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HFSP J. 2010 April; 4(2): 43–51.
Published online 2010 April 21. doi:  10.2976/1.3385660
PMCID: PMC2931296

Spatiotemporal organization of Ca2+ dynamics: a modeling-based approach

Abstract

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 Ca2+, these signals are highly organized both in time and space to ensure reliability and specificity. This review deals with the spatiotemporal organization of the Ca2+ signaling pathway in electrically nonexcitable cells in which InsP3 receptors are by far the most important Ca2+ 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 InsP3 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 InsP3 metabolism and Ca2+ oscillations will be considered. Finally, we discuss the relationships between stochastic openings of the Ca2+ 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.

Calcium is one of the most widespread second messenger in cellular biology. It mediates vital physiological responses as, for example, fertilization, secretion, gene expression, or apoptosis (Berridge et al., 2003; Combettes et al., 2004). Importantly, these Ca2+ signals are highly organized in time and space (Dupont et al. 2007). When cells are activated, particularly through receptors leading to the synthesis of inositol 1,4,5-trisphosphate (InsP3), the level of Ca2+ often oscillates with a frequency that varies with agonist concentration. These frequency-encoded Ca2+ pulses then convey information as to the nature and the extent of the stimulated physiological response. Moreover, each Ca2+ pulse spreads throughout the cytosol in the form of a wave. It is now well established that InsP3-induced Ca2+ oscillations originate from the repetitive opening of InsP3 receptors (InsP3R), which are Ca2+ channels embedded in the membrane of the endoplasmic reticulum (ER) that behaves as a huge Ca2+ store. When open, the InsP3R allows for a flux of Ca2+ down its electrochemical gradient. The concentration of Ca2+ in the ER [[Ca2+]ER≈500 μM, see Chatton et al. (1995) and Meldolesi and Pozzan (1998)] is indeed considerably higher than in the cytosol ([Ca2+]i≈0.1 μM). Cytosolic Ca2+ also regulates the Ca2+ releasing activity of this channel. InsP3 and Ca2+ thus act as coagonists of the InsP3R. Intermediate levels of cytosolic Ca2+ activate Ca2+ release from the ER while high levels of Ca2+ ultimately lead to channel closing (Bezprozvanny et al., 1991). Models indicate that this biphasic regulation of InsP3 receptors by Ca2+ can explain both oscillations and waves much resembling experimental observations.

Calcium release is initiated by inositol 1,4,5-trisphosphate (InsP3), generated in response to the external stimulus through a well-characterized signaling cascade (Berridge, 1993). Oscillations are then maintained as long as InsP3 remains elevated in the cytoplasm. As expected, oscillations in [Ca2+]ER are observed in antiphase to [Ca2+]i oscillations (Ishii et al., 2006). Although Ca2+ oscillations generally take the form of regular spikes, more complex Ca2+ oscillations have also been observed. In particular, bursting behavior similar to that displayed by electrically excitable cells has been reported. One of the best known example of Ca2+ bursting is the complex oscillatory pattern seen in hepatocytes after stimulation by specific agonists such as cAMP or ATP (Dixon et al., 1993; Green et al., 1997). The mechanism underlying this type of Ca2+ oscillations remains poorly understood from an experimental point of view and has mainly been investigated by several theoretical approaches (Kummer et al., 2000; Marhl et al., 2000).

It is often assumed that the potency of the cell to control so diverse physiological processes with a compound as simple as Ca2+ ions results from the large versatility of the signal-induced Ca2+ changes. The existence of different InsP3R isoforms is probably an important factor allowing such a diversity of responses [see Taylor et al. (1999) and Vermassen et al. (2004) for reviews]. The interplay between InsP3 metabolism and Ca2+ signaling is another factor that can modulate signal-induced Ca2+ rises depending on the cell type or other internal factors such as the specific cytoplasmic locus or the developmental state of the cell (Dupont and Erneux, 1997). In this review, we will first give a brief outline about models for Ca2+ oscillations based on the dynamics of the InsP3R (Deterministic models for Ca2+ oscillations section). After that, we will emphasize how this basic oscillatory mechanism can be affected either by differences at the regulatory level due to specific receptor isoforms (Impact of the InsP3 receptor isoform on Ca2+ oscillations section), or by interplay with InsP3 metabolism (Interplay between Ca2+ oscillations and InsP3 metabolism section).

