Transient inositol 1,4,5-trisphosphate-induced Ca2+ release: a model based on regulatory Ca(2+)-binding sites along the permeation pathway

The interplay between associated proteins, redox state and Ca2+ in the intraluminal ER compartment regulates the IP3 receptor

There have been in the last three decades repeated publications indicating that the inositol 1,4,5-trisphosphate receptor (IP3R) is regulated not only by cytosolic Ca2+ but also by intraluminal Ca2+. Although most studies indicated that a decreasing intraluminal Ca2+ level led to an inhibition of the IP3R, a number of publications reported exactly the opposite effect, i.e. an inhibition of the IP3R by high intraluminal Ca2+ levels. Although intraluminal Ca2+-binding sites on the IP3Rs were reported, a regulatory role for them was not demonstrated. It is also well known that the IP3R is regulated by a vast array of associated proteins, but only relatively recently proteins were identified that can be linked to the regulation of the IP3R by intraluminal Ca2+. The first to be reported was annexin A1 that is proposed to associate with the second intraluminal loop of the IP3R at high intraluminal Ca2+ levels and to inhibit the IP3R. More recently, ERdj5/PDIA19 reductase was described to reduce an intraluminal disulfide bridge of IP3R1 only at low intraluminal Ca2+ levels and thereby to inhibit the IP3R. Annexin A1 and ERdj5/PDIA19 can therefore explain most of the experimental results on the regulation of the IP3R by intraluminal Ca2+. Further studies are needed to provide a fuller understanding of the regulation of the IP3R from the intraluminal side. These findings underscore the importance of the state of the endoplasmic reticulum in the control of IP3R activity.

A mathematical model of calcium dynamics in HSY cells

Saliva is an essential part of activities such as speaking, masticating and swallowing. Enzymes in salivary fluid protect teeth and gums from infectious diseases, and also initiate the digestion process. Intracellular calcium (Ca2+) plays a critical role in saliva secretion and regulation. Experimental measurements of Ca2+ and inositol trisphosphate (IP3) concentrations in HSY cells, a human salivary duct cell line, show that when the cells are stimulated with adenosine triphosphate (ATP) or carbachol (CCh), they exhibit coupled oscillations with Ca2+ spike peaks preceding IP3 spike peaks. Based on these data, we construct a mathematical model of coupled Ca2+ and IP3 oscillations in HSY cells and perform model simulations of three different experimental settings to forecast Ca2+ responses. The model predicts that when Ca2+ influx from the extracellular space is removed, oscillations gradually slow down until they stop. The model simulation of applying a pulse of IP3 predicts that photolysis of caged IP3 causes a transient increase in the frequency of the Ca2+ oscillations. Lastly, when Ca2+-dependent activation of PLC is inhibited, we see an increase in the oscillation frequency and a decrease in the amplitude. These model predictions are confirmed by experimental data. We conclude that, although concentrations of Ca2+ and IP3 oscillate, Ca2+ oscillations in HSY cells are the result of modulation of the IP3 receptor by intracellular Ca2+, and that the period is modulated by the accompanying IP3 oscillations.

Interplay Between Intracellular Ca(2+) Oscillations and Ca(2+)-stimulated Mitochondrial Metabolism

Oscillations of cytosolic Ca(2+) concentration are a widespread mode of signalling. Oscillatory spikes rely on repetitive exchanges of Ca(2+) between the endoplasmic reticulum (ER) and the cytosol, due to the regulation of inositol 1,4,5-trisphosphate receptors. Mitochondria also sequester and release Ca(2+), thus affecting Ca(2+) signalling. Mitochondrial Ca(2+) activates key enzymes involved in ATP synthesis. We propose a new integrative model for Ca(2+) signalling and mitochondrial metabolism in electrically non-excitable cells. The model accounts for (1) the phase relationship of the Ca(2+) changes in the cytosol, the ER and mitochondria, (2) the dynamics of mitochondrial metabolites in response to cytosolic Ca(2+) changes, and (3) the impacts of cytosol/mitochondria Ca(2+) exchanges and of mitochondrial metabolism on Ca(2+) oscillations. Simulations predict that as expected, oscillations are slowed down by decreasing the rate of Ca(2+) efflux from mitochondria, but also by decreasing the rate of Ca(2+) influx through the mitochondrial Ca(2+) uniporter (MCU). These predictions were experimentally validated by inhibiting MCU expression. Despite the highly non-linear character of Ca(2+) dynamics and mitochondrial metabolism, bioenergetics were found to be robust with respect to changes in frequency and amplitude of Ca(2+) oscillations.

An allosteric model of the inositol trisphosphate receptor with nonequilibrium binding

The inositol trisphosphate receptor (IPR) is a crucial ion channel that regulates the Ca(2+) influx from the endoplasmic reticulum (ER) to the cytoplasm. A thorough study of the IPR channel contributes to a better understanding of calcium oscillations and waves. It has long been observed that the IPR channel is a typical biological system which performs adaptation. However, recent advances on the physical essence of adaptation show that adaptation systems with a negative feedback mechanism, such as the IPR channel, must break detailed balance and always operate out of equilibrium with energy dissipation. Almost all previous IPR models are equilibrium models assuming detailed balance and thus violate the dissipative nature of adaptation. In this article, we constructed a nonequilibrium allosteric model of single IPR channels based on the patch-clamp experimental data obtained from the IPR in the outer membranes of isolated nuclei of the Xenopus oocyte. It turns out that our model reproduces the patch-clamp experimental data reasonably well and produces both the correct steady-state and dynamic properties of the channel. Particularly, our model successfully describes the complicated bimodal [Ca(2+)] dependence of the mean open duration at high [IP3], a steady-state behavior which fails to be correctly described in previous IPR models. Finally, we used the patch-clamp experimental data to validate that the IPR channel indeed breaks detailed balance and thus is a nonequilibrium system which consumes energy.

