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

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.

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.

Modelling the effect of specific inositol 1,4,5-trisphosphate receptor isoforms on cellular Ca2+ signals

BACKGROUND INFORMATION

Oscillations of cytosolic Ca2+ are well-known to rely on the regulatory properties of the InsP3R (inositol 1,4,5-trisphosphate receptor). Three isoforms of this channel have been identified. They differ in their regulatory properties by Ca2+ and InsP3. Experiments in different cell types clearly indicate that the relative amounts of each isoform affect the time course of Ca2+ changes after agonist stimulation. In the present study, we investigate whether different steady-state curves for the open probability of the InsP3Rs as a function of Ca2+ imply different dynamical behaviours when these receptors are present in a cellular environment. We therefore describe by a specific phenomenological model the three main types of curves that have been reported: (i) the classical bell-shaped curve, (ii) the bell-shaped curve that is shifted towards higher Ca2+ concentrations when InsP3 is increased, and (iii) a monotonous increasing function of cytosolic Ca2+.

RESULTS

We show that, although these types of curves can be ascribed to slight differences in the channel regulation by Ca2+ and InsP3, they can indicate important variations as to the receptor role in cellular Ca2+ control. Thus the receptor associated with the classical bell-shaped curve appears to be the most robust Ca2+ oscillator. If the steady-state curve is supposed to be a monotonous increasing function of cytosolic Ca2+, the modelled receptor cannot sustain Ca2+ oscillations in the absence of Ca2+ exchanges with the extracellular medium. When the bell-shaped curve is shifted towards higher Ca2+ concentrations with increasing InsP3 levels, the model predicts that the receptor is less robust to changes in density; this receptor, however, provides a finer control of the steady-state level of Ca2+ when varying the InsP3 concentration.

CONCLUSIONS

Our model allows us to propose an explanation for the experimental observations about the effect of selectively expressing or down-regulating InsP3R isoforms, as well as to make theoretical predictions.

Calcium oscillations

Calcium signaling results from a complex interplay between activation and inactivation of intracellular and extracellular calcium permeable channels. This complexity is obvious from the pattern of calcium signals observed with modest, physiological concentrations of calcium-mobilizing agonists, which typically present as sequential regenerative discharges of stored calcium, a process referred to as calcium oscillations. In this review, we discuss recent advances in understanding the underlying mechanism of calcium oscillations through the power of mathematical modeling. We also summarize recent findings on the role of calcium entry through store-operated channels in sustaining calcium oscillations and in the mechanism by which calcium oscillations couple to downstream effectors.

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.

Spatiotemporal organization of Ca dynamics: a modeling-based approach

Calcium is a ubiquitous second messenger that mediates vital physiological responses such as fertilization, secretion, gene expression, or apoptosis. Given this variety of processes mediated by Ca(2+), these signals are highly organized both in time and space to ensure reliability and specificity. This review deals with the spatiotemporal organization of the Ca(2+) signaling pathway in electrically nonexcitable cells in which InsP(3) receptors are by far the most important Ca(2+) channels. We focus on the aspects of this highly regulated dynamical system for which an interplay between experiments and modeling is particularly fruitful. In particular, the importance of the relative densities of the different InsP(3) receptor subtypes will be discussed on the basis of a modeling approach linking the steady-state behaviors of these channels in electrophysiological experiments with their behavior in a cellular environment. Also, the interplay between InsP(3) metabolism and Ca(2+) oscillations will be considered. Finally, we discuss the relationships between stochastic openings of the Ca(2+) releasing channels at the microscopic level and the coordinated, regular behavior observed at the whole cell level on the basis of a combined experimental and modeling approach.

What can we learn from the irregularity of Ca2+ oscillations?

In most cells, Ca(2+) increases in response to external stimulation are organized in the form of oscillations and waves that sometimes propagate from one cell to another. Numerous experimental and theoretical studies reveal that this spatiotemporal organization contains a non-negligible level of stochasticity. In this study, we extend the previous work based on a statistical analysis of experimental Ca(2+) traces in isolated, hormone-stimulated hepatocytes and on stochastic simulations of Ca(2+) oscillations based on the Gillespie's algorithm. Comparison of the coefficients of variation in the periods of experimental and simulated Ca(2+) spikes provides information about the clustering and the specific subtypes of the Ca(2+) channels. In hepatocytes coupled by gap junctions, the global perfusion with a hormone leads to successive Ca(2+) responses, giving the appearance of an intercellular wave. Statistical analysis of experimental Ca(2+) oscillations in coupled hepatocytes confirms that this coordinated Ca(2+) spiking corresponds to a phase wave but suggests the existence of an additional coupling mechanism.