Besides the observation of global Ca2+ signals, an impressive number of experimental studies have been devoted to the imaging of the Ca2+ releasing activity of a small number of InsP3 receptors in vivo. This activity can occur either spontaneously or at submaximal InsP3 concentrations. As can be expected, these events appear to be stochastic. Their amplitude and the interval between them vary significantly under the same conditions, allowing only for a statistical description. The smallest observed events, called blips, provoke a rise in cytosolic Ca2+ of about 40 nM and last in average 70 ms. These data correspond to successive openings of a single InsP3 receptor in the cytosol, which is possible due to the poor diffusing properties of Ca2+ in the cytoplasm allowing for rapid rebinding of Ca2+ on the channels activating sites, as indicated by quantitative models (Swillens et al., 1998; Shuai and Jung, 2002; Thul and Falcke, 2004; Nguyen et al., 2005; Cooper et al., 2009). Alternatively, the openings of one InsP3R may trigger other InsP3Rs within a cluster to generate slightly larger Ca2+ increases known as Ca2+ puffs. The rise in Ca2+ is then of about 200 nM and lasts ~300 ms. Blips and puffs have been extensively studied in HeLa cells (Thomas et al., 2000; Bootman et al., 2002) andXenopus oocytes (Marchant et al., 1999; Smith and Parker, 2009). These elementary events have been modeled using a stochastic description of the dynamics of the InsP3R (Swillens et al., 1999; Williams et al., 2008; Smith et al., 2009). Given that (irregular) Ca2+ puffs and (much more regular) oscillations can be observed in the same cell for different levels of stimulus, the study of Ca2+ dynamics offers the fascinating possibility to study the transition from a stochastic to a deterministic regime. This transition corresponds to the emergence of a macroscopic self-organized system, which is a central concern in biology (Nicolis and Prigogine, 1977). The current development in this field, in relation with Ca2+ dynamics, will be briefly reviewed in the Stochastic aspects of Ca2+ oscillations section.

DETERMINISTIC MODELS FOR Ca2+ OSCILLATIONS

Most models for InsP3-induced Ca2+ oscillations rely on a mathematical description of both the Ca2+ transfer processes and the regulatory mechanisms known to play an essential role in this periodic behavior. In particular, the rapid activation-slow inhibition of the InsP3R by cytosolic Ca2+ clearly plays a crucial role in the onset of oscillations (Li and Rinzel, 1994). Thus, models need to reproduce this regulation, and the associated well-known bell-shaped curve for the stationary open probability of the InsP3R as a function of cytosolic Ca2+ (Bezprozvanny et al., 1991). One of these models is schematized in Fig. Fig.1.1. As shown in the left panel, it is based on the mutually exclusive activation-inhibition of the InsP3 receptor by Ca2+. It is assumed that InsP3 and Ca2+-mediated activation are instantaneous, whereas Ca2+-induced inhibition develops slowly. The binding of Ca2+ to the activating and inhibitory sites are supposed to be mutually exclusive. If activation and inhibition are considered as cooperative processes characterized by Hill coefficients na and ni, respectively, the change in the fraction of inactive receptors (Ri) obeys the following equation [see Dupont and Swillens (1996) for a detailed description of the equations]:

dRidt=k+Cni1Ri1+(CKact)nakRi,
(1)

where k+ and k are the kinetic constants of Ca2+ association to and dissociation from the inhibitory Ca2+ binding site of the receptor, C represents the concentration of cytosolic Ca2+, and Kact is the dissociation constant of Ca2+ binding to the activating Ca2+ binding site of the InsP3R. The time evolution of the concentration of cytosolic Ca2+ is given by

dCdt=k1(b+IRa)[CtotC(α+1)]VMPC2KP2+C2
(2)

in which IRa represents the fraction of active (i.e., open) InsP3R and is given by

IRa=IRableCnaKactna+Cna,
(3)

where IRable is the fraction of receptors that can be activated (“activatable”). This fraction, which represents the fraction of noninhibited receptors bound to InsP3, can also be evaluated from an algebraic equation.

