The relative contributions of store‐operated and voltage‐gated Ca2+ channels to the control of Ca2+ oscillations in airway smooth muscle

Agonist‐dependent oscillations in the concentration of free cytosolic calcium are a vital mechanism for the control of airway smooth muscle contraction and thus are a critical factor in airway hyper‐responsiveness. Using a mathematical model, closely tied to experimental work, we show that the oscillations in membrane potential accompanying the calcium oscillations have no significant effect on the properties of the calcium oscillations. In addition, the model shows that calcium entry through store‐operated calcium channels is critical for calcium oscillations, but calcium entry through voltage‐gated channels has much less effect. The model predicts that voltage‐gated channels are less important than store‐operated channels in the control of airway smooth muscle tone.


Introduction
Airway hyper-responsiveness (AHR) (i.e. the excessive contraction of airway smooth muscle) is characteristic of asthma and is frequently the key mechanism responsible for difficulties in breathing. The contraction of airway smooth muscle cells (ASMC) reduces the airway diameter Ressmeyer et al. 2010) and this dramatically increases airway resistance. Because airway smooth muscle contraction is primarily dependent on increases in intracellular Ca 2+ concentration, it is essential to understand the dynamics of these Ca 2+ signals in ASMC if we are to address the etiology of AHR and develop new therapeutic approaches. The major mechanism that initiates airway smooth muscle contraction is stimulation with agonists released from neurons or mast cells. These include ACh, 5-hydroxytryptamine, histamine and leukotrienes, which bind to G-protein-coupled receptors in the plasma membrane . This leads to the activation of 1-phosphatidylinositol-4,5-bisphosphate phosphodiesterase β, followed by the production of inositol (1,4,5)-trisphosphate (InsP 3 ), which diffuses into the cytosol and binds to InsP 3 receptors (IPR) in the membrane between cytosol and sarcoplasmic reticulum (SR). Upon binding, InsP 3 increases the open probability of the IPR to release Ca 2+ from the SR into the cytosol.
This Ca 2+ release occurs in the form of Ca 2+ oscillations that propagate as Ca 2+ waves along the ASMC. Each Ca 2+ oscillation results from periodic Ca 2+ release via the IPR followed by Ca 2+ reuptake into the SR via Ca 2+ ATPase pumps (SERCA). The frequency of these Ca 2+ oscillations is proportional to the agonist concentration and ranges from 20 to 60 min −1 (Perez and Sanderson, 2005). These Ca 2+ oscillations activate myosin light chain kinase which, in turn, phosphorylates regulatory myosin light chains to allow actin/myosin cross-bridge cycling and the generation of force. Importantly, the frequency of the Ca 2+ oscillations determines the extent of contraction (Perez and Sanderson, 2005;Bai and Sanderson, 2006;Marhl et al. 2006;Delmotte et al. 2010).
Although most of the Ca 2+ released during each Ca 2+ oscillation is returned to the SR, the cell loses some Ca 2+ to the extracellular fluid as a result of Ca 2+ ATPase activity in the plasma membrane. Thus, to sustain Ca 2+ oscillations and the associated airway contraction, the ASMC requires a Ca 2+ influx across the plasma membrane. A variety of major Ca 2+ influx pathways have been identified in the ASMC, including (but not limited to) store-operated, receptor-operated and voltage-gated Ca 2+ channels . Non-specific Ca 2+ channels, as well as Na + /Ca 2+ exchangers, may also contribute to Ca 2+ influx . However, it remains highly controversial as to which Ca 2+ influx pathways are utilized by ASMC to maintain contraction. Because membrane depolarization is a significant regulator of contraction in vascular smooth muscle, and because membrane hyperpolarization, as mediated by localized Ca 2+ signals and Ca 2+ -dependent K + channels, has been proposed as a mechanism mediating smooth muscle relaxation, voltage-gated Ca 2+ channels (VGCC) have been proposed as a major Ca 2+ influx pathway in ASMC. By contrast, agonist-induced Ca 2+ oscillations in ASMC are based on Ca 2+ release from the SR (Perez and Sanderson, 2005;Ressmeyer et al. 2010) which, more recently, was found to be strongly regulated by store-operated Ca 2+ channels (SOCC). The identification of this Ca 2+ influx pathway is extremely important because it is a key mechanism contributing to AHR and therefore has the potential to serve as a novel therapeutic target.
