Mechanisms underlying angiotensin II-induced calcium oscillations

Abstract

To gain insight into the mechanisms that underlie angiotensin II (ANG II)-induced cytoplasmic Ca²⁺ 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 (IP₃) production and imposing a 90% blockade of K⁺ channels. We investigated two configurations, one in which ryanodine receptors (RyR) and IP₃ receptors (IP₃R) occupy a common store and a second in which they reside on separate stores. Our results suggest that Ca²⁺ 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 Ca²⁺-induced Ca²⁺ release (CICR) threshold of RyR is exceeded, the ensuing Ca²⁺ release is limited by Ca²⁺ reuptake into stores and export across the plasmalemma. If sarco(endo)plasmic reticulum Ca²⁺-ATPase (SERCA) pumps do not remain saturated and sarcoplasmic reticulum Ca²⁺ 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 IP₃R and RyR stores are separate because, in that case, Ca²⁺ released by RyR during CICR can enhance filling of adjacent IP₃ 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.

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 Ca²⁺ concentration ([Ca]_cyt) elevation (36, 52, 56), activation of Ca²⁺-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 (IP₃) generation, followed by repetitive cycles of Ca²⁺-induced Ca²⁺ 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, IP₃ receptor (IP₃R)- and RyR-mediated release of ER/SR Ca²⁺, activity of ER/SR ATP-dependent Ca²⁺ [sarco(endo)plasmic reticulum Ca²⁺-ATPase (SERCA)] pumps, variation of compartmental ion concentrations, and Ca²⁺ 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 Ca²⁺ 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 IP₃ generation and variation of Ca²⁺ store configuration. Two possibilities are considered, one in which release of Ca²⁺ via RyR and IP₃R occurs from a common space and a second in which RyR and IP₃R reside on separate stores. The model predicts that oscillations are more prone to occur when IP₃R and RyR stores are separate because, in that case, Ca²⁺ released by RyR during CICR enhances filling of adjacent IP₃ stores to yield a high subsequent leak rate that generates repetitive CICR events. Filling of IP₃R stores during RyR-induced CICR leads to the prediction that repetitive cycles of IP₃R 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 IP₃R share a common Ca²⁺ store, whereas the separate-store model posits the existence of two functionally distinct Ca²⁺ stores, the first expressing only IP₃R 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.

Fig. 1.

Schematic representation of the cell, with its 3 compartments: bulk cytosol (Cyt), microdomains (MD), and sarcoplasmic reticulum (SR). Not shown are K_Ca, K_ir, K_ATP, and K_v channels and the chelating agents calmodulin and calsequestrin. A: common-store model. B: separate-store model.

The equations that describe ionic currents across SERCA pumps, RyR, IP₃R, 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.

Table 1.