IRable=(1Ri)IP3KIP3+IP3.
(4)

IP stands for the intracellular concentration in InsP3. k1 is the kinetic constant governing the flux of Ca2+ from the lumen into the cytosol and k1b represents the basal flux in the absence of InsP3. KIP is the dissociation constant of InsP3 binding to its receptor. In agreement with experimental results, activation by InsP3 is taken as a cooperative process with a Hill coefficient equal to 3. Ctot represents the total intracellular concentration of free Ca2+ (not bound to buffers). α is a factor defined as the ratio between the volume of the ER and the volume of the cytosol. As total Ca2+ is assumed to remain constant (the model does not consider possible Ca2+ exchanges with the extracellular medium), the term between brackets in Eq. 2 represents the difference in Ca2+ concentration between the ER and the cytosol. The second term of the right-hand side of Eq. 2 stands for Ca2+ pumping back from the cytosol into the ER, a process which is known to be mediated by SERCA pumps, which are well-characterized ATPases (Lytton et al., 1992).

Figure 1
Schematic representation of a model forCa 2+ oscillations based on the regulatory properties of theInsP3receptor/Ca2+channel.

Various mathematical models based on this core oscillatory mechanism have been proposed (see Thul et al. (2008) for a recent review). While, as in the example above, some models use an empirical description of the successive positive and negative feedbacks leading to oscillations (Atri et al., 1993; Dupont and Swillens, 1996), mechanistic models describing in a more detailed way the transitions between the states of the receptor channel have also been proposed (De Young and Keizer, 1992; Tang et al., 1996). As these models describe the evolution of the various states of the receptors, they usually contain a large number of variables. However, quasi-steady-state assumptions on some receptor states can drastically simplify the dynamical description and reduce the system to a two-variable system whose equations are similar to the Hodgkin–Huxley equations (Li and Rinzel, 1994). It should be noted that besides InsP3R regulation, some models take into account other phenomena related to InsP3 and/or Ca2+ dynamics that markedly influence Ca2+ oscillations and waves in some cell types. Thus, Ca2+ feedbacks at the level of InsP3 synthesis or metabolism into InsP2 or InsP4 can markedly influence the time evolution of Ca2+ and InsP3 (Meyer and Stryer, 1991; Dupont and Erneux, 1997; Politi et al., 2006). Because Ca2+ is electrically charged, ER membrane potential oscillations can also play a role (Marhl et al., 2000). The fact that Ca2+ enters the cytosol both from the ER and from the external medium makes the understanding of the observed Ca2+ dynamics significantly more difficult and has also been considered in some models (Sneyd et al., 2004). Finally, Ca2+-activated protein kinase C (Kummer et al., 2000; Kang and Othmer, 2007) or phosphorylation of the InsP3R (LeBeau et al., 1999) have been shown to affect the rate of InsP3 synthesis or Ca2+ dynamics after stimulation of some types of phospholipase (PLC)-linked external receptors.

IMPACT OF THE InsP3 RECEPTOR ISOFORM ON Ca2+ OSCILLATIONS

Three isoforms of the InsP3R have been identified (InsP3R1, InsP3R2, and InsP3R3) and their levels of expression are largely tissue specific. Experiments where the levels of expression of these isoforms have been modified (overexpress or knock-down) clearly indicate that their proportions significantly affect the time course of cytosolic Ca2+ concentration (Miyakawa et al., 1999; Morel et al., 2003; Hattori et al., 2004). The three receptor subtypes are distributed inhomogeneously throughout the cytoplasm. For example, in hepatocytes, InsP3-induced Ca2+ signals begin sooner in the apical region where InsP3R2 are concentrated (Pusl and Nathanson, 2004). The various ways of regulating Ca2+ dynamics lead to different functions. For example, the type 3 isoform is specifically involved in the induction of apoptosis by preferentially transmitting Ca2+ signals into mitochondria (Mendes et al., 2005). The level of expression of the different subtypes is also developmental and conditions specific. For example, in cardiomyocytes, the levels of type 1 and 2 receptors are increased, but differently, by stressors such as cold exposure and hypoxia (Krizanova et al., 2008).

