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¹ Department of Neurosciences, Case Western Reserve University, Cleveland, OH 44106
² Department of Mathematics, University of Auckland, Auckland, New Zealand

Correspondence to David Friel: ddf2{at}case.edu

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

 
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.

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.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

 
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

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

 
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 40x 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 (I_Ca) were made under voltage clamp in fura-2 AM–loaded cells using the perforated patch technique. Patch electrodes (1–2 M) 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 M. 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 (, , , e.g., in nmol/s) and volumes (v_i, v_MT, v_ER, e.g., in li) scaled by buffering factors (_i', _MT', _ER'):

(M1)

where

(M2)

and

(M3)

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

(M4)

where

(M5)

and

(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 () and the cytoplasmic volume (v_i):

(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 (v_Cell) estimated from the membrane capacitance (C_m), assuming a specific capacitance of 1 µF/cm², and the fraction (f_i) of the total cell volume occupied by cytoplasm. Since cells plated on polylysine-coated glass coverslips assumed an approximately spherical shape, we calculated v_Cell as () (C_m x 10^–6)^3/2 cm³. Sources of error in our estimate of v_i include deviations from spherical geometry and uncertainty in the precise value of f_i. We chose not to estimate f_i explicitly but instead to include it as a factor in the buffering parameters to be estimated. Errors in our estimate of v_i arising from deviations from spherical geometry would be expected to influence the estimated buffering parameters through their dependence on f_i (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 to the cytoplasmic volume scaled by a buffering factor (_i'):

(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 _i' is related to the well-known differential buffering factor _i and includes contributions from both endogenous (_Endog) and exogenous buffers (_fura):

(M9)

where the j^th endogenous buffer is present at concentration B_total,j and binds Ca²⁺ with dissociation constants K_d,j. In our experiments, the only exogenous buffer is fura-2, present at concentration B_total,fura (estimated in different cells to be 50–100 µM) with a dissociation constant K_d,fura assumed to be 224 nM. Albrecht et al. (2002) provided a description of the [Ca²⁺]_i dependence of _i' in sympathetic neurons and estimates of _Endog and _fura near resting [Ca²⁺]_i. In the following, we treat _Endog 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 _i' to fura-2. Accordingly, _i' depends on [Ca²⁺]_i in a way that can be described by two lumped parameters, (1+ _Endog) and B_total,fura. However, since _i' appears in the equations describing [Ca²⁺]_i dynamics scaled by f_i (see Eq. M8 above), it is convenient to define the lumped parameters P₁ = f_i(1 + _Endog) and P₂ = f_iB_total,fura, which can be determined from measured quantities:

(M10)

In the following, we drop the prime with the understanding that _i differs from the more common definition by unity. To estimate the two parameters P₁ and P₂, we obtained two independent measurements of f_ii 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 f_ii was made near resting [Ca²⁺]_i as the ratio of I_Ca/2Fv_Cell to –d[Ca²⁺]_i/dt during the early phase of the response, when [Ca²⁺]_i increases approximately linearly with time. Since I_Ca is time dependent, we used the average of I_Ca 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 f_ii was obtained from the ratio of I_Ca/2Fv_Cell to the instantaneous change in –d[Ca²⁺]_i/dt (d[Ca²⁺]_i/dt) after repolarization (Fig. 1). Given the two estimates of f_ii 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 f_ii. However, since the two f_ii measurements from which P₁ and P₂ were determined included errors arising from I_Ca averaging and calculations of d[Ca²⁺]_i/dt from noisy [Ca²⁺]_i signals, and because they represent estimates of f_ii at only two values of [Ca²⁺]_i, P₁ and P₂ were treated as initial estimates of the parameters of f_ii, which were then optimized by fitting to measured responses.

View larger version (11K): [in this window] [in a new window] [Download PPT slide]   Figure 1. Method used to estimate cytoplasmic Ca2+ buffering strength. Buffering strength was characterized quantitatively based on measurements of ICa and changes in [Ca2+]i that follow depolarization and repolarization. (A) Illustration of the Ca2+ current elicited by stepping the membrane potential from –70 to –35 mV. Dotted lines indicate average ICa during the intervals indicated by arrows. (B) [Ca2+]i response elicited by this depolarization. Arrows indicate the initial rate of rise of [Ca2+]i after depolarization, and the change in d[Ca2+]i/dt after repolarization. These measurements were used to describe Ca2+ buffering strength near resting [Ca2+]i (in this case 52 nM) and at the peak of the response (623 nM), which made it possible to estimate the two parameters P1 and P2 of fii according to Eq. M10.

