively. To see this figure in color, go online.

Figure 6

Snapshots of genus-5 vesicles with the ADE potential at (a–d) $r_{\text{po}}/R_{\text{ves}}=0.077$ and (e and f) $r_{\text{po}}/R_{\text{ves}}=0.2$. (a) $V^{\ast }=0.6$ and $\Delta a_0=-0.5$. (b) $V^{\ast }=0.6$ and $\Delta a_0=1.5$. (c) $V^{\ast }=0.6$ and $\Delta a_0=2$. (d) $V^{\ast }=0.2$ and $\Delta a_0=1.5$. (e) $V^{\ast }=0.6$ and $\Delta a_0=1.5$. (f) $V^{\ast }=0.6$ and $\Delta a_0=2$. In the front view of (f), the front-half is removed and the cross sections are indicated by the thick (blue) lines. To see this figure in color, go online.

Figure 7

Snapshots of genus-8 vesicles at $r_{\text{po}}/R_{\text{ves}}=0.077$ (a and b) without and (c and d) with the ADE potential. (a) No volume constraint. (b) $V^{\ast }=0.4$. (c) $V^{\ast }=0.5$. (d) $\Delta a_0=1.5$. To see this figure in color, go online.