Here we analyze a drift-diffusion model on the half-line. The drift is linked to exchanges at 0. This corresponds to the modeling of filament generation following protein exchanges between the membrane and the cytoplasm of a cell. This model has a critical mass and the asymptotic behaviors can be divided into three categories. In subcritical mass regime, diffusive behavior dominates and convergence in self-similar variables can be obtain with parabolic scaling. In supercritical mass regime, drift compensates for diffusion and convergence towards equilibrium distributions occurs. In the critical case, convergence towards 0 for the drift occurs over a long period of time, the behavior exhibits a distinct relevant scaling. We quantify the convergences in the relevant variables in the three regimes.