Self-induced stochastic resonance for Brownian ratchets under load

Abstract

Abstract. We consider a Brownian ratchet model where the particle on the ratchet is coupled to a cargo. We show that in a distinguished limit where the diffusion coefficient of the cargo is small, and the amplitude of thermal fluctuations is small, the system becomes completely coherent: the times at which the particle jumps across the teeth of the ratchet become deterministic. We also show that the dynamics of the ratchet-cargo system do not depend on the fine structure of the Brownian ratchet. These results are relevant in the context of molecular motors transporting a load, which are often modeled as a ratchet-cargo compound. They explain the regularity of the motor gait that has been observed in numerical experiments, as well as justify the coarsening into Markov jump processes which is commonly done in the literature.Key words. stochastic resonance, self-induced stochastic resonance, Brownian ratchets, molecular motors AMS subject classifications. 60H10, 60F10, 92C10, 60G35, 34E13 The simplest example of a Brownian ratchet is the "perfect" ratchet. Here a particle is moving on a one-dimensional track on which it diffuses freely except at a set of discrete locations (called the "teeth"). At these teeth, the particle is only allowed to pass in one direction (say, from left to right), inducing an average drift of the particle to the right. Another, slightly more sophisticated, example of a Brownian ratchet is that of a particle diffusing in a potential with a sequence of local minima, such that each local minimum has lower energy than the one to its left. The bias encoded in the potential also induces an average drift to the right in the particle's motion.While a particle in a Brownian ratchet moves with a nonzero mean velocity, its position also has a significant variance. For instance, in the second model above, if the noise is small, each minimum of the potential becomes metastable, and jumps occur amongst minima following Arrhenius' law, i.e. the times between these jumps are approximately independent and exponentially distributed.The purpose of this paper is to show that a simple modification of Brownian ratchets makes them much more regular. Specifically, we show that if we couple the particle in the ratchet to a heavy cargo which applies a force to this particle, the compound moves deterministically in a distinguished limit when (i) the diffusion coefficient of the cargo is much smaller than that of the particle in the ratchet, and (ii) the energy due to thermal effects is much less than the energy barrier to move the particle from one ratchet tooth to the next while keeping the cargo fixed. Under these

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