Decay to equilibrium of the filament end density along the leading edge of the lamellipodium

A model for the dynamics of actin filament ends along the leading edge of the lamellipodium is analyzed. It contains accounts of nucleation by branching, of deactivation by capping, and of lateral flow along the leading edge by polymerization. A nonlinearity arises from a Michaelis-Menten type modeling of the branching process. For branching rates large enough compared to capping rates, the existence and stability of nontrivial steady states is investigated. The main result is exponential convergence to nontrivial steady states, proven by investigating the decay of an appropriate Lyapunov functional. Acknowledgements: This work has been supported by the Austrian Science Fund (FWF) through the doctoral school Dissipation and Dispersion in Nonlinear PDEs (project W1245) as well as the Vienna Science and Technology Fund (WWTF) (project LS13/029). The authors are grateful to J. Vic Small for many hours of discussions and for the permission to use the drawing in Fig. 1.