3D cell neighbour dynamics in growing pseudostratified epithelia

During morphogenesis, epithelial sheets remodel into complex geometries. How cells dynamically organise their contact with neighbouring cells in these tightly packed tissues is poorly understood. We have used light-sheet microscopy of growing mouse embryonic lung explants, three-dimensional cell segmentation, and physical theory to unravel the principles behind 3D cell organisation in growing pseudostratified epithelia. We find that cells have highly irregular 3D shapes and exhibit numerous neighbour intercalations along the apical-basal axis as well as over time. Despite the fluidic nature, the cell packing configurations follow fundamental relationships previously described for apical epithelial layers, that is, Euler's polyhedron formula, Lewis' law, and Aboav-Weaire's law, at all times and across the entire tissue thickness. This arrangement minimises the lateral cell-cell surface energy for a given cross-sectional area variability, generated primarily by the distribution and movement of nuclei. We conclude that the complex 3D cell organisation in growing epithelia emerges from simple physical principles.

Arithmetic proof of the multiplicity-weighted Euler characteristic for symmetrically arranged space-filling polyhedra

The puzzling observation that the famous Euler's formula for three-dimensional polyhedra V - E + F = 2 or Euler characteristic χ = V - E + F - I = 1 (where V, E, F are the numbers of the bounding vertices, edges and faces, respectively, and I = 1 counts the single solid itself) when applied to space-filling solids, such as crystallographic asymmetric units or Dirichlet domains, are modified in such a way that they sum up to a value one unit smaller (i.e. to 1 or 0, respectively) is herewith given general validity. The proof provided in this paper for the modified Euler characteristic, χ_m = V_m - E_m + F_m - I_m = 0, is divided into two parts. First, it is demonstrated for translational lattices by using a simple argument based on parity groups of integer-indexed elements of the lattice. Next, Whitehead's theorem, about the invariance of the Euler characteristic, is used to extend the argument from the unit cell to its asymmetric unit components.

Multiplicity-weighted Euler's formula for symmetrically arranged space-filling polyhedra

For many tested cases of identical space-filling polyhedra, such as the space-group-specific asymmetric units or Dirichlet domains, the numbers of their faces (Fn), edges (En) and vertices (Vn), in each case normalized by division by the multiplicity of their (potentially special) symmetry position, fulfill a modified Euler’s formula Fn − En + Vn = 1.

The famous Euler’s rule for three-dimensional polyhedra, F − E + V = 2 (F, E and V are the numbers of faces, edges and vertices, respectively), when extended to many tested cases of space-filling polyhedra such as the asymmetric unit (ASU), takes the form Fn − En + Vn = 1, where Fn, En and Vn enumerate the corresponding elements, normalized by their multiplicity, i.e. by the number of times they are repeated by the space-group symmetry. This modified formula holds for the ASUs of all 230 space groups and 17 two-dimensional planar groups as specified in the International Tables for Crystallography, and for a number of tested Dirichlet domains, suggesting that it may have a general character. The modification of the formula stems from the fact that in a symmetrical space-filling arrangement the polyhedra (such as the ASU) have incomplete bounding elements (faces, edges, vertices), since they are shared (in various degrees) with the space-filling neighbors.