Buffer regulation of calcium puff sequences

Puffs are localized Ca(2 +) signals that arise in oocytes in response to inositol 1,4,5-trisphosphate (IP3). They are the result of the liberation of Ca(2 +) from the endoplasmic reticulum through the coordinated opening of IP3 receptor/channels clustered at a functional release site. The presence of buffers that trap Ca(2 +) provides a mechanism that enriches the spatio-temporal dynamics of cytosolic calcium. The expression of different types of buffers along the cell's life provides a tool with which Ca(2 +) signals and their responses can be modulated. In this paper we extend the stochastic model of a cluster of IP3R-Ca(2 +) channels introduced previously to elucidate the effect of buffers on sequences of puffs at the same release site. We obtain analytically the probability laws of the interpuff time and of the number of channels that participate of the puffs. Furthermore, we show that under typical experimental conditions the effect of buffers can be accounted for in terms of a simple inhibiting function. Hence, by exploring different inhibiting functions we are able to study the effect of a variety of buffers on the puff size and interpuff time distributions. We find the somewhat counter-intuitive result that the addition of a fast Ca(2 +) buffer can increase the average number of channels that participate of a puff.

A data-driven model of a modal gated ion channel: the inositol 1,4,5-trisphosphate receptor in insect Sf9 cells

The inositol 1,4,5-trisphosphate (IP(3)) receptor (IP(3)R) channel is crucial for the generation and modulation of intracellular Ca(2+) signals in animal cells. To gain insight into the complicated ligand regulation of this ubiquitous channel, we constructed a simple quantitative continuous-time Markov-chain model from the data. Our model accounts for most experimentally observed gating behaviors of single native IP(3)R channels from insect Sf9 cells. Ligand (Ca(2+) and IP(3)) dependencies of channel activity established six main ligand-bound channel complexes, where a complex consists of one or more states with the same ligand stoichiometry and open or closed conformation. Channel gating in three distinct modes added one complex and indicated that three complexes gate in multiple modes. This also restricted the connectivity between channel complexes. Finally, latencies of channel responses to abrupt ligand concentration changes defined a model with specific network topology between 9 closed and 3 open states. The model with 28 parameters can closely reproduce the equilibrium gating statistics for all three gating modes over a broad range of ligand concentrations. It also captures the major features of channel response latency distributions. The model can generate falsifiable predictions of IP(3)R channel gating behaviors and provide insights to both guide future experiment development and improve IP(3)R channel gating analysis. Maximum likelihood estimates of the model parameters and of the parameters in the De Young-Keizer model yield strong statistical evidence in favor of our model. Our method is simple and easily applicable to the dynamics of other ion channels and molecules.

Permeant calcium ion feed-through regulation of single inositol 1,4,5-trisphosphate receptor channel gating

The ubiquitous inositol 1,4,5-trisphosphate (InsP(3)) receptor (InsP(3)R) Ca(2+) release channel plays a central role in the generation and modulation of intracellular Ca(2+) signals, and is intricately regulated by multiple mechanisms including cytoplasmic ligand (InsP(3), free Ca(2+), free ATP(4-)) binding, posttranslational modifications, and interactions with cytoplasmic and endoplasmic reticulum (ER) luminal proteins. However, regulation of InsP(3)R channel activity by free Ca(2+) in the ER lumen ([Ca(2+)](ER)) remains poorly understood because of limitations of Ca(2+) flux measurements and imaging techniques. Here, we used nuclear patch-clamp experiments in excised luminal-side-out configuration with perfusion solution exchange to study the effects of [Ca(2+)](ER) on homotetrameric rat type 3 InsP(3)R channel activity. In optimal [Ca(2+)](i) and subsaturating [InsP(3)], jumps of [Ca(2+)](ER) from 70 nM to 300 µM reduced channel activity significantly. This inhibition was abrogated by saturating InsP(3) but restored when [Ca(2+)](ER) was raised to 1.1 mM. In suboptimal [Ca(2+)](i), jumps of [Ca(2+)](ER) (70 nM to 300 µM) enhanced channel activity. Thus, [Ca(2+)](ER) effects on channel activity exhibited a biphasic dependence on [Ca(2+)](i). In addition, the effect of high [Ca(2+)](ER) was attenuated when a voltage was applied to oppose Ca(2+) flux through the channel. These observations can be accounted for by Ca(2+) flux driven through the open InsP(3)R channel by [Ca(2+)](ER), raising local [Ca(2+)](i) around the channel to regulate its activity through its cytoplasmic regulatory Ca(2+)-binding sites. Importantly, [Ca(2+)](ER) regulation of InsP(3)R channel activity depended on cytoplasmic Ca(2+)-buffering conditions: it was more pronounced when [Ca(2+)](i) was weakly buffered but completely abolished in strong Ca(2+)-buffering conditions. With strong cytoplasmic buffering and Ca(2+) flux sufficiently reduced by applied voltage, both activation and inhibition of InsP(3)R channel gating by physiological levels of [Ca(2+)](ER) were completely abolished. Collectively, these results rule out Ca(2+) regulation of channel activity by direct binding to the luminal aspect of the channel.

Unitary Ca(2+) current through recombinant type 3 InsP(3) receptor channels under physiological ionic conditions

