Media

Hanks’ balanced salt solution with Hepes (HBSS-H) contained 137 mM NaCl, 5.4 mM KCl, 1.3 mM CaCl₂, 0.41 mM MgSO₄, 0.49 mM MgCl₂, 0.34 mM Na₂HPO₄, 0.44 mM KH₂PO₄, 5.5 mM glucose, 20 mM Hepes-NaOH (pH 7.4). Intracellular-like medium (ICM) contained 125 mM KCl, 19 mM NaCl, 10 mM Hepes-KOH (pH 7.3), 1mM EGTA, and appropriate concentrations of CaCl₂ (330 μM CaCl₂ for 50 nM free Ca²⁺).

Cell culture

HSY-EA1 cells were cultured in Dulbecco’s Eagle’s medium nutrient mixture F-12 Ham (Sigma) supplemented with 10% newborn calf serum, 2 mM glutamine, and 100 U/ml penicillin and 100 μg/ml streptomycin, as previously described. These cells were grown in fibronectin-coated experimental chambers consisting of plastic cylinders (7 mm in diameter) glued to round glass coverslips. For monitoring changes in intracellular [Ca²⁺], cells were incubated with 2 μM Fluo-3 acetoxymethyl ester (Dojin Chemicals, Kumamoto, Japan) in HBSS-H HBSS-H containing 1% bovine serum albumin (BSA) for 30 min at room temperature.

Measurement of fluorescence

Experiments were carried out on a TE2000 inverted fluorescence microscope (Nikon Tokyo, Japan) with dual-source illumination system, in which two different light sources, a 150 W xenon arc light source U7773 (Hamamatsu photonics, Shizuoka, Japan) and a 100 W mercury light source (Nikon) were equipped. An experimental chamber was placed on the microscope, and fluorescence images were captured using a imaging system consisting of a C9100-13 EM-CCD camera (Hamamatsu photonics) and were analysed with AQUACOSMOS 2.6 software (Hamamatsu photonics).

IP₃-induced Ca²⁺ oscillations were examined with cell-permeable caged IP₃, iso-Ins(1,4,5)P₃/PM(caged), (Alexks Biochemicals, Grunberg, Germany). In this experiments, cells were incubated with cell-permeable caged IP₃ (2–10 μM) and Fluo-3 (2 μM) in HBSS-H containing 1% BSA for 30 min at room temperature. During monitoring Ca²⁺ responses with 490 nm light from the Nikon mercury light source, cells were also illuminated with 400 nm light that was derived from U7773 for the photolysis of cell-permeable caged IP₃. These illumination lights were reflected by a 500 nm long pass dichroic beamsplitter, and exposed the chamber through a Nikon Plan Fluor 40x (numerical aperture, 1.3) or 60x objective oil immersion lens (numerical aperture, 1.2). In this experiment, 400 nm light was continuously exposed to a small area including several cells, and the strength of the light was controlled with the grating monochromator in U7773 and neutral density filters, so that the Ca²⁺ oscillations was induced in a constant frequency. The emitted light of Fluo-3 can passed the dichroic beamsplitter and 535 nm emotion filter, and detected by the EM-CCD camera. ATP, CCh, and U73122 were directly added to the chamber during the measurements.

The calcium model

Many mathematical models have been developed to explain intracellular Ca²⁺ dynamics of various cell types. The details of these models are different, as each aims to describe the precise dynamics in a certain cell, but they share a primary goal of explaining how Ca²⁺ dynamics is regulated. For recent reviews of computational models of Ca²⁺ oscillations, see Dupont et al. [24] and Schuster et al. [25]. There are two main hypotheses that are thought to be involved in the formation of Ca²⁺ oscillations: the biphasic regulation of the IPR by Ca²⁺, and oscillations in IP₃ concentration ([IP₃]) driven by cross-coupling between Ca²⁺ and IP₃. In the former hypothesis, Ca²⁺ activation and inhibition of the IPR regulate the periodic opening of the receptors, giving rise to Ca²⁺ oscillations in the absence of oscillating [IP₃] [26, 27]. Cytosolic Ca²⁺ can cause the release of Ca²⁺ from the ER through a Ca²⁺-induced Ca²⁺ release (CICR) mechanism, by binding onto the activating site of the IPR on a relatively fast timescale. Subsequently, Ca²⁺ can also bind onto the different sites of the IPR on a slower timescale to inhibit the receptors. Numerous studies employed this mechanism to describe Ca²⁺ oscillations and waves observed in different cell types, including Xenopus laevis oocytes [27, 28], hepatocytes [29], and airway smooth muscle cells [30]. The models that generate Ca²⁺ oscillations through Ca²⁺ feedback on the IPR are called Class I models. The latter hypothesis assumes that Ca²⁺ is involved in the formation and degradation of IP₃, leading to coupled oscillations of [Ca²⁺] and [IP₃] [31, 32]. As positive feedback, Ca²⁺ can activate the enzyme that produces IP₃, phospholipase C (PLC), to increase [IP₃], which in turn stimulates the IPR to release Ca²⁺ from the ER. However, Ca²⁺ can also be involved in the degradation of IP₃, by activating IP₃ 3-kinase (IP₃K), the enzyme that degrades IP₃ to IP₄. This forms a negative feedback on the Ca²⁺ release through the IPR. Other negative feedback processes may involve Ca²⁺ inhibition of the IPR or suppressed production of IP₃ by protein kinase C (PKC) [32–34]. The models that incorporate positive and negative Ca²⁺ feedback on IP₃ as the main oscillatory mechanism are called Class II models. Despite the distinctive difference in the fundamental mechanisms underlying Ca²⁺ oscillations, it is very hard to experimentally distinguish oscillations generated by one mechanism from those induced by the other.

Recent development of IP₃ biosensors has enabled the monitoring of changes in [IP₃] during Ca²⁺ oscillations, which led to the discovery of concurrently oscillating [Ca²⁺] and [IP₃] in some cells [22, 33, 35, 36]. Several studies thoroughly investigated the effects of cross-coupling between Ca²⁺ and IP₃ on intracellular Ca²⁺ oscillations, both experimentally and theoretically. Dupont et al. [35] proposed that IP₃ oscillations observed in hepatocytes, presumably induced by Ca²⁺-dependent IP₃ metabolism, are not required for Ca²⁺ oscillations. Furthermore, Bartlett et al. [34] observed that photolysis of caged IP₃ can elicit Ca²⁺ oscillations in hepatocytes, without any contribution from PLC. Politi et al. [32] constructed a model of Ca²⁺ dynamics with positive and negative Ca²⁺ feedback on IP₃ regulation, and found that these feedback mechanisms extend the range of oscillation frequencies. Interestingly, Gaspers et al. [33] found that positive Ca²⁺ feedback on IP₃ is essential for generating low-frequency Ca²⁺ oscillations. These results are readdressed in the Discussion in more detail, as they are closely related to the results of this paper.

As experimentally observed intracellular Ca²⁺ behaviours exhibit stochastic events, it is becoming increasingly important to include stochasticity in Ca²⁺ models. Indeed, a number of studies have used stochastic models to explain intracellular Ca²⁺ dynamics [28, 37–40]. For a recent review of stochastic Ca²⁺ models, see Rdiger [41]. Although stochastic models can be useful to study intrinsically random activities of the IPR, Cao et al. [30] showed that deterministic models can make equally valid predictions about intracellular Ca²⁺ dynamics as stochastic models can. Thus, we aim to study Ca²⁺ dynamics in HSY cells using a deterministic model.