(M11)

This method follows from the idea that the total Ca²⁺ flux is the sum of an FCCP-sensitive flux (J_uni), and an FCCP-resistant flux (J_i,+FCCP = J_Extru) that depends on [Ca²⁺]_i at each time point during the recovery (Colegrove et al., 2000a). J_uni was described using Eqs. A11 and A12. The rate of Ca²⁺ release via the mitochondrial Na/Ca exchanger (J_Na/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. J_Na/Ca was determined by subtracting the CGP-resistant flux (J_i,+CGP) from the total flux at corresponding [Ca²⁺]_i levels during the recovery:

(M12)

This makes use of the observation that J_i,+CGP is a function of [Ca²⁺]_i at each time during the recovery (Colegrove et al., 2000a). J_NaCa was described using Eqs. A20–A22. Since Na⁺ was included in the pipette solution in these experiments, it is possible that J_i,+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 J_Na/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 (J_MT) was measured as the FCCP-sensitive component of J_i in cells where mitochondrial uptake and release were both active. J_MT was determined by subtracting the FCCP-resistant flux (J_i,+FCCP) from the total flux at corresponding [Ca²⁺]_i levels during the recovery:

(M13)

Here, the FCCP-sensitive flux gives J_uni + J_NaCa instead of only J_uni 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, D_Endog, mitochondrial fractional volume, v_MT/v_Cell). To describe I_Ca 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 J_Extru to measurements from responses over a [Ca²⁺]_i range that did not extend far beyond 300 nM, we constrained EC_50,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

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

 
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 I_Ca 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 (I_Ca; 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, I_Ca underwent rapid deactivation and [Ca²⁺]_i recovered slowly toward its prestimulation level.

View larger version (35K): [in this window] [in a new window] [Download PPT slide]   Figure 2. Analysis of a [Ca2+]i response elicited by weak depolarization in the absence of Ca2+ uptake and release by internal stores (Case 1). (A) Depolarization-evoked Ca current ICa. (B) Measured [Ca2+]i response (dots). Gray trace shows the result of integrating Eq. 2 using a description of fii derived from "on" and "off" responses obtained from the same cell. Dark trace shows integrated response after optimizing the parameters of fii. Traces are nearly superimposable. (C) Measurement of total free Ca2+ flux during the response onset and recovery (JiOnset, JiRecov). Numbers refer to transition points between onset and recovery (see Results). Shaded region shows the contribution from Ca2+ entry (JVSCC) to the free Ca2+ flux during the onset. (D) Measurement of (volume-normalized) net Ca2+ fluxes during the response onset and recovery. Shaded region shows the contribution from Ca2+ entry (JVSCC*) to the net flux during the onset. Net fluxes per unit cell volume are enclosed by quotes. (E) [Ca2+]i dependence of Ca2+ buffering strength. Filled squares show measurements from the cell illustrated and the thick line shows the buffering curve calculated from Eq. M10 using parameters deduced by fitting the spatially uniform model to the [Ca2+]i response in B. Open symbols and interrupted curves show measurements and calculations from two other cells under the same condition. (F) Comparison between the time course of the net Ca2+ flux underlying the dynamics of [Ca2+]i (thick trace) and its components representing Ca2+ entry and extrusion (thin traces). Parameter values for this cell were as follows: Vmax,Extru = 30.3 nM/s, Kd,Extru = 350 nM, nExtru = 2.5, fiEndog = 95.5, fiBtotal,Fura = 64.7 µM, kleak = 1.3 x 10–7 s–1.

The total cytoplasmic Ca²⁺ flux (J_i, 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, J_i 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 (_i) 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 (, e.g., in nmol/s).

Fig. 2 C plots J_i versus [Ca²⁺]_i throughout the response, distinguishing between the onset and recovery phases of the response. During the onset, J_i first increases rapidly to become a large inward flux (transition 1®2, see downward "On" arrow; note that inward fluxes are negative). J_i 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, J_i 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 J_i that occur during the early moments of the onset and recovery that parallel I_Ca activation and deactivation, accounting for the apparent discontinuities in J_i between points 1,2, and 3,4.