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.

A method for determining the dependence of calcium oscillations on inositol trisphosphate oscillations

In some cell types, oscillations in the concentration of free intracellular calcium ([Ca2+]) are accompanied by oscillations in the concentration of inositol 1,4,5-trisphosphate ([IP3]). However, in most cell types it is still an open question as to whether oscillations in [IP3] are necessary for Ca2+ oscillations in vivo, or whether they merely follow passively. Using a wide range of models, we show that the response to an artificially applied pulse of IP3 can be used to distinguish between these two cases. Hence, we show that muscarinic receptor-mediated, long-period Ca2+ oscillations in pancreatic acinar cells depend on [IP3] oscillations, whereas short-period Ca2+ oscillations in airway smooth muscle do not.

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.

Involvement of ryanodine receptors in IP3-mediated calcium signalling in neurons. A modelling approach

Phospholipase C (PLC)-coupled metabotropic receptors trigger the release of intracellular Ca2+ through activation of IP3 receptors (IP3Rs). Increasing evidence suggests that they can also and perhaps more efficiently mobilize Ca2+ through ryanodine receptors (RyRs). We constructed a model allowing a variable PLC stimulation level (via the parameter gamma) as well as a variable involvement of RyRs (via the parameter A). The sole presence of RyRs (A not = 0) affected the basal Ca2+ concentration [Ca2+]i. To keep Ca2+ below 160 nM, we fixed the upper limit of A to 0.2, a value that is compatible with the numerical ratio between RyRs and IP3Rs in cerebellar Purkinje neurons. Metabotropic responses were simulated by abruptly raising the value of gamma to various levels. In the absence of RyRs, the model starts to oscillate with gamma=0.4. For lower levels of PLC stimulation (gamma< or =0.3), the presence of RyR is capable of triggering an oscillatory signal. When A< or =0.18, the frequency of the Ca2+ oscillations augments from 0.1 to 1.3 Hz as a function of gamma. Conversely, as the frequency increases, the amplitude of the oscillations is reduced from 1 microM to 50 nM. With higher values of A, the oscillating pattern is definitively inhibited. It is concluded that RyRs have the potentiality to strikingly affect the temporal pattern of the Ca2+ signalling triggered by IP3-related metabotropic responses.

Simulation of calcium waves in ascidian eggs: insights into the origin of the pacemaker sites and the possible nature of the sperm factor

Fertilization triggers repetitive waves of cytosolic Ca2+ in the egg of many species. The mechanism involved in the generation of Ca2+ waves has been studied in much detail in mature ascidian eggs, by raising artificially the level of inositol 1,4,5-trisphosphate [Ins(1,4,5)P3] or of its poorly metabolizable analogue, glycero-myo-phosphatidylinositol 4,5-bisphosphate [gPtdIns(4,5)P2]. Here, we use this strategy and the experimental results it provides to develop a realistic theoretical model for repetitive Ca2+ wave generation and propagation in mature eggs. The model takes into account the heterogeneous spatial distribution of the endoplasmic reticulum. Our results corroborate the hypothesis that Ca2+ wave pacemakers are associated with cortical accumulations of endoplasmic reticulum. The model is first tested and validated by the adequate match between its theoretical predictions and the observed effects of localized injections of massive amounts of Ins(1,4,5)P3 analogues. In a second step, we use the model to make some propositions about the possible characteristics of the sperm factor. We find that to account for the spatial characteristics of the first series of Ca2+ waves seen at fertilization in ascidian eggs, it has to be assumed that, if the sperm factor is a phospholipase C, it is Ca2+-sensitive and highly diffusible. Although the actual state of knowledge does not allow us to explain the observed relocalization of the Ca2+ wave pacemaker site, the model corroborates the assumption that PtdIns(4,5)P2, the substrate for phospholipase C is distributed over the entire egg. We also predict that the dose of sperm factor injected into the egg should modulate the temporal characteristics of the first, long-lasting fertilization wave.

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.