Tanimura and Turner observed that permeabilised HSY cells can exhibit repetitive Ca²⁺ release and re-uptake by intracellular stores, suggesting that Ca²⁺ oscillations are directly generated by Ca²⁺ feedback on IPR [42, 43]. Their work was later supported by Tojyo et al. [23], who presented evidence that Ca²⁺ oscillations in HSY cells, in the form of baseline spikes, can arise without activating IP₃ formation. Subsequently, Tanimura et al. [22] monitored emission ratios of LIBRAvIIS and fura-2, biosensors of IP₃ and Ca²⁺, respectively, and found that they exhibit coupled oscillations upon agonist stimulation with different concentrations of ATP (see Fig 1). This suggests that the model should contain Ca²⁺ feedback on IP₃ production and/or degradation. Interestingly, the coupled oscillations showed a pattern where peaks of Ca²⁺ spikes preceded those of IP₃ spikes. However, it was not known whether the IP₃ oscillations were necessary in order for Ca²⁺ oscillations to exist, or whether they were simply a passive reflection of the Ca²⁺ oscillations. In this section, we construct a mathematical model of Ca²⁺ dynamics in HSY cells based on the experimental observations. The basic assumption of our model is that the oscillations arise via a Class I mechanism, in which the IP₃ oscillations are simply a passive reflection of the Ca²⁺ oscillations. As we shall see, this assumption is upheld by experimental testing of model predictions.

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Fig 1. Estimated [IP₃] (in black) and excitation ratio of fura-2 (in red) during ATP-induced Ca²⁺ oscillations in HSY cells.

Traces are representative of 31 of 55 oscillating cells. The axis on the left is for [IP₃] and the one on the right is for the ratio of fura-2. The black bar at the top shows the dosage of ATP used to stimulate the HSY cells. The triangles at the top indicate the peak of each spike. This figure was adapted from Tanimura et al. [22].

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Calcium dynamics.

We separate the cell domain into three compartments: inside the ER, a small region (microdomain) near a cluster of IPR, and the cytosol (see Fig 2); the Ca²⁺ concentration in each region is denoted by C_ER, C_b and C, respectively. The total free Ca²⁺ concentration within a cell, C_t, is given by where γ₁ and γ₂ are volume ratios between different compartments, as given in Table 1. Thus, C_ER = γ₂(C_t − C − C_b/γ₁). IPR interact with C_b and the [Ca²⁺] at the mouth of an open channel (denoted by C_p); thus there is no direct interplay between the open probability of the receptors and the cytosolic Ca²⁺. The idea of the microdomain around the mouth of an IP₃ receptor, or around a cluster of IPR, was first introduced by Swillens et al. [44], where they assumed that this domain, which is separate from the cytosol and the ER lumen, contains the receptor’s activating and inhibiting Ca²⁺-binding sites. Assuming that the receptor activity is modulated by the [Ca²⁺] in the microdomain, they were able to reproduce experimental observations where increments of IP₃ induce repeated transient release of Ca²⁺. Dupont and Swillens further showed that the model could account for Ca²⁺ oscillations as well [45]. In addition, they discovered that one of the outcomes of the presence of a microdomain is an increase in the oscillation period.

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Fig 2. Schematic diagram of calcium fluxes in a cell.

The directions of the fluxes are indicated with the arrows. When an IP₃ receptor is opened, Ca²⁺ is first released from the ER to a nearby domain (J_IPR), then diffuses to the rest of the cytosol (J_diff). Eventually, Ca²⁺ is pumped back into the ER (J_SERCA). There are fluxes across the plasma membrane (J_in and J_pm), and a small leak from the ER to the cytosol as well (J_leak).

https://doi.org/10.1371/journal.pcbi.1005275.g002

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Table 1. Parameter values of the Ca²⁺ fluxes.

https://doi.org/10.1371/journal.pcbi.1005275.t001

When Ca²⁺ is released through IPR, C_b increases to a high concentration essentially instantaneously. Ca²⁺ in the microdomain then diffuses to the cytosol, and eventually gets pumped back into the ER via sarco/endoplasmic reticulum Ca²⁺-ATPase (SERCA) pumps or removed from the cell by the plasma membrane Ca²⁺ ATP-ase (PMCA) pumps. Based on the schematic diagram of calcium dynamics in Fig 2, we write the following ODEs, (1) (2) (3) The functional form for each flux is specified in the next section.

IP₃ receptor model

A crucial part of the model is our choice of the model for the IPR. We use the model of Cao et al. [30], which is an improved version of the Siekmann IPR model from [51]. The model consists of two modes: the drive mode and park mode; see Fig 3. The receptors are mostly open when they are in the drive mode, and closed when in the park mode. The drive mode has one open state (O₆) and one closed state (C₂), with transition rates q₂₆ and q₆₂ as shown in Table 2. Hence, the open probability of the drive mode is q₂₆/(q₂₆ + q₆₂) (≈ 70%). The park mode has one closed state (C₄), with an open probability close to zero. Cao et al. denoted the transition rates between the modes as q₂₄ and q₄₂, given by If D is defined to be the proportion of the IPR that are in the drive mode, then the IPR have an open probability as follows: The opening kinetics of the receptors is governed by the following: where and are the quasi-equilibrium values of m₄₂ and h₄₂, respectively, and the λ’s are the rates at which the quasi-equilibria are approached, We assume that λ_m42, L, and H are relatively small, as shown in Table 2, corresponding to slow opening kinetics of the receptor; see the Discussion for more detail about the dynamics of the IPR model. The closing kinetics is controlled by m₂₄ and h₂₄, which are assumed to be at their quasi-equilibrium values, respectively. C_p = C_p0(C_ER/680) denotes the [Ca²⁺] at the pore. The expressions for the V’s, a’s and k’s are For full details of the IPR model derivation, refer to Cao et al. [30].

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Fig 3. The structure of the IPR model.

The model is comprised of two modes. The drive mode has one open state and one closed state, and has an open probability of q₂₆/(q₂₆ + q₆₂). The park mode has one closed state with an open probability close to zero. q₂₄ and q₄₂ are the transition rates between the modes.

https://doi.org/10.1371/journal.pcbi.1005275.g003

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Table 2. Parameter values for the IPR model.

https://doi.org/10.1371/journal.pcbi.1005275.t002

Ca²⁺ feedback mechanisms on IP₃ dynamics and IPR

As mentioned before, Class I models assume that Ca²⁺ feedback on the opening and closing kinetics of IPR is the main driving force behind Ca²⁺ oscillations. Intracellular Ca²⁺ can increase or decrease the open probability of an IPR by binding to its different binding sites. In this case, time-dependent Ca²⁺ feedback on IPR is necessary to generate Ca²⁺ oscillations. Conversely, Class II models assume that Ca²⁺ oscillations are the result of Ca²⁺ feedback on the production and degradation of IP₃. As a positive feedback, Ca²⁺ can activate phospholipase C (PLC), and increase the rate of IP₃ production. Ca²⁺ can also have a negative feedback on IP₃ by activating IP₃ 3-kinase (IP₃K) to remove IP₃. The increases and decreases in [IP₃] are coupled with the open probability of the IPR, leading to oscillations in [Ca²⁺]. Whereas a Class II model requires oscillating [IP₃] in order to give rise to Ca²⁺ oscillations, a Class I model does not, and hence Ca²⁺ oscillations can occur even at a constant [IP₃].