Although all the InsP3R subtypes display similar ion permeation properties, they differ significantly in their regulatory properties due to different interactions with accessory proteins, various modes of regulation by ATP or phosphorylation (Taylor and Laude, 2002; Patterson et al., 2004). The exact nature of these differences, as well as their molecular origin, remains controversial. It is clear that they are differently regulated by Ca2+ and InsP3. In particular, different open probabilities at steady-state have been reported. Although consistent experimental data are lacking, the classical bell-shaped curve most probably corresponds to type 2 InsP3R. This curve can be explained by a change in the effect of cytosolic Ca2+ on channel activity, which passes from activating to inhibitory when increasing Ca2+ concentration. Another subtype, presumably type 3, displays similar activation at low Ca2+ concentration but inhibition occurs for higher Ca2+ levels in the range of a few micromolars. Finally, a displacement of the bell-shaped curve with InsP3 has also been reported. In this case, which corresponds to the InsP3R1, the maximum of the bell-shaped curve shifts to the right when increasing the InsP3 concentration (Kaftan et al., 1997; Moraru et al., 1999). The differences among the subtypes concerning regulation by InsP3 are much less controversial: type 2 has the greatest affinity for InsP3, followed by types 1 and 3 successively (Vermassen et al., 2004).

On the basis of the observed steady-state open probabilities, modeling has allowed to link the changes in the overall Ca2+ dynamics observed after modification of the levels of expression of the various subtypes to their different regulations by InsP3 and Ca2+ (Dupont and Combettes, 2006). Thus, type 2 receptor, which shows the sharpest dependence on cytosolic Ca2+ and has the highest affinity for InsP3 is the main oscillatory unit, as shown in DT40 cells (Miyakawa et al., 1999) and myocytes (Morel et al., 2003). Stimulation of type 1 receptor can also lead to oscillations but most often damped rather than sustained (Miyakawa et al., 1999; Hattori et al., 2004). In contrast, type 3 receptor tends to suppress oscillations. This surprising result is due to the fact that InsP3R3, which are not inhibited at the Ca2+ level reached during oscillations, provide a constant flux of Ca2+ without providing the negative feedback necessary for oscillations to occur. Thus, when all three InsP3 receptor subtypes are present, type 3 receptor constantly releases Ca2+, which “pushes” types 1 and 2 receptors into the regime where cytosolic Ca2+ inhibits further Ca2+ release. Modeling however predicts that this suppression of oscillations occurs only at relatively high receptor density, as is illustrated in Fig. Fig.22 (left panels). In a cell type where the total receptor density (InsP3R1+InsP3R2+InsP3R3) is low (Fig. (Fig.2,2, right panels), the constant Ca2+ flux provided by type 3 receptors could activate types 1 and 2 and thereby favor Ca2+ oscillations (Dupont and Combettes, 2006). This prediction might be related to the observation performed in pancreatic acinar cells that the addition of both InsP3R2 and InsP3R3 support Ach and CCK-induced Ca2+ oscillations (Futatsugi et al., 2005).

Figure 2
Simulations of the possibly antagonistic effects of the expression ofInsP3receptors of type 3 onCa2+oscillations.

InsP3 receptors mostly exist in the form of tetramers. These functional channels are composed of subunits of the same (homotetramers) or different isoforms (heterotetramers), which adds another level of complexity to the present modeling approach. In the same line, the fact that InsP3Rs form macromolecular signaling complexes with other proteins such as calmodulin-kinase II or phosphatases 1 and 2A should not be overlooked (Mikoshiba, 2007). These accessory proteins indeed regulate channel opening and thereby affect Ca2+ signaling. Little is known about the possibly distinct abilities of the different isoforms of the InsP3Rs to form these signaling complexes, as most studies have been performed in cerebellar tissues, which mostly express type 1 InsP3R.

