Calcium oscillations

Changes in cellular calcium concentration control a wide range of physiological processes, from the subsecond release of synaptic neurotransmitters, to the regulation of gene expression over months or years. Calcium can also trigger cell death through both apoptosis and necrosis, and so the regulation of cellular calcium concentration must be tightly controlled through the concerted action of pumps, channels and buffers that transport calcium into and out of the cell cytoplasm. A hallmark of cellular calcium signalling is its spatiotemporal complexity: stimulation of cells by a hormone or neurotransmitter leads to oscillations in cytoplasmic calcium concentration that can vary markedly in time course, amplitude, frequency, and spatial range. In this chapter we review some of the biological roles of calcium, the experimental characterisation of complex dynamic changes in calcium concentration, and attempts to explain this complexity using computational models. We consider the "toolkit" of cellular proteins which influence calcium concentration, describe mechanistic models of key elements of the toolkit, and fit these into the framework of whole cell models of calcium oscillations and waves. Finally, we will touch on recent efforts to use stochastic modelling to elucidate elementary calcium signal events, and how these may evolve into global signals.

Sloshing and slamming oscillations in collapsible channel flow

We consider laminar high-Reynolds-number flow through a finite-length planar channel, where a portion of one wall is replaced by a thin massless elastic membrane that is held under longitudinal tension T and subject to an external pressure distribution. The flow is driven by a fixed pressure drop along the full length of the channel. We investigate the global stability of two-dimensional Poiseuille flow using a method of matched local eigenfunction expansions, which is compared to direct numerical simulations. We trace the neutral stability curve of the primary oscillatory instability of the system, illustrating a transition from high-frequency ‘sloshing’ oscillations at high T to vigorous ‘slamming’ motion at low T . Small-amplitude sloshing at high T can be captured using a low-order eigenmode truncation involving four surface-based modes in the compliant segment of the channel coupled to Womersley flow in the rigid segments. At lower tensions, we show that hydrodynamic modes contribute increasingly to the global instability and we demonstrate a change in the mechanism of energy transfer from the mean flow, with viscous effects being destabilising. Simulations of finite-amplitude oscillations at low T reveal a generic slamming motion, in which the the flexible membrane is drawn close to the opposite rigid wall before rapidly recovering. A simple model is used to demonstrate how fluid inertia in the downstream rigid channel segment, coupled to membrane curvature downstream of the moving constriction, together control slamming dynamics.