Simulation of intracellular Ca2+ oscillation in a sympathetic neurone

Combined computational and experimental approaches to understanding the Ca(2+) regulatory network in neurons

Ca(2+) is a ubiquitous signaling ion that regulates a variety of neuronal functions by binding to and altering the state of effector proteins. Spatial relationships and temporal dynamics of Ca(2+) elevations determine many cellular responses of neurons to chemical and electrical stimulation. There is a wealth of information regarding the properties and distribution of Ca(2+) channels, pumps, exchangers, and buffers that participate in Ca(2+) regulation. At the same time, new imaging techniques permit characterization of evoked Ca(2+) signals with increasing spatial and temporal resolution. However, understanding the mechanistic link between functional properties of Ca(2+) handling proteins and the stimulus-evoked Ca(2+) signals they orchestrate requires consideration of the way Ca(2+) handling mechanisms operate together as a system in native cells. A wide array of biophysical modeling approaches is available for studying this problem and can be used in a variety of ways. Models can be useful to explain the behavior of complex systems, to evaluate the role of individual Ca(2+) handling mechanisms, to extract valuable parameters, and to generate predictions that can be validated experimentally. In this review, we discuss recent advances in understanding the underlying mechanisms of Ca(2+) signaling in neurons via mathematical modeling. We emphasize the value of developing realistic models based on experimentally validated descriptions of Ca(2+) transport and buffering that can be tested and refined through new experiments to develop increasingly accurate biophysical descriptions of Ca(2+) signaling in neurons.

Calcium oscillations in neurons

Oscillations in the cytosolic free Ca2+ concentration ([Ca2+]i) have been described in a variety of cells. In some cases, [Ca2+]i oscillations reflect cycles of membrane depolarization and voltage-dependent Ca2+ entry. In others, they are caused by periodic Ca2+ uptake and release by internal stores, with little immediate requirement for external Ca2+. A third type of [Ca2+]i oscillation is typified by caffeine-induced oscillations in sympathetic neurons. Here, the oscillations depend on the interplay between Ca2+ transport across the plasma membrane and transport by a caffeine-sensitive store. These oscillations can occur at a steady membrane potential and are blocked by ryanodine (1 microM), indicating that they do not result from voltage-dependent changes in Ca2+ entry but do require Ca(2+)-induced Ca2+ release. Entry of Ca2+ from the external medium is important during all phases of the oscillatory cycle except the rapid upstroke, which is dominated by Ca2+ release from an internal store. It is proposed that caffeine-induced [Ca2+]i oscillations are cyclic perturbations of [Ca2+]i caused by exchange of Ca2+ between the cytosol and the caffeine-sensitive store: net Ca2+ loss from the store increases [Ca2+]i transiently above its steady-state value ([Ca2+]ss), whereas net accumulation of Ca2+ by the store transiently depresses [Ca2+]i below [Ca2+]ss. The effects of rapid removal of Ca2+ and caffeine on the rate of change of [Ca2+]i (d[Ca2+]i/dt) provide estimates of the rates of net Ca2+ entry and (caffeine-sensitive) Ca2+ release and information on the way these rates vary during the oscillatory cycle.

Complex intracellular calcium oscillations. A theoretical exploration of possible mechanisms

Intracellular Ca(2+) oscillations are commonly observed in a large number of cell types in response to stimulation by an extracellular agonist. In most cell types the mechanism of regular spiking is well understood and models based on Ca(2+)-induced Ca(2+) release (CICR) can account for many experimental observations. However, cells do not always exhibit simple Ca(2+) oscillations. In response to given agonists, some cells show more complex behaviour in the form of bursting, i.e. trains of Ca(2+) spikes separated by silent phases. Here we develop several theoretical models, based on physiologically plausible assumptions, that could account for complex intracellular Ca(2+) oscillations. The models are all based on one- or two-pool models based on CICR. We extend these models by (i) considering the inhibition of the Ca(2+)-release channel on a unique intracellular store at high cytosolic Ca(2+) concentrations, (ii) taking into account the Ca(2+)-activated degradation of inositol 1,4,5-trisphosphate (IP(3)), or (iii) considering explicity the evolution of the Ca(2+) concentration in two different pools, one sensitive and the other one insensitive to IP(3). Besides simple periodic oscillations, these three models can all account for more complex oscillatory behaviour in the form of bursting. Moreover, the model that takes the kinetics of IP(3) into account shows chaotic behaviour.

Dynamic and static calcium gradients inside large snail (Helix aspersa) neurones detected with calcium-sensitive microelectrodes

We have used quartz Ca²⁺-sensitive microelectrodes (CASMs) in large voltage-clamped snail neurones to investigate the inward spread of Ca²⁺ after a brief depolarisation. Both steady state and [Ca²⁺]_i transients changed with depth of penetration. When the CASM tip was within 20 μm of the far side of the cell the [Ca²⁺]_i transient time to peak was 4.4 ± 0.5 s, rising to 14.7 ± 0.7 s at a distance of 80 μm. We estimate that the Ca²⁺ transients travelled centripetally at an average speed of 6 μm² s⁻¹ and decreased in size by half over a distance of about 45 μm. Cyclopiazonic acid had little effect on the size and time to peak of Ca²⁺ transients but slowed their recovery significantly. This suggests that the endoplasmic reticulum curtails rather than reinforces the transients. Injecting the calcium buffer BAPTA made the Ca²⁺ transients more uniform in size and increased their times to peak and rates of recovery near the membrane. We have developed a computational model for the transients, which includes diffusion, uptake and Ca²⁺ extrusion. Good fits were obtained with a rather large apparent diffusion coefficient of about 90 ± 20 μm² s⁻¹.This may assist fast recovery by extrusion.

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.

Mechanics of cytogels I: Oscillations in Physarum

The contractility of actomyosin gels is the basis for a variety of cellular motility phenomena. We present here a mechanical analysis of contractile gels. By making certain hypotheses on the chemical regulation of cytogel contraction we formulate a model for the rhythmic contractions of plasmodia in the slime mold Physarum polycephalum which is in accord with a number of experimental observations.