Tanimura and Turner [43] permeabilised the plasma membrane of HSY cells, and applied IP₃ to monitor changes in the [Ca²⁺] in the IP₃-sensitive store. The addition of IP₃ caused subsequent release of Ca²⁺ from the store, followed by re-uptake of Ca²⁺ via SERCA pumps. This Ca²⁺ release and re-uptake were repeatedly observed in the absence of Ca²⁺ buffer (EGTA) but disappeared in the presence of EGTA. Their result suggests that Ca²⁺ oscillations in HSY cells are mainly generated by feedback effects of Ca²⁺ on IPR dynamics. It was also hypothesised in Tanimura et al. [22] that, although there are IP₃ oscillations in HSY cells, they are not crucial for Ca²⁺ oscillations. These observations indicate that the oscillations can be reproduced in a Class I model. Additionally, we want to model IP₃ dynamics with positive and/or negative feedback from Ca²⁺ to ensure there are coupled oscillations. The order of [Ca²⁺] and [IP₃] peaks, in which [Ca²⁺] peaks are followed by [IP₃] peaks (see Fig 1), suggests that Ca²⁺ has positive feedback on IP₃. For this reason, we conjecture that in HSY cells, the production rate of IP₃ is an increasing function of [Ca²⁺], and thus each peak in [IP₃] is caused by the preceding peak in [Ca²⁺]. In order to construct a model as simple as possible, and yet still capture the essence of the experimental data, our model includes Ca²⁺ feedback on the production of IP₃ only, with a constant rate of IP₃ degradation.

The production of IP₃ can be processed by many different forms of PLC [52]. There is little information about the PLCs expressed in HSY cells. However, Nezu et al. [53] proposed that HSY cells exhibit two distinct pathways for IP₃ generation. They studied HSY cells that were mechanically stimulated by poking the cell surface with a glass micropipette. They also studied the neighboring cells that were not directly stimulated with the micropipette. This type of stimulation may induce both Ca²⁺ release from the ER and extracellular Ca²⁺ entry through channels on the plasma membrane [54]. The cells showed waves of Ca²⁺ and IP₃ upon stimulation, which started from the site of stimulation and propagated to the neighboring cells. The Ca²⁺ wave was observed in the absence of extracellular Ca²⁺. When the cells were pretreated with suramin, a blocker that inhibits the receptors on the plasma membrane that are activated by external agonists to stimulate PLC, the responses in the neighboring cells were abolished. This observation suggest that the responses in the neighboring HSY cells were induced by the generation of intracellular Ca²⁺ and IP₃ via a pathway that depends on external agonists. The responses in the cells with direct stimulation were not affected by suramin, which indicates that mechanical stimulation induces the generation of IP₃ that is independent of external agonist. When the cells were pretreated with a PLC inhibitor, U73122, the mechanical-stimulation-induced Ca²⁺ and IP₃ responses were suppressed. Additionally, HSY cells exhibited Ca²⁺ oscillations when stimulated with external agonist [22, 53]. These results suggested that there are at least two types of PLCs in HSY cells; one that is primarily independent of external agonist, and the other that is activated by the addition of agonist. When modeling PLC regulation in our model, we assume that HSY cells express PLCδ, a PLC that is regulated by Ca²⁺, and PLCβ, another PLC that is activated by the addition of external agonist. Both the Ca²⁺ released from the ER and the Ca²⁺ that enters from the extracellular space contribute towards the activation of PLCδ. In our model, the concentration of applied agonist is denoted by ν. We model the IP₃ production rate generated from PLCβ as a saturating function, so that the rate of production increases as the agonist concentration increases, before reaching the maximum production rate [55]. The Ca²⁺ activation of PLC (PLCδ) is expressed with a Hill function [32, 50]. We model the production rate of IP₃ as a sum of the rates generated from PLCβ and PLCδ, (4) where ψ₁ and ψ₂ are the maximum production rate induced by PLCβ and PLCδ, respectively. When ψ₂ = 0 μM s^–1, IP₃ production rate is solely from the agonist; when ψ₂ ≠ 0 μM s^–1, an increase in [Ca²⁺] also increases IP₃ production rate, leading to a positive feedback loop. We assume a constant degradation rate of IP₃, where r_deg is the rate of decay. Thus, the ODE for [IP₃] is, (5) with parameter values shown in Table 3.

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Table 3. Parameter values for Eq (5).

https://doi.org/10.1371/journal.pcbi.1005275.t003

The full model

Based on the above, we write the full HSY cell calcium model with six ODEs, (6) where the choice ε = 0 or 1 determines whether the model is a closed-cell model or an open-cell model. We nondimensionalised the system (with ε = 1), not only to make the system dimensionless, but, more importantly, to identify the timescale of each variable. The nondimensionalisesd system is included in the S1 Appendix. As a result, we found that there are at least three different timescales in the system. C_b evolves on the fastest timescale, while C_t evolves on the slowest. The other variables have timescales between the fastest and the slowest.

Model validation

Experimentally, HSY cells were stimulated with different concentrations of ATP. Higher agonist concentrations caused faster Ca²⁺ oscillations. We model ATP application by increasing ν, the concentration of applied agonist. With the parameter values in Tables 1–3, the model generates oscillations of [Ca²⁺] and [IP₃] at ν = 15 μM (Fig 4A). Specifically, Fig 4B shows that the model captures the correct relative timing of the Ca²⁺ and IP₃ oscillations in HSY cells, with a Ca²⁺ spike peak coming just before an IP₃ spike peak. Furthermore, as shown in Fig 4C, the model displays faster Ca²⁺ oscillations when stronger stimulation is applied. When the system is stimulated with ν = 15 μM, the resulting Ca²⁺ oscillations have a period of 62 seconds, while ν = 20 μM produces oscillations with a period of 35 seconds. These model simulations are in good agreement with the experimental data in Fig 1. Also, we find that a larger ν causes a longer delay within a pair of [Ca²⁺] and [IP₃] peaks. At ν = 15 μM, a peak of [IP₃] is about 0.63 seconds behind a [Ca²⁺] peak. This delay is extended to 0.67 seconds at ν = 20 μM. When ν is decreased to 5 μM or increased to 30 μM, no oscillations are found.

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Fig 4. Oscillations in Ca²⁺ and IP₃ concentrations, C and P, in the HSY cell model, system (6).

A: When ε = 1, the system is stimulated with ν = 15 μM for t > 0. [Ca²⁺] is shown in red; [IP₃] is in black. The period is about 62 seconds. B: Enlargement of A for 85 < t < 95, showing a pair of Ca²⁺ and IP₃ spikes. The peak of the Ca²⁺ spike is followed by the peak of the IP₃ spike. C: Starting from the initial condition with ν = 0 μM for t < 60 s, the system with ε = 1 is perturbed by ν = 15 μM for 60 s < t < 240 s and ν = 20 μM for 240 s < t < 360 s. The oscillations are faster at a higher agonist concentration. D: Coupled oscillations of [Ca²⁺] and [IP₃] in the closed-cell model, system (6) with ε = 0 and C_t = 68 μM. The system is stimulated with ν = 15 μM for t > 0.

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As discussed in the previous section, experimental results suggest that plasma membrane fluxes are not necessary to produce Ca²⁺ oscillations in HSY cells [23, 56]. We simulate this by studying the closed-cell model, which can be obtained by setting ε = 0 in system (6) (dC_t/dt = 0 μM s^–1). Physiologically, a closed cell does not have influx or efflux of Ca²⁺ across the plasma membrane, and thus maintains a constant intracellular [Ca²⁺]. For the purpose of this model simulation, C_t is no longer treated as a variable, but rather as a constant parameter. Thus, the closed-cell model forms a five-dimensional system, system (6) without the equation for C_t. Preliminary analysis of the closed-cell model revealed that in order to generate Ca²⁺ oscillations in the closed-cell model, ν and C_t need to balance each other so that the system is within its oscillatory regime. In other words, even if the system is stimulated with large ν, the system cannot produce oscillations if C_t is too low. In order to generate Ca²⁺ oscillations from the closed-cell model stimulated with ν = 15 μM, we need to set the parameter C_t to a value between 65 μM and 78 μM.

