  The optimal shapes attained by contractile cells on adhesive substrates are
determined by the interplay between intracellular forces and adhesion with the
extracellular matrix. We model the cell as a contractile film bounded by an
elastic cortex and connected to the substrate via elastic links. When the
adhesion sites are continuously distributed, optimal cell shape is constrained
by the adhesion geometry, with a spread area sensitively dependent on the
substrate stiffness and contractile tension. For discrete adhesion sites,
equilibrium cell shape is convex at weak contractility, while developing local
concavities at intermediate values of contractility. Increasing contractility
beyond a critical value, controlled by mechanical and geometrical properties of
adhesion, cell boundary undergoes a discontinuous transition to a star-shaped
configuration with cusps and protrusions, accompanied by a region of
bistability and hysteresis.
