Molecular motors, Brownian ratchets, and reflected diffusions

Molecular motors are protein structures that play a central role in accomplishing mechanical work inside a cell. While chemical reactions fuel this work, it is not exactly known how this chemical-mechanical conversion occurs. Recent advances in microbiological techniques have enabled at least indirect observations of molecular motors which in turn have led to significant effort in the mathematical modeling of these motors in the hope of shedding light on the underlying mechanisms involved in intracellular transport. Kinesin which moves along microtubules that are spread throughout the cell is a prime example of the type of motors that are studied in this work. The motion is linked to the presence of a chemical, ATP, but how the ATP is involved in motion is not clearly understood. One commonly used model for the dynamics of kinesin in the biophysics literature is the Brownian ratchet mechanism. In this work, we give a precise mathematical formulation of a Brownian ratchet (or more generally a diffusion ratchet) via an infinite system of stochastic differential equations with reflection. This formulation is seen to arise in the weak limit of a natural discrete space model that is often used to describe motor dynamics in the literature. Expressions for asymptotic velocity and effective diffusivity of a biological motor modeled via a Brownian ratchet are obtained. Linearly progressive biomolecular motors often carry cargos via an elastic linkage. A two-dimensional coupled stochastic dynamical system is introduced to model the dynamics of the motor-cargo pair. By proving that an associated two dimensional Markov process has a unique stationary distribution, it is shown that the asymptotic velocity of a motor pulling a cargo is well defined as a certain Law of Large Number limit, and finally an expression for the asymptotic velocity in terms of the invariant measure of the Markov process is obtained.