Rotating lamellipodium waves in polarizing cells

Cellular protrusion- and lamellipodium waves are widespread for both non-motile and moving cells and observed for many cell types. They are involved in the cell’s exploration of the substrate, its internal organization, as well as for the establishment of self-polarization prior to the onset of motion. Here we apply the recently developed phase field approach to model shape waves and their competition on the level of a whole cell, including all main physical effects (acto-myosin, cell membrane, adhesion formation and substrate deformation via traction) but ignoring specific biochemistry and regulation. We derive an analytic description of the emergence of a single wave deformation, which is of Burgers/Fisher-Kolmogorov type. Finally, we develop an amplitude equation approach to study multiple competing rotational waves and show how they allow the cell to transition from a non-moving state towards a polarized, steady moving state.