A membrane potential model with counterions for cytosolic calcium oscillations

Measuring steady-state and dynamic endoplasmic reticulum and Golgi Zn2+ with genetically encoded sensors

Zn(2+) plays essential roles in biology, and cells have adopted exquisite mechanisms for regulating steady-state Zn(2+) levels. Although much is known about total Zn(2+) in cells, very little is known about its subcellular distribution. Yet defining the location of Zn(2+) and how it changes with signaling events is essential for elucidating how cells regulate this essential ion. Here we create fluorescent sensors genetically targeted to the endoplasmic reticulum (ER) and Golgi to monitor steady-state Zn(2+) levels as well as flux of Zn(2+) into and out of these organelles. These studies reveal that ER and Golgi contain a concentration of free Zn(2+) that is 100 times lower than the cytosol. Both organelles take up Zn(2+) when cytosolic levels are elevated, suggesting that the ER and Golgi can sequester elevated cytosolic Zn(2+) and thus have the potential to play a role in influencing Zn(2+) toxicity. ER Zn(2+) homeostasis is perturbed by small molecule antagonists of Ca(2+) homeostasis and ER Zn(2+) is released upon elevation of cytosolic Ca(2+) pointing to potential exchange of these two ions across the ER. This study provides direct evidence that Ca(2+) signaling can influence Zn(2+) homeostasis and vice versa, that Zn(2+) dynamics may modulate Ca(2+) signaling.

Quantitative cellular description of gastric slow wave activity

Interstitial cells of Cajal (ICC) are responsible for the spontaneous and omnipresent electrical activity in the stomach. A quantitative description of the intracellular processes whose coordinated activity is believed to generate electrical slow waves has been developed and is presented here. In line with recent experimental evidence, the model describes how the interplay between the mitochondria and the endoplasmic reticulum in cycling intracellular Ca(2+) provides the primary regulatory signal for the initiation of the slow wave. The major ion channels that have been identified as influencing slow wave activity have been modeled according to data obtained from isolated ICC. The model has been validated by comparing the simulated profile of the slow waves with experimental recordings and shows good correspondence in terms of frequency, amplitude, and shape in both control and pharmacologically altered conditions.

Spatio-temporal modelling explains the effect of reduced plasma membrane Ca2+ efflux on intracellular Ca2+ oscillations in hepatocytes

In many non-excitable eukaryotic cells, including hepatocytes, Ca(2+) oscillations play a key role in intra- and intercellular signalling, thus regulating many cellular processes from fertilisation to death. Therefore, understanding the mechanisms underlying these oscillations, and consequently understanding how they may be regulated, is of great interest. In this paper, we study the influence of reduced Ca(2+) plasma membrane efflux on Ca(2+) oscillations in hepatocytes. Our previous experiments with carboxyeosin show that a reduced plasma membrane Ca(2+) efflux increases the frequency of Ca(2+) oscillations, but does not affect the duration of individual transients. This phenomenon can be best explained by taking into account not only the temporal, but also the spatial dynamics underlying the generation of Ca(2+) oscillations in the cell. Here we divide the cell into a grid of elements and treat the Ca(2+) dynamics as a spatio-temporal phenomenon. By converting an existing temporal model into a spatio-temporal one, we obtain theoretical predictions that are in much better agreement with the experimental observations.

‘Quantal’ Ca2+release reassessed - a clue to oscillation and synchronization

Ca(2+) release from intracellular Ca(2+) stores, a pivotal event in Ca(2+) signaling, is a 'quantal' process; it terminates after a rapid release of a fraction of stored Ca(2+). To explain the 'quantal' nature, 'all-or-none' model and 'steady-state' model were proposed. This article shortly reviews these hypotheses and considers a recently proposed mechanism, 'luminal potential' model, in which the membrane potential of Ca(2+) store regulates Ca(2+) efflux. By reassessing the 'quantal' nature, other important features of Ca(2+) signaling, oscillation and synchronization, are highlighted. The mechanism for 'quantal' Ca(2+) release may underlie the temporal and spatial control of Ca(2+) signaling.