Ca2+ oscillations in hepatocytes do not require the modulation of InsP3 3-kinase activity by Ca2+

Receptor-mediated production of inositol 1,4,5-trisphosphate (InsP(3)) initiates Ca(2+) release and is responsible for cytosolic Ca(2+) oscillations. InsP(3) oscillations have also been observed in some cells. One of the enzymes controlling InsP(3) catabolism, the InsP(3) 3-kinase, is stimulated by Ca(2+); this regulation is presumably part of the reason for InsP(3) oscillations that have been observed in some cells. Here, we investigate the possible role of Ca(2+)-activated InsP(3) catabolism on the characteristics of the InsP(3)-induced Ca(2+) oscillations. Numerical simulations show that if it is assumed that the Ca(2+)-independent InsP(3) catabolism is predominant, Ca(2+) oscillations remain qualitatively unchanged although the relative amplitude of the oscillations in InsP(3) concentrations becomes minimal. We tested this prediction in hepatocytes by masking the Ca(2+)-dependent InsP(3) catabolism by 3-kinase through the injection of massive amounts of InsP(3) 5-phosphatase, which is not stimulated by Ca(2+). We find that in such injected hepatocytes, Ca(2+) oscillations generated by modest agonist levels are suppressed, presumably because of the decreased dose in InsP(3), but that at higher doses of agonist, oscillations reappear, with characteristics similar to those of untreated cells at low agonist doses. Altogether, these results suggest that oscillations in InsP(3) concentration due to Ca(2+)-stimulated InsP(3) catabolism do not play a major role for the oscillations in Ca(2+) concentration.

On the role of stochastic channel behavior in intracellular Ca2+ dynamics

I present a stochastic model for intracellular Ca(2+) oscillations. The model starts from stochastic binding and dissociation of Ca(2+) to binding sites on a single subunit of the IP(3)-receptor channel but is capable of simulating large numbers of clusters for many oscillation periods too. I find oscillations with variable periods ranging from 17 s to 120 s and a standard deviation well in the experimentally observed range. Long period oscillations can be perceived as nucleation phenomenon and can be observed for a large variety of single channel dynamics. Their period depends on the geometric characteristics of the cluster array. Short periods are in the range of the time scale of channel dynamics. Both long and short period oscillations occur for parameters with a nonoscillatory deterministic regime.

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.

Excitable calcium wave propagation in the presence of localized stores

We study the propagation of calcium waves in the presence of a discrete distribution of calcium stores. Calcium-induced calcium release coupled to diffusion can be used to produce a criterion for wave propagation across connected clusters of stores. The velocity of the resulting wave and its relationship to the frequency of the excitatory stimulus can then be described using percolation theory. Simulations show a homogeneous and a fractal regime and are in agreement with both experiments and theory.

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.

Agonist-dependent phosphorylation of the inositol 1,4,5-trisphosphate receptor: A possible mechanism for agonist-specific calcium oscillations in pancreatic acinar cells

The properties of inositol 1,4,5-trisphosphate (IP3)-dependent intracellular calcium oscillations in pancreatic acinar cells depend crucially on the agonist used to stimulate them. Acetylcholine or carbachol (CCh) cause high-frequency (10-12-s period) calcium oscillations that are superimposed on a raised baseline, while cholecystokinin (CCK) causes long-period (>100-s period) baseline spiking. We show that physiological concentrations of CCK induce rapid phosphorylation of the IP3 receptor, which is not true of physiological concentrations of CCh. Based on this and other experimental data, we construct a mathematical model of agonist-specific intracellular calcium oscillations in pancreatic acinar cells. Model simulations agree with previous experimental work on the rates of activation and inactivation of the IP3 receptor by calcium (DuFour, J.-F., I.M. Arias, and T.J. Turner. 1997. J. Biol. Chem. 272:2675-2681), and reproduce both short-period, raised baseline oscillations, and long-period baseline spiking. The steady state open probability curve of the model IP3 receptor is an increasing function of calcium concentration, as found for type-III IP3 receptors by Hagar et al. (Hagar, R.E., A.D. Burgstahler, M.H. Nathanson, and B.E. Ehrlich. 1998. Nature. 396:81-84). We use the model to predict the effect of the removal of external calcium, and this prediction is confirmed experimentally. We also predict that, for type-III IP3 receptors, the steady state open probability curve will shift to lower calcium concentrations as the background IP3 concentration increases. We conclude that the differences between CCh- and CCK-induced calcium oscillations in pancreatic acinar cells can be explained by two principal mechanisms: (a) CCK causes more phosphorylation of the IP3 receptor than does CCh, and the phosphorylated receptor cannot pass calcium current; and (b) the rate of calcium ATPase pumping and the rate of calcium influx from the outside the cell are greater in the presence of CCh than in the presence of CCK.

