INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B REFERENCES

Transient microscopic elevations in cytosolic Ca²⁺ concentration, termed "Ca²⁺ sparks," were first discovered in ventricular cardiac muscle cells labeled with the Ca²⁺-sensitive indicator dye fluo-3 (Cheng et al., 1993). Subsequently, Ca²⁺ sparks were also observed in other striated and smooth muscle cells (Tsugorka et al., 1995; Klein et al., 1996). The spatio-temporal summation of individual Ca²⁺ sparks can explain the cell-wide Ca²⁺ transient during activation of cardiac muscle cells (Cannell et al., 1994; Bridge et al., 1999). Ca²⁺ sparks occur at the location of t-tubules (Shacklock et al., 1995; Cheng et al., 1996a), which is where the dyadic junctions between closely opposed transverse tubular and sarcoplasmic reticulum (SR) membranes occur (Jorgensen et al., 1993; Soeller and Cannell, 1997; Franzini-Armstrong et al., 1999). The discovery of these local events has helped to explain how cardiac excitation-contraction coupling can exhibit high amplification of the trigger Ca²⁺ influx across the sarcolemma without becoming unstable (Stern, 1992; Cannell et al., 1995; Santana et al., 1996).

Recent studies suggest that cardiac Ca²⁺ sparks arise from the opening of a small number of SR release channels or ryanodine receptors (RyRs) (Bridge et al., 1999; Lukyanenko et al., 2000; Wang et al., 2001). Because the fluorescence image of a Ca²⁺ spark reflects the gating of RyRs in their native environment, analysis of Ca²⁺ sparks can provide insight into the regulation and gating of RyRs in situ. The magnitude of the SR Ca²⁺ release flux underlying Ca²⁺ sparks is the product of the number of RyRs present, their single-channel flux, and their open probability and can be used to provide insight into these variables (Cannell and Soeller, 1998). For example, if just the single-channel current is known, the minimum number of RyRs involved in generating a Ca²⁺ spark can be estimated. If the number of channels is also known (for a fixed spark site under steady-state conditions), the open probability of the RyRs can be estimated.

Unfortunately, extraction of the Ca²⁺ release flux underlying Ca²⁺ sparks is complicated by the recorded fluorescence transient not being a direct measure of the SR release flux. This problem arises from the diffusion and binding of Ca²⁺ to the indicator (and other Ca²⁺ binding species) and the low signal-to-noise ratio of the recorded Ca²⁺ spark. Although flux reconstruction methods have been used to estimate SR release fluxes underlying sparks (Blatter et al., 1997; Rios et al., 1999) and analyze modulation of RyR gating (Lukyanenko et al., 1998), a number of serious problems and uncertainties remain. First, the ability to extract the release flux underlying Ca²⁺ sparks has not yet been experimentally tested. Second, it is known that significant dye binding to immobile structures occurs in the cytosol (Harkins et al., 1993) and a recent study of Ca²⁺-indicator signals from muscle has shown that a detailed model of indicator binding is required to explain the experimental observations (Hollingworth et al., 2000). Nevertheless, all previous reconstruction procedures have been based on "simple" dye binding models where the binding of the fluorescent indicator dye to immobile intracellular structures was not considered in detail (but approximated using a lower effective diffusion coefficient). A third problem arises from the sensitivity of the flux reconstruction algorithm to noise in the input data, which, in combination with the noise associated with Ca²⁺ sparks records, renders estimation of release fluxes problematic. Although averaging of Ca²⁺ spark records could provide a way to improve the signal-to-noise ratio of the input data, simply averaging from randomly selected sites is complicated by the site-to-site variation in Ca²⁺ spark properties (Parker and Wier, 1997; Bridge et al., 1999). Finally, it is unknown whether the average reconstructed release flux is proportional to the release flux reconstructed from an averaged Ca²⁺ spark (i.e., if the method is linear). Without such linearity, the flux reconstructed from average sparks will not reflect the flux underlying individual Ca²⁺ sparks.

