Periodic three-dimensional assemblies of polyhedral lipid vesicles

We theoretically study the structure of periodic bulk assemblies of identical lipid vesicles. In our model, each vesicle is represented as a convex polyhedron with flat faces, rounded edges, and rounded vertices. Each vesicle carries an elastic and an adhesion energy and in the limit of strong adhesion, the minimal-energy shape of cells minimizes the weighted total edge length. We calculate exactly the shape of the rounded edge and show that it can be well described by a cylindrical surface. By comparing several candidate space-filling polyhedra, we find that the oblate shapes are preferred over prolate shapes for all volume-to-surface ratios. We also study periodic assemblies of vesicles whose adhesion strength on lateral faces is different from that on basal or apical faces. The anisotropy needed to stabilize prolate shapes is determined and it is shown that, at any volume-to-surface ratio, the transition between oblate and prolate shapes is very sharp. The geometry of the model vesicle assemblies reproduces the shapes of cells in certain simple animal tissues.

Figure 1

An example of the model periodic three-dimensional vesicle assembly, which consists of a regular arrangement of identical rounded polyhedra. Shown here is a part of two adjacent layers of a stack of prolate regular hexagonal prisms. Each vesicle is a convex space-filling polyhedron with rounded edges and vertices (top inset). The lower inset shows the cross section of a regular 3-valent edge.

Figure 2

Three shapes derived from a rhombic dodecahedron. The first prolate shape shown in panel b is obtained by cutting the rhombic dodecahedron (panel a) along the mirror plane perpendicular to the three-fold axis (red line labeled by the scissor symbol) and inserting a regular hexagonal prism. This produces a prolate shape whose reduced volume depends on the height of the inserted prism. The second prolate shape (panel d) is constructed by cutting the rhombic dodecahedron in panel c along a plane perpendicular to the four-fold axis (red line labeled by the scissor symbol) and inserting a square prism. The oblate shape based on the rhombic dodecahedron is derived from the prolate shape with four-fold axis extended such that the four hexagonal lateral sides are regular (panel e). This shape is cut along two parallel planes equidistant from the mirror plane containing the four-fold axis. The central part is removed and the two caps form the oblate shape whose reduced volume is controlled by the thickness of the central part (panel f). For simplicity we refer to shapes in panels b, d, and f as 3-elongated rhombic dodecahedron, 4-elongated rhombic dodecahedron, and flattened rhombic dodecahedron, respectively.

Figure 3

Cross section of a 3-valent edge of arbitrary geometry parametrized by the angles among the contact zones ${\varphi }_1,{\varphi }_2$, and ${\varphi }_3=2\pi -{\varphi }_1-{\varphi }_2$. Sections corresponding to the contact zones (DA, EB, and FC) are straight lines and $H_{\mathit{Zi}}$ denotes the corresponding Hamiltonians. The Hamiltonians of the noncontact zones (AB, BC, and CA) are denoted by $H_i$.

Figure 4

Cross- sections of the rounded edges for a few sets of (${\varphi }_1$,${\varphi }_2$): ${\varphi }_1={90}^{°},{\varphi }_2={90}^{°}-{135}^{°}$ (steps of ${15}^{°})$ (left panel); ${\varphi }_1={120}^{°},{\varphi }_2={70}^{°}-{120}^{°}$ (steps of ${10}^{°}$) (middle panel); ${\varphi }_1={150}^{°},{\varphi }_2={65}^{°}-{105}^{°}$ (steps of ${10}^{°}$) (right panel). The cross sections are color coded to indicate the fractional increase of the edge energy per unit length relative to the regular 3-valent edge with ${\varphi }_1={\varphi }_2={120}^{°}$. As long as none of the angles is small, the energy increase due to a deviation from the regular configuration is moderate: the T-shaped ${90}^{°}-{90}^{°}-{180}^{°}$ edge carries an energy 27.3% larger than the regular edge whereas the ${60}^{°}-{120}^{°}-{180}^{°}$ edge is just short of 40% above the regular edge. The magnitude of the reduced adhesion strength $\gamma$ affects only the length scale of the curved noncontact arcs of edge cross section but not their shape. Thus the cross sections shown here are the same in all edges of a given geometry specified by ${\varphi }_1$ and ${\varphi }_2$: The shape of an edge is independent of $\gamma$ and so is the fractional increase of energy.

Figure 5

Close-up of the energy landscape of a 3-valent edge in the vicinity of the regular configuration with ${\varphi }_1={\varphi }_2={120}^{°}$. For deviations smaller than ${10}^{°}$, the fractional increase of the edge energy does not exceed 4.5%.

Figure 6

Reduced edge length $\lambda$ as a function of reduced volume $v$ for the different polyhedra examined: elongated equilateral triangular prism (a), elongated square prism (b), 4-elongated rhombic dodecahedron (c), elongated hexagonal prism (d), 3-elongated rhombic dodecahedron (e), flattened equilateral triangular prism (f), flattened square prism (g), flattened truncated octahedron (h), flattened rhombic dodecahedron (i), flattened hexagonal prism (j), and rhombic dodecahedron (k). Shape k is represented in the graph as a point at $v=\sqrt{\pi }/2^{1/4}\sqrt{3}\simeq 0.861$ and $\lambda =4.215$. The polyhedron with maximal reduced volume $v$ is the truncated octahedron with $v=0.868$. The oblate shapes are energetically favorable across the whole range of $v$; the hexagonal prism is the overall optimal shape. The inset shows the reduced edge length of the three shapes with almost identical energy: flattened hexagonal prism (j), flattened rhombic dodecahedron (i), and flattened truncated octahedron (h). Note that the energies of shapes b and d are only slightly lower than those of shapes c and e, respectively.

Figure 7

Phase diagram of prismatic shapes as a function of adhesion anisotropy $\alpha$ and reduced volume $v$. Oblate hexagonal prisms are optimal at small values of adhesion anisotropy (shaded region), whereas prolate prisms are stable at large $\alpha$.

Figure 8

(a) The aspect ratio $\xi$ of stable prismatic shapes of $v=$ 0.25, 0.45, 0.65, and 0.777 as a function of adhesion anisotropy $\alpha$ showing that the magnitude of discontinuity in $\xi$ decreases with reduced volume. (b) The aspect ratio of the oblate ${\xi }_o$ and the prolate ${\xi }_p$ vesicles stable in the small and large adhesion anisotropy regimes, respectively. As $v$ is increased, the difference in $\xi$ between the two branches decreases; shapes at large $v$ are fairly isometric, neither flattened nor elongated.

Figure 9

Scanning electron microscope (SEM) image of hepatocytes in mammalian liver: The roughly isometric cells are polyhedral but not identical in shape. The dilations between some cells are the bile canaliculi. Despite the irregularity of the polyhedral shape of cells, their columnar stacking demonstrates a certain degree of cell positional order. A column of cells is outlined and colored (shaded) to emphasize the polyhedral shapes and should merely serve as a guide to the eye. Image courtesy of Dr. R. Wagner, University of Delaware.