Consequently, to evaluate the relative contributions of these two Ca 2+ influx pathways, we have constructed a mathematical model of agonist-induced Ca 2+ dynamics in ASMC based on a wealth of experimental data obtained from responses of ASMC in mouse precision-cut lung slices (PCLS), as described in full in the companion paper (Chen and Sanderson, 2017). Importantly, in addition to the SR ion channels and the Ca 2+ ATPase pumps responsible for Ca 2+ oscillations, this model incorporates the major plasma membrane ion channels that are assumed to contribute to establishment of membrane potential and that respond to changes in [Ca 2+ ]. Thus, the model not only simulates the Ca 2+ oscillation response to agonist stimulation, but also predicts the changes in membrane potential associated with these Ca 2+ oscillations and the consequences that this has for Ca 2+ influx either via SOCC or VGCC.
The outcome of our model simulations, many of which have been experimentally verified, is that, during agonist-induced Ca 2+ oscillations, the membrane potential oscillates as a result of Ca 2+ -activated Cl − and K + channels, although in a voltage range that has little effect on the opening of the VGCC. Thus, Ca 2+ influx via VGCC is mainly the result of their baseline activity. By contrast, SOCC current increases with Ca 2+ oscillations to mediate a Ca 2+ influx that is similar in magnitude to that provided by the VGCC. Thus, ongoing agonist-induced Ca 2+ oscillations are maintained by similar, but small, Ca 2+ influxes via SOCC and VGCC. Perhaps more importantly, our model simulations predict experimental outcomes when VGCC or SOCC are inhibited. Consistent with the experimental results, our model simulations predict that the inhibition of SOCC results in a complete cessation of the Ca 2+ oscillations, whereas the inhibition of VGCC results only in a slowing of Ca 2+ oscillations. This behaviour is explained by the the ability of ASMC to increase SOCC activity and compensate for the loss of the Ca 2+ influx via VGCC, whereas VGCC cannot compensate for the loss of SOCC activity. Thus, in terms of limiting or reversing AHR, store-operated Ca 2+ entry via SOCC is the most important contributor and thereby a target for novel bronchodilators.

Methods
All experimental methods relevant to the use of PCLS and the measurement of Ca 2+ changes in ASMC to test model predictions are described in detail in the companion paper (Chen and Sanderson, 2017).

Mathematical model
A diagrammatic representation of our mathematical model of the ASMC is shown in Fig. 1. Importantly, the model includes components that describe the Ca 2+ signalling of the ASMC, as well as components to model ionic current flow across the cell membrane.

The calcium signalling submodel
Our Ca 2+ signalling submodel is an extension of the model of Cao et al. (2014) and is based on the following assumptions: r The IPR are modelled as in Cao et al. (2014), omitting any consideration of the stochastic opening of IPR. Our model is thus deterministic. Although localized Ca 2+ increases (i.e. Ca 2+ puffs) originating from IPR cluster activity are inherently random, Cao et al. (2014) showed that a deterministic model of the periodic Ca 2+ oscillations resulting from the combined behaviour of Agonist binding to G-protein-coupled receptors stimulates the production of InsP 3 , which diffuses into the cytosol. Calcium is released from the SR through IPR and RyR. Reuptake is via ATPase pumps (SERCA). Calcium effux is via the plasma membrane ATPase (PM) and influx is through ROCC, SOCC and L-type VGCC Ca 2+ channels. The model includes Ca 2+ -activated K + and Cl − channels (KCa) and (ClCa), the delayed rectifier K + currents (Kdr), basal Na + (bNa) and K + currents (bK) and the Na + -K + exchanger (NaK). Calcium fluxes and currents are denoted by J, whereas currents of Na + and K + are denoted by I. [Colour figure can be viewed at wileyonlinelibrary.com] multiple puffs will make the same predictions as a stochastic model, and thus has almost the same predictive power.
