Many models have been developed to account for stimulus-evoked [Ca2+] responses, but few address how responses elicited in specific cell types are defined by the Ca2+ transport and buffering systems that operate in the same cells. In this study, we extend previous modeling studies by linking the time course of stimulus-evoked [Ca2+] responses to the underlying Ca2+ transport and buffering systems. Depolarization-evoked [Ca2+]i responses were studied in sympathetic neurons under voltage clamp, asking how response kinetics are defined by the Ca2+ handling systems expressed in these cells. We investigated five cases of increasing complexity, comparing observed and calculated responses deduced from measured Ca2+ handling properties. In Case 1, [Ca2+]i responses were elicited by small Ca2+ currents while Ca2+ transport by internal stores was inhibited, leaving plasma membrane Ca2+ extrusion intact. In Case 2, responses to the same stimuli were measured while mitochondrial Ca2+ uptake was active. In Case 3, responses were elicited as in Case 2 but with larger Ca2+ currents that produce larger and faster [Ca2+]i elevations. Case 4 included the mitochondrial Na/Ca exchanger. Finally, Case 5 included ER Ca2+ uptake and release pathways. We found that [Ca2+]i responses elicited by weak stimuli (Cases 1 and 2) could be quantitatively reconstructed using a spatially uniform model incorporating the measured properties of Ca2+ entry, removal, and buffering. Responses to strong depolarization (Case 3) could not be described by this model, but were consistent with a diffusion model incorporating the same Ca2+ transport and buffering descriptions, as long as endogenous buffers have low mobility, leading to steep radial [Ca2+]i gradients and spatially nonuniform Ca2+ loading by mitochondria. When extended to include mitochondrial Ca2+ release (Case 4) and ER Ca2+ transport (Case 5), the diffusion model could also account for previous measurements of stimulus-evoked changes in total mitochondrial and ER Ca concentration.

Introduction

Ionized free calcium (Ca2+) is a ubiquitous signaling ion that serves both as an important charge carrier and as a chemical intermediate that links a variety of physiological stimuli to their intracellular effectors (Berridge, 1998; Carafoli et al., 2001). One of the central problems in Ca2+ signaling research is to understand how Ca2+ handling mechanisms function together in intact cells to define the spatiotemporal properties of stimulus-evoked [Ca2+] responses. In its signaling capacity, Ca2+ acts by binding to specific sites whose occupancy translates the Ca2+ signal into downstream cellular effects. Since binding site occupancy depends on the history of the free Ca2+ concentration ([Ca2+]), Ca2+ dynamics are critical. As a result, [Ca2+] is tightly regulated by various Ca2+ channels, pumps, exchangers (here collectively referred to as transporters), and buffers that determine the proportion of Ca2+ that is free vs. bound.

Mathematical modeling provides a useful tool for investigating the basis of Ca2+ dynamics, making it possible to address topics that would be difficult to investigate in other ways. For example, model calculations can be used to test if a particular combination of transport and buffering systems can support a qualitative mode of Ca2+ dynamics, such as Ca2+ oscillations (Goldbeter et al., 1990; Dupont and Goldbeter, 1993). Modeling can also clarify how individual transport systems contribute to evoked Ca2+ responses (Colegrove et al., 2000b). This can be difficult to do simply by observing the effects of specific inhibitors on evoked Ca2+ responses because of secondary changes in the activity of other Ca2+ handling systems that arise because of their dependence on [Ca2+]. A third area well suited to modeling is the investigation of how quantitative changes in the functional properties of Ca2+ regulatory systems influence Ca2+ dynamics in genetic disease. Here it can be asked if a Ca2+ regulatory syndrome is consistent with the operation of a single defective gene product, or requires consideration of compensatory mechanisms as well. Finally, realistic models of Ca2+ dynamics can be used to test in silico how selective pharmacological perturbations of Ca2+ transport and buffering are expected to impact Ca2+ signaling in intact cells.

There is a long and productive history of modeling in Ca2+ signaling research. For example, mathematical models have been used to examine qualitative properties of Ca2+ dynamics under the control of various Ca2+ handling systems whose properties are specified based on literature values from multiple cell types. This approach has helped build intuition about Ca2+ signal generation, and has clarified the conditions required for generation of Ca2+ oscillations and waves (Keener and Sneyd, 1998). There has been less progress, however, in understanding how the specific Ca2+ regulatory mechanisms that function together in intact cells define stimulus-evoked [Ca2+] elevations, although important contributions have been made in this area by studies in muscle cells (Kovacs et al., 1983; Sipido and Wier, 1991; Schuhmeier and Melzer, 2004). There are several reasons for this, all of which represent a lack of experimental data for constraining model development, rather than a lack of mathematical tools: (a) technical difficulties in measuring and characterizing the relevant Ca2+ fluxes in intact cells, particularly those representing Ca2+ uptake and release by intracellular stores; nevertheless, several groundbreaking studies clarified how this can be done (Herrington et al., 1996; Sipido and Wier, 1991; Kovacs et al., 1983); (b) lack of information about the important “independent” variables that control the activity of these regulatory systems in their normal intracellular milieu, and how activity depends on these variables; and (c) poor consensus regarding the functional role of Ca2+ regulatory events on small vs. large space and time scales. Given that comprehensive descriptions of Ca2+ regulation that apply over all distance and time scales are currently unavailable, descriptions that apply over intermediate ranges of distance and time would be of considerable practical value. For example, they would serve as important limiting cases for models that address signaling in microdomains of space and time, and facilitate rigorous testing of hypotheses derived from experimental studies. Indeed, one of the most revealing ways to evaluate the importance of a proposed Ca2+ handling mechanism for Ca2+ signal generation is to determine if it is necessary to reconstruct measured Ca2+ responses. Of course, this requires a “complete” description of the regulatory mechanisms operating in the cell under study, since exclusion of any critical pathway would render such a reconstruction impossible.

Our previous work has described approaches for measuring and characterizing Ca2+ fluxes and buffering strength in sympathetic neurons (Colegrove et al., 2000a, Albrecht et al., 2002). The purpose of the present study is to determine if these characterizations are sufficient, when taken together, to account quantitatively for depolarization-induced Ca2+ responses in these cells. We found that when these characterizations are incorporated into a model that assumes cytoplasmic Ca2+ is uniformly distributed, it was possible to account quantitatively for responses to weak depolarization, but not responses to stronger depolarization when radial [Ca2+]i gradients are expected to be steep. However, we could account for these responses using a more general model that explicitly considers radial differences in Ca2+ concentration. The two models yielded very similar [Ca2+]i trajectories for weak depolarization, indicating that the uniform model describes a limiting case of the diffusion model when [Ca2+]i gradients are shallow. Overall, the results provide a core description of neuronal Ca2+ regulation that can be applied to other cells as more information becomes available regarding cell type–specific regulatory mechanisms and their cellular distribution.

Materials And Methods

Cell Preparation

Bullfrog sympathetic neurons were dissociated enzymatically and placed in culture for up to 1 wk as described previously (Colegrove et al., 2000a). All procedures conform to guidelines established by our Institutional Animal Care and Use Committee.

Cytosolic Calcium Measurements

Cells were incubated with 3 μM fura-2 AM (Invitrogen) for 40 min at room temperature with gentle agitation. Fura-2 AM was dispensed from a 1 mM stock solution in DMSO containing 25% (wt/wt) pluronic F127 (BASF Corporation) that was stored at −20 C°. Cells were rinsed and recordings began after ∼20 min to allow time for de-esterification of the Ca2+ indicator. Since Fura-2 was loaded into cells as the AM ester, the cytoplasmic concentration of the indicator was not measured directly but was estimated based on measured changes in Ca2+ concentration after depolarization and repolarization (see below, Fig. 1). Culture dishes with adherent cells were placed on the stage of an inverted microscope (Nikon Diaphot TMD) and superfused continuously (∼5 ml/min) with normal Ringer's solution. Drug application was accomplished by changing the solution bathing individual cells (∼200 ms) using a system of microcapillaries (Drummond microcaps, 20 μl) mounted on a micromanipulator.

To measure [Ca2+]i, neurons were illuminated by light from a 150 W xenon lamp that was first passed through excitation filters (350 ± 5 nm, 380 ± 5 nm) mounted on a filter wheel rotating at 40–100 Hz and then focused with a 40× objective (Nikon, Fluor, NA 1.3). Emitted light passed through a long-pass dichroic mirror (400 nm) and emission filter (510 ± 10 nm) and was detected with a photomultiplier tube (Thorn EMI 9124). A spectrophotometer (Cairn Research Limited) was used to control the filter wheel and measure the spatially averaged fluorescence intensity at the two excitation wavelengths. Fluorescence measurements were made at 4–5 Hz and saved on a laboratory computer. [Ca2+]i was calculated according to the method of Grynkiewicz et al. (1985) as described previously (Colegrove et al., 2000a).

Voltage Clamp

Simultaneous measurements of depolarization-evoked [Ca2+]i elevations and voltage-sensitive Ca2+ currents (ICa) were made under voltage clamp in fura-2 AM–loaded cells using the perforated patch technique. Patch electrodes (1–2 MΩ) were pulled (Sutter Instruments P-97) and tips were filled with a solution containing (in mM) 125 CsCl, 5 MgCl2, 10 HEPES, and 0 or 6.5 mM Na+ (the latter with equimolar reduction in CsCl), pH 7.3 with CsOH. After filling tips, pipettes were back-filled with the same solution supplemented with 520 μM amphotericin B, dispensed from concentrated aliquots (12 mg/100 μl DMSO). Fresh amphotericin B–containing internal solutions were made and kept on ice and used within 2 h. For the cells included in this study, after achieving a high resistance seal, series resistance declined over 5–10 min to <10 MΩ. Cells were exposed to an extracellular solution containing (in mM) 130 TEACl, 10 HEPES, 10 glucose, 2 CaCl2, 1 MgCl2, pH 7.3 with TEAOH. Currents were measured with an Axopatch 200A voltage clamp (Molecular Devices) using series resistance compensation (∼90%) and were filtered at 5 kHz. Neurons were held at −70 mV and depolarized to voltages between −35 and −10 mV while current and fluorescence intensity were measured at 2–5 kHz just before (10–100 ms) and after (100–200 ms) changes in voltage, and at 4–5 Hz otherwise, and saved on a laboratory computer. Currents were corrected for a linear leak based on responses to small hyperpolarizing voltage steps. [Ca2+]i elevations evoked under voltage clamp were somewhat larger than those elicited by high K+ at comparable membrane potentials, presumably because of more rapid depolarization and more efficient Ca2+ channel activation under voltage clamp. However, [Ca2+]i recovery kinetics after repolarization were similar for the two techniques.

Spatially Uniform Model

To investigate how depolarization-evoked [Ca2+]i elevations are defined by the Ca2+ handling systems that operate in sympathetic neurons, we characterized these systems experimentally and obtained analytical expressions that described their activity. We then asked if these expressions, when used as the defining rate equations in a model of Ca2+ dynamics, make it possible to reconstruct observed depolarization-evoked [Ca2+]i responses. We began with responses elicited under relatively simple experimental conditions in which all but a few transport pathways are blocked. Once an adequate description of [Ca2+]i dynamics was in hand, we examined responses under conditions where additional transport mechanisms were enabled. At each stage, it was asked if the differences in evoked [Ca2+]i responses observed after enabling a particular system can be understood in terms of the properties of that system in the functional context provided by the other systems. We began by incorporating the measured descriptions of Ca2+ handling into a spatially uniform model of Ca2+ dynamics, and asked if such a model can account for measured responses. We found that the uniform model was adequate to account for responses to weak depolarization, but not responses to strong depolarization, when intracellular Ca2+ gradients are expected to be steep. We then turned to a diffusion model that incorporated the same transport and buffering descriptions but considered spatially heterogeneous Ca2+ signals. Since the uniform and nonuniform models are similar, we give an overview of the uniform model here and describe the diffusion model in the Appendix.

The spatially uniform model includes three cellular compartments, cytoplasm, mitochondria, and the ER with free Ca2+ concentrations [Ca2+]i, [Ca2+]MT, and [Ca2+]ER, respectively. These dynamical variables change at a rate that depends on the intercompartmental net Ca2+ fluxes (

\({\tilde{J}}_{i}\)
⁠,
\({\tilde{J}}_{MT}\)
,
\({\tilde{J}}_{ER}\)
, e.g., in nmol/s) and volumes (vi, vMT, vER, e.g., in li) scaled by buffering factors (κi′, κMT′, κER′):

 
\begin{eqnarray*}&&\frac{d[Ca^{2{+}}]_{i}}{dt}{=}{-}\frac{{\tilde{J}}_{\mathrm{i}}}{v_{i}{\kappa}{^\prime}_{i}}\\&&\frac{d[Ca^{2{+}}]_{MT}}{dt}{=}\frac{{\tilde{J}}_{\mathrm{MT}}}{v_{MT}{\kappa}{^\prime}_{MT}}\\&&\frac{d[Ca^{2{+}}]_{ER}}{dt}{=}\frac{{\tilde{J}}_{\mathrm{ER}}}{v_{ER}{\kappa}{^\prime}_{ER}},\end{eqnarray*}
(M1)

where

 
\[{\tilde{J}}_{i}{=}{\tilde{J}}_{\mathrm{PM}}{+}{\tilde{J}}_{\mathrm{MT}}{+}{\tilde{J}}_{\mathrm{ER}}\]
(M2)

and

 
\begin{eqnarray*}&&{\tilde{J}}_{PM}{=}{\tilde{J}}_{VSCC}{+}{\tilde{J}}_{Extru}\\&&{\tilde{J}}_{MT}{=}{\tilde{J}}_{uni}{+}{\tilde{J}}_{NaCa}\\&&{\tilde{J}}_{ER}{=}{\tilde{J}}_{SERCA}{+}{\tilde{J}}_{\mathrm{Re}lease}.\end{eqnarray*}
(M3)

These equations can be rewritten so that they are more easily related to measured cytoplasmic Ca2+ fluxes:

 
\begin{eqnarray*}&&\frac{d[Ca^{2{+}}]_{i}}{dt}{=}{-}J_{\mathrm{i}}\\&&\frac{d[Ca^{2{+}}]_{MT}}{dt}{=}J_{\mathrm{MT}}\frac{v_{i}{\kappa}{^\prime}_{i}}{v_{MT}{\kappa}{^\prime}_{MT}}\\&&\frac{d[Ca^{2{+}}]_{ER}}{dt}{=}J_{\mathrm{ER}}\frac{v_{i}{\kappa}{^\prime}_{i}}{v_{ER}{\kappa}{^\prime}_{ER}},\end{eqnarray*}
(M4)

where

 
\begin{eqnarray*}&&J_{i}{=}\frac{{\tilde{J}}_{\mathrm{i}}}{v_{i}{\kappa}{^\prime}_{i}}\\&&{=}\frac{{\tilde{J}}_{\mathrm{PM}}{+}{\tilde{J}}_{\mathrm{MT}}{+}{\tilde{J}}_{\mathrm{ER}}}{v_{i}{\kappa}{^\prime}_{i}}\\&&{=}J_{\mathrm{PM}}{+}J_{\mathrm{MT}}{+}J_{\mathrm{ER}}\end{eqnarray*}
(M5)

and

 
\begin{eqnarray*}&&J_{PM}{=}\frac{{\tilde{J}}_{PM}}{v_{i}{\kappa}{^\prime}_{i}}\\&&J_{MT}{=}\frac{{\tilde{J}}_{MT}}{v_{i}{\kappa}{^\prime}_{i}}\\&&J_{ER}{=}\frac{{\tilde{J}}_{ER}}{v_{i}{\kappa}{^\prime}_{i}}.\end{eqnarray*}
(M6)

These fluxes will be referred to as free Ca2+ fluxes. In the following, we describe how components of the model were characterized experimentally. The equations that were used to represent the fluxes are listed in the Appendix.

