Stochastic Transport in Complex and Dynamic Geometries
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Stochastic Transport in Complex and Dynamic Geometries


Stochastic transport is a widely studied phenomenon among physicists. This includes diffusive processes like Brownian motion which have helped describe numerous systems ranging from the spreading of dye molecules in a liquid to the spreading of human populations. Transport behavior can be affected by properties such as the local curvature of a surface or the dynamics of a network on which the transport takes place. A quantitative characterization of these factors is critical for a deeper understanding of transport in such cases and is of much interest to the study of random walk theory, stochastic processes, and anomalous diffusion in general. In this dissertation, we aim to accomplish this aim by focusing on two specific cases - (i) anomalous diffusion of a random walker on curved surfaces and (ii) transport of cargo on dynamic filament networks.Levy walks are a class of anomalous diffusion studied in Euclidean space. In many cases of interest, transport takes place on surfaces with non-zero Gaussian curvature. We take the first steps towards studying how surface curvature affects anomalous transport described by Levy walk statistics. We develop a computational model to simulate Levy walks along geodesics in Euclidean, spherical, and hyperbolic spaces. By comparing our numerical results to a Taylor expansion of the mean-squared displacement (MSD) in powers of curvature around the Euclidean case, we can establish the validity of a generalized expression for MSD of anomalous diffusion with curvature corrections. The transport of cargo within cells is a critical physiological process. Many new studies consider the impact on transport of the morphology of the networks of filaments. One aspect that has received less attention is the growth/shrinkage and dynamic turnover of these networks. We study transport of cargo carried by myosin motors on dynamic actin network. Use a stochastic simulation model accounting for both active and passive transport and incorporate the dynamics of the actin network. We show how treadmilling speed of actin filament affect cargo transport, motor attachment/detachment rates and network density. We show the existence of filament dynamics in physiological regimes that optimize cargo transport and how it can be tuned.

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