Analysis of the Underlying Ca²⁺ Handling Systems.
Inspection of Fig. 2 C shows that J_i declines during the onset between points 2 and 3. This is due to at least two factors. First, I_Ca 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 I_Ca 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 J_Extru, is given by the rate at which [Ca²⁺]_i declines during the recovery:

(1)

Our previous work has shown that during the recovery, J_Extru 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 J_Extru.

We hypothesized that during the response, J_i is the sum of two flux components, voltage-sensitive Ca²⁺ entry (J_VSCC) and J_Extru. According to this idea, J_i is influenced by both J_VSCC and J_Extru during depolarization, but only J_Extru during the recovery (or rather that portion of the recovery after complete Ca²⁺ channel deactivation, when J_VSCC = 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

(2)

Reconstruction of the [Ca²⁺]_i Response.
We reasoned that if J_Extru has the same [Ca²⁺]_i dependence during depolarization as it does during the recovery, then measurement of J_Extru 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:

(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 J_Extru, 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 f_ii, 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 f_ii (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 f_ii 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 f_ii 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 V_max,Extru = 29.8 ± 1.5 nM/s, n_Extru = 3.0 ± 0.3, f_iEndog = 63.0 ± 18.0, and f_iB_total,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 J_Extru 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 J_Extru was obtained from analysis of the recovery, when the membrane potential was clamped at –70 mV, while during the onset V_m = –35 mV. This indicates that the description of J_Extru 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 J_i representing Ca²⁺ entry during the response onset. This component (J_VSCC) can be calculated by subtracting J_Extru from J_i at corresponding values of [Ca²⁺]_i (Fig. 2 C, top, shaded region and bottom trace), which follows directly from the description of J_i as the sum of J_VSCC and J_Extru and the functional dependence of J_Extru on [Ca²⁺]_i. After depolarization, J_VSCC undergoes a rapid increase in magnitude when it dominates J_i (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 I_Ca inactivates. However, since J_VSCC 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 f_ii, the net Ca²⁺ fluxes can be obtained by multiplying the free Ca²⁺ fluxes by f_ii 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, and both decay monotonically, in contrast to the free Ca²⁺ fluxes. Therefore, J_VSCC does not parallel I_Ca during the response onset because it is the ratio of two declining quantities, and f_ii, with the latter falling more rapidly than the former. This relationship between the time dependence of and the [Ca²⁺]_i dependence of f_ii 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 . is a function of time, expressing the time dependence of I_Ca at constant voltage, while is a function of [Ca²⁺]_i that varies because [Ca²⁺]_i is dynamic. 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 I_Ca(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.

View larger version (31K): [in this window] [in a new window] [Download PPT slide]   Figure 3. Analysis of Ca2+ responses elicited by weak depolarization when mitochondrial Ca2+ uptake is enabled (Case 2). (A) Depolarization-evoked Ca current ICa (dark trace; light trace shows ICa from Case 1 response for comparison). (B) [Ca2+]i response (dots). Gray trace shows integrated form of Eq. 5 based on initial description of fii (E, open squares). Dark trace shows integrated results with optimized buffering parameters; dots and dark trace overlap. Dashed trace shows the result of integrating after setting Juni = 0. (C) Measurement of the free Ca2+ fluxes during the response onset and recovery and their components. Shaded region shows contribution from the Ca2+ uniporter (Juni) during the recovery. (D) Measurement of net Ca2+ fluxes during the response onset and recovery and their components. Shaded region shows contribution from the Ca2+ uniporter during the recovery. Single quotes designate net fluxes per unit cell volume. (E) [Ca2+]i dependence of Ca2+ buffering strength. Open squares show measurements for the cell illustrated and the buffering curve calculated based on fitting the spatially uniform model to the [Ca2+]i response in B. Open circles and triangles and interrupted curves show analogous measurements and calculations from two other cells. Filled squares show measurements from the same cell in the presence of FCCP. (F) Comparison between the time course of the net Ca2+ flux underlying the dynamics of [Ca2+]i (thick trace) and its component representing Ca2+ entry and removal (thin traces). Parameter values for the calculated response: fiEndog = 48.2, fiBtotal,Fura = 91.2 µM, kleak = 1.3 x 10–7 s–1, A = 7.8 x 10–4, nuni = 0.8. JPMCA parameters were the same as in Case 1.

