Abstract of thesisCalcium ions are an important second messenger in living cells. Indeed calcium signals in the form of waves have been the subject of much recent experimental interest. A fundamental approach for studying cellular signalling is the combination of state of the art experimental techniques, in particular high resolution fluorescence imaging, with spatio-temporal mathematical models of intracellular calcium regulation. Experimental findings can be incorporated into mathematical models and, vice versa, model predictions can be directly tested in experiments. This approach provides a powerful tool to clarify the very complex mechanisms involved in cellular Ca2+ signalling.
The aim of this thesis is to provide insight into oscillations and waves of cytosolic Ca2+ in both single and multi-cellular systems from a mathematical perspective. We focus on two models of Ca2+ release for a systematic mathematical and numerical analysis of Ca2+ dynamics. One of them is a biophysically detailed model which we study using tools from bifurcation theory, numerical continuation and numerical simulation. The other is a much simpler minimal model of Ca2+ dynamics that emphasises the fundamental space and time scales of cellular Ca2+ dynamics and allows for exact mathematical analysis. For the detailed biophysical model we calculate the speed and stability of travelling waves as a function of physiologically significant parameters. The minimal model of Ca2+ dynamics is obtained via a systematic reduction of the biophysical model and its analytically obtained behaviour is shown to be in excellent agreement with the original biophysical model. This minimal model is then used to gain insight into the effects of spatial heterogeneity and biologically realistic sources of noise on intra- and inter-cellular cell signalling. In particular we pursue issues of wave propagation, wave propagation failure and the role of noise in generating coherent whole cell rhythms.
Keywords: calcium, puffs/sparks, Fire-diffuse-fire model, noise, stochastic propagation, intracellular and intercellular waves, non-equilibrium phase-transition.