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Modification of cytosolic calcium signaling by subplasmalemmal microdomains

¹Department of Chemical and Biological Engineering, Tufts University, Medford, Massachusetts; and ²Departments of Medicine and Physiology, University of Maryland School of Medicine, Baltimore, Maryland

Submitted 28 September 2006 ; accepted in final form 13 February 2007

	ABSTRACT

TOP ABSTRACT MODEL RESULTS DISCUSSION GRANTS REFERENCES

 
To investigate the hypothesis that Na⁺ concentration in subplasmalemmal microdomains regulates Ca²⁺ 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⁺/Ca²⁺ exchange (NCX), and the distributions of Na⁺-K⁺-ATPase ₁- and ₂-isoforms in the plasma membrane. Ca²⁺ release from SR stores is assumed to occur via ryanodine (RyR) and inositol trisphosphate (IP₃R) 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 Ca²⁺ signaling is verified. In the resting state, NCX operates in "forward mode," with Na⁺ entry and Ca²⁺ 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 Ca²⁺ into SR stores and [Ca²⁺]_cyt variation.

electrochemical model; descending vasa recta; pericytes; ionic currents

It is known that the relationship between Ca²⁺ stores and [Ca²⁺]_cyt is complex. The extent of filling of Ca²⁺ stores influences the magnitude of agonist-induced Ca²⁺ release (3, 34). In turn, regulation of the filling of Ca²⁺ 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⁺/Ca²⁺ exchanger (NCX) to inhibit its function, thereby directing more Ca²⁺ to fill SR stores, a sequence of events often referred to as the Blaustein hypothesis (6). The ₂-₄ isoforms of Na⁺-K⁺-ATPase are targeted to the microdomains, whereas the 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 Ca²⁺ export by NCX is not the only means through which OLF influence Ca²⁺ 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-PK₁ cells, downstream PLC1 activation was shown to generate inositol trisphosphate (IP₃) and induce [Ca²⁺]_cyt elevation (48).

Motivated by the importance of Ca²⁺ trafficking, we formulated a mathematical simulation of ion concentration changes and IP₃ 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 [Ca²⁺]_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 Ca²⁺ into stores and [Ca²⁺]_cyt variation.

View larger version (13K): [in this window] [in a new window]   Fig. 1. Diagram of the cell, with its 3 compartments: bulk cytosol (subscript cyt), microdomains (md), and sarcoplasmic reticulum (SR). Ji, diff is the electrodiffusive flux of ion i between the microdomains and the bulk cytosol, where i = K+, Na+, or Ca2+. Not shown are KCa, Kir, KATP, and Kv channels, and the chelating agents calmodulin (CM), troponin (Trpn), and calsequestrin (Calseq). ER, endoplasmic reticulum; IP3R and RyR: inositol trisphosphate and ryanodine receptor, respectively; SERCA, sarcoplasmic-endoplasmic reticulum Ca2+-ATPase; SOC, store-operated, nonselective cation channel.

	MODEL

TOP ABSTRACT MODEL RESULTS DISCUSSION GRANTS REFERENCES

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/cm² (27), the average capacitive membrane surface area is estimated as 1.21 x 10^–5 cm².

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 x 10^–5 cm²) = 1.7 x 10^–6 cm², and that the total volume of the microdomains (vol_md) is about (1.7 x 10^–6 cm²)(19 x 10^–7 cm) = 3.2 x 10^–12 cm³ 3 x 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 Ca²⁺ (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 ₂-isoforms of Na⁺-K⁺-ATPase pumps (I_{NaK, 2}) are expressed exclusively in the region of the cell membrane that lies directly above the microdomains (23), whereas the ₁-isoforms (I_{NaK, 1}) are restricted to the region of the cell membrane directly above the bulk cytosol (10).

We also assume that all the Na⁺/Ca²⁺ 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 Ca²⁺ ions.

The interfaces between the SR and cytosol and the SR and microdomains are both populated by Ca²⁺-ATPase pumps (SERCA; I_SERCA), ryanodyne receptors (RyR; I_RyR), and inositol trisphosphate receptors (IP₃R; 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 Ca²⁺. 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 P₁ and P₂ denote the two exponential components of the channel activation process, and _P1 and _P2 denote the respective time constants. The parameter _Kv, which is voltage dependent, represents the steady-state value of P₁ and P₂.