Repetitive application of caffeine sensitizes caffeine-induced Ca2+ release in bullfrog sympathetic ganglion neurons

Cytosolic free calcium concentration ([Ca(2+)](i)) was recorded from cultured bullfrog sympathetic ganglion cells loaded with the Ca(2+)-indicator Fura-2 or Fura-6F. Repetitive application of caffeine at a low concentration, which either failed to produce any [Ca(2+)](i) elevation or induced a small gradual increase in [Ca(2+)](i) at first challenge, produced a drastic increase in the amplitude of Ca(2+) release (caffeine response). The caffeine response eventually reached peak amplitude and then remained constant even if caffeine application were continued. This augmentation was maintained for up to 2 h, and was achieved not only by repetitive application but also by a long exposure of caffeine. However, this augmentation was neither achieved by repetitive administration of high K(+)-solution, nor caused by inhibition of phosphodiesterase by caffeine. The repetitive or sustained application of caffeine is suggested to increase the caffeine sensitivity of the calcium release channel to calcium, thus causing the potentiation of the caffeine response.

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.

Coupled intra- and extracellular Ca2+ dynamics in recurrent seizure-like events

We considered the evolution of Ca2+ oscillation dynamics in recurrent seizure-like events. Dynamic system behaviour was characterized in the state space reconstructed from intra- and extracellular [Ca2+] fluctuations simultaneously measured in cultured rat hippocampal slices under low-[Mg2+] conditions. When associated in the seizure-like event, these fluctuations occurred on a restricted set, the attractor, embedded in the full state space with less than five degrees of freedom. Instantaneous relative phase differences indicated field potential-driven phase jumps locked onto seizure-like events. To account for recurrent dynamics, calculations were performed on different extensions of a model for Ca2+ oscillation. These identified bidirectional, asymmetrical coupling of extracellular with intracellular (cytosolic, Ca2+ store, mitochondrial) Ca2+ dynamics as critical in its development.

Low-[Mg2+]-induced Ca2+ fluctuations in organotypic hippocampal slice cultures

We show here by whole field monitoring of free intracellular Ca2+ ([Ca2+]i), locally recorded field potential (fp) and external [Ca2+], that low-[Mg2+] induces seizure like events (SLEs) accompanied by simultaneous fluctuations of [Ca2+]i and [Ca2+]e in cultured hippocampal slices. Within a SLE, complex [Ca2+]e fluctuations are seen throughout phases of Ca2+ depletion (tonic) and Ca2+ recovery (clonic) of the extracellular space. Information theory entropy-based analyses revealed strong asymmetric associations of [Ca2+]i and [Ca2+]e kinetics. By contrast, signal-associations between SLEs were found to be weak and of symmetric nature distinguishing seizure-like and interictal events by extensive coupling and decoupling of [Ca2+]i and [Ca2+]e fluctuations, respectively.

Modes of propagation of Ca(2+)-induced Ca2+ release in bullfrog sympathetic ganglion cells

How depolarization-induced Ca2+ entry or caffeine activates Ca(2+)-induced Ca2+ release (CICR) in the cytoplasm and nucleoplasm was studied by recording intracellular Ca2+ ([Ca2+]i) with a confocal microscope in cultured bullfrog sympathetic ganglion cells. The amplitude and propagation speed of voltage pulse-induced rises in [Ca2+]i were greater in the submembrane (< 5 microns depth) region than in the core region, and delayed and smaller, but significant, in the nucleus. Ryanodine and dantrolene reduced the rises in [Ca2+]i in both the cytoplasm and nucleus. A rapid application of high K+ solution induced global rises in [Ca2+]i in both the cytoplasm and nucleoplasm, which were decreased by dantrolene. Caffeine produced a slow, small rise in [Ca2+]i which grew into a global, regenerative rise both in the cytoplasm and nucleoplasm with some inward gradient in the cytoplasm. Each of the high [Ca2+]i phases during caffeine-induced [Ca2+]i oscillation began in the submembrane region, while low [Ca2+]i phases started in the core region. These results suggest that CICR activated by Ca2+ entry or caffeine occurs predominantly in the submembrane region causing an inwardly spreading Ca2+ wave or [Ca2+]i oscillations, and that the nuclear envelope can cause CICR in the nucleoplasm, which is delayed due to Ca2+ diffusion barrier at the nuclear pores.

Caffeine-induced [Ca2+] oscillations in neurones of frog sympathetic ganglia

1. Single cell fluorimetry was used to monitor caffeine-induced oscillations of cytosolic [Ca2+] in frog sympathetic ganglion neurones in 2.0 mM K+ Ringer solution. 2. [Ca2+] oscillations decreased in frequency and exhibited three different amplitude patterns after the first large peak of [Ca2+]: (a) a series of big oscillations (BOs) of constant large amplitude (300-400 nM), (b) a series of much smaller oscillations (SOs) (40-60 nM), or (c) a series of decaying oscillations (DOs) of rapidly decreasing amplitude. 3. A model in which the oscillation amplitude was determined by the Ca2+ content of the endoplasmic reticulum (ER) whereas the oscillation frequency was controlled by how rapidly the cytosolic [Ca2+] reached the threshold for Ca2+-induced Ca2+ release (CICR) was able to simulate each observed pattern by varying the level of activity of the ER Ca2+ pump (SERCA), CICR and release-activated Ca2+ transport (RACT). A cumulative, cytosolic Ca2+-dependent inactivation of the plasma membrane (PM) Ca2+ influx or of the Ca2+-sensitive leak coefficient of the ryanodine receptors caused the oscillation frequency to decrease in the model. 4. Transitions between BOs and SOs and changes in [Ca2+] oscillations caused by ryanodine, thapsigargin, lanthanum and FCCP could also be simulated. 5. We conclude that RACT, SERCA, CICR and Ca2+-dependent PM Ca2+ influx are major mechanisms underlying [Ca2+] oscillations in these neurones.