In this study, we have tested some of the implicit assumptions of previous flux reconstruction methods and have developed new methods that are applied to sparks recorded with an averaging protocol. By generating artificial Ca²⁺ sparks with known underlying fluxes (with two-photon excited spot photolysis of caged Ca²⁺, Soeller and Cannell, 1999b) we have obtained the first experimental test of a flux reconstruction method. In addition, we have investigated how a more complicated dye binding model (Harkins et al., 1993) improves flux estimation. Our results highlight the errors associated with an oversimplification of the processes shaping the Ca²⁺ spark and suggest that the calculated time course of release flux inactivation based on the conventional flux reconstruction algorithm (Lukyanenko et al., 1998) is in error (because it depends on the amplitude of release flux). Our spark reconstructions calculated with the new algorithm suggest peak fluxes of 7-12 pA, suggesting that sparks must arise from clusters of >15 RyRs. Some of these results have been presented in abstract form (Soeller and Cannell, 2001).

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B REFERENCES

Numerical modeling

A numerical model of Ca²⁺ sparks was used to generate simulated fluorescence data for some numerical tests of the flux reconstruction algorithms. To accurately model the binding of the fluorescent indicator fluo-3 to Ca²⁺ and proteins (Hollingworth et al., 2000), the scheme suggested by Harkins et al. (1993) was used. This scheme accounts for dye binding to immobile structures and the reduced apparent Ca²⁺ affinity of fluo-3 that is observed in intact cells. Figure 3 A shows the binding scheme, and the various on- and off-rates are summarized in Table 1 (adjusted from 16°C to 22°C using a Q₁₀ of 2, cf. Hollingworth et al., 2000). Only the protein-free fluo-3 (F3) and fluo-3-Ca complex (F3Ca) are free to diffuse in this scheme. The value of the free diffusion coefficient (D_F3) was estimated from published values of the diffusion coefficients of fluorescein and rhodamine 6G in water at 25°C (5.1 and 2.8 · 10⁶ cm²/s, respectively, (Ko et al., 1997)). Assuming approximate spherical symmetry of the diffusing molecules and a van der Waals volume directly proportional to molecular weight (Ko et al., 1997) fluo-3 should be 1.37 and 1.2 times larger than fluorescein and rhodamine 6G, respectively. Thus, D_F3 should lie between 2.3 and 3.7 · 10⁶ cm²/s in water. We therefore assumed a value of 3 · 10⁶ cm²/s for diffusion in water. Because the viscosity of the cytosol is approximately twice that of a simple salt solution (Kushmerick and Podolsky, 1969) a value of 1.5 · 10⁶ cm²/s was used for modeling and flux reconstruction in cell experiments. The resulting reaction-diffusion equations for the fluo-3-Ca complex (free and protein bound) are listed in the Appendix (Eqs. A1 and A2), and corresponding equations describe the evolution of Ca²⁺ free fluo-3 concentration.

The reaction diffusion equation for Ca²⁺ is given in Eq. A5 in the Appendix. Binding to various Ca²⁺ buffers in the cytosol gives rise to a flux q_buf (Eq. A6), SR uptake causes an additional removal flux (q_rem) and the principal source of Ca²⁺ influx is the Ca²⁺ release flux (q_Ca). With first-order buffer reactions,
Immobile buffers included troponin and Ca²⁺ binding sites on the outer SR membrane and the inner sarcolemma. Membrane binding sites were treated as homogeneously distributed throughout the cytosol (Smith et al., 1998). Diffusible buffers included adenosine triphosphate and calmodulin. As previously used in a model of Ca²⁺ movements in skeletal muscle (Baylor and Hollingworth, 1998), we used effective adenosine triphosphate Ca²⁺ binding rates in the presence of 0.5 mM Mg²⁺ to reduce the number of differential equations (i.e., we assume constant [Mg²⁺]). Binding rates, total concentrations and diffusion coefficients for the various species (with references) are given in Table 2.