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 _PF and _PS denote the respective time constants. The steady-state open probability of the channel is given by _KCa.

I_NaK. As described above, we assume that the ₁-isoform of the pump is expressed only in the plasma membrane above the bulk cytosol, whereas the 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 and , respectively), and _m and _h are the associated time constants.

I_NaCa. As described above, we assume that all Na⁺/Ca²⁺ 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 Ca²⁺ 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 _d and _f are the time constants for activation and inactivation, respectively; the symbols _L and _L denote the steady-state values of d_L and f_F.

I_SOC. The general form of the equation for the SOC Ca²⁺ current is that proposed by LeBeau et al. (24):

(40)

This expression accounts for the fact that a decrease in SR Ca²⁺ 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 Ca²⁺ currents in compartment j:

(41)

In the absence of experimental measurements, the maximum Ca²⁺ conductance of cytosolic SOC is taken as 20 pS, so as to yield a three- to fourfold increase in [Ca²⁺]_cyt when all SERCA pumps are inhibited (3). The maximum Ca²⁺ 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 Ca²⁺ permeability (that is, _{RyR, max}) is the same at the SR-cytosol and SR-microdomain interfaces.

IP₃R. We use the kinetic model for IP₃R 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 IP₃ binding site (denoted as site 1) and two Ca²⁺ 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 S₁₁₀ (corresponding to the binding of one IP₃ and one activating Ca²⁺) for the IP₃R channel to be open. Assuming that the binding kinetics of IP₃ and the activation of the receptors by Ca²⁺ are fast processes relative to Ca²⁺ 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 IP₃ binding, (where k = 1 if = 0, and k = 3 if = 1). In particular, the open probability of IP₃R depends on the fraction of receptors in the S₁₁₀ state:

(52)

To determine the concentration of IP₃ in a given compartment j (j = cyt or md), denoted by [IP₃]_j, we also use the model of De Young and Keizer (16), which includes Ca²⁺ feedback on the concentration of IP₃:

(53a)

where I_r is the rate constant for loss of IP₃ (taken as 1 s^–1), ₄ is the maximum rate of IP₃ production, and k₄ is the dissociation constant (taken as 1.1 µM). If ₄ = 0, there is no Ca²⁺ feedback; the other extreme is ₄ = 1. We assume a baseline value ₄ = 0.5. Since the value of 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 [IP₃]^eq is the equilibrium concentration of IP₃, taken as 240 nM (16). The baseline value of ₄ is chosen such that the basal level of IP₃ in the cytosol is [IP₃]_cyt = 240 nM. With ₄ = 0.5, this implies that ₄ = 1.85 s^–1.

The IP₃R release current in compartment j (j = cyt, md) is calculated as:

(54)

where _{IP3R, max} is the maximum Ca²⁺ permeability through the IP₃ receptors. In the absence of data, we assume that _{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 Ca²⁺ 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 Ca²⁺ 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 Ca²⁺ 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 Ca²⁺ 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 (d), where d is the vessel circumference that is enveloped by the pericyte (d = 13 µm), and is the gap between the plasmalemma and the superficial SR (19 nm). The whole-cell diffusivity of calcium is taken as 0.3 x 10^–5 cm²/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

TOP ABSTRACT MODEL RESULTS DISCUSSION GRANTS REFERENCES

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 [Ca²⁺]_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 Ba²⁺, DVR pericytes were depolarized from –68 to –63 and –57 mV, respectively. The discrepancy may due to the fact that Ba²⁺ becomes internalized and substitutes for Ca²⁺ 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 [Ca²⁺]_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 x 10^–3.

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 Ca²⁺ 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.