A Ca2+-induced Ca2+ release mechanism involved in asynchronous exocytosis at frog motor nerve terminals

The extent to which Ca2+-induced Ca2+ release (CICR) affects transmitter release is unknown. Continuous nerve stimulation (20-50 Hz) caused slow transient increases in miniature end-plate potential (MEPP) frequency (MEPP-hump) and intracellular free Ca2+ ([Ca2+]i) in presynaptic terminals (Ca2+-hump) in frog skeletal muscles over a period of minutes in a low Ca2+, high Mg2+ solution. Mn2+ quenched Indo-1 and Fura-2 fluorescence, thus indicating that stimulation was accompanied by opening of voltage-dependent Ca2+ channels. MEPP-hump depended on extracellular Ca2+ (0.05-0.2 mM) and stimulation frequency. Both the Ca2+- and MEPP-humps were blocked by 8-(N, N-diethylamino)octyl3,4,5-trimethoxybenzoate hydrochloride (TMB-8), ryanodine, and thapsigargin, but enhanced by CN-. Thus, Ca2+-hump is generated by the activation of CICR via ryanodine receptors by Ca2+ entry, producing MEPP-hump. A short interruption of tetanus (<1 min) during MEPP-hump quickly reduced MEPP frequency to a level attained under the effect of TMB-8 or thapsigargin, while resuming tetanus swiftly raised MEPP frequency to the previous or higher level. Thus, the steady/equilibrium condition balancing CICR and Ca2+ clearance occurs in nerve terminals with slow changes toward a greater activation of CICR (priming) during the rising phase of MEPP-hump and toward a smaller activation during the decay phase. A short pause applied after the end of MEPP- or Ca2+-hump affected little MEPP frequency or [Ca2+]i, but caused a quick increase (faster than MEPP- or Ca2+-hump) after the pause, whose magnitude increased with an increase in pause duration (<1 min), suggesting that Ca2+ entry-dependent inactivation, but not depriming process, explains the decay of the humps. The depriming process was seen by giving a much longer pause (>1 min). Thus, ryanodine receptors in frog motor nerve terminals are endowed with Ca2+ entry-dependent slow priming and fast inactivation mechanisms, as well as Ca2+ entry-dependent activation, and involved in asynchronous exocytosis. Physiological significance of CICR in presynaptic terminals was discussed.

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.

Amplification of the thapsigargin-evoked increase in the cytosolic free Ca2+ concentration by acetylcholine in acutely isolated mouse submandibular acinar cells

The intracellular Ca2+ concentration was measured in single, acutely isolated, mouse submandibular acinar cells loaded with fura-2 AM. All experiments were performed in the absence of extracellular Ca2+ in order to eliminate Ca2+ influx. The microsomal ATPase inhibitor, thapsigargin, was used to release Ca2+ from intracellular stores and simultaneously prevent re-uptake into the stores. Sequential application of thapsigargin (2 microM) and the Ca2+ ionophore ionomycin (500 nM) indicated that thapsigargin was able to mobilize practically all intracellular Ca2+. Furthermore, in comparison with results obtained following inhibition of the plasma membrane Ca(2+)-ATPase by La3+ (2 mM), it may be shown that slowly unloading the intracellular Ca2+ stores using thapsigargin does not normally cause a massive, cytotoxic, increase in the cytosolic Ca2+ concentration, because Ca2+ is rapidly extruded from the cell across the plasma membrane. Application of a submaximal dose of acetylcholine (500 nM) during the rising phase of the response to thapsigargin caused a 3-4-fold increase in the amplitude of the rise in the cytosolic Ca2+ concentration without any significant alteration of the time course of the response. As thapsigargin alone is capable of mobilizing all releasable Ca2+, this increase in amplitude is most likely the result of inhibition of the Ca2+ extrusion process by acetylcholine.

Modeling slowly bursting neurons via calcium store and voltage-independent calcium current

Recent experiments indicate that the calcium store (e.g., endoplasmic reticulum) is involved in electrical bursting and [Ca2+]i oscillation in bursting neuronal cells. In this paper, we formulate a mathematical model for bursting neurons, which includes Ca2+ in the intracellular Ca2+ stores and a voltage-independent calcium channel (VICC). This VICC is activated by a depletion of Ca2+ concentration in the store, [Ca2+]cs. In this model, [Ca2+]cs oscillates slowly, and this slow dynamic in turn gives rise to electrical bursting. The newly formulated model thus is radically different from existing models of bursting excitable cells, whose mechanism owes its origin to the ion channels in the plasma membrane and the [Ca2+]i dynamics. In addition, this model is capable of providing answers to some puzzling phenomena, which the previous models could not (e.g., why cAMP, glucagon, and caffeine have ability to change the burst periodicity). Using mag-fura-2 fluorescent dyes, it would be interesting to verify the prediction of the model that (1) [Ca2+]cs oscillates in bursting neurons such as Aplysia neuron and (2) the neurotransmitters and hormones that affect the adenylate cyclase pathway can influence this oscillation.

Ethanol actions on the mechanisms of Ca2+ mobilization in rat hippocampal cells are mediated by protein kinase C

The effects of ethanol on intracellular free Ca(2+) concentration, [Ca](i), were studied in cultured rat hippocampal neurons using fluo-3 and confocal microscopy. Ethanol application transiently elevAted [Ca](i) due to Ca(2+)-induced Ca(2+) release from internal stores since the effect was observed also in solutions containing zero Ca(2+) or 0.3 mM La(3+) and restoration of external Ca(2+) content led to secondary response in presence of ethanol. The sites of highest [Ca]i increases correlated well with those obtained after Ca(2+) release from caffeine-and IP3-sensitive internal stores. After single ethanol exposure the caffeine-evoked [Ca](i) transients were potentiated whereas Ca(2+) release induced by IP(3)-mobilizing agonists was suppressed. Similar effects were observed by activation of protein kinase C (PKC) by phorbol esters which also occluded ethanol actions. Ethanol increased fluorescence of Rim-1, a PKC indicator dye. The data obtained are consistent with ethanol activation of PKC whereby Ca(2+) release via ryanodine receptors is potentiated and IP(3) receptors are down-modulated. Since the effects of both ethanol and phorbol esters were mimicked by cytochalasins B and D, PKC-induced cytoskeleton phosphorylation and its subsequent rearrangements can be responsible for observed effects.

Characterization of spontaneous calcium transients in nerve growth cones and their effect on growth cone migration

This study examines the mechanisms of spontaneous and induced [Ca2+]i spiking in nerve growth cones and the effect of spikes on growth cone migration. Over a 10-20 min observation period, 29% of DRG growth cones undergo spontaneous and transient elevations in physiological extracellular Ca2+ ((Ca2+)o; 2 mM), whereas 67% of growth cones exposed to 20 mM (Ca2+)o exhibit similar [Ca2+]i spikes. Spontaneous [Ca2+]i spiking was not observed in neuronal cell bodies or nonneuronal cells. Ca2+ influx through non-voltage-gated Ca2+ channels was required for spontaneous [Ca2+]i spikes in growth cones, since removal of (Ca2+)o, or addition of the general Ca2+ channel blockers La3+ or Ni2+, reversibly blocked [Ca2+]i spiking, while blockers of the voltage-gated Ca2+ channels did not. Experiments using agents that influence intracellular Ca2+ stores suggest that Ca2+ stores may buffer and release Ca2+ during growth cone [Ca2+]i spikes. Growth cone migration was immediately and transiently inhibited by [Ca2+]i spikes, but eventually returned to prespike rates.

