Stochastic model of elongation-shortening-switching kinetics of a polymeric filament: Critical slowing down of first-passage and finite-size regularization

Several filamentous biopolymers exhibit unusual polymerization-depolymerization kinetics where continued elongation (shortening) is possible only in an active' (inactive’) state while occasional random transition (switching) between the active and inactive states takes place. Here, we present a minimal stochastic model capturing only the key features of random elongation-shortening-switching kinetics of a generic filament. The structure and/or function of the filament ends when its length hits a critical value during its shortening; the time to reach this critical length can be formulated as a first-passage time (FPT) and for its calculation the critical length is treated as an absorbing boundary. Moreover, the lengths of the filaments that inspire our model are constrained to a finite maximum value $\mathcal{N}$; therefore, in our model we impose a reflecting boundary beyond which the filament cannot grow. We determined the distribution of the FPT and derived its first two moments. We identify a critical point where the drift of the tip of the filament vanishes making its movement purely diffusive; at this critical point, in the limit $\mathcal{N}\to \mathrm{\infty }$, the fluctuations in the FPT diverges and MFPT becomes infinite indicating `critical slowing down’. Our analytical results will serve as the testing ground for the accuracy of the approximate theoretical results of more realistic models for polymerization-depolymerization kinetics of specific bio-filaments.