  It is now widely recognized that mechanical interactions between cells play a
crucial role in epithelial morphogenesis, yet understanding the mechanisms
through which stress and deformation affect cell behavior remains an open
problem due to the complexity inherent in the mechanical behavior of cells and
the difficulty of direct measurement of forces within tissues. Theoretical
models can help by focusing experimental studies and by providing the framework
for interpreting measurements. To that end, "vertex models" have introduced an
approximation of epithelial cell mechanics based on a polygonal tiling
representation of planar tissue. Here we formulate and analyze an Active
Tension Network (ATN) model, which is based on the same polygonal
representation of epithelial tissue geometry, but in addition i) assumes that
mechanical balance is dominated by cortical tension and ii) introduces tension
dependent local remodeling of the cortex, representing the active nature of
cytoskeletal mechanics. The tension-dominance assumption has immediate
implications for the geometry of cells, which we demonstrate to hold in certain
types of Drosophila epithelial tissues. We demonstrate that stationary
configurations of an ATN form a manifold with one degree of freedom per cell,
corresponding to "isogonal" - i.e. angle preserving - deformations of cells,
which dominate the dynamic response to perturbations. We show that isogonal
modes account for approximately 90% of experimentally observed deformation of
cells during the process of ventral furrow formation in Drosophila. Other
interesting properties of our model include the exponential screening of
mechanical stress and a negative Poisson ratio response to external uniaxial
stress. We also provide a new approach to the problem of inferring local
cortical tensions from the observed geometry of epithelial cells in a tissue