r Ryanodine receptors (RyR) are modelled by an algebraic function of [Ca 2+ ] (Friel, 1995), and extended with a store-dependent term by Shannon et al. (2004) and Wang et al. (2010). r Stimulation by agonists causes an increased Ca 2+ influx from outside that is neither voltage-gated, nor dependent on the Ca 2+ concentration in the SR. This current, often called a receptor-operated Ca 2+ current (ROCC) is modelled as previously (Sneyd et al. 2004;Wang et al. 2010). However, in ASMC, previous experimental/modelling work indicates that the ROCC current is considerably smaller than SOCC (Croisier et al. 2013).
r Voltage-gated Ca 2+ entry through L-type Ca 2+ channels is modelled by an activation gating variable and a Goldman-Hodgkin-Katz driving force (LeBeau et al. 1997;Wang et al. 2010). The activation gating variable is assumed to be at instantaneous steady-state because, outside of microdomains, the time scale of changes in Ca 2+ concentration is typically orders of magnitude slower than the kinetics of ion channel gating variables, and this simplification does not affect the qualitative dynamics (Boie et al. 2016). Furthermore, the L-type Ca 2+ channel is not modelled as being dependent on the Ca 2+ concentration, as it is, for example, in cardiac cells (Jafri et al. 1998;Greenstein et al. 2006;Hinch et al. 2006). This simplification is justified by the fact that L-type channels in ASMC are not situated, as they are in cardiac cells, in diadic clefts, where the Ca 2+ concentration becomes very high very quickly.
r The plasma membrane Ca 2+ ATPase pumps are modelled by a Hill equation (Wang et al. 2010).
r Store-operated Ca 2+ entry is modelled by a steady-state open probability (modelled by a Hill function) and a dynamic variable that results in a delay between depletion of the SR and opening of the SOCC (Croisier et al. 2013).
r We do not model in detail the steps between agonist binding and InsP 3 production. Instead, InsP 3 is simply assumed to be an increasing function of agonist stimulation, and is usually treated as constant. Thus, as suggested by the experimental evidence (Sneyd et al. 2006), we ignore any possible effects of Ca 2+ on InsP 3 production or degradation.

The voltage submodel
The Ca 2+ submodel is coupled to a model of ionic current flow across the cell membrane. In accordance with previous work by Roux et al. (2006), we have included Na + , K + and Cl − channels. The large-conductance Ca 2+ -activated K + channels (I KCa ) are an important J Physiol 595.10 contributor to membrane hyperpolarization (Kume, 2014). The model for I KCa is adapted from Roux et al. (2006) and has been parametrized for canine tracheal cells. The delayed-rectifier K + channel (I Kdr ) further hyperpolarizes the cell. The model is taken from Roux et al. (2006) based on work perfomed in porcine and canine airway smooth muscle cells (Kotlikoff, 1990;Boyle et al. 1992). The Ca 2+ -activated Cl − channel (I ClCa ) depolarizes the cell (Gallos and Emala, 2014). The model has been constructed using experimental data of rat tracheal myocytes (Roux et al. 2001). The Na + -K + ATPases (I NaK ) used in Roux et al. (2006) was inherited from OXSOFT cardiac models (Noble et al. 1999). Furthermore, contributions of background Na + (I bNa ) and K + (I bK ) channels are included.

Model validation
The model is validated by ensuring that it can reproduce the following experimental results from PCLS: r An agonist induces Ca 2+ oscillations with a period of ß1-5 s via the production of InsP 3 (Computations not shown, but see figs 6 and 7C in Perez and Sanderson, 2005).  r In Ca 2+ -free solutions, oscillations can be triggered by an agonist but cease after a short time (computations shown in Fig. 7A, which should be compared with figs 8E and F in Perez and Sanderson, 2005).