SR uptake was approximated according to the equation,

(1)

where k_SR,max is 2 · 10⁴ Ms¹, m is 3.9 and K_D,SR is 184 nM (Bassani et al., 1994). Strictly speaking, a SR leak that matches the uptake activity at resting [Ca²⁺] should be included. However, this flux was so small that we could neglect it on the time scale of these simulations (<30 ms).

The Ca²⁺ release flux q_Ca time course was either a step lasting for 10 ms or an exponentially decaying function with a time constant of 7 ms. The amplitude of the release flux was varied between 0.5 and 16.5 pA (see Results). The release flux was spread uniformly over a spherical region 200 nm in radius resembling the diameter of junctional diads in cardiac muscle (see Langer and Peskoff, 1996; Soeller and Cannell, 1997).

The Ca²⁺ diffusion coefficient in the cytosol (D_Ca) was 3 · 10⁶ cm²s¹ to account for the higher viscosity of the cytosol as compared to simple salt solutions (Kushmerick and Podolsky, 1969). All simulations were started from a uniform equilibrium distribution of all species at a resting Ca²⁺ concentration of 100 nM. The model equations were implemented using FACSIMILE (Chance et al., 1977) on a uniform grid with spherical symmetry. The radius of the simulation space was 10 µm, with a grid spacing of 67 nm.

Simulation of line scan images

Protein-bound and protein-free forms of the Ca-fluo-3 complex were assumed to contribute equally to emitted fluorescence (Harkins et al., 1993). The contribution of Ca²⁺-free fluo-3 to measured fluorescence was neglected because it fluoresces about 200 times less strongly than Ca-fluo-3 (Harkins et al., 1993). Records of normalized fluorescence (F/F₀) were generated by dividing the computed distribution of total fluo-3-Ca complex ([TF3Ca], see Eq. A3) by the equilibrium value calculated at the resting [Ca²⁺]. Optical blurring by a confocal microscope was simulated by convolving the calculated fluorescence distribution with a point spread function (PSF) that was the product of Gaussians in x, y, and z directions, respectively (where z was chosen parallel to the optical axis). Gaussians with a full width at half maximum (FWHM) of 0.4 µm in x and y and 0.9 µm in z were chosen to mimic the resolution that we routinely achieve in experiments (as measured with subresolution fluorescent beads). Finally, line scan images (Cannell et al., 1994) were simulated by extracting the time course of blurred and normalized fluorescence along a line located at a chosen distance to the center of the calculated spark (to simulate possible out-of-focus recording).

Generation of artificial model sparks

Artificial "model sparks" with known underlying release waveform were generated using two-photon excited-flash photolysis (TPEFP) of the Ca²⁺ cage DM-Nitrophen (Kaplan and Ellis-Davies, 1988). Solutions for flash photolysis were prepared in a basic solution containing 140 mM KCl and 20 mM HEPES, pH 7.4. To this solution 0.8 mM DM-Nitrophen (Calbiochem, Alexandria, Australia) and 100 µM Fluo-3 (Molecular Probes, Eugene, OR) were added. CaCl₂ was added to this solution to set the free Ca²⁺ concentration to 100 nM. Illumination for two-photon excitation was provided by a modelocked Ti:Sapphire (Ti:S) laser tuned to a wavelength of 730 nm. After passing through a Pockels cell (Conoptics, Danbury, CT) the light was coupled into a modified confocal microscope (Zeiss LSM 410, Jena, Germany) and focused to a stationary spot parfocal with the confocal image plane. The PSF of photorelease was determined by uncaging immobilized caged fluorescein-dextran (0.5 µm in plane and 1.4 µm axially; Soeller and Cannell, 1999b). The distance between the focus and the cover slip was ~20 µm to avoid edge effects. Fluorescence changes during the spot photolysis were monitored using the 488-nm line of an argon ion laser (Uniphase, San Jose, CA) in the confocal line-scanning mode of the microscope. The pinhole was set to a diameter of ~1.5 Airey units, resulting in a PSF with an FWHM diameter of 400 nm in plane and 900 nm in the axial direction. Accurate axial alignment between the photolysis spot (near-infrared light from Ti:S laser) and the recording plane (488 nm light from the Ar⁺ laser) was assured by adjustment of a lens in the Ti:S beam path, which corrected longitudinal chromatic aberration (Soeller and Cannell, 1999b). During line-scan imaging, the scan line either passed through the center of the photolysis spot (in-focus recording) or was offset from the center by 600 nm (out-of-focus recording). Flash intensities were kept low (<1 mW at the sample) to prevent photobleaching of fluo-3 and cage depletion at the site of photolysis (confirmed by the square dependence of released Ca²⁺ on flash intensity). The fluo-3 fluorescence excited by the stationary uncaging beam was negligible in comparison to that excited by the 488-nm monitoring beam (<2%). Between 30 and 480 responses were averaged to improve the signal-to-noise ratio of line-scan images of model sparks. Due to the two-photon nature of the excitation process, the photolysis rate is expected to be proportional to the square of the flash intensity. To measure the squared flash time course (for comparison with the reconstructed release time course) we also measured two-photon excited fluorescence in a dilute solution (100 µM) of fluorescein in response to flashes that were used to generate model sparks.