View larger version (28K): [in this window] [in a new window]   Fig. 2. A: effects of inhibiting cytosolic SERCA pumps at t = 300 s on resting concentrations in the bulk cytosol (subscript cyt), the microdomains (md), and the SR, as a function of time (in s). B: effects of inhibiting cytosolic SERCA pumps at t = 300 s, shown on a 3-s time scale to delineate the early transient. C: effects of inhibiting cytosolic SERCA pumps at t = 300 s on currents (given in pA), as a function of time (in s). The solid and dashed lines denote cytosolic and microdomain currents, respectively. I(NaK): current through Na+-K+-ATPase pumps; I(NaCa): current through Na+/Ca2+ exchangers; I(SOC): current through store-operated channels; I(SERCA): current through SERCA pumps; I(RyR); current through ryanodyne receptors; I(IP3R): current through IP3 receptors. D: effects of inhibiting cytosolic SERCA pumps at t = 300 s on fluxes contributing to [Ca]cyt variations (brackets denote concentration). Ca2+ fluxes are given in µM/s, and a positive flux indicates Ca2+ export. The flux J corresponding to current I is determined as J = I/(2·F·volcyt, Ca). Not shown is the negligible voltage-dependent Ca2+ flux corresponding to ICa, L. The troponin flux includes binding to both high- and low-affinity sites.

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. 2C, Ca²⁺ release into the cytosol through IP₃R decreases with increasing [Ca]_cyt. Our model is based on that of De Young and Keiser (16), which describes Ca²⁺ release via IP₃R. Their model predicts a biphasic dependence of IP₃R activity on [Ca]. When [Ca] is less than 0.25 µM, [Ca] elevation increases the open probability of IP₃R; above 0.25 µM, however, elevating [Ca] decreases the open probability.

Shown in Fig. 2D 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 IP₃R and RyR occur within 2 s following cytosolic SERCA inhibition (Fig. 2B), whereas the other Ca²⁺ 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 Ca²⁺ stores also increases Ca²⁺ and Na⁺ entry through cytosolic SOCs, thereby raising [Na]_cyt.

View larger version (15K): [in this window] [in a new window]   Fig. 3. Effect of inhibiting microdomain SERCA pumps at t = 300 s on resting concentrations.

Roles of RyR and IP₃R. We then examined the roles of RyR and IP₃R 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 IP₃R, however, has significant effects on resting Ca²⁺ and Na⁺ concentrations. Removing IP₃R at the SR-cytosol interface (Fig. 4A) decreases [Ca]_cyt by 15% and raises [Ca]_sr by 15%. The [Ca]_sr elevation augments RyR- and IP₃R-mediated Ca²⁺ release into the microdomains, thereby raising [Ca]_md by 10%. Stimulation of NCX above the microdomains also raises [Na]_md.

View larger version (14K): [in this window] [in a new window]   Fig. 4. A: effect of inhibiting IP3R at SR-cytosol interface at t = 300 s. B: effect of inhibiting IP3R at SR-microdomain interface at t = 300 s.

Inhibition of NCX and SR Ca²⁺ loading. Under resting conditions, "forward mode" NCX extrudes Ca²⁺ 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 Ca²⁺ export by NCX to permit loading of the SR with Ca²⁺ (6, 9, 10). Figure 5 verifies the feasibility of that contention. Note that [Ca]_sr rises as SERCA uptake increases after NCX is blocked.

View larger version (14K): [in this window] [in a new window]   Fig. 5. Effect of inhibiting Na+/Ca2+ exchange (NCX) at t = 300 s on resting concentrations.

View larger version (15K): [in this window] [in a new window]   Fig. 6. Effect of lowering [Na]out sequentially from 140 to 125, 100, and 50 mM every 150 s, starting at t = 300 s.

View larger version (14K): [in this window] [in a new window]   Fig. 7. Effect of inhibiting (at t = 300 s) microdomain Na+-K+-ATPase (A) or both microdomain and cytosolic Na+-K+-ATPase (B) on NCX current and [Ca]md.

View larger version (12K): [in this window] [in a new window]   Fig. 8. A: effect of inhibiting cytosol SOC channels at t = 300 s on resting concentrations. B: effect of inhibiting microdomain SOC channels at t = 300 s on resting concentrations.

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 Ca²⁺ concentrations. The results are summarized in Table 6.

