Biomolecular motors and diffusion ratchets

Biomolecular motors are proteins, or structures of multiple proteins, that play a central role in accomplishing mechanical work in the interior of a cell. While chemical reactions fuel this work, it is not exactly known how this chemical-mechanical conversion occurs. Recent advances in molecular biological techniques have enabled at least indirect observations of molecular motors which in turn have led to significant research 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 motor that will be discussed in this thesis. On one end of the motor, there are twin heads that move step by step on the microtubule. The other end consists of a long amino acid chain which attaches itself to cargo that must be transported. 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 Kinesin in the biophysics literature is the Brownian ratchet mechanism. In this thesis, a precise mathematical formulation of a Brownian ratchet (or more generally a diffusion ratchet) will be given via an infinite system of stochastic differential equations with reflection. It will be proved that this formulation arises in the weak limit of a natural discrete space pure jump Markov process that is used to describe motor dynamics in the literature. Using renewal theory it will be shown that the asymptotic velocity of the motor exists in an almost sure sense. We will also establish a functional central limit theorem in order to quantify fluctuations about the asymptotic velocity. This result will yield the effective diffusivity of the motor. Numerical techniques will be provided to compute asymptotic quantities such as asymptotic velocity, effective diffusivity, and the randomness parameter for this model and other closely related models.; Linearly progressive biomolecular motors often carry cargos via an elastic linkage. A two-dimensional coupled stochastic dynamical system will be introduced to model the dynamics of the motor-cargo pair. Weak convergence results will be established in order to relate the models with the natural discrete space pure jump Markov model for the dynamics. Using ergodic theory for Markov processes, it will be shown that the asymptotic velocity exists in the sense of convergence in probability. Numerical results for computing the associated asymptotic quantities will be presented. Frequently in experiments, the motor is too small to be tracked; only the cargo which is much larger can be dynamically observed. Filtering algorithms to infer the position of the motor and to estimate model parameters based on cargo observations will be presented.