Fig 4D shows the model simulation of the closed-cell model, with a fixed C_t = 68 μM. At t = 0 s, the system is at its steady state without any stimulation; agonist is then applied with ν = 15 μM for t > 0. This shows that the closed-cell model can exhibit Ca²⁺ oscillations, which agrees with the experimental data that suggest Ca²⁺ oscillations in HSY cells do not require fluxes across the plasma membrane.

Thus, our model suggests that IP₃ oscillations in HSY cells do not seem to be essential in generating Ca²⁺ oscillations. We hypothesise that the main mechanism for generating Ca²⁺ oscillations in HSY cells is Ca²⁺ feedback on IPR. We aim to validate this hypothesis through proposing model predictions, and comparing them to experimental data.

Model predictions and experimental verification

Ca²⁺-free medium.

The Ca²⁺-free medium experiment in Tojyo et al. [23] stimulated HSY cells with ATP, which generated Ca²⁺ oscillations, and repeatedly interchanged external solutions between Ca²⁺-containing and Ca²⁺-free mediums. Similarly, Liu et al. [56] used CCh to stimulate HSY cells, and studied the Ca²⁺ response to removal and re-addition of external Ca²⁺. However, the question of long-term behaviour in Ca²⁺-free conditions was not addressed in those experiments. We thus study this question both theoretically and experimentally.

For the model simulation of the Ca²⁺-free medium experiment, J_in is set to 0 μM s^–1 to model the situation that there is no Ca²⁺ influx from the extracellular domain into the cytosol. System (6) is stimulated with ν = 20 μM, then J_in is turned off for t > 120 s. Fig 5A shows the corresponding model simulation. As explained before, our model has slow Ca²⁺ fluxes across the plasma membrane. Thus, when J_in is removed at t = 120 s, the model behaves much like a closed-cell model for a transient period. The model predicts that removing J_in does not stop the Ca²⁺ oscillations immediately. Physiologically speaking, this means that there is enough Ca²⁺ in the ER to fire Ca²⁺ spikes, even without any Ca²⁺ contribution from outside of the cell. However, the spike amplitude decreases and the interspike interval gets longer after each spike until the oscillations terminate eventually. C_t is a slow variable in the system, and it decreases slowly over time to the point where the system is no longer in the oscillatory range. With J_in = 0, the rate of change for C_t becomes negative (dC_t/dt = −J_pm < 0). Consequently, C_t in the model eventually decreases to 0 μM, which indicates that the cell becomes completely depleted in Ca²⁺.

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Fig 5. Model response to Ca²⁺-free medium and corresponding experimental data.

A: Response of the HSY cell model, system (6) with ε = 1, to elimination of J_in. The system is stimulated with ν = 20 μM, which generated oscillations of [Ca²⁺]. J_in is then removed from the system (J_in = 0 μM s^–1) for t > 120 s. As a result, the oscillations persist for about 20 mins. The oscillations show a decrease in the spike amplitude and an increase in the interspike interval after each spike. At t ≈ 1400 s (21 min), the system stops producing Ca²⁺ spikes. B: A representative Ca²⁺ response in HSY cells in a Ca²⁺-free medium. 26 out of 40 oscillating cells has a qualitatively similar response. Initially, the cells were placed in Ca²⁺-containing medium. CCh was used to generate Ca²⁺ oscillations. The external solution was changed to a Ca²⁺-free medium for the time indicated by the blue bar. The oscillations persisted for about 20 minutes in the absence of extracellular Ca²⁺.

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A representative experimental trace of [Ca²⁺] in a HSY cell that was placed in Ca²⁺-free solution is shown in Fig 5B. Initially, there was free Ca²⁺ in the extracellular domain, as the cells were surrounded by Ca²⁺-containing medium. CCh was used to stimulate the cells and induce Ca²⁺ oscillations. After some time, the cells were perfused with Ca²⁺-free solution, to wash away all the extracellular Ca²⁺. As we predicted from the model, Ca²⁺ spikes persisted for a while, before they eventually stopped. 26 out of 40 oscillating cells showed similar responses. In particular, the model predicts that it takes about 20 minutes for the oscillations to disappear, which is confirmed by the representative response in Fig 5B. Over the time, the cell must have been losing Ca²⁺ across the plasma membrane, until eventually intracellular [Ca²⁺] was too low to allow another Ca²⁺ spike. Also, in the absence of the external Ca²⁺, each interspike interval was longer than the previous one, as predicted by the model.

IP₃ pulses.

The [IP₃] in the cytosol affects the open probability of the IPR, and hence governs the majority of Ca²⁺ release from the ER. In order to investigate the response of intracellular [Ca²⁺] to an increase of [IP₃], experimentalists often use caged IP₃, which is biologically inactive, and then release IP₃ by flashing UV light. This photoreleased IP₃, which has the same function as IP₃, binds to the IPR and opens them. However, photoreleased IP₃ metabolises more slowly than IP₃, and therefore stays longer in a cell [57].

We model the release of caged IP₃ by adding another variable that represents the concentration of photoreleased IP₃, denoted by P_s. Since photoreleased IP₃ behaves like IP₃, every P in the IPR model and in J_ROCC in system (6) is rewritten as P + P_s, in order to include the effect of photoreleased IP₃ on IPR and ROCC. Experimentally, a flash of UV light causes a sharp rise in the concentration of photoreleased IP₃. Based on this, the production rate of photoreleased IP₃ is modeled as in [58], (7) where H is the Heaviside function M is the pulse magnitude, t₀ is the time at which the pulse starts and △ is the pulse duration. The production rate of photoreleased IP₃ for t < t₀ and t > t₀ + △ is 0 μM s^–1. Similarly to IP₃, we assume a constant degradation rate of photoreleased IP₃, (8) However, since photoreleased IP₃ metabolises more slowly than IP₃, we assume that r_{s_deg} < r_deg. The equation for P_s is (9) Fig 6A shows a model simulation with ν = 15 μM, t₀ = 200 s, M = 0.04 μM s^–1, △ = 0.7 s and r_{s_deg} = 0.006 s^–1. The black curve in the figure represents the sum of P and P_s, the total effective concentration of IP₃. A pulse of P_s increases the rate of the following series of Ca²⁺ spikes. As P_s gradually decreases, the system slowly recovers its former Ca²⁺ spiking rate. In this model simulation, the pulse causes an 84% increase in the oscillation frequency.

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Fig 6. Model response to a pulse of photoreleased [IP₃] and corresponding experimental data.