The ubiquitous inositol 1,4,5-trisphosphate (InsP(3)) receptor (InsP(3)R) channel, localized primarily in the endoplasmic reticulum (ER) membrane, releases Ca(2+) into the cytoplasm upon binding InsP(3), generating and modulating intracellular Ca(2+) signals that regulate numerous physiological processes. Together with the number of channels activated and the open probability of the active channels, the size of the unitary Ca(2+) current (i(Ca)) passing through an open InsP(3)R channel determines the amount of Ca(2+) released from the ER store, and thus the amplitude and the spatial and temporal nature of Ca(2+) signals generated in response to extracellular stimuli. Despite its significance, i(Ca) for InsP(3)R channels in physiological ionic conditions has not been directly measured. Here, we report the first measurement of i(Ca) through an InsP(3)R channel in its native membrane environment under physiological ionic conditions. Nuclear patch clamp electrophysiology with rapid perfusion solution exchanges was used to study the conductance properties of recombinant homotetrameric rat type 3 InsP(3)R channels. Within physiological ranges of free Ca(2+) concentrations in the ER lumen ([Ca(2+)](ER)), free cytoplasmic [Ca(2+)] ([Ca(2+)](i)), and symmetric free [Mg(2+)] ([Mg(2+)](f)), the i(Ca)-[Ca(2+)](ER) relation was linear, with no detectable dependence on [Mg(2+)](f). i(Ca) was 0.15 +/- 0.01 pA for a filled ER store with 500 microM [Ca(2+)](ER). The i(Ca)-[Ca(2+)](ER) relation suggests that Ca(2+) released by an InsP(3)R channel raises [Ca(2+)](i) near the open channel to approximately 13-70 microM, depending on [Ca(2+)](ER). These measurements have implications for the activities of nearby InsP(3)-liganded InsP(3)R channels, and they confirm that Ca(2+) released by an open InsP(3)R channel is sufficient to activate neighboring channels at appropriate distances away, promoting Ca(2+)-induced Ca(2+) release.

Designing dynamical output feedback controllers for store-operated Ca²+ entry

Store-operated calcium entry (SOCE) has been proposed as the main process controlling Ca²+ entry in non-excitable cells. Although recent breakthroughs in experimental studies of SOCE have been made, its mathematical modeling has not been developed. In the present work, SOCE is viewed as a feedback control system subject to an extracellular agonist disturbance and an extracellular calcium input. We then design a dynamic output feedback controller to reject the disturbance and track Ca²+ resting levels in the cytosol and the endoplasmic reticulum (ER). The constructed feedback control system is validated by published experimental data and its global asymptotic stability is proved by using the LaSalle's invariance principle. We then simulate the dynamic responses of STIM1 and Orai1, two major components in the operation of the store-operated channels, to the depletion of Ca²+ in the ER with thapsigargin, which show that: (1) Upon the depletion of Ca²+ in the ER, the concentrations of activated STIM1 and STIM1-Orai1 cluster are elevated gradually, indicating that STIM1 is accumulating in the ER-PM junctions and that the cytosolic portion of the active STIM1 is binding to Orai1 and driving the opening of CRAC channels for Ca²+ entry; (2) after the extracellular Ca²+ addition, the concentrations of both STIM1 and STIM1-Orai1 cluster decrease but still much higher than the original levels. We also simulate the system responses to the agonist disturbance, which show that, when a sequence of periodic agonist pulses is applied, the system returns to its equilibrium after each pulse. This indicates that the designed feedback controller can reject the disturbance and track the equilibrium.

Inositol trisphosphate receptor and ion channel models based on single-channel data

The inositol trisphosphate receptor (IPR) plays an important role in controlling the dynamics of intracellular Ca(2+). Single-channel patch-clamp recordings are a typical way to study these receptors as well as other ion channels. Methods for analyzing and using this type of data have been developed to fit Markov models of the receptor. The usual method of parameter fitting is based on maximum-likelihood techniques. However, Bayesian inference and Markov chain Monte Carlo techniques are becoming more popular. We describe the application of the Bayesian methods to real experimental single-channel data in three ion channels: the ryanodine receptor, the K(+) channel, and the IPR. One of the main aims of all three studies was that of model selection with different approaches taken. We also discuss the modeling implications for single-channel data that display different levels of channel activity within one recording.

Mode switching is the major mechanism of ligand regulation of InsP3 receptor calcium release channels

The inositol 1,4,5-trisphosphate (InsP₃) receptor (InsP₃R) plays a critical role in generation of complex Ca²⁺ signals in many cell types. In patch clamp recordings of isolated nuclei from insect Sf9 cells, InsP₃R channels were consistently detected with regulation by cytoplasmic InsP₃ and free Ca²⁺ concentrations ([Ca²⁺]_i) very similar to that observed for vertebrate InsP₃R. Long channel activity durations of the Sf9-InsP₃R have now enabled identification of a novel aspect of InsP₃R gating: modal gating. Using a novel algorithm to analyze channel modal gating kinetics, InsP₃R gating can be separated into three distinct modes: a low activity mode, a fast kinetic mode, and a burst mode with channel open probability (P_o) within each mode of 0.007 ± 0.002, 0.24 ± 0.03, and 0.85 ± 0.02, respectively. Channels reside in each mode for long periods (tens of opening and closing events), and transitions between modes can be discerned with high resolution (within two channel opening and closing events). Remarkably, regulation of channel gating by [Ca²⁺]_i and [InsP₃] does not substantially alter channel P_o within a mode. Instead, [Ca²⁺]_i and [InsP₃] affect overall channel P_o primarily by changing the relative probability of the channel being in each mode, especially the high and low P_o modes. This novel observation therefore reveals modal switching as the major mechanism of physiological regulation of InsP₃R channel activity, with implications for the kinetics of Ca²⁺ release events in cells.

Rapid ligand-regulated gating kinetics of single inositol 1,4,5-trisphosphate receptor Ca2+ release channels

The ubiquitous inositol 1,4,5-trisphosphate receptor (InsP(3)R) intracellular Ca(2+) release channel is engaged by thousands of plasma membrane receptors to generate Ca(2+) signals in all cells. Understanding how complex Ca(2+) signals are generated has been hindered by a lack of information on the kinetic responses of the channel to its primary ligands, InsP(3) and Ca(2+), which activate and inhibit channel gating. Here, we describe the kinetic responses of single InsP(3)R channels in native endoplasmic reticulum membrane to rapid ligand concentration changes with millisecond resolution, using a new patch-clamp configuration. The kinetics of channel activation and deactivation showed novel Ca(2+) regulation and unexpected ligand cooperativity. The kinetics of Ca(2+)-mediated channel inhibition showed the single-channel bases for fundamental Ca(2+) release events and Ca(2+) release refractory periods. These results provide new insights into the channel regulatory mechanisms that contribute to complex spatial and temporal features of intracellular Ca(2+) signals.