[Ca2+]i oscillations in sympathetic neurons: an experimental test of a theoretical model

[Ca2+]i oscillations have been described in a variety of cells. This study focuses on caffeine-induced [Ca2+]i oscillations in sympathetic neurons. Previous work has shown that these oscillations require Ca2+ entry from the extracellular medium and Ca(2+)-induced Ca2+ release from a caffeine- and ryanodine-sensitive store. The aim of the study was to understand the mechanism responsible for the oscillations. As a starting point, [Ca2+]i relaxations were examined after membrane depolarization and exposure to caffeine. For both stimuli, post-stimulus relaxations could be described by the sum of two decaying exponential functions, consistent with a one-pool system in which Ca2+ transport between compartments is regulated by linear Ca2+ pumps and leaks. After modifying the store to include a [Ca2+]i-sensitive leak, the model also exhibits oscillations such as those observed experimentally. The model was tested by comparing measured and predicted net Ca2+ fluxes during the oscillatory cycle. Three independent fluxes were measured, describing the rates of 1) Ca2+ entry across the plasma membrane, 2) Ca2+ release by the internal store, and 3) Ca2+ extrusion across the plasma membrane and uptake by the internal store. Starting with estimates of the model parameters deduced from post-stimulus relaxations and the rapid upstroke, a set of parameter values was found that provides a good description of [Ca2+]i throughout the oscillatory cycle. With the same parameter values, there was also good agreement between the measured and simulated net fluxes. Thus, a one-pool model with a single [Ca2+]i-sensitive Ca2+ permeability is adequate to account for many of the quantitative properties of steady-state [Ca2+]i oscillations in sympathetic neurons. Inactivation of the intracellular Ca2+ permeability, cooperative nonlinear Ca2+ uptake and extrusion mechanisms, and functional links between plasma membrane Ca2+ transport and the internal store are not required.

Calcium transients in growth cones and axons of cultured Helisoma neurons in response to conditioning factors

Accumulating evidence indicates that cytosolic calcium levels regulate growth cone motility and neurite extension. The purpose of this study was to determine if intracellular calcium levels also influence the initiation of neurite extension induced by growth-promoting factors. An in vitro preparation of axotomized neurons that can be maintained in the absence of growth-promoting factors was utilized. The distal axons of cultured Helisoma neurons plated into defined medium do not extend neurites until they are exposed to Helisoma brain-conditioned medium. This provided the opportunity to study the intracellular changes associated with neurite extension. Cytosolic calcium levels were monitored with the calcium-sensitive dye fura 2 at the distal axon. In control medium calcium levels in the distal axon were constant. However, transient elevations in cytosolic calcium in the axonal growth cone occurred after addition of conditioned medium and coincident with the initiation of neurite extension. Application of calcium channel blockers showed that the transients resulted from calcium influx across the neuronal membrane. The transients, however, were not required for neurite extension, although they did influence the rate and extent of neurite outgrowth. Simultaneous extracellular patch recordings demonstrated that the calcium transients were correlated temporally with an increase in rhythmic spontaneous electrical activity of cells, suggesting that conditioned medium influences ionic membrane properties of these neurons.

Properties of intracellular Ca2+ waves generated by a model based on Ca(2+)-induced Ca2+ release

Cytosolic Ca2+ waves occur in a number of cell types either spontaneously or after stimulation by hormones, neurotransmitters, or treatments promoting Ca2+ influx into the cells. These waves can be broadly classified into two types. Waves of type 1, observed in cardiac myocytes or Xenopus oocytes, correspond to the propagation of sharp bands of Ca2+ throughout the cell at a rate that is high enough to permit the simultaneous propagation of several fronts in a given cells. Waves of type 2, observed in hepatocytes, endothelial cells, or various kinds of eggs, correspond to the progressive elevation of cytosolic Ca2+ throughout the cell, followed by its quasi-homogeneous return down to basal levels. Here we analyze the propagation of these different types of intracellular Ca2+ waves in a model based on Ca(2+)-induced Ca2+ release (CICR). The model accounts for transient or sustained waves of type 1 or 2, depending on the size of the cell and on the values of the kinetic parameters that measure Ca2+ exchange between the cytosol, the extracellular medium, and intracellular stores. Two versions of the model based on CICR are considered. The first version involves two distinct Ca2+ pools sensitive to inositol 1,4,5-trisphosphate (IP3) and Ca2+, respectively, whereas the second version involves a single pool sensitive both to Ca2+ and IP3 behaving as co-agonists for Ca2+ release. Intracellular Ca2+ waves occur in the two versions of the model based on CICR, but fail to propagate in the one-pool model at subthreshold levels of IP3. For waves of type 1, we investigate the effect of the spatial distribution of Ca(2+)-sensitive Ca2+ stores within the cytosol, and show that the wave fails to propagate when the distance between the stores exceeds a critical value on the order of a few microns. We also determine how the period and velocity of the waves are affected by changes in parameters measuring stimulation, Ca2+ influx into the cell, or Ca2+ pumping into the stores. For waves of type 2, the numerical analysis indicates that the best qualitative agreement with experimental observations is obtained for phase waves. Finally, conditions are obtained for the occurrence of "echo" waves that are sometimes observed in the experiments.

Cyclic ADP-ribose modulates Ca2+ release channels for activation by physiological Ca2+ entry in bullfrog sympathetic neurons

Although Ca(2+)-induced Ca2+ release (CICR) via ryanodine receptors has been found to occur in intact neurons, little is known about the physiological processes that regulate it. We studied the effects of cyclic ADP-ribose (cADPR) on CICR in cultured bullfrog sympathetic neurons by fura-2 fluorescence recording and patch-clamp techniques. cADPR applied through a patch pipette augmented action potential- or depolarizing pulse-induced rises in intracellular Ca2+ without a change in Ca2+ entry initiating the responses, but not in the presence of ryanodine. Likewise, cADPR enhanced a single or oscillatory rise(s) in intracellular Ca2+ induced by caffeine. These results strongly suggest that cADPR can be an endogenous modulator of ryanodine receptors in neurons.

