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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.

Molecular motors, Brownian ratchets, and reflected diffusions