Dynamics of Cytoplasmic Total Ca Concentration

According to the spatially uniform model, Ca2+ is uniformly distributed within the cytoplasm at all times. Such a description is expected to provide a reasonable approximation to measured [Ca2+] responses as long as the rate of net Ca2+ transport between compartments is slow compared with the rate of Ca2+ diffusion within compartments. The cytoplasmic total Ca concentration [Ca]i changes at a rate that depends on both the net Ca flux into the cytoplasm (

\({\tilde{J}}_{i}\)
⁠) and the cytoplasmic volume (vi):

 
\begin{eqnarray*}&&\frac{d[Ca]_{i}}{dt}{=}{-}\frac{{\tilde{J}}_{\mathrm{i}}}{v_{i}}\\&&{=}{-}\frac{{\tilde{J}}_{\mathrm{i}}}{v_{Cell}f_{i}},\end{eqnarray*}
(M7)

where, by convention, fluxes that lead to an increase in [Ca]i are negative. Cytoplasmic volume was taken as the product of total cell volume (vCell) estimated from the membrane capacitance (Cm), assuming a specific capacitance of 1 μF/cm2, and the fraction (fi) of the total cell volume occupied by cytoplasm. Since cells plated on polylysine-coated glass coverslips assumed an approximately spherical shape, we calculated vCell as (

\(1/6\sqrt{{\pi}}\)
⁠) (Cm × 10−6)3/2 cm3. Sources of error in our estimate of vi include deviations from spherical geometry and uncertainty in the precise value of fi. We chose not to estimate fi explicitly but instead to include it as a factor in the buffering parameters to be estimated. Errors in our estimate of vi arising from deviations from spherical geometry would be expected to influence the estimated buffering parameters through their dependence on fi (see below).

Dynamics of Cytoplasmic Free Ca Concentration

The free cytoplasmic Ca concentration ([Ca2+]i) changes at a rate that depends on the ratio of

\({\tilde{J}}_{i}\)
to the cytoplasmic volume scaled by a buffering factor (κi′):

 
\begin{eqnarray*}&&\frac{d[Ca^{2{+}}]_{i}}{dt}{=}{-}\frac{{\tilde{J}}_{\mathrm{i}}}{v_{i}{\kappa}{^\prime}_{i}}\\&&{=}{-}\frac{{\tilde{J}}_{\mathrm{i}}}{v_{Cell}(f_{i}{\kappa}{^\prime}_{i})},\end{eqnarray*}
(M8)

This uses the fast buffering approximation, according to which Ca2+ binding to buffers is fast compared with changes in [Ca2+]i. This is a reasonable approximation for fura-2, which achieves equilibrium binding to Ca2+ with a time constant that is <10 ms (Kao and Tsien, 1988), and the changes in [Ca2+]i that we describe occur on a time scale of 100 ms to minutes. For the endogenous buffers, the approximation requires experimental validation. One approach to validating the fast buffer approximation is to calculate [Ca2+]i trajectories using the approximation and then compare with measured responses.

Cytoplasmic Ca2+ Buffering

The quantity κi′ is related to the well-known differential buffering factor κi and includes contributions from both endogenous (κEndog) and exogenous buffers (κfura):

 
(M9)

where the jth endogenous buffer is present at concentration Btotal,j and binds Ca2+ with dissociation constants Kd,j. In our experiments, the only exogenous buffer is fura-2, present at concentration Btotal,fura (estimated in different cells to be ∼50–100 μM) with a dissociation constant Kd,fura assumed to be 224 nM. Albrecht et al. (2002) provided a description of the [Ca2+]i dependence of κi′ in sympathetic neurons and estimates of κEndog and κfura near resting [Ca2+]i. In the following, we treat κEndog as a [Ca2+]i-independent constant to be determined, for example, describing contributions from low affinity endogenous buffers, and attribute the [Ca2+]i dependence of κi′ to fura-2. Accordingly, κi′ depends on [Ca2+]i in a way that can be described by two lumped parameters, (1+ κEndog) and Btotal,fura. However, since κi′ appears in the equations describing [Ca2+]i dynamics scaled by fi (see Eq. M8 above), it is convenient to define the lumped parameters P1 = fi(1 + κEndog) and P2 = fiBtotal,fura, which can be determined from measured quantities:

 
\begin{eqnarray*}&&f_{i}{\kappa}{^\prime}_{i}{=}f_{i}(1{+}{\kappa}_{Endog}){+}\frac{f_{i}B_{total,fura}K_{d,fura}}{([Ca^{2{+}}]_{i}{+}K_{d,fura})^{2}}\\&&{=}P_{1}{+}P_{2}\left[\frac{K_{d,fura}}{([Ca^{2{+}}]_{i}{+}K_{d,fura})^{2}}\right].\end{eqnarray*}
(M10)

In the following, we drop the prime with the understanding that κi differs from the more common definition by unity. To estimate the two parameters P1 and P2, we obtained two independent measurements of fiκi at low and high [Ca2+]i from the initial changes in [Ca2+]i that follow depolarization and repolarization, respectively (Fig. 1). The first measurement of fiκi was made near resting [Ca2+]i as the ratio of ICa/2FvCell to −d[Ca2+]i/dt during the early phase of the response, when [Ca2+]i increases approximately linearly with time. Since ICa is time dependent, we used the average of ICa over a short time interval that included the peak (at −35 mV, from 50 to 150 ms after the onset of depolarization; at −20 mV, from 3–51 ms after the onset of depolarization). The initial rate of rise of [Ca2+]i (d[Ca2+]i/dt) was measured by fitting a line to the initial rising phase of the [Ca2+]i response (−35 mV, during a 450-ms interval starting at the instant of depolarization; −10 mV, from 3–51 ms after depolarization onset). The second measurement of fiκi was obtained from the ratio of ICa/2FvCell to the instantaneous change in −d[Ca2+]i/dt (Δd[Ca2+]i/dt) after repolarization (Fig. 1). Given the two estimates of fiκi near resting and peak [Ca2+]i, P1 and P2 were determined algebraically or by fitting the measurements with Eq. M10, which provided an initial characterization of the [Ca2+]i dependence of fiκi. However, since the two fiκi measurements from which P1 and P2 were determined included errors arising from ICa averaging and calculations of d[Ca2+]i/dt from noisy [Ca2+]i signals, and because they represent estimates of fiκi at only two values of [Ca2+]i, P1 and P2 were treated as initial estimates of the parameters of fiκi, which were then optimized by fitting to measured responses.

Measuring Components of the Cytoplasmic Ca2+ Flux

In the following, the free cytoplasmic Ca2+ flux (Ji, e.g., in nM/s) represents the rate at which Ca2+ enters or leaves the cytoplasm (

\({\tilde{J}}_{i}\)
⁠, nmol/s) divided by viκi. Components of Ji representing different pathways were measured as follows. The component representing Ca2+ entry through voltage-sensitive Ca2+ channels (JVSCC) was taken as the ratio of the associated net Ca2+ flux (
\({\tilde{J}}_{VSCC}{=}I_{Ca}/2F\)
) and viκi, where F is the Faraday constant. The rate of Ca2+ extrusion across the plasma membrane (JExtru) was measured as the total cytoplasmic Ca2+ flux (−d[Ca2+]i/dt) during the recovery after depolarization-induced [Ca2+]i elevations in cells treated with carbonyl cyanide p-trifluoromethoxyphenylhydrazone (FCCP) and thapsigargin (Tg) to inhibit Ca2+ uptake by mitochondria and the ER, respectively (Colegrove et al., 2000a). JExtru was described by Eqs. A4 and A5 (see Appendix). The rate of Ca2+ uptake by the mitochondrial Ca2+ uniporter (Juni) was measured as the FCCP-sensitive component of the total flux under conditions where mitochondrial Ca2+ release via the Na/Ca exchanger was inhibited, either by using Na+-free pipette solutions or treating cells with the specific Na/Ca exchange inhibitor CGP 37157 (CGP). Specifically, Juni was measured by subtracting the FCCP-resistant flux (Ji,+FCCP) from the total flux at corresponding [Ca2+]i levels during the recovery:

 
\[J_{uni}(t){=}J_{i}(t){-}J_{i,{+}FCCP}([Ca^{2{+}}]_{i}(t)).\]
(M11)

This method follows from the idea that the total Ca2+ flux is the sum of an FCCP-sensitive flux (Juni), and an FCCP-resistant flux (Ji,+FCCP = JExtru) that depends on [Ca2+]i at each time point during the recovery (Colegrove et al., 2000a). Juni was described using Eqs. A11 and A12. The rate of Ca2+ release via the mitochondrial Na/Ca exchanger (JNa/Ca) was measured as the CGP-sensitive component of the total Ca2+ flux in cells where the exchanger was enabled by including Na+ in the pipette solution. JNa/Ca was determined by subtracting the CGP-resistant flux (Ji,+CGP) from the total flux at corresponding [Ca2+]i levels during the recovery:

 
\[J_{Na/Ca}(t){=}J_{i}(t){-}J_{i,{+}CGP}([Ca^{2{+}}]_{i}(t)).\]
(M12)

This makes use of the observation that Ji,+CGP is a function of [Ca2+]i at each time during the recovery (Colegrove et al., 2000a). JNaCa was described using Eqs. A20A22. Since Na+ was included in the pipette solution in these experiments, it is possible that Ji,+CGP contains a component representing Ca2+ entry via the plasma membrane Na/Ca exchanger (NCX) operating in reverse mode (external solutions were Na+ free). Such a component would not interfere with our measurement of JNa/Ca as long as the rate of Ca2+ entry via NCX is a single-valued function of [Ca2+]i at fixed membrane potential. This is expected based on the kinetic properties of NCX (Weber et al., 2001), assuming that in our experiments [Na+]i is approximately constant. The rate of mitochondrial Ca2+ transport (JMT) was measured as the FCCP-sensitive component of Ji in cells where mitochondrial uptake and release were both active. JMT was determined by subtracting the FCCP-resistant flux (Ji,+FCCP) from the total flux at corresponding [Ca2+]i levels during the recovery:

 
\[J_{MT}(t){=}J_{i}(t){-}J_{i,{+}FCCP}([Ca^{2{+}}]_{i}(t)).\]
(M13)

Here, the FCCP-sensitive flux gives Juni + JNaCa instead of only Juni because the Na/Ca exchanger is enabled by including Na+ in the pipette solutions. Finally, descriptions of ER Ca2+ uptake and release fluxes were taken from Albrecht et al. (2002).

Curve Fitting

Analysis was performed using IgorPro (Wavemetrics). Individual Ca2+ transport and buffering systems were described mathematically by fitting equations to measured quantities (see Appendix). While the equations were mechanistically motivated, they can be considered as empirical descriptions of the way the activity of each system depends on the corresponding independent variables. The differential equations were numerically integrated using the built-in IntegrateODE function in IgorPro (fourth order Runge-Kutta method with adaptive step size). In Cases 1 and 2, we fit the integrated responses by optimizing the two parameters (P1, P2) that describe cytoplasmic Ca2+ buffering. In the other cases, we examined how the agreement between calculated and simulated responses varied over a range of the parameter values that were not completely constrained by experiment (e.g., diffusion coefficient of endogenous buffer, DEndog, mitochondrial fractional volume, vMT/vCell). To describe ICa during numerical integration, we used the measured leak-subtracted Ca2+ current during the period of depolarization, and zeroed the current at other times to avoid contributions from baseline current fluctuations and drift during the recovery, which occurred over hundreds of seconds. In fitting the equation for JExtru to measurements from responses over a [Ca2+]i range that did not extend far beyond ∼300 nM, we constrained EC50,Extru to 350 nM, near the population average, and fit to obtain the remaining parameter values.

Drugs

CGP 37157 was a gift from Anna Suter (Novartis). Unless indicated otherwise, all other compounds were obtained from Sigma-Aldrich.

Results

We measured depolarization-evoked [Ca2+]i responses and Ca2+ currents simultaneously in fura-2 AM–loaded sympathetic neurons under voltage clamp, using the perforated patch technique to reduce disruption of the intracellular milieu. Neurons were depolarized from a holding potential of −70 mV to voltages ranging from −35 to −10 mV to stimulate Ca2+ entry through predominantly N-type Ca2+ channels (Jones and Mark, 1989) at different rates, and the resulting [Ca2+]i responses were recorded. We examined five cases of increasing complexity. In each case, we compared measured [Ca2+]i responses with calculated responses deduced from quantitative descriptions of Ca2+ transport and buffering obtained under the same experimental conditions, using the measured Ca2+ current to describe the rate of Ca2+ entry during the response. To minimize complications arising from cell-to-cell variability, we compared responses elicited in the simplest cases (Cases 1–3) in the same cells. While this required long and stable ICa and [Ca2+]i recordings, it made it feasible to address the following questions. (a) Given a description of Ca2+ handling that accounts for [Ca2+]i responses elicited by weak depolarization when Ca2+ uptake and release by internal stores is inhibited (Case 1), is it possible to account for responses elicited by the same stimulus when mitochondrial Ca2+ uptake is enabled (Case 2) by simply adding a functional description of the mitochondrial uniporter? (b) Does a description of Ca2+ handling that accounts for responses to weak depolarization also account for responses to strong depolarization if the larger Ca2+ current is used as the “input” in the calculations? After addressing Cases 1–3, we analyzed contributions from Ca2+ release by the mitochondrial Na/Ca exchanger (Case 4) and Ca2+ uptake and release by the ER (Case 5). Finally we compared published measurements of depolarization-induced changes in mitochondrial and ER total [Ca] levels with calculated changes deduced from the model incorporating all the measured Ca2+ handling systems.