A: Response of the HSY cell model, system (6) with ε = 1, to a pulse of P_s, Eq (9). The system is stimulated with ν = 15 μM at t = 0 s. The pulse is applied at t = 200 s, indicated by the arrow. Ca²⁺ concentration, C, is in red; the total IP₃ concentration, including photoreleased [IP₃], is in black. The pulse increases the frequency of the Ca²⁺ oscillations by 84%. As photoreleased IP₃ degrades slower than IP₃, the oscillations return only slowly to the original frequency. B: A representative Ca²⁺ response to photolysis of caged IP₃ in HSY cells. 20 out of 26 oscillating cells had a qualitatively similar response. The cells were prepared with ATP and caged IP₃, and then a flash of UV light was applied at the time indicated by the arrow and the dashed line. The cells exhibited an increase in their oscillation frequencies. The frequency increase in the representative cell was about 85%.

https://doi.org/10.1371/journal.pcbi.1005275.g006

Fig 6B shows a representative Ca²⁺ response to photolysis of caged IP₃ in HSY cells. The cells were initially stimulated with ATP, and then treated with caged IP₃ to generate Ca²⁺ oscillations. The cells were exposed to a flash of UV light, while they were actively firing Ca²⁺ spikes. The experimental traces show that a sudden increase in photoreleased [IP₃] accelerated ATP-induced Ca²⁺ oscillations. However, the accelerated rate did not last long, and the oscillations slowed to the rate before the photolysis. IP₃ concentration was not measured in this experiment; nevertheless, we expect that [IP₃] would be oscillating as well, with its peaks preceded by [Ca²⁺] peaks. 20 out of 26 oscillating cells showed a transient increase in their oscillation frequencies in response to the photolysis of caged IP₃. Fig 7 shows the box plot of the percentage increase in oscillation frequencies, measured from 20 cells. On average, the photolysis increased the oscillation frequency by approximately 77%. The transient increase in oscillation frequency was accurately predicted by the model.

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Fig 7. Changes in oscillation frequency induced by the photolysis of caged IP₃.

Photolysis of caged IP₃ induced a transient increase in the oscillation frequency (20 out of 26 oscillating cells). For each cell, the oscillation frequencies before and after the photolysis were measured for comparison. On average, the photolysis increased the oscillation frequency by 77%.

https://doi.org/10.1371/journal.pcbi.1005275.g007

Inhibition of PLC.

Our next model simulations are designed to study how IP₃ dynamics affects Ca²⁺ oscillations. As shown in Eq (4), the system has two stimuli that activate PLC: agonist and cytosolic [Ca²⁺]. The feedback of Ca²⁺ on IP₃ production ensures coupled oscillations, which was one of the main features of the observed data in Fig 1 that we aimed to reproduce. However, because evidence suggests that IP₃ oscillations are not necessary for Ca²⁺ oscillations to exist, we wished to study what effect the IP₃ oscillations have, and how they affect the Ca²⁺ responses. One would expect that when positive Ca²⁺ feedback on PLC is removed, [IP₃] no longer oscillates, as the coupling between the two is broken.

Experimentally, PLC can be inhibited by applying a PLC inhibitor, U73122. This compound has been shown to have inhibitory effects on PLC in various cell types, including smooth muscle cells [59] and pancreatic acinar cells [60]. However, U73122 is not a selective inhibitor that targets PLC activated by Ca²⁺; instead, it inhibits overall PLC, including the PLC activated by the external agonist. Thus, if cells were stimulated with external agonist, applying U73122 would terminate Ca²⁺ oscillations as there is no other way to produce IP₃. In order to overcome this constraint, we use caged IP₃ to introduce photoreleased IP₃ intracellularly, which is independent of PLC. Continuously applying a small amount of UV light to cells that have caged IP₃ produces photoreleased IP₃ at a constant rate. As the degradation rate of photoreleased IP₃ is also constant, the concentration of photoreleased IP₃ is expected to stabilise at some steady state concentration. Thus, the cells now have a constant photoreleased [IP₃] that can induce Ca²⁺ oscillations, without any contribution from PLC. In addition, some studies show that U73122 is not a selective inhibitor of PLC, and that it has other effects, independent of its effect on PLC [61–63]. Based on evidence that U73122 inhibits SERCA pumps, the model simulation includes this effect as well [64].

To model a constant production rate of photoreleased IP₃ from low-level continuous UV light, we use Eq (9) with a constant V_{S_plc}, The equation for photoreleased [IP₃] (P_s) is now (10) which has a steady state at . In the model simulations, k_{s_plc} needs to be large enough to bring P_s above the minimum required for Ca²⁺ oscillations. When the Ca²⁺ concentration exceeds the PLC activation threshold, positive feedback on PLC in Eq (4) comes into play and starts generating IP₃ oscillations. We then remove Ca²⁺ feedback on PLC, by decreasing the coupling strength between Ca²⁺ and IP₃, ψ₂. Alternatively, we can increase K_plc, which would have the same effect on the model. As ψ₂ decreases, the term gets close to 0, thus decreasing the effect that Ca²⁺ has on IP₃ production. Additionally, we take into account that the PLC inhibitor could also block SERCA pumps; we model this by decreasing the maximum capacity flux of SERCA, V_S.

Fig 8A shows the model simulation of applying continuous UV light, and then adding PLC inhibitor to remove Ca²⁺ activation on PLC. The system is stimulated with k_{s_plc} = 0.0006 μM s^–1, without any external agonist (ν = 0 μM). Photoreleased [IP₃] slowly accumulates to its steady state concentration, μM. Once there is enough photoreleased IP₃ and Ca²⁺, the system starts generating Ca²⁺ spikes (at t ≈ 350 s). At t = 600 s, ψ₂ is decreased to 0.4 μM s^–1 which halves the coupling strength between Ca²⁺ and IP₃, and V_S is decreased to 9.5 μM s^–1. The model predicts that the effects of PLC inhibition are a decrease in the Ca²⁺ spike amplitude and an increase in the oscillation frequency. The frequency increase in the model simulation shown in Fig 8A is about 22%. However, numerical computations indicate that the long-term increase in the oscillation frequency is about 18%; see Table 4. At t = 900 s, ψ₂ and V_S are further decreased to 0 μM s^–1 and 9 μM s^–1, respectively, to amplify the effects of applying a PLC inhibitor, and eliminate Ca²⁺ feedback on PLC. As a results, the oscillations are almost destroyed. Table 4 shows the periods of the stable periodic orbits generated from system (6), with different combinations of parameter values for ψ₂ and V_S. It is clear that diminishing Ca²⁺ feedback on PLC alone induces a substantial decrease in the oscillation period. When only V_S is decreased, the oscillations get slower, but the impact on the oscillation period is greater when both ψ₂ and V_S are decreased.

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Fig 8. Model response to inhibition of PLC and corresponding experimental data.

A: Response of the HSY cell model, system (6) with ε = 1, to the inhibition of positive Ca²⁺ feedback on IP₃, as well as SERCA pumps. The system is stimulated with slowly increasing P_s, Eq (10) with k_{s_plc} = 0.0006 μM s^–1. At t = 600 s, the magnitude of the Ca²⁺ feedback on PLC is halved, and the maximum SERCA capacity is also decreased. The oscillation frequency increases by 22%. At t = 900 s, the coupling between Ca²⁺ and IP₃ is completely removed, and the SERCA capacity is decreased further. B: A representative Ca²⁺ response to injection of U73122 in HSY cells. 13 out of 21 oscillating cells has a qualitatively similar response. The cells were prepared with caged IP₃ and continuously exposed to low-level UV light. photoreleased IP₃ then generated Ca²⁺ oscillations. 2 μM of U73122 was applied to the cells for the time indicated by the thin blue bar, then the dose was increased to 5 μM for the time indicated by the thicker blue bar. The addition of U73122 increased the oscillation frequency. The frequency increase in the representative cell was about 46%.

https://doi.org/10.1371/journal.pcbi.1005275.g008

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Table 4. Periods of the long-term oscillations in system (6), with different values of ψ₂ and V_S.

https://doi.org/10.1371/journal.pcbi.1005275.t004

The experimental test of this model prediction is shown in Fig 8B, which shows a representative Ca²⁺ response in an HSY cell to the PLC inhibitor U73122. 13 out of 21 oscillating cells showed qualitatively similar responses. The cells were treated with caged IP₃, and were then exposed to low UV light continuously. As a result, photoreleased IP₃ generated Ca²⁺ oscillations. The cells were then subjected to 2 μM of U73122 to inhibit PLC. Subsequently, the dose of U73122 was increased to 5 μM to enhance the effect of U73122. The result shows that impeding PLC decreased the amplitude of the Ca²⁺ spikes, while increasing their frequency. When a higher concentration of U73122 was applied, the oscillations were abolished. These results verify the model prediction. Fig 9 shows the box plot of the percentage increases in oscillation frequency induced by the addition of 2 μM of U73122, measured from 13 cells. On average, the cells exhibited a 50% increase in the oscillation frequency.