Models of the inositol trisphosphate receptor

The inositol (1,4,5)-trisphosphate receptor (IPR) plays a crucial role in calcium dynamics in a wide range of cell types, and is often a central feature in quantitative models of calcium oscillations and waves. We review deterministic and stochastic mathematical models of the IPR, from the earliest ones of the 1970s and 1980s, to the most recent. The effects of IPR stochasticity on Ca2+ dynamics are briefly discussed.

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.

A model of IP3 receptor with a luminal calcium binding site: stochastic simulations and analysis

We have constructed a stochastic model of the inositol 1,4,5-trisphosphate receptor-Ca2+ channel that is based on quantitative measurements of the channel's properties. It displays the observed dependence of the open probability of the channel with cytosolic [Ca2+] and [IP3] and gives values for the dwell times that agree with the observations. The model includes an explicit dependence of channel gating with luminal calcium. This not only explains several observations reported in the literature, but also provides a possible explanation of why the open probabilities and shapes of the bell-shaped curves reported in [Nature 351 (1991) 751] and in [Proc. Natl. Acad. Sci. U.S.A. 269 (1998) 7238] are so different.

Spontaneous channel activity of the inositol 1,4,5-trisphosphate (InsP3) receptor (InsP3R). Application of allosteric modeling to calcium and InsP3 regulation of InsP3R single-channel gating

The InsP3R Ca2+ release channel has a biphasic dependence on cytoplasmic free Ca2+ concentration ([Ca2+]i). InsP3 activates gating primarily by reducing the sensitivity of the channel to inhibition by high [Ca2+]i. To determine if relieving Ca2+ inhibition is sufficient for channel activation, we examined single-channel activities in low [Ca2+]i in the absence of InsP3, by patch clamping isolated Xenopus oocyte nuclei. For both endogenous Xenopus type 1 and recombinant rat type 3 InsP3R channels, spontaneous InsP3-independent channel activities with low open probability Po ( approximately 0.03) were observed in [Ca2+]i < 5 nM with the same frequency as in the presence of InsP3, whereas no activities were observed in 25 nM Ca2+. These results establish the half-maximal inhibitory [Ca2+]i of the channel to be 1.2-4.0 nM in the absence of InsP3, and demonstrate that the channel can be active when all of its ligand-binding sites (including InsP3) are unoccupied. In the simplest allosteric model that fits all observations in nuclear patch-clamp studies of [Ca2+]i and InsP3 regulation of steady-state channel gating behavior of types 1 and 3 InsP3R isoforms, including spontaneous InsP3-independent channel activities, the tetrameric channel can adopt six different conformations, the equilibria among which are controlled by two inhibitory and one activating Ca2+-binding and one InsP3-binding sites in a manner outlined in the Monod-Wyman-Changeux model. InsP3 binding activates gating by affecting the Ca2+ affinities of the high-affinity inhibitory sites in different conformations, transforming it into an activating site. Ca2+ inhibition of InsP3-liganded channels is mediated by an InsP3-independent low-affinity inhibitory site. The model also suggests that besides the ligand-regulated gating mechanism, the channel has a ligand-independent gating mechanism responsible for maximum channel Po being less than unity. The validity of this model was established by its successful quantitative prediction of channel behavior after it had been exposed to ultra-low bath [Ca2+].

Kinetic model of the inositol trisphosphate receptor that shows both steady-state and quantal patterns of Ca2+ release from intracellular stores

The release of Ca(2+) from intracellular stores via InsP(3) receptors shows anomalous kinetics. Successive additions of low concentrations of InsP(3) cause successive rapid transients of Ca(2+) release. These quantal responses have been ascribed to all-or-none release from stores with differing sensitivities to InsP(3) or, alternatively, to a steady-state mechanism where complex kinetic properties of the InsP(3) receptor allow partial emptying of all the stores. We present here an adaptive model of the InsP(3) receptor that can show either pattern, depending on the imposed experimental conditions. The model proposes two interconvertible conformational states of the receptor: one state binds InsP(3) rapidly, but with low affinity, whereas the other state binds slowly, but with high affinity. The model shows repetitive increments of Ca(2+) release in the absence of a Ca(2+) gradient, but more pronounced incremental behaviour when released Ca(2+) builds up at the mouth of the channel. The sensitivity to Ins P (3) is critically dependent on the density of InsP(3) receptors, so that different stores can respond to different concentration ranges of Ins P (3). Since the model generates very high Hill coefficients (h approximately 7), it allows all-or-none release of Ca(2+) from stores of differing receptor density, but questions the validity of the use of h values as a guide to the number of InsP(3) molecules needed to open the channel. The model presents a mechanism for terminating Ca(2+) release in the presence of positive feedback from released Ca(2+), thereby providing an explanation of why elementary Ca(2+) signals ('blips' and 'puffs') do not inevitably turn into regenerative waves.

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.

Hierarchical organization of calcium signals in hepatocytes: from experiments to models

The proper working of the liver largely depends on the fine tuning of the level of cytosolic Ca(2+) in hepatocytes. Thanks to the development of imaging techniques, our understanding of the spatio-temporal organization of intracellular Ca(2+) in this - and other - cell types has much improved. Many of these signals are mediated by a rise in the level of inositol 1,4,5-trisphosphate (InsP(3)), a second messenger which can activate the release of Ca(2+) from the endoplasmic reticulum. Besides the now well-known hepatic Ca(2+) oscillations induced by hormonal stimulation, intra- and intercellular Ca(2+) waves have also been observed. More recently, subcellular Ca(2+) increases associated with the coordinated opening of a few Ca(2+) channels have been reported. Given the complexity of the regulations involved in the generation of such processes and the variety of time and length scales necessary to describe those phenomena, theoretical models have been largely used to gain a precise and quantitative understanding of the dynamics of intracellular Ca(2+). Here, we review the various aspects of the spatio-temporal organization of cytosolic Ca(2+) in hepatocytes from the dual point of view provided by experiments and modeling. We first focus on the description and the mechanism of intracellular Ca(2+) oscillations and waves. Second, we investigate in which manner these repetitive Ca(2+) increases are coordinated among a set of hepatocytes coupled by gap junctions, a phenomenon known as 'intercellular Ca(2+) waves'. Finally, we focus on the so-called elementary Ca(2+) signals induced by low InsP(3) concentrations, leading to Ca(2+) rises having a spatial extent of a few microns. Although these small-scale events have been mainly studied in other cell types, we theoretically infer general properties of these localized intracellular Ca(2+) rises that could also apply to hepatocytes.

