## Abstract

To gain insight into the mechanisms that underlie angiotensin II (ANG II)-induced cytoplasmic Ca^{2+} concentration ([Ca]_{cyt}) oscillations in medullary pericytes, we expanded a prior model of ion fluxes. ANG II stimulation was simulated by doubling maximal inositol trisphosphate (IP_{3}) production and imposing a 90% blockade of K^{+} channels. We investigated two configurations, one in which ryanodine receptors (RyR) and IP_{3} receptors (IP_{3}R) occupy a common store and a second in which they reside on separate stores. Our results suggest that Ca^{2+} release from stores and import from the extracellular space are key determinants of oscillations because both raise [Ca] in subplasmalemmal spaces near RyR. When the Ca^{2+}-induced Ca^{2+} release (CICR) threshold of RyR is exceeded, the ensuing Ca^{2+} release is limited by Ca^{2+} reuptake into stores and export across the plasmalemma. If sarco(endo)plasmic reticulum Ca^{2+}-ATPase (SERCA) pumps do not remain saturated and sarcoplasmic reticulum Ca^{2+} stores are replenished, that phase is followed by a resumption of leak from internal stores that leads either to [Ca]_{cyt} elevation below the CICR threshold (no oscillations) or to elevation above it (oscillations). Our model predicts that oscillations are more prone to occur when IP_{3}R and RyR stores are separate because, in that case, Ca^{2+} released by RyR during CICR can enhance filling of adjacent IP_{3} stores to favor a high subsequent leak that generates further CICR events. Moreover, the existence or absence of oscillations depends on the set points of several parameters, so that biological variation might well explain the presence or absence of oscillations in individual pericytes.

- mathematical model
- sarcoplasmic reticulum stores
- ryanodine receptors
- inositol trisphosphate receptors
- medullary pericytes

descending vasa recta (DVR) arise from juxtamedullary efferent arterioles and provide the sole route through which blood flow reaches the renal medulla. DVR are surrounded by smooth remnants (pericytes) that react to vasoconstrictors and vasodilators to modulate luminal diameter (36). In a series of prior studies, we showed that angiotensin II (ANG II) stimulation of DVR pericytes induces cytoplasmic Ca^{2+} concentration ([Ca]_{cyt}) elevation (36, 52, 56), activation of Ca^{2+}-dependent Cl^{−} channels (ClCa) (34–36), inhibition of K^{+} channels (9, 34), and depolarization. In addition, during voltage clamp experiments, a large fraction of pericytes responded to ANG II with spiking oscillations of inward Cl^{−} current, synchronized to underlying [Ca]_{cyt} oscillations (35, 52). The inward currents were inhibited by blockade of Cl^{−} channels (niflumic acid), ryanodine receptor (RyR) stimulation (ryanodine, caffeine), or nonselective cation channel inhibition (SKF-96365) (52). Together, these observations suggested that initiation of [Ca]_{cyt} elevation by inositol trisphosphate (IP_{3}) generation, followed by repetitive cycles of Ca^{2+}-induced Ca^{2+} release (CICR) and endoplasmic reticulum/sarcoplasmic reticulum (ER/SR) store refilling, underlies ANG II-induced pericyte oscillations.

The reasons that pericyte ANG II responses are sometimes monophasic and sometimes oscillatory remain unclear. Many events must participate in the genesis of oscillations, including openings and closings of various ion channels, IP_{3} receptor (IP_{3}R)- and RyR-mediated release of ER/SR Ca^{2+}, activity of ER/SR ATP-dependent Ca^{2+} [sarco(endo)plasmic reticulum Ca^{2+}-ATPase (SERCA)] pumps, variation of compartmental ion concentrations, and Ca^{2+} buffering. To delineate those roles, we expanded on a previously formulated mathematical simulation of ion fluxes that is largely based on the dimensions and conductances of DVR pericytes (12, 13). The model accounts for plasma membrane (PM) and ER/SR pumps, exchangers and ion channels, cytosolic and store Ca^{2+} buffering, and the privileged exchange between the ER/SR and PM via subplasmalemmal microdomain spaces (7). We revised the model to account for ANG II stimulation of IP_{3} generation and variation of Ca^{2+} store configuration. Two possibilities are considered, one in which release of Ca^{2+} via RyR and IP_{3}R occurs from a common space and a second in which RyR and IP_{3}R reside on separate stores. The model predicts that oscillations are more prone to occur when IP_{3}R and RyR stores are separate because, in that case, Ca^{2+} released by RyR during CICR enhances filling of adjacent IP_{3} stores to yield a high subsequent leak rate that generates repetitive CICR events. Filling of IP_{3}R stores during RyR-induced CICR leads to the prediction that repetitive cycles of IP_{3}R and RyR store filling and emptying occur out of phase.

## MODEL EQUATIONS

Throughout this study we compare two models: the common-store model assumes that RyR and IP_{3}R share a common Ca^{2+} store, whereas the separate-store model posits the existence of two functionally distinct Ca^{2+} stores, the first expressing only IP_{3}R and the second expressing only RyR. In the latter model, we assume that the two stores each communicate with both the cytosol and the microdomains; the SERCA pumps that refill each store at the cytosol-SR interface and the microdomain-SR interface are also distinguished. The model and its two hypotheses are illustrated in a schematic diagram of the cell in Fig. 1.

The equations that describe ionic currents across SERCA pumps, RyR, IP_{3}R, and store-operated channels (SOC), as well as those describing all Cl^{−} currents, are summarized below. Other model equations are in the appendix. All of the Cl^{−} transport equations are new additions to the model. Other ionic transport equations are identical to those employed in our previous study (12), except for the RyR open probability (*Eqs. 4*–*6*), the inward rectifier potassium (K_{ir}) conductance (*Eqs. A3*–*A5*), and the voltage-operated sodium (VONa) inactivation variable (*Eq. A28*). Parameter values are given in Table 1 and resting concentrations in Table 2.

