Tutorials in Mathematical Biosciences II

Living tissues are complex organizations of individual cells and to perform their specific functions the activity of each cell within the tissue must be regulated in a coordinated manner. The mechanisms through which this regulation occurs can be equally complex, but a common way to exert control is via neural transmission or hormonal stimulation. Irrespective of the organization of the extracellular control system, the regulatory signals need to be translated into an intracellular messenger that can modulate the cellular processes. Again, there are a variety of intracellular messengers that achieve this aim, including cAMP, cGMP and NO, but here we focus on the calcium ion as the internal messenger. The objective of this article is to provide an overview of the basic mechanisms of how Ca²⁺ serves as a signaling messenger. For greater detail, the reader must refer to the many extensive reviews (for example, Berridge et al., 2003; Berridge et al., 2002). The details of the individual mechanisms are extremely important since they can confer specificity on the signaling model. As a result, model simulations of Ca²⁺ signaling are most useful when the model is designed for a specific cell type and sufficient experimental detail can be incorporated.

Calcium is critically important for a vast array of cellular functions, as discussed in detail in Chap. 1. There are a number of Ca²⁺ control mechanisms operating on different levels, all designed to ensure that Ca²⁺ is present in sufficient quantity to perform its necessary functions, but not in too great a quantity in the wrong places. Prolonged high concentrations of Ca²⁺ are toxic. For instance, it is known that cellular Ca²⁺ overload can trigger apoptotic cell death, a process in which the cell kills itself. Indeed, control of Ca²⁺ homeostasis is so crucial that even just disruptions in the normal Ca²⁺ fluxes can lead to initiation of active cell death. There are many reviews of Ca²⁺ physiology in the literature: in 2003 an entire issue of Nature Reviews was devoted to the subject and contains reviews of Ca²⁺ homeostasis (Berridge et al., 2003), extracellular Ca²⁺ sensing (Hofer and Brown, 2003), Ca²⁺ signaling during embryogenesis (Webb and Miller, 2003), the Ca²⁺-apoptosis link (Orrenius et al., 2003), and the regulation of cardiac contractility by Ca²⁺ (MacLennan and Kranias, 2003). Other useful reviews are Berridge, (1997) and Carafoli (2002).

Cardiac myocyte excitation-contraction coupling (ECC, Table 1) is an intricate process by which many proteins and substances interact to form a complex but well-tuned system. The regulation of this system is essential to modulation of contractile activity.

Intracellular calcium (Ca²⁺) concentration plays an important regulatory role in a number of cellular processes. Cellular influx of Ca²⁺ activates intracellular signaling pathways that in turn regulate gene expression. Studies have identified over 300 genes and 30 transcription factors which are regulated by intracellular Ca²⁺ [1,2]. Fluctuation of intracellular Ca²⁺ levels is also known to regulate intracellular metabolism by activation of mitochondrial matrix dehydrogenases. The subsequent effects on the tri-carboxylic acid cycle increase the supply of reducing equivalents (NADH, FADH2), stimulating increased flux of electrons through the respiratory chain [3]. Most importantly, Ca²⁺ is a key signaling molecule in excitation-contraction (EC) coupling, the process by which electrical activation of the cell is coupled to mechanical contraction and force generation.

The forces involved in muscle contraction result from the contractile proteins, myosin and actin. Myosin captures the free energy available from the hydrolysis of adenosine triphosphate (ATP), and via interaction with actin, generates the force and motion necessary for the survival of higher organisms. How this protein-mediated conversion of chemical energy into mechanical energy occurs remains a fundamental, unresolved question in physiology and biophysics. As a problem in thermodynamics, mathematical modeling of this chemomechanical free energy transduction has played an important role in helping to organize the experimental database into a coherent framework. In this chapter, I will discuss basic models that have been used to analyze this really quite remarkable process – the generation of force and motion from a protein-protein interaction involving the ancillary biochemical reaction of nucleotide hydrolysis.

When exposed to odorants, olfactory receptor neurons respond with the generation of action potentials. This conversion of odorous information in the inhaled air into electrical nerve impulses is accomplished by an intracellular enzymatic cascade, which leads to the opening of ion channels and the generation of a receptor current. The resulting depolarisation of the neuron activates voltage-gated ion channels to trigger action potentials, which are conveyed to the olfactory bulb in the brain. This review summarises the information gained over recent years about the details of olfactory signal transduction, including many biophysical parameters helpful for a quantitative description of olfactory signalling.

The synapse is the storehouse of memories, both short-term and long-term, and is the location at which learning takes place. There are trillions of synapses in the brain, and in many ways they are one of the fundamental building blocks of this extraordinary organ. As one might expect for such an important structure, the inner workings of the synapse are quite complex. This complexity, along with the small size of a typical synapse, poses many experimental challenges. It is for this reason that mathematical models and computer simulations of synaptic transmission have been used for more than two decades. Many of these models have focused on the presynaptic terminal, particularly on the role of Ca²⁺ in gating transmitter release (Parnas and Segel, 1981; Simon and Llinás, 1985; Fogelson and Zucker, 1985; Yamada and Zucker, 1992; Aharon et al., 1994; Heidelberger et al., 1994; Bertram et al., 1996; Naraghi and Neher, 1997; Bertram et al., 1999a; Tang et al., 2000; Matveev et al., 2002). The terminal is where neurotransmitters are released, and is the site of several forms of short-term plasticity, such as facilitation, augmentation, and depression (Zucker and Regehr, 2002). Mathematical modeling has been used to investigate the properties of various plasticity mechanisms, and to refine understanding of these mechanisms (Fogelson and Zucker, 1985; Yamada and Zucker, 1992; Bertram et al., 1996; Klingauf and Neher, 1997; Bertram and Sherman, 1998; Tang et al., 2000; Matveev et al., 2002). Importantly, modeling has in several cases been the motivation for new experiments (Zucker and Landò, 1986; Hochner et al., 1989; Kamiya and Zucker, 1994; Winslow et al., 1994; Tang et al., 2000). In this chapter, we describe some of the mathematical models that have been developed for transmitter release and presynaptic plasticity, and discuss how these models have shaped, and have been shaped by, experimental studies.