Abstract
Biomolecular motors are molecules that transform energy to perform physiological functions. The operation of molecular motors has a number of interesting physical properties that have attracted a great deal of investigation. One of the most compelling theories describes molecular motors as Brownian motion on a multidimensional free-energy potential landscape. Approximation methods are required to be able to compare this theory with experimental results.
In this thesis, we develop two tight-binding methods to systematically transform the continuous equation describing Brownian motion on a potential with deep wells to a much simpler discrete master equation that can be used to facilitate comparison with experiments. Both methods are based on an expansion in a basis of localized states. One of them, the generalized Wannier-state method, exploits the periodicity of the system to define localized states in terms of Bloch eigenfunctions. The other method, analogous to the atomic-orbital approach of quantum mechanics, defines localized states in terms of the eigenfunctions of local metastable potentials around individual wells.
For simplicity, we focus on the tight-coupling regime where the average diffusion dynamics on the multidimensional potential landscape can be approximated by the diffusion along a one-dimensional coupling channel.
We show that the tight-binding methods connect the master equation to properties of the potential landscape and allow the regime of validity of the master equation to be determined. Thus, in contrast to more phenomenological treatments, our work provides a link between discrete master equations and an underlying theory of molecular motor operation.
Finally, applying our method to particular model potentials, we find that the theory predicts qualitatively different discrete operation for molecular motors in different regimes that could be observed in current single-molecule experiments.