The concentrations of *ion i* (*i* = K^{+}, Na^{+}, Cl^{−}, and Ca^{2+}) in the cytosol, the microdomains, and the extracellular environment are denoted by [*i*]_{cyt}, [*i*]_{md}, and [*i*]_{out}, respectively. In the common-store model, the SR Ca^{2+} store concentration is denoted by [Ca]_{SR}; in the separate-store model, the Ca^{2+} concentration of the store expressing only IP_{3}R and that of the store expressing only RyR are denoted by [Ca]_{IP3R-SR} and [Ca]_{RyR-SR}, respectively. We also distinguish between the transmembrane potential above the bulk cytosol (*V*) and that over the microdomains (*V*). The fraction of the cell membrane that lies directly above the cytosol (*f*^{cyt}) is taken as 0.858 and that above the microdomains (*f*^{md}) as 0.142 (25).

The Nernst potential of *ion i* (valence *z*_{i}) is calculated based on the concentration difference between the extracellular environment and the intracellular *compartment j* (*j* = cyt or md): (1) where R is the gas constant, T is the temperature, and *F* is the Faraday constant. The extracellular concentrations of K^{+}, Na^{+}, Cl^{−}, and Ca^{2+} are taken as 5.4, 140, 110, and 2 mM, respectively.

### SERCA Pump Current, I_{SERCA}

We assume that the SERCA pumps are located at both the cytosol-SR interface and the microdomain-SR interface (Fig. 1). The uptake current from *compartment j* (*j* = cyt or md) into the SR Ca^{2+} stores is given by (39): (2) where *K*_{mf} and *K*_{mr} are saturation constants, and H is the Hill coefficient.

In the separate-store model, we distinguish between the pumps that refill the Ca^{2+} store expressing only IP_{3}R and those that refill the Ca^{2+} store expressing only RyR. The term [Ca]_{SR} in *Eq. 2* is then replaced accordingly by [Ca]_{IP3R-SR} or [Ca]_{RyR-SR}.

As indicated by *Eq. 2*, the SERCA pumping direction can be reversed if [Ca]_{SR} is large enough; that is, *I*< 0 if ([Ca]_{j}/*K*_{mf}) < ([Ca]_{SR}/*K*_{mr}). Backflux through the pump has been described in SR membrane vesicles and intact myocytes (40). Conversely, SERCA pumps can be saturated if [Ca]_{j} is large enough; that is, *I*≈ *I*_{SERCA,max}f^{j} if ([Ca]_{j}/*K*_{mf}) ≫ ([Ca]_{SR}/*K*_{mr}).

The maximum current *I*_{SERCA,max} is chosen as 200 pA in the common-store model (51) and as 100 pA each for IP_{3}R-SERCA and RyR-SERCA in the separate-store model.

### RyR Current, I_{RyR}

The RyR model is that developed by Keiser and Levine, with parameters from the recent study of Ventura and Sneyd (47). The RyR-mediated release current into *compartment j* (*j* = cyt or md) is given by: (3) where ν_{RyR} is the Ca^{2+} conductivity of RyR, vol_{SR} is the SR volume (0.07 pl), and [Ca]_{SR} equals [Ca]_{RyR-SR} in the separate-store model. The open probability of RyR (*P*) is calculated as: (4) (5) (6) where *K*_{a}, *K*_{b}, *K*_{c}, and *K* are constants and ω is the fraction of channels in the two open states and the dominant closed state.

### IP_{3}R Current, I_{IP3R}

The IP_{3}R model is that developed by De Young and Keiser (11). The IP_{3}R-mediated release current in *compartment j* (*j* = cyt or md) is calculated as: (7) where ν_{IP3R} is the Ca^{2+} conductivity of IP_{3}R, *x*_{110} is the fraction of receptors bound by one activating Ca^{2+} and one IP_{3} (calculated as described below), and [Ca]_{SR} equals [Ca]_{IP3R-SR} in the separate-store model. The concentration of IP_{3} in *compartment j* (*j* = cyt or md) is given by: (8) where ν_{ip3} is the maximum rate of IP_{3} production, *I*_{r} is the rate constant for IP_{3} consumption, [IP] is a constant, and α_{4} determines the strength of the Ca^{2+} feedback on IP_{3} production.

The kinetic model for IP_{3}R developed by De Young and Keiser (11) assumes that three equivalent and independent subunits are involved in conduction and that each subunit has one IP_{3} binding site (denoted as *site 1*) and two Ca^{2+} binding sites, one for activation (*site 2*) and the other for inhibition (*site 3*). The fraction of receptors in state *S* is denoted by *x*(*i*_{j} equals 0 or 1), where the *j*th binding site is occupied if *i*_{j} = 1. All three subunits must be in the state *S*_{110} (corresponding to the binding of 1 IP_{3} and 1 activating Ca^{2+}) for the IP_{3}R channel to be open. Assuming rapid equilibrium for IP_{3} binding, *x* = ([IP_{3}]_{j}/d_{k})*x* (where *k* = 1 if β = 0 and *k* = 3 if β = 1). In particular, the open probability of IP_{3}R depends on the fraction of receptors in the *S*_{110} state, *x* = ([IP_{3}]_{j}/d_{1}). The conservation equations for the fractions at the SR-microdomain interface (*j* = md) or at the SR-cytosol interface (*j* = cyt) can be written as: (9) (10) (11) (12)

### Store-Operated Channel Current, I_{SOC}

The Ca^{2+} current through SOC into *compartment j* (*j* = cyt or md) is calculated as: (13) where [Ca]_{SR} is taken as the average of [Ca]_{IP3R-SR} and [Ca]_{RyR-SR} in the separate-store model and *G*_{CaSOC,max} is maximum Ca^{2+} SOC current conductance. The [Ca]_{SR} dependence of *I*_{Ca,SOC} illustrates the fact that decreases in SR Ca^{2+} load stimulate SOC activity. The Goldman-Huxley-Katz current equation is used to relate the SOC Na^{+} and Ca^{2+} currents in *compartment j*: (14)

The Ca^{+}-to-Na^{+} SOC permeability ratio (/*P*) is taken to be equal to 8 (10).