Case 1. Cytoplasmic [Ca2+] Dynamics Regulated by Ca2+ Entry, Extrusion, and Buffering

We start with the simplest case in which [Ca2+]i dynamics depend on the interplay between Ca2+ transport across the surface membrane and cytoplasmic Ca2+ buffering, without contributions from internal stores. For these measurements, cells were treated with FCCP (1 μM) and Tg (20–200 nM) to inhibit Ca2+ uptake by mitochondria and the ER, respectively. Fig. 2 (A and B) illustrates a representative response. Weak depolarization from −70 to −35 mV evoked a Ca2+ current (ICa; Fig. 2 A) that led to an increase in [Ca2+]i from a resting level of ∼60 nM to slightly over 600 nM. After repolarization, ICa underwent rapid deactivation and [Ca2+]i recovered slowly toward its prestimulation level.

Measurement and Characterization of the Total Ca2+ Flux.

To develop a quantitative description of Ca2+ dynamics, we characterized the Ca2+ handling systems that are expected to define how [Ca2+]i changes during and after stimulated Ca2+ entry. It was then asked if these characterizations, when taken together, can account for the observed [Ca2+]i responses.

The total cytoplasmic Ca2+ flux (Ji, nM/s) was measured throughout the response by calculating −d[Ca2+]i/dt at each point in time. Assuming that (a) Ca2+ transport by stores is completely inhibited, (b) cytoplasmic Ca2+ is uniformly distributed, and that (c) Ca2+ binding to cytoplasmic buffers reaches equilibrium rapidly compared with [Ca2+]i relaxations induced by Ca2+ entry, Ji has the following simple physical interpretation. It is the net flux of Ca2+ across the surface membrane per unit cytoplasmic volume scaled by a [Ca2+]i-dependent buffering factor (κi) that describes the relationship between changes in free and total Ca concentration (Eqs. M4 and M5). These fluxes will be referred to as “free Ca2+ fluxes” (e.g., in nM/s) and represented by J's, to distinguish them from the net Ca2+ fluxes representing the amount of Ca2+ transported per unit time, which will be designated by a tilde (

\({\tilde{J}}\)
⁠, e.g., in nmol/s).

Fig. 2 C plots Ji versus [Ca2+]i throughout the response, distinguishing between the onset and recovery phases of the response. During the onset, Ji first increases rapidly to become a large inward flux (transition 1®2, see downward “On” arrow; note that inward fluxes are negative). Ji then hovers at ∼−50 nM/s while [Ca2+]i <200 nM before declining in magnitude during the remainder of the onset as [Ca2+]i approaches its peak value (Point 3). After repolarization, Ji changes sign to become an outward flux (transition 3®4, see upward “Off” arrow) and then declines toward zero as [Ca2+]i approaches its resting level (Point 5). We were unable to resolve the rapid changes in Ji that occur during the early moments of the onset and recovery that parallel ICa activation and deactivation, accounting for the apparent discontinuities in Ji between points 1,2, and 3,4.

Analysis of the Underlying Ca2+ Handling Systems.

Inspection of Fig. 2 C shows that Ji declines during the onset between points 2 and 3. This is due to at least two factors. First, ICa undergoes partial inactivation (Fig. 2 A). Second, an outward flux develops as [Ca2+]i rises, which reduces the impact of Ca2+ entry on [Ca2+]i. This flux is revealed after repolarization and ICa deactivation (Fig. 2 C, upward Off arrow) and is responsible for the subsequent [Ca2+]i recovery. It represents the activity of all Ca2+ extrusion systems that operate during the recovery (Herrington et al., 1996; Wennemuth et al., 2003), including plasma membrane Ca2+ ATPase (PMCA), as well as a Ca2+ leak. Contributions from the plasma membrane Na/Ca exchanger are minimal because internal and external solutions are Na+ free. This flux, which will be referred to as JExtru, is given by the rate at which [Ca2+]i declines during the recovery:

 
\begin{eqnarray*}&&\frac{d[Ca^{2{+}}]_{i}(t)}{dt}\left|\right._{\mathrm{Recov}}{=}{-}J_{i}^{\mathrm{Recov}}(t)\\&&{=}{-}J_{Extru}([Ca^{2{+}}]_{i}(t)).\end{eqnarray*}
(1)

Our previous work has shown that during the recovery, JExtru depends solely on [Ca2+]i, indicating that the underlying Ca2+ transport systems adjust rapidly to changes in [Ca2+]i. Fig. 2 C (top, smooth curve) illustrates the [Ca2+]i dependence of JExtru.

We hypothesized that during the response, Ji is the sum of two flux components, voltage-sensitive Ca2+ entry (JVSCC) and JExtru. According to this idea, Ji is influenced by both JVSCC and JExtru during depolarization, but only JExtru during the recovery (or rather that portion of the recovery after complete Ca2+ channel deactivation, when JVSCC = 0). Extending our description of [Ca2+]i dynamics to include the period of depolarization, and describing cytoplasmic Ca2+ buffering as in Materials and methods, we have

 
\begin{eqnarray*}&&\frac{d[Ca^{2{+}}]_{i}(t)}{dt}{=}{-}J_{i}(t)\\&&{=}{-}[J_{VSCC}(t){+}J_{Extru}(t)]\\&&{=}{-}\left[\frac{I_{Ca}(t)}{2Fv_{Cell}f_{i}{\kappa}_{i}([Ca^{2{+}}]_{i}(t))}{+}J_{Extru}([Ca^{2{+}}]_{i}(t))\right].\end{eqnarray*}
(2)

Reconstruction of the [Ca2+]i Response.

We reasoned that if JExtru has the same [Ca2+]i dependence during depolarization as it does during the recovery, then measurement of JExtru during the recovery will provide a reliable description of Ca2+ removal during depolarization. In this case, it should be possible to reconstruct the entire [Ca2+]i response by numerically integrating Eq. 2 subject to the initial condition:

 
\[[Ca^{2{+}}]_{i}(0){=}[Ca^{2{+}}]_{i,rest},\]
(3)

using the measured current as a time-dependent input to describe the rate of voltage-sensitive Ca2+ entry at each point in time.

To describe the [Ca2+]i dependence of JExtru, we used a look-up table consisting of the flux measurements at each [Ca2+]i, using linear interpolation to obtain values at each time step during numerical integration, or alternatively a smooth curve fit to the measurements based on Eqs. A4, and A5 (Fig. 2 C). These approaches gave essentially identical results. To describe the [Ca2+]i dependence of fiκi, we used On and Off responses to estimate the strength of Ca2+ buffering at low and high [Ca2+]i (Fig. 1) and used these measurements to obtain initial estimates of the two parameters of fiκi (P1, P2) (see Eq. M10).

Using these descriptions of Ca2+ entry, extrusion, and buffering, we found that numerical integration of Eq. 2 provided a remarkably good description of the [Ca2+]i response, both during the onset and the recovery, without any free parameters (Fig. 2 B, gray trace). Since the [Ca2+]i dependence of fiκi is based on measurements at only two [Ca2+]i levels (near resting and peak [Ca2+]i), we asked if a better approximation of the [Ca2+]i response could be obtained by fitting the response, treating P1 and P2 as adjustable parameters. It was found that after fitting, the integrated response even more closely approximated the measured one (Fig. 2 B, dark trace). Moreover, the [Ca2+]i dependence of fiκi calculated based on the optimized values of P1 and P2 was consistent with the initial estimates obtained from analysis of the On and Off responses (Fig. 2 E, compare thick curve with filled squares). The same analysis was performed on three cells stimulated with the same protocol, yielding similar results (Fig. 2 E). For these cells, the parameter values describing extrusion and buffering were Vmax,Extru = 29.8 ± 1.5 nM/s, nExtru = 3.0 ± 0.3, fiκEndog = 63.0 ± 18.0, and fiBtotal,Fura = 66.4 ± 12.2 μM.

What conclusions can be drawn from these results? While reconstruction of the [Ca2+]i recovery follows trivially from the description of JExtru as −d[Ca2+]i/dt and the value of [Ca2+]i at the instant of repolarization, reconstruction of the response onset does not. The description of JExtru was obtained from analysis of the recovery, when the membrane potential was clamped at −70 mV, while during the onset Vm = −35 mV. This indicates that the description of JExtru obtained from analysis of the recovery is also adequate to describe Ca2+ extrusion during the onset when the membrane potential is depolarized. Thus, the Ca2+ extrusion systems that are responsible for restoring [Ca2+]i in the aftermath of depolarization do not appear to be strongly voltage dependent, at least over the range −70 to −35 mV. Moreover, the results are consistent with the idea that for a given (spatially averaged) [Ca2+]i level, these systems generate a similar Ca2+ flux during the onset and recovery, as if the submembrane [Ca2+]i levels that regulate extrusion during these phases of the response are comparable.

Properties of the Ca2+ Flux Components.

Having validated our description of Ca2+ dynamics in Case 1, it is possible to obtain a description of the component of Ji representing Ca2+ entry during the response onset. This component (JVSCC) can be calculated by subtracting JExtru from Ji at corresponding values of [Ca2+]i (Fig. 2 C, top, shaded region and bottom trace), which follows directly from the description of Ji as the sum of JVSCC and JExtru and the functional dependence of JExtru on [Ca2+]i. After depolarization, JVSCC undergoes a rapid increase in magnitude when it dominates Ji (Fig. 2 C, transition 1 2), which is followed by a secondary increase before declining to ∼−30 nM/s as [Ca2+]i approaches its peak value. The secondary increase might seem paradoxical, given that it occurs while ICa inactivates. However, since JVSCC depends on the rate Ca2+ entry and the strength of Ca2+ buffering (Fig. 2 E), it is necessary to consider both sources of variation when evaluating changes in this flux during the onset.

Given measurements of the free Ca2+ fluxes and the [Ca2+]i dependence of fiκi, the net Ca2+ fluxes can be obtained by multiplying the free Ca2+ fluxes by fiκi at corresponding values of [Ca2+]i. The resulting fluxes are normalized by cell volume (not cytoplasmic volume, see Eq. M8) and will be designated by quotes to distinguish them from the un-normalized net fluxes. Fig. 2 D shows the volume-normalized net Ca2+ fluxes corresponding to the free Ca2+ fluxes in Fig. 2 C. These fluxes are larger in magnitude than the free Ca2+ fluxes (note difference in the scale, μM/s vs. nM/s) and show a different dependence on [Ca2+]i. Most notably, after their initial rapid increase following depolarization,

\({\tilde{J}}_{i}\)
and
\({\tilde{J}}_{VSCC}\)
both decay monotonically, in contrast to the free Ca2+ fluxes. Therefore, JVSCC does not parallel ICa during the response onset because it is the ratio of two declining quantities,
\({\tilde{J}}_{VSCC}\)
and fiκi, with the latter falling more rapidly than the former. This relationship between the time dependence of
\({\tilde{J}}_{VSCC}\)
and the [Ca2+]i dependence of fiκi is important, because it ultimately determines how [Ca2+]i changes in response to depolarization.

Fig. 2 F shows how the net Ca2+ fluxes vary with time, illustrating how the rate of net Ca2+ transport depends continuously on the relative rates of Ca2+ entry and extrusion. Comparing plots of the net fluxes vs. [Ca2+]i (Fig. 2 D) and time (Fig. 2 F) also illustrates an important difference between the independent variables that control the components of

\({\tilde{J}}_{i}\)
⁠.
\({\tilde{J}}_{VSCC}\)
is a function of time, expressing the time dependence of ICa at constant voltage, while
\({\tilde{J}}_{Extru}\)
is a function of [Ca2+]i that varies because [Ca2+]i is dynamic.
\({\tilde{J}}_{VSCC}\)
may also depend on [Ca2+]i, e.g., through [Ca2+]i-dependent modulation of N-type Ca2+ channel activity (Liang et al., 2003), but it is not necessary to explicitly include such effects in our description of Ca2+ entry since their impact is included in the direct measurements of ICa(t).

These results demonstrate that in the absence of Ca2+ uptake and release by internal stores, [Ca2+]i responses elicited by weak depolarization can be accurately described by a spatially uniform system endowed with [Ca2+]i-dependent Ca2+ extrusion and buffering mechanisms subject to time-dependent Ca2+ entry.

Case 2. Contributions from the Mitochondrial Ca2+ Uniporter during Weak Depolarization

This section describes the added contribution of the mitochondrial Ca2+ uniporter to depolarization-evoked [Ca2+]i responses, using the same depolarization protocol as in the preceding section. Qualitatively, mitochondrial Ca2+ uptake is expected to increase the rate of cytoplasmic Ca2+ removal to an extent that increases steeply with [Ca2+]i. To study this directly, the same cells that were depolarized in the presence of FCCP were also stimulated in its absence to enable the uniporter. Since these cells were voltage clamped using a pipette solution that contained no added Na+, the FCCP-sensitive flux provides a description of Ca2+ uptake via the uniporter, without contamination from the Na/Ca exchanger, which requires internal Na+ for its operation.

Fig. 3 (A and B) illustrates a response elicited under these conditions from the same cell described in Fig. 2 before it was exposed to FCCP. While the Ca2+ current elicited by the voltage step was very similar to that observed in the presence of FCCP (compare dark and light traces in Fig. 3 A), [Ca2+]i increased more slowly during the onset, showed a smaller peak, and recovered faster, consistent with the expected effects of enabling mitochondrial Ca2+ uptake.

To investigate the free cytoplasmic Ca2+ flux during this response, we again calculated Ji = −d[Ca2+]i/dt during the onset and recovery (Fig. 3 C). As in Case 1, Ji showed an initial rapid increase in magnitude followed by a slow secondary rise and ultimately a decline toward zero as [Ca2+]i approached a steady elevated level (Point 3). After repolarization and ICa deactivation, Ji changed sign to become an outward flux, which then declined toward zero as [Ca2+]i approached its resting level.

Analysis of the Ca2+ Handling Systems.

Our previous work has shown that the decline in [Ca2+]i during the recovery under these conditions of stimulation is influenced by the combined actions of Ca2+ extrusion and mitochondrial Ca2+ uptake by the uniporter (Colegrove et al., 2000a). Like JExtru, the rate of Ca2+ uptake by the uniporter (Juni) depends on [Ca2+]i but not its history. Therefore, since both components of the free Ca2+ flux are defined by [Ca2+]i, so is their sum:

 
\begin{eqnarray*}&&\frac{d[Ca^{2{+}}]_{i}(t)}{dt}\left|\right._{\mathrm{Recov}}{=}{-}J_{i}^{\mathrm{Recov}}(t)\\&&{=}{-}\left[J_{Extru}([Ca^{2{+}}]_{i}(t)){+}J_{uni}([Ca^{2{+}}]_{i}(t))\right]{\vert}_{\mathrm{Recov}}.\end{eqnarray*}
(4)

(see Fig. 3 C). Extending this description to include the period of depolarization, we have

 
\begin{eqnarray*}&&\frac{d[Ca^{2{+}}]_{i}(t)}{dt}{=}\\&&{-}\left[\frac{I_{Ca}(t)}{2Fv_{Cell}f_{i}{\kappa}_{i}([Ca^{2{+}}]_{i}(t))}{+}\left[J_{Extru}([Ca^{2{+}}]_{i}(t)){+}J_{uni}([Ca^{2{+}}]_{i}(t))\right]\right],\end{eqnarray*}
(5)

subject to initial condition (Eq. 3).