Note that, in requiring our model to agree with these experimental data, we are implicitly requiring that our model is valid for Ca 2+ oscillations in ASMC in PCLS. Other experimental data (e.g. from isolated airway smooth muscle cells) are not taken into account. This is because isolated ASMC can behave quite differently compared to those in PCLS. Because the reasons for this remain unclear, it would be unrealistic to try and construct a single model that can explain both isolated ASMC, as well as ASMC in PCLS. This set of experiments, all from PCLS, contains the major results that have been used previously to construct and validate models of Ca 2+ signalling in ASMC from PCLS, and thus constitutes a set of results that must be satisfied for any new model to be at least as good as previous models. This is not to say that there are no other experiments from ASMC in PCLS that are not also important, nor to say that the model necessarily fits all these other possible data. However, to validate a model, it is necessary to choose which results are considered most important in determining the qualitative model structure.
Here, we simply base our choice on previous work to ensure that, at the very least, our model can reproduce earlier theoretical results, and is thus not a step backwards.

Parameter values
Although it might appear that the model contains a very large number of parameters, which are thus relatively ill-determined, allowing for great variability in model behaviour, this is not the case. Rather, the majority of the parameters have been well determined by previous studies.There are three principal groups of parameters.
1. The parameters from Cao et al. (2014) control the detailed behaviour of the IPR and the calcium signalling, and determine such things as the period of the Ca 2+ oscillations and the statistics of Ca 2+ puffs. These parameters were determined by detailed modelling of ASMC Ca 2+ oscillations, as well as from single-channel data from IPR, and are regarded as known and fixed; with the exception of small changes required to combine the models of Wang et al. (2010), Croisier et al. (2013) and Cao et al. (2014). The work of Cao et al. (2014) was itself based on previous detailed models of the IPR (Siekmann et al., 2011(Siekmann et al., , 2012 and on a detailed stochastic model of Ca 2+ signalling in ASMC (Cao et al. 2013). In addition, some parameters of the RyR were taken from Wang et al. (2010) who constructed a detailed model of the interaction of IPR and RyR Ca 2+ release in ASMC. All of these parameters are treated as known and fixed; they cannot be changed significantly without significant and unacceptable changes to the properties of the underlying Ca 2+ signalling. 2. The parameters from Roux et al. (2006) and Roux et al. (2001) control the electrical properties of the cell. The model of Roux et al. (2006) was designed specifically for ASMC and incorporated a wide array of previous experimental data, mostly collected from tracheal smooth muscle. Where data from airway smooth muscle was not available, data from gastric smooth muscle were used, or the parameters were determined by requiring physiological reasonable steady-states.
The model of Roux et al. (2006) is, by some way, the most detailed and reliable electrical model of an airway smooth muscle cell, and we take these parameters to be known and fixed, with the exception of the potassium concentrations, [K int ] and [K ext ], and the maximum current through the plasma membrane pump. 3. Finally, we treated a group of parameters as unknown, and adjusted them to obtain the correct behaviour as seen experimentally (Chen and Sanderson, 2017). The parameter modifications were performed manually (we did not fit parameters algorithmically by a Markov chain Monte Carlo procedure or similar method); after each modification, we ran simulations and checked whether the model satisfies all of the model validation criteria as outlined above.
r These parameters were those controlling the relative magnitudes of the Ca 2+ fluxes through SOCC, ROCC, VGCC and RyR. Finally, the ratio of Ca 2+ transport across the plasma membrane to Ca 2+ transport across the ER membrane was considered to be an adjustable parameter. Also, two parameters related to the maximal SERCA pump rate, as well as the plasma membrane pump rate, were modified. We were not able to find significantly different parameter sets that comply with experimental results.
