Keratocytes are skin cells that are capable of persistent motion at constant velocity both in vivo and in vitro. In analogy to traveling wave solutions to reaction diffusion equations (constant profile moving at fixed speed) this behavior may be modeled by a traveling wave solution to an elliptic-parabolic free boundary problem where the differential equations describe stress and flow of the cytoskeleton inside the cell and the variable domain models the cell shape.
We shall motivate the model formulation, discuss its relation to the classical Keller-Segel model for chemotaxis, prove the existence of radially symmetric steady states (corresponding to a resting cell) and show how traveling wave solutions bifurcate from these steady states. Moreover, the issue of multiple steady states and possible finite time blow up will be touched, and biological interpretations are given.
Finally, the crucial differences to models for tumor growth and traveling waves occuring therein will be commented on.