Mechanism of receptor-oriented intercellular calcium wave propagation in hepatocytes

Intercellular calcium signals are propagated in multicellular hepatocyte systems as well as in the intact liver. The stimulation of connected hepatocytes by glycogenolytic agonists induces reproducible sequences of intracellular calcium concentration increases, resulting in unidirectional intercellular calcium waves. Hepatocytes are characterized by a gradient of vasopressin binding sites from the periportal to perivenous areas of the cell plate in hepatic lobules. Also, coordination of calcium signals between neighboring cells requires the presence of the agonist at each cell surface as well as gap junction permeability. We present a model based on the junctional coupling of several hepatocytes differing in sensitivity to the agonist and thus in the intrinsic period of calcium oscillations. In this model, each hepatocyte displays repetitive calcium spikes with a slight phase shift with respect to neighboring cells, giving rise to a phase wave. The orientation of the apparent calcium wave is imposed by the direction of the gradient of hormonal sensitivity. Calcium spikes are coordinated by the diffusion across junctions of small amounts of inositol 1,4, 5-trisphosphate (InsP(3)). Theoretical predictions from this model are confirmed experimentally. Thus, major physiological insights may be gained from this model for coordination and spatial orientation of intercellular signals.-Dupont, G., Tordjmann, T., Clair, C., Swillens, S., Claret, M., Combettes, L. Mechanism of receptor-oriented intercellular calcium wave propagation in hepatocytes.

From calcium blips to calcium puffs: theoretical analysis of the requirements for interchannel communication

In the cytoplasm of cells of different types, discrete clusters of inositol 1,4,5-trisphosphate-sensitive Ca(2+) channels generate Ca(2+) signals of graded size, ranging from blips, which involve the opening of only one channel, to moderately larger puffs, which result from the concerted opening of a few channels in the same cluster. These channel clusters are of unknown size or geometrical characteristics. The aim of this study was to estimate the number of channels and the interchannel distance within such a cluster. Because these characteristics are not attainable experimentally, we performed computer stochastic simulations of Ca(2+) release events. We conclude that, to ensure efficient interchannel communication, as experimentally observed, a typical cluster should contain two or three tens of inositol 1,4,5-trisphosphate-sensitive Ca(2+) channels in close contact.

Inositol trisphosphate receptors: Ca2+-modulated intracellular Ca2+ channels

The three subtypes of inositol trisphosphate (InsP3) receptor expressed in mammalian cells are each capable of forming intracellular Ca2+ channels that are regulated by both InsP3 and cytosolic Ca2+. The InsP3 receptors of many, though perhaps not all, tissues are biphasically regulated by cytosolic Ca2+: a rapid stimulation of the receptors by modest increases in Ca2+ concentration is followed by a slower inhibition at higher Ca2+ concentrations. Despite the widespread occurrence of this form of regulation and the belief that it is an important element of the mechanisms responsible for the complex Ca2+ signals evoked by physiological stimuli, the underlying mechanisms are not understood. Both accessory proteins and Ca2+-binding sites on InsP3 receptors themselves have been proposed to mediate the effects of cytosolic Ca2+ on InsP3 receptor function, but the evidence is equivocal. The effects of cytosolic Ca2+ on InsP3 binding and channel opening, and the possible means whereby the effects are mediated are discussed in this review.

Ca2+ feedback on "quantal" Ca2+ release involving ryanodine receptors

The influence of luminal and cytoplasmic Ca2+ on the ability of ryanodine-sensitive stores to undergo multiple partial ("quantal") releases has been assessed. Increased luminal Ca2+ levels do indeed modulate sarcoplasmic reticulum Ca2+ release by lowering the threshold agonist concentration required to elicit release, but the decrease in luminal Ca2+ that accompanies a partial release is not sufficient by itself to terminate release. Similarly, an increase in cytoplasmic Ca2+ lowers the threshold agonist concentration required to elicit release; thus, the bulk cytoplasmic Ca2+ levels attained during a release would only stimulate further release, not terminate it before it reached completion. Very high cytoplasmic Ca2+ levels (1-3 mM) also triggered release but were unable to terminate release before reaching completion. Thus, even the high local cytoplasmic Ca2+ concentration that might accompany release would also not terminate release. It is concluded that Ca2+ feedback can modulate release through ryanodine receptors but that it does not account for the properties of quantal release. The low affinity inhibitor tetracaine induces a decrease in the extent of release that cannot be explained solely by heterogeneous caffeine sensitivity of the stores. The results are interpreted in terms of a scheme that includes (i) heterogeneous sensitivity of stores, conferred in part by differences in luminal Ca2+ content and (ii) adaptive behavior on the part of individual ryanodine receptors.

Calcium signalling: how do IP3 receptors work?

The intracellular receptor for inositol 1,4,5-trisphosphate (IP3) is responsible for generation and control of very complex Ca2+ signals. New experimental approaches to studying the kinetics of the IP3 receptor are now beginning to give some insight into the mechanisms behind its rather bizarre properties.