The adjustable parameters are tightly constrained by the data. For example, the requirement that blockage of VGCC slows the oscillations by a small amount, whereas blockage of SOCC eliminates oscillations entirely, sets the ratio of the relative contributions of VGCC and SOCC. Similarly, the requirement that oscillations persist in the absence of external Ca 2+ , but persist for longer when SOCC is blocked, means that the ratio of currents through SOCC and ROCC can take only a narrow range of values. We hypothesize that, as long as the adjustable parameters are chosen so as to give more or less the same balance of Ca 2+ fluxes, the model behaviour will remain qualitatively the same. In other words, a change in one parameter can be compensated for by a change in another parameter, although only as long as the ratios of the various Ca 2+ fluxes remain unchanged. It is the ratio of the Ca 2+ fluxes that matter, not the actual parameter values. Deviations of ß10% in the relative Ca 2+ fluxes through VGCC and SOCC, at the same time as keeping the total current through both channel types fixed (at rest), fails to reproduce critical experimental results (specifically the experimental results shown in fig. 4C in Wang et al. (2010)). Thus, it is not the total inward Ca 2+ current that matters, it is the ratio of the VGCC current to the SOCC current.

Store-operated Ca 2+ entry is critical for agonist-induced oscillations
In ASMC, SOCC are activated when the SR Ca 2+ concentration is low (Peel et al. 2006;Prakash et al. 2006;Billington et al. 2014). Furthermore, it has been argued that the Ca 2+ influx during agonist-induced oscillations is largely a result of SOCC (Sweeney et al. 2002;Billington et al. 2014).
In a Ca 2+ -free medium, Ca 2+ oscillations terminate as the cell becomes progressively depleted of Ca 2+ (Fig. 2A). The model predicts that blocking SOCC has a similar effect, although the Ca 2+ oscillations persist for longer (Fig. 2B). Furthermore, the model predicts that, in the absence of SOCC, it takes about three times as long for the oscillations to disappear compared to in the absence of extracellular Ca 2+ . This difference results from the fact that blocking SOCC does not block all Ca 2+ influx. Ca 2+ can still enter the cell via VGCC and ROCC, but removing Ca 2+ from outside the cell prevents any Ca 2+ influx. Thus, it is predicted that blockage of SOCC takes longer to eliminate oscillations than does removal of extracellular Ca 2+ .
An experimental test of this prediction is shown in Fig. 2C and D. Oscillations were stimulated with 400 nM methacholine (MCh) and then eliminated by exposing the lung slice to zero extracellular Ca 2+ (Fig. 2C) or by adding the SOCC blocker GSK-7975A (Fig. 2D). In the first case, removal of extracellular Ca 2+ results in disappearance of the oscillations after ß40 s, as predicted by the model, whereas blocking SOCC results in the disappearance of the oscillations after ß80 s.

Voltage-gated Ca 2+ entry is less important than SOCC for agonist-induced oscillations
By experimentally blocking VGCC and observing the effect on Ca 2+ oscillations, we obtain an indirect measure of the relative contribution of VGCC and SOCC to total influx. We stimulate ASMC by 400 nM MCh, which causes fast Ca 2+ oscillations (period of ß1.3 s) ( Fig. 3B and  C). Then, we block VGCC using 100 μM nifedipine. The oscillations gradually slow down to a period of ß2.3 s. Hence, the frequency has reduced by around one half after blocking VGCC. The corresponding model simulations are shown in Fig. 3A. In both model and experiment, the nifedipine-induced change in frequency is slow, taking over 100 s to change the frequency appreciably.

Depolarization-induced Ca 2+ oscillations entry are dependent on VGCC but not SOCC
Conversely, depolarization-induced Ca 2+ oscillations are critically dependent on Ca 2+ entry through VGCC. We stimulate ASMC using 50 mM KCl, which J Physiol 595.10 causes depolarization of the plasma membrane. This depolarization increases the influx through VGCC. Sustained Ca 2+ influx cause internal stores to overfill and release Ca 2+ through RyR (Fig. 4A). Reuptake of Ca 2+ by SERCA pumps then leads to refilling of the SR, which overfills again (because the VGCC are still letting in Ca 2+ ). Thus, oscillations arise from a cycle of SR overfilling followed by release of Ca 2+ .
