Stochastic Modeling of Reversible Biochemical Reaction-Diffusion Systems and High-Resolution Shock-Capturing Methods for Fluid Interfaces
Del Razo Sarmina, Mauricio J.
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My thesis contains two parts, both of which are motivated by biological problems. One is on stochastic reaction-diffusion for biochemical systems and the other on shock-capturing methods for fluid interfaces. In both parts, conservation laws are key to determine the dynamics and effective numerical methods. The first part is motivated by the need for quantitative mathematical models for cell-scale biological systems. Such a mathematical description must be inherently stochastic where the chancy reaction process is mediated by diffusion encounter. Diffusion-influenced reaction theory describes this coupling between diffusion and reaction. We apply this theory to theoretical and numerical kinetic Monte Carlo studies of the robustness of fluorescence correlation spectroscopy (FCS) theory, a widely used experimental method to determine chemical rate constants and diffusion coefficients of stochastic reaction-diffusion systems. We found that current FCS theory can produce significant errors at cell-scales. In addition, we developed a framework to understand diffusion-influenced reaction theory from a stochastic perspective. For irreversible bimolecular reactions, the theory is derived by introducing absorbing boundary conditions to overdamped Brownian motion theory. This provides a clear stochastic interpretation that describes the probability distribution dynamics and the stochastic sample trajectories. However, the stochastic interpretation is not clear for reversible reactions modeled with a back-reaction boundary condition. In order to address this, we developed a discrete stochastic model that conserves probability and recovers the classical equations in the continuous limit. In the case of reversible reactions, it recovers the back-reaction boundary condition and provides an accurate stochastic interpretation. We also explore extensions of this model and its relation to nonequilibrium stochastic processes as well as extensions into volume reactivity using coupled-diffusion processes. The second part was inspired by a collaboration with experimentalists at Seattle's Veterans Administration (VA) Hospital, who are studying the underlying biological mechanisms behind blast-induced traumatic brain injury (TBI). To better understand the effect of shock waves on the brain, we have investigated an in vitro model in which blood-brain barrier endothelial cells are grown in fluid-filled transwell vessels, placed inside a shock tube and exposed to shocks. As it is difficult to experimentally measure the forces inside the transwell, we developed a computational model of the experimental setup to measure them. First, we implemented a one-dimensional model using Euler equations coupled with a Tammann equation of state (EOS) to model the different materials and interfaces within the experimental setup. From this model, we learned that we can neglect very thin interfaces in our computations. Using this result, we implemented a three-dimensional wave propagation framework modeled with two-dimensional axisymmetric Euler equations and a Tammann EOS. In order to solve these equations, we used high-resolution conservative methods and implemented new Riemann solvers into the Clawpack software in a mixed Eulerian/Lagrangian frame of reference. We found that pressures can fall below vapor pressure due to the interaction of reflecting and diffracting shock waves, suggesting that cavitation bubbles could be a damage mechanism. We also show extensions of this model that allow the implementation of mapped grids and adaptive mesh refinement.
- Applied mathematics