Continuum solution of a many-filament Brownian ratchet

I will present a derivation of a two-filament Brownian ratchet in continuous space, that generalises to N filaments. Here, two inhomogeneous filaments grow and contract, and interact with a drift-diffusing membrane by exclusion. This system approaches a steady state whereby the membrane has a net drift over time induced by a `ratcheting' mechanism. By taking the continuum limit of a more intuitive lattice-based system, I derive and solve a diffusion equation and set of boundary conditions for this system, and calculate the steady state velocity of the system as a function of the filament and membrane properties.

This derivation in fact generalises to an arbitrary number of filaments, as well as when there are additional restoring forces between filaments and membrane, or between neighbouring filaments. From this, we can draw parallels between this system and the dynamics of actin filament networks that move the leading edge of eukaryotic cells.