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UBC Theses and Dissertations

A homotopy-minimization method for parameter estimation in differential equations and its application in unraveling the reaction mechanism of the Min system Carlquist, William Christopher


A mathematical model of a dynamical process, often in the form of a system of differential equations, serves to elucidate underlying dynamical structure and behavior of the process that may otherwise remain opaque. However, model parameters are often unknown and may need to be estimated from data for a model to be informative. Numerical-integration-based methods, which estimate parameters in a differential equation model by fitting numerical solutions to data, can demand extensive computation, especially for large stiff systems that require implicit methods for stability. Non-numerical integration methods, which estimate parameters in a differential equation model by fitting solution approximations to data, do not provide an impartial measure of how well a model fits data, a measure required for the testability of a model. In this dissertation, I propose a new method that steps back from a numerical-integration-based method, and instead allows an optimal data-fitting numerical solution to emerge as part of an optimization process. This method bypasses the need for implicit solution methods, which can be computationally intensive, seems to be more robust than numerical-integration-based methods, and, interestingly, admits conservation principles and integral representations, which allow me to gauge the accuracy of my optimization. The Escherichia coli Min system is one of the simplest known biological systems that demonstrates diverse complex dynamic behavior or transduces local interactions into a global signal. Various mathematical models of the Min system show behaviors that are qualitatively similar to dynamic behaviors of the Min system that have been observed in experiments, but no model has been quantitatively compared to time-course data. In this dissertation, I extract time-course data for model fitting from experimental measurements of the Min system and fit established and novel biochemistry-based models to the time-course data using my parameter estimation method for differential equations. Comparing models to time-course data allows me to make precise distinctions between biochemical assumptions in the various models. My modeling and fitting supports a novel model, which suggests that a regular, ordered, stability-switching mechanism underlies the emergent, dynamic behavior of the Min system.

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