### Chloride Currents

The Cl^{−} transport pathways expressed by DVR pericytes have not been fully characterized. Following the approach of Hund and Rudy (22), we consider the following types of Cl^{−} fluxes: the background Cl^{−} current (*I*_{Cl,b}), the Ca^{2+}-dependent Cl^{−} current (*I*_{Cl,Ca}), and the Cl^{−} fluxes through Na^{+}-Cl^{−} cotransporters (*I*_{NaCl}) and K^{+}-Cl^{−} cotransporters (*I*_{KCl}). We also include the Cl^{−} flux through Na^{+}-K^{+}-2Cl^{−} cotransporters (*I*_{NKCC}), based on the model of Novotny and Jakobsson (32).

#### Cl^{−} background current.

(15)

#### Cl^{−} flux through Na^{+}-Cl^{−} cotransporters.

(16) where Δ_{CT} is a Cl^{−} cotransport parameter.

#### Cl^{−} flux through K^{+}-Cl^{−} cotransporters.

(17)

#### Cl^{−} flux through Na^{+}-K^{+}-2Cl^{−} cotransporters.

(18)

#### Cl^{−} current through Ca^{2+}-dependent channels.

Pallone and colleagues showed that DVR pericytes express a 16.8-pS Ca^{2+}-dependent Cl^{−} channel (55). The behavior of Ca^{2+}-dependent Cl^{−} channels is modeled with a kinetic scheme proposed by Kuruma and Hartzell (24). This model assumes the existence of four closed states (*C*_{0}–*C*_{3}) and three open states (*O*_{1}–*O*_{3}). The channel is able to open from each of the Ca^{2+}-bound closed states (*C*_{1}–*C*_{3}); transitions between the three open states are forbidden. If *y*, *y*, *y*, and *y* denote the fraction of channels in states *C*_{0}, *C*_{1}, *C*_{2}, and *C*_{3}, respectively, and *y*, *y*, and *y* denote the fraction of channels in states *O*_{1}, *O*_{2}, and *O*_{3}, respectively, the model yields the following kinetic equations: (19) (20) (21) (22) (23)

The channel opening rates (α_{k})_{k=1–3} are assumed to be Ca^{2+}- and voltage independent and to increase with the number of Ca^{2+} ligands bound to the closed states. The channel closing rates (β_{k})_{k=1–3} are expressed as β_{k}(*V*) = λ_{β}exp(*V*_{1} + *V*_{2}·*V*). The current through the Ca^{2+}-activated Cl^{−} channels above *compartment j* (*j* = cyt, md) is then given by: (24)

## RESULTS

### IP_{3}R- and RyR-Mediated Ca^{2+} Release

As described below, interaction between RyR- and IP_{3}R-mediated Ca^{2+} release from the SR into the cytosol and microdomains plays a fundamental role in model predictions of calcium signaling. To illustrate and contrast the salient properties, we begin by plotting the open probability (*P*_{o}) of RyR and IP_{3}R at steady state, as a function of Ca^{2+} concentration. The open probability of IP_{3}R was calculated assuming that [IP_{3}] = 240 nM. As shown in Fig. 2, when [Ca] is ∼100 nM (i.e., under resting conditions), *P* is significantly lower than *P* (5 × 10^{−3} vs. 2 × 10^{−2}). As [Ca] increases to ∼200 nM, *P* and *P* rise with [Ca] at a similar rate. Beyond the 200 nM threshold, however, an important departure ensues: *P* decreases because IP_{3}R is inhibited by high [Ca], whereas *P* increases rapidly to approach unity when [Ca] ∼ 10 μM. Consequently, in the cytosol, *I*_{RyR} is generally negligible compared with *I*_{IP3R}, while in the microdomains, where [Ca]_{md} can reach very high levels, *I*_{RyR} significantly surpasses *I*_{IP3R}. As shown below, CICR via activation of RyR is a critical feature that accounts for oscillatory behavior. Because of the high [Ca] therein, CICR is readily triggered within microdomains but leads to asynchronous alteration of [Ca] in various compartments of the cell.

### Effects of Elevating External KCl

We first simulated the effects of elevating extracellular KCl concentration to 100 mM, akin to the common experimental maneuver of depolarizing the membrane without exposing the cell to specific ligands. As observed experimentally (56), the model predicts a [Ca]_{cyt} increase after the step change (Fig. 3). The general motivation for this experimental maneuver is to test whether voltage-operated Ca^{2+} (VOCa) influx will arise. The consequences of raising external KCl concentration, however, are predicted to be more complex and dependent on the Ca^{2+} store configuration.

The cascade of events is summarized in Fig. 3*D*. Membrane depolarization (from −79 to +3 mV above the cytosol) affects net Ca^{2+} entry in two ways. First, it gates opening of VOCa channels that mediate Ca^{2+} influx into cytoplasm and microdomains, favoring elevation of [Ca]_{cyt} and [Ca]_{md}. Second, it provides an additional contribution to the increase in [Ca]_{md} by reducing driving forces that favor the export of Ca^{2+}, in exchange for 3 Na^{+}, by Na^{+}/Ca^{2+} exchangers (NCX). When [Ca]_{md} exceeds the threshold for stimulation of RyR-mediated CICR, an additional marked elevation of [Ca]_{md} ensues, accounting for the ability of KCl to induce an early [Ca]_{cyt} peak. Overall, KCl depolarization is predicted to raise [Ca]_{md} into the micromolar range and saturate microdomain SERCA pumps. The large rise in [Ca]_{md} that follows CICR eventually drives a sufficiently high rate of Ca^{2+} export via NCX to halt further rise of [Ca]_{md}. In addition, partial membrane repolarization (following K^{+} efflux and increase in electrogenic Na^{+}-K^{+}-ATPase activity) reduces Ca^{2+} influx into the cell via VOCa channels, accounting for the subsequent fall of [Ca]_{cyt}.