Our simulations suggest that the parameters that determine the rate of IP₃R-mediated Ca²⁺ release also significantly affect Na⁺ and Ca²⁺ concentrations. A 100% increase in the equilibrium value of [IP₃] ([IP₃]^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 [IP₃]^eq reduces [Ca]_md by 40% and raises [Ca]_sr by 50%. In the baseline case, we assumed that Ca²⁺ exerts a moderate feedback effect on the production of IP₃. By following the approach of De Young and Keiser (16), the parameter that characterizes this interaction is denoted by ₄ (see Eq. 53b). To investigate the effects of [Ca] feedback on Ca²⁺ signaling, we compared the resting [Ca] values when ₄ = 0 (no feedback), ₄ = 0.5 (baseline case), and ₄ = 1 (maximal feedback). The related parameter ₄ was adjusted in each case so that the resting value of [IP₃]_cyt was equal to 240 nM. In the baseline case (₄ = 0.5), [Ca]_md is greater than [Ca]_cyt, so that the resting value of [IP₃]_md (250 nM) is higher than that of [IP₃]_cyt. In the absence of feedback (₄ = 0), [IP₃]_md = [IP₃]_cyt = 240 nM, and the flux of Ca²⁺ released by IP₃R 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 ₄ = 1, so that maximal IP₃ production is achieved through Ca²⁺ stimulation, high [Ca]_md augments IP₃ 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 Ca²⁺ permeability of RyR (_{RyR, max}) does not significantly affect Ca²⁺ 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 Ca²⁺ 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 ₂:₁ 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 Ca²⁺ 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 Ca²⁺, 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

TOP ABSTRACT MODEL RESULTS DISCUSSION GRANTS REFERENCES

 
Cardiotonic steroids such as digitalis and ouabain inhibit transport by Na⁺-K⁺-ATPase by binding to the first extracellular NH₂-terminal loop of the -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 Ca²⁺ 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 ₁-isoform of Na⁺-K⁺-ATPase, which performs a general housekeeping function in rodents, is ouabain insensitive. In contrast, other rodent isoforms (₂–₄) 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 Ca²⁺ trafficking between the plasma membrane and cellular stores. Colocalization of ₂ Na⁺ pumps with SR protrusions that abut the plasma membrane supports that contention (7, 10). More recently, the existence of an NH₂-terminal sorting motif that tethers ₂ Na⁺ pumps to microdomains has been defined (42).

Much functional evidence tends to confirm that Ca²⁺ signaling is modulated through ₂–₄ isoforms of Na⁺-K⁺-ATPase. Their inhibition or reduction of expression enhances agonist-induced Ca²⁺ 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 Ca²⁺ 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 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 Ca²⁺ 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 Ca²⁺ 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 Ca²⁺ 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 IP₃ 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 Ca²⁺-activated Cl^– channels but, due to voltage dependence, their contribution is small near the resting potential.

The release of Ca²⁺ via IP₃R was modeled using the model of De Young and Keiser (16). A more recent model of IP₃R-mediated Ca²⁺ release was developed by Sneyd and Dufour (41). Using the I_IP3R equations given in the latter study, the [Ca]_cyt profile predicted following 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 Ca²⁺ stores are organized into compartments, which release calcium in response to cyclopiazonic acid (IP₃R), caffeine (RyR), or both. Given that the density of RyR and IP₃R at each interface (microdomain-SR and bulk cytosol-SR) is unknown, we assumed that the time constants for Ca²⁺ release through RyR and IP₃R, 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 Ca²⁺ 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 Ca²⁺ loading in the SR and Ca²⁺ 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, IP₃R, 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 Ca²⁺ extrusion through NCX, triggers a rapid burst of Ca²⁺ release via RyR and IP₃R, (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 IP₃R and RyR decreases.

Our results also suggest an important role of NCX in Ca²⁺ signaling. In the resting state, NCX is predicted to operate in "forward mode," with Na⁺ entry and Ca²⁺ 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 Ca²⁺ signaling by transport events in subplasmalemmal microdomains is a high level of sequestration from the cytosol.

	GRANTS

TOP ABSTRACT MODEL RESULTS DISCUSSION GRANTS REFERENCES

 
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

 

Address for reprint requests and other correspondence: A. Edwards, Dept. of Chemical and Biological Engineering, Tufts Univ., 4 Colby St., Medford, MA 02155 (e-mail: aurelie.edwards{at}tufts.edu)

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

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