A model for shape generation by strain and cell-cell adhesion in the epithelium of an arthropod leg segment

We present a model for the energetic factors determining the most stable shape of a tubular epithelium such as the hypodermis of an arthropod leg segment. The model uses the analysis by Steinberg (1963) of rearrangement of cells in aggregates under the influence of differential adhesion, combining this analysis with the assumption that the epithelium behaves as an elastic sheet. The epithelium is assumed to consist of blocks of cells with different adhesive affinities, which remain unmixed in a quilt pattern. Rearrangement of cells within each block can adjust the shape of the tube by changing the shapes of the blocks. By means of such rearrangements the tube develops that shape which minimizes a free energy. The free energy is the difference between the energy of mechanical strain due to bending of the epithelium and the work of adhesion among cells. Minimization of the free energy for a cylindrical segment yields a scaling relation involving the length and radius of the segment. Leg segments of Drosophila conformed approximately to this relation, with deviations which suggest that a whole-limb pattern of adhesive affinities modulates the shaping effects of an adhesive pattern repeated in each leg segment. The model also predicts a transient deformation in an epithelium following a grafting operation. For example, deleting a slab of tissue from a tubular segment and reuniting the cut ends should produce a constriction of the tube at the host-graft junction. We propose that patterns of strain and adhesion can provide positional information which regulates subsequent development. Local increases in strain or adhesive disparity may stimulate mitoses; the resulting changes in distribution of cells will affect morphogenesis.