Abstract:

Reduced models for enzymatic mechanisms, widely used in real-world biochemical applications, are often governed by parametric restrictions that determine their accuracy and applicability. The work in this thesis focuses on addressing challenges in analyzing reduced models of two cornerstone enzymatic mechanisms: Michaelis–Menten (MM) mechanism and Intermolecular Autocatalytic Zymogen Activation (IAZA) mechanism. For the MM model, we investigate the parametric conditions under which the standard quasi-steady-state approximation (QSSA) is not only valid but also predominant among competing reductions, a distinction that guides optimal model selection for experimental design. Using the theory of fences and anti-funnels, we derive sharp error bounds for the standard QSSA and introduce a new and more restrictive qualifier that ensures its predominance. For the IAZA model, we analyze a perturbation regime that exhibits a dynamic transcritical bifurcation. Since classical reduction theory fails at bifurcations, we implement the blow-up method to estimate the solutions and validate the well-known QSSA(s) near the bifurcation.

A second focus of this thesis is the development of a gradient-based optimization framework for fitting noisy differential equation models of neuronal sleep-wake behavior to discrete sleep score data for rodents. Leveraging automatic differentiation and maximum mean discrepancy, we optimize a physiologically motivated flip-flop model of interacting neuronal populations by incorporating exogenous noise. Our results show that modeling noise in both neural firing activity and the homeostatic sleep drive is essential to replicate the complex, polyphasic structure of rodent sleep.