Ionic current parameters

Parameter	Description	Value	Reference
ISERCA,max	Sarco(endo)plasmic reticulum Ca2+-ATPase (SERCA) maximum current	200/100 pA	See text
Kmf	Ca2+-ATPase constant for cytosolic or microdomain Ca2+	310 nM†	14
Kmr	Ca2+-ATPase constant for SR Ca2+	1.7 mM	41
nH	SR Ca2+-ATPase Hill coefficient	2†	37
νRyR	Ryanodine receptor (RyR) Ca2+ conductivity	12 s−1†	This study
Ka	RyR constant	0.3722 μM*	47
Kb	RyR constant	0.6360 μM*	47
Kc	RyR constant	0.0571 μM*	47
Kc−	RyR constant	0.1 s−1*	47
νIP3R	IP3R Ca2+ conductivity	10 s−1†	This study
νip3	Rate constant for IP3 production	1.85 s−1	12
[IP3ref]	Reference IP3 concentration	240 nM	11
α4	Feedback constant for Ca2+ stimulation of IP3 production	0.5	12
k4	Dissociation constant for Ca2+ stimulation of IP3 production	1.1 μM	11
Ir	Rate constant for IP3 consumption	1 s−1	11
d1	IP3 receptor dissociation constant	0.13 μM	11
a4	IP3 receptor binding constant	0.2 μM−1·s−1	11
d4	IP3 receptor dissociation constant	144.5 nM	11
a5	IP3 receptor binding constant	20 μM−1·s−1	11
d5	IP3 receptor dissociation constant	82.34 nM	11
GCaSOC,max	Maximum Ca2+ SOC current conductance	87 pS†	This study¶
KSOC	ISOC SR Ca2+ constant	100 μM	12
ΔCT	Chloride cotransport parameter	87.8251 mV*	22
JNaCl,max	Maximum flux though Na+-Cl− cotransporter	6.0 × 10−17 mol/s*	This study¶
JKCl,max	Maximum flux though K+-Cl− cotransporter	3.5 × 10−18 mol/s*	22
JNKCC,max	Maximum flux though Na+-K+-2Cl− cotransporter (NKCC)	6.0 × 10−17 mol/s*	This study¶
KNaNKCC	Na+ half-saturation constant for NKCC	46 mM*	43
KKNKCC	K+ half-saturation constant for NKCC	4 mM*	43
KCl,1NKCC	Cl− first half-saturation constant for NKCC	78 mM*	43
KCl,2NKCC	Cl− second half-saturation constant for NKCC	78 mM*	43
kon	Ca-Cl channel on-rate	3 μM−1·s−1*	24
koff	Ca-Cl channel off-rate	50 s−1*	24
α1	Ca-Cl kinetic constant	10 s−1*	24
α2	Ca-Cl kinetic constant	30 s−1*	24
α3	Ca-Cl kinetic constant	300 s−1*	24
β1, β2, β3	Ca-Cl kinetic constants	224 exp(−3.8 − 0.01Vmj) s−1*	24
GClCa,max	Ca-Cl channel maximum conductance	12 nS*	34
GK,b	K+ background current conductance	0.010 nS	51
GNa,b	Na+ background current conductance	0.010 nS	51
GCa,b	Ca+ background current conductance	0.012 nS	51
GCl,b	Cl− background current conductance	0.010 nS*	This study
GKir,max	Inward rectifier conductance	0.3 nS†	8
[K]ref	Reference K+ concentration	5.4 mM	51
GKv,max	Maximum IK,v conductance	9.8325 nS	51
GKATP,maxPATP	Maximum IK,ATP conductance at resting ATP concentration	1.036 nS	9
nATP	IK,ATP constant	0.463	9
GKCa,max	Maximum IK,Ca conductance	0.5 nS	51
τPF	Fast time constant for IK,Ca	0.5 ms	51
τPS	Slow time constant for IK,Ca	11.5 ms	51
INaK,α1,max	Maximum Na+/K+-ATPase α1-isoform current	1.95 μA/μF†	This study¶
INaK,α2,max	Maximum Na+/K+-ATPase α2-isoform current	0.30 mA/μF†	This study¶
Km,K	K+ half-saturation constant for Na+-K+-ATPase (both isoforms)	1.5 mM	12
Km,Na,α1	Na+ half-saturation constant for Na+-K+-ATPase α1-isoform	12 mM	12
Km,Na,α2	Na+ half-saturation constant for Na+-K+-ATPase α2-isoform	22 mM	12
INaCa,max	Maximum Na+/Ca2+ exchange current	14.13 μA/μF†	41
KmCai	Internal Ca2+ half-saturation constant for Na+/Ca2+ exchange	3.59 μM	41
KmCao	External Ca2+ half-saturation constant for Na+/Ca2+ exchange	1.3 mM	41
KmNai	Internal Na+ half-saturation constant for Na+/Ca2+ exchange	12.29 mM	41
KmNao	External Na+ half-saturation constant for Na+/Ca2+ exchange	87.5 mM	41
ksat	Na+/Ca2+ exchange saturation factor at negative potentials	0.27	41
γ	Constant for voltage dependence of Na+/Ca2+ exchange	0.35	41
KmCa,act	Constant for Na+/Ca2+ exchange	0.125 μM†	22
GVONa,max	Maximum IVONa conductance	2.86 nS	54
τm	IVONa activation time constant	0.1 mS	54
τh	IVONa inactivation time constant	1 mS	54
ICa,P,max	Maximum sarcolemmal Ca2+ pump current	1 μA/μF†	This study¶
Km,Ca,P	Half-saturation constant for sarcolemmal Ca2+ pump	0.17 μM†	15
GCaL,max	Maximum ICa,L conductance	1.4151 nS	51
[CM]tot	Total concentration of calmodulin binding sites available for Ca2+ binding in cytosol or microdomains	0.1 mM†	51
kCMon	Ca2+ on-rate for calmodulin	34 μM−1·s−1	41
kCMoff	Ca2+ off-rate for calmodulin	8.84 s−1†	Based on calmodulin dissociation constant (51)
[Bf]tot	Total concentration of other Ca2+-binding proteins in cytosol or microdomains	0.4 mM†	51
kBfon	Ca2+ on-rate for other binding proteins	100 μM−1·s−1	41
kBfoff	Ca2+ off-rate for other binding proteins	300 s−1†	Based on caltropin dissociation constant (29)
[Calseq]SRtot	Total concentration of calsequestrin sites available for Ca2+ binding in SR	140 μM	41
kCalseqon	Ca2+ on-rate for calsequestrin	100 μM−1·s−1	41
kCalseqoff	Ca2+ off-rate for calsequestrin	80,000 s−1†	Based on calsequestrin dissociation constant (48)

