Realistic extensions of a Brownian ratchet for protein translocation

We study a model for the translocation of proteins across membranes through a nanopore using a ratcheting mechanism. When the protein enters the nanopore it diffuses in and out of the pore according to a Brownian motion. Moreover, it is bound by ratcheting molecules which hinder the diffusion of the protein out of the nanopore, i.e. the Brownian motion is reflected such that no ratcheting molecule exits the pore. New ratcheting molecules bind at rate gamma. Extending our previous approach (Depperschmidt and Pfaffelhuber, 2010) we allow the ratcheting molecules to dissociate (at rate delta) from the protein (Model I). We also provide an approximate model (Model II) which assumes a Poisson equilibrium of ratcheting molecules on one side of the current reflection boundary. Using analytical methods and simulations we show that the speed of both models are approximately the same. Our analytical results on Model II give the speed of translocation by means of a solution of an ordinary differential equation.

💡 Research Summary

This paper develops and analyzes two stochastic models for the translocation of proteins across biological membranes through a nanopore, incorporating the action of ratcheting molecules that bind to the protein and prevent its backward diffusion. The biological motivation stems from observations that molecular chaperones such as mtHsp70 in mitochondria or BiP in the endoplasmic reticulum act as ratchets, either actively pulling the substrate or passively blocking back‑sliding. Earlier theoretical work often assumed that once a ratchet molecule binds it never dissociates, an assumption that is unrealistic for many cellular contexts.

The authors therefore introduce a “broken Brownian ratchet” in which ratchet molecules bind at a constant rate γ along the length of the protein and dissociate independently at rate δ. The protein’s position Xₜ is modeled as a one‑dimensional Brownian motion reflected at the location Rₜ of the nearest bound ratchet molecule. Two distinct formulations are presented.

Model I (γ/δ‑broken Brownian ratchet) treats each binding event as a point (τ, r, z) in a three‑dimensional Poisson point process N_{γ,δ}. Here τ is the binding time, r the binding location (always behind the current protein tip), and z the random lifetime of that ratchet molecule, exponentially distributed with parameter δ. The reflection boundary Rₜ jumps forward when a new ratchet binds ahead of the current one, and jumps backward when the active ratchet dissociates. The construction is recursive (equations (2.1)–(2.3)) and yields a continuous reflected Brownian path. Theorem 2.1 proves that the long‑time average speed a_{γ,δ}=lim_{t→∞}Xₜ/t exists and is bounded between positive constants. Moreover, a scaling relation a_{γ,δ}=γ^{1/3}·a_{1,δ/γ^{2/3}} is derived, showing that the speed grows like the cube‑root of the binding rate, while the dissociation rate only influences the prefactor.

Although mathematically rigorous, Model I is non‑local: a single Poisson point may become active multiple times, creating intricate dependencies that hinder explicit analysis.

Model II (γ/δ‑broken Brownian ratchet, approximation) circumvents this difficulty by assuming that, at any moment, the set of potential ratchet sites below the active one is already in its stationary Poisson distribution with intensity γ/δ. Consequently, when the active ratchet dissociates, the new reflection boundary jumps backward by an independent exponential amount Eₙ∼Exp(γ/δ). Forward jumps occur when a new ratchet binds in the interval between the current boundary and the protein tip, at rate γ. This yields a simpler Markovian description (equations (2.4)–(2.6)).

The central analytical result for Model II is Theorem 2.2, which states that the asymptotic speed equals –A′(0)/(2A(0)), where A(z) and B(z) solve the coupled ordinary differential equations

 A″(z)=−2δ B(z)+2γ z A(z), B′(z)=−A′(z)−(γ/δ) B(z),

with boundary conditions A(0)=½, B(0)=0, A decreasing to 0 as z→∞. Solving this system (numerically) yields a unique value of A′(0) and thus an explicit expression for the translocation speed as a function of γ and δ.

Simulation study compares the two models across a range of δ values while fixing γ (thanks to the scaling properties). The simulations confirm that both models produce virtually identical speeds. For small δ, Model II is slightly faster because the equilibrium assumption supplies more “ready” ratchet sites, whereas for large δ Model I appears faster due to a conservative implementation that fixes the reflection boundary at zero, thereby under‑estimating the true speed.

Implications The work provides a mathematically rigorous framework linking microscopic ratchet kinetics (γ, δ) to macroscopic translocation velocity. The cube‑root scaling with γ suggests that increasing the concentration of ratchet molecules yields diminishing returns, a prediction that can be tested experimentally. Moreover, the ODE‑based speed formula from Model II offers a computationally cheap tool for fitting experimental data without resorting to extensive stochastic simulations.

In summary, the paper extends earlier Brownian‑ratchet models by incorporating realistic binding‑and‑unbinding dynamics, proves positivity and scaling of the translocation speed, and supplies an analytically tractable approximation that matches full simulations. These results advance our quantitative understanding of passive protein translocation mechanisms and lay groundwork for future experimental validation and extensions to more complex, force‑dependent scenarios.