Reconstruction of the [Ca2+]i Response.

To determine if Eq. 5 accurately describes [Ca2+]i dynamics when mitochondrial Ca2+ uptake is enabled, we integrated this equation and compared the results with measured responses. The [Ca2+]i dependence of JExtru+Juni was described using a look-up table as in Case 1, or by fitting an equation to the Ji measurements during the recovery (Fig. 3 C). For fitting, the same equation and parameters were used for JExtru as in Case 1 and Juni was described using Eq. A12. We started with the characterization of Ca2+ buffering obtained from the same cell in the presence of FCCP, since the measured strength of Ca2+ buffering was similar in the two cases (Fig. 3 E, compare filled and open squares). With these descriptions of Ca2+ handling, we found that the calculated response closely resembled the measured response (Fig. 3 B, compare gray trace with dots), although it underestimated the [Ca2+]i measurements during the period of depolarization. After optimizing the buffering parameters, the integrated [Ca2+]i response was essentially indistinguishable from the measured one (Fig. 3 B, dark trace), and the [Ca2+]i dependence of fiki (Fig. 3 E, thick curve) was consistent with the measured values in the presence and absence of FCCP. The same comparison between responses elicited in the presence and absence of FCCP was performed in two other cells, giving comparable results (Fig. 3 E). Analysis of four additional cells stimulated under Case 2 conditions gave similar results. Overall, for these seven cells fiκEndog = 53.9 ± 10.2, fiBtotal,Fura = 65 ± 13 μM, A = 4 ± 2 × 10−4, and nuni = 1.0 ± 0.2.

Properties of the Ca2+ Flux Components.

Analysis of the Ca2+ flux components (Fig. 3 C) shows how JVSCC varies with [Ca2+]i during the response onset. JVSCC increases rapidly after depolarization (Fig. 3 C, top, transition 1 2) when it dominates Ji, and continues to increase despite inactivation of ICa. As in Case 1, this occurs because there are quantitative differences in the way ICa and fiκi change during the onset. While ICa shows a time course that is very similar to that observed in the presence of FCCP, fiκi changes more dramatically because [Ca2+]i is restricted to a lower range, where fiκi is more sensitive to [Ca2+]i. As a result, the ratio ICa/fiκi increases throughout the onset, despite the decline in ICa due to inactivation. This complex behavior in not observed with the (volume-normalized) net Ca2+ fluxes obtained by multiplying the free Ca2+ fluxes by fiκi at corresponding [Ca2+]i levels (Fig. 3 D). In particular, both

\({\tilde{J}}_{i}\)
and
\({\tilde{J}}_{VSCC}\)
decline monotonically as [Ca2+]i rises throughout the depolarization (Fig. 3 D).

Perhaps the most striking difference observed when [Ca2+]i responses are elicited while mitochondrial Ca2+ uptake is enabled is that [Ca2+]i elevations are smaller and rates of Ca2+ removal are higher at all [Ca2+]i levels. For example, when mitochondrial Ca2+ uptake is enabled, the rate of net Ca2+ removal after repolarization, when [Ca2+]i ∼350 nM, exceeds 4 μM/s, roughly twice the rate observed at the corresponding [Ca2+]i level when mitochondrial Ca2+ uptake is disabled (Fig. 2 D and Fig. 3 D). In addition, when the uniporter is enabled, Ca2+ entry and removal rates are in balance when [Ca2+]i reaches a much lower level than that attained during depolarization in Case 1. When this same [Ca2+]i level is reached in Case 1 (Fig. 2), there is still a large imbalance between Ca2+ entry and extrusion rates that favors a continued rise in [Ca2+]i. Mitochondrial Ca2+ uptake is not expected to influence steady-state [Ca2+]i elevations when Ca2+ release by the Na/Ca exchanger is enabled and operates at less than its maximal rate. However, when release is inhibited, or saturated after a large Ca2+ load, mitochondria are expected to behave as “absorbing” organelles that depress [Ca2+]i. Consistent with this idea, we found that in three cells with low internal Na+, which would be expected to promote continuous mitochondrial Ca2+ uptake, exposure to FCCP increased resting [Ca2+]i (unpublished data). In intact cells with normal [Na+]i, this is not observed (Friel and Tsien, 1994).

We quantified the rate of mitochondrial Ca2+ uptake via the uniporter (Juni) at each point in time (t) during the recovery by subtracting from the total flux Ji(t) the FCCP-resistant flux evaluated at [Ca2+]i(t) (Eq. M11). This flux component is illustrated by the shaded region in Fig. 3 C; the corresponding (volume-normalized) net Ca2+ flux is shown in panel D. Fig. 3 F compares the time courses of

\({\tilde{J}}_{VSCC}\)
and
\({\tilde{J}}_{Extru}{+}{\tilde{J}}_{uni}\)
(thin traces) and their sum
\({\tilde{J}}_{i}\)
(thick trace) throughout the response. As in Case 1,
\({\tilde{J}}_{i}\)
depends on the imbalance between Ca2+ entry and removal rates, the former depending explicitly on time (through the time dependence of ICa), the latter depending on [Ca2+]i (through the [Ca2+]i dependence of Ca2+ removal rates and buffering strength).

It should be noted that previous work suggests that mitochondrial Ca2+ uptake does not influence [Ca2+]i dynamics appreciably when stimuli are weak and [Ca2+]i is low (Thayer and Miller, 1990; Friel and Tsien, 1994). However, comparison of Figs. 2 and 3 shows that [Ca2+]i responses elicited by weak depolarization were considerably larger when mitochondrial Ca2+ uptake is inhibited. Two factors contribute to this difference. First, our previous results indicate that FCCP has little effect on [Ca2+]i responses in intact sympathetic neurons when [Ca2+]i ≤ ∼200 nM (Friel and Tsien, 1994), a lower range than that achieved by 20-s depolarization to −35 mV under voltage clamp, which exceeds 350 nM. Second, the responses described here were elicited in cells with low internal Na+. By suppressing Na+-dependent mitochondrial Ca2+ release, this would leave uptake unopposed, increasing the overall rate of mitochondrial Ca2+ accumulation and exaggerating the difference between responses elicited in the presence and absence of FCCP.

Another way to illustrate the impact of the uniporter on [Ca2+]i responses evoked by weak depolarization is to integrate Eq. 5 after setting A = 0 in Eq. A12 to eliminate contributions from the uniporter in the calculations. This produced a response (Fig. 3 B, dashed trace) that was very similar to that observed in the presence of FCCP (Fig. 2 B), the small differences being accounted for by differences in ICa, resting [Ca2+]i, and buffering strength (unpublished data).

Case 3. [Ca2+]i Responses Elicited by Strong Depolarization

We next asked if the Ca2+ handling systems that define responses to weak depolarization in the preceding case can also account for [Ca2+]i elevations elicited by strong depolarization. To examine this, cells were depolarized for 2–3 s from −70 to −10 mV. A response elicited by such a stimulus in the same cell examined in Cases 1 and 2 above is presented in Fig. 4 (A and B). The most obvious differences are the larger Ca2+ current (peak ∼−3.3 nA) and [Ca2+]i elevation (∼1 μM). Two observations indicate that responses like those shown in Fig. 4 B are within the dynamic range of fura-2. First, larger [Ca2+]i elevations were observed after exposure to FCCP (unpublished data). Second, measurements in other cells showed that stronger depolarizations produce even larger [Ca2+]i responses (Fig. 4 F).

Fig. 4 C shows Ji calculated during the onset and recovery phases of the response. Qualitatively, Ji changes during depolarization and after repolarization as with the weaker stimuli in Cases 1 and 2 (see Fig. 2 C and Fig. 3 C). However, the magnitude of the flux is much larger. We asked if the same equation (Eq. 5) that describes cytoplasmic Ca2+ handling in Case 2 can also describe [Ca2+]i responses elicited by stronger depolarization if the larger Ca2+ current is used as the time-dependent input. As in Case 2, measurements of Ji during the recovery were used to characterize the overall rate of Ca2+ removal (Fig. 4 C). Panel D shows the region enclosed by the dashed box in panel C on an expanded scale. The rate of Ca2+ removal increases steeply with [Ca2+]i, and shows a [Ca2+]i dependence very similar to that observed after weak depolarization over the common range of [Ca2+]i (open symbols).

Analysis of the Underlying Ca2+ Handling Systems.

We measured the component of Ji representing Ca2+ uptake by the uniporter by subtracting JExtru from Ji at corresponding values of [Ca2+]i (shaded region in Fig. 4 D) and described the [Ca2+]i dependence of Juni using Eq. A12. Based on analysis of three cells, A = 5.6 ± 2.8 × 10−6 and nuni = 1.8 ± 0.2, which is somewhat different from the parameter values obtained by fitting over the lower range of [Ca2+]i in Case 2 (see Appendix). Measurements of Ca2+ buffering strength near resting [Ca2+]i were consistent with analysis of the same cell during weak depolarization in Cases 1 and 2 (Fig. 4 E), even though the rate of Ca2+ entry was >10 times higher. However, the apparent buffering strength measured at high [Ca2+]i levels was approximately twofold higher than would be predicted based on extrapolations of the fiki curve deduced from responses to weak depolarization. Although these estimates of buffering strength are derived from noisy measurements, a similar discrepancy was seen in each of the three cells studied in Cases 1–3 (Fig. 4 E, see box). This contrasts with the tightly clustered values of buffering strength near resting [Ca2+]i measured from On responses to strong depolarization.

Comparison between Measured Responses and Calculations Based on the Spatially Uniform Model.

Calculations based on Eq. 5 were performed using the Ca2+ buffering parameters obtained from the same cells under Case 1 conditions, reasoning that buffering strength should be similar, at least for overlapping ranges of [Ca2+]i. While the simulated responses closely paralleled the measured [Ca2+]i responses during the recovery, as they must, given the way Ca2+ removal rate is defined in Eq. 5, the calculated values of [Ca2+]i during depolarization were much larger than the measured ones (Fig. 4 B, compare continuous trace with dots). This was paralleled by a nearly twofold overestimation of Ji during the onset (Fig. 4 C, bottom traces). As shown above, this difference cannot be accounted for by fura-2 saturation (Fig. 4 F). Overall, we were unable to find a simple description of fiκi that made it possible to reconstruct [Ca2+]i responses elicited by strong depolarization that also agreed with measurements obtained from responses elicited by weak depolarization. Therefore, we considered alternative explanations of the systematic difference between measured and calculated [Ca2+]i during strong depolarization.

Reconstruction of the [Ca2+]i Response Using a Diffusion Model.

One possibility is that, in contrast to weak depolarization, the rate of Ca2+ entry during strong depolarization is sufficiently high compared with the rate of radial diffusion that [Ca2+]i levels beneath the plasma membrane greatly exceed the spatial average. This would cause Ca2+ removal rates to be higher than expected based on spatially averaged [Ca2+]i levels. While this would be expected to include a higher rate of Ca2+ extrusion, the increase in JExtru required to account for the upward shift in Ji during the onset (Fig. 4 C) is >25 times larger than the estimated maximal rate of extrusion. Increased mitochondrial Ca2+ uptake is more likely to play an important role. Given the steep [Ca2+] dependence of Ca2+ uptake by the uniporter, large increases in [Ca2+]i in outer cytoplasmic shells would lead to disproportionately higher rates of mitochondrial Ca2+ uptake in those same shells. Indeed, measurements of total mitochondrial Ca concentration indicate that this occurs (Pivovarova et al., 1999; Hongpaisan et al., 2001; see below). One consequence would be that a larger fraction of the Ca2+ that enters the cytoplasm during depolarization would be taken up by peripheral mitochondria than expected based on the magnitude of spatially averaged [Ca2+]i. As a result, a smaller fraction of the Ca2+ that enters the cell would be available to bind fura-2, in essence, hiding it from the fluorescence measurements.

One approach to investigating this possibility is to incorporate the measured descriptions of Ca2+ handling into a model that takes into consideration spatial variations of [Ca2+]i within the cytoplasm. To do this, the model cell was divided into 49 concentric shells, with a small central sphere. The same equations were used to describe Ca2+ movements across the plasma membrane as before, and the movements of Ca2+ and buffers between shells were described by a diffusion equation assuming fast buffering (Eq. A27). Finally, we assumed that mitochondria are distributed uniformly throughout the cell interior and express Juni with the same [Ca2+]i dependence as that determined during the recovery, when [Ca2+]i is expected to be more nearly uniform (see below). In this diffusion model, there are three additional parameters beyond those used in the uniform model: the diffusion coefficients of Ca2+ (DCa) and of the endogenous and exogenous Ca2+ buffers (DEndog and Dfura2).

Fig. 5 illustrates the main findings from the diffusion model. Fig. 5 A shows the measured current and panel B illustrates the measured [Ca2+]i response along with the following simulated quantities: [Ca2+]i in the central and outermost shells, the spatial average of [Ca2+]i calculated according to Eq. A32, and finally the spatial average calculated based on the fura-2 calibration parameters used in the original experiments (Eqs. A31 and A33). The calculations shown in Fig. 5 B were made with DEndog = 15 μm2/s. Smaller and larger values systematically under- and overestimate the measured spatial average of [Ca2+]i during depolarization, respectively (Fig. 5, C and D). Indeed, there is a distinct minimum in the error of the calculated [Ca2+]i responses near this value of DEndog (Fig. 5, D and E). In three cells, the optimal value of DEndog fell within the range 3–15 μm2/s, consistent with an endogenous buffer having low mobility.

We also tested how the agreement between simulated and measured responses depends jointly on DEndog and DFura by varying each in the range 0–50 μm2/s, while holding DCa constant at 300 μm2/s. We found that the optimal combination was DEndog= 15 and DFura= 50 μm2/s. When DEndog was held at 15 μm2/s and DFura was varied from 0 to 50 μm2/s, spanning the range of values used in several previous studies (see McHugh and Kenyon, 2004), the error declined slowly and monotonically as DFura approached 50 μm2/s. Thus, over the range of expected values for DFura, the optimal value is 50 μm2/s. Changing DCa over the range 50–300 μm2/s, while holding DEndog = 15 and DFura = 50, had relatively little effect, with sizable effects on the error being evident only when DCa < 50 μm2/s.

It should be noted that in Case 3 the radial nonuniformity of [Ca2+]i that is observed during depolarization collapses shortly after repolarization (Fig. 5 B). Thus, analysis of (spatially averaged) [Ca2+]i during the recovery is expected to provide information about the [Ca2+]i dependence of Ca2+ buffering and transport rates that extends to Ca2+ handling during the onset.

