Mathematical Models for Facilitated Diffusion and the Brownian Ratchet

A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy.In this dissertation, mathematical models for two biophysically related topics are investigated: facilitated diffusion and the Brownian ratchet. Both phenomena exhibit counterintuitive behavior: in the case of facilitated diffusion, transport of a fast-moving ligand (e.g. oxygen) is enhanced through reversible association with a slow-moving macromolecule (e.g. hemoglobin); in the case of the Brownian ratchet, a diffusing particle with random fluctuations among several internal states can yield directed motion against an external force. In Part I, a chemical kinetic model for facilitated transport of a ligand mediated by a carrier molecule is developed. This is an alternative formulation of the problem that has been extensively studied using partial differential equations, by J.D. Murray and J. Wyman, among many others. The chemical kinetic model illuminates the open, driven nature of the system: a fixed gradient in the ligand drives the transport, and the presence of the carrier molecule introduces a "parallel pathway" guaranteeing an enhanced rate of transport for the ligand. A general model for N ligands is also presented. Analytical results from the models are compared to previous experimental measurements and theoretical results from Wyman-Murray facilitated diffusion of oxygen and carbon monoxide in muscle. In Part II, a continuous diffusion formalism Brownian ratchet model is introduced as an alternative formulation of the model proposed by C.S. Peskin, G.M. Odell, and G. Oster to describe the actin polymerization driven stochastic movement of the bacteria Listeria monocytogenes. It is shown that an attractive force between the polymer and the bacterium has the same effect as an increased resistant force on the bacterium. The growth of a bundle consisting of N identical polymer filaments is also considered. The bundle grows as a single leading filament with similar velocity to that of single polymer in the absence of a load, but the bundle can oppose N times the external force under the stalling condition. Relationships are obtained for the velocity of the bacterium movement and its apparent diffusivity as functions of the resistant force and the number of filaments in the bundle. The theoretical study provides new insights for statistical data analysis in future experiments