INTERPLAY BETWEEN Ca2+ OSCILLATIONS AND InsP3 METABOLISM

Membrane-bound PLC is responsible for InsP3 synthesis via hydrolysis of PIP2. This enzyme possesses several isoforms (β, γ, δ, ε, and ζ), differing by the signaling pathway in which they are involved and by their Ca2+ sensitivity. PLCβ are activated by protein-G coupled receptors, PLCγ by tyrosine kinases, PLCδ by PIP2 and Ca2+, PLCε by Ras, and PLCζ by Ca2+. Theoretically, stimulation of PLC activity by Ca2+ in the appropriate concentration range can lead to concomitant Ca2+ and InsP3 oscillations with a period that depends on the rate of InsP3 synthesis by PLC (Meyer and Stryer, 1991). Concerning its degradation, InsP3 can be either dephosphorylated by the Ins(1,4,5)P3 5-phosphatase to yield Ins(1,4) bisphosphate or phosphorylated by the Ins(1,4,5)P3 3-kinase into Ins(1,3,4,5) tetrakisphosphate (Shears, 1992). The 3-kinase is stimulated by an increase in Ca2+ as binding of Ca2+/calmodulin enhances its activity. The stimulation factor varies from 2 to 10 depending on the isoform (Takazawa et al., 1990; Sims and Allbritton, 1998). Importantly, the product of InsP3 phosphorylation by 3-kinase, InsP4, is a competitive inhibitor of the other InsP3-metabolizing enzyme, the 5-phosphatase (see Fig. Fig.3).3). Much care should thus be taken when interpreting experiments where the levels of 3-kinase have been modified, as these changes are, at least in part, counteracted by the changes in activity of 5-phosphatase. As expected intuitively, the incorporation of such a regulation in a mathematical model can also lead to InsP3 oscillations as each Ca2+ spike provokes an enhanced degradation of InsP3 (Dupont and Erneux, 1997). However, in contrast to the stimulation of PLC activity by Ca2+, in this case, InsP3 oscillations passively follow Ca2+ oscillations (i.e., this regulation cannot on its own give rise to sustained oscillations and the peak of InsP3 slightly follows that of Ca2+). Before turning to the question of the relevance of these regulations of InsP3 metabolism by Ca2+ in real cells, it should be mentioned that it is not possible to exclude that InsP3 oscillations result from other regulatory influences on InsP3 synthesis, more specifically at a step located upstream of PLC activation. For hepatocytes, for example, a specific regulation at the level of the Gα subunit of the G-protein activation cascade leading to PLCβ activation might play such a role (Kummer et al., 2000). Also, Ca2+ and InsP3 oscillations induced by the activation of mGluR5 receptors rely on a cyclical inactivation of these receptors due to the phosphorylation by PKC (Nash et al., 2002).

Figure 3
Schematic representation of the interplay between intracellularCa2+dynamics andInsP3metabolism.

Although it is since long known that InsP3 3-kinase is activated by Ca2+, it took some time to clearly state that this regulation leads to InsP3 oscillations. However, based on a careful quantitative evaluation of the kinetics of InsP3 metabolism by the kinase and the phosphatase, the modeling suggests that InsP3 oscillations resulting from this regulation have a tiny amplitude (Dupont and Erneux, 1997). Cellular conditions are indeed such that the major part of the InsP3 pool is catabolized by 5-phosphatase. This conclusion holds with the observation performed in CHO cells that overexpression of 5-phosphatase clearly abolishes any Ca2+ oscillatory activity in response to stimulation while a similar 3-kinase overexpression has a negligible effect on internal Ca2+ mobilization (De Smedt et al., 1997).