On the other hand, the model predicts that KCl-induced oscillations remain mostly unaffected by blockage of SOCC (Fig. 5A) and this prediction is also confirmed experimentally (Fig. 5B). This is because the SR is overfilled with Ca 2+ during KCl-induced oscillations (thus activating Ca 2+ release through RyR), in which case SOCC is minimized.

SOCC current adapts to blockage of VGCC
During depolarization-induced oscillations, the flux through the VGCC is the largest Ca 2+ influx (Fig. 6A) because the depolarization has increased the current through the VGCC nearly threefold compared to the current at rest. However, during agonist-induced oscillations, the contributions as a result of VGCC and SOCC are of similar magnitude (Fig. 6B), with the VGCC current being slightly smaller on average.
Initially, the fact that SOCC and VGCC currents are of similar magnitude during agonist-induced oscillations would appear to be inconsistent with blockage of SOCC eliminating agonist-induced oscillations, whereas blockage of VGCC does not. The explanation for this can be seen in Fig. 6C.
Upon addition of nifedipine (in the presence of MCh), a slow depletion of the SR leads to a slow increase in SOCC,   which partially counterbalances the decreased Ca 2+ influx as a result of the blockage of VGCC. Conversely, upon blockage of SOCC, VGCC are unable to adapt in the same way because they are not dependent on the SR [Ca 2+ ]. Thus, because SOCC adapt to blockage of VGCC, whereas VGCC do not adapt to blockage of SOCC, blocking the two different channels results in different qualitative effects.

Voltage oscillations have little effect on agonist-induced or KCl-induced Ca 2+ oscillations
Agonist-induced Ca 2+ oscillations are accompanied by oscillations in the membrane voltage (Fig. 7A). During Ca 2+ release from the SR, the cell hyperpolarizes to almost −56 mV and, during the reuptake of Ca 2+ , the cell depolarizes to around −43 mV. The range over which the voltage oscillates is consistent with experimental recordings by ZhuGe et al. (2010).
However, these oscillations in membrane potential do not lead to large changes in the current through the VGCC (Fig. 6B). Furthermore, the voltage oscillations have little effect on either the amplitude or the frequency of the Ca 2+ oscillations. Figure 7B and C compares the amplitude and frequency of a model with oscillating membrane voltage (blue curves) against a model with voltage fixed near the maximum of the voltage during agonist-induced oscillations (V 0 = −43.0 mV; red curves). Both the amplitude and frequency are almost unchanged, as indicated by the red and blue lines barely being distinguishable over a large range of the parameter values.
Of course, changes in voltage affect the driving force for ion transport across the plasma membrane. Consequently, changes in voltage change the currents through all ion channels. In particular, the current through SOCC is dependent on the voltage across the plasma membrane (Parekh and Putney, 2005;Vig et al. 2006  During KCl-induced oscillations, the membrane potential also oscillates, although now it is around a more depolarized average level (Fig. 7D). However, just as with the agonist-induced oscillations, forcing the membrane potential to remain constant at its average value results in little change in the amplitude (Fig. 7E) or the frequency (Fig. 7F) of the oscillations.

Discussion
We have constructed and studied a model of Ca 2+ oscillations in ASMC, which includes a detailed treatment of electrical currents. Our model is thus able to determine how much the Ca 2+ oscillations are affected by the voltage oscillations, and vice versa. The model itself is based on the previous models of Roux et al. (2006), Wang et al. (2010) and Croisier et al. (2013); electrical aspects of the model, particularly the models of the K + , Na + and Cl − channels, are based on Roux et al. (2006), whereas the models of the Ca 2+ dynamics, VGCC and ROCC, are adapted from Wang et al. (2010) and Roux et al. (2006). Croisier et al. (2013) did not model the Ca 2+ oscillations directly but, instead, mimicked the release of Ca 2+ from internal stores to focus on studying the voltage dynamics of ASMC. By contrast, we use a model for Ca 2+ handling that allows Ca 2+ oscillations based on Ca 2+ release through InsP 3 receptors (Cao et al. 2014) and RyR (Wang et al. 2010) and reuptake by the SR through pumps.