Parameters for flux reconstruction from artificial sparks

Reconstruction of the fluxes underlying artificial sparks generated by two-photon excited-spot photolysis of DM-Nitrophen required estimation of its binding and transport parameters. The diffusion coefficient of DM-Nitrophen (D_DM) in water was estimated as 4 · 10⁶ cm²/s using a similar procedure to that used to estimate D_F3 above. On and off rates for DM-Nitrophen were 3 · 10⁷ M¹s¹ and 0.144 s¹, respectively (Escobar et al., 1997). The on and off rates for Ca²⁺ binding to fluo-3 in the salt solution used were 2 · 10⁸ M¹s¹ and 150 s¹, respectively, resulting in a K_d of 0.75 µM (similar to that determined by Escobar et al., 1997). Note that proteins were not present in our flash solutions and D_F3 was, accordingly, 3 · 10⁶ cm²/s.

Implementation of flux reconstruction algorithms

The first method of reconstructing the SR Ca²⁺ release flux that was implemented was similar to that used by others (Blatter et al., 1997; Rios et al., 1999). Three different algorithms with two different fluo-3 binding schemes were implemented. Figure 1 shows an overview of the dye reaction models underlying these algorithms. The first algorithm (called the "simple binding algorithm" or SBA) corresponded to a simplified fluo-3 binding scheme that takes fluo-3 binding to immobile structures into account by using a lower apparent diffusion coefficient (D_F3,app) and changed binding rate constants (to reflect the reduced apparent Ca²⁺ affinity resulting from protein binding in the cytosol). Algorithms of the SBA type have been used in previous studies (Blatter et al., 1997; Lukyanenko et al., 1998; Rios et al., 1999). The second algorithm (called the "nonequilibrium binding algorithm" or NBA) was based on the reaction scheme given by Harkins et al. (1993, see also Fig. 3 A and Table 1). The NBA considers separate freely diffusing and protein bound forms of the indicator. Furthermore, the kinetics of reversible protein binding was explicitly modeled. The third algorithm (called the "fast binding algorithm" or FBA) was an approximation to the NBA, which also considered freely diffusing and bound forms of the indicator but additionally assumed that protein binding was instantaneous as described by the respective dissociation constants of the Harkins scheme (cf. Table 1). To test the approximation of instantaneous protein binding used in the derivation of the FBA spark, simulations were run using the nonequilibrium dye binding model (corresponding to the NBA), and the binding ratios of fluo-3 and fluo-3-ca to protein were calculated (i.e., [PF3]/([P][F3]) and [PF3Ca]/([P][F3Ca]), respectively). Using a release current amplitude of 10 pA and starting from a resting [Ca²⁺] of 100 nM, both ratios remained within <4% of their equilibrium values. The NBA and FBA were developed as part of this work, and a detailed description is given in the Appendix.