Caffeine induces periodic oscillations of Ca(2+)-activated K+ current in pulmonary arterial smooth muscle cells

The periodic oscillations of outward currents were studied in smooth muscle cells of the rabbit pulmonary artery. The combined stimuli of superfusion with 1 mM caffeine and depolarization of the membrane potential to 0 mV evoked periodic oscillations of outward currents with fairly uniform amplitudes and intervals. The oscillating outward currents induced by caffeine were dependent on intracellular Ca2+ concentration ([Ca2+]i) and had a reversal potential near to the equilibrium potential for K+. So the oscillating outward currents are carried by K+ through Ca(2+)-dependent K+ channels (IK(Ca)), and may reflect the oscillations of [Ca2+]i. The oscillating outward currents were abolished, or their frequency reduced, by lowering external [Ca2+], Ca2+ channel blockers, or by 1 microM ryanodine, indicating that: (1) there is a continuous influx of Ca2+ through the plasma membrane at a holding potential of 0 mV; (2) the periodic transient increases of [Ca2+]i are ascribed to the rhythmic release of Ca2+ from ryanodine-sensitive intracellular store by the mechanism of Ca(2+)-induced Ca2+ release (CICR). On the basis of the above results, we simulated the oscillation of [Ca2+]i induced by caffeine, which is known to lower the threshold of CICR. The patterns of peak amplitude histograms of spontaneous transient outward currents (STOC) in the oscillating cells were different from those in non-oscillating cells. The amplitudes of STOC in the latter were more variable than those in the former. The oscillating outward currents were modulated by 1 microM forskolin and 1 microM sodium nitroprusside, but STOC were little affected. The above differences between STOC and oscillating outward currents suggest that the two currents are activated by the Ca2+ originating from different intracellular Ca2+ stores which are functionally heterogeneous.

Transitions of latency time and oscillation phase on parameter surfaces from models of intracellular calcium ion dynamics

The dynamics of two classical elementary compartmental models stimulating intracellular calcium ion oscillatory behavior are examined in terms of parameter surfaces. It has been found that, along certain lines of instability on surfaces defined by model parameters, the highly non-linear nature of these models produces sharp transitions in the latency time which determines the phase of oscillations once they commence. This sensitivity to initial conditions in deterministic models, along with the stochastic variance inevitably present in actual biological systems, illustrates how two seemingly identical cells activated by identical synchronous stimulation can exhibit oscillatory responses which are out of phase with respect to each other.

Characteristics of Ca2+ release induced by Ca2+ influx in cultured bullfrog sympathetic neurones

1. A rise in intracellular Ca2+ ([Ca2+]i) and a Ca2+ current (ICa) induced by a depolarizing pulse were simultaneously recorded by fura-2 or indo-1 fluorescence and whole-cell patch clamp techniques in cultured bullfrog sympathetic ganglion cells. 2. [Ca2+]i (calculated from the ratio of fura-2 fluorescences excited at 380 and 340 nm and recorded with a photomultiplier at > 492 nm) rose regeneratively (in most cells) during a command pulse (from -60 to 0 mV, 100 ms), continued to rise thereafter, peaked at 666 ms (on average) and decayed slowly with a half-decay time of 22.8 s. 3. Scanning a single horizontal line across the cytoplasm with an ultraviolet argon ion laser (351 nm) and recording indo-1 fluorescences at two wavelengths (peaked at 410 and 475 nm) with a confocal microscope demonstrated that [Ca2+]i beneath the cell membrane rose much faster than that in the deeper cytoplasm. The time course of the spatial integral of [Ca2+]i, however, corresponded well with that recorded with fura-2 fluorescence using a photomultiplier. 4. [Ca2+]i measured by fura-2 fluorescence ratio using a photomultiplier did not increase during a strong depolarizing pulse (-60 to +80 mV), but sometimes rose after the pulse. A depolarization-induced rise in [Ca2+]i ([Ca2+]i transient) was blocked in a Ca(2+)-free, EGTA solution, reduced by lowering the extracellular Ca2+ concentration ([Ca2+]o) to 0.45 or 0.9 mM and enhanced by raising [Ca2+]o to 7.2 or 14.4 nM. 5. The extracellular Ca2+ dependence was non-linear when long depolarizing pulses (up to 500 ms) were applied; the amplitude of [Ca2+]i transient/Ca2+ entry (unit [Ca2+]i transient) increased with an increase in Ca2+ entry. 6. Increasing the duration of depolarization (-50 or -60 to 0 mV) from 20 to 500 ms enhanced asymptotically the integral of ICa (due to inactivation), and progressively the magnitude of [Ca2+]i transients, leading to the apparent non-linear dependence of unit [Ca2+]i transient on Ca2+ entry as well as on the duration of membrane depolarization. The peak time of [Ca2+]i transient was unchanged for pulse durations up to 300 ms, but prolonged with an increase in pulse duration to 500 ms. 7. Inhibitors of Ca2+ release from intracellular Ca2+ reservoirs, dantrolene (10 microM) and ryanodine (50 microM), blocked the [Ca2+]i transient to 56 and 30%, respectively, of the control. 8. The higher the basal [Ca2+]i level, the greater was the magnitude of the [Ca2+]i transients.(ABSTRACT TRUNCATED AT 400 WORDS)

One-pool model for Ca2+ oscillations involving Ca2+ and inositol 1,4,5-trisphosphate as co-agonists for Ca2+ release

Experimental observations indicate that Ca(2+)-induced Ca2+ release (CICR) may underlie Ca2+ oscillations in a variety of cells. In its original version, a theoretical model for signal-induced Ca2+ oscillations based on CICR assumed the existence of two types of pools, one sensitive to inositol 1,4,5-trisphosphate (IP3) and the other one sensitive to Ca2+. Recent experiments indicate that Ca2+ channels may sometimes be sensitive to both IP3 and Ca2+. Such a regulation may be viewed as Ca(2+)-sensitized IP3-induced Ca2+ release or, alternatively, as a form of IP3-sensitized CICR. We show that sustained oscillations can still occur in a one-pool model, provided that the same Ca2+ channels are sensitive to both Ca2+ and IP3 behaving as co-agonists. This model and the two-pool model based on CICR both account for a number of experimental observations but differ in some respects. Thus, while in the two-pool model the latency and period of Ca2+ oscillations are of the same order of magnitude and correlate in a roughly linear manner, latency in the one-pool model is always brief and remains much shorter than the period of oscillations. Moreover, the first Ca2+ spike is much larger than the following ones in the one-pool model. These distinctive properties might provide an explanation for the differences in Ca2+ oscillations observed in various cell types.