Biophysical re-equilibration of Ca2+ fluxes as a simple biologically plausible explanation for complex intracellular Ca2+ release patterns

Physiological regulation of Ca(2+) release from the endoplasmic reticulum (ER) is critical for cell function. Recent direct measurements of free [Ca(2+)] inside the ER ([Ca(2+)](ER)) revealed that [Ca(2+)](ER) itself is a key regulator of ER Ca(2+) handling. However, the role of this new regulatory process in generating various patterns of Ca(2+) release remains to be elucidated in detail. Here, we incorporate the recently quantified experimental correlations between [Ca(2+)](ER) and Ca(2+) movements across the ER membrane into a mathematical model ER Ca(2+) handling. The model reproduces basic experimental dynamics of [Ca(2+)](ER). Although this was not goal in model design, the model also exhibits mechanistically unclear experimental phenomena such as "quantal" Ca(2+) release, and "store charging" by increasing resting cytosolic [Ca(2+)]. While more complex explanations cannot be ruled out, on the basis of our data we propose that "quantal release" and "store charging" could be simple re-equilibration phenomena, predicted by the recently quantified biophysical dynamics of Ca(2+) movements across the ER membrane.

Role of voltage-dependent modulation of store Ca2+ release in synchronization of Ca2+ oscillations

Slow waves are rhythmic depolarizations that underlie mechanical activity of many smooth muscles. Slow waves result through rhythmic Ca(2+) release from intracellular Ca(2+) stores through inositol 1,4,5-trisphosphate (IP(3)) sensitive receptors and Ca(2+)-induced Ca(2+) release. Ca(2+) oscillations are transformed into membrane depolarizations by generation of a Ca(2+)-activated inward current. Importantly, the store Ca(2+) oscillations that underlie slow waves are entrained across many cells over large distances. It has been shown that IP(3) receptor-mediated Ca(2+) release is enhanced by membrane depolarization. Previous studies have implicated diffusion of Ca(2+) or the second messenger IP(3) across gap junctions in synchronization of Ca(2+) oscillations. In this study, a novel mechanism of Ca(2+) store entrainment through depolarization-induced IP(3) receptor-mediated Ca(2+) release is investigated. This mechanism is significantly different from chemical coupling-based mechanisms, as membrane potential has a coupling effect over distances several orders of magnitude greater than either diffusion of Ca(2+) or IP(3) through gap junctions. It is shown that electrical coupling acting through voltage-dependent modulation of store Ca(2+) release is able to synchronize oscillations of cells even when cells are widely separated and have different intrinsic frequencies of oscillation.

Modelling of calcium handling in airway myocytes

Airway myocytes are the primary effectors of airway reactivity which modulates airway resistance and hence ventilation. Stimulation of airway myocytes results in an increase in the cytosolic Ca(2+) concentration ([Ca(2+)](i)) and the subsequent activation of the contractile apparatus. Many contractile agonists, including acetylcholine, induce [Ca(2+)](i) increase via Ca(2+) release from the sarcoplasmic reticulum through InsP(3) receptors. Several models have been developed to explain the characteristics of InsP(3)-induced [Ca(2+)](i) responses, in particular Ca(2+) oscillations. The article reviews the modelling of the major structures implicated in intracellular Ca(2+) handling, i.e., InsP(3) receptors, SERCAs, mitochondria and Ca(2+)-binding cytosolic proteins. We developed theoretical models specifically dedicated to the airway myocyte which include the major mechanisms responsible for intracellular Ca(2+) handling identified in these cells. These biocomputations pointed out the importance of the relative proportion of InsP(3) receptor isoforms and the respective role of the different mechanisms responsible for cytosolic Ca(2+) clearance in the pattern of [Ca(2+)](i) variations. We have developed a theoretical model of membrane conductances that predicts the variations in membrane potential and extracellular Ca(2+) influx. Stimulation of this model by simulated increase in [Ca(2+)](i) predicts membrane depolarisation, but not great enough to trigger a significant opening of voltage-dependant Ca(2+) channels. This may explain why airway contraction induced by cholinergic stimulation does not greatly depend on extracellular calcium. The development of such models of airway myocytes is important for the understanding of the cellular mechanisms of airway reactivity and their possible modulation by pharmacological agents.

