## Abstract

To investigate the hypothesis that Na^{+} concentration in subplasmalemmal microdomains regulates Ca^{2+} concentrations in cellular microdomains ([Ca]_{md}), the cytosol ([Ca]_{cyt}), and sarcoplasmic reticulum (SR; [Ca]_{sr}), we modeled transport events in those compartments. Inputs to the model were obtained from published measurements in descending vasa recta pericytes and other smooth muscle cells. The model accounts for major classes of ion channels, Na^{+}/Ca^{2+} exchange (NCX), and the distributions of Na^{+}-K^{+}-ATPase α_{1}- and α_{2}-isoforms in the plasma membrane. Ca^{2+} release from SR stores is assumed to occur via ryanodine (RyR) and inositol trisphosphate (IP_{3}R) receptors. The model shows that the requisite existence of a significant Na^{+} concentration difference between the cytosol ([Na]_{cyt}) and microdomains ([Na]_{md}) necessitates restriction of intercompartmental diffusion. Accepting the latter, the model predicts resting ion concentrations that are compatible with experimental measurements and temporal changes in [Ca]_{cyt} similar to those observed on NCX inhibition. An important role for NCX in the regulation of Ca^{2+} signaling is verified. In the resting state, NCX operates in “forward mode,” with Na^{+} entry and Ca^{2+} extrusion from the cell. Inhibition of NCX respectively raises and reduces [Ca]_{cyt} and [Na]_{cyt} by 40 and 30%. NCX translates variations in Na^{+}-K^{+}-ATPase activity into changes in [Ca]_{md}, [Ca]_{sr}, and [Ca]_{cyt}. Taken together, the model simulations verify the feasibility of the central hypothesis that modulation of [Na]_{md} can influence both the loading of Ca^{2+} into SR stores and [Ca^{2+}]_{cyt} variation.

- electrochemical model
- descending vasa recta
- pericytes
- ionic currents

contractility of resistance vessels is regulated by the variation of intracellular Ca^{2+} concentration ([Ca^{2+}]_{cyt}) in both endothelium and smooth muscle. Within those cells, spatial and temporal variations of [Ca^{2+}]_{cyt} are tightly controlled by exchanging Ca^{2+} with both the extracellular space and intracellular storage sites, including endoplasmic/sarcoplasmic reticulum (SR) and mitochondria (4). Free [Ca^{2+}]_{cyt} concentration is also modulated through chelation by proteins such as calmodulin, calsequestrin, and calreticulin. Measurement of [Ca^{2+}]_{cyt}, with Ca^{2+}-sensitive fluorescent probes such as fura 2 and fluo 4, generally leads to the conclusion that globally averaged resting [Ca^{2+}]_{cyt} lies in the range of 50–100 nM. It is now recognized that highly localized elevations of Ca^{2+} concentration that affect functions of enzymes, ion channels, and transporters occur through quantum transport events that require sophisticated optical sectioning and rapid image acquisition to be observed (11, 21). Ca^{2+}-sensitive fluorescent probes may fail to resolve the spatial and temporal details of such signaling if they are not properly targeted, if their Ca^{2+} affinity differs substantially from microdomain Ca^{2+} concentrations, or if their detection is contaminated by light from outside the focal plane of interest. It seems likely that some highly localized variations of [Ca^{2+}]_{cyt} cannot be experimentally resolved. We reasoned that mathematical simulations might be used to examine the mechanisms that underlie control of compartmental Ca^{2+} concentrations within cells and verify the biophysical feasibility of cytoplasmic-microdomain interactions.

It is known that the relationship between Ca^{2+} stores and [Ca^{2+}]_{cyt} is complex. The extent of filling of Ca^{2+} stores influences the magnitude of agonist-induced Ca^{2+} release (3, 34). In turn, regulation of the filling of Ca^{2+} stores probably occurs in microdomains formed through close association of the SR with the overlying plasma membrane (8, 10). A longstanding hypothesis has been that reduction of microdomain Na^{+}-K^{+}-ATPase activity elevates Na^{+} concentration near the Na^{+}/Ca^{2+} exchanger (NCX) to inhibit its function, thereby directing more Ca^{2+} to fill SR stores, a sequence of events often referred to as the Blaustein hypothesis (6). The α_{2}-α_{4} isoforms of Na^{+}-K^{+}-ATPase are targeted to the microdomains, whereas the α_{1}-isoform is more diffusely distributed for “housekeeping” functions. In rodents, the former but not the latter is sensitive to endogenous ouabain-like factors (OLF) that influence myocyte contractility (5). A pivotal assumption of that hypothesis is that diffusional exchange of microdomain Na^{+} with the “bulk” cytoplasm is limited. Inhibition of Ca^{2+} export by NCX is not the only means through which OLF influence Ca^{2+} signaling. Xie and colleagues (43, 45) elegantly demonstrated that, independent of Na^{+} pump function, ouabain binds to Na^{+}-K^{+}-ATPase and stimulates tyrosine phosphorylation through Src kinase. In LLC-PK_{1} cells, downstream PLCγ1 activation was shown to generate inositol trisphosphate (IP_{3}) and induce [Ca^{2+}]_{cyt} elevation (48).

Motivated by the importance of Ca^{2+} trafficking, we formulated a mathematical simulation of ion concentration changes and IP_{3} release within the bulk cytoplasm and microdomains. As shown in Fig. 1, the model accounts for the characteristics of the channels and transporters that exchange ions among the plasma membrane, SR stores, and cytosol. We sought to determine which factors are most likely to serve as principal determinants of [Ca^{2+}]_{cyt} and delineate the mechanisms through which the bulk cytosol, microdomains, and SR communicate. Taken together, the simulations verify the feasibility of the central hypothesis that modulation of microdomain Na^{+} concentration can influence both the loading of Ca^{2+} into stores and [Ca^{2+}]_{cyt} variation.

## MODEL

### Geometric Parameters

We consider three distinct compartments within the cell: the bulk cytosol (subscript or superscript “cyt”), subplasmalemmal microdomains (subscript or superscript “md”), and the SR (subscript or superscript “sr”).

The average cell capacitance of the descending vasa recta (DVR) pericyte has been measured as 12.1 ± 0.7 pF (13). Assuming a specific membrane capacity of 1 μF/cm^{2} (27), the average capacitive membrane surface area is estimated as 1.21 × 10^{−5} cm^{2}.

