Balance of Isotropic and Directed Forces Determines Cell Shape

The shape of a cell membrane is largely defined by the underlying actin cytoskeleton and membrane mechanics. The actin cytoskeleton asserts contractile forces on the membrane that can be divided in isotropic and directed forces. We present a theory which is an extension of the Young-Laplace equation. It models cell edges as parts of one uniform ellipse, which changes from cell to cell. The ellipse parameters are characterized by the ratio of isotropic to directed contractility of the cell. We demonstrate the capabilities of this model using fibroblasts seeded on an elastic micro-pillar array. In this way adhesion forces exerted by the cell at single adhesion sites are measured. We show that isotropic and directed forces balance the line tension in cortical actin. Furthermore, for cells with homogeneous contractile forces and a single orientation of stress-fibers any part of the cell edge follows a universal ellipse, enabling us to calculate the magnitude of isotropic and directed contractility in a single cell. We show that in 3T3 fibroblasts the directed contractility is about three times as strong as the isotropic contractility. If myosin motors are inhibited, however, directed contractility decreases, effectively disabling forces generated by stress-fibers, and the elliptical cell cortex turns into a circular shape predicted for an isotropic contractile cytoskeleton. Our analysis shows that a simple two-parameter model explains polarity, shape of the cell cortex and cellular forces as experimentally observed. Potentially this model can be used to predict stresses and forces on the extracellular matrix and tissue. Biological and Soft Matter Physic