Unless otherwise indicated, parameter values are identical to those of the original model (12). SR, sarcoplasmic reticulum; VONa, voltage-activated sodium.

^¶Parameters were chosen so as to yield resting concentrations compatible with experimental observations.

^*New parameter;

^†revised parameter value.

Table 2.

Resting values

Variable	Resting Value	Units
Vmcyt	−79	mV
Vmmd	−79	mV
[K]cyt	144	mM
[K]md	144	mM
[Na]cyt	8.6	mM
[Na]md	15.9	mM
[Cl]cyt	27	mM
[Cl]md	24	mM
[Ca]cyt	107	nM
[Ca]md	153	nM
[Ca]SR	0.54 (common store)	mM
	0.52/0.56 (separate store)*

cyt, Cytosol; md, microdomain.

^*First and second values correspond to the IP₃R and RyR stores, respectively.

The concentrations of ion i (i = K⁺, Na⁺, Cl⁻, and Ca²⁺) 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 Ca²⁺ store concentration is denoted by [Ca]_SR; in the separate-store model, the Ca²⁺ concentration of the store expressing only IP₃R 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 Ca²⁺ 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 Ca²⁺ 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 Ca²⁺ store expressing only IP₃R and those that refill the Ca²⁺ 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,maxf^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 IP₃R-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 Ca²⁺ 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.

IP₃R Current, I_IP3R

The IP₃R model is that developed by De Young and Keiser (11). The IP₃R-mediated release current in compartment j (j = cyt or md) is calculated as:

(7)

where ν_IP3R is the Ca²⁺ conductivity of IP₃R, x₁₁₀ is the fraction of receptors bound by one activating Ca²⁺ and one IP₃ (calculated as described below), and [Ca]_SR equals [Ca]_IP3R-SR in the separate-store model. The concentration of IP₃ in compartment j (j = cyt or md) is given by:

(8)

where ν_ip3 is the maximum rate of IP₃ production, I_r is the rate constant for IP₃ consumption, [IP] is a constant, and α₄ determines the strength of the Ca²⁺ feedback on IP₃ production.