Lee et al. (25) found that in vascular smooth muscle, 14.2% of the membrane is closely associated with the superficial ER, and the average distance between the adjacent membranes is 19 nm. This suggests that the membrane area directly over the microdomains is about (0.142)(1.2 × 10^{−5} cm^{2}) = 1.7 × 10^{−6} cm^{2}, and that the total volume of the microdomains (vol_{md}) is about (1.7 × 10^{−6} cm^{2})(19 × 10^{−7} cm) = 3.2 × 10^{−12} cm^{3} ∼ 3 × 10^{−3} pl. Based on cellular geometry, we estimate that the intracellular volume of pericytes (vol_{cyt}) is ∼0.5 pl. Using the same volume ratios as Yang et al. (46), we therefore assume that the intracellular volume available to free Ca^{2+} (vol_{cyt, Ca}) is (0.7)(0.5 pl) = 0.35 pl, and that the volume of the entire SR compartment (vol_{sr}) is (0.14)(0.5 pl) = 0.07 pl.

### Ionic Channel Distributions

Our representative model is shown in Fig. 1. We assume the following distribution of channels, pumps, and exchangers (corresponding currents are denoted in parentheses).

Uniformly distributed over the plasma membrane are inward rectifier potassium channels (*I*_{K, ir}), delayed rectifier potassium channels (*I*_{K, v}), ATP-activated potassium channels (*I*_{K, ATP}), calcium-activated potassium channels (*I*_{K, Ca}), voltage-activated sodium channels (*I*_{VONa}), calcium pumps (*I*_{Ca, P}), and L-type voltage-dependent calcium channels (*I*_{Ca, L}).

We assume that the α_{2}-isoforms of Na^{+}-K^{+}-ATPase pumps (*I*_{NaK, α2}) are expressed exclusively in the region of the cell membrane that lies directly above the microdomains (23), whereas the α_{1}-isoforms (*I*_{NaK, α1}) are restricted to the region of the cell membrane directly above the bulk cytosol (10).

We also assume that all the Na^{+}/Ca^{2+} exchangers (*I*_{NaCa}) are localized above the microdomains (8, 22, 30).

We assume that store-operated, nonselective cation channels (*I*_{SOC}) are present above both bulk cytosol and microdomains. The SOC channels are assumed to be permeable to both Na^{+} and Ca^{2+} ions.

The interfaces between the SR and cytosol and the SR and microdomains are both populated by Ca^{2+}-ATPase pumps (SERCA; *I*_{SERCA}), ryanodyne receptors (RyR; *I*_{RyR}), and inositol trisphosphate receptors (IP_{3}R; *I*_{IP3R}). We assume that the proportion of SERCA pumps at the SR-microdomain interface is 14.2%.

### Ionic Currents

We distinguish between the transmembrane potential above the bulk cytosol (*V*_{m}^{cyt}) and that over the microdomains (*V*_{m}^{md}) and among the concentrations of potassium, sodium, and calcium in the bulk cytosol and in the microdomains. As described above, we assume that 85.8% of the cell membrane lies directly above the cytosol (i.e., f^{cyt} = 0.858) and 14.2% above the microdomains (i.e., f^{md} = 0.142). Whenever possible, we use the pericyte data obtained by Pallone and colleagues, and the equations given in the model of Yang et al. (46) as it was developed for vascular smooth muscle cells.

The convention adopted in this study is that the exit of a positive charge from the cell is a positive current. Conversely, the entry of a positive charge into the cell is a negative current.

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

For potassium, sodium, and calcium, the Nernst potential is calculated based on the concentration difference between the extracellular compartment and the cell interior. In each case, the interior ion concentration is that of the compartment where the simulated channels reside: (1) The background currents are then calculated as: (2)

#### I_{K, ir}.

The current flowing across K_{ir} channels lying above the bulk cytosol and above the microdomains is expressed as (46): (3a) (3b)

#### I_{K, v}.

The current flowing across K_{v} channels lying above each compartment (*j* = cyt, md) is calculated as (46): (4) (5) (6) (7) (8) (9) (10) 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. The parameter *P̄*_{Kv}, which is voltage dependent, represents the steady-state value of *P*_{1} and *P*_{2}.

#### I_{K,ATP}.

Since the model of Yang et al. (46) doesn't include K_{ATP} channels, we employ the formulation of Shaw and Rudy (40), which does not consider the kinetics of channel opening and closing: (11a) (11b) where *P*_{ATP} is the fraction of K_{ATP} channels available at a given ATP concentration and is given by a Hill equation. We assume that the ATP concentration remains fixed, so that *P*_{ATP} is a constant in our model.

#### I_{K, Ca}.

*I*_{K, Ca} is taken to depend on the calcium concentration in the compartment where the channels are located. In each compartment (*j* = cyt, md), *I*_{K, Ca} is calculated as (46): (12) (13) (14) (15) (16) 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}.

#### I_{NaK}.

As described above, we assume that the α_{1}-isoform of the pump is expressed only in the plasma membrane 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 follows (27): (17) (18) (19) (20)

#### I_{VONa}.

Based on pericyte experimental data (51), we use a model with a single inactivation time constant: (21) (22) (23) (24) (25) 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.

#### I_{NaCa}.

As described above, we assume that all Na^{+}/Ca^{2+} exchangers are preferentially localized in the membrane regions that are close to the SR. We use the equations of Shannon et al. (39), which account for the asymmetric affinities of the exchangers on the internal and external sides of the membrane: (26) (27) (28) (29) (30)

#### I_{Ca, P}.

The cytosolic and microdomain *I*_{Ca, P} currents are expressed as (46): (31)

#### I_{Ca,L}.

The currents flowing across voltage-activated Ca^{2+} channels are calculated as (46): (32) (33) (34) (35) (36) (37) (38) (39) 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, respectively; the symbols *d̄*_{L} and *f̄*_{L} denote the steady-state values of *d*_{L} and *f*_{F}.

#### I_{SOC}.

The general form of the equation for the SOC Ca^{2+} current is that proposed by LeBeau et al. (24): (40) This expression accounts for the fact that a decrease in SR Ca^{2+} load stimulates SOC activity, as observed experimentally. We assume a Ca:Na permeability ratio *P*_{Ca}^{SOC}/*P*_{Na}^{SOC} = 8 (2, 15) and use the Goldman-Huxley-Katz current equation (19) to relate the SOC Na^{+} and Ca^{2+} currents in compartment *j*: (41) In the absence of experimental measurements, the maximum Ca^{2+} conductance of cytosolic SOC is taken as 20 pS, so as to yield a three- to fourfold increase in [Ca^{2+}]_{cyt} when all SERCA pumps are inhibited (3). The maximum Ca^{2+} conductance of microdomain SOC is taken as 3.5 pS to obtain agreement with measurements of microdomain-to-cytosol Na^{+} concentration gradients (see below).