Calcium wave propagation by calcium-induced calcium release: an unusual excitable system

We discuss in detail the behaviour of a model, proposed by Goldbeter et al. (1990. Proc. natn. Acad. Sci. 87, 1461-1465), for intracellular calcium wave propagation by calcium-induced calcium release, focusing our attention on excitability and the propagation of waves in one spatial dimension. The model with no diffusion behaves like a generic excitable system, and threshold behaviour, excitability and oscillations can be understood within this general framework. However, when diffusion is included, the model no longer behaves like a generic excitable system; the fast and slow variables are not distinct and previous results on excitable systems do not necessarily apply. We consider a piecewise linear simplification of the model, and construct travelling pulse and periodic plane wave solutions to the simplified model. The analogous behaviour in the full model is studied numerically. Goldbeter's model for calcium-induced calcium release is an excitable system of a type not previously studied in detail.

A membrane model for cytosolic calcium oscillations. A study using Xenopus oocytes

Cytosolic calcium oscillations occur in a wide variety of cells and are involved in different cellular functions. We describe these calcium oscillations by a mathematical model based on the putative electrophysiological properties of the endoplasmic reticulum (ER) membrane. The salient features of our membrane model are calcium-dependent calcium channels and calcium pumps in the ER membrane, constant entry of calcium into the cytosol, calcium dependent removal from the cytosol, and buffering by cytoplasmic calcium binding proteins. Numerical integration of the model allows us to study the fluctuations in the cytosolic calcium concentration, the ER membrane potential, and the concentration of free calcium binding sites on a calcium binding protein. The model demonstrates the physiological features necessary for calcium oscillations and suggests that the level of calcium flux into the cytosol controls the frequency and amplitude of oscillations. The model also suggests that the level of buffering affects the frequency and amplitude of the oscillations. The model is supported by experiments indirectly measuring cytosolic calcium by calcium-induced chloride currents in Xenopus oocytes as well as cytosolic calcium oscillations observed in other preparations.

Oscillations and waves of cytosolic calcium: insights from theoretical models

Oscillations in cytosolic Ca2+ occur in a wide variety of cells, either spontaneously or as a result of external stimulation. This process is often accompanied by intracellular Ca2+ waves. A number of theoretical models have been proposed to account for the periodic generation and spatial propagation of Ca2+ signals. These models are reviewed and their predictions compared with experimental observations. Models for Ca2+ oscillations can be distinguished according to whether or not they rely on the concomitant, periodic variation in inositol 1,4,5-trisphosphate. Such a variation, however, is not required in models based on Ca(2+)-induced Ca2+ release. When Ca2+ diffusion is incorporated into these models, propagating waves of cytosolic Ca2+ arise, with profiles and rates comparable to those seen in the experiments.

Phase-dependent contributions from Ca2+ entry and Ca2+ release to caffeine-induced [Ca2+]i oscillations in bullfrog sympathetic neurons

Sympathetic neurons display robust [Ca2+]i oscillations in response to caffeine and mild depolarization. Oscillations occur at constant membrane potential, ruling out voltage-dependent changes in plasma membrane conductance. They are terminated by ryanodine, implicating Ca(2+)-induced Ca2+ release. Ca2+ entry is necessary for sustained oscillatory activity, but its importance varies within the oscillatory cycle: the slow interspike rise in [Ca2+]i requires Ca2+ entry, but the rapid upstroke does not, indicating that it reflects internal Ca2+ release. Sudden alterations in [Ca2+]o, [K+]o, or [caffeine]o produce immediate changes in d[Ca2+]i/dt and provide information about the relative rates of surface membrane Ca2+ transport as well as uptake and release by internal stores. Based on our results, [Ca2+]i oscillations can be explained in terms of coordinated changes in Ca2+ fluxes across surface and store membranes.

Basal Ca2+ and the oscillation of Ca2+ in caffeine-treated bullfrog sympathetic neurones

1. Effects of caffeine on the intracellular free Ca2+ concentration ([Ca2+]i) in single bullfrog sympathetic neurones in excised tissue were studied by recording Fura-2 fluorescence excited at 340, 361 or 380 nm and taking their ratios (R340/380 or R361/380). 2. Caffeine (3-10 mM) produced oscillation of [Ca2+]i and an 'apparent' decrease in the basal level of [Ca2+]i during a period between phasic rises. The mechanism of the latter effect was analysed in relation to the mechanism of the former. 3. Caffeine (3-10 mM) increased Fura-2 fluorescence in a range of excitation wavelength from 330 to 390 nm. The ratios of fluorescences, R340/380 and R361/380, however, were not significantly affected by caffeine. These results suggest that the 'apparent' reduction in the basal [Ca2+]i seen as a decrease in R340/380 or R361/380 results from a true decrease in [Ca2+]i. 4. Caffeine-induced decrease in [Ca2+]i persisted for every period between phasic rises of [Ca2+]i during [Ca2+]i oscillation, and after the blockade of [Ca2+]i oscillation by ryanodine. The decrease in the latter condition lasted for more than 20 min. 5. The decrease in the basal [Ca2+]i depended on the external Ca2+ concentration and was not mimicked by the action of cyclic nucleotides. 6. Possible mechanisms underlying the decrease in the basal [Ca2+]i produced by caffeine (effects on Ca2+ transport at the cell or Ca(2+)-storing organelle membrane) and their significance in relation to the [Ca2+]i oscillation were discussed.

Acetylcholine- and caffeine-evoked repetitive transient Ca(2+)-activated K+ and C1- currents in mouse submandibular cells

1. Resting and acetylcholine-induced membrane currents were measured in single mouse submandibular acinar cells using the patch-clamp whole-cell current recording technique. 2. Micromolar ACh activated a large, sustained outward, Ca(2+)-dependent K+ current and a single transient inward Ca(2+)-dependent C1-current. 3. Nanomolar ACh induced a series of transients in both the K+ and C1- currents; C1- current activation was now observed throughout the period of agonist application. We consider this repetitive transient current activation better able to support sustained fluid and electrolyte secretion than the response elicited by a high dose of agonist. 4. Repetitive K+ and C1- current transients were also induced by 1 mM-caffeine, consistent with caffeine-induced Ca2+ release from the Ca(2+)-sensitive Ca2+ stores which are thought to comprise part of the pathway for activation of secretion. 5. The ACh-induced current transients were inhibited by 10 mM-caffeine, 100 microM-IBMX and 10 microM membrane-permeable cyclic AMP. Therefore, it seems likely that caffeine is able to inhibit agonist-induced calcium mobilization via a cyclic AMP-dependent pathway.

