Lexicographic Sensitivity Functions for Nonsmooth Models in Mathematical Biology

Systems of ordinary differential equations (ODEs) may be used to model a wide variety of real-world phenomena in biology and engineering. Classical sensitivity theory is well-established and concerns itself with quantifying the responsiveness of such models to changes in parameter values. By performing a sensitivity analysis, a variety of insights can be gained into a model (and hence, the real-world system that it represents); in particular, the information gained can uncover a system\u27s most important aspects, for use in design, control or optimization of the system. However, while the results of such analysis are desirable, the approach that classical theory offers is limited to the case of ODE systems whose right-hand side functions are at least once continuously differentiable. This requirement is restrictive in many real-world systems in which sudden changes in behavior are observed, since a sharp change of this type often translates to a point of nondifferentiability in the model itself. To contend with this issue, recently-developed theory employing a specific class of tools called lexicographic derivatives has been shown to extend classical sensitivity results into a broad subclass of locally Lipschitz continuous ODE systems whose right-hand side functions are referred to as lexicographically smooth. In this thesis, we begin by exploring relevant background theory before presenting lexicographic sensitivity functions as a useful extension of classical sensitivity functions; after establishing the theory, we apply it to two models in mathematical biology. The first of these concerns a model of glucose-insulin kinetics within the body, in which nondifferentiability arises from a biochemical threshold being crossed within the body; the second models the spread of rioting activity, in which similar nonsmooth behavior is introduced out of a desire to capture a tipping point behavior where susceptible individuals suddenly begin to join a riot at a quicker rate after a threshold riot size is crossed. Simulations and lexicographic sensitivity functions are given for each model, and the implications of our results are discussed

Unravelling the insulin signalling pathway using mechanistic modelling

Type two diabetes affects 5% of the world's population and is increasing in prevalence. A key precursor to this disease is insulin resistance, which is characterised by a loss of responsiveness to insulin in liver, muscle and adipose tissue. This thesis focuses on understanding insulin signalling using the 3T3-L1 adipocyte cell model. Computational modelling was used to generate quantitative predictions in the signalling pathways of the adipocyte, many of which are mediated by enzymatic reactions. This study began by comparing existing enzyme kinetic models and evaluating their applicability to insulin signalling in particular. From this understanding, we developed an improved enzyme kinetic model, the differential quasi-steady state model (dQSSA), that avoids the reactant stationary assumption used in the Michaelis Menten model. The dQSSA was found to more accurately model the behaviours of enzymes in large in silico systems, and in various coenzyme inhibited and non-inhibited reactions in vitro. To apply the dQSSA, the SigMat software package was developed in the MATLAB environment to construct mathematical models from qualitative descriptions of networks. After the robustness of the package was verified, it was used to construct a basic model of the insulin signalling pathway. This model was trained against experimental temporal data at 1 nM and 100 nM doses of insulin. It revealed that the simple description of Akt activation, which displays an overshoot behaviour, was insufficient to describe the kinetics of substrate phosphorylation, which does not display the overshoot behaviour. The model was expanded to include Akt translocation and the individual phosphorylation at the 308 and 473 residues. This model resolved the discrepancy and predicts that Akt substrates are only accessible to Akt localised in the cytosol and that PIP3 sequestration of cytosolic Akt acts as a negative feedback

Responsive Hydrogels for Label-Free Signal Transduction within Biosensors

Hydrogels have found wide application in biosensors due to their versatile nature. This family of materials is applied in biosensing either to increase the loading capacity compared to two-dimensional surfaces, or to support biospecific hydrogel swelling occurring subsequent to specific recognition of an analyte. This review focuses on various principles underpinning the design of biospecific hydrogels acting through various molecular mechanisms in transducing the recognition event of label-free analytes. Towards this end, we describe several promising hydrogel systems that when combined with the appropriate readout platform and quantitative approach could lead to future real-life applications