The kinetic model for IP₃R developed by De Young and Keiser (11) assumes that three equivalent and independent subunits are involved in conduction and that each subunit has one IP₃ binding site (denoted as site 1) and two Ca²⁺ 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 jth binding site is occupied if i_j = 1. All three subunits must be in the state S₁₁₀ (corresponding to the binding of 1 IP₃ and 1 activating Ca²⁺) for the IP₃R channel to be open. Assuming rapid equilibrium for IP₃ binding, x = ([IP₃]_j/d_k)x (where k = 1 if β = 0 and k = 3 if β = 1). In particular, the open probability of IP₃R depends on the fraction of receptors in the S₁₁₀ state, x = ([IP₃]_j/d₁). 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 Ca²⁺ 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 Ca²⁺ SOC current conductance. The [Ca]_SR dependence of I_Ca,SOC illustrates the fact that decreases in SR Ca²⁺ load stimulate SOC activity. The Goldman-Huxley-Katz current equation is used to relate the SOC Na⁺ and Ca²⁺ 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 Ca²⁺-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 Ca²⁺-dependent channels.

Pallone and colleagues showed that DVR pericytes express a 16.8-pS Ca²⁺-dependent Cl⁻ channel (55). The behavior of Ca²⁺-dependent Cl⁻ channels is modeled with a kinetic scheme proposed by Kuruma and Hartzell (24). This model assumes the existence of four closed states (C₀–C₃) and three open states (O₁–O₃). The channel is able to open from each of the Ca²⁺-bound closed states (C₁–C₃); transitions between the three open states are forbidden. If y, y, y, and y denote the fraction of channels in states C₀, C₁, C₂, and C₃, respectively, and y, y, and y denote the fraction of channels in states O₁, O₂, and O₃, 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 Ca²⁺- and voltage independent and to increase with the number of Ca²⁺ ligands bound to the closed states. The channel closing rates (β_k)_k=1–3 are expressed as β_k(V) = λ_βexp(V₁ + V₂·V). The current through the Ca²⁺-activated Cl⁻ channels above compartment j (j = cyt, md) is then given by:

(24)

RESULTS

IP₃R- and RyR-Mediated Ca²⁺ Release

As described below, interaction between RyR- and IP₃R-mediated Ca²⁺ 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 IP₃R at steady state, as a function of Ca²⁺ concentration. The open probability of IP₃R was calculated assuming that [IP₃] = 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 IP₃R 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.

Fig. 2.

Open probability (P_o) of ryanodine receptors (RyR) and inositol trisphosphate receptors (IP₃R) at steady state as a function of calcium concentration ([Ca], in M). In these calculations, [IP₃] is taken as 240 nM.

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 Ca²⁺ (VOCa) influx will arise. The consequences of raising external KCl concentration, however, are predicted to be more complex and dependent on the Ca²⁺ store configuration.

Fig. 3.

Effects of external KCl elevation on Ca²⁺ concentrations in cytosol ([Ca]_cyt, A), microdomains ([Ca]_md, B), and SR ([Ca]_SR, C) as a function of time (in s). For clarity of presentation, [K]_out is raised from 5.4 to 100 mM at t = 100 s for the common-store model and at t = 120 s for the separate-store model. D: signaling pathway after external KCl elevation.

The cascade of events is summarized in Fig. 3D. Membrane depolarization (from −79 to +3 mV above the cytosol) affects net Ca²⁺ entry in two ways. First, it gates opening of VOCa channels that mediate Ca²⁺ 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 Ca²⁺, in exchange for 3 Na⁺, by Na⁺/Ca²⁺ 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 Ca²⁺ 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 Ca²⁺ 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 IP₃R/RyR stores then becomes apparent. In the common-store model, on KCl depolarization SR Ca²⁺ remains depleted by CICR. As a consequence, SERCA pumps continue to transport Ca²⁺ 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. 3A); a plateau [Ca]_cyt elevation is not achieved. In contrast, the separate-store model yields a different prediction (Fig. 3B): the marked rise in [Ca]_md, related to depletion of RyR stores during CICR, stimulates SERCA-driven Ca²⁺ uptake into the IP₃R stores, where the Ca²⁺ content rises. Thus the Ca²⁺ contents of RyR and IP₃R stores simultaneously fall and rise, respectively, in opposition (Fig. 3C). Stated simply, the high [Ca]_md achieved during CICR greatly stimulate microdomain IP₃R-SERCA pumps to load IP₃R stores at the expense of RyR stores. In fact, the IP₃R store Ca²⁺ is predicted to increase so much that the cytosolic IP₃R-SERCA pump reverses direction, carrying Ca²⁺ into the cytosol to elevate [Ca]_cyt (Fig. 4).

