# Solvable model of a many-filament Brownian ratchet

## TL;DR

This paper presents an exactly solvable model of a many-filament Brownian ratchet, capturing the stochastic growth and contraction of filaments and their forces on a membrane, with explicit solutions for steady-state behavior and velocity.

## Contribution

The authors develop and solve an N-dimensional diffusion model for heterogeneous filaments interacting with a membrane, providing exact results for system velocity and critical parameters.

## Key findings

- Membrane velocity scales with the square root of the force constant under restoring forces.
- Introducing surface tension reduces membrane velocity.
- Exact steady-state distributions are obtained for linear and quadratic potentials.

## Abstract

We construct and exactly solve a model of an extended Brownian ratchet. The model comprises an arbitrary number of heterogeneous, growing and shrinking filaments which together move a rigid membrane by a ratchet mechanism. The model draws parallels with the dynamics of actin filament networks at the leading edge of the cell. In the model, the filaments grow and contract stochastically. The model also includes forces which derive from a potential dependent on the separation between the filaments and the membrane. These forces serve to attract the filaments to the membrane or generate a surface tension that prevents the filaments from dispersing. We derive an N -dimensional diffusion equation for the N filament-membrane separations, which allows the steady-state probability distribution function to be calculated exactly under certain conditions. These conditions are fulfilled by the physically relevant cases of linear and quadratic interaction potentials. The exact solution of the diffusion equation furnishes expressions for the average velocity of the membrane and critical system parameters for which the system stalls and has zero net velocity. In the case of a restoring force, the membrane velocity grows as the square root of the force constant, whereas it decreases once a surface tension is introduced.