#### SR SERCA pumps.

We assume that the SERCA pumps are located at both the cytosol-SR interface and the microdomains-SR interface and that the uptake current depends on concentrations on the internal and external sides of the SR membrane (39): (42)

#### RyR.

We distinguish between the RyRs situated at the cytosol-SR interface and those situated at the microdomains-SR interface. We use the Keiser-Levine model as modified by Jafri et al. (20), which takes into account the fact that RyR can be exposed to large [Ca]_{md} values, in excess of 10 μM. We assume four different states of RyRs, two open states (fractions *P*_{O1} and *P*_{O2}, respectively) and two closed states (fractions *P*_{C1} and *P*_{C2}, respectively). The multistate kinetic model of RyR activation yields the following differential equations for each compartment: (43) (44) (45) (46) The RyR release current is then given by: (47) We assume that the maximum RyR Ca^{2+} permeability (that is, ν_{RyR, max}) is the same at the SR-cytosol and SR-microdomain interfaces.

#### IP_{3}R.

We use the kinetic model for IP_{3}R developed by De Young and Keiser (16). The model 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*), the other for inhibition (*site 3*). The fraction of receptors in state *S*_{i1i2i3} is denoted by *x*_{i1i2i3}(*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 one IP_{3} and one activating Ca^{2+}) for the IP_{3}R channel to be open. Assuming that the binding kinetics of IP_{3} and the activation of the receptors by Ca^{2+} are fast processes relative to Ca^{2+} inhibition, the number of receptor subunit states in the model is reduced from eight to four. Accordingly, the conservation equations for the fractions *x*_{0i2i3}^{j} at the SR-microdomain interface (*j* = md) or at the SR-cytosol interface (*j* = cyt) can be written as: (48) (49) (50) (51) Assuming rapid equilibrium for IP_{3} binding, (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: (52) To determine the concentration of IP_{3} in a given compartment *j* (*j* = cyt or md), denoted by [IP_{3}]_{j}, we also use the model of De Young and Keizer (16), which includes Ca^{2+} feedback on the concentration of IP_{3}: (53a) where *I*_{r} is the rate constant for loss of IP_{3} (taken as 1 s^{−1}), ν_{4} is the maximum rate of IP_{3} production, and *k*_{4} is the dissociation constant (taken as 1.1 μM). If α_{4} = 0, there is no Ca^{2+} feedback; the other extreme is α_{4} = 1. We assume a baseline value α_{4} = 0.5. Since the value of ν_{4} is given in s^{−1} in the study of De Young and Keiser (16), we have modified the previous equation as follows: (53b) where [IP_{3}]^{eq} is the equilibrium concentration of IP_{3}, taken as 240 nM (16). The baseline value of ν_{4} is chosen such that the basal level of IP_{3} in the cytosol is [IP_{3}]_{cyt} = 240 nM. With α_{4} = 0.5, this implies that ν_{4} = 1.85 s^{−1}.

The IP_{3}R release current in compartment *j* (*j* = cyt, md) is calculated as: (54) where ν_{IP3R, max} is the maximum Ca^{2+} permeability through the IP_{3} receptors. In the absence of data, we assume that ν_{IP3R, max} is the same at the SR-cytosol and SR-microdomain interfaces.

### Calcium Buffers

As in the study of Jafri et al. (20), we account for chelation of Ca^{2+} by the high- and low-affinity binding sites of troponin in the bulk cytosol, calmodulin in both the microdomains and the cytosol, and calsequestrin in the SR. As described immediately below, rather than relying on rapid buffering approximations, we use full equations to describe the effect of those buffers on free Ca^{2+} concentration.

We model the binding of calcium to calmodulin in both the bulk cytosol and the microdomains as follows. Let [CM]_{j} ^{tot} denote the total concentration of calmodulin binding sites available for Ca^{2+} binding in compartment *j* (*j* = cyt, md), and [CM·Ca]_{j} the concentration of calcium-bound calmodulin sites in that compartment. Assuming that calcium buffering can be described as first-order dynamic processes (46), we have: (55) Similarly, let [Ltr]_{cyt}^{tot} and [Htr]_{cyt}^{tot} denote the total cytosolic concentration of low-affinity site troponin and high-affinity site troponin, respectively. If [Ltr·Ca]_{cyt} and [Htr·Ca]_{cyt} represent the cytosolic concentration of calcium-bound low affinity sites and calcium-bound high-affinity sites, respectively, buffering by troponin can be described by: (56) (57) Last, the buffering of calcium by calsequestrin sites in the SR is modeled as: (58) where [Calseq]_{sr}^{tot} denotes the total concentration of calsequestrin binding sites available for Ca^{2+} binding in the SR, and [Calseq·Ca]_{sr} is the concentration of calcium-bound calsequestrin sites in that compartment.

### Electrodiffusive Flux from Microdomain to Cytosol

The electrodiffusive flux of ion *i* (valence *z*_{i}) is calculated as follows, such that *J*_{i, diff} yields a positive current into the cytosol when cations flow down their electrochemical gradient from microdomain to cytosol: (59) (60) (61) where *A* is the cross-sectional area at the interface between the microdomains and the cytosol, *D*_{i} is the diffusivity of ion *i*, *L* is the distance from the center of the microdomains to the center of the cytosol, and *h* is a hindrance factor, which lumps together steric and charge-related effects.

The distance *L* is estimated as half the width of a DVR pericyte, 0.5 μm. The area *A* is roughly approximated as (π*d*)δ, where π*d* is the vessel circumference that is enveloped by the pericyte (*d* = 13 μm), and δ is the gap between the plasmalemma and the superficial SR (19 nm). The whole-cell diffusivity of calcium is taken as 0.3 × 10^{−5} cm^{2}/s (35). 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, respectively, i.e., the ratios of the diffusivities in dilute solution (19, 36).

### Change in Internal Concentrations

(62) (63) (64) (65) (66) (67) (68)

### Change in Membrane Potential

The sum of the currents flowing into the bulk cytosol is given by: (69) The sum of the currents flowing into the microdomains is given by: (70) The time dependence of *V*_{m}^{cyt} and *V*_{m}^{md} is given by: (71) (72)

### Numerical Methods

The complete model consists of 44 state variables, the initial values of which are given in Table 1. Parameters related to cell geometry, ionic currents, and buffers are summarized in Tables 2, 3, and 4, respectively. The system of ordinary differential equations was programmed with MATLAB and solved numerically on a personal computer with an Intel-based processor.