The critical difference between predictions based on common versus separate IP_{3}R/RyR stores then becomes apparent. In the common-store model, on KCl depolarization SR Ca^{2+} remains depleted by CICR. As a consequence, SERCA pumps continue to transport Ca^{2+} from the cytosol into the SR at a high rate, causing [Ca]_{cyt} to remain below the resting value that existed before external KCl elevation (Fig. 3*A*); a plateau [Ca]_{cyt} elevation is not achieved. In contrast, the separate-store model yields a different prediction (Fig. 3*B*): the marked rise in [Ca]_{md}, related to depletion of RyR stores during CICR, stimulates SERCA-driven Ca^{2+} uptake into the IP_{3}R stores, where the Ca^{2+} content rises. Thus the Ca^{2+} contents of RyR and IP_{3}R stores simultaneously fall and rise, respectively, in opposition (Fig. 3*C*). Stated simply, the high [Ca]_{md} achieved during CICR greatly stimulate microdomain IP_{3}R-SERCA pumps to load IP_{3}R stores at the expense of RyR stores. In fact, the IP_{3}R store Ca^{2+} is predicted to increase so much that the cytosolic IP_{3}R-SERCA pump reverses direction, carrying Ca^{2+} into the cytosol to elevate [Ca]_{cyt} (Fig. 4).

### Effects of Exposure to ANG II: Common-Store Model

ANG II is a potent constrictor of microvessels. In particular, it rapidly contracts the DVR pericytes, on which many of the characteristics of this model are based (12, 13). The effects of ANG II binding to the AT_{1} receptor include IP_{3} generation, inhibition of several classes of K^{+} channels (20, 21, 30, 38), and elevation of [Ca]_{cyt}. For simplicity, we simulated those effects as a 100% step increase in ν_{ip3}, the maximum rate of IP_{3} production in *Eq. 8*, and 90% inhibition of all types of K^{+} channels.

The predictions of the common-store model are shown in Fig. 5. The increase in IP_{3} stimulates IP_{3}R-mediated Ca^{2+} release from the SR, while K^{+} channel inhibition results in membrane depolarization above both the cytosol and microdomains. The Ca^{2+} release combines with depolarization-induced gating of inward VOCa currents to raise [Ca]_{cyt} and [Ca]_{md}. In turn, the [Ca]_{md} increase triggers RyR-mediated CICR in the microdomains to further raise [Ca]_{md}. The large release of Ca^{2+} from RyR significantly depletes the common SR Ca^{2+} store and saturates microdomain SERCA pumps (Fig. 6). The subsequent increase in Ca^{2+} export via NCX limits the rise in [Ca]_{md}, while increased pumping of Ca^{2+} into the SR partly repolarizes the membrane and lowers [Ca]_{cyt}. As in the case of external KCl elevation, the SR Ca^{2+} stores remain depleted so that the rate of Ca^{2+} uptake from the cytosol remains elevated, reducing [Ca]_{cyt} to ∼30% below pre-ANG II values (Fig. 5).

Some investigators have observed that ANG II also raises the conductance of VOCa channels, denoted by *G*_{Ca,L} (33, 39). We therefore simulated the combined effects of 90% inhibition of K^{+} channels, 100% increase in IP_{3} production, and 100% increase in *G*_{Ca,L} at *t* = 100 s. The results (not shown) suggest that the qualitative effects on Ca^{2+} concentrations are similar, but the [Ca]_{cyt} and [Ca]_{md} peaks are higher because of the larger influx of Ca^{2+} via VOCa channels. In particular, [Ca]_{cyt} increases up to ∼ 1,200 nM (vs. 310 nM without the step change in *G*_{Ca,L}; see Fig. 5) and stabilizes around 100 nM.

If, however, the extent of inhibition of K^{+} channels is less (i.e., ≤75% rather than 90%), a twofold increase in ν_{ip3} yields [Ca]_{cyt} oscillations (Fig. 5). In that case, the accompanying membrane depolarization is smaller (29 mV lower than that which occurs with 90% inhibition), [Ca]_{md} elevation following CICR is less, SERCA pumps do not saturate, and SR Ca^{2+} avoids continuous depletion. Once [Ca]_{md} has dropped back to ∼100 nM and the stores are replenished, the ensuing slow, IP_{3}R-mediated leak of Ca^{2+} from the SR into the microdomains gradually raises [Ca]_{md} to the CICR threshold, triggering repetitive oscillations.

The generation of oscillations is also dependent on the extent of IP_{3} generation. For oscillations to occur, ν_{ip3} must increase by at least 100% or 85% if the degree of K^{+} channel inhibition (inhib_{K}) is 75% or 50%, respectively. When VOCa conductance (*G*_{Ca,L}) is assumed to increase in conjunction with K^{+} channel inhibition, the CICR-induced [Ca]_{md} peak is higher and the model fails to predict oscillations when inhib_{K} = 75%. This prediction is insensitive to IP_{3} generation; it is unaltered even if ν_{ip3} increases by more than a factor of 5.

### Effects of Exposure to ANG II: Separate-Store Model

In contrast to the common-store model, the separate-store model predicts that [Ca]_{cyt} oscillations can follow the combined effects of 90% K^{+} channel inhibition and 100% increase in IP_{3} production (Fig. 7). The underlying mechanisms can be dissected by simultaneously analyzing Ca^{2+} concentrations (Fig. 8) and Ca^{2+} currents (Fig. 9). As in the common-store model, the sudden increase in IP_{3} and depolarization-mediated VOCa activation raise both [Ca]_{cyt} and [Ca]_{md}, triggering RyR-mediated CICR in the microdomains. In contrast to the common-store model, however, when the RyR store is depleted by CICR the IP_{3}R store fills because of stimulation of microdomain IP_{3}R-SERCA pump activity by high [Ca]_{md} (Fig. 9). The overall consequence is that a fraction of the Ca^{2+} released from the RyR-SR store is transferred through microdomains to the IP_{3}R-SR store.

In concert with CICR, the sharp [Ca]_{md} increase stimulates Ca^{2+} export from the microdomains via NCX. Both [Ca]_{md} and [Ca]_{cyt} then fall, favored by NCX activity and inhibition of VOCa by membrane repolarization, respectively. Thus, after CICR, the Ca^{2+} loads in the RyR and IP_{3}R stores reach their respective minima and maxima and then reverse course. The reversal occurs because the RyR-SR store is eventually replenished by RyR-SERCA pumps that transport Ca^{2+} at a rate that exceeds RyR-mediated losses. Conversely, the IP_{3}R store is depleted as the rate of Ca^{2+} uptake by the microdomain IP_{3}R-SERCA pumps declines with decreasing [Ca]_{md}.

