The vertex model accounts qualitatively and quantitatively for the observed packing geometry in the wing disc and its response to perturbation by laser ablation. Epithelial packing geometry is a consequence of both physical cellular properties and the disordering influence of proliferation. The occurrence of T2 transitions during network growth suggests that elimination of cells from the proliferating disc epithelium may be the result of junctional force balances.

Planar cell polarity (PCP) proteins form polarized cortical domains that govern polarity of external structures such as hairs and cilia in both vertebrate and invertebrate epithelia. The mechanisms that globally orient planar polarity are not understood, and are investigated here in the Drosophila wing using a combination of experiment and theory. Planar polarity arises during growth and PCP domains are initially oriented toward the well-characterized organizer regions that control growth and patterning. At pupal stages, the wing hinge contracts, subjecting wing-blade epithelial cells to anisotropic tension in the proximal-distal axis. This results in precise patterns of oriented cell elongation, cell rearrangement and cell division that elongate the blade proximo-distally and realign planar polarity with the proximal-distal axis. Mutation of the atypical Cadherin Dachsous perturbs the global polarity pattern by altering epithelial dynamics. This mechanism utilizes the cellular movements that sculpt tissues to align planar polarity with tissue shape.

Subdividing proliferating tissues into compartments is an evolutionarily conserved strategy of animal development [1-6]. Signals across boundaries between compartments can result in local expression of secreted proteins organizing growth and patterning of tissues [1-6]. Sharp and straight interfaces between compartments are crucial for stabilizing the position of such organizers and therefore for precise implementation of body plans. Maintaining boundaries in proliferating tissues requires mechanisms to counteract cell rearrangements caused by cell division; however, the nature of such mechanisms remains unclear. Here we quantitatively analyzed cell morphology and the response to the laser ablation of cell bonds in the vicinity of the anteroposterior compartment boundary in developing Drosophila wings. We found that mechanical tension is approximately 2.5-fold increased on cell bonds along this compartment boundary as compared to the remaining tissue. Cell bond tension is decreased in the presence of Y-27632 [7], an inhibitor of Rho-kinase whose main effector is Myosin II [8]. Simulations using a vertex model [9] demonstrate that a 2.5-fold increase in local cell bond tension suffices to guide the rearrangement of cells after cell division to maintain compartment boundaries. Our results provide a physical mechanism in which the local increase in Myosin II-dependent cell bond tension directs cell sorting at compartment boundaries.

Epithelia are sheets of cells that are dynamically remodelled by cell division and cell death during development. Here we describe the cell shapes and packings as networks of polygons: stable and stationary network configurations obey force balance and are represented as local minima of a potential function. We characterize the physical properties of this vertex model, including the set of ground states, and the energetics of topological rearrangements. We furthermore discuss a quasistatic description of cell division that allows us to study the mechanics and dynamics of tissue remodelling during growth. The biophysics of cells and their rearrangements can account for the morphology of cell packings observed in experiments.

Our results argue that natural selection acts predominantly on cell size and indirectly influences the spindle through the scaling of the spindle with cell size. Previous studies have shown that the spindle also scales with cell size during early development. Thus, the scaling of the spindle with cell size controls its variation over both ontogeny and phylogeny.