  Recent experiments in epithelial wound healing have demonstrated the
necessity of Mitogen-activated protein kinase (MAPK) for coordinated cell
movement after damage. This MAPK activity is characterized by two wave-like
phenomena. One MAPK "wave" that originates immediately after injury, propagates
deep into the cell layer, and then rebounds back to the wound interface. After
this initial MAPK activity has largely disappeared, a second MAPK front
propagates slowly from the wound interface and continues into the tissue,
maintaining a sustained level of MAPK activity throughout the cell layer. It
has been suggested that the first wave is initiated by reactive oxygen species
(ROS) generated at the time of injury. In this paper, we develop a minimal
mechanistic diffusion-convection model that reproduces the observed behavior.
The main ingredients of our model are a competition between ligand (e.g.,
Epithelial Growth Factor) and ROS for the activation of Epithelial Growth
Factor Receptor (EGFR) and a second MAPK wave that is sustained by stresses
induced by the slow cell movement that closes the wound. We explore the
mathematical properties of the model in connection with the bistability of the
MAPK cascade and look for traveling wave solutions consistent with the
experimentally observed MAPK activity patterns.