In the next phase, Ca^{2+} concentrations in the cytosol and the SR tend to stabilize while RyR- and IP_{3}R-mediated Ca^{2+} leak into microdomains slowly rises. Because of the leak, [Ca]_{md} increases progressively, until it reaches the threshold needed to generate a repeat cycle of CICR (Fig. 8).

As expected, oscillations vanish after elimination of either IP_{3}R-mediated or RyR-mediated Ca^{2+} release into the microdomains (Fig. 7*D*). In contrast, block of cytosolic IP_{3}R or cytosolic RyR reduces the magnitude of oscillations but does not affect their frequency (Fig. 7*D*). These results indicate that, together, IP_{3} and CICR events in the microdomains control the appearance of Ca^{2+} oscillations in the entire cell.

As a baseline, we assumed that store-operated cation channel activity is regulated by the average of Ca^{2+} concentrations in the IP_{3} and RyR storage pools. If SOC activity were controlled by [Ca]_{IP3R-SR} alone, which is lower than the average SR Ca^{2+} concentration during oscillations (Fig. 8), Ca^{2+} influx via SOC channels would increase along with the frequency of oscillations (0.11 vs. 0.09 Hz). Conversely, if SOC channels were controlled by [Ca]_{RyR-SR}, the frequency of oscillations would decrease to 0.05 Hz. There would be no significant effect on the amplitude of oscillations in either case.

### Role of IP_{3}R and RyR in ANG II-Mediated Oscillations

A comparison between the common-store and separate-store models suggests that the key to generation of oscillations after the initial IP_{3}-mediated stimulation is *1*) an increase in [Ca]_{md} that is sufficient to reach the CICR threshold and *2*) sufficient reloading of SR Ca^{2+} stores following CICR-induced [Ca]_{md} elevation. In the single-pool model, SR Ca^{2+} stores are predicted to remain depleted by the first CICR event if membrane depolarization is too large. This does not occur in the dual-pool model, however, because as one type of store is depleted the other is replenished. In both models, the upward drift of [Ca]_{md} requires sufficient [IP_{3}]_{md} elevation to release IP_{3}R-SR store Ca^{2+} into the microdomains. Consistent with that interpretation, the separate-store model predicts that oscillations would not occur if IP_{3} production following ANG II rose by 50% instead of 100% (Fig. 10*A*) because the upward drift of [Ca]_{md} would be too small to trigger CICR anew. More specifically, the model predicts that the ANG II-mediated increase in IP_{3} production must be ≥70% to generate [Ca]_{cyt} oscillations. When VOCa conductance (*G*_{Ca,L}) is assumed to increase in conjunction with 90% K^{+} channel inhibition, that threshold is lower: if *G*_{Ca,L} increases by a factor of 2, ANG II-mediated increase in IP_{3} production must be >40% to generate [Ca]_{cyt} oscillations. Note that because of the feedback activation by [Ca] on [IP_{3}] production (see parameter α_{4} in *Eq. 8*), [IP_{3}] is predicted to oscillate in tandem with [Ca] (Fig. 10*B*).

The following predictions also underscore the deterministic roles of IP_{3}, IP_{3}R, and RyR: in both models, a twofold increase in IP_{3} production is sufficient to generate [Ca]_{cyt} oscillations in the absence of K^{+} channel inhibition (results not shown). In contrast, isolated 90% inhibition of K^{+} channels (i.e., without the 100% increase in IP_{3} production) at *t* = 100 s does not induce oscillations, because the resulting membrane depolarization does not raise [Ca]_{md} sufficiently to trigger CICR in the microdomains. In addition, assuming that ANG II raises IP_{3} production by 100% and inhibits K^{+} channels by 90%, predicted oscillations with the separate-store model vanish if ν_{IP3R}, the Ca^{2+} conductivity of IP_{3}R (*Eq. 7*), is decreased from its baseline level of 10 s^{−1} to below 9 s^{−1}. Oscillations also vanish if ν_{RyR}, the Ca^{2+} conductivity of RyR (*Eq. 3*), is decreased from 12 to <10 s^{−1}.

### Role of Ca^{2+} Influx and SERCA Uptake in ANG II-Mediated Oscillations

In this section and the one that follows, we focus on the separate-store model, which predicts [Ca]_{cyt} oscillations more readily than the common-store model. The qualitative effects of the parameters examined below on oscillations are similar for both models.

The rates of Ca^{2+} uptake into the cell and SR also play key roles in the genesis of Ca^{2+} oscillations. If the overall conductance of SOC to both Na^{+} and Ca^{2+} is lowered by 15% or more, simulations suggest that the presumed effects of ANG II (a 2-fold increase in IP_{3} production and 90% K^{+} channel inhibition) do not elicit [Ca]_{cyt} oscillations. Similarly, if the background Ca^{2+} conductance (*G*_{Ca,b}, see *Eq. A1*) is reduced by 25% or more, oscillations are eliminated. This is because [Ca]_{md} rises too slowly after the first CICR event to reach the threshold needed to initiate sequential CICR. Thus, along with RyR and IP_{3}R leak (see above), Ca^{2+} influx into the cell must contribute to the upward drift of [Ca]_{md} toward the CICR threshold.

SERCA uptake of Ca^{2+} from microdomains to stores also governs the rate and extent of rise of [Ca]_{md} after the first CICR event. A 10% increase in the maximum SERCA current abolishes [Ca]_{cyt} oscillations, because the rate of Ca^{2+} uptake from the microdomains into the SR is too large for [Ca]_{md} to rise to the CICR threshold. Likewise, a 10% increase in the rate of Ca^{2+} export from the cell (10% increase in *I*_{CaP,max}, *Eq. A33*) prevents a sufficient rise in [Ca]_{md} and abolishes oscillations.

### Main Determinants of Oscillation Frequency and Amplitude

We next investigated the effects of changes in IP_{3}R and RyR conductivity on the frequency and amplitude of oscillations elicited by a twofold increase in IP_{3} production and 90% K^{+} channel inhibition, using the separate-store model. As illustrated in Fig. 11, oscillations are maintained over a large range of ν_{IP3R} values (i.e., >1 order of magnitude), whereas ν_{RyR}, *G*_{Ca,b}, *G*_{CaSOC,max}, *I*_{SERCA,max}, and *I*_{CaP,max} must be confined to narrower ranges. The model predicts that ν_{RyR} has the largest effects on frequency (Fig. 11*A*), meaning that the Ca^{2+} conductivity of RyR predominantly controls the interval between sequential CICR events. Conversely, the amplitude of oscillations depends primarily on the maximum SERCA current (Fig. 11*B*), that is, the rate at which Ca^{2+} is pumped back into the SR.