The major results from our model can be summarized as:   Our conclusions are consistent with those of Roux et al. (2006) who predict that cholinergic stimulation (e.g. stimulation by an agonist) does not lead to substantial Ca 2+ influx through VGCC.
However, our results are in contrast to a conclusion made by Zhang et al. (2013), who measured the effect of L-type VGCC blockers in isolated mouse ASMC, and found that VGCC blockers are able to reverse the MCh-induced contraction of ASMC. From this finding, they deduce that VGCC are the major contributor to Ca 2+ influx during agonist-induced contraction.
There are two possible explanations for this discrepancy. First, Zhang et al. (2013) performed their experiments in isolated ASMC, which do not have the same properties as ASMC in PCLS, such as we used in our experiments. Second, Zhang et al. (2013) employed a protocol of depolarization (by 20 mM KCl) combined with MCh. Such a depolarization upregulates VGCC and shifts the balance to a stronger contribution of VGCC compared to SOCC. Under such conditions, where VGCC has been increased above its resting level, blockage of VGCC is predicted to have a greater effect on agonist-induced Ca 2+ oscillations.
Our results, both theoretical and experimental, have the corollary that modulation of VGCC is not, in general, an effective way to control ASMC tone. If the cell is initially depolarized, then blockage of VGCC will cause a significant reduction of Ca 2+ influx and thus relax the cell. However, in cases where the cell is not depolarized (i.e. at rest, or during agonist-induced oscillations) there is little Ca 2+ coming in through VGCC, in which case blockage of VGCC has little effect on contraction.
In other muscle cells, such as striated muscle (Flucher and Franzini-Armstrong, 1996), vascular smooth muscle (Janssen et al. 2001), gastrointestinal smooth muscle (Bolton et al. 1999) and heart muscle (Bers, 2002), depolarization and voltage-gated Ca 2+ entry leads to influx of Ca 2+ , followed by calcium-induced Ca 2+ release. Local Ca 2+ release from the SR through the RyR can activate big-conductance Ca 2+ -activated K + channels, resulting in a spontaneous transient outward current that hyperpolarizes the cell. This hyperpolarization reduces the conductivity of the VGCC and decreases the Ca 2+ influx, which may then lead to relaxation (ZhuGe et al. 2010;Lifshitz et al. 2011;Janssen, 2012;Zhang et al. 2013;Kume, 2014). Lifshitz et al. (2011) demonstrated the spatial proximity of two subtypes of RyR to K + channels and ZhuGe et al. (2010) report experimental evidence that local Ca 2+ release (sparks) lead to biphasic responses in the membrane potential.
However, our results suggest that this potential mechanism does not have a significant effect on Ca 2+ signalling in ASMC in PCLS; the contribution of VGCC to the overall Ca 2+ influx during agonist-induced oscillations is not sufficiently large for VGCC inhibition to affect Ca 2+ signalling significantly. Furthermore, Ca 2+ sparks are not observed experimentally in ASMC from PCLS. Although inhibition of VGCC has a significant effect on depolarization-induced Ca 2+ oscillations, these are unphysiological. Thus, any physiological effect that VGCC inhibition might have on relaxation probably occurs via other mechanisms that are independent of Ca 2+ .
Spatial inhomogeneities may affect depolarization and hyperpolarization during Ca 2+ oscillations. The hyperpolarization is mainly established by the big-conductance K + and the delayed-rectifier K + channels. Depolarization is mainly induced by the activity of the Ca 2+ -activated Cl − . In our case, these channels 'see' the cytosolic Ca 2+ concentration, whereas, in reality, the Ca 2+ at their respective channel mouths may be very different. Nevertheless, the time course of the membrane potential in our model is consistent with recordings by ZhuGe et al. (2010).