The influence of different InsP(3) receptor isoforms on Ca(2+) signaling in tracheal smooth muscle cells

In airway myocytes, like in many cells, Ca(2+) signaling is controlled by inositol 1,4,5-trisphosphate (InsP(3)) via InsP(3) receptors (InsP(3)R) located in the sarco-endoplasmic reticulum. Three types of InsP(3)R exist, labeled Types 1, 2, and 3, which differ in their gating kinetics. We analyze a possible impact of the different gating kinetics of Type 1 and Type 3 InsP(3)R on the time course of cytosolic Ca(2+) concentration in tracheal smooth muscle cells upon agonist stimulation. Previous experimental data in rat tracheal myocytes showed that upon gradually increased stimulation with acetylcholine (ACh), a contractile agonist that acts via InsP(3) production, signal spikes, several spikes with declining maxima, and sustained oscillations appear. Our model reproduces the time courses of cytosolic Ca(2+) measured in tracheal myocytes. Moreover, by postulating slight variations in the model parameters which determine the total number of receptors expressed and the ratio between Type 1 and Type 3 InsP(3)R, it offers an explanation to the experimental observation of qualitatively different responses of cells within a presumably homogeneous tissue.

Modelling of simple and complex calcium oscillations. From single-cell responses to intercellular signalling

This review provides a comparative overview of recent developments in the modelling of cellular calcium oscillations. A large variety of mathematical models have been developed for this wide-spread phenomenon in intra- and intercellular signalling. From these, a general model is extracted that involves six types of concentration variables: inositol 1,4,5-trisphosphate (IP3), cytoplasmic, endoplasmic reticulum and mitochondrial calcium, the occupied binding sites of calcium buffers, and the fraction of active IP3 receptor calcium release channels. Using this framework, the models of calcium oscillations can be classified into 'minimal' models containing two variables and 'extended' models of three and more variables. Three types of minimal models are identified that are all based on calcium-induced calcium release (CICR), but differ with respect to the mechanisms limiting CICR. Extended models include IP3--calcium cross-coupling, calcium sequestration by mitochondria, the detailed gating kinetics of the IP3 receptor, and the dynamics of G-protein activation. In addition to generating regular oscillations, such models can describe bursting and chaotic calcium dynamics. The earlier hypothesis that information in calcium oscillations is encoded mainly by their frequency is nowadays modified in that some effect is attributed to amplitude encoding or temporal encoding. This point is discussed with reference to the analysis of the local and global bifurcations by which calcium oscillations can arise. Moreover, the question of how calcium binding proteins can sense and transform oscillatory signals is addressed. Recently, potential mechanisms leading to the coordination of oscillations in coupled cells have been investigated by mathematical modelling. For this, the general modelling framework is extended to include cytoplasmic and gap-junctional diffusion of IP3 and calcium, and specific models are compared. Various suggestions concerning the physiological significance of oscillatory behaviour in intra- and intercellular signalling are discussed. The article is concluded with a discussion of obstacles and prospects.