Incremental Ca2+ mobilization by inositol trisphosphate receptors is unlikely to be mediated by their desensitization or regulation by luminal or cytosolic Ca2+

The kinetics of Ins(1,4,5)P3 (InsP3)-stimulated Ca2+ release from intracellular stores are unusual in that submaximal concentrations of InsP3 rapidly release only a fraction of the InsP3-sensitive Ca2+ stores. By measuring unidirectional 45Ca2+ efflux from permeabilized rat hepatocytes, we demonstrate that such quantal responses to InsP3 occur at all temperatures between 2 and 37 degrees C, but at much lower rates at the lower temperatures. Preincubation with submaximal concentrations of InsP3, which themselves evoked quantal Ca2+ release, had no effect on the sensitivity of the stores to further additions of InsP3. The final Ca2+ content of the stores was the same whether they were stimulated with two submaximal doses of InsP3 or a single addition of the sum of these doses. Such incremental responses and the persistence of quantal behaviour at 2 degrees C indicate that InsP3-evoked receptor inactivation is unlikely to be the cause of quantal Ca2+ mobilization. Reducing the Ca2+ content of the intracellular stores by up to 45% did not affect their sensitivity to InsP3, but substantially reduced the time taken for each submaximal InsP3 concentration to exert its full effect. These results suggest that neither luminal nor cytosolic Ca2+ regulation of InsP3 receptors are the determinants of quantal behaviour. Our results are not therefore consistent with incremental responses to InsP3 depending on mechanisms involving attenuation of InsP3 receptor function by cytosolic or luminal Ca2+ or by InsP3 binding itself. We conclude that incremental activation of Ca2+ release results from all-or-nothing emptying of stores with heterogeneous sensitivities to InsP3. These characteristics allow rapid graded recruitment of InsP3-sensitive Ca2+ stores as the cytosolic InsP3 concentration increases.

Cooperative activation of IP3 receptors by sequential binding of IP3 and Ca2+ safeguards against spontaneous activity

BACKGROUND

Ca2+ waves allow effective delivery of intracellular Ca2+ signals to cytosolic targets. Propagation of these regenerative Ca2+ signals probably results from the activation of intracellular Ca2+ channels by the increase in cytosolic [Ca2+] that follows the opening of these channels. Such positive feedback is potentially explosive. Mechanisms that limit the spontaneous opening of intracellular Ca2+ channels are therefore likely to have evolved in parallel with the mechanism of Ca2+-induced Ca2+ release.

RESULTS

Maximal rates of 45Ca2+ efflux from permeabilised hepatocytes superfused with medium in which the [Ca2+] was clamped were cooperatively stimulated by inositol 1,4,5-trisphosphate (IP3). A minimal interval of approximately 400 msec between IP3 addition and the peak rate of Ca2+ mobilisation indicate that channel opening does not immediately follow binding of IP3. Although the absolute latency of Ca2+ release was unaffected by further increasing the IP3 concentration, it was reduced by increased [Ca2+].

CONCLUSIONS

We propose that the closed conformation of the IP3 receptor is very stable and therefore minimally susceptible to spontaneous activation; at least three (probably four) IP3 molecules may be required to provide enough binding energy to drive the receptor into a stable open conformation. We suggest that a further defence from noise is provided by an extreme form of coincidence detection. Binding of IP3 to each of its four receptor subunits unmasks a site to which Ca2+ must bind before the channel can open. As IP3 binding may also initiate receptor inactivation, there may be only a narrow temporal window during which each receptor subunit must bind both of its agonists if the channel is to open rather than inactivate.

The effects of inositol 1,4,5-trisphosphate (InsP3) analogues on the transient kinetics of Ca2+ release from cerebellar microsomes. InsP3 analogues act as partial agonists

An investigation of the effects of a number of inositol trisphosphate analogues on the transient kinetics of Ca2+ release from cerebellar microsomes was undertaken. All the analogues investigated could release the total Ca2+ content of the inositol 1, 4,5-trisphosphate (Ins(1,4,5)P3) mobilizable Ca2+ store; however, their potencies were substantially reduced compared to Ins(1,4,5)P3. The concentration required to induce half-maximal Ca2+ mobilization was 0.14 microM for Ins(1,4,5)P3, 1.8 microM for 3-deoxyinositol 1,4, 5-trisphosphate (3-deoxyInsP3), 1.0 microM for 2,3-dideoxyinositol 1, 4,5-trisphosphate (2,3-dideoxyInsP3), 24 microM for 2,3, 6-trideoxyinositol 1,4,5-trisphopshate (2,3,6-trideoxyInsP3), and 2.9 microM for inositol 2,4,5-trisphosphate (Ins(2,4,5)P3). In all cases and for all concentrations tested, the inositol trisphosphate analogues induced biphasic transient release of Ca2+, which could fit to a biexponential equation assuming two independent processes. The rate constants calculated for the release process were much larger for Ins(1,4,5)P3 than the other inositol trisphosphates (the fast phase rate constant varying from 0.3 to 1.6 s-1 and the slow phase from 0.01-0.5 s-1, at concentrations between 0.03 and 20 microM Ins(1,4,5)P3). The rate constants for all other inositol trisphosphates did not appear to exceed 0.4 s-1 for the fast phase and 0.1 s-1 for the slow phase at their highest concentrations tested. The maximum amplitudes for Ca2+ release by the two phases appeared to be similar for all inositol trisphosphates (approximately 45% for the fast phase and approximately 55% for the slow phase). On comparing the rate constants for Ca2+ release at inositol trisphosphate concentrations for the analogues which all induced the same extent of Ca2+ release, it was apparent that the rates of release were independent of the extent of Ca2+ release. As the extent of Ca2+ release can be related to degree of occupancy of the binding sites, it is evident that different analogues which occupy the binding site of the receptor to the same extent can induce Ca2+ to be released at different rates. We explain this conclusion in terms of partial agonism where inositol phosphates can induce two (or more) occupied states of the channel.