## RESULTS

The objective of this study was to investigate the main determinants of ion concentrations in the bulk cytosol, microdomains and SR, as well as the mechanisms by which Ca^{2+} is transferred between those compartments. Our model differs from previous theoretical studies of Ca^{2+} signaling in that we focus on the role of sequestered microdomains in the modulation of [Ca]_{cyt} signaling and examine the feasibility of the Blaustein hypothesis that interaction between Na^{+}-K^{+}-ATPase and NCX therein modulates SR Ca^{2+} content and [Ca]_{cyt} (6). The microdomains represent a very small fraction (0.6%) of cell volume and are significantly isolated from the bulk cytosol; we therefore distinguish among the K^{+}, Na^{+}, and Ca^{2+} concentrations in the cytosol and microdomains. We account for the specific distributions of Na^{+}-K^{+}-ATPase α_{1}- and α_{2}-isoforms and NCX in the plasma membrane. We also simulate Ca^{2+} release from SR stores via both the RyR and IP_{3}R.

### Model Validation

We first sought to validate our model by comparing predictions with available experimental data. As reviewed by Meldolesi and Pottan (29), reported values of [Ca^{2+}]_{sr} vary between 5 μM and 5 mM. In particular, they have been measured as 160 μM in unstimulated myocytes (8), and as 500 μM in hepatocytes (38). Our model predicts an equilibrium value of 258 μM, well within that broad experimental range.

#### Effect of inhibition of K_{ATP} or K_{ir} channels.

To mimic the effect of the K_{ATP} blocker glybenclamide (Glb), we simulated the complete inhibition of K_{ATP} channels. The predicted effect is to increase *V*_{m}^{cyt} by +6.0 mV; the average Glb-induced depolarization measured by Cao et al. (13) in DVR pericytes was +4 mV. Similarly, we simulated the inhibition of K_{ir} channels and predicted a +1.0-mV depolarization. When Cao et al. (14) inhibited K_{ir} channels with 10 and 30 μM Ba^{2+}, DVR pericytes were depolarized from −68 to −63 and −57 mV, respectively. The discrepancy may due to the fact that Ba^{2+} becomes internalized and substitutes for Ca^{2+} to affect other cellular processes besides K_{ir} activity.

#### Depolarization induced by K^{+} substitution.

We then examined the membrane depolarization induced by K^{+} substitution. As observed by Pallone et al. (31), an increase in the extracellular K^{+} concentration from 5 to 100 mM raises the transmembrane potential to a value that is ∼1 mV below the theoretical Nernst equilibrium potential for the K^{+} ion (*E*_{K}). Similarly, our results indicate that increasing [K]_{out} from 5.4 to 100 mM raises *V*_{m}^{cyt} from −71.2 to within 2 mV of *E*_{K}^{cyt}.

### Diffusion Between Cytosol and Microdomains

A critical feature of this model is the assumption that Na^{+} and other ions do not freely diffuse between the microdomains and bulk cytosol. Near-membrane Na^{+} gradients have been observed in cardiac myocytes by Wendt-Gallitelli et al. (44), verifying that plausibility. The average resting [Na]_{md} was measured as 18 ± 4.5 mM in the subplasmalemmal regions vs. 10.5 ± 4.3 mM in the center of the cell. As described below, without restriction of diffusion, maintenance of such microdomain-to-cytosol Na^{+} gradients cannot be sustained, and the model fails to predict the interactions between SR stores and [Ca^{2+}]_{cyt} that have been observed experimentally.

If diffusion between the microdomains and the cytosol is not significantly restricted, the hindrance factor *h* (*Eq. 61*) should be on the order of unity. Conversely, if microdomains are completely isolated, the hindrance factor is zero. ⇑Table 5 shows predictions of resting concentrations for different values of *h*. Since microdomain concentrations depend significantly on SOC maximum conductances, whenever possible we adjusted *G*_{Ca, SOC,}^{md}_{max} so as to yield [Na]_{md} ∼15 nM. This was not possible, however, for *h* ≤ 1 × 10^{−3}.

If *h* > 10^{−2}, the model predicts that the difference between [Na]_{md} and [Na]_{cyt} does not exceed 3 mM, independently of the value of *G*_{Ca, SOC,}^{md}_{max} (all results not shown). It is only when *h* ≤ 10^{−2} that [Na]_{md} becomes significantly larger than [Na]_{cyt}. When *h* ≤ 10^{−4}, the predicted [Na]_{cyt} is below 5 mM, given that the Na^{+} electrodiffusive flux is small and there is no NCX above the cytosol. This suggests that the microdomains and the bulk cytosol must be significantly, but not entirely, isolated. To predict a [Na]_{md}-to-[Na]_{cyt} concentration gradient that is consistent with experimental observations (44), the baseline value of *h* is taken as 2.5 × 10^{−3} in the remainder of this study.

### Contribution of SERCA Pumps, SR Receptors, and Ion Channels to Resting Values

To understand the specific contribution of the channels, pumps, and receptors involved in Ca^{2+} signaling, we examined the selective effects of removing each one.

#### Role of SERCA pumps.

Figure 2 shows the selective effect of eliminating those SERCA pumps located at the SR-cytosol interface. As summarized in Table 1, the resting values in the baseline case (before SERCA inhibition) are *V*_{m}^{cyt} = −71.2 mV, *V*_{m}^{md} = −73.0 mV, [K]_{cyt} = 97 mM, [K] _{md} = 104 mM, [Na]_{cyt} = 5.9 mM, [Na] _{md} = 15.1 mM, [Ca]_{cyt} = 92 nM, [Ca]_{md} = 156 nM, and [Ca]_{sr} = 258 μM.

Our model predicts that removing cytosolic SERCA pumps raises [Ca]_{cyt} by a factor of 3.8, and reduces [Ca]_{sr} by ∼80% (Fig. 2, *A* and *B*). [Na]_{cyt} increases because the decrease in [Ca]_{sr} stimulates Na^{+} entry via SOCs (Fig. 2*C*). The depletion of SR Ca^{2+} stores reduces IP_{3}R- and RyR-mediated Ca^{2+} release into the microdomains, thereby lowering [Ca]_{md}. The NCX current above the microdomains briefly changes sign as NCX transports in “reverse mode,” with Ca^{2+} entry, Na^{+} export, and consequent reduction in [Na]_{md}. Although the decrease in [Na]_{md} reduces microdomain Na^{+}-K^{+}-ATPase activity (i.e., decreases K^{+} import and Na^{+} export), [K]_{md} increases slightly because of membrane hyperpolarization.

