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Über dieses Buch

This book discusses the ways in which mathematical, computational, and modelling methods can be used to help understand the dynamics of intracellular calcium. The concentration of free intracellular calcium is vital for controlling a wide range of cellular processes, and is thus of great physiological importance. However, because of the complex ways in which the calcium concentration varies, it is also of great mathematical interest.This book presents the general modelling theory as well as a large number of specific case examples, to show how mathematical modelling can interact with experimental approaches, in an interdisciplinary and multifaceted approach to the study of an important physiological control mechanism.

Geneviève Dupont is FNRS Research Director at the Unit of Theoretical Chronobiology of the Université Libre de Bruxelles; Martin Falcke is head of the Mathematical Cell Physiology group at the Max Delbrück Center for Molecular Medicine, Berlin; Vivien Kirk is an Associate Professor in the Department of Mathematics at the University of Auckland, New Zealand; James Sneyd is a Professor in the Department of Mathematics at The University of Auckland, New Zealand.



Basic Theory


Chapter 1. Some Background Physiology

Calcium physiology is a vast field, far too large for us to do it justice in this book. Here, instead, we shall be concerned almost entirely with the physiology and dynamical behaviour of ionised intracellular calcium (Ca2+).

Geneviève Dupont, Martin Falcke, Vivien Kirk, James Sneyd

Chapter 2. The Calcium Toolbox

The toolbox concept that was introduced in the previous chapter is a particularly useful way to approach Ca2+ modelling.

Geneviève Dupont, Martin Falcke, Vivien Kirk, James Sneyd

Chapter 3. Basic Modelling Principles: Deterministic Models

The task of combining toolbox components into an overall model is neither simple nor easy.

Geneviève Dupont, Martin Falcke, Vivien Kirk, James Sneyd

Chapter 4. Hierarchical and Stochastic Modelling

As we have already pointed out (Section 1.3), the basic building block of a Ca2+ response is a stochastic event at the level of an individual Ca2+ channel, whether an IP3 receptor, a ryanodine receptor, or a voltage-gated Ca2+ channel. The behaviours that we call oscillations, or spiking, or waves, or microdomain transients, are merely emergent properties of the interaction of channel-level stochastic processes.

Geneviève Dupont, Martin Falcke, Vivien Kirk, James Sneyd

Chapter 5. Nonlinear Dynamics of Calcium

The focus of the earlier chapters of this book has been on the construction of mathematical models based on information about the underlying physiology, and on comparison of model predictions with experimental data. A recurrent theme has been that a relatively small number of broad principles are applicable in the construction of a wide variety of different Ca2+ models in different physiological contexts. This chapter has a different focus, and instead looks at some mathematical methods that have proved useful for the analysis of Ca2+ models. However, there is an analogous theme, which is that a relatively small number of mathematical methods underlie much of current practice in model analysis.

Geneviève Dupont, Martin Falcke, Vivien Kirk, James Sneyd

Specific Models


Chapter 6. Nonexcitable Cells

In the remainder of this book we present a small number of examples, from a range of specific cell types, in more depth. Of course, it is not possible to give a complete overview of all the Ca2+ models that have been constructed, and so our choice here is based on a mixture of convenience and personal preference. However, the selected examples cover a broad range of modelling styles, in many qualitatively different types of cells, and should give a reasonably comprehensive picture of the types of models that have proved useful.

Geneviève Dupont, Martin Falcke, Vivien Kirk, James Sneyd

Chapter 7. Muscle

In all three types of muscle cells – skeletal, cardiac, and smooth – Ca2+ plays a major role in excitation-contraction (EC) coupling, i.e., the sequence of events that links electrical stimulation to contraction (or, in nonexcitable smooth muscle, the events that link agonist stimulation to contraction). In skeletal and cardiac muscle, an action potential arriving from a neuron is propagated along the membrane of the muscle cell, and penetrates deep into the interior of the cell via invaginations of the cell membrane, called T-tubules. This action potential causes the release of Ca2+ from the sarcoplasmic reticulum, which then allows the crossbridge cycle to develop force (Huxley, 1957; Bers, 2001).

Geneviève Dupont, Martin Falcke, Vivien Kirk, James Sneyd

Chapter 8. Neurons and Other Excitable Cells

Calcium plays a major role in every neuron. Not only does it play a part in controlling the membrane potential, but it is also a crucial ingredient of both the pre-synaptic and post-synaptic terminals. At the pre-synaptic terminal the secretionsecretionof neurotransmitter of neurotransmitter is controlled by Ca2+, as is short-term plasticityplasticity, while in the post-synaptic terminal Ca2+ is both necessary and sufficient for long-term synaptic plasticity, i.e., long-term depression, long-term potentiation, and spike-timing-dependent plasticity. In neuroendocrine cells from the hypothalamus and pituitary, as well as in the endocrine pancreas, Ca2+ controls the membrane potential, the shape of the bursts of electrical spiking, and the secretionsecretionof hormones of hormones. In photoreceptorsphotoreceptors and olfactory receptorsolfactory receptors, Ca2+ is at the centre of the biochemical networks that control adaptationadaptation to a maintained stimulus, while in interstitial cells of Cajalinterstitial cells of Cajal Ca2+ controls the membrane potential in ways that we do not yet fully understand. There are many other examples of the importance of Ca2+ in neurons and excitable cells.

Geneviève Dupont, Martin Falcke, Vivien Kirk, James Sneyd


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