In contrast to simulated responses to strong depolarization, responses to weak depolarization (Fig. 5, right) show little spatial heterogeneity during the stimulus and are not very sensitive to DEndog over the range 0–25 μm2/s (Fig. 5, G–J). In fact, responses to weak depolarization based on the diffusion model are very similar to those deduced from the spatially uniform model. Thus, when Ca2+ entry rates are low, free diffusion is adequate to ensure that [Ca2+]i is essentially uniform.

While these results suggest that steep [Ca2+]i gradients coupled with nonlinear Ca2+ uptake by mitochondria may contribute significantly to the difference between measured and calculated [Ca2+]i during strong depolarization, another possible contribution arises from [Ca2+]i estimations based on spatially averaged fluorescence measurements (McHugh and Kenyon, 2004). In brief, the equation used to calculate [Ca2+]i from fluorescence measurements depends nonlinearly on fluorescence intensity (see Eqs. A31 and A33). As a result, averaging contributions to fluorescence intensity from regions with different [Ca2+]i levels can systematically underestimate the true spatial average. To examine possible contributions from this effect, we calculated the ratio of the spatially averaged fluorescence intensities at two excitation wavelengths (350 and 380 nm) based on calibration parameters from the original experiments. We then used this ratio to estimate the average [Ca2+]i that would be reported by fura-2 fluorescence measurements. The results are illustrated in Fig. 5 B by the red dashed curve, which is slightly lower than the “true” spatial average of [Ca2+]i calculated according to Eq. A32 (red continuous curve). These results suggest that a combination of low mobility buffers, nonlinear Ca2+ uptake by near-membrane mitochondria, and underestimation of the spatial average of [Ca2+]i resulting from spatially averaged fluorescence measurements accounts for the discrepancy between calculations based on the spatially uniform model and measured responses.

Case 4. Inclusion of the Mitochondrial Na/Ca Exchanger

In Cases 2 and 3, mitochondrial Ca2+ release was inhibited because the pipette solution did not contain added Na+, a necessary substrate of the Na/Ca exchanger. In these cases, mitochondria can be viewed as Ca2+-absorbing organelles, as long as the driving force for uptake is maintained. In this section, we examine the added contribution to Ca2+ dynamics from the Na/Ca exchanger, which was enabled by including Na+ (6.5 mM) in the pipette solution. Under these conditions, depolarization from −70 to −10 mV leads to [Ca2+]i currents and [Ca2+]i elevations (Fig. 6, A and B) like those seen in the preceding section. However, recoveries display a slow plateau phase lasting many seconds, which is prolonged by Ca2+ release from mitochondria that are loaded during depolarization (Thayer and Miller, 1990; Friel and Tsien, 1994).

Measurement and Characterization of the Ca2+ Flux Generated by the Na/Ca Exchanger.

To investigate Ca2+ release via the Na/Ca exchanger, we depolarized Tg-treated cells in the presence and absence of CGP 37157 (CGP), a specific inhibitor of the exchanger (Cox and Matlib, 1993; Colegrove et al., 2000a). JNaCa was then measured as the CGP-sensitive component of the total cytoplasmic Ca2+ flux (Eq. M12, Fig. 6 C). This flux component represents Ca2+ release during most of the recovery (after vertical dotted line in Fig. 6 B; see shaded region in C). Flux measurements over this period are shown on an expanded scale in Fig. 6 D. We focused on the CGP-sensitive flux during this phase of the recovery, instead of during the initial rapid phase, when it is difficult to measure reliably. This has little consequence, since the CGP-sensitive flux constitutes a small fraction of the total flux during the initial phase of recovery, when [Ca2+]i is high and the uniporter dominates Ca2+ removal. In contrast, this flux component represents a substantial fraction of Ji during the late, slow, phase of recovery and therefore strongly influences recovery kinetics (Colegrove et al., 2000b).

Analysis of JNaCa is more complicated than any of the other fluxes described above because it depends on intramitochondrial free Ca2+ concentration ([Ca2+]MT), which was not directly measured in these experiments. In the cases analyzed above, each Ca2+ flux (except for JVSCC) could be described as a function of measured [Ca2+]i. To overcome this difficulty, we used an indirect measure of [Ca2+]MT that was derived from measurements of [Ca2+]i. This method makes use of the integrated mitochondrial Ca2+ flux during the recovery as a way to describe changes in intramitochondrial Ca2+ concentration during the recovery, viewed from the vantage point of the cytoplasm. This quantity is represented by Δ[Ca2+]MT(0) (Eqs. A17 and A18). JNaCa was calculated as the difference between Ji and Ji,+CGP (shaded region in Fig. 6 D), plotted vs. Δ[Ca2+]MT(0) (Fig. 6 E), and then fit with a saturable function of [Ca2+]MT expressed in terms of Δ[Ca2+]MT(0) (Eq. A22; see smooth curve), which provided estimates of the parameters of JNaCa.

Reconstruction of the [Ca2+]i Response.

We then asked if this description of JNaCa, together with the other Ca2+ handling systems discussed in connection with Cases 2 and 3 makes it possible to reproduce measured [Ca2+]i responses. The response calculated based on measurements of Ca2+ removal and release rates in the same cell, and buffering parameters from Cases 1–3 above, is shown by the gray curve in Fig. 6 B. This response has the same shape as the measured response during the recovery, but slightly overestimates [Ca2+]i. However, small changes in the buffering parameters (<20%) led to better agreement with the measured response (Fig. 6 B, dark curve). Similar results were obtained in another cell stimulated and analyzed using the same protocol.

Fig. 6 F shows the calculated time course of [Ca2+]MT during the response. Note that [Ca2+]MT (t) is related to the measured quantity Δ[Ca2+]MT(0)(t) but also depends on the mitochondrial volume fraction and buffering strength (Eq. A18). This is simply because a given net Ca2+ flux delivered to a small mitochondrial compartment would produce a larger change in total mitochondrial Ca concentration than if the same flux were deposited into a larger volume. In addition, the impact of this change in [Ca]MT on intraluminal free Ca2+ levels depends on the strength of intramitochondrial Ca2+ buffering. The calculation illustrated in Fig. 6 F assumes a mitochondrial volume fraction of 0.01 and buffering strength of 4,000, according to which [Ca2+]MT reaches a peak of ∼7 μM during depolarization. Larger or smaller values of volume fraction or buffering strength would give proportionally different values.

The simulation illustrated in Fig. 6 B is based on the spatially uniform model and overshoots measured [Ca2+]i during the period of depolarization as in Case 3. Fig. 7 shows simulations that make use of the same descriptions of Ca2+ transport and buffering but in the context of the diffusion model. Fig. 7 B compares the calculated spatial average of [Ca2+]i that would be expected from fluorescence measurements and the measured values, which are in reasonably good agreement if DEndog ∼5 μm2/s.

A noteworthy feature of diffusion simulations that include mitochondrial Ca2+ uptake is the relatively long-lasting spatial nonuniformity of [Ca2+]MT during the recovery (Fig. 7 C), which contrasts with the virtually complete dissipation of [Ca2+]i gradients after repolarization (Fig. 7 B). The mitochondrial Ca2+ gradients are sustained longer than the cytoplasmic gradients because Ca2+ does not diffuse between mitochondria, and because the rate of release is slow compared with cytoplasmic diffusion.

Case 5. Effects of ER Ca2+ Uptake and Release

Fig. 8 illustrates a diffusion simulation using the same stimulus and parameter values as in Case 4, but with the addition of a second internal compartment representing the ER. We used descriptions of ER Ca2+ uptake and release from our previous work (Albrecht et al., 2002; Eqs. A2426) that are based on measurements from multiple cells, since we have not obtained a complete set of flux measurements describing mitochondrial and ER Ca2+ transport in the same cell.

There are several interesting properties of these [Ca2+] responses. Inclusion of the ER does not strongly influence either cytoplasmic or mitochondrial [Ca2+] dynamics, consistent with the idea that under these conditions, mitochondria are more influential than the ER. The ER acts as a Ca2+ source during the early phase of the response, when [Ca2+]i is highest, and as a sink during most of the recovery. This is consistent with the biphasic [Ca2+]i dependence of the direction of net ER Ca2+ transport described previously in these cells (Albrecht et al., 2002). The initial decline in [Ca2+]ER is more pronounced in the outer shells, where [Ca2+]i is highest, compared with the central shells. However, in contrast to [Ca2+]MT, ER Ca2+ levels are spatially much more uniform than [Ca2+]MT, showing only ∼5% difference between central and peripheral regions. [Ca2+]ER displays a prolonged overshoot during the recovery that continues even after mitochondrial [Ca2+]MT is close to its prestimulation level. This agrees with measurements of total ER Ca concentration during the recovery after depolarization in the same cells (Hongpaisan et al., 2001) as well as in neurons in cultured hippocampal slices after field stimulation (Pozzo-Miller et al., 1997). Simulations over longer times demonstrate that this overshoot is transient (unpublished data).

Comparison between Simulated and Measured Changes in Total Mitochondrial and ER Ca Concentrations.

Our previous work has described depolarization-induced changes in total mitochondrial and ER Ca concentration in sympathetic neurons (Pivovarova et al., 1999; Hongpaisan et al., 2001). We found that Ca2+ entry leads to a time-dependent increase in [Ca]MT and a biphasic change in [Ca]ER, where these measurements represent total organelle Ca concentration averaged over the entire cell. In addition, we described the spatial profile of total Ca concentration in these organelles at fixed time points during depolarization (Hongpaisan et al., 2001). It was found that mitochondria near the plasma membrane accumulate Ca at a higher rate than those situated further away. ER Ca transport is also more pronounced near the plasma membrane, but transport by this organelle favors net Ca2+ release. The distinct spatial variations in total organellar Ca concentration described in those studies presumably represents the combined effects of spatial variations in [Ca2+]i during stimulation and intrinsic differences in Ca2+ handling by mitochondria and the ER. Overall, these measurements provide another opportunity to evaluate our model by comparing measured and calculated total mitochondrial and ER Ca concentration. Before presenting the results, it is important to describe how changes in intraluminal Ca concentration are calculated in the model and how these calculations can be related to measured quantities.

Calculating Changes in Intraluminal Total Ca Concentration.

Intraluminal total Ca concentration ([Ca]) changes at a rate that depends on the rate of net Ca2+ transport between organelle and cytoplasm, as well as organelle volume. However, all transport rates were measured based on changes in cytoplasmic Ca2+ concentration. For example, we measured the mitochondrial free Ca2+ flux (JMT) as the FCCP-sensitive component of the cytoplasmic Ca2+ flux. This flux can be interpreted as the net Ca2+ flux between mitochondria and the cytoplasm (

\({\tilde{J}}_{MT}\)
⁠) divided by the product of the cytoplasmic volume vi and ki (Eq. M6):

 
\[J_{MT}{=}\frac{{\tilde{J}}_{MT}}{v_{i}{\kappa}_{i}}\]

To calculate changes in [Ca]MT during stimulation, a different quantity is needed,

\({\tilde{J}}_{MT}/v_{MT}\)
⁠, which can be integrated between any two times to give the change in [Ca]MT between those times. However, given an estimate of vMT/vCell,
\({\tilde{J}}_{MT}/v_{MT}\)
can be determined from the measured quantities JMT and fiκi as follows:

 
\begin{eqnarray*}&&\frac{{\tilde{J}}_{MT}}{v_{MT}}{=}J_{MT}\left[{\kappa}_{i}\left(\frac{v_{i}}{v_{MT}}\right)\right]\\&&{=}J_{MT}\left[f_{i}{\kappa}_{i}\left(\frac{v_{Cell}}{v_{MT}}\right)\right].\end{eqnarray*}
(6)

An analogous approach can be used to estimate the fluxes that lead to changes in total ER Ca concentration. These considerations underscore the importance of organellar volume fraction in calculating changes in total Ca concentration from measured free cytoplasmic Ca2+ fluxes. Volume fraction can be estimated from the fractional area occupied by the respective organelles in fixed sections (Weibel and Elias, 1967). Recent estimates in sympathetic neurons give 0.01–0.04 as the range for mitochondria (Andrews, S.B., personal communication), which is lower than the value we used previously (0.1).

Measurements of ER volume fraction are more difficult using this approach. Our previous estimate was ∼0.05–0.10 (Albrecht et al., 2002). A complementary approach is to use the model to determine the ER volume fraction for which the calculated resting [Ca]ER agrees with the measured value (∼3.6 mM; Hongpaisan et al., 2001). Using κER = 20 (Mogami et al., 1999), this approach yielded a value of 0.043. Such a method for estimating mitochondrial volume fraction is not possible because resting [Ca]MT is difficult to measure (Pivovarova et al., 1999).

With estimates of mitochondrial and ER volume fractions in hand, we asked if calculated and measured spatial averages of [Ca]MT and [Ca]ER agree at 45 s during depolarization. To make this comparison, it was necessary to construct an appropriate ICa for use in the model calculations. This is because the total Ca concentration was measured in sections from cells in ganglia that were rapidly frozen at different times during stimulation with 50 mM K+, making simultaneous measurement of ICa impractical. However, control experiments show that under these conditions of stimulation, the membrane potential is steady at ∼−20 mV (Friel and Tsien, 1992). In lieu of direct measurements of ICa, we fit a representative Ca2+ current elicited by depolarization to the same potential with a triexponential function plus a constant. The steady-state value of the fitted curve was constrained to give a simulated [Ca2+]i level at 45 s that agrees with measured values in cells exposed to 50 mM K+ (618 ± 86 nM, n = 4 cells). For these simulations, the fura-2 concentration was set to zero because cells in which elemental analysis was performed were not loaded with a Ca2+ indicator. This caused simulated [Ca2+]i to increase more rapidly during stimulation, but as expected, the steady level approached during maintained stimulation was not altered (unpublished data).

Comparison between Measured and Calculated Spatially Averaged Total Ca Concentrations.

Fig. 9 (A–D) shows the time courses of ICa, [Ca2+]i, [Ca]MT, and [Ca]ER simulated using parameters for plasma membrane, mitochondrial, and ER Ca2+ transport from Case 5 above. According to the model, the spatially averaged total mitochondrial Ca concentration increases throughout the period of Ca2+ entry, attaining a value at 45 s (21.6 mM; Fig. 9 C, red trace) that is between the measured values determined by EDX (8.7 mM) and EELS (29.7 mM) (Hongpaisan et al., 2001), where the latter average was calculated from the spatially resolved measurements according to Eq. A32. Total ER Ca concentration declined from its resting level (3.6 mM) under the influence of graded net Ca2+-induced Ca2+ release (CICR), reaching 3.5 and 3.0 mM at 10 and 45 s, respectively. This can be compared with EDX measurements at these times (2.9 and 3.2 mM; Hongpaisan et al., 2001). As expected, the calculated changes in [Ca]MT and [Ca]ER are inversely proportional to the respective volume fractions. Fig. 9 E (left) shows how for mitochondria the absolute difference between the measured (EELS) and calculated spatial average of [Ca]MT at 45 s depends on volume fraction. There is a distinct minimum at a volume fraction of 0.0065, where the calculated and measured values are nearly equal.

