Stable Length Distributions in Colocalized Polymerizing and Depolymerizing Protein Filaments

A model for the dynamics of the length distribution in co-localized groups of polar polymer filaments is presented. It considers nucleation, polymerization at plus-ends, and depolymerization at minus-ends and is derived as a continuous macroscopic limit from a discrete description. Its main feature is a nonlinear coupling due to competition of the depolymerizing ends for the limited supply of a depolymerization agent. The model takes the form of an initial-boundary value problem for a one-dimensional nonlinear hyperbolic conservation law, subject to a nonlinear, nonlocal boundary condition. Besides existence and uniqueness of entropy solutions, convergence to a steady state is proven. Technical difficulties are caused by the fact that the prescribed boundary data are not always assumed by entropy solutions.