Our results also suggest that oscillation frequency and amplitude generally increase with increasing rates of Ca^{2+} entry into the microdomains and cytosol (i.e., with increasing ν_{IP3R}, ν_{RyR}, *G*_{Ca,b}, or *G*_{CaSOC,max}) and decrease with increasing rates of Ca^{2+} uptake into the SR or export out of the cell (i.e., with increasing *I*_{SERCA,max} or *I*_{CaP,max}).

### Comparison with Experimental Data

The frequency of oscillations predicted in the baseline case with the separate-store model is 0.09 Hz (Fig. 7), in good agreement with the dominant frequency measured in DVR pericytes exposed to ANG II during voltage clamp (52). Although the average frequency reported in that study was 1.15 ± 3.3 Hz (mean ± SD), the majority of values were near 0.09 Hz. These data are reproduced in Fig. 12*A*, along with an expanded histogram (Fig. 12*B*) to show that the average frequency was skewed by a few, very high values. Frequencies on the order of 0.1 Hz have also been measured in other vascular smooth muscle cells stimulated by ANG II (2, 19). In contrast, the predicted frequency with the common-store model (when K^{+} channel inhibition is low enough that oscillations do occur) ranges from 0.03 Hz (if inhib_{K} = 75%) to 0.04 Hz (if inhib_{K} = 0%).

Model predictions of membrane current in cells exposed to ANG II during voltage clamp were also compared with measurements in pericytes (52). With the separate-store model, the predicted frequency of the membrane current is 0.12 Hz (Fig. 13), which is close to the most common value of 0.09 Hz illustrated in the expanded frequency histogram (Fig. 12, *A* and *B*). The predicted amplitude of the spontaneous transient inward current (STIC), on the order of −10 pA, is substantially lower than the mean value reported in pericytes (52). Again, expansion of the associated histogram reveals that the most common STIC amplitude measured in the pericytes was, in fact, close to the predictions of this model (Fig. 12, *C* and *D*). In contrast, the common-store model did not yield any oscillations during voltage clamp, a prediction that is insensitive to the degree of K^{+} channel inhibition.

We also simulated the effects of 100 nM ouabain on ANG II-stimulated pericytes. The inhibition of ouabain-sensitive Na^{+}-K^{+}-ATPase isoforms, as well as the effects on IP_{3} production and on IP_{3}R conductance, were modeled as in our previous study (13). The separate-store model predicts that 100 nM ouabain raises the oscillation frequency slightly, from 0.09 to 0.13 Hz, and has a <10% effect on amplitude. Experimentally, the effects of ouabain on ANG II-associated oscillations were similarly negligible (52). The common-store model predicts that ouabain abolishes oscillations. Experimentally, nanomolar ouabain did not abolish oscillations (unpublished data).

## DISCUSSION

Growing evidence favors the notion that the architecture of ER/SR stores is complex and varies with cell type. Studies of Ca^{2+} release in mesenteric myocytes (6) and astrocytes (18) favor the existence of subplasmalemmal microdomains that are populated by ouabain-sensitive Na^{+}-K^{+}-ATPase, NCX, and SOC to facilitate localized signaling and routes for refilling of ER/SR stores (7). Controversy also surrounds the possibility that the spatial locations of RyR and IP_{3}R on the SR stores are variable. Blaustein and colleagues (6) showed that cyclopiazonic acid and serotonin release Ca^{2+} from a RyR-insensitive store in mesenteric myocytes. Similar observations in astrocytes also favor independence of stores occupied by RyR and IP_{3}R Ca^{2+} release channels (17). More complex configurations may also exist, in which a common store that expresses both IP_{3}R and RyR resides alongside a store dominated by IP_{3}R (3) or RyR (14). Regional variation along the microcirculation may occur. Studies in pulmonary arterial myocytes favor coincidence of IP_{3}R and RyR stores in resistance but not conduit vessels (46).

Investigations into the nature of the cross talk between IP_{3}R and RyR have yielded similarly varied results. In particular, controversy surrounds the ability of agonist-induced Ca^{2+} release via IP_{3}R to stimulate adjacent CICR via RyR or IP_{3}R as a mechanism to propagate intracellular Ca^{2+} waves or generate oscillations (27, 50, 53). Together, these observations lead to the conclusion that the varied complexity of ER/SR architecture, transporter expression, and interactions tailors cellular responses to various inputs such as mechanical stimulation, neural activity, or ligand binding to G protein-coupled receptors (5).

To gain insight into the complex relationships between cellular responses, we expanded a previously formulated model (12, 13) to simulate ANG II stimulation. Apart from inclusion of explicit equations to account for Cl^{−} conductance, a new feature that yields possible insight into the genesis of oscillations concerns putative interactions between RyR and IP_{3}R and separation of the stores on which they reside. We simulated the effects of ANG II by raising the maximum rate of IP_{3} production (ν_{ip3} in *Eq. 8*) by 100% and assuming 90% blockade of all K^{+} channels. Those inputs are based on experimental measurements obtained in DVR pericytes and other cells. ANG II has been found to inhibit several types of K^{+} channels in pericytes (9, 34) and other cells: K_{ATP} (20), K_{Ca} (30), K_{ir} (38), and K_{v} (21). With respect to IP_{3} generation, ANG II stimulated IP_{3} production in cultured rat aortic cells about twofold (see Fig. 1 in Ref. 1).