Comparison between Measured and Calculated Spatially Resolved Total Ca Concentrations.

Finally, we examined the spatial profiles of [Ca2+]i, [Ca]MT, and [Ca]ER at different times during depolarization (Fig. 9, F–H). As in previous studies (Neher and Augustine, 1992; Hua et al., 1993), the spatial profile of [Ca2+]i shows steep gradients during the first few seconds of depolarization that dissipate with time (Fig. 9 F). Over the same period, peripheral [Ca]MT increases more rapidly than in more central regions (Fig. 9 G) because [Ca2+]i is higher near the plasma membrane and because the rate of Ca2+ uptake by the uniporter increases steeply with [Ca2+]i. The measured spatial profiles of [Ca]MT (EELS) and [Ca]ER (EDX) at 45 and 10 s, respectively, are shown in Fig. 9 (G and H, symbols) (Hongpaisan et al., 2001). The calculated changes in total organelle Ca concentration were inversely proportional to volume fraction, and the steepness of the spatial profile depended strongly on DEndog, with small values favoring steeper gradients of both [Ca]MT and [Ca]ER. However, the spatial averages of [Ca]MT and [Ca]ER at fixed times were relatively insensitive to DEndog (unpublished data). Assuming a mitochondrial volume fraction of 1%, the RMS error between the calculated and measured spatial profile of [Ca]MT was minimal when DEndog= 0 (Fig. 9 E, right; see Fig. 9 G, dashed curve). We found that with mitochondrial volume fraction set to 0.65–1% and DEndog= 0–5 μm2/s, model calculations and experimental measurements were very similar (Fig. 9 G). Calculations of [Ca]ER were in good agreement with measurement in all regions except the outer shell, where the measurements suggest more pronounced Ca release from the ER during the first 10 s of stimulation. Overall, these results indicate that the diffusion model provides a good quantitative description of Ca2+ handling by mitochondria and the ER in sympathetic neurons, except that it appears to underestimate the magnitude of evoked changes in ER total Ca concentration within 2–4 μm of the plasma membrane.

Discussion

This work describes the reconstruction of depolarization-evoked Ca2+ responses based on measured characterizations of surface membrane, mitochondrial, and ER Ca2+ transport and cytoplasmic Ca2+ buffering in the same cell type. When incorporated into a diffusion model with compartments representing the cytoplasm, mitochondria, and ER, these characterizations make it possible to account quantitatively for many features of depolarization-induced Ca2+ responses. These include the time course of spatially averaged cytoplasmic Ca2+ concentration during and after depolarization, the spatial averages of intraluminal mitochondrial and ER total Ca concentration at a fixed time (45 s) after depolarization, and the spatial profile of [Ca]MT at the same time point. Overall, these results indicate that the main Ca2+ handling systems operating in these neurons have been taken into consideration, and that they have been described with sufficient accuracy to account for the measurements. Our model can be used to study responses to other stimuli in the same cells, and a similar approach could be used to investigate Ca2+ dynamics in other cells, once the specific Ca2+ handling systems expressed in these cells are described.

Approximations Used in the Model

We started with a model in which cytoplasmic Ca2+ was assumed to be uniformly distributed, and found that this model could account for responses elicited by weak but not strong depolarization. However, responses to strong depolarization could be accounted for if the Ca2+ handling descriptions were included in a diffusion model that takes into consideration radial variations in [Ca2+]i. The two models agree in the case of weak depolarization, as expected if the uniform model describes a limiting case of the more general diffusion model when radial [Ca2+]i gradients are shallow.

Our model cell was assumed to be spherically symmetrical with a radially uniform distribution of organelles. The finding that model calculations were close to measured responses indicates that deviations from spherical symmetry and radially uniform organelle distribution are not critical for the spatiotemporal properties of the measured [Ca2+]i, [Ca]ER, and [Ca]MT signals. Nevertheless, there was a hint that our description of Ca2+ handling near the plasma membrane was not sufficiently detailed. While the model reproduced the decline in [Ca]ER observed near the plasma membrane, it underestimated the magnitude of this decline measured at 10 s. This points to a need to consider additional Ca2+ regulatory processes in this subcellular region (see below).

Another simplifying assumption made in constructing the model is that within both the mitochondria and the ER, free intraluminal Ca concentration is proportional to total Ca concentration. Evaluation of this assumption was not possible because, although we made use of previous measurements of total organelle Ca concentration (free + bound), we did not measure intraluminal free Ca2+ concentration. There is evidence that Ca2+ buffering inside mitochondria is complex (David et al., 2003; Nicholls and Chalmers, 2004). By regulating intraluminal free Ca2+ levels, this system could influence the rate of mitochondrial Ca2+ release, thereby contributing to the level and duration of the slow [Ca2+]i recovery that follows strong stimulation. This, in turn, would be expected to affect all processes that are sensitive to [Ca2+]i.

As a stimulus, we used Ca2+ currents elicited by step depolarizations from a hyperpolarizing membrane potential. While not a naturally occurring physiological stimulus, this does provide a convenient way to perturb Ca2+ regulation so that contributions from different regulatory systems can be investigated. Other, more physiological, model inputs are possible, including measured action potential waveform-induced Ca2+ currents, whose effects could then be compared with measured [Ca2+]i responses.

Stimulus-induced Changes in Mitochondrial and ER Total Ca Concentrations

Our previous results showed that Ca2+ entry induced by strong depolarization stimulates mitochondrial Ca2+ accumulation and net ER Ca2+ release. Both of these observations were reproduced by the model. Difference in the direction of net Ca transport by these organelles can be explained by differences in the intrinsic [Ca2+]i sensitivity of the pathways by which they take up and release Ca2+ (Friel, 2003). High [Ca2+]i levels generated near sites of Ca2+ entry during stimulation drive powerful Ca2+ accumulation by peripheral mitochondria because at these concentrations, Ca2+ uptake via the uniporter is much faster than Ca2+ release via the Na/Ca exchanger. Net Ca2+ release by the ER occurs because at these same [Ca2+]i levels the rate of passive Ca2+ release exceeds the rate of uptake by SERCAs. At high [Ca2+]i during strong depolarization, net Ca2+ transport by mitochondria dominates, accounting for the major role it plays in shaping [Ca2+]i responses under these conditions (Herrington et al., 1996). On the other hand, at the lower [Ca2+]i levels reached during weak depolarization, the ER accumulates Ca2+ and is more influential than mitochondria (Albrecht et al., 2002).

To compare calculated changes in mitochondrial and ER total Ca concentrations with measured values, we used Ca2+ transport descriptions obtained from cytoplasmic Ca2+ measurements. To calculate intraluminal total Ca concentration, it was necessary to specify the mitochondrial and ER volume fractions. Information about these parameters is incomplete, which introduces uncertainty into calculated intraluminal total Ca concentrations. However, it is possible to estimate ranges for these parameters. One approach is to measure the fraction of the cellular area occupied by the respective organelles in tissue sections (Weibel and Elias, 1967). For mitochondria this gives ∼1–4%. For the ER, expected values are in the range 4–5%. We found that for volume fractions within these ranges, the calculated values of total mitochondrial and ER Ca concentration during depolarization closely resemble the measured ones; smaller or larger volume fractions give proportionately different calculated values. For mitochondria, the measured spatial average of [Ca]MT at 45 s during a 50 mM K+ depolarization (29.7 mM, based on EELS measurements) matched the calculated value when volume fraction is 0.65%, slightly below the estimated range (1–4%). For the ER, a value of 4.3% gave an identity between the measured and calculated resting [Ca]ER. It should be noted that because mitochondrial volume fraction estimates are based on measurements of the area enclosed by mitochondria in tissue sections, they include regions outside the mitochondrial matrix and therefore tend to overestimate the desired quantity. This may help explain why the optimal value (0.65%) was <1%.

We found that the model could account for elevations in [Ca2+]i during strong depolarization, as long as endogenous cytoplasmic buffers are assumed to have low mobility (DEndog ∼5–15 μm2/s). This also led to agreement between the calculated and measured spatial profile of [Ca]MT at a fixed time during stimulation, although in this case simulations agreed more closely with measurements when DEndog ∼0–5 μm2/s. In contrast to the high sensitivity of the spatial profile of [Ca]MT to DEndog over the range 0–15 μm2/s, there was a comparatively low sensitivity to changes in the diffusion coefficients of fura-2 (DFura) and Ca2+ (DCa) within the ranges 20–50 and 100–300 μm2/s.

Comparison between Measured and Calculated Near-Membrane Total Organelle Ca Concentrations

In addition to providing a framework for testing hypotheses regarding basic mechanisms, Ca2+ regulatory models can also help point out inadequacies in our understanding. There was a discrepancy between calculated and measured total ER Ca concentration in the outer 2–4-μm shell, where [Ca2+]i is expected to undergo the highest, and fastest, elevations in responses to Ca2+ entry. There was also a hint that the calculated value of [Ca]MT underestimates the measured value in this region, unless it is assumed that DEndog ∼0, which falls outside the range of values (5–15 μm2/s) that were optimal for reconstructing [Ca2+]i responses to strong depolarization. The basis for these discrepancies is not clear, but it is possible that they reflect inaccuracies or omissions in the description of Ca2+ handling in sympathetic neurons. For example, previous work has described a rapid mode of mitochondrial Ca2+ uptake (RaM; Buntinas et al., 2001) that was not explicitly included in the model. Such a mechanism would be expected to enhance Ca2+ loading by peripheral mitochondria, although it would also diminish the rate of ER Ca2+ release by reducing [Ca2+]i elevations in the same region. It is also possible that the spatial distribution of Ca2+ handling systems was not accurately represented in the model. For example, previous reports have suggested that the distribution of ryanodine receptors falls off with distance from the plasma membrane (Akita and Kuba, 2000; McDonough et al., 2000). A higher density of Ca2+ release sites near sites of Ca2+ entry could promote local regenerative Ca2+ release that amplifies near-membrane [Ca2+]i elevations, which would secondarily enhance Ca2+ loading by peripheral mitochondria. Including more realistic descriptions of the spatial distribution of organelles is an important area for model refinement (Means et al., 2006). However, it should be noted that, despite the assumption that organelles are distributed uniformly, the differences between measured and calculated values of [Ca]MT and [Ca]ER were in most cases within the level of uncertainty in the measurements. This supports the conclusion that the major features of Ca regulation have been included. Measurements of Ca2+ levels at higher spatial and temporal resolution will provide the basis for developing and evaluating more refined models of these and other cells, and accounting for such measurements will require consideration of more detailed features of cell structure and transporter distribution.

Characterization of Ca2+ Fluxes in Intact Cells

A novel feature of this study is that it provides enough information about Ca2+ handling in a particular cell type to permit quantitative comparisons between measured and calculated Ca2+ responses. This required the following: (a) a “complete” set of flux descriptions, such that the sum of the individual flux components equals the total flux driving changes in Ca2+ concentration; (b) measurement and characterization of the individual fluxes. We did this by suppressing transport by one or more pathway(s) using specific inhibitors and measuring the drug-sensitive and -resistant components of the total flux. It was also necessary to (c) identify the “independent” variables that control the activity of the transport systems responsible for the measured fluxes. While multiple variables influence Ca2+ transport (e.g., temperature, pH, ATP levels, and [Ca2+]) only some of these change appreciably during an evoked Ca2+ response. This distinguishes between quantities whose variation contributes to changes in transport rate during responses, from those that are approximately constant and can be regarded as parameters. Another requirement was (d) a description of the functional dependence of transport rate on the independent variables. Ideally, this step would use analytical expressions derived from mechanistic considerations, but empirical descriptions are also valid as long as they adequately describe the relationship between the fluxes and the independent variables. In the case of Juni, we started with a sigmoidal function of [Ca2+]i that is expected to be centered around ∼10–20 μM (for review see Gunter and Gunter, 1994). Since unique parameter values cannot be obtained for this equation when [Ca2+]i is well below the midpoint for activation (<1 μM), we switched to a power function approximation that is valid at low [Ca2+]i. This provided a good description of the way Juni varies with [Ca2+]i but gave different parameter values when fit over low and high ranges of [Ca2+]i, suggesting a more complex [Ca2+]i dependence of the uniporter, or functionally dependent parameters (see Appendix). However, despite this, the power function description of the [Ca2+]i dependence of Juni was adequate to describe the contribution from the uniporter to [Ca2+]i dynamics. Finally, (e) it was necessary to incorporate the rate descriptions into a model with a defined compartmental structure, geometry, and transporter distribution.

Relating measured flux components to known transport pathways was facilitated by the use of blockers under conditions that favor inhibition of a single pathway. Juni was measured as the FCCP-sensitive component of the total cytoplasmic Ca2+ flux in cells where Ca2+ release was inhibited using Na+-free pipette solutions. In this case, the FCCP-resistant flux provided a description of Ca2+ extrusion (JExtru). To enable Ca2+ release via the Na/Ca exchanger (Case 4), Na+-containing pipette solutions were used. In this case, JNa/Ca was taken as the CGP-sensitive component of the total flux, and the total mitochondrial Ca2+ flux (JMT) was measured as the FCCP-sensitive flux. Here, the CGP-resistant flux was used to describe Ca2+ removal (JExtru + Juni). In this case, JExtru may have included an additional component representing Ca2+ entry via NCX, operating in reverse, that would not have been present when experiments were performed with low internal Na+. As described in Materials and methods, this would not be expected to interfere with our measurement of JNa/Ca as long as NCX contributes equally to the total flux and the CGP-resistant component of this flux. Contributions from NCX in Case 4 would impart a voltage dependence to Ca2+ extrusion, introducing an error in the Ca2+ removal rate during depolarization when calculated based on measurements during the recovery. However, simulated responses agreed with measurements during both the response onset and recovery, arguing that this source of error is small. Under physiological conditions, the presence of extracellular Na+ would cause NCX to act as an extrusion system (Herrington et al., 1996; Suzuki et al., 2002; Wanaverbecq et al., 2003). More detailed analysis of the components of JExtru, including NCX in the presence and absence of extracellular Na+, will require future experiments.