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 J_Extru, the rate of Ca²⁺ uptake by the uniporter (J_uni) 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:

(4)

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

(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 J_Extru+J_uni was described using a look-up table as in Case 1, or by fitting an equation to the J_i measurements during the recovery (Fig. 3 C). For fitting, the same equation and parameters were used for J_Extru as in Case 1 and J_uni 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 f_ik_i (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 f_iEndog = 53.9 ± 10.2, f_iB_total,Fura = 65 ± 13 µM, A = 4 ± 2 x 10^–4, and n_uni = 1.0 ± 0.2.

Properties of the Ca²⁺ Flux Components.
Analysis of the Ca²⁺ flux components (Fig. 3 C) shows how J_VSCC varies with [Ca²⁺]_i during the response onset. J_VSCC increases rapidly after depolarization (Fig. 3 C, top, transition 1 2) when it dominates J_i, and continues to increase despite inactivation of I_Ca. As in Case 1, this occurs because there are quantitative differences in the way I_Ca and f_ii change during the onset. While I_Ca shows a time course that is very similar to that observed in the presence of FCCP, f_ii changes more dramatically because [Ca²⁺]_i is restricted to a lower range, where f_ii is more sensitive to [Ca²⁺]_i. As a result, the ratio I_Ca/f_ii increases throughout the onset, despite the decline in I_Ca due to inactivation. This complex behavior in not observed with the (volume-normalized) net Ca²⁺ fluxes obtained by multiplying the free Ca²⁺ fluxes by f_ii at corresponding [Ca²⁺]_i levels (Fig. 3 D). In particular, both and 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 (J_uni) at each point in time (t) during the recovery by subtracting from the total flux J_i(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 and (thin traces) and their sum (thick trace) throughout the response. As in Case 1, depends on the imbalance between Ca²⁺ entry and removal rates, the former depending explicitly on time (through the time dependence of I_Ca), 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 I_Ca, 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).

View larger version (23K): [in this window] [in a new window] [Download PPT slide]   Figure 4. Analysis of Ca2+ responses elicited by strong depolarization when mitochondrial Ca2+ uptake is enabled (Case 3): spatially uniform model. A. Depolarization-evoked Ca current, ICa. (B) [Ca2+]i response (dots). Gray trace shows integration of Eq. 5 based on description of fii from On and Off measurements in Case 1 from the same cell (filled squares, E). Parameter values for the calculated response: A = 10–5, nuni = 1.6. JPMCA and buffering parameters were the same as in Case 1. Calculations of JExtru and Juni at [Ca2+]i levels above the measured peak were based on extrapolations of the flux curves to higher levels. Insets in A and B show ICa and [Ca2+]i during depolarization on an expanded scale. (C) Measurement of the free Ca2+ flux during the response onset and recovery. Shaded region shows contribution from the Ca2+ uniporter during the recovery (Juni). (D) [Ca2+]i dependence of the fluxes from C (dashed box) shown on an expanded scale. Measurements of Ji during the recovery overlap measurements from the same cell after weak depolarization (compare filled and open circles). (E) [Ca2+]i dependence of Ca2+ buffering strength in Cases 1–3 from this cell. Open circles show measurements derived from the response shown in A and B. Triangles show measurements from two other cells studied with the same protocol. While measurements at low [Ca2+]i agree with those from the same cell in Cases 1 and 2, the measurements at high [Ca2+]i deviate systematically from extrapolated curves deduced from responses to weak depolarization. (F) [Ca2+]i responses from another cell elicited by 2-s-long depolarizing pulses of different magnitudes, illustrating the dynamic range of the fura-2 measurements.

Analysis of the Underlying Ca²⁺ Handling Systems.
We measured the component of J_i representing Ca²⁺ uptake by the uniporter by subtracting J_Extru from J_i at corresponding values of [Ca²⁺]_i (shaded region in Fig. 4 D) and described the [Ca²⁺]_i dependence of J_uni using Eq. A12. Based on analysis of three cells, A = 5.6 ± 2.8 x 10^–6 and n_uni = 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 f_ik_i 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 J_i 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 f_ii 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 J_Extru required to account for the upward shift in J_i 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.