Models of Calcium Dynamics: Analysis Using Physiologically Accurate Simulations and Non-Standard Geometric Singular Perturbation Theory

Abstract

This thesis considers models of intracellular calcium ions (Ca²⁺). We aim to show how mathematical modelling can help us understand Ca²⁺ dynamics and how the investigation of Ca²⁺ dynamics models can motivate the development of new mathematical tools. The first part of the thesis presents a model of Ca²⁺ dynamics in parotid acinar cells. This model is simulated using a finite element method on an anatomically accurate reconstruction of a cluster of cells. Parotid acinar cells are exocrine cells; therefore, the Ca²⁺ model is coupled with a fluid flow model. From simulations, we gathered three main results. Firstly, the structure of the cell determines which of the possible mechanisms can create the observed Ca²⁺ concentration oscillations. Secondly, a wave propagation mechanism is needed to transport the Ca²⁺ oscillation from the apical to the basal region; we propose a mechanism based on calcium-induced calcium-release channels. Finally, there is a strong co-dependence between fluid secretion and Ca²⁺ dynamics; therefore, it is necessary to model fluid secretion alongside Ca²⁺ dynamics. Geometric singular perturbation theory (GSPT) in its classical form, which assumes that each variable is associated with a distinct timescale, has previously been used to study Ca²⁺ dynamics problems with multiple timescales. However, this association is not valid in general and particularly for models of Ca²⁺ dynamics; instead, a non-standard form of GSPT, which does not rely on the separation of variables by timescale, is more appropriately used for the analysis of Ca²⁺ models. We applied non-standard GSPT to a simplified canonical model of Ca²⁺ dynamics to explain the structure of its relaxation oscillations. We linked timescales to distinct physiological processes underlying different terms in the model, making possible a physiological interpretation of the analysis. Our approach overcomes problems that arise when using classical GSPT. Specifically, we were able to study models that exhibit more timescales than variables and in which a variable can be characterised as either fast or slow depending on the position in phase space. Our strategy of identifying timescales in a model based on careful consideration of the underlying physiology is quite general and is expected to be useful for other Ca²⁺ dynamics models or process-based models with multiple timescales.