View larger version (38K): [in this window] [in a new window]   FIGURE 1   Comparison of the algorithms used for flux reconstruction. The dye binding models the algorithms are based upon are shown schematically in this figure. The SBA is based on a simplified binding scheme that takes dye binding into account by using a reduced apparent diffusion coefficient and reduced binding rate constants. The dye binding model underlying the NBA and the PFA explicitly considers both freely diffusing and protein-bound immobile forms of the indicator. Additionally, it takes the nonequilibrium properties of reversible dye binding to proteins into account. The FBA uses an approximation of the dye-binding scheme underlying the NBA. In this scheme, free and protein-bound forms of the indicator are also explicitly accounted for, but the reversible binding of fluo-3 to immobile proteins is always in local equilibrium. F3, freely diffusing Ca-free fluo-3; F3Ca, freely diffusing fluo-3-Ca complex; PF3, protein-bound Ca-free fluo-3; PF3Ca, protein-bound fluo-3-Ca complex; P, immobile protein; D, diffusion coefficient of respective form of indicator; DF3, free diffusion coefficient of fluo-3 in the cytoplasm; DF3,app, reduced apparent diffusion coefficient of fluo-3 in the cytoplasm.

All algorithms involve three main steps. First, an estimate of the distribution of Ca²⁺-bound fluo-3 was calculated from the normalized fluorescence data (F/F₀). F/F₀ was obtained by dividing the fluorescence transient by the resting fluorescence recorded before the spark. Next, the underlying Ca²⁺ distribution was computed (using Eqs. A9 (SBA) or A13 and A17-A19 when using the NBA and FBA, respectively). Finally, the Ca²⁺ release flux (q_Ca) was calculated using both the fluo-3-Ca complex data and the previously calculated Ca²⁺ distribution (according to Eq. A10 for the SBA or Eqs. A7 and A20 for the FBA and NBA). A more detailed description of the procedure is contained in the Appendix.

The calculation of the time course of [Ca²⁺] and q_Ca required numerical derivatives of image data with respect to time and space (cf. e.g., Eqs. A4 and A7). Numerical derivatives were computed using standard finite differencing schemes on a pixel-by-pixel basis when the input data was essentially noise free (e.g., when using simulated data). The pixel spacing in time and space was typically 1 ms by 0.15 µm, and time and space derivatives were estimated using backward and centered differences, respectively. Because numerical differentiation is very sensitive to noise in the input data, filters were used to minimize noise amplification when dealing with noisy experimental (confocal) input data. We used Savitzky-Golay filters, a class of polynomial smoothing filters that have been shown to be useful for smoothing and differentiating noisy experimental data (Bromba and Ziegler, 1981; Ziegler, 1981; Press et al., 1992). These filters are characterized by the order of the smoothing polynomial, and the width N of the filter (which is equal to the number of filter coefficients). For signals that have a Gaussian shape, the optimal width of a Savitzky-Golay filter of order 4 is 1.72 times wider than the FWHM of the signal (Bromba and Ziegler, 1981). With a typical spark width of ~2 µm and a sampling resolution of 0.15 µm, the spark FWHM equals ~13.3 pixels and the optimal N is thus ~23. We chose slightly weaker filtering to preserve more high-frequency content and a filter of order 4 and width 19 was used in experimental reconstructions.

Because the line scan images only contain data along one spatial dimension, an assumption was necessary to allow calculation of the Laplacians in the diffusion/reaction equations. Assuming an approximately spherically symmetric fluorescence distribution, we used the Laplacian in spherical coordinates,

(2)

The center of the spark was determined by fitting a Gaussian to the spark profile at the peak of the fluorescence transient. The singular behavior of the spherical Laplacian in Eq. 2 close to the center of the spark (r = 0) was avoided by using the Cartesian form there:

(3)

This approach is different from that used by Blatter et al. (1997), who applied the latter form of the Laplacian everywhere.

Numerical integration of diffusion equations during the reconstruction procedure was performed using FACSIMILE. In some cases, the calculated Ca²⁺ distributions were post-processed by a Gaussian smoothing filter (FWHM 0.25 µm × 1.5 ms) before the release flux was determined. Buffer species concentrations, binding parameters, and diffusion coefficients used in the calculations were identical