In recent studies we showed that ANG II stimulation of DVR pericytes leads to oscillations of [Ca]_{cyt} and synchronous variations of membrane potential (35) or Cl^{−} currents in voltage-clamped pericytes (52). Those oscillations occurred in only about half of the pericytes; the changes in the remaining half were characterized by monophasic elevations of [Ca]_{cyt} and Cl^{−} conductance. Our model points to a possible role for the spatial arrangement of ER/SR stores as a determinant of oscillations; we compared two configurations, a common-store model (Fig. 1*A*) and a separate-store model (Fig. 1*B*) in which RyR and IP_{3}R release Ca^{2+} from different stores. The model predicts that oscillations are more prone to occur in the separate-store case because of cross talk between stores via their respective IP_{3}R or RyR: as one type of store is depleted the other is replenished, thereby sustaining Ca^{2+} release into the microdomains as a cyclic phenomenon punctuated by periodic CICR events (Figs. 7–10).

More specifically, our results suggest that two key determinants of [Ca]_{cyt} oscillations are *1*) release of Ca^{2+} from stores and *2*) entry of Ca^{2+} from the extracellular space, both of which raise [Ca] in the vicinity of RyR. If the rise exceeds the threshold for CICR by RyR, a burst of CICR occurs that is followed by SERCA-mediated reuptake into stores and export across the plasmalemma by Na^{+}/Ca^{2+} exchange and Ca^{2+}-ATPase activity. If those events have reduced intracellular Ca^{2+} near RyR to a low level, the release phase begins again as a trickle of leak current from the internal stores. Either stabilization of intracellular Ca^{2+} is achieved (no oscillations) or it reaches CICR levels to generate another burst of RyR-mediated Ca^{2+} release (oscillations). The model readily predicts oscillations when IP_{3}R and RyR channels are on separate SR stores that face a common cytoplasmic domain because Ca^{2+} released by RyR stores during CICR can “overfill” the adjacent IP_{3} stores to favor a high subsequent leak rate that subsequently triggers the RyR-mediated CICR of the next oscillation. As might be expected from that scenario, the magnitude of IP_{3} generation that favors the leak from IP_{3}R stores is another key determinant of oscillations.

Other predictions of the model also favor separation of RyR and IP_{3}R stores. In medullary pericytes, KCl elevation causes a sustained increase in [Ca]_{cyt} (56), as expected from the separate-store model, whereas a transient increase, without a stable plateau, is predicted by the common-store model (Fig. 3). As described above, the calculated frequency of oscillations is closer to experimental values in pericytes (52) with the separate-store model. Abolition of oscillations by ouabain, predicted only for the common-store model, does not occur experimentally. The predicted mechanisms underlying oscillations are also consistent with a number of prior experimental findings: blockade of Ca^{2+} influx with SKF-96365 (a nonselective cation channel inhibitor expected to interfere with SOC activity) and exposure to ryanodine or to caffeine inhibited oscillations (52).

This study also underscores two key aspects of SERCA pump function. One is the importance of SERCA saturation: if SERCA currents were able to increase in parallel with RyR and IP_{3}R currents, they would prevent a sufficient rise of [Ca]_{md} to elicit repetitive CICR. The other is SERCA pump reversal. Backflux through SERCA pumps has been described by several investigators (23, 28, 40, 44). Our model predicts that backflux through cytosolic SERCA associated with the IP_{3}R-SR stores occurs after KCl elevation, allowing for sustained elevation of [Ca]_{cyt}.

Many permutations of configurations, IP_{3}R and RyR isoform variations, spatial distributions, and PM-ER/SR-cytosolic interactions, are conceptually possible and, moreover, likely to exist in living cells (3, 5, 14, 17, 18, 27, 45, 46, 50). As such, we recognize that the predictions of this model represent a limited exploration, guided largely by measurements obtained from DVR pericytes. This model has been successful in simulating microdomain-dependent ouabain interactions (13) and now yields realistic simulations of pericyte [Ca]_{cyt} responses to agonist (ANG II) stimulation (36, 52, 56) and KCl depolarization (36, 56) when IP_{3}R and RyR stores are separated. Other authors have investigated the minimal model requirements for generation of oscillations and concluded that RyR participation is not a requisite feature (4, 16). Moreover, because of the activation and inhibition achieved through binding of Ca^{2+} ions to IP_{3}R, oscillations of IP_{3} concentration may not be needed (16, 49).

This model has not been constructed to explore the minimal configurations needed to generate oscillations but instead was devised to account for a current state of knowledge that represents a particular cell type. It simulates the expressions of classes of anion and cation channels, Na^{+}-K^{+}-ATPase, Ca^{2+}-ATPase, RyR, and IP_{3}R at interfaces (PM-microdomain-ER/SR, PM-cytosol-ER/SR) as well as realistic compartmental volumes and Ca^{2+} buffering. As such, numerical integration rather than “closed” solution of the governing differential equations is required. It is satisfying that oscillations, observed in vivo, readily emerge as a prediction of the effort. One benefit of the complexity of this model is that modes of operation can be explored to define ranges over which observed behaviors, such as oscillations, are predicted. As shown in Fig. 11, the existence or absence of oscillations is a natural consequence of modifying the “set points” of key parameters. Similar variations in biological systems seem likely to exist and may well explain the varied presence of oscillations in ANG II-stimulated pericytes (52).

## APPENDIX

### Ionic Currents

The convention adopted in this study is that exit of positive charge from the cell is a positive current. Conversely, exit of negative charge is a negative current. Unless otherwise noted, current equations are taken from the model developed by Yang et al. (51) for vascular smooth muscle cells.

#### Background currents for K^{+}, Na^{+}, Cl^{−}, and Ca^{2+}.

The background current of *ion i* in *compartment j* (*j* = cyt or md) is calculated as: (A1)

#### Inward rectifier potassium current, I_{K,ir}.

The current flowing across K_{ir} channels is expressed as (26): (A2) (A3) (A4) (A5) where the parameters were obtained by fitting DVR pericyte data (8). The expression for the K_{ir} conductance has been revised to obtain better agreement with the data.

#### Delayed rectifier potassium current, I_{K,v}.

The current flowing across K_{v} channels lying above *compartment j* is calculated as: (A6) (A7) (A8) (A9) (A10) (A11) (A12) where *P*_{1} and *P*_{2} denote the two exponential components of the channel activation process and τ_{P1} and τ_{P2} denote the respective time constants (in ms). The parameter *P̄*_{Kv}, which is voltage dependent, represents the steady-state value of *P*_{1} and *P*_{2}.