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Fig 9. Changes in oscillation frequency induced by the addition of 2 μM of U73122.

13 out of 21 oscillating cells showed an increase in oscillation frequency. For each cell, the frequencies before and after the addition of U73122 were measured for comparison. On average, applying 2 μM of U73122 increased the oscillation frequency by about 50%.

https://doi.org/10.1371/journal.pcbi.1005275.g009

With this model, we can also investigate the effects of the photolysis of caged IP₃, while PLC activity is being inhibited. Fig 10A shows the model simulation of the addition of U73122, followed by continuous UV light. Initially, system (6) with Eq (10) is simulated with ε = 1, ψ₂ = 0 μM s^–1, and V_S = 9 μM s^–1 to model the addition of U73122. At t = 200 s, k_{s_plc} is increased to 0.00056 μM s^–1 to describe a constant increase in P_s, which mimics the application of continuous UV light after the inhibition of PLC. As a result, the system starts generating Ca²⁺ spikes at around t = 500 s. The model predicts that photolysis of caged IP₃ can elicit Ca²⁺ oscillations in HSY cells, even under inhibited PLC activity.

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Fig 10. Model response to continuous photolysis of caged IP₃, after the inhibition of PLC, and corresponding experimental data.

A: Response of the HSY cell model, system (6) with ε = 1, to the addition of P_s, Eq (10), while Ca²⁺ feedback on PLC is completely inhibited. System (6) is simulated with ψ₂ = 0 μM s^–1 and V_S = 9 μM s^–1, to include the effects of the addition of U73122, indicated by the lower blue bar. From t = 200 s, k_{s_plc} is increased to 0.0006 μM s^–1, to model the continuous photolysis of caged IP₃, indicated by the upper blue line. The system starts generating Ca²⁺ spikes at around t = 500 s. B: A representative Ca²⁺ response to the addition of U73122, followed by photolysis of caged IP₃. 13 out of 20 oscillating cells has a qualitatively similar response. The cells were prepared with caged IP₃ and U73122, and then exposed to UV light during the time indicated by the upper blue bar.

https://doi.org/10.1371/journal.pcbi.1005275.g010

This model prediction was tested through experiments, with the result shown in Fig 10B. Initially, 20 HSY cells were prepared with caged IP₃. Then they were applied with U73122, in order to inhibit any PLC activities. Subsequently, they were continuously exposed to UV light for the photolysis of caged IP₃ at a constant rate. As a result, 13 cells exhibited a series of Ca²⁺ spikes, which indicates that Ca²⁺ oscillations can be generated from photoreleased IP₃, without any contribution from PLC activities. The data show that the oscillations emerged as soon as the photolysis took place, whereas in the model simulation, it takes about 5 mins before the first oscillation is generated. This discrepancy is addressed in more detail in the Discussion.

As U73122 inhibits both PLC and SERCA, it is difficult to discern the effects of inhibiting PLC only, independent of SERCA activities. However, it can be studied through model simulations. Fig 11 shows the model simulation of completely removing Ca²⁺ feedback on PLC. From t = 0 s, the system is stimulated with k_{s_plc} = 0.0006 μM s^–1, which induces Ca²⁺ oscillations at around t = 300 s. At t = 600s, ψ₂ is decreased from 0.8 μM s^–1 to 0 μM s^–1, to simulate the complete inhibition of positive Ca²⁺ feedback on PLC. In this simulation ψ₂ is decreased to 0 μM s^–1 while all the other parameters are kept the same. Setting ψ₂ = 0 μM s^–1 essentially removes the coupling between Ca²⁺ and IP₃. Without any PLC activation from Ca²⁺ or external agonist, [IP₃] decreases to 0 μM. This allows the total cytosolic [IP₃], which is the sum of [IP₃] and photoreleased [IP₃], to relax to its steady state, μM; see Fig 11B. The numerical computations indicate that when the coupling between Ca²⁺ and IP₃ is completely eliminated, the oscillation frequency increases by about 24%; see Table 4.

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Fig 11. Model simulations showing Ca²⁺ and IP₃ responses to inhibition of Ca²⁺ feedback on PLC.

System (6) with ε = 1 is stimulated with slowly increasing P_s, Eq (10) with k_{s_plc} = 0.0006 μM s^–1. At t = 600 s, ψ₂ is decreased to 0 μM s^–1 to completely break the coupling between Ca²⁺ and IP₃. All the other parameters are unchanged. The resulting oscillations have a smaller amplitude and a higher frequency. The change in the oscillation frequency is not obvious in this figure. However, numerical computations indicate that when ψ₂ is decreased from 0.8 μM s^–1 to 0 μM s^–1, the increase in the frequency is about 24%; see Table 4. A: The red traces show the Ca²⁺ response in the model simulation. B: The black traces show the sum of [IP₃] and photoreleased [IP₃]. Without any Ca²⁺ feedback on PLC, [IP₃] no longer oscillates, and the total effective concentration of IP₃ stabilises at a constant level.

https://doi.org/10.1371/journal.pcbi.1005275.g011

Quantitative variation

Our model is a deterministic model, which assumes that IPR clusters are continuously distributed per unit cell volume, and that they behave synchronously: either all open or all closed. Also, the frequency and the amplitude of Ca²⁺ oscillations are pre-determined by the model parameters. However, the experimental data exhibit stochastic Ca²⁺ events, with spontaneous spikes that vary in frequency and amplitude. They suggest that the number of clusters with open IPR for each Ca²⁺ spike is determined through a stochastic process. In a stochastic model, the number of IPR involved in the spike initiation and the interspike interval form distributions, and random Ca²⁺ release through an individual IP₃ receptor plays an important role in generating global Ca²⁺ signals.

However, Cao et al. [30] compared the Ca²⁺ dynamics in a model with stochastic kinetics of the receptors with that of a model with deterministic receptor kinetics and showed that deterministic models make qualitatively accurate predictions about whole-cell Ca²⁺ dynamics. We follow Cao et al. in believing that our deterministic model is sufficient to study qualitative features of Ca²⁺ dynamics in HSY cells. Introducing stochastic components to the model may reduce the quantitative discrepancies between the model results and the experimental data, but would require far more extensive and difficult computations.

