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.