In ASMC, there is evidence that SOCC are activated when SR Ca 2+ concentration is sufficiently depleted Pabelick et al. 2004;Prakash et al. 2006). We predict that partially blocking SOCC leads to a reduced frequency of Ca 2+ oscillations. Furthermore, blocking SOCC entirely leads to a complete stop of Ca 2+ oscillations. Our model shows that fully blocking SOCC stops Ca 2+ oscillations, although it takes around three times longer compared to oscillations with all Ca 2+ influx prevented. This model prediction is confirmed experimentally, although the time courses do not agree exactly. Experimentally, blockage of SOCC eliminates Ca 2+ oscillations after ß80 s, whereas, in the model, the oscillations take ß100 s to disappear after SOCC are blocked. This result is consistent with that of Sweeney et al. (2002), who show that SOCC contribute significantly to agonist-induced contraction in rat and human ASMC.
There are multiple possible reasons why the model gives the correct qualitative prediction about the disappearance of oscillations but is quantitatively inaccurate. Most importantly, the model is a highly simplified model of an ASMC, taking into account neither the spatial inhomogeneities of the cell, nor the stochastic properties of the Ca 2+ oscillations, particularly at low agonist or Ca 2+ concentrations. At such low concentrations, the stochastic nature of InsP 3 receptor kinetics becomes more important, leading to longer and more irregular interspike intervals in the tail when the ASMC is in a calcium-free medium or when SOCC are blocked (Fig. 2). Our model is unable, a priori, to reproduce such stochastic variability. Similarly, spatial inhomogeneity in the distributions of IPR and RyR may affect the time it takes until oscillations cease.
It is also possible that our model is underestimating the relative size of the SOCC current, and that Ca 2+ influx through VGCC is even less important than predicted by our model. If so, this would lead to faster disappearance of the model oscillations upon blockage of SOCC, and a lower decrease in oscillation frequency upon blockage J Physiol 595.10 of VGCC. However, this simply reinforces our overall conclusion that SOCC is the important influx pathway during agonist-induced oscillations; our model certainly has quantitative inaccuracies, although they probably understate, rather than overstate, our point.
There are many other quantitative discrepancies between our model and the data. The model oscillations have shapes that are more or less correct but incorrect in detail, whereas the baseline Ca 2+ concentrations, both for oscillations and for the steady concentrations (e.g. compare Fig. 2A and 2C or Fig. 4A and 4C) are not always in good agreement. Some of this is no doubt a result of inaccuracies caused by our neglect of detailed Ca 2+ buffering, our neglect of the dynamics of the Ca 2+ fluorescent dye, and the probable effects of Ca 2+ microdomains and spatial heterogeneities. It is also possible that some of these quantitative discrepancies might be removed by different choices of parameter values (although, given the difficulty that Ca 2+ models have had, for over two decades, regarding reproducing exact oscillation shapes and baselines, we consider this to be highly improbable).
The caution noted by Roux et al. (2006) applies to our mathematical model as well, which is constructed using data from different species. The models for the ion channels are constructed and parametrized for canine, porcine, rat and mice. Similarly, the model for RyR is based on bullfrog sympathetic neurons and the model for IPR is parametrized by IPR data obtained in DT40 cells . There is no dataset available characterizing the main ion channels and Ca 2+ handling pathways for a single species. Thus, quantitative predictions should be treated with caution.
However, the qualitative predictions of the mathematical model agree with all available experimental data (from PCLS) of Ca 2+ measurements during agonist-induced and depolarization-induced oscillations, and yield testable predictions about the importance (or lack thereof) of voltage dynamics on the intracellular Ca 2+ handling, the effect of (partially) blocking SOCC and the time it takes until oscillations stop in the absence of influx through SOCC. The qualitative predictions are not sensitive to parameter fine tuning, and have been confirmed experimentally, which gives us confidence in the predictive power of the model.