Analytical steady-state solution to the rapid buffering approximation near an open Ca2+ channel

We derive an analytical steady-state solution for the Ca2+ profile near an open Ca2+ channel based on a transport equation which describes the buffered diffusion of Ca2+ in the presence of rapid stationary and mobile Ca2+ buffers (Wagner and Keizer, 1994). This steady-state rapid buffering approximation gives an upper bound on local Ca2+ elevations such as Ca2+ puffs or sparks when conditions for the validity of the rapid buffering approximation are met and is an alternative to approximations that assume that mobile buffers are unsaturable. This result also provides an analytical estimate of the cytosolic Ca2+ domain concentration ([Ca2+]d) near a channel pore and shows the dependence of [Ca2+]d on moderate concentrations of endogenous mobile buffer, Ca2+ indicator dye, and bulk cytosolic Ca2+. Assuming a simple relationship between [Ca2+]d and the lumenal depletion domain of an intracellular Ca2+ channel, lumenal and cytosolic Ca2+ profiles are matched to give an implicit analytical expression for the effect of bulk lumenal Ca2+ on [Ca2+]d.

Quantal release, incremental detection, and long-period Ca2+ oscillations in a model based on regulatory Ca2+-binding sites along the permeation pathway

Quantal release, incremental detection, and oscillations are three types of Ca2+ responses that can be obtained in different conditions, after stimulation of the intracellular Ca2+ stores by submaximum concentrations of inositol 1,4,5-triphosphate (InsP3). All three phenomena are thought to occur through the regulatory properties of the InsP3 receptor/Ca2+ channel. In the present study, we perform further analysis of the model (Swillens et al., 1994, Proc. Natl. Acad. Sci. USA. 91:10074-10078) previously proposed for transient InsP3-induced Ca2+ release, based on the bell-shaped dependence of the InsP3 receptor activity on the Ca2+ level and on the existence of an intermediate Ca2+ domain located around the mouth of the channel. We show that Ca2+ oscillations also arise in the latter model. Conditions for the occurrence of the various behaviors are investigated. Numerical simulations also show that the existence of an intermediate Ca2+ domain can markedly increase the period of oscillations. Periods on the order of 1 min can indeed be accounted for by the model when one assigns realistic values to the kinetic constants of the InsP3 receptor, which, in the absence of a domain, lead to oscillations with periods of a few seconds. Finally, theoretical support in favor of a positive cooperativity in the regulation of the InsP3 receptor by Ca2+ is presented.

Quantal Ca2+ release and inactivation in a model of the inositol 1,4,5-trisphosphate receptor involving transformation of the ligand by the receptor

A model explaining quantal Ca2+ release as an intrinsic property of the inositol 1,4,5-triphosphate (IP3) receptor has been put forward. The model is based on the hypothesis that the IP3 receptor can catalyze a transformation of the IP3 molecule differing from its conventional metabolism. A simple kinetic mechanism is considered, in which IP3-induced Ca2+ channel opening is followed by the step of IP3 conversion and channel closure. Examination of the resulting mathematical model shows that it can reproduce well both partial release of stored Ca2+ and the same responsiveness to subsequent IP3 additions. On incorporation of an additional closed state of the channel, the model describes also a time-dependent channel inactivation at a high IP3 dose. Temperature sensitivity of the catalytic step accounts for the reported elimination of quantal responses and inactivation at low temperature. The transformation product is surmised to be a positional or stereo isomer of IP3.

Regulation of inositol trisphosphate receptors by luminal Ca2+ contributes to quantal Ca2+ mobilization

The quantal behaviour of inositol trisphosphate (InsP3) receptors allows rapid graded release of Ca2+ from intracellular stores, but the mechanisms are unknown. In Ca2+-depleted stores loaded with Fura 2, InsP3 caused concentration dependent increases in the rates of fluorescence quench by Mn2+ that were unaffected by prior incubation with InsP3, indicating that InsP3 binding did not cause desensitization. When Fura 2 was used to report the luminal free [Ca2+] after inhibition of further Ca2+ uptake, submaximal concentrations of InsP3 caused rapid, partial decreases in fluorescence ratios. Subsequent addition of a maximal InsP3 concentration caused the fluorescence to fall to within 5% of that recorded after ionomycin. Addition of all but the lowest concentrations of InsP3 to stores loaded with the lower affinity indicator, Calcium Green-5N, caused almost complete emptying of the stores at rates that increased with InsP3 concentration. The lowest concentration of InsP3 (10 nM) slowly emptied approximately 80% of the stores, but within 3 min the rate of Ca2+ release slowed leaving approximately 7 microM Ca2+ within the stores, which was then rapidly released by a maximal InsP3 concentration. In stores co-loaded with both indicators, InsP3-evoked Ca2+ release appeared quantal with Fura 2 and largely non-quantal with Calcium Green-5N; the discrepancy is not, therefore, a direct effect of the indicators. The fall in luminal [Ca2+] after activation of InsP3 receptors may, therefore, cause their inactivation, but only after the Ca2+ content of the stores has fallen by approximately 95% to < or = 10 microM.

Synergistic effects of inositol 1,3,4,5-tetrakisphosphate on inositol 2,4,5-triphosphate-stimulated Ca2+ release do not involve direct interaction of inositol 1,3,4,5-tetrakisphosphate with inositol triphosphate-binding sites

We have previously found that for permeabilized L1210 cells, low micromolar concentrations of Ins(1,3,4,5)P4 added prior to Ins(2,4,5)P3 enhance the effects of suboptimal concentrations of Ins(2,4,5)P3 in causing Ca2+ release from InsP3-sensitive Ca2+ stores [Cullen, Irvine and Dawson (1990) Biochem J. 271, 549-553]. If this was due either to some conversion of added Ins(1,3,4,5)P4 into Ins(1,4,5)P3 by the 3-phosphatase, or to Ins(1,3,4,5)P4 acting as a weak (or partial) agonist on the InsP3 receptor it would be expected that,in the presence of thimerosal to sensitize the InsP3 receptor, the dose-response curve to Ins(1,3,4,5)P4 would be left-shifted by the same extent as that of Ins(1,4,5)P3. This was found not to be the case; the dose-response curve to Ins(1,3,4,5)P4 was not shifted at all by thimerosal. Furthermore, L-Ins(1,3,4,5)P4, which can displace radiolabelled D-Ins(1,3,4,5)P4 but not D-Ins(1,4,5)P3 from their respective high-affinity binding sites, mimicked the effects of D-Ins(1,3,4,5)P4 in enhancing the slow phase of Ins(2,4,5)P3-stimulated Ca2+ release. Ins(1,3,4,5)P4 caused an increase in magnitude of the slow phase of InsP3-stimulated Ca2+ release leaving the magnitude of the fast phase unaltered, in contrast to increasing Ins(2,4,5)P3 concentrations which increased the size of both phases. In addition, Ins(1,3,4,5)P4 decreased the rate constant for the slow phase of Ca2+ release. These findings point strongly to the conclusion that InsP4 is not working directly via the InsP3 receptor but indirectly via an InsP4 receptor.