Two roles of Ca2+ in agonist stimulated Ca2+ oscillations

We propose a mechanism for agonist-stimulated Ca2+ oscillations that involves two roles for cytosolic Ca2+: (a) inhibition of inositol-1,4,5-trisphosphate (IP3) stimulated Ca2+ release from the endoplasmic reticulum (ER) and (b) stimulation of the production of IP3 through its action on phospholipase C (PLC), via a Gq protein related mechanism. Relying on quantitative experiments by Parker, I., and I. Ivorra (1990. Proc. Natl. Acad. Sci. USA. 87:260-264) on the inhibition of Ca2+ release from the ER using caged-IP3, we develop a kinetic model of inhibition that allows us to simulate closely their experiments. The model assumes that the ER IP3 receptor is a tetramer of independent subunits that can bind both Ca2+ and IP3. Upon incorporation of the action of Ca2+ on PLC that leads to production of IP3, we observe in-phase-oscillations of Ca2+ and IP3 at intermediate values of agonist stimulation. The oscillations occur on a time scale of 10-20 s, which is comparable to the time scale for inhibition in Xenopus oocytes. Analysis of the mechanism shows that Ca(2+)-inhibition of IP3-stimulated Ca2+ release from the ER is an essential step in the mechanism. We also find that the effect of Ca2+ on PLC can lead to an indirect increase of cytosolic Ca2+, superficially resembling "Ca(2+)-induced Ca(2+)-release." The mechanism that we propose appears to be consistent with recent experiments on REF52 cells by Harootunian, A. T., J. P. Y. Kao, S. Paranjape, and R. Y. Tsien. (1991. Science [Wash. DC]. 251:75-78.) and we propose additional experiments to help test its underlying assumptions.

Two-dimensional model of calcium waves reproduces the patterns observed in Xenopus oocytes

Biological excitability enables the rapid transmission of physiological signals over distance. Using confocal fluorescence microscopy, we previously reported circular, planar, and spiral waves of Ca2+ in Xenopus laevis oocytes that annihilated one another upon collision. We present experimental evidence that the excitable process underlying wave propagation depends on Ca2+ diffusion and does not require oscillations in inositol (1,4,5)trisphosphate (IP3) concentration. Extending an existing ordinary differential equation (ODE) model of Ca2+ oscillations to two spatial dimensions, we develop a partial differential equation (PDE) model of Ca2+ excitability. The model assumes that cytosolic Ca2+ couples neighboring Ca2+ release sites. This simple PDE model qualitatively reproduces our experimental observations.

Some new observations on caffeine-induced rhythmic hyperpolarization in frog sympathetic ganglion cells

Unstimulated bullfrog sympathetic ganglia were studied in vitro by intracellular and extracellular recording methods. In 80% of the cells impaled with K citrate microelectrodes, caffeine caused initial hyperpolarization (ICH) followed by rhythmic membrane hyperpolarization (RMH). Four different patterns of rhythmicity were observed, the most common being a regular beating pattern. RMH frequency depended on both caffeine and Ca2+. Tetraethylammonium reduced RMH amplitude, but did not affect frequency. Caffeine effects on cyclic AMP are not responsible for RMH since neither dibutyryl cyclic AMP nor phosphodiesterase inhibitors elicited RMH. However, the anion in the microelectrode filling solution is critical to both the incidence and amplitude of RMH, the order of effectiveness being: citrate much much greater than glutamate, acetate and chloride. In cells impaled by electrodes filled with K thiocyanate or K iodide, caffeine also caused large amplitude hyperpolarizing oscillations of membrane potential, suggesting that the effectiveness of citrate is not due to Ca2+ chelation. High gain extracellular DC recording revealed no sign of caffeine ICH, RMH or any hyperpolarizing effects. The absence of signs of caffeine hyperpolarization with extracellular recording has several interpretations, and these are discussed.

The brain ryanodine receptor: a caffeine-sensitive calcium release channel

The release of stored Ca2+ from intracellular pools triggers a variety of important neuronal processes. Physiological and pharmacological evidence has indicated the presence of caffeine-sensitive intracellular pools that are distinct from the well-characterized inositol 1,4,5,-trisphosphate (IP3)-gated pools. Here we report that the brain ryanodine receptor functions as a caffeine- and ryanodine-sensitive Ca2+ release channel that is distinct from the brain IP3 receptor. The brain ryanodine receptor has been purified 6700-fold with no change in [3H]ryanodine binding affinity and shown to be a homotetramer composed of an approximately 500 kd protein subunit, which is identified by anti-peptide antibodies against the skeletal and cardiac muscle ryanodine receptors. Our results demonstrate that the brain ryanodine receptor functions as a caffeine-sensitive Ca2+ release channel and thus is the likely gating mechanism for intracellular caffeine-sensitive Ca2+ pools in neurons.

Signal-induced Ca2+ oscillations: properties of a model based on Ca(2+)-induced Ca2+ release

We consider a simple, minimal model for signal-induced Ca2+ oscillations based on Ca(2+)-induced Ca2+ release. The model takes into account the existence of two pools of intracellular Ca2+, namely, one sensitive to inositol 1,4,5 trisphosphate (InsP3) whose synthesis is elicited by the stimulus, and one insensitive to InsP3. The discharge of the latter pool into the cytosol is activated by cytosolic Ca2+. Oscillations in cytosolic Ca2+ arise in this model either spontaneously or in an appropriate range of external stimulation; these oscillations do not require the concomitant, periodic variation of InsP3. The following properties of the model are reviewed and compared with experimental observations: (a) Control of the frequency of Ca2+ oscillations by the external stimulus or extracellular Ca2+; (b) correlation of latency with period of Ca2+ oscillations obtained at different levels of stimulation; (c) effect of a transient increase in InsP3; (d) phase shift and transient suppression of Ca2+ oscillations by Ca2+ pulses, and (e) propagation of Ca2+ waves. It is shown that on all these counts the model provides a simple, unified explanation for a number of experimental observations in a variety of cell types. The model based on Ca(2+)-induced Ca2+ release can be extended to incorporate variations in the level of InsP3 as well as desensitization of the InsP3 receptor; besides accounting for the phenomena described by the minimal model, the extended model might also account for the occurrence of complex Ca2+ oscillations.

