Elastic energy of polyhedral bilayer vesicles

In recent experiments [M. Dubois, B. Demé, T. Gulik-Krzywicki, J.-C. Dedieu, C. Vautrin, S. Désert, E. Perez, and T. Zemb, Nature (London) 411, 672 (2001)] the spontaneous formation of hollow bilayer vesicles with polyhedral symmetry has been observed. On the basis of the experimental phenomenology it was suggested [M. Dubois, V. Lizunov, A. Meister, T. Gulik-Krzywicki, J. M. Verbavatz, E. Perez, J. Zimmerberg, and T. Zemb, Proc. Natl. Acad. Sci. USA 101, 15082 (2004)] that the mechanism for the formation of bilayer polyhedra is minimization of elastic bending energy. Motivated by these experiments, we study the elastic bending energy of polyhedral bilayer vesicles. In agreement with experiments, and provided that excess amphiphiles exhibiting spontaneous curvature are present in sufficient quantity, we find that polyhedral bilayer vesicles can indeed be energetically favorable compared to spherical bilayer vesicles. Consistent with experimental observations we also find that the bending energy associated with the vertices of bilayer polyhedra can be locally reduced through the formation of pores. However, the stabilization of polyhedral bilayer vesicles over spherical bilayer vesicles relies crucially on molecular segregation of excess amphiphiles along the ridges rather than the vertices of bilayer polyhedra. Furthermore, our analysis implies that, contrary to what has been suggested on the basis of experiments, the icosahedron does not minimize elastic bending energy among arbitrary polyhedral shapes and sizes. Instead, we find that, for large polyhedron sizes, the snub dodecahedron and the snub cube both have lower total bending energies than the icosahedron.

Figure 1

Side view of a ridge with dihedral angle ${\alpha }_i$ bending (a) over an arc length $d$ of a cylinder with radius $R_1$ and (b) over an arc length comparable to the small-scale cutoff $b$. The red and blue amphiphile species represent myristic acid and CTAOH, which are negatively and positively charged, respectively. The arrows in panel (b) denote bond vectors connecting adjacent amphiphiles.

Figure 2

Illustration of a polyhedral vertex with face angle ${\beta }_j$. As in Fig. 1b, the arrows represent bond vectors connecting adjacent amphiphiles, but now with the bond vectors being parallel rather than perpendicular to ridges.

Figure 3

Schematic illustrations of two models of vertex pores for an amphiphile monolayer thickness $m$ and an amphiphile headgroup thickness $h$. (a) Cross section of half of a pore around the tip of a cone (inset) with apex angle $\pi -2\theta$ and radius $r$. (b) Side view (left panel) and top-down view (right panel) of a pore with radius $r$ composed of straight edges of length $s_j$ along each face which bend through an angle ${\gamma }_j$ from one face to a neighboring face.

Figure 4

Side view of a ridge with dihedral angle ${\alpha }_i$ and perfect segregation of excess amphiphiles in the outer bilayer leaflet. Note that, compared to Fig. 1, the neutral plane of bending is shifted from the mid-plane of the bilayer to the amphiphile head-tail interface of the inner membrane leaflet.

Figure 5

Total bending energy of a conical pore in Eq. (14), rim contribution in Eq. (32), and loop contribution in Eq. (33) versus pore radius $r$ for [7] $m=2$ nm, $h=0.5$ nm, and $H_0^{\star }=0$ nm${}^{-1}$, with (a) $\theta =0$ and (b) $\theta =0.4\pi$.

Figure 6

Bending energy of a conical pore in Eq. (14) versus pore radius $r$ for [7] $m=2$ nm, $h=0.5$ nm, and (a) $\theta =0$ and (b) $\theta =0.4\pi$ for the indicated values of the monolayer spontaneous curvature $H_0^{\star }$.

Figure 7

Edge tension $\lambda$ in Eq. (34) for a conical pore versus pore radius $r$ for $m=2$ nm and (a) $\theta =0$ and $h=0.5$ nm with $K_b^{\star }=10k_{B}T$, (b) $\theta =0$ and $h=0.5$ nm with $K_b^{\star }=20k_{B}T$, (c) $\theta =0.4\pi$ and $h=0.5$ nm with $K_b^{\star }=10k_{B}T$, and (d) $\theta =0$ and $h=0.8$ nm with $K_b^{\star }=10k_{B}T$ using $H_0^{\star }=0$ nm${}^{-1}$ for the green (upper) curves, $H_0^{\star }=0.15$ nm${}^{-1}$ for the blue (middle) curves, and $H_0^{\star }=0.3$ nm${}^{-1}$ for the red (lower) curves in each panel. The shaded regions of the plots correspond to typical measured values [2] of the edge tension of amphiphile bilayers.

Figure 8

Elastic bending energy of a polygonal pore in Eq. (21) versus pore radius $r$ for the icosahedron, and ratio of conical and polygonal pore energies (inset), for [7] $m=2$ nm and $h=0.5$ nm, using $H_0^{\star }=0$ nm${}^{-1}$ for the green curves (upper curves in the large-$r$ regime), $H_0^{\star }=0.15$ nm${}^{-1}$ for the blue curves (middle curves in the large-$r$ regime), and $H_0^{\star }=0.3$ nm${}^{-1}$ for the red curves (lower curves in the large-$r$ regime). The polygonal pore energy is calculated by noting that, for the icosahedron, five ridges meet at each vertex with the face angle ${\beta }_j=\pi /3$. The corresponding conical pore energy is obtained using the vertex angle $\Omega =2\pi -5\mathit{arcsin}(2/3)\approx 2.6$ for the icosahedron, which gives $\theta \approx 0.2\pi$. The horizontal black line in the inset denotes the ratio of the circumference of conical and polygonal pores, which is equal to $\pi /(5\sin \frac{\pi }{6})\approx 1.3$ for the icosahedron.