Birhythmicity, trirhythmicity and chaos in bursting calcium oscillations

We have analyzed various types of complex calcium oscillations. The oscillations are explained with a model based on calcium-induced calcium release (CICR). In addition to the endoplasmic reticulum as the main intracellular Ca2+ store, mitochondrial and cytosolic Ca2+ binding proteins are also taken into account. This model was previously proposed for the study of the physiological role of mitochondria and the cytosolic proteins in gene rating complex Ca2+ oscillations [1]. Here, we investigated the occurrence of different types of Ca2+ oscillations obtained by the model, i.e. simple oscillations, bursting, and chaos. In a bifurcation diagram, we have shown that all these various modes of oscillatory behavior are obtained by a change of only one model parameter, which corresponds to the physiological variability of an agonist. Bursting oscillations were studied in more detail because they express birhythmicity, trirhythmicity and chaotic behavior. Two different routes to chaos are observed in the model: in addition to the usual period doubling cascade, we also show intermittency. For the characterization of the chaotic behavior, we made use of return maps and Lyapunov exponents. The potential biological role of chaos in intracellular signaling is discussed.

Complex calcium oscillations and the role of mitochondria and cytosolic proteins

Intracellular calcium oscillations, which are oscillatory changes of cytosolic calcium concentration in response to agonist stimulation, are experimentally well observed in various living cells. Simple calcium oscillations represent the most common pattern and many mathematical models have been published to describe this type of oscillation. On the other hand, relatively few theoretical studies have been proposed to give an explanation of complex intracellular calcium oscillations, such as bursting and chaos. In this paper, we develop a new possible mechanism for complex calcium oscillations based on the interplay between three calcium stores in the cell: the endoplasmic reticulum (ER), mitochondria and cytosolic proteins. The majority ( approximately 80%) of calcium released from the ER is first very quickly sequestered by mitochondria. Afterwards, a much slower release of calcium from the mitochondria serves as the calcium supply for the intermediate calcium exchanges between the ER and the cytosolic proteins causing bursting calcium oscillations. Depending on the permeability of the ER channels and on the kinetic properties of calcium binding to the cytosolic proteins, different patterns of complex calcium oscillations appear. With our model, we are able to explain simple calcium oscillations, bursting and chaos. Chaos is also observed for calcium oscillations in the bursting mode.

Inositol 1,4,5-trisphosphate and basal Ca2+ release is affected by the cytoplasmic concentration of Cl- in endothelial cells

The effects of varying Cl- concentration in the intracellular bathing medium, on IP3-induced 45Ca2+ release from internal stores, were examined in saponin-permeabilised bovine aortic endothelial (BAE) cells. Results from this study show that the release of Ca2+ from the internal stores is affected by the cytoplasmic concentration of Cl- ions. Complete replacement of Cl- with gluconate augmented IP3 (3 microM)-induced 45Ca2+ release by 33 +/- 8%. Replacement of both Cl- and K+ with gluconate and NMG, respectively, had no significant effect on 45Ca2+ release. However, resting levels of internal 45Ca2+ were found to be affected by Cl- removal. These data suggest that in BAE cells, IP3 and also basal 45Ca2+ release may be regulated by the physiological intracellular Cl- concentration.

Modeling the interrelations between the calcium oscillations and ER membrane potential oscillations

A refined electrochemical model accounting for intracellular calcium oscillations and their interrelations with oscillations of the potential difference across the membrane of the endoplasmic reticulum (ER) or other intracellular calcium stores is established. The ATP dependent uptake of Ca2+ from the cytosol into the ER, the Ca2+ release from the ER through channels following a calcium-induced calcium release mechanism, and a potential-dependent Ca2+ leak flux out of the ER are included in the model and described by plausible rate laws. The binding of calcium to specific proteins such as calmodulin is taken into account. The quasi-electroneutrality condition allows us to express the transmembrane potential in terms of the concentrations of cytosolic calcium and free binding sites on proteins, which are the two independent variables of the model. We include monovalent ions in the model, because they make up a considerable portion in the balance of electroneutrality. As the permeability of the endoplasmic membrane for these ions is much higher than that for calcium ions, we assume the former to be in Nernst equilibrium. A stability analysis of the steady-state solutions (which are unique or multiple depending on parameter values) is carried out and the Hopf bifurcation leading from stable steady states to self-sustained oscillations is analysed with the help of appropriate mathematical techniques. The oscillations obtained by numerical integration exhibit the typical spike-like shape found in experiments and reasonable values of frequency and amplitude. The model describes the process of switching between stationary and pulsatile regimes as well as changes in oscillation frequency upon parameter changes. It turns out that calcium oscillations can arise without a permanent influx of calcium into the cell, when a calcium-buffering system such as calmodulin is included.