Spatial Heterogeneity of Evoked Ca Signals

Our flux characterizations were based on analysis of spatially averaged [Ca2+]i measurements during the recovery after stimulation. These characterizations, in the context of a spatially uniform model, provided a good description of measured responses to weak depolarization. This indicates that spatial nonuniformities of intracellular Ca2+ levels are not highly influential in shaping responses to weak depolarization. We suggest that under these conditions, Ca2+ entry is slow enough compared with diffusion that pronounced Ca2+ gradients do not develop. On the other hand, we found that strong depolarization elicits responses that are not consistent with the uniform model, but are consistent with a diffusion model that includes the same Ca2+ handling descriptions but considers radial variations in intracellular [Ca2+] levels. This suggests that with the higher Ca2+ entry rates observed during strong depolarization, diffusion is not fast enough to ensure near-spatial uniformity of [Ca2+]i. According to this model, cytoplasmic [Ca2+] gradients virtually collapse after Ca2+ entry ceases. Thus, measurements of average [Ca2+]i during the recovery would be expected to provide a good estimate of local [Ca2+]i near sites of extrusion and uptake, which can then be used to characterize the [Ca2+]i dependence of transport via these pathways, even after strong depolarization.

The diffusion model suggests that in contrast to cytoplasmic [Ca2+]i, the spatial nonuniformity of mitochondrial total Ca concentration that develops during depolarization does not dissipate rapidly after repolarization, but persists for hundreds of seconds. While we do not have direct experimental verification of this at present, this would mean that during the recovery, mitochondrial Ca2+ release rate is spatially nonuniform. This raises two questions. First, how can radial nonuniformities of intramitochondrial Ca levels persist while [Ca2+]i is nearly uniform? Second, how would persistence of [Ca]MT gradients impact our measurements of mitochondrial Ca2+ fluxes?

Mitochondrial Ca2+ fluxes were calculated based on measurements of spatially averaged [Ca2+]i during the recovery, which according to the model, becomes nearly uniform shortly after Ca2+ entry ends. However, the model also predicts that mitochondrial [Ca] gradients are more persistent. This is because the only pathway for dissipating these gradients, the Na/Ca exchanger, releases Ca2+ slowly compared with the rate of cytoplasmic Ca2+ diffusion. As a result, at each point in time during the recovery, our measurement of JNaCa must be interpreted as a spatial average of the release rate that can be related to (spatially averaged) [Ca2+]i at that time. Our description of JNaCa was adequate to describe how mitochondrial Ca2+ release contributed to the slow plateau phase of recovery seen in previous studies (Thayer and Miller, 1990; Colegrove et al., 2000a). Our results also suggest that spatial heterogeneity of JNa/Ca has less impact on the recovery of [Ca2+]i than of intramitochondrial Ca levels, given the similarity of calculated recoveries using the spatially uniform and nonuniform models (compare Figs. 6 and 7). Nonetheless, the recoveries are not identical and methods for functionally characterizing JNaCa with subcellular resolution would be useful.

Regarding the functional impact of spatially nonuniform Ca2+ loading by mitochondria, it has been shown that Ca2+ can positively regulate mitochondrial ATP production (Robb-Gaspers et al., 1998). One possibility is that Ca2+ uptake by near-membrane mitochondria leads to an increase in the local availability of ATP, helping to fuel ion transport near the surface membrane in the aftermath of stimulation and aid in restoring the resting state.

Role of CICR

Our results support a role for CICR in shaping depolarization-induced Ca2+ responses, although owing to the relatively slow rate of release compared with the rates of transport by mitochondria, the impact on [Ca2+]i responses elicited by strong depolarization is small. This is supported by the weak effect of thapsigargin on [Ca2+]i responses elicited by strong depolarization (Hongpaisan et al., 2001) and the subtle difference between simulated responses in the presence and absence of ER Ca2+ transport (Figs. 7 and 8). Two recent studies have suggested that [Ca2+]i elevations elicited by action potential–induced Ca2+ entry in amphibian sympathetic neurons depends almost entirely on CICR (Akita and Kuba, 2000; Cseresnyes and Schneider, 2004). This conclusion is based on the observation that inhibiting ER Ca2+ transport by exposure to sarco/endoplasmic reticulum Ca2+ ATPase (SERCA) inhibitors reduces or eliminates AP-induced increases in [Ca2+]. While our measurements do not address this point specifically, it is important to reconcile experimental and modeling results performed on different space and time scales. This should help answer important questions regarding the role of microheterogeneity in Ca2+ regulation, e.g., near plasma membrane sites of Ca2+ entry, and the relative role of brief/local elevations in [Ca2+]i vs. more prolonged and global increases in regulating Ca2+-dependent processes within the cell.

Appendix

Surface Membrane Ca2+ Transport: JVSCC, JExtru

When Ca2+ uptake and release by mitochondria and the ER are inhibited, the only remaining components of

\({\tilde{J}}_{i}\)
represent the net Ca2+ flux across the plasma membrane (
\({\tilde{J}}_{PM}\)
). We distinguished two components of
\({\tilde{J}}_{PM}\)
: Ca2+ entry through voltage-gated Ca2+ channels (
\({\tilde{J}}_{VSCC}\)
), and Ca2+ extrusion (
\({\tilde{J}}_{Extru}\)
), representing contributions from all other surface membrane Ca2+ transport pathways, including pumps and leaks. Since these pathways operate in parallel, their contributions to
\({\tilde{J}}_{i}\)
are additive. Therefore,

 
\begin{eqnarray*}&&\frac{d[Ca^{2{+}}]_{i}}{dt}{=}{-}\frac{{\tilde{J}}_{\mathrm{i}}}{v_{Cell}(f_{i}{\kappa}_{i})}\\&&{=}{-}\frac{{\tilde{J}}_{VSCC}{+}{\tilde{J}}_{Extru}}{v_{Cell}(f_{i}{\kappa}_{i})}.\end{eqnarray*}
(A1)

This equation can be recast into a form that is more easily related to measured quantities:

 
\[\frac{d[Ca^{2{+}}]_{i}(t)}{dt}{=}{-}\left[\frac{I_{Ca}(t)}{2Fv_{Cell}f_{i}{\kappa}_{i}}{+}J_{Extru}\right]\]
(A2)

using the following relations:

 
\begin{eqnarray*}&&{\tilde{J}}_{VSCC}{=}\frac{I_{Ca}}{2F}\\&&J_{Extru}{=}\frac{{\tilde{J}}_{Extru}}{v_{Cell}(f_{i}{\kappa}_{i})}\end{eqnarray*}
(A3)

and F is the Faraday constant. We plotted JExtru vs. [Ca2+]i and fit with the following equation, which describes a parallel pump/leak system whose overall transport rate depends on [Ca2+]i:

 
\[J_{Extru}{=}J_{PMCA}{+}J_{leak},\]
(A4)

where

 
\begin{eqnarray*}&&J_{PMCA}{=}\frac{V_{\mathrm{max},PMCA}}{\left(1{+}\left[\frac{K_{d,PMCA}}{[Ca^{2{+}}]_{i}}\right]^{n_{PMCA}}\right)}\\&&J_{leak}{=}k_{leak}([Ca^{2{+}}]_{i}{-}[Ca^{2{+}}]_{o}).\end{eqnarray*}
(A5)

This equation incorporates the finding that in sympathetic neurons treated with FCCP and Tg, the rate of Ca2+ removal at each point in time during the recovery depends on the [Ca2+]i level at that time. Thus, the time dependence of JExtru during the recovery is derived from the time dependence of [Ca2+]i:

 
\[J_{Extru}(t){=}J_{Extru}([Ca^{2{+}}]_{i}(t)).\]
(A6)

It should be noted that in some cells (e.g., human T cells) Ca2+ extrusion via the PMCA is time dependent (Bautista et al., 2002). When fitting Eqs. A4 and A5 to JExtru measurements during the recovery (as in Fig. 2 C), we reduced the number of free parameters by requiring that JExtru = 0 when [Ca2+]i is at its resting level, making it possible to solve for kleak in terms of the remaining parameters.

Mitochondrial Ca2+ Uniporter: Juni

To investigate the contribution from mitochondrial Ca2+ uptake via the uniporter in isolation from Ca2+ release by the mitochondrial Na/Ca exchanger, responses were also elicited in the absence of FCCP, using either Na+-free pipette solutions or application of CGP to inhibit mitochondrial Ca2+ release. In this case,

\({\tilde{J}}_{i}\)
includes a third component representing the uniporter:

 
\[\frac{d[Ca^{2{+}}]_{i}}{dt}{=}{-}\frac{{\tilde{J}}_{VSCC}{+}{\tilde{J}}_{Extru}{+}{\tilde{J}}_{uni}}{v_{Cell}(f_{i}{\kappa}_{i})}.\]
(A7)

As in Case 1, this equation can be recast into a form that is more easily related to measured quantities:

 
\[\frac{d[Ca^{2{+}}]_{i}}{dt}{=}{-}\left[\frac{I_{Ca}}{2Fv_{Cell}(f_{i}{\kappa}_{i})}{+}(J_{Extru}{+}J_{uni})\right],\]
(A8)

where

 
\[J_{uni}{=}\frac{{\tilde{J}}_{uni}}{v_{Cell}(f_{i}{\kappa}_{i})}.\]
(A9)

We characterized the sum (JExtru + Juni) quantitatively as described in Colegrove et al. (2000a). In brief, this flux was measured by calculating −d[Ca2+]i/dt during the recovery after depolarization-induced [Ca2+]i elevations in Tg-treated cells in the absence of FCCP under conditions where mitochondrial Na/Ca exchange is inhibited. During the recovery, ICa = 0, so that [Ca2+]i changes under the influence of a net flux (JExtru+Juni) that can be measured directly by calculating −d[Ca2+]i/dt. We plotted (JExtru+Juni) vs. [Ca2+]i during the recovery and fit with the following equation:

 
\[\frac{d[Ca^{2{+}}]_{i}}{dt}{=}{-}(J_{PMCA}{+}J_{leak}{+}J_{uni}),\]
(A10)

where JPMCA and Jleak are as described in Case 1 and

 
\[J_{uni}{=}\frac{V_{\mathrm{max},uni}[Ca^{2{+}}]_{i}}{\left(1{+}\left[\frac{K_{d,uni}}{[Ca^{2{+}}]_{i}}\right]^{n_{uni}}\right)},\]
(A11)

which was approximated by the following equation:

 
\begin{eqnarray*}&&J_{uni}{\cong}\left[\frac{V_{\mathrm{max},uni}}{K_{d,uni}^{n_{uni}}}\right][Ca^{2{+}}]_{i}^{n_{uni}{+}1}\\&&{=}A[Ca^{2{+}}]_{i}^{n_{uni}{+}1},\end{eqnarray*}
(A12)

where A is a constant, which is valid as long as [Ca2+]i ≪ Kd,uni. Based on available information, Kd,uni ∼ 10–20 μM, which is 5–10 times the largest measured [Ca2+]i elevation shown in this study. This approximation was used because it was not possible to obtain unique parameters for Eq. A11 when [Ca2+]i is low compared with Kd.uni. While Eq. A12 provided a good description of the [Ca2+]i dependence of Juni, the parameter values obtained from different sets of measurements ranged widely (Table I). This could be due to the high sensitivity of the parameters to subtle differences in the shape of Juni and/or a functional dependence of A on nuni (see Eq. A12).

The equations describing Ca2+ removal are consistent with the finding that in sympathetic neurons treated with Tg and subjected to blockade of the Na/Ca exchanger, the rate of Ca2+ removal at each instant in time during the recovery depends on the [Ca2+]i level at that time. Thus, under these conditions, the time dependence of (JExtru+Juni) during the recovery is derived from the time dependence of [Ca2+]i:

 
\[J_{Extru}(t){+}J_{uni}(t){=}J_{Extru}([Ca^{2{+}}]_{i}(t)){+}J_{uni}([Ca^{2{+}}]_{i}(t)).\]
(A13)

Mitochondrial Na+/Ca2+ Exchanger: JNaCa

To study mitochondrial Ca2+ release via the Na+/Ca2+ exchanger, cells were stimulated under the same conditions as in Case 3 but with pipette solutions containing added Na+.

\({\tilde{J}}_{i}\)
now includes a fourth component (
\({\tilde{J}}_{NaCa}\)
):

 
\begin{eqnarray*}&&\frac{d[Ca^{2{+}}]_{i}}{dt}{=}{-}\frac{{\tilde{J}}_{VSCC}{+}{\tilde{J}}_{Extru}{+}{\tilde{J}}_{uni}{+}{\tilde{J}}_{NaCa}}{v_{Cell}(f_{i}{\kappa}_{i})}\\&&\frac{d[Ca^{2{+}}]_{MT}}{dt}{=}\frac{{\tilde{J}}_{uni}{+}{\tilde{J}}_{NaCa}}{v_{MT}{\kappa}_{MT}}.\end{eqnarray*}
(A14)

The equation for d[Ca2+]i/dt can be rewritten as follows:

 
\[\frac{d[Ca^{2{+}}]_{i}}{dt}{=}{-}\left[\frac{I_{Ca}}{2Fv_{Cell}(f_{i}{\kappa}_{i})}{+}(J_{Extru}{+}J_{uni}{+}J_{NaCa})\right],\]
(A15)

where

 
\[J_{NaCa}{=}\frac{{\tilde{J}}_{NaCa}}{v_{Cell}(f_{i}{\kappa}_{i})}.\]
(A16)

Analysis of JNaCa and its regulation by Ca2+ is more complicated than JExtru or Juni because