Fig. 4.

Effects of external KCl elevation on microdomain and cytosolic Ca²⁺ currents as a function of time (in s), using the separate-store model. At t = 120 s, [K]_out is raised from 5.4 to 100 mM. A: IP₃R current is low because of its inhibition by [Ca]_md elevation while RyR current is large because of [Ca]_md-mediated stimulation. B: [Ca]_md elevation stimulates both IP₃R store- and RyR store-sarco(endo)plasmic reticulum Ca²⁺-ATPase (SERCA) pumps facing the microdomains to the point that they become saturated. NCX, Na⁺/Ca²⁺ exchanger. C: the increase in IP₃R-SR store Ca²⁺ leads to reversal of pumping by IP₃R-SERCA that face the cytosol (negative values mean flux from the SR to the cytosol). The RyR-SERCA facing the cytosol take up Ca²⁺ from the cytosol into the RyR-SR store. D: the large increase in [Ca]_SR-IP3R leads to elevation of Ca²⁺ current through those IP₃R that face the cytosol, and the rise in [Ca]_cyt stimulates elevation of current through the RyR that face the cytosol.

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 AT₁ receptor include IP₃ 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 IP₃ 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 IP₃ stimulates IP₃R-mediated Ca²⁺ release from the SR, while K⁺ channel inhibition results in membrane depolarization above both the cytosol and microdomains. The Ca²⁺ 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 Ca²⁺ from RyR significantly depletes the common SR Ca²⁺ store and saturates microdomain SERCA pumps (Fig. 6). The subsequent increase in Ca²⁺ export via NCX limits the rise in [Ca]_md, while increased pumping of Ca²⁺ into the SR partly repolarizes the membrane and lowers [Ca]_cyt. As in the case of external KCl elevation, the SR Ca²⁺ stores remain depleted so that the rate of Ca²⁺ uptake from the cytosol remains elevated, reducing [Ca]_cyt to ∼30% below pre-ANG II values (Fig. 5).

Fig. 5.

Effects of 90% and 75% K⁺ channel inhibition and 100% increase in IP₃ production at t = 100 s on [Ca]_cyt (A), [Ca]_md (B), and [Ca]_SR (C), with the common-store model.

Fig. 6.

Effects of 90% K⁺ channel inhibition and 100% increase in IP₃ production at t = 100 s on microdomain (A) and cytosolic (B) Ca²⁺ currents, with the common-store model. The scale in B has been reduced for clarity; the cytosolic SERCA and voltage-operated Ca²⁺ (VOCa) currents peak at +73 and −103 pA, respectively.

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 IP₃ production, and 100% increase in G_Ca,L at t = 100 s. The results (not shown) suggest that the qualitative effects on Ca²⁺ concentrations are similar, but the [Ca]_cyt and [Ca]_md peaks are higher because of the larger influx of Ca²⁺ 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 Ca²⁺ avoids continuous depletion. Once [Ca]_md has dropped back to ∼100 nM and the stores are replenished, the ensuing slow, IP₃R-mediated leak of Ca²⁺ 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 IP₃ 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 IP₃ 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 IP₃ production (Fig. 7). The underlying mechanisms can be dissected by simultaneously analyzing Ca²⁺ concentrations (Fig. 8) and Ca²⁺ currents (Fig. 9). As in the common-store model, the sudden increase in IP₃ 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 IP₃R store fills because of stimulation of microdomain IP₃R-SERCA pump activity by high [Ca]_md (Fig. 9). The overall consequence is that a fraction of the Ca²⁺ released from the RyR-SR store is transferred through microdomains to the IP₃R-SR store.