Differential modulation of the phases of a Ca2+ spike by the store Ca2+-ATPase in human umbilical vein endothelial cells

1. Histamine-stimulated cytosolic free Ca2+ ([Ca2+]i) oscillations in human umbilical vein endothelial cells (HUVECs) comprise repetitive spikes generated by pulsatile release from stores. We have investigated the roles of the store Ca2+-ATPases in regulating both the upstroke and downstroke of a Ca2+ spike. 2. The sarco-endoplasmic reticulum Ca2+-ATPase (SERCA) inhibitor cyclopiazonic acid (CPA) dramatically affected oscillations whereas inhibition of the plasma membrane Ca2+-ATPase (PMCA) with La3+ had little effect. This and other evidence suggested that the downstroke of a spike is predominantly mediated by SERCA. 3. Artificial [Ca2+]i spiking generated by repetitive pulsatile application of 0.3 microM histamine in Ca2+-free medium did not cause net loss of Ca2+ from the cell whereas repetitive pulsatile application of 1 and 10 microM histamine did, with the higher concentration being more effective. We conclude that there is an inverse relationship between stimulus intensity and relative SERCA activity. 4. For a Ca2+ transient, the initiation of release was suppressed by SERCA during either the lag phase or the interspike period (ISP) since: (i) the ISP was shortened by low CPA concentrations, (ii) higher concentrations of CPA stimulated an explosive Ca2+ release when applied during the ISP but not when applied in the absence of agonist, and (iii) CPA synchronized the initial Ca2+ response to a low histamine dose (even recruiting silent, histamine-unresponsive cells). 5. Two aspects of the regenerative upstroke of a spike were differently affected by SERCA inhibition: Ca2+ wave velocity was entirely unaffected by CPA whereas the local rate of rise was increased. 6. The [Ca2+]i at which a Ca2+ spike terminated depended on SERCA since CPA dose dependently enhanced the peak [Ca2+]i. 7. We conclude that SERCA plays a powerful and dynamic role in regulating [Ca2+]i oscillations in HUVECs. SERCA differentially modulates the phases of Ca2+ release in addition to bringing about the falling phase of a Ca2+ spike.

Stochastic simulation of a single inositol 1,4,5-trisphosphate-sensitive Ca2+ channel reveals repetitive openings during 'blip-like' Ca2+ transients

Confocal microscope studies with fluorescent dyes of inositol 1,4,5-trisphosphate (InsP3)-induced intracellular Ca2+ mobilization recently established the existence of 'elementary' events, dependent on the activity of individual InsP3-sensitive Ca2+ channels. In the present work, we try by theoretical stochastic simulation to explain the smallest signals observed in those studies, which were referred to as Ca2+ 'blips' [Parker I., Yao Y. Ca2+ transients associated with openings of inositol trisphosphate-gated channels in Xenopus oocytes. J Physiol Lond 1996; 491: 663-668]. For this purpose, we assumed a simple molecular model for the InsP3-sensitive Ca2+ channel and defined a set of parameter values accounting for the results obtained in electrophysiological bilayer experiments [Bezprozvanny I., Watras J., Ehrlich B.E. Bell-shaped calcium-response curves of Ins(1,4,5)P3- and calcium-gated channels from endoplasmic reticulum of cerebellum. Nature 1991; 351: 751-754; Bezprozvanny I., Ehrlich B.E. Inositol (1,4,5)-trisphosphate (InsP3)-gated Ca channels from cerebellum: conduction properties for divalent cations and regulation by intraluminal calcium. J Gen Physiol 1994; 104: 821-856]. With a stochastic procedure which considered cytosolic Ca2+ diffusion explicitly, we then simulated the behaviour of a single channel, placed in a realistic physiological environment. An attractive result was that the simulated channel exhibited bursts of activity, arising from repetitive channel openings, which were responsible for transient rises in Ca2+ concentration and were reminiscent of the relatively long-duration experimental Ca2+ blips. The influence of the values chosen for the various parameters (affinity and diffusion coefficient of the buffers, luminal Ca2+ concentration) on the kinetic characteristics of these theoretical blips is analyzed.