Despite this observation, one might speculate that in other cell types, the 3-kinase might play a more important role, and, more especially, allow for a rather long time interval between two Ca2+ spikes. If each Ca2+ spike provokes the catabolism of a significant amount of InsP3, it could take some time for InsP3 to reach a level sufficient to trigger Ca2+ release. In this view, the long time interval between two successive Ca2+ spikes would correspond to the time necessary to rebuild the level of InsP3 necessary to activate Ca2+ release through the receptor. This possibility has been indirectly tested in hepatocytes (Dupont et al., 2003). The strategy used in this study aimed at masking Ca2+-dependent InsP3 catabolism by 3-kinase through the injection of massive amounts of 5-phosphatase, which is not stimulated by Ca2+. In such injected hepatocytes, Ca2+ oscillations generated by modest agonist doses are suppressed because of the resulting low level of InsP3. At higher doses of agonist, oscillations reappear. Importantly, the characteristics of these oscillations are similar to those of untreated cells at low agonist dose, despite the fact that InsP3 oscillations due to 3-kinase stimulation by Ca2+ do not occur (nearly all the InsP3 is metabolized by the 5-phosphatase, which is much more abundant because of the injection). This study thus confirms that the oscillations of InsP3 that would result from Ca2+-regulation of the InsP3 3-kinase do not play an active role in the control of Ca2+ oscillations. Similar conclusions have been reached recently in HSY-EA1 cells, together with the imaging of InsP3 levels (Tanimura et al., 2009).

Modeling has also been used to define experimental tests that could discriminate between an InsP3R-based or an InsP3 metabolism-based mechanism. Sneyd et al. (2006) proposed to perturb agonist-induced Ca2+ oscillations by the direct, artificial release of InsP3 in the cytoplasm (flash photolysis of caged InsP3). In a model where InsP3 metabolism is at the basis of Ca2+ oscillations, this sudden increase in InsP3 will provoke a delay in the occurrence of the next Ca2+ spike, which corresponds to the time required for the level of InsP3 to go back to its normal range of concentrations during oscillatory cycles. Once this is done, the situation is similar to the prepulse one, and no change in frequency is observed. The situation is drastically different for Ca2+ oscillations occurring with a constant level of InsP3 due to the sequential activation and inhibition of the InsP3R. In the framework of such a mechanism, the frequency of Ca2+ oscillations directly depends on the (constant) level of InsP3. Thus, a sudden increase in InsP3 during agonist-induced Ca2+ oscillations provokes a transient rise in frequency. The interspike interval then progressively decreases to the period of the unperturbed system. The number of spikes necessary to the resettlement of the prepulse periodicity increases with the amount of InsP3 released into the cell [(Sneyd et al., 2006); see also (Chatton et al., 1998)].

STOCHASTIC ASPECTS OF Ca2+ OSCILLATIONS

In experiments, a rise in the level of InsP3 transforms random, elementary Ca2+ increases into regular Ca2+ increases propagating as waves in the cytoplasm. This transition can be explained by the fact that the rise in InsP3 leads to an increase in the number of channels participating in the Ca2+ dynamics. The effect of fluctuations is then expected to become rather small, allowing for a deterministic description. To test this hypothesis, we have performed a statistical analysis of the regularity of Ca2+ oscillations in noradrenaline-stimulated hepatocytes and found that the coefficient of variation in the period lies between 10% and 15% (Dupont and Combettes, 2009). Stochastic simulations taking into account realistic numbers of InsP3Rs (about 6000 in a typical hepatocyte) accounted for such a low variability, if the receptors are assumed to be grouped in clusters of a few tens of channels (Dupont et al., 2008; Dupont and Combettes, 2009). This corroborates the idea that repetitive Ca2+ spiking can be described as a deterministic oscillator. As the number of clusters is rather low, this oscillator is however perturbed by noise leading to the 10–15% variation in the period. In agreement with this view, this coefficient of variation decreases as the level of InsP3 rises since the number of active channels increases (Fig. (Fig.4).4). In a small number of cases, very noisy oscillations (CV~30%) are observed at low levels of stimulation that in most cells do not induce any Ca2+ increase. Simulations using Gillespie’s algorithm strongly suggest that this situation corresponds to a regime of stochastic resonance. In such a regime, the level of InsP3 is just below the Hopf bifurcation point (i.e., the threshold that delimitates oscillatory from nonoscillatory states in a fully deterministic regime). As this corresponds to an excitable regime, fluctuations can then sometimes lead the system above the excitation threshold and induce the occurrence of a whole Ca2+ spike. Such dynamics lead to highly variable interspike intervals. Both in the simulations and in experiments, this situation is rarely encountered and a slight increase in InsP3 leads to much more regular Ca2+ oscillations, which can be interpreted as a noisy deterministic behavior.