Fig. 7.

Effects of 90% K⁺ channel inhibition and 100% increase in IP₃ production at t = 100 s on Ca²⁺ concentrations in cytosol (A), microdomains (B), and SR stores (C), with the separate-store model. The additional effects of IP₃R and RyR block on [Ca]_cyt are shown in D.

Fig. 8.

Effects of 90% K⁺ channel inhibition and 100% increase in IP₃ production at t = 100 s on [Ca]_cyt (nM), [Ca]_md (nM), and [Ca]_SR (×10² mM), with the separate-store model. Note that the [Ca]_md peak is off-scale. Phase A: RyR-mediated Ca²⁺-induced Ca²⁺ release (CICR) in microdomains, sharp [Ca]_md ascent. Subsequent depletion of RyR stores and filling up of IP₃R stores. Rapid membrane depolarization and repolarization, [Ca]_cyt peak. Phase B: activation of NCX current, sharp [Ca]_md descent. Ca²⁺ content in RyR and IP₃R stores reaches minima and maxima. Phase C: slower [Ca]_md descent. Subsequent replenishment of RyR stores and depletion of IP₃R stores as I_RyR^md and I_IP3R^md-SERCA decrease. Phase D: slowly rising RyR- and IP₃R-mediated Ca²⁺ release into microdomains, progressive [Ca]_md increase. Stabilized Ca²⁺ concentration in cytosol and SR stores.

Fig. 9.

Effects of 90% K⁺ channel inhibition and 100% increase in IP₃ production at t = 100 s on microdomain and cytosolic Ca²⁺ currents, with the separate-store model. A: microdomain RyR current rises up to 60 pA during CICR, whereas microdomain IP₃R current remains low because its inhibition by [Ca]_md elevation. B: CICR-mediated [Ca]_md increase stimulates IP₃R store- and RyR store-SERCA pumps facing the microdomains, as well as Ca²⁺ export from the microdomains via NCX (minimal value, −14 pA). C: immediately after CICR, the IP₃R-SERCA and RyR-SERCA that face the cytosol both take up Ca²⁺ from the cytosol to replenish SR stores. Soon thereafter, the increase in IP₃R-SR store Ca²⁺ leads to reversal of pumping by IP₃R-SERCA that face the cytosol. D: cytosolic IP₃R current rises and falls in conjunction with [Ca]_SR-IP3R. Cytosolic RyR current is stimulated during the depolarization-induced rise in [Ca]_cyt.

In concert with CICR, the sharp [Ca]_md increase stimulates Ca²⁺ 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 Ca²⁺ loads in the RyR and IP₃R 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 Ca²⁺ at a rate that exceeds RyR-mediated losses. Conversely, the IP₃R store is depleted as the rate of Ca²⁺ uptake by the microdomain IP₃R-SERCA pumps declines with decreasing [Ca]_md.

In the next phase, Ca²⁺ concentrations in the cytosol and the SR tend to stabilize while RyR- and IP₃R-mediated Ca²⁺ 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 IP₃R-mediated or RyR-mediated Ca²⁺ release into the microdomains (Fig. 7D). In contrast, block of cytosolic IP₃R or cytosolic RyR reduces the magnitude of oscillations but does not affect their frequency (Fig. 7D). These results indicate that, together, IP₃ and CICR events in the microdomains control the appearance of Ca²⁺ oscillations in the entire cell.

As a baseline, we assumed that store-operated cation channel activity is regulated by the average of Ca²⁺ concentrations in the IP₃ and RyR storage pools. If SOC activity were controlled by [Ca]_IP3R-SR alone, which is lower than the average SR Ca²⁺ concentration during oscillations (Fig. 8), Ca²⁺ 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 IP₃R 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 IP₃-mediated stimulation is 1) an increase in [Ca]_md that is sufficient to reach the CICR threshold and 2) sufficient reloading of SR Ca²⁺ stores following CICR-induced [Ca]_md elevation. In the single-pool model, SR Ca²⁺ 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 [IP₃]_md elevation to release IP₃R-SR store Ca²⁺ into the microdomains. Consistent with that interpretation, the separate-store model predicts that oscillations would not occur if IP₃ production following ANG II rose by 50% instead of 100% (Fig. 10A) 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 IP₃ 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 IP₃ production must be >40% to generate [Ca]_cyt oscillations. Note that because of the feedback activation by [Ca] on [IP₃] production (see parameter α₄ in Eq. 8), [IP₃] is predicted to oscillate in tandem with [Ca] (Fig. 10B).