Inhibition of the cerebellar inositol 1,4,5-trisphosphate-sensitive Ca2+ channel by ethanol and other aliphatic alcohols

The effects of ethanol and other aliphatic alcohols on the endoplasmic reticulum Ca2+ pump and the inositol 1,4,5-trisphosphate (InsP3)-sensitive Ca2+ channel were studied in pig cerebellar microsomes. Methanol, ethanol and propanol all stimulated ATP-dependent Ca2+ uptake, whereas butanol inhibited this process. Ethanol inhibited InsP3-induced Ca2+ release [half-maximal inhibition at 3.5%, v/v (600 mM)]. However, ethanol affected only the amount of InsP3-releasable Ca2+, without affecting the concentration of InsP3 required to induce half-maximal release. Other alcohols of longer chain length were more potent than ethanol at inhibiting InsP3-induced Ca2+ release, but none of the alcohols tested affected [3H]InsP3 binding to its receptor. Using stopped-flow techniques, measurements of the rate of InsP3-induced Ca2+ release in the preparation of pig cerebellar microsomes used in this study showed the kinetics to be monophasic, with a rate constant of 0.93s-1 at 20 microM InsP3. This rate constant was dependent upon InsP3 concentration, decreasing to 0.38s-1 at 0.25 microM InsP3. Ethanol was shown to reduce the fractional amount of InsP3-induced Ca2+ release without significantly affecting the rate constant for this process.

Quantal responses to inositol 1,4,5-trisphosphate are not a consequence of Ca2+ regulation of inositol 1,4,5-trisphosphate receptors

Submaximal concentrations of inositol 1,4,5-trisphosphate (InsP3) rapidly release only a fraction of the InsP3-sensitive intracellular Ca2+ stores, despite the ability of further increases in InsP3 concentration to evoke further Ca2+ release. The mechanisms underlying such quantal Ca2+ mobilization are not understood, but have been proposed to involve regulatory effects of cytosolic Ca2+ on InsP3 receptors. By examining complete concentration-effect relationships for InsP3-stimulated 45Ca2+ efflux from the intracellular stores of permeabilized hepatocytes, we demonstrate that, at 37 degrees C, responses to InsP3 are quantal in Ca(2+)-free media heavily buffered with either EGTA or BAPTA [1,2-bis-(2-aminophenoxy)ethane-N,N,N',N'-tetra-acetic acid]. Lower concentrations of InsP3 were used to examine the kinetics of Ca2+ mobilization at 2 degrees C, because at the lower temperature the stores were more sensitive to InsP3: the concentration of InsP3 causing half-maximal Ca2+ release (EC50) after a 30 s incubation decreased from 281 +/- 37 nM at 37 degrees C to 68 +/- 3 nM at 2 degrees C. At 2 degrees C, the EC50 for InsP3-stimulated Ca2+ mobilization decreased as the duration of exposure to InsP3 was increased: the EC50 was 68 +/- 3 nM after 30 s, and 29 +/- 2 nM after 420 s. InsP3-stimulated Ca2+ mobilization is therefore non-quantal at 2 degrees C: InsP3 concentration determines the rate, but not the extent, of Ca2+ release. By initiating quantal responses to InsP3 at 37 degrees C and then simultaneously diluting and chilling cells to 2 degrees C, we demonstrated that the changes that underlie quantal responses do not rapidly reverse at 2 degrees C. At both 37 degrees C and 2 degrees C, modest increases in cytosolic Ca2+ increased the sensitivity of the stores to InsP3, whereas further increases were inhibitory; both Ca2+ effects persisted after prior removal of ATP. We conclude that the effects of Ca2+ on InsP3 receptors are unlikely either to be enzyme-mediated or to underlie the quantal pattern of Ca2+ release evoked by InsP3.

InsP3-induced Ca2+ excitability of the endoplasmic reticulum

Oscillations in intracellular Ca2+ can be induced by a variety of cellular signalling processes (Woods et al., 1986; Berridge 1988; Jacob et al., 1988) and appear to play a role in secretion (Stojilković et al., 1994), fertilization (Miyazaki et al., 1993), and smooth muscle contraction (Iino and Tsukioka, 1994). Recently, great progress has been made in understanding the mechanisms involved in a particular class of Ca2+ oscillation, associated with the second messenger inositol 1,4,5-trisphosphate (InsP3) (Berridge, 1993). Working in concert with intracellular Ca2+, InsP3 controls Ca2+ release via the InsP3 receptor in the endoplasmic reticulum (ER) (Berridge and Irvine, 1989). The IP3 receptor is regulated by its coagonists InsP3 and Ca2+, which both activate and inhibit Ca2+ release (Finch et al., 1991; Bezprozvanny et al., 1991; De Young and Keizer, 1992). These processes, together with the periodic activation of Ca2+ uptake into the ER, have been identified as key features in the mechanism of InsP3-induced Ca2+ oscillations in pituitary gonadotrophs (Li et al., 1994), Xenopus laevis oocytes (Lechleiter and Clapham, 1992; Atri et al., 1993), and other cell types (Keizer and De Young, 1993). Earlier discussions and models of InsP3-induced Ca2+ oscillations focused on the nature and number of internal releasable pools of Ca2+ (Goldbeter et al., 1990; Swillens and Mercan, 1990; Somogyi and Stucki, 1991), the importance of oscillations in InsP3 (Meyer and Stryer, 1988), and other issues not based on detailed experimental findings in specific cells types.