  The interface dynamics of a 3D cell immersed in a 3D extracellular matrix is
investigated. We suggest a 3D generalization of a known 2D minimal phase field
model suggested in [1] for the description of keratocyte motility. Our model
consists of two coupled evolution equations for the order parameter and a
three-dimensional vector field describing the actin network polarization
(orientation). We derive a closed evolutionary integro-differential equation
governing the interface dynamics of a 3D cell. The equation includes the normal
velocity of the membrane, its curvature, cell volume relaxation, and a
parameter that is determined by the non-equilibrium effects in the
cytoskeleton. This equation can be considered as a 3D generalization of the 2D
case that was derived in [2].