Figure 9

Image and net representations of (a) the icosahedron, (b) the snub dodecahedron, (c) the snub cube, (d) the great rhombicosidodecahedron, (e) the triangular prism, (f) the square antiprism, (g) the gyroelongated pentagonal birotunda, and (h) the pentagonal hexecontahedron. Polyhedron (a) is a Platonic solid, polyhedra (b), (c), and (d) are Archimedean solids, polyhedron (e) is a prism, polyhedron (f) an antiprism, polyhedron (g) a Johnson solid, and polyhedron (h) a Catalan solid. The Catalan solid in (h) is the dual of the Archimedean solid in (b), and the polyhedra in (b), (c), (g), and (h) are chiral.

Figure 10

Total bending energies of the convex polyhedra with regular faces, normalized by the bending energy of the icosahedron, $G_i$, for homogeneous bilayers with (a) the vertex energy $G_v^{(h)}$ in Eq. (10) and the ridge energy $G_r^{(h)}$ in Eq. (7), (b) the polygonal pore energy $G_p^{(p)}$ in Eq. (21) with $r=0$ and the ridge energy $G_r^{(h)}$ in Eq. (7), (c) the polygonal pore energy $G_p^{(p)}$ in Eq. (21) with $r=0$ and the upper bound $Y/K_b=10$ nm${}^{-2}$ on the ridge energy $G_r^{(\mathit{LW})}$ in Eq. (31), and (d) the polygonal pore energy $G_p^{(p)}$ in Eq. (21) with $r=0$ and the lower bound $Y/K_b={10}^{-3}$ nm${}^{-2}$ on the ridge energy $G_r^{(\mathit{LW})}$ in Eq. (31). We use the parameter values [6, 7, 30] $m=2$ nm, $h=0.5$ nm, $K_b^{\star }=K_b/100$, and $H_0^{\star }=0$ nm${}^{-1}$. The bold black curve denotes the bending energy of the sphere, and the colored (gray) curves denote the bending energies of bilayer polyhedra, where the bold curve minimizing polyhedral bending energy in the large-$R_p$ regime corresponds to the snub dodecahedron.

Figure 11

Ridge energies of the 13 Catalan solids, normalized by the ridge energy of the icosahedron, $G_i$, with (a) the ridge energy $G_r^{(h)}$ in Eq. (7), and (b) the upper bound $Y/K_b=10$ nm${}^{-2}$ on the ridge energy $G_r^{(\mathit{LW})}$ in Eq. (31).

Figure 12

Total bending energies of the convex polyhedra with regular faces, normalized by the bending energy of the icosahedron, $G_i$, with segregation of excess amphiphiles at vertices but not at ridges. The energy curves are obtained with the ridge energy $G_r^{(h)}$ in Eq. (7) [panels (a), (c), and (e)] and the upper bound $Y/K_b=10$ nm${}^{-2}$ on the ridge energy $G_r^{(\mathit{LW})}$ in Eq. (31) [panels (b), (d), and (f)] with pores of radius (a), (b) $r=0$, (c), (d) $r=20$ nm, and (e), (f) $r=40$ nm at each vertex. The bold black curve denotes the bending energy of the sphere, and the colored (gray) curves denote the bending energies of bilayer polyhedra, where the bold curve minimizing polyhedral bending energy in the large-$R_p$ regime corresponds to the snub dodecahedron.

Figure 13

Total bending energies of the convex polyhedra with regular faces, normalized by the bending energy of the icosahedron, $G_i$, with segregation of excess amphiphiles at ridges but not pores [panel (a)], and at ridges and pores [panels (b)–(f)]. For panel (a) we use the pore energy $G_p^{(p)}$ in Eq. (21) with $r=0$ and the ridge energy $G_r^{(s)}$ in Eq. (22) with the parameter values [6, 7, 30] $m=2$ nm, $h=0.5$ nm, $K_b^{\star }=K_b/100$, and $H_0^{\star }=0$ nm${}^{-1}$. The remaining panels are obtained using only the ridge energy $G_r^{(s)}$ in Eq. (22) with pores of radius (b) $r=0$, (c) $r=1$ nm, (d) $r=5$ nm, (e) $r=20$ nm, and (f) $r=40$ nm. The bold black curve denotes the bending energy of the sphere, and the colored (gray) curves denote the bending energies of bilayer polyhedra, where the bold curve minimizing polyhedral bending energy in the large-$R_p$ regime corresponds to the snub dodecahedron.

Figure 14

Theoretical estimates of the optimal amphiphile imbalance $r_I$, defined in Eq. (24), for the convex polyhedra with regular faces as a function of the polyhedron radius $R_p$ with [7] $m=2$ nm for (a) $r=0$ nm and (b) $r=20$ nm.

Figure 15

Image and net representations of the gyroelongated square dipyramid.