Repetitive calcium transients in hamster oocytes

Golden hamster oocytes show repetitive Ca2+ transients at fertilization: a propagating Ca2+ rise from the sperm attachment site in the first 2-3 responses and synchronous Ca2+ rise in the entire egg in the succeeding responses. Cyclic Ca2+ rises are produced in unfertilized eggs by an injection of GTP gamma S or continuous injection of inositol 1,4,5-trisphosphate (InsP3). Both InsP3-induced Ca2+ release (IICR) and Ca(2+)-induced Ca2+ release (CICR) are observed in hamster eggs, associated with a refractory period of 1-2 min after a Ca2+ release. In addition, external Ca2+ is a prerequisite for maintaining the repeated Ca2+ transients. The conditions that are expected to alter Ca2+ influx affect the frequency of Ca2+ transients with little effect on each response. The fertilizing sperm causes an increase in Ca2+ permeability of the egg plasma membrane and an increase in Ca2+ sensitivity of CICR. Feedback inhibition through protein kinase C is observed in G-protein-mediated Ca2+ transients but this inhibition seems to operate rather tonically. A model of Ca2+ oscillation is proposed: basically a second messenger-controlled oscillator model. InsP3 as the rigger of Ca2+ release is continuously supplied while an elevated basal [Ca2+]i level due to Ca2+ influx provides a favourable condition for IICR and CICR as well as for recharging the Ca2+ pools ready to release Ca2+ again.

InsP3- and Ca2(+)-induced Ca2+ release in single mouse oocytes

To better understand the mechanism of intracellular Ca2+ mobilization, mouse oocytes were micro-injected with 'caged'-inositol-1,4,5 triphosphate caged-InsP3) together with the Ca2+ indicator Fluo-3 to directly induce and monitor Ca2+ redistribution. Photo-released InsP3 elicits [Ca2+]i changes exhibiting several kinetic phases and threshold behaviour. Often Ca2+ oscillations were induced after a single InsP3 pulse. Autoregenerative Ca2+ transients could also be induced by injections of Ca2+ itself, demonstrating unequivocally the presence of a Ca2(+)-induced Ca2(+)-release mechanism in these cells.

Latency correlates with period in a model for signal-induced Ca2+ oscillations based on Ca2(+)-induced Ca2+ release

Oscillations in cytosolic Ca2+ develop in a variety of cells after an induction phase, called latency, the duration of which depends on the magnitude of external stimulation. Experiments in hepatocytes indicate that the period and latency of Ca2+ oscillations both decrease as the level of the stimulus increases. We analyze the correlation between period and latency in a model recently proposed for signal-induced Ca2+ oscillations. We show that the linear relationship between period and latency observed in the experiments arises naturally in this model as a result of the mechanism of Ca2(+)-induced Ca2+ release on which it is based.

Allosteric regulation, cooperativity, and biochemical oscillations

Allosteric regulation is associated with a number of periodic phenomena in biochemical systems. The cooperative nature of such regulatory interactions provides a source of nonlinearity that favors oscillatory behavior. We assess the role of cooperativity in the onset of biochemical oscillations by analyzing two specific examples. First, we consider a model for a product-activated allosteric enzyme which has previously been proposed to account for glycolytic oscillations. While enzyme cooperativity plays an important role in the occurrence of oscillations, we show that these may nevertheless occur in the absence of cooperativity when the reaction product is removed in a Michaelian rather than linear manner. The second model considered was recently proposed to account for signal-induced oscillations of intracellular calcium. This phenomenon originates from a nonlinear process of calcium-induced calcium release. Here also, the cooperative nature of that positive feedback favors the occurrence of oscillations but is not absolutely required for periodic behavior. Besides underlining the importance of cooperativity, the results highlight the role of diffuse nonlinearities distributed over several steps within a regulated system: even in the absence of cooperativity, such mild nonlinearities (e.g., of the Michaelian type) may combine to raise the overall degree of nonlinearity up to the level required for oscillations.

Subcellular calcium transients visualized by confocal microscopy in a voltage-clamped vertebrate neuron

Confocal laser-scanned microscopy and long-wavelength calcium (Ca2+) indicators were combined to monitor both sustained and rapidly dissipating Ca2+ gradients in voltage-clamped sympathetic neurons isolated from the bullfrog. After a brief activation of voltage-dependent Ca2+ channels, Ca2+ spreads inwardly, and reaches the center of these spherical cells in about 300 milliseconds. Although the Ca2+ redistribution in the bulk of the cytosol could be accounted for with a radial diffusion model, local nonlinearities, suggesting either nonuniform Ca2+ entry or spatial buffering, could be seen. After electrical stimulation, Ca2+ signals in the nucleus were consistently larger and decayed more slowly than those in the cytosol. A similar behavior was observed when release of intracellular Ca2+ was induced by caffeine, suggesting that in both cases large responses originate from Ca2+ release sites near or within the nucleus. These results are consistent with an amplification mechanism involving Ca2(+)-induced Ca2+ release, which could be relevant to activity-dependent, Ca2(+)-regulated nuclear events.

Minimal model for signal-induced Ca2+ oscillations and for their frequency encoding through protein phosphorylation

In a variety of cells, hormonal or neurotransmitter signals elicit a train of intracellular Ca2+ spikes. The analysis of a minimal model based on Ca2(+)-induced Ca2+ release from intracellular stores shows how sustained oscillations of cytosolic Ca2+ may develop as a result of a rise in inositol 1,4,5-trisphosphate (InsP3) triggered by external stimulation. This rise elicits the release of a certain amount of Ca2+ from an InsP3-sensitive intracellular store. The subsequent rise in cytosolic Ca2+ in turn triggers the release of Ca2+ from a second store insensitive to InsP3. In contrast to the model proposed by Meyer and Stryer [Meyer, T. & Stryer, L. (1988) Proc. Natl. Acad. Sci. USA 85, 5051-5055], the present model, which contains only two variables, predicts the occurrence of periodic Ca2+ spikes in the absence of InsP3 oscillations. Such results indicate that repetitive Ca2+ spikes evoked by external stimuli do not necessarily require the concomitant, periodic variation of InsP3. The model is closely related to that proposed by Kuba and Takeshita [Kuba, K. & Takeshita, S. (1981) J. Theor. Biol. 93, 1009-1031] for Ca2+ oscillations in sympathetic neurones, based on Ca2(+)-induced Ca2+ release. We extend their results by showing the minimal conditions in which the latter process gives rise to periodic behavior and take into account the role of the rise in InsP3 caused by external stimulation. The analysis further shows how signal-induced Ca2+ oscillations might be effectively encoded in terms of their frequency through the phosphorylation of a cellular substrate by a protein kinase activated by cytosolic Ca2+.