Fig. 10.

Effects of 90% K⁺ channel inhibition and either 100% or 50% increase in IP₃ production at t = 100 s on microdomain Ca²⁺ (A) and IP₃ (B) concentrations, with the separate-store model. Note that the [Ca]_md peak is off-scale.

The following predictions also underscore the deterministic roles of IP₃, IP₃R, and RyR: in both models, a twofold increase in IP₃ 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 IP₃ 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 IP₃ production by 100% and inhibits K⁺ channels by 90%, predicted oscillations with the separate-store model vanish if ν_IP3R, the Ca²⁺ conductivity of IP₃R (Eq. 7), is decreased from its baseline level of 10 s⁻¹ to below 9 s⁻¹. Oscillations also vanish if ν_RyR, the Ca²⁺ conductivity of RyR (Eq. 3), is decreased from 12 to <10 s⁻¹.

Role of Ca²⁺ 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 Ca²⁺ uptake into the cell and SR also play key roles in the genesis of Ca²⁺ oscillations. If the overall conductance of SOC to both Na⁺ and Ca²⁺ is lowered by 15% or more, simulations suggest that the presumed effects of ANG II (a 2-fold increase in IP₃ production and 90% K⁺ channel inhibition) do not elicit [Ca]_cyt oscillations. Similarly, if the background Ca²⁺ 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 IP₃R leak (see above), Ca²⁺ influx into the cell must contribute to the upward drift of [Ca]_md toward the CICR threshold.

SERCA uptake of Ca²⁺ 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 Ca²⁺ 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 Ca²⁺ 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 IP₃R and RyR conductivity on the frequency and amplitude of oscillations elicited by a twofold increase in IP₃ 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. 11A), meaning that the Ca²⁺ conductivity of RyR predominantly controls the interval between sequential CICR events. Conversely, the amplitude of oscillations depends primarily on the maximum SERCA current (Fig. 11B), that is, the rate at which Ca²⁺ is pumped back into the SR.

Fig. 11.

Effects of parameter variations on the frequency (A) and amplitude (B) of oscillations induced by 90% K⁺ channel inhibition and 100% increase in IP₃ production. Parameters are given relative to their baseline value. Results are obtained for the separate-store configuration. Oscillations persist when Ca²⁺ conductivity of IP₃R (ν_IP3R) equals up to 18 times its baseline value.

Our results also suggest that oscillation frequency and amplitude generally increase with increasing rates of Ca²⁺ 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 Ca²⁺ 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. 12A, along with an expanded histogram (Fig. 12B) 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%).

Fig. 12.

A and B: frequency histograms of 2,062 spontaneous transient inward currents (STICs) recorded in 73 cells exposed to 10 nM ANG II (52). Most values are well below 1 Hz. The reason that the mean is 1.15 Hz is that a few frequencies are as high as 25, so that SD is very large and the mean is skewed away from the predominant frequency, near 0.09 Hz. C and D: amplitude histograms of the same data from Ref. 52. The predominant amplitude was near −10 pA, but the mean is much larger because of rare cells that exhibited large currents.

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.

Fig. 13.

Predicted membrane current during voltage clamp at −80 mV and exposure to angiotensin II (ANG II), with the separate-store model. We assume that exposure to ANG II results in 90% K⁺ channel inhibition and 100% increase in IP₃ production. A: current above cytosol. B: current above microdomains. C: overall current, calculated as the sum of the currents crossing the cytosol and the microdomains, and the leak current (taken as −8 pA, assuming a 10-GΩ seal).