Caffeine mediates cation influx and intracellular Ca2+ release in leech P neurones

We investigated the effect of caffeine on the intracellular free Ca2+ concentration ([Ca2+]i) of leech P neurones by using the fluorescent indicator Fura-2. Caffeine induced a [Ca2+]i increase that was strongly reduced, but not abolished, in Ca(2+)-free solution. The effect of caffeine on [Ca2+]i was dose-dependent: while 5 mM caffeine evoked a persistent [Ca2+]i increase that could be elicited repetitively, 10 mM caffeine or more induced a transient [Ca2+]i increase that was strongly reduced upon subsequent applications at the same concentration. Surprisingly, the cells remained fully responsive to a moderately increased caffeine concentration. The caffeine-induced [Ca2+]i increase was not blocked by millimolar concentrations of La3+, Mg2+, Cd2+, Zn2+, Co2+, Ni2+, or Mn2+. While La3+ and Mg2+ had no effect on the caffeine response, the other cations caused irreversible changes in the Fura-2 fluorescence. The inhibitors of intracellular Ca2+ pumps-thapsigargin, cyclopiazonic acid (CPA), and 2,5-di-(t-butyl)-1,4-hydroquinone (BHQ)--had no effect on the caffeine-induced [Ca2+]i increase at normal extracellular Ca2+ concentration, but they reduced it in Ca(2+)-free solution. Ryanodine had no effect on the caffeine-induced [Ca2+]i increase at normal extracellular Ca2+ concentration, and also in Ca(2+)-free solution it seemed to be largely ineffective. Caffeine evoked complete fluctuations of the membrane potential. The effect in Ca2+ free and in Na(+)-free solution suggests that the depolarizing response components were mainly due to Na+ influx, while Ca2+ reduced the Na+ influx and/or activated mechanisms which re- or hyperpolarize the cells. It is concluded that leech P neurones possess caffeine-sensitive intracellular Ca2+ stores, as well as caffeine-sensitive ion channels, in the plasma membrane that are activated by a voltage-independent mechanism. The plasma membrane channels are permeable to various divalent cations including Ca2+, and possibly also to Na+.

Simulations of the effects of inositol 1,4,5-trisphosphate 3-kinase and 5-phosphatase activities on Ca2+ oscillations

Inositol 1,4,5-trisphosphate (Ins-1,4,5-P3) is responsible for Ca2+ mobilization in response to external stimulation in many cell types. The latter phenomenon often occurs as repetitive Ca2+ spikes. In this study, the effect of the two Ins-1,4,5-P3 metabolizing enzymes (Ins-1,4,5-P3 3-kinase and 5-phosphatase) on the temporal pattern of Ca2+ oscillations has been investigated. On the basis of the well-documented Ins-1,4,5-P3 3-kinase stimulation by the Ca2+/calmodulin complex and of the experimentally-determined kinetic characteristics of these enzymes, we predict that 5-phosphatase primarily controls the levels of Ins-1,4,5-P3 and, thereby, the occurrence and frequency of Ca2+ oscillations. Consequently, the model reproduces the experimental observation performed in Chinese hamster ovary cells that 5-phosphatase overexpression has a much more pronounced effect on the pattern of Ca2+ oscillations than 3-kinase overexpression. We also investigated, in more detail, under which conditions a similar effect could be observed in other cell types expressing various Ins-1,4,5-P3 3-kinase activities.

A general computational framework for modeling cellular structure and function

The "Virtual Cell" provides a general system for testing cell biological mechanisms and creates a framework for encapsulating the burgeoning knowledge base comprising the distribution and dynamics of intracellular biochemical processes. It approaches the problem by associating biochemical and electrophysiological data describing individual reactions with experimental microscopic image data describing their subcellular localizations. Individual processes are collected within a physical and computational infrastructure that accommodates any molecular mechanism expressible as rate equations or membrane fluxes. An illustration of the method is provided by a dynamic simulation of IP3-mediated Ca2+ release from endoplasmic reticulum in a neuronal cell. The results can be directly compared to experimental observations and provide insight into the role of experimentally inaccessible components of the overall mechanism.