On the roles of Ca2+ diffusion, Ca2+ buffers, and the endoplasmic reticulum in IP3-induced Ca2+ waves

We have investigated the effects of Ca2+ diffusion, mobile and stationary Ca2+ buffers in the cytosol, and Ca2+ handling by the endoplasmic reticulum on inositol 1,4,5-trisphosphate-induced Ca2+ wave propagation. Rapid equilibration of free and bound Ca2+ is used to describe Ca2+ sequestration by buffers in both the cytosol and endoplasmic reticulum (ER) lumen. Cytosolic Ca2+ regulation is based on a kinetic model of the inositol 1,4,5-trisphosphate (IP3) receptor of De Young and Keizer that includes activation and inhibition of the IP3 receptor Ca2+ channel in the ER membrane and SERCA Ca2+ pumps in the ER. Diffusion of Ca2+ in the cytosol and the ER and the breakdown and diffusion of IP3 are also included in our calculations. Although Ca2+ diffusion is severely limited because of buffering, when conditions are chosen just below the threshold for Ca2+ oscillations, a pulse of IP3 or Ca2+ results in a solitary trigger wave that requires diffusion of Ca2+ for its propagation. In the oscillatory regime repetitive wave trains are observed, but for this type of wave neither the wave shape nor the speed is strongly dependent on the diffusion of Ca2+. Local phase differences lead to waves that are predominately kinematic in nature, so that the wave speed (c) is related to the wavelength (lambda) and the period of the oscillations (tau) approximately by the formula c = lambda/tau. The period is determined by features that control the oscillations, including [IP3] and pump activity, which are related to recent experiments. Both solitary waves and wave trains are accompanied by a Ca2+ depletion wave in the ER lumen, similar to that observed in cortical preparations from sea urchin eggs. We explore the effect of endogenous and exogenous Ca2+ buffers on wave speed and wave shape, which can be explained in terms of three distinct effects of buffering, and show that exogenous buffers or Ca2+ dyes can have considerable influence on the amplitude and width of the waves.

The inositol trisphosphate receptor of Xenopus oocytes

Xenopus laevis oocytes (stages V and VI) are a widely used model system for the study of Ca2+ signaling. The properties of the Xenopus oocyte InsP3 receptor (InsP3R) are of paramount importance for our thinking about this system and for our efforts to model Ca2+ dynamics in the oocyte cytosol. The recent data regarding the molecular structure, the regulation and the functional properties of the Xenopus oocyte InsP3R are summarized in this review. The main properties of the Xenopus oocyte InsP3R are compared with the properties of the cerebellar InsP3R and are shown to be remarkably similar. The density of the InsP3R in Xenopus oocyte cytoplasm is estimated to a value between 1.1-4.1 x 10(14) tetrameric InsP3R/l. The use of these numbers in a quantitative model of Ca2+ wave propagation leads to values of Ca2+ wave amplitude (0.8-1.5 microM Ca2+) and velocity of the wave propagation (12-24 microns/s) that are in excellent agreement with the values observed experimentally. The density of InsP3Rs in Purkinje cells of the cerebellum is estimated to be about 20,000-fold higher, but in other types of neurons and in peripheral tissues the InsP3R density is estimated to be of the same order of magnitude as, or up to 20-fold higher than, in Xenopus oocytes. The implications of differences in InsP3R density for Ca2+ signaling are discussed.