Transport in Biology, from Theory to Application
This dissertation focuses on applications of partial differential equations with an emphasis on chemotaxis. Chemotaxis is a response of motile cells or organisms in which the direction of movement is affected by the gradient of a diffusible substance. In multicellular organisms, chemotaxis is critical to development as well as normal function. We also include a specific application that utilizes differential equations and stochastic simulation to model placental transmission of Cytomegalovirus (CMV) with experimental data.Our methods include regularity estimates, maximal and comparison principles, probabilistic techniques, and stochastic processes. This work resulted in several conclusions: We prove quantitatively chemotaxis enhances reaction in 1D setting: the upper bound for reaction time with chemotaxis is smaller than the lower bound for reaction time without chemotaxis.
We have also analyzed a random searcher in shear flow and chemotaxis; on the rigorous level, we are able to establish that the very large shear rates are a dimension reduction mechanism: the expected search time converges to the one of the corresponding one-dimensional problem. Numerically, we see fast decrease of the expected hitting time for shear and chemotactic coupling; we discover there is an optimal shear rate range where the searching time is minimal.
We also explore a model aiming to describe the origin of chemotactic ability of cells, in particular proving global regularity.
We present an end-to-end model of placental CMV transmission that allow us to study different type of infections, the timing of inoculation, the effect of immune suppression on the risk of placental transmission, and the effect of treatment.
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