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 IP₃ production and on IP₃R 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 Ca²⁺ 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 IP₃R on the SR stores are variable. Blaustein and colleagues (6) showed that cyclopiazonic acid and serotonin release Ca²⁺ from a RyR-insensitive store in mesenteric myocytes. Similar observations in astrocytes also favor independence of stores occupied by RyR and IP₃R Ca²⁺ release channels (17). More complex configurations may also exist, in which a common store that expresses both IP₃R and RyR resides alongside a store dominated by IP₃R (3) or RyR (14). Regional variation along the microcirculation may occur. Studies in pulmonary arterial myocytes favor coincidence of IP₃R and RyR stores in resistance but not conduit vessels (46).

Investigations into the nature of the cross talk between IP₃R and RyR have yielded similarly varied results. In particular, controversy surrounds the ability of agonist-induced Ca²⁺ release via IP₃R to stimulate adjacent CICR via RyR or IP₃R as a mechanism to propagate intracellular Ca²⁺ 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 IP₃R and separation of the stores on which they reside. We simulated the effects of ANG II by raising the maximum rate of IP₃ 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 IP₃ generation, ANG II stimulated IP₃ 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. 1A) and a separate-store model (Fig. 1B) in which RyR and IP₃R release Ca²⁺ 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 IP₃R or RyR: as one type of store is depleted the other is replenished, thereby sustaining Ca²⁺ 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 Ca²⁺ from stores and 2) entry of Ca²⁺ 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⁺/Ca²⁺ exchange and Ca²⁺-ATPase activity. If those events have reduced intracellular Ca²⁺ 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 Ca²⁺ is achieved (no oscillations) or it reaches CICR levels to generate another burst of RyR-mediated Ca²⁺ release (oscillations). The model readily predicts oscillations when IP₃R and RyR channels are on separate SR stores that face a common cytoplasmic domain because Ca²⁺ released by RyR stores during CICR can “overfill” the adjacent IP₃ 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 IP₃ generation that favors the leak from IP₃R stores is another key determinant of oscillations.

Other predictions of the model also favor separation of RyR and IP₃R 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 Ca²⁺ 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 IP₃R 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 IP₃R-SR stores occurs after KCl elevation, allowing for sustained elevation of [Ca]_cyt.

Many permutations of configurations, IP₃R 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 IP₃R 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 Ca²⁺ ions to IP₃R, oscillations of IP₃ 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, Ca²⁺-ATPase, RyR, and IP₃R at interfaces (PM-microdomain-ER/SR, PM-cytosol-ER/SR) as well as realistic compartmental volumes and Ca²⁺ 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).

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).

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 Ca²⁺.

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 P₁ and P₂ 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 P₁ and P₂.

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,maxP_ATP is a constant.

Calcium-activated potassium current, I_K,Ca.

The current flowing across Ca²⁺-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 α₁-isoform of the Na⁺-K⁺-ATPase pump is expressed only in the PM above the bulk cytosol, whereas the α₂-isoform is confined to the membrane region that is above the microdomains. The respective currents are determined as (26):

(A20)

(A21)

(A22)

(A23)

Na⁺/Ca²⁺ exchanger current, I_NaCa.

We assume that all Na⁺/Ca²⁺ 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 Ca²⁺ pumps is expressed as:

(A33)

Voltage-dependent Ca²⁺ current, I_Ca,L.

The current flowing across voltage-activated Ca²⁺ 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 Ca²⁺ by calmodulin and other Ca²⁺-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 Ca²⁺ 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 Ca²⁺-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 Ca²⁺ binding in the SR and [Calseq·Ca]_SR the concentration of Ca²⁺-bound calsequestrin sites in that compartment. In the separate-store model, we distinguish between the Ca²⁺ binding sites in the SR stores expressing only IP₃R 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 μm²), 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 cm²/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 Ca²⁺ 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 Ca²⁺, 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 Ca²⁺ 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.