#### ATP-activated potassium current, I_{K,ATP}.

The current flowing across K_{ATP} channels is modeled as (42): (A13) where P_{ATP} is the fraction of K_{ATP} channels available at a given ATP concentration; we assume that the intracellular ATP concentration remains fixed, so that the product *G*_{KATP,max}P_{ATP} is a constant.

#### Calcium-activated potassium current, I_{K,Ca}.

The current flowing across Ca^{2+}-activated K^{+} channels is calculated as: (A14) (A15) (A16) (A17) (A18) (A19) where *P*_{F} and *P*_{S} denote the fast and slow components of the channel activation process, respectively, and τ_{PF} and τ_{PS} denote the respective time constants. The steady-state open probability of the channel is given by *P̄*_{KCa}.

#### Na^{+}-K^{+}-ATPase pump current, I_{NaK}.

As described previously (12), our model posits that the α_{1}-isoform of the Na^{+}-K^{+}-ATPase pump is expressed only in the PM above the bulk cytosol, whereas the α_{2}-isoform is confined to the membrane region that is above the microdomains. The respective currents are determined as (26): (A20) (A21) (A22) (A23)

#### Na^{+}/Ca^{2+} exchanger current, I_{NaCa}.

We assume that all Na^{+}/Ca^{2+} exchangers (NCX) are preferentially localized in the microdomains (6, 31). The microdomain NCX current is given by (22): (A24) (A25) (A26) (A27)

#### Voltage-activated sodium current, I_{VONa}.

The current flowing through voltage-activated Na^{+} channels is calculated based on DVR pericyte data (54): (A28) (A29) (A30) (A31) (A32) where *m*_{VONa} and *h*_{VONa} are the activation and inactivation components (with steady-state values *m̄* and *h̄*, respectively) and τ_{m} and τ_{h} are the associated time constants. The current is taken to be proportional to the first power of *h*_{VONa} instead of the third power (12), so as to obtain better agreement with the experimental data (54).

#### Calcium pump current, I_{Ca,P}.

The current through plasmalemmal Ca^{2+} pumps is expressed as: (A33)

#### Voltage-dependent Ca^{2+} current, I_{Ca,L}.

The current flowing across voltage-activated Ca^{2+} channels is calculated as: (A34) (A35) (A36) (A37) (A38) (A39) (A40) (A41) where *d*_{L} is the activation variable, *f*_{L} and *f*_{F} are inactivation variables, and τ_{d} and τ_{f} are the time constants for activation and inactivation (in ms), respectively; *d̄*_{L} and *f̄*_{F} denote the steady-state values of *d*_{L} and *f*_{F}.

### Calcium Buffers

We account for the chelation of Ca^{2+} by calmodulin and other Ca^{2+}-binding proteins in the cytosol and the microdomains and by calsequestrin in the SR. Let [CM] denote the total concentration of calmodulin sites available for Ca^{2+} binding in *compartment j* (*j* = cyt or md), and [CM·Ca]_{j} the concentration of calcium-bound calmodulin sites in that compartment. Similarly, let [Bf] denote the total concentration of other Ca^{2+}-binding proteins in *compartment j* and [Bf·Ca]_{j} the concentration of the calcium-bound sites of these other buffering elements. Assuming that calcium buffering can be described as first-order dynamic processes (51), we have: (A42) (A43) Finally, the buffering of calcium by calsequestrin in the SR is modeled as: (A44) where [Calseq] denotes the total concentration of calsequestrin sites available for Ca^{2+} binding in the SR and [Calseq·Ca]_{SR} the concentration of Ca^{2+}-bound calsequestrin sites in that compartment. In the separate-store model, we distinguish between the Ca^{2+} binding sites in the SR stores expressing only IP_{3}R and those in the SR stores expressing only RyR; [Ca]_{SR} in *Eq. A44* is replaced accordingly by [Ca]_{IP3R-SR} or [Ca]_{RyR-SR}.

### Electrodiffusive Flux from Microdomain to Cytosol

The electrodiffusive flux of *ion i* (valence *z*_{i}) is defined such that *J*_{i,diff} yields a positive current into the cytosol when cations flow down their electrochemical gradient from microdomains to cytosol (12): (A45) (A46) (A47) where *A* is the cross-sectional area at the interface between the microdomains and the cytosol (taken as 0.776 μm^{2}), *D*_{i} is the diffusivity of *ion i*, *L* is the distance from the center of the microdomains to that of the cytosol (taken as 0.5 μm), and *h* is a hindrance factor, which lumps together steric and charge-related effects (taken as 1 × 10^{−3}). As described previously (12), the whole cell diffusivity of calcium is taken as 0.3 × 10^{−5} cm^{2}/s, and the sodium-to-calcium and potassium-to-calcium diffusivity ratios are assumed to be equal to 1.33/0.79 and 1.96/0.79, i.e., the ratios of the diffusivities in dilute solution.

### Time Variations in Internal Concentrations

The concentrations of K^{+}, Na^{+}, Cl^{−}, and Ca^{2+} as a function of time are determined by integrating the following differential equations: (A48) (A49) (A50) (A51) (A52) (A53) (A54) (A55) In the common-store model: (A56) In the separate-store model: (A57) (A58) where vol_{cyt} (0.5 pl), vol_{cyt,Ca} (0.35 pl), and vol_{md} (0.003 pl) denote the total volume of the cytosol, the volume of the cytosol that is available to Ca^{2+}, and the volume of the microdomains, respectively.

### Time Variations in Membrane Potential

The net sum of the currents flowing out of the bulk cytosol is given by: (A59) The net sum of the currents flowing out of the microdomains is given by: (A60) The time dependence of *V* and *V* is given by: (A61) (A62) where *C*_{m} is the membrane capacitance (1.21 × 10^{−5} μF). Neglected in *Eqs. A61* and *A62* is the movement of ions other than Ca^{2+} across the SR. Given the lack of experimental data, we cannot currently account for all the pathways (pumps, channels, and transporters) by which charge flows into and out of the SR.

## GRANTS

This work was supported by National Institutes of Health Grants DK-53775 (A. Edwards), DK-42495, HL-78870, and DK-67621 (T. L. Pallone).

## Footnotes

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- Copyright © 2008 the American Physiological Society