Analytical aspects of the Brownian motor effect in randomly flashing ratchets

TL;DR

This paper rigorously analyzes the Brownian ratchet model of molecular motors, demonstrating how unidirectional transport can occur under various potential symmetries and transition rates, with explicit conditions derived for the transport direction.

Contribution

It provides a mathematical proof that unidirectional transport arises in the randomly flashing ratchet model, including cases with symmetric potentials and asymmetric transition rates, expanding understanding of motor protein mechanisms.

Findings

Transport direction determined by potential minima and diffusive mean

Unidirectional transport possible with symmetric potentials and asymmetric rates

Explicit conditions for transport direction derived

Abstract

The muscle contraction, operation of ATP synthase, maintaining the shape of a cell are believed to be secured by motor proteins, which can be modelled using the Brownian ratchet mechanism. We consider the randomly flashing ratchet model of a Brownian motor, where the particles can be in two states, only one of which is sensitive the applied spatially periodic potential (the mathematical setting is a pair of weakly coupled reaction-diffusion and Fokker-Planck equations). We prove that this mechanism indeed generates unidirectional transport by showing that the amount of mass in the wells of the potential decreases/increases from left to right. The direction of transport is unambiguously determined by the location of each minimum of the potential with respect to the so-called diffusive mean of its adjacent maxima. The transport can be generated not only by an asymmetric potential, but also by a symmetric potential and asymmetric transition rates, and as a consequence of the general result we derive explicit conditions when the latter happens. When the transitions are localized on narrow active sites in the protein conformation space, we find a more explicit characterization of the bulk transport direction, and infer that some common preconditions of the motor effect are redundant.