The model consists of a set of ODEs, and hence assumes homogeneous cytosolic [Ca²⁺] and [IP₃]. In other words, Ca²⁺ spikes in the model represent global and simultaneous increase and decrease in the cytosolic [Ca²⁺]. Practically, this is an incorrect assumption, as it does not take into account that the cytosol is spatially inhomogeneous. When the ER starts to release Ca²⁺, the [Ca²⁺] near a dense group of IPR clusters is unlikely to be the same as in some other parts of the cytosol, where the clusters are sparse, particularly since Ca²⁺ diffusion is relatively slow. Mathematically, we can incorporate spatial dimensions in our ODE model by extending it to a partial differential equation (PDE) model, and reproduce Ca²⁺ waves across the domain. With a PDE model, we can specify the spatial distribution of the receptor clusters, and hence we can include coupled reactions between the neighbouring clusters. For instance, a small Ca²⁺ release from a cluster can trigger a series of releases from the adjacent clusters, and consequently, lead to global Ca²⁺ waves. In order to build a Ca²⁺ model with PDEs, we need high resolution images of intracellular Ca²⁺ dynamics, so that we can observe Ca²⁺ waves and accurately analyse the wave speed, direction, and amplitude. At this stage, no such data are available from HSY cells, making it difficult to construct a quantitative version of such a model. Thus, although it is possible that spatial aspects might explain some of the quantitative differences between the present model and the data, we cannot be entirely confident in the ability of a spatial model to do so.

Numerous studies have reported the possibility of mitochondria serving as a Ca²⁺ buffer [65–67]. Given the fact that mitochondrial Ca²⁺ uptake sites can be localised near Ca²⁺ release channels, it is possible that mitochondria could influence the rate of increase and the amplitude of a Ca²⁺ spike during agonist-induced Ca²⁺ oscillations. Also, mitochondria potentially increase the time it takes for a Ca²⁺ spike to reach its baseline [Ca²⁺] from the peak, as mitochondria release Ca²⁺ back into the cytosol during the decay phase of the spike. However, there is not enough information about the regulation of mitochondria in HSY cells for us to implement it in our model.

Activation kinetics of the IPR

The original IPR model of Cao et al. [30] was used to explain Ca²⁺ oscillations in airway smooth muscle cells, which exhibit fast oscillations, with periods on the order of a few seconds. Although many of the parameters of the IPR model were determined by fitting to single-channel data, not all of the transition rates could be determined from the data. In particular, λ_h42, the rate at which h₄₂ responds to changes in [Ca²⁺], was not determined by the single-channel data, and thus L and H were chosen by Cao et al. so as to give a model that could reproduce the fast Ca²⁺ oscillations in airway smooth muscle cells. However, there is no reason to believe that these parameters (which are essentially phenomenological, rather than based directly on known biophysical processes) should be the same in HSY cells. Thus, based on the slow timescale of Ca²⁺ oscillations in HSY cells, we assume that L and H are smaller in HSY cells than in airway smooth muscle cells. Extensive efforts to reproduce the long periods seen in HSY cells by varying other model parameters were unsuccessful; although the period can be changed a small amount by changes in other parameters, changes in L and H are by far the most effective at doing so.

In effect, we are predicting that the period of Ca²⁺ oscillations can be most effectively manipulated by changing how fast the rate of IPR activation by Ca²⁺ responds to changes in [Ca²⁺]. However, we have not tested this prediction directly, and so this proposed mechanism for generating slow oscillations in HSY cells remain hypothetical.

Modeling continuous photolysis of caged IP₃

In order to study the effects of Ca²⁺-activated PLC on Ca²⁺ oscillations in HSY cells, a PLC inhibitor, U73122, was applied to the cells. For the purpose of this experiment, the cells were pre-treated with caged IP₃, so that their Ca²⁺ oscillations could be initiated by photoreleased IP₃, in a PLC-independent manner. According to our model simulations, it takes a certain amount of time (about 5 mins) to trigger Ca²⁺ oscillations from the slowly increasing photoreleased [IP₃]; see Figs 8A and 10A. On the other hand, the corresponding experimental data suggest that the continuous photolysis of caged IP₃ almost immediately induces Ca²⁺ spikes, as shown in Figs 8B and 10B. We suspect that this discrepancy is caused by the simplification in our method of modeling the continuous photolysis of caged IP₃. Our model assumes that the continuous photolysis of caged IP₃ would initially give a constant increase in photoreleased [IP₃], and hence would be equivalent to having constant rates of photoreleased IP₃ production and degradation. However, the actual biological process may be quite different from our model assumption. For instance, at the beginning of the photolysis, there could be a sharp increase in photoreleased [IP₃], followed by a decrease to a saturated level. This type of reaction would explain the immediate Ca²⁺ responses to the continuous photolysis of caged IP₃.

Although there is still room for improvement in modeling the continuous photolysis of caged IP₃ with the correct timescale, our model accurately predicts that the photolysis can trigger Ca²⁺ oscillations, even when PLC is inhibited. Furthermore, it is not our main purpose to model the accurate biological process of the continuous photolysis. Thus, we are not concerned about the time that it takes for the model to generate Ca²⁺ oscillations with constant rates of photoreleased IP₃ production and degradation.

Ca²⁺ feedback on PLC

The experimental data from Tanimura et al. [22] shows that there are coupled oscillations of [Ca²⁺] and [IP₃] in HSY cells, which suggests the inclusion of positive and/or negative feedback in the model. Specifically, peaks of IP₃ spikes being preceded by those of Ca²⁺ spikes strongly suggests the presence of positive Ca²⁺ feedback on the formation of IP₃. Dupont et al. [35] pointed out that Ca²⁺ feedback on IP₃ degradation could explain Ca²⁺ oscillations with relatively low frequency. In this case, each Ca²⁺ spike would cause a subsequent decrease in [IP₃], and there would be a latency before the next Ca²⁺ spike as [IP₃] would need to build up to a certain level to activate the IPR again. We studied a case where both positive and negative feedback coexist in the model. For this case, both production and degradation rates of IP₃ were modeled as functions of [Ca²⁺]. The main conflict between the model with positive and negative feedback and the data from [22] was that the model could not reproduce the order of [Ca²⁺] and [IP₃] peaks. In fact, the model generated coupled oscillations with a IP₃ spike peak occurring just before a Ca²⁺ spike peak, which is not the case for HSY cells. For this reason, we decided not to include negative feedback in the model. However, we note that the phase shift in peak order could be relevant to other cellular systems in which Ca²⁺ feedback on the degradation of IP₃ leads to a system with negative feedback, such as the models of Meyer and Stryer [31] and Dupont and Erneux [68].

Nezu et al. [53] observed Ca²⁺ and IP₃ responses in HSY cells, induced by mechanical stimulation. Their results suggested the existence of a family of PLCs in HSY cells that is independent of external agonist such as ATP and CCh. We formulated three different equations for the production rate of IP₃:

1. (a). PLC by agonist + PLC by Ca²⁺
2. (b). PLC by agonist + PLC by agonist and Ca²⁺ + PLC by Ca²⁺
3. (c). PLC by agonist and Ca²⁺ + PLC by Ca²⁺

For cases (b) and (c), we assumed that PLC that gets activated by both agonist and Ca²⁺ is expressed as a product of some functions, f(ν) ⋅ g(C). If we express it as a sum, f(ν) + g(C), the structure of the PLC equations for these cases would not be any different from that of case (a). We studied the model responses with each expression, and observed that they showed no qualitative difference. Our model simulations show that case (a), which is the simplest form of all three, correctly predicts Ca²⁺ responses in all the experimental settings.