Figure 4
Simulations ofCa2+oscillations by a stochastic Gillespie’s algorithm taking into account the numbers ofCa2+ions andInsP3receptors present in a typical hepatocyte.

It has also been proposed that Ca2+ dynamics are always stochastic. The reason for that would lie in the assumed spatial arrangement of the InsP3Rs in clusters spaced from each other by several microns. As Ca2+ is a poorly diffusible messenger in the cytosol, the Ca2+ rise occurring at one puff site would be unable to activate release from adjacent sites. This absence of communication would prevent global signaling. Thus, Ca2+ waves could be initiated only if, by chance, a sufficient number of clusters of InsP3Rs became active at the same time. This process, called nucleation, would lead to a Ca2+ increase that is large enough to activate all the InsP3-bound InsP3Rs and generate a Ca2+ spike. Under these circumstances, the variation in the period is of the order of the period itself (Falcke, 2004; Skupin et al., 2008), which is much larger than in noradrenaline-stimulated hepatocytes (Dupont et al., 2008). Interestingly, in this framework, spike initiation—that corresponds to the time required for the concomitant opening of a few adjacent cluster sites—can take very long as it is a random event. This would provide a possible explanation to the long periodicity of Ca2+ oscillations as compared to the characteristic times of the InsP3R dynamics.

CONCLUSIONS

Together with electrophysiology, intracellular Ca2+ dynamics is one of the field of cellular physiology where the interplay between experiments and modeling has been the most fruitful. One of the reasons for this is that, as other rhythmic phenomena, they rely on specific, nonlinear feedback processes and can thus hardly be fully approached without mathematical modeling and numerical simulations. In the same manner, cAMP oscillations, circadian rhythms, cell cycle-related variations in the activity of cyclin associated kinases or the tumor-associated p53/mdm2 loop are other oscillatory phenomena in biology whose investigation largely benefits from a modeling approach (Goldbeter, 2008). Given the accessibility of a variety of quantitative data, modeling not only allows the investigation of the molecular mechanisms responsible for Ca2+ oscillations but can also lead to a detailed investigation of the cell to cell variability. This can be ascribed to subtle differences in the characteristics of the three existing isoforms of the InsP3 receptors that are expressed at different levels in the different cell types. Kinetic characteristics of InsP3 metabolism also modulate Ca2+ oscillations.

Surprisingly, many factors that are known to alter Ca2+ oscillations have not yet been deeply considered in models. For example, despite the large number of experimental studies devoted to the mechanism of Ca2+ entry (Putney and Bird, 1998), most models for Ca2+ oscillations in nonexcitable cells consider a closed system where Ca2+ exchanges are limited to fluxes between the cytosol and the ER. Mitochondrial Ca2+ handling is also known to alter cellular Ca2+ signals in the cytosol (Halestrap, 2009). Although some models have been developed to explain the pumping and releasing properties of suspensions of mitochondria (Selivanov et al., 1998), their implication in intact cells have rarely been investigated theoretically (Marhl et al., 2000; Fall and Keizer, 2001).

Finally, much remains to be done in the field of Ca2+ dynamics as to the link between elementary Ca2+ events due to the random opening of a few Ca2+ channels at low levels of InsP3 and coordinated, regular Ca2+ oscillations. This question pertains to the emergence of a macroscopic self-organized system from a large number of microscopic random entities, which is a central concern in biology. As this transition is observed in experiments just by increasing the level at which the cell is stimulated, Ca2+ dynamics can be seen as an ideal system to investigate its physicochemical foundations.

ACKNOWLEDGMENTS

G.D. is “Maître de Recherche” at the Belgian F.R.S.-F.N.R.S. We acknowledge support from the Fonds de la Recherche Scientifique Médicale (Grant No. 3.4636.04), the European Union through the Network of Excellence BioSIM (Contract No. LSHB-CT-2004-005137), and the Belgian Program of Interuniversity Attraction Poles, initiated by the Belgian Federal Science Policy Office, Project No. P6/22 (BIOMAGNET).

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