The changes in [Na]_{cyt} and [K]_{cyt} occur over ∼100–200 s and are smooth and slow compared with the variations in [Ca]_{cyt} and microdomain concentrations. Indeed, the microdomain-to-cytosol volume ratio is 3:500, so that the currents through cytosolic K^{+} channels and Na^{+}-K^{+}-ATPase slowly adjust to these variations.

As illustrated in Fig. 2*C*, Ca^{2+} release into the cytosol through IP_{3}R decreases with increasing [Ca]_{cyt}. Our model is based on that of De Young and Keiser (16), which describes Ca^{2+} release via IP_{3}R. Their model predicts a biphasic dependence of IP_{3}R activity on [Ca]. When [Ca] is less than ∼0.25 μM, [Ca] elevation increases the open probability of IP_{3}R; above 0.25 μM, however, elevating [Ca] decreases the open probability.

Shown in Fig. 2*D* are the individual fluxes that contribute to [Ca]_{cyt} variations (see *Eq. 66*). The relaxation times of the buffer reactions (<1 s) are significantly faster than those of the other currents (note the different scale of the *x*-axis). The sharp peaks corresponding to CICR through IP_{3}R and RyR occur within 2 s following cytosolic SERCA inhibition (Fig. 2*B*), whereas the other Ca^{2+} currents reach equilibrium within tens of seconds.

We then investigated the effects of selectively blocking those SERCA pumps located at the SR-microdomain interface. Figure 3 shows that the SERCA inhibition raises [Ca]_{md} from 156 to 321 nM and lowers [Ca]_{sr} by ∼30%. The resulting stimulation of microdomain NCX and SOC significantly raises [Na]_{md}. The consequent export of [Na]_{md} in exchange for extracellular K^{+} by Na^{+}-K^{+}-ATPase is not sufficient to raise [K]_{md} above preinhibition levels because of membrane depolarization. The depletion of SR Ca^{2+} stores also increases Ca^{2+} and Na^{+} entry through cytosolic SOCs, thereby raising [Na]_{cyt}.

Numerous studies have shown that addition of nonselective SERCA pump inhibitors (e.g., cyclopiazonic acid, thapsigargin) leads to very large increases in cytoplasmic Ca^{2+}. Our model predicts that blocking both cytosol and microdomain SERCA pumps raises [Ca]_{cyt} to ∼400 nM, that is, a factor of 4, in general agreement with experimental measurements in vascular smooth muscle and endothelia (2, 3, 33).

#### Roles of RyR and IP_{3}R.

We then examined the roles of RyR and IP_{3}R at the SR-microdomain and SR-cytosol interfaces. Since the baseline values of *I*_{RyR}^{md} and *I*_{RyR}^{cyt} are small (∼0.25 and 0.16 pA, respectively), removing RyR has a negligible effect.

Removing IP_{3}R, however, has significant effects on resting Ca^{2+} and Na^{+} concentrations. Removing IP_{3}R at the SR-cytosol interface (Fig. 4*A*) decreases [Ca]_{cyt} by 15% and raises [Ca]_{sr} by ∼15%. The [Ca]_{sr} elevation augments RyR- and IP_{3}R-mediated Ca^{2+} release into the microdomains, thereby raising [Ca]_{md} by ∼10%. Stimulation of NCX above the microdomains also raises [Na]_{md}.

In the absence of IP_{3}R at the SR-microdomain interface (Fig. 4*B*), the resting value of [Ca]_{md} is about half that in the baseline case, and [Ca]_{sr} is ∼40% higher. As a consequence, IP_{3}R release more Ca^{2+} from the SR into the cytosol, raising [Ca]_{cyt}. Reduction of [Ca]_{md} leads to a reduction of the NCX current above the microdomains so that [Na]_{md} falls in parallel.

#### Inhibition of NCX and SR Ca^{2+} loading.

Under resting conditions, “forward mode” NCX extrudes Ca^{2+} stoichometrically coupled to entry of three Na^{+} ions. Thus inhibiting NCX raises [Ca]_{cyt} and [Ca]_{md}, and lowers [Na]_{cyt} and [Na]_{md} (Fig. 5). It is a central tenant of the Blaustein hypothesis that elevation of [Na]_{md} favors reduction of Ca^{2+} export by NCX to permit loading of the SR with Ca^{2+} (6, 9, 10). Figure 5 verifies the feasibility of that contention. Note that [Ca]_{sr} rises as SERCA uptake increases after NCX is blocked.

#### Effect of lowering [Na]_{out}.

In the simulations shown in Fig. 6, [Na]_{out} is decreased sequentially from 140 to 125, 100, and 50 mM every 150 s. Consequently, the NCX current is sharply reduced, thereby lowering [Na]_{cyt} and [Na]_{md} and increasing [Ca]_{md} and [Ca]_{cyt} (and thus [Ca]_{sr}, via SERCA pumps). The NCX-mediated increase in [Ca]_{md} and [Ca]_{cyt} evokes a rapid CICR via RyR and IP_{3}R, which gives rise to [Ca]_{md} and [Ca]_{cyt} transients. After the initial CICR, SERCA activation partially lowers [Ca]_{md} and [Ca]_{cyt} toward plateau levels that continue to exceed the prior baseline where [Na]_{out} = 140 mM. Such a “peak and plateau” pattern of [Ca]_{cyt} elevation after reduction of [Na]_{out} has been experimentally observed (34).

#### Inhibition of Na^{+}-K^{+}-APTase.

Shown in Fig. 7 are the effects of blocking either the α_{2}-isoform of the Na^{+}-K^{+}-APTase only (Fig. 7*A*), or both α_{1}- and α_{2}-isoforms (Fig. 7*B*). Complete inhibition of the microdomain Na^{+}-K^{+}-ATPase (α_{2}) current reduces NCX activity, thereby raising [Ca]_{md} by ∼20% and subsequently elevating [Ca]_{cyt} by a comparable factor. When the cytosolic Na^{+}-K^{+}-ATPase (α_{1}) is inhibited as well, [Na]_{cyt} rises progressively, leading to a secondary increases in [Na]_{md} (via *J*_{Na, diff}) and in [Ca]_{md} (via NCX). The subsequent elevation of [Ca]_{sr} raises [Ca]_{cyt} by a factor of ∼2.3. These simulations support the general contention of the Blaustein hypothesis that blockade of Na^{+}-K^{+}-ATPase translates to changes in intracellular calcium through modulation of NCX activity.