Positive and negative Ca²⁺ feedback on IP₃ was extensively studied in Politi et al. [32] and Gaspers et al. [33], both theoretically and experimentally; they observed qualitatively different behaviours in oscillations. Politi et al. built two different Ca²⁺ models, one with Ca²⁺ activation on PLC (positive feedback), and the other with Ca²⁺ activation on IP₃K (negative feedback). Both models exhibited Ca²⁺ oscillations with sharp spikes from the basal Ca²⁺ level. However, the shapes of the IP₃ oscillations in the models were different. In the positive feedback model, the shape of the IP₃ spikes was similar to that of the Ca²⁺ spike, with a sharp rise from the basal line to the peak and a fast decrease down to the basal level. On the other hand, IP₃ oscillations in the negative feedback model had a zig-zag pattern, where a Ca²⁺ spike caused a sudden decrease in [IP₃], followed by a slow increase until the next Ca²⁺ spike. Politi et al. also found that both positive and negative Ca²⁺ feedback on IP₃ regulation extend the range of oscillation frequencies.

Gaspers et al. compared the effects of introducing an IP₃ buffer in models with positive or negative Ca²⁺ feedback on IP₃. In the absence of buffer, the models exhibited agonist-induced Ca²⁺ oscillations. When an IP₃ buffer was added to the negative feedback model, it responded with decreased oscillation frequency and increased latency before the first spike. The oscillations persisted even at high concentrations of IP₃ buffer. In contrast, when IP₃ buffer was added to the positive feedback model, the oscillations were abolished. In this model, the Ca²⁺ flux across the plasma membrane was assumed to be relatively small compared to the flux across the ER membrane, and hence was neglected from the model. However, when the model had substantial Ca²⁺ fluxes across the plasma membrane, the inclusion of IP₃ buffer slowed Ca²⁺ oscillations. They concluded that positive feedback of Ca²⁺ on the production of IP₃ is essential for the generation of long-period, baseline-separated Ca²⁺ oscillations and waves. We have not tested the effects of introducing an IP₃ buffer in our model. However, the results of our current work is similar to the findings of Gaspers et al., whereby positive Ca²⁺ feedback on IP₃ regulation in HSY cells is shown to induce Ca²⁺ oscillations with lower frequency.

Bartlett et al. [34] investigated the characteristics of Ca²⁺ oscillations in hepatocytes, primarily induced by photolysis of caged IP₃. One of their experiments involved pre-treating hepatocytes with U73122, followed by applying UV light. They observed that uncaging of IP₃ can elicit oscillatory Ca²⁺ behaviours even after PLC is inhibited. Based on their experimental results, they concluded that Ca²⁺ oscillations induced by uncaging of IP₃ in hepatocytes (in the absence of external stimulation) do not require PLC activities, and are generated from CICR. Also, their data indicated that positive Ca²⁺ feedback on PLC is not a key player in Ca²⁺ oscillations elicited by photolysis of caged IP₃. Our experiments and model simulations of intracellular Ca²⁺ dynamics in HSY cells lead to observations that are parallel to those of Bartlett et al., where both HSY cells and hepatocytes show oscillatory Ca²⁺ responses to photolysis of caged IP₃, that seem to be independent of PLC activation. We then further investigated the contribution of Ca²⁺ activation on PLC to caged IP₃-induced oscillations. The model simulations revealed that although positive Ca²⁺ feedback on PLC may not be necessary to trigger the oscillations in HSY cells, it has some effects on the oscillation frequency. It would be interesting to test whether Ca²⁺ activation on PLC in hepatocytes has the same effects on the oscillations generated from photolysis of caged IP₃, without any external stimulation.

PLC inhibition and the frequency of Ca²⁺ oscillations

We wanted to study Ca²⁺ oscillations without Ca²⁺ feedback on PLC. Mathematically, this can be achieved by decreasing the parameter ψ₂ from 0.8 μM s^–1 to a smaller value, so that the role of Ca²⁺ on PLC is minimised. However, once cells are stimulated with external agonist, we cannot selectively inhibit Ca²⁺ feedback on PLC, as U73122 inhibits overall PLC, including agonist-activated PLC. It was necessary for the model to have Ca²⁺ oscillations in the absence of external agonist, as a simple decrease of ψ₂ alone would prevent the formation of IP₃, thus preventing any oscillatory activity. We thus simulated the situation where ψ₂ is decreased at the same time as IP₃ is photoreleased by continuous application of low-level UV light. This lets us predict the Ca²⁺ responses under the conditions where PLC is not being stimulated by Ca²⁺, but with photoreleased IP₃ in the background as a primary stimulus of the oscillations.

When HSY cells are pre-treated with caged IP₃ and exposed to continuous UV light, the concentration of photoreleased IP₃ is expected to reach a steady state, due to the constant rates of production and degradation. During Ca²⁺ oscillations, additional IP₃ is introduced on top of the baseline level of photoreleased IP₃. Thus, the average concentration of IP₃ during Ca²⁺ oscillations is lower when Ca²⁺-activated PLC is eliminated. In many cell types, it is generally the case that, within an appropriate range, higher concentrations of IP₃ lead to faster oscillations. However, the model simulation of inhibiting PLC while simultaneously photo-releasing IP₃, showed an increase in oscillation frequency. Although this result seems somewhat counterintuitive from a biophysical point of view, it is in fact consistent with the theory of oscillations that are generated by positive feedback. Tsai et al. [69] studied the effects of positive feedback in a number of biophysical models that contain both positive and negative feedback mechanisms. One of their main conclusions is that ranges of oscillation frequency are wider when the models have stronger positive feedback, and that an increase in the strength of the positive feedback will commonly lead to a decrease in oscillation frequency.

When we computed periods of the Ca²⁺ oscillations in the model with different ψ₂ values (see Fig 12), it was clear that at a given k_{s_plc} (i.e., at a given production rate of photoreleased IP₃), having larger ψ₂ generates slower oscillations. For this computation, the value of parameter V_S was kept the same (V_S = 10 μM s^–1), to confirm that the elimination of the cross-coupling between Ca²⁺ and IP₃ is responsible for the increase in oscillation frequency, without any input from modified SERCA parameters. The maximum oscillation period with ψ₂ = 0.8 μM s^–1 was about 2 minutes, whereas the maximum period with ψ₂ = 0 μM s^–1 was 1.5 minutes. This result agrees with what was found in Politi et al. [32], where eliminating positive Ca²⁺ feedback on PLC caused oscillations to have shorter periods. We treat Ca²⁺- and agonist-stimulated PLC as separate activities, which is different from the work of Politi et al., where Ca²⁺ activation is applied to agonist-induced PLC. However, our results resemble the findings of Politi et al., whereby the cross-coupling between Ca²⁺ and IP₃ is shown to increase the range of oscillation frequencies. This indicates that Ca²⁺ activation on PLC in some cells, whether it is agonist-dependent or independent, may modulate oscillation frequency. Interestingly, the experimental data showed a similar result (see Fig 8B). When U73122, a PLC inhibitor, was applied to HSY cells that had oscillating [Ca²⁺], there was a clear decrease in the oscillation period and amplitude. Given the agreement between the model prediction and the experimental result, this is strong evidence that Ca²⁺ oscillations in HSY cells do not depend on simultaneous IP₃ oscillations, but that, when IP₃ oscillations also occur, their function is to increase the oscillation period.

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Fig 12. Periods of Ca²⁺ oscillations in the HSY cell model, system (6) with ε = 1 and two different values of ψ₂.

The blue and red curves show the periods of the long-term stable oscillations generated by the system with ψ₂ = 0.8 μM s^–1 and 0 μM s^–1, respectively. The system is stimulated with slowly increasing P_s, Eq (10). For fixed k_{s_plc}, oscillations with ψ₂ = 0 μM s^–1 have shorter period than those with ψ₂ = 0.8 μM s^–1.

https://doi.org/10.1371/journal.pcbi.1005275.g012