\({\tilde{J}}_{NaCa}\)
depends on the intramitochondrial Ca2+ concentration ([Ca2+]MT), which was not directly measured in this study. However, Colegrove et al. (2000b) described an approach to measuring a quantity that can be directly related to [Ca2+]MT, as long as [Ca2+]MT is proportional to the total concentration, which is assumed here. This approach makes use of the net mitochondrial Ca2+ flux (JMT). With information about the time course of JMT, it is possible to obtain information about the dynamics of intramitochondrial Ca concentration. The starting point is provided by the following equation:

 
\begin{eqnarray*}&&[Ca^{2{+}}]_{MT}(t){=}[Ca^{2{+}}]_{MT}(0){+}{\Delta}[Ca^{2{+}}]_{MT}(t)\\&&{=}[Ca^{2{+}}]_{MT}(0){+}{{\int}_{0}^{t}}\frac{{\tilde{J}}_{MT}(t)}{v_{MT}{\kappa}_{MT}}dt,\end{eqnarray*}
(A17)

where [Ca2+]MT(t) is the mitochondrial Ca2+ concentration at time t, [Ca2+]MT(0) is the resting level of [Ca2+]MT, and Δ[Ca2+]MT(t) is the difference between [Ca2+]MT(t) and [Ca2+]MT(0). As shown below, integration of JMT provides a measure of intramitochondrial Ca2+ levels during the recovery that can be used to characterize the functional dependence of JNa/Ca on [Ca2+]MT. If mitochondrial Ca2+ buffering is fast and low affinity and can be described by a constant buffering factor κMT, then [Ca2+]MT can be related to measured quantities as follows:

 
\begin{eqnarray*}&&[Ca^{2{+}}]_{MT}(t){=}[Ca^{2{+}}]_{MT}(0){+}{{\int}_{0}^{t}}\frac{{\tilde{J}}_{MT}(t)}{v_{MT}{\kappa}_{MT}}dt\\&&{=}[Ca^{2{+}}]_{MT}(0){-}{{\int}_{t}^{{\infty}}}\frac{{\tilde{J}}_{MT}(t)}{v_{MT}{\kappa}_{MT}}dt\\&&{=}[Ca^{2{+}}]_{MT}(0){-}{{\int}_{t}^{{\infty}}}J_{MT}(t)\frac{v_{i}{\kappa}_{i}([Ca^{2{+}}]_{i})}{v_{MT}{\kappa}_{MT}}dt\\&&{=}[Ca^{2{+}}]_{MT}(0){-}\frac{v_{i}}{v_{MT}{\kappa}_{MT}}{{\int}_{t}^{{\infty}}}J_{MT}(t){\kappa}_{i}([Ca^{2{+}}]_{i})dt\\&&{=}[Ca^{2{+}}]_{MT}(0){+}\frac{v_{i}}{v_{MT}{\kappa}_{MT}}{\Delta}[Ca^{2{+}}]_{MT}^{(0)}(t),\end{eqnarray*}
(A18)

where Δ[Ca2+]MT(0)(t) is the negative integral of JMTκi from t to ∞, and the negative integral from t to ∞ has been substituted for the integral from 0 to t, making use of Eq. A19, which is valid as long as stimulus-evoked changes in [Ca2+]MT are reversible (i.e., if [Ca2+]MT eventually returns to its resting level in the aftermath of a stimulus):

 
\begin{eqnarray*}&&0{=}{{\int}_{0}^{{\infty}}}\frac{{\tilde{J}}_{MT}(t)}{v_{MT}{\kappa}_{MT}}dt\\&&{=}{{\int}_{0}^{t}}\frac{{\tilde{J}}_{MT}(t)}{v_{MT}{\kappa}_{MT}}dt{+}{{\int}_{t}^{{\infty}}}\frac{{\tilde{J}}_{MT}(t)}{v_{MT}{\kappa}_{MT}}dt.\end{eqnarray*}
(A19)

An analogous approach was used to evaluate changes in ER Ca2+ levels influencing the driving force for Ca2+ release via RyRs (Albrecht et al., 2002). To describe the relationship between JNa/Ca and [Ca2+]MT we used the following equation:

 
\[J_{NaCa}([Ca^{2{+}}]_{MT}){=}{-}\frac{{\tilde{V}}_{\mathrm{max},NaCa}}{\left[1{+}\frac{EC_{50,NaCa}}{[Ca^{2{+}}]_{MT}}\right]{\kappa}_{i}}.\]
(A20)

Substituting for [Ca2+]MT from Eq. A18 gives a relationship between the measured quantities JNaCa, κi, and Δ[Ca2+]MT(0)(t):

 
\begin{eqnarray*}&&J_{NaCa}([Ca^{2{+}}]_{MT}(t)){=}\\&&{-}\frac{{\tilde{V}}_{\mathrm{max},NaCa}}{\left[1{+}\left[\frac{EC_{50,NaCa}}{[Ca^{2{+}}]_{MT}(0){+}\frac{v_{i}}{v_{MT}{\kappa}_{MT}}{\Delta}[Ca^{2{+}}]_{MT}^{(0)}(t)}\right]\right]{\kappa}_{i}}.\end{eqnarray*}
(A21)

This equation includes four parameters,

\({\tilde{V}}_{\mathrm{max},NaCa}\)
⁠, EC50,NaCa, [Ca2+]MT(0), and vi/(vMTκMT), but can be converted to an equation with three lumped parameters that is more convenient for comparison with data:

 
\begin{eqnarray*}&&J_{NaCa}([Ca^{2{+}}]_{MT}(t)){=}\\&&{-}\frac{{\tilde{V}}_{\mathrm{max},NaCa}}{\left[1{+}\left[\frac{\left\{EC_{50,NaCa}\frac{v_{MT}{\kappa}_{MT}}{v_{i}}\right\}}{\left\{[Ca^{2{+}}]_{MT}(0)\frac{v_{MT}{\kappa}_{MT}}{v_{i}}\right\}{+}{\Delta}[Ca^{2{+}}]_{MT}^{(0)}(t)}\right]\right]{\kappa}_{i}}\\&&{=}{-}\frac{P_{3}}{\left[1{+}\left[\frac{P_{4}}{P_{5}{+}{\Delta}[Ca^{2{+}}]_{MT}^{(0)}(t)}\right]\right]{\kappa}_{i}},\end{eqnarray*}
(A22)

where

 
\begin{eqnarray*}&&P_{3}{=}{\tilde{V}}_{\mathrm{max},NaCa}\\&&P_{4}{=}EC_{50,NaCa}\frac{v_{MT}{\kappa}_{MT}}{v_{i}}\\&&P_{5}{=}[Ca^{2{+}}]_{MT}(0)\frac{v_{MT}{\kappa}_{MT}}{v_{i}}.\end{eqnarray*}
(A23)

We plotted JNaCa vs. Δ[Ca2+]MT(0)(t) and fit the data with Eq. A22 to obtain estimates of P3–P5, treating this equation as a function of two independent variables, Δ[Ca2+]MT(0) and κi([Ca2+]i), that depend parametrically on time. Given (κMTvMT)/vi and the parameters P4 and P5, estimates of EC50,NaCa and [Ca2+]MT(0) can be obtained by dividing the parameters by (κMTvMT)/vi. The latter calculation gave an initial value of [Ca2+]MT for simulations.

ER Ca2+ Uptake and Release: JSERCA, JRyR

To include contributions from ER Ca2+ transport, we used descriptions of Ca2+ uptake via SERCAs (JSERCA) and release (JRelease) from Albrecht et al. (2002) and inserted these into Eqs. M1M3.

 
\[{\tilde{J}}_{SERCA}([Ca^{2{+}}]_{i}(t)){=}\frac{{\tilde{V}}_{\mathrm{max},SERCA}}{\left[1{+}\frac{EC_{50,SERCA}}{[Ca^{2{+}}]_{i}}\right]^{n_{SERCA}}}\]
(A24)
 
\[{\tilde{J}}_{\mathrm{Re}lease}([Ca^{2{+}}]_{i},[Ca^{2{+}}]_{ER}){=}{\tilde{P}}_{ER}([Ca^{2{+}}]_{i}{-}[Ca^{2{+}}]_{ER}),\]
(A25)

where

 
\[{\tilde{P}}_{ER}{=}{\tilde{P}}_{ER,Basal}{+}\frac{{\tilde{P}}_{\mathrm{max},RyR}}{\left[1{+}\frac{EC_{50,RyR}}{[Ca^{2{+}}]_{i}}\right]^{n_{RyR}}}.\]
(A26)

The parameters of JSERCA have the conventional meanings.

\({\tilde{P}}_{ER}\)
describes the Ca2+ permeability of the ER, which consists of a constant, Ca2+-insensitive component (
\({\tilde{P}}_{ER,Basal}\)
), and a Ca2+-sensitive component that represents the permeability imparted to the ER by ryanodine receptors (RyRs). The latter component has a maximal value
\({\tilde{P}}_{\mathrm{max},RyR}\)
, [Ca2+]i for half maximal activation EC50, RyR, and cooperativity factor nRyR.

Diffusion Model

We developed a radial model of Ca2+ diffusion in a spherical cell consisting of multiple concentric shells and a central sphere (Sala and Hernández-Cruz, 1990; Nowycky and Pinter, 1993; see also Connor and Nikolakopoulou, 1982). Calcium enters the cell across the plasma membrane and diffuses into the interior shells while also being transported by the organelles within each shell. We adopted the fast-buffer approximation (Keener and Sneyd, 1998), which assumes that buffering occurs instantaneously, which is reasonable given the long time scales and continuous stimuli we are considering (Strier et al., 2003). Using this simplification, diffusion leads to changes in Ca2+ concentration in the nth shell ([Ca2+]i,n) described by Eq. A27:

 
\begin{eqnarray*}&&\frac{{\partial}[Ca^{2{+}}]_{i,n}}{{\partial}t}{=}\\&&\frac{(D_{Ca}{+}D_{Endog}{\kappa}_{Endog}{+}D_{Fura}{\kappa}_{Fura})}{1{+}{\kappa}_{Endog}{+}{\kappa}_{Fura}}{\nabla}^{2}[Ca^{2{+}}]_{i,n}\\&&{-}\frac{2D_{Fura}{\kappa}_{Fura}}{1{+}{\kappa}_{Endog}{+}{\kappa}_{Fura}}{\vert}{\nabla}[Ca^{2{+}}]_{i,n}{\vert}^{2},\end{eqnarray*}
(A27)

where DCa, DEndog, and DFura are the diffusion coefficients of cytoplasmic Ca2+, the endogenous buffer, and fura-2, respectively; Ca2+ transport by organelles is described by additional terms that are not shown for simplicity. Within each shell, the concentration of free and total calcium in organelles was assumed to be spatially uniform, and direct intraluminal diffusion between shells was not allowed. The gradient was calculated by taking radially symmetric derivatives. Our measurements of cytoplasmic buffering strength provided estimates of P1 = fi(1 + κEndog) and P2 = fiBtotal,fura, which were used to describe buffering in the uniform model (see Eq. M10). For the diffusion model, descriptions of κEndog and κFura were needed. Instead of estimating fi, which based on a combined mitochondrial and ER volume fraction of ∼0.05 would give fi ∼ 0.95, and calculating κEndog and κFura, we used the measured parameters directly (equivalent to setting fi = 1), which may lead to an underestimation of the buffering parameters by ∼5%. The similarity between calculations based on the diffusion model (with fi set to unity) and the uniform model (which uses the measured values P1 and P2 and makes no assumptions about fi) when ICa is small argues that this approximation does not introduce appreciable error.

We integrated the model numerically in a spherically symmetric coordinate system using a partially implicit approach. In this method, the diffusion rates are calculated using the concentration at the next time step, while the fluxes are calculated using the calcium concentration at the previous time step. To check our model, we first made sure that with ICa = 0, resting [Ca2+]i was steady in all shells. As another check, we made sure calcium was conserved during diffusion. During all of our simulations we calculated the total calcium concentration in the cell in two ways: first by adding the number of moles of calcium in the cytoplasm and all organelles, and second by calculating the starting concentration and then integrating all of the fluxes entering and leaving the cell. We found that these two calculations were within 1% of each other for the diffusion parameters we used, and that this error could be reduced with smaller time and space steps.

Each shell included two organellar compartments that occupy a constant volume fraction of that shell. Since all fluxes were measured based on the impact of the corresponding transport pathway on [Ca2+]i, to describe the impact of these fluxes on organellar Ca2+ levels, we multiplied the equations describing organelle transport by vCellfiκi/vMTκMT (for mitochondria) and vCellfiκi/vERκER (for ER). fiκi was obtained from cytoplasmic Ca2+ buffering measurements, and the lumped parameters vCell/vMTκMT and vCell/vERκER were calculated based on estimates of volume fraction (vMT/vCell, vER/vCell) and buffering strength (κMT, κER).

Simulated Fluorescence-based [Ca2+]i Measurements

We based fluorescence simulations on those of McHugh and Kenyon (2004). In brief, we calculated the fluorescence arising from the nth shell at two excitation wavelengths:

 
\[F_{350,n}{=}S_{f,350}([Fura]_{n}){+}S_{b,350}[CaFura]_{n}),\]
(A28)
 
\[F_{380,n}{=}S_{f,380}([Fura]_{n}){+}S_{b,380}[CaFura]_{n}),\]
(A29)

where [Fura] and [CaFura] are the concentrations of free and bound fura-2, and Sf,350 = 4.97, Sf,380 = 11.3, Sb,350 = 11, Sb,380 = 1 are parameters describing contributions to fluorescence intensity from free and bound forms of fura-2 satisfying the experimentally determined relations Rmax = (Sb,350/Sb,380) = 11, Rmin = (Sf,350/Sf,380) = 0.44, and (Sf,380/Sb,380) = 11.3. The concentration of bound fura-2 in the nth compartment ([CaFura]n) is:

 
\[[CaFura]_{n}{=}\frac{[Ca^{2{+}}]_{i,n}B_{total,fura}}{K_{d,Fura}{+}[Ca^{2{+}}]_{i,n}},\]
(A30)

where Kd,Fura = 224 nM and Btotal,fura is the total concentration of fura-2. We calculated the concentration of unbound fura-2 by subtracting [CaFura]n from Btotal,fura, which is assumed to be constant in all shells. From F350,n and F380,n we calculated the fluorescence ratio in each shell Rn = F350,n/F380,n, which can be directly converted to [Ca2+]i,n by the equation (Grynkiewicz et. al., 1985):

 
\[[Ca^{2{+}}]_{i,n}{=}K_{D,Fura}\left(\frac{R_{n}{-}R_{\mathrm{min}}}{R_{\mathrm{max}}{-}R_{n}}\right)\frac{S_{f,380}}{S_{b,380}}.\]
(A31)

Average [Ca2+]i ([Ca2+]i,avg) was calculated in two ways (McHugh and Kenyon, 2004). For the first approach,

 
\[[Ca^{2{+}}]_{i,avg}{=}\frac{{{\sum}_{n{=}0}^{N}}[Ca^{2{+}}]_{i,n}vol_{n}}{vol_{total}}.\]
(A32)

For the second approach, the total fluorescence was calculated at each of the two excitation wavelengths by summing the fluorescence from each shell, and the ratio was calculated:

 
\[R_{avg}{=}\frac{{\sum}_{n{=}0}^{N}(S_{f,350}([Fura]_{n}){+}S_{b,350}[CaFura]_{n})vol_{n}}{{\sum}_{n{=}0}^{N}(S_{f,380}([Fura]_{n}){+}S_{b,380}[CaFura]_{n})vol_{n}}.\]
(A33)

Ravg was then inserted into Eq. A31 to give [Ca2+]i,avg.

Acknowledgments

The authors thank Meredith A. Albrecht and Stephen L. Colegrove for their early contributions to this work and S. Brian Andrews for discussions regarding organellar volume fractions.

This work was supported by a grant from the National Institutes of Health/National Institute of Neurological Disorders and Stroke (NS 33514) to D.D. Friel.

Kenneth C. Holmes served as editor.

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M. Patterson's present address is Department of Neurobiology, Duke University, Durham, NC.

Abbreviations used in this paper: CICR, Ca2+-induced Ca2+ release; FCCP, carbonyl cyanide p-trifluoromethoxyphenylhydrazone; NCX, plasma membrane Na/Ca exchanger; PMCA, plasma membrane Ca2+ ATPase; SERCA, sarco/endoplasmic reticulum Ca2+ ATPase; Tg, thapsigargin.