#### Role of SOC channels.

The effects of removing store-operated channels from the plasmalemma above the bulk cytosol and the microdomains are illustrated in Fig. 8, *A* and *B*, respectively. The model predicts that inhibition of cytosol SOC causes both [Ca]_{cyt} and [Na]_{cyt} to drop, leading to secondary reduction of [Ca]_{sr} via cytosolic SERCA pumps. [Ca]_{md} falls in parallel with [Ca]_{sr}, as the currents across microdomain SERCA pumps, IP_{3}R, and RyR adjust to the variations in SR Ca^{2+} stores. The decrease in [Na]_{md} is coupled to that in [Ca]_{md} via NCX.

Inhibition of microdomain SOC causes [Ca]_{md} and [Na]_{md} to drop, thereby reducing [Ca]_{sr} via microdomain SERCA pumps (Fig. 8*B*). As receptor-mediated Ca^{2+} release into the cytosol is subsequently reduced, [Ca]_{cyt} is lowered by ∼10%.

### Sensitivity Analysis

To determine which parameter variations have the greatest effect on model predictions, we performed a sensitivity analysis, focusing on those most likely to affect Na^{+} and Ca^{2+} concentrations. The results are summarized in Table 6.

Important parameters include those that govern the rate of SERCA Ca^{2+} uptake (Fig. 3), the rate of IP_{3}R-mediated Ca^{2+} release (Fig. 4), the NCX current (Fig. 5), and the conductance of the SOC channels (Fig. 8). The rate of SERCA Ca^{2+} uptake is a function of the maximum rate, *I*_{SERCA, max}, and of the saturation constants *K*_{mf} and *K*_{mr} and the Hill coefficient H (*Eq. 42*). Variations in the latter two parameters did not affect intracellular concentrations very significantly (results not shown). However, a factor of 2 increase in *K*_{mf} is predicted to lower [Ca]_{sr} by ∼40%, to raise [Ca]_{cyt} from 92 to 127 nM, and to increase [Na]_{cyt} from 5.9 to 6.9 mM (Table 6). Conversely, a factor of 2 decrease in *K*_{mf} reduces [Ca]_{cyt} to 69 nM.

Our simulations suggest that the parameters that determine the rate of IP_{3}R-mediated Ca^{2+} release also significantly affect Na^{+} and Ca^{2+} concentrations. A 100% increase in the equilibrium value of [IP_{3}] ([IP_{3}]^{eq}; see *Eq. 53b*) is predicted to raise the resting value of [Ca]_{md} by 75% and reduce that of [Ca]_{sr} by ∼60%. Conversely, a 50% decrease in [IP_{3}]^{eq} reduces [Ca]_{md} by 40% and raises [Ca]_{sr} by ∼50%. In the baseline case, we assumed that Ca^{2+} exerts a moderate feedback effect on the production of IP_{3}. By following the approach of De Young and Keiser (16), the parameter that characterizes this interaction is denoted by α_{4} (see *Eq. 53b*). To investigate the effects of [Ca] feedback on Ca^{2+} signaling, we compared the resting [Ca] values when α_{4} = 0 (no feedback), α_{4} = 0.5 (baseline case), and α_{4} = 1 (maximal feedback). The related parameter ν_{4} was adjusted in each case so that the resting value of [IP_{3}]_{cyt} was equal to 240 nM. In the baseline case (α_{4} = 0.5), [Ca]_{md} is greater than [Ca]_{cyt}, so that the resting value of [IP_{3}]_{md} (250 nM) is higher than that of [IP_{3}]_{cyt}. In the absence of feedback (α_{4} = 0), [IP_{3}]_{md} = [IP_{3}]_{cyt} = 240 nM, and the flux of Ca^{2+} released by IP_{3}R receptors at the SR-microdomain interface (*I*_{IP3R}^{md}) is lower than in the baseline case (4.3 vs. 4.6 pA). Consequently, [Ca]_{md} is slightly lower and [Ca]_{sr} is higher than at baseline. When α_{4} = 1, so that maximal IP_{3} production is achieved through Ca^{2+} stimulation, high [Ca]_{md} augments IP_{3} production in the microdomains more than [Ca]_{cyt} does in the cytosol. As a consequence, the resting value of [Ca]_{md} is predicted to rise by a factor of ∼9.

In contrast, simulations show that a twofold increase or decrease in maximum Ca^{2+} permeability of RyR (ν_{RyR, max}) does not significantly affect Ca^{2+} concentrations.

Another essential determinant of intracellular calcium concentrations is the maximum current that can be achieved by Na^{+}-K^{+}-ATPase, *I*_{NaK, max}. Increasing *I*_{NaK, max} enhances Na^{+} extrusion from the cell, lowering its concentration. This favors an increase in Ca^{2+} extrusion by NCX, significantly lowering intracellular calcium concentrations. Conversely, decreasing the Na^{+}-K^{+}-ATPase current reduces the NCX current and raises [Ca]_{md} and [Ca]_{cyt}. The Na^{+}-K^{+}-ATPase α_{2}:α_{1} ratio has not been measured in vascular smooth muscle cells, to the best of our knowledge. Our model suggests that this parameter also affects Na^{+} and Ca^{2+} resting concentrations (Table 6), albeit to a lesser extent than *I*_{NaK, max}.

Literature estimates of the maximum current through NCX, *I*_{NaCa, max}, vary by several orders of magnitude; Luo and Rudy (27) use a value of 2,000 μA/μF, whereas the corresponding parameter in the study of Shannon et al. (39) is equal to 14.1 μA/μF. We therefore assumed an intermediate value, 200 μA/μF. Our simulations indicate that resting concentrations and *I*_{NaCa} are relatively insensitive to variations in *I*_{NaCa, max} (Table 6). A 10-fold increase has no significant effect, whereas a 10-fold decrease lowers [Na] and raises [Ca] by <5–10%. Analysis of each of the terms in *Eq. 26* reveals that these slight changes in [Na] and [Ca] are nevertheless sufficient to increase the numerator on the right-hand side of the equation by a factor of ∼6, which partly compensates for the 10-fold decrease in *I*_{NaCa, max}. We also performed simulations in which we assumed that some fraction of NCX is expressed above the bulk cytosol. As expected, [Na]_{cyt} and [Ca]_{cyt} are then predicted to be higher and lower, respectively, than in the baseline (Table 6).

The distribution and maximum conductance of SOC have not been determined experimentally in DVR pericytes. A 100% increase in *G*_{SOC, Ca, max}^{cyt} is predicted to raise [Na]_{cyt} from 5.9 to 7.2 mM and [Ca]_{cyt} from 92 to 118 nM. The subsequent elevation in [Ca]_{sr} allows [Ca]_{md} to rise as well, which stimulates NCX and increases Na^{+} import into the microdomains (Table 6). A 100% increase in the SOC Na:Ca permeability ratio raises the concentrations of not only Na^{+} but also Ca^{2+}, by reducing the activity of NCX (*I*_{NaCa} = −0.70 pA in the baseline, and −0.52 pA if *P*_{Na}^{SOC}:*P*_{Ca}^{SOC} is doubled).

Equally uncertain is the distribution of SERCA pumps. In the baseline case, we assumed that the fraction of SERCA pumps at the SR-microdomain interface is 14.2% (i.e., the fractional membrane area above the microdomains). As shown in Table 6, this parameter has a significant effect on [Ca]_{md}, and therefore on [Na]_{md}: a 50% decrease is predicted to raise [Ca]_{md} by >20%. However, corresponding variations in [Ca]_{cyt} and [Ca]_{sr} remain smaller.

We also varied the fractional membrane area above the microdomains (and adjusted the microdomain volume accordingly). Our simulations suggest that an increase in f^{md} reduces *V*_{m}^{md} (i.e., *V*_{m}^{md} becomes more negative), thereby lowering [Ca]_{md}. The subsequent reduction in NCX activity lowers [Na]_{md} as well as the microdomain-to-cytosol Na^{+} electrodiffusive flux (*J*_{Na, diff}); hence the [Na]_{cyt} reduction. Conversely, a decrease in f^{md} raises both *V*_{m}^{md} (from −73.0 to −72.2 mV) and [Ca]_{md}. As [Na]_{md} subsequently increases via NCX, the driving force for *J*_{Na, diff} is significantly augmented, and [Na]_{cyt} increases, too (Table 6).

## DISCUSSION

Cardiotonic steroids such as digitalis and ouabain inhibit transport by Na^{+}-K^{+}-ATPase by binding to the first extracellular NH_{2}-terminal loop of the α-subunit. The observation that the ouabain binding site is conserved in evolution, and that ouabain enhances myocyte contractility, prompted Blaustein and colleagues (6, 9, 10) to hypothesize that inhibition of Na^{+} export is coupled to elevation of intracellular Ca^{2+} through NCX. The demonstration that endogenous ouabain-like factors (OLF) are synthesized by the adrenal gland and hypothalamus and circulate in nanomolar concentration lent credence to the existence of an important physiological role (18, 47). An observation that complicated acceptance of the hypothesis is that the α_{1}-isoform of Na^{+}-K^{+}-ATPase, which performs a general housekeeping function in rodents, is ouabain insensitive. In contrast, other rodent isoforms (α_{2}–α_{4}) retain ouabain sensitivity but are less abundant (5, 10, 26). That paradox was explained by postulating that the ouabain-sensitive isoforms are targeted to cellular microdomains that accommodate Ca^{2+} trafficking between the plasma membrane and cellular stores. Colocalization of α_{2} Na^{+} pumps with SR protrusions that abut the plasma membrane supports that contention (7, 10). More recently, the existence of an NH_{2}-terminal sorting motif that tethers α_{2} Na^{+} pumps to microdomains has been defined (42).

Much functional evidence tends to confirm that Ca^{2+} signaling is modulated through α_{2}–α_{4} isoforms of Na^{+}-K^{+}-ATPase. Their inhibition or reduction of expression enhances agonist-induced Ca^{2+} release from cellular stores in smooth muscle and endothelium (3, 34) and increases resting [Ca]_{cyt} and myogenic tone in isolated mesenteric arterioles (50). A hurdle to acceptance of the hypothesis that reduction of Na^{+}-K^{+}-ATPase activity leads to inhibition of Ca^{2+} export from myocytes is the need for Na^{+} concentration to rise near the cytoplasmic face of NCX. Stated another way, the putative microdomain within which α_{2} Na^{+} pumps associate with NCX must be sequestered in such a manner that diffusional exchange of Na^{+} and other ions with the bulk cytoplasm is severely limited (7, 10). Accepting that the latter can occur, we felt it relevant to mathematically simulate the putative system associated with the Blaustein hypothesis (6, 9), based on cellular geometry and the known characteristics of channels, transporters, and Ca^{2+} binding proteins, to assess its feasibility. The model strongly supports the concept that changes in NCX activity and [Na]_{md} affect loading of SR stores with Ca^{2+} and that [Ca]_{sr} changes can modulate both resting and agonist-stimulated levels of [Ca]_{cyt}.

To generate the model, we incorporated the seminal approaches of prior investigators (27, 40), several of whom considered transport events in small subspaces (sometimes referred to as clefts) under the plasma membrane (20, 39). To test the Blaustein hypothesis, we carried that further to severely restrict diffusional exchange between microdomain and cytosol, except through interactions with the SR (Fig. 1). The precise cellular substructure that might facilitate such compartmental isolation is uncertain; however, several possibilities exist. The distance between SR and overlying plasma membrane that envelopes the putative microdomain volume has been measured as ∼19 nm (25). Within that region, cytoskeletal elements, binding proteins, channels, and transporters must be present in high concentration so that water is partially excluded and a high density of fixed charges on amino acid residues exists. Those factors alone might be sufficient to limit lateral diffusion. It is also possible that close apposition or fusion of the lipid bilayer of the SR and plasmalemma near the border with the cytosol prevents the escape of diffusible solutes from microdomains. Whatever the explanation, it is clear that the model will not predict control of [Ca]_{cyt} via changes in the microdomains unless there is a high degree of isolation that restricts diffusive equilibration with the cytosol. Stated another way, isolation of microdomains is a fundamental premise that enables a cogent model to predict ion concentrations that are significantly different from those present in the bulk cytosol. As described in association with Table 6, a pivotal parameter, poorly defined in the literature, is the conductance of SOC channels. When the latter was chosen to yield [Na]_{md} − [Na]_{cyt} ≈ 8 mM, to agree with measurements obtained by electron probe in myocytes (44), we predict that [Ca]_{md} equals 156 nM (vs. 92 in the bulk cytosol), and [K]_{md} 104 mM (vs. 97 in the bulk cytosol).

An additional feature of interest concerns the need for nonzero fluxes of K^{+} and Na^{+} ions between microdomains and cytosol to obtain realistic predictions. Given the absence of cytosolic NCX (8, 22, 30), simulations suggest that [Na]_{cyt} would remain below 5 mM if the microdomain-to-cytosol Na^{+} electrodiffusive flux (*J*_{Na, diff}) were negligible (Table 5). Hence, while the model predicts the need for diffusional sequestration so that [Na]_{md} can significantly exceed [Na]_{cyt}, limited trafficking of Na^{+} and K^{+} between the microdomains and the bulk cytosol is nevertheless required.

To obtain inputs for the model, we used recent measurements of cell geometry and membrane conductance from contractile DVR pericytes of the renal medulla (13, 14, 31) and studies of cerebrovascular (46) and other vascular smooth muscle cells (20, 27, 39). A limitation of our model stems from the fact that the kinetics and [Ca]_{sr} dependence of SOC currents have not been entirely defined (1, 12, 28). Consequently, the equations characterizing the Ca^{2+} flux through SOC in this study were taken from the neuronal model of LeBeau et al. (24). This model assumes that SOC conductance is regulated by ER/SR subcompartments, as observed experimentally. However, it does not include gating kinetics, nor does it account for possible activation of SOC via direct coupling with IP_{3} receptors (24). In addition, stretch-sensitive ion channels have not been included in our model. Finally, we have not included equations that account for anion, particularly chloride, movement. Experiments have shown that pericytes and other smooth muscle cells possess Cl^{−} channels. However, little information is available to fully account for the combined entry and exit pathways utilized to achieve homeostasis. Changes in [Ca]_{cyt} modulate conductance of Ca^{2+}-activated Cl^{−} channels but, due to voltage dependence, their contribution is small near the resting potential.

The release of Ca^{2+} via IP_{3}R was modeled using the model of De Young and Keiser (16). A more recent model of IP_{3}R-mediated Ca^{2+} release was developed by Sneyd and Dufour (41). Using the *I*_{IP3R} equations given in the latter study, the [Ca]_{cyt} profile predicted following α_{2} inhibition was inconsistent with experimental observations, and the model predicted an enormous [Ca]_{md} peak following NCX inhibition (∼200 μM), which seemed unrealistic.

Measurements by Blaustein et al. (8) suggest that vascular smooth muscle cell SR Ca^{2+} stores are organized into compartments, which release calcium in response to cyclopiazonic acid (IP_{3}R), caffeine (RyR), or both. Given that the density of RyR and IP_{3}R at each interface (microdomain-SR and bulk cytosol-SR) is unknown, we assumed that the time constants for Ca^{2+} release through RyR and IP_{3}R, respectively, are identical at both interfaces. Similarly, the number of SERCA pumps at each interface is unknown, and our sensitivity analysis shows it to be a significant determinant of [Ca]_{md} (Table 6).

Despite these limitations, some confidence is derived from comparison of our predictions with trends in experimental data. Resting [Ca]_{cyt} of ∼100 nM and [Na]_{cyt} of ∼6 mM are reasonable, as is the prediction of [Ca]_{sr}, ∼260 μM. It is also encouraging that experimentally observed biphasic elevations of [Ca]_{cyt} are predicted to occur on inhibition of NCX-mediated Ca^{2+} export through reduction of [Na]_{out} (Fig. 6) (34). This model supports CICR as the explanation for the peak phase transients of the associated [Ca]_{cyt} responses.

In summary, this study describes a mathematical simulation designed to test feasibility of the hypothesis that transport events in sequestered cellular microdomains regulate Ca^{2+} loading in the SR and Ca^{2+} signaling in the cytosol. The model predicts resting ion concentrations that are compatible with experimental measurements and predicts temporal changes in [Ca]_{cyt} that have been observed with NCX inhibition. Our results show the relative importance of microdomain transporters in the setting of [Ca]_{md}. In the absence of current through microdomain SERCA pumps or NCX, the resting value of [Ca]_{md} would increase by >50%. Simulations also suggest that cytosolic and microdomain SERCA pumps, IP_{3}R, Na^{+}-K^{+}-ATPase, NCX, and SOC are important determinants of [Ca]_{sr}. [Ca]_{sr} is generally predicted to be 200–400 μM, but would fall below 100 μM if SERCA pumps were removed from the SR-cytosol interface. Our sensitivity analysis also suggests that relative variations in [Ca]_{cyt} are generally smaller than those in [Ca]_{md} (Table 6), in part because the volume of the cytosol is much larger than that of the microdomains.

The small microdomain-to-cytosol volume ratio (equal to 3:500) also explains why microdomain concentration changes are predicted to occur rapidly compared with the cytosol. The cytosolic concentrations of Na^{+} and K^{+} in particular are predicted to adjust to step changes over several minutes, whereas microdomain concentrations generally equilibrate within 60 s (Fig. 6).

Simulations indicate that a rapid increase in [Ca]_{md} or [Ca]_{cyt}, such as that following a reduction in Ca^{2+} extrusion through NCX, triggers a rapid burst of Ca^{2+} release via RyR and IP_{3}R, (i.e., CICR) as long as [Ca]_{sr} is not simultaneously decreasing (see Figs. 5 and 6). As a result, [Ca]_{md} or [Ca]_{cyt} peaks within a few seconds, then drops significantly as the current through IP_{3}R and RyR decreases.

Our results also suggest an important role of NCX in Ca^{2+} signaling. In the resting state, NCX is predicted to operate in “forward mode,” with Na^{+} entry and Ca^{2+} extrusion from the cell. As shown in Fig. 5, complete inhibition of NCX is predicted to raise [Ca]_{md} from 156 to 248 nM, [Ca]_{cyt} from 92 to 138 nM, and to lower [Na]_{md} from 15.1 to 5.6 mM, and [Na]_{cyt} from 5.9 to 4.0 mM. In addition, NCX translates variations in Na^{+}-K^{+}-ATPase current into variations of [Ca]_{md}, [Ca]_{sr}, and [Ca]_{cyt} (see Fig. 7 and Table 6), supporting feasibility of the Blaustein hypothesis (6, 10). We conclude that a pivotal supposition necessary for modulation of Ca^{2+} signaling by transport events in subplasmalemmal microdomains is a high level of sequestration from the cytosol.

## GRANTS

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

## Acknowledgments

We thank the reviewers, whose helpful comments led to many improvements in this manuscript.

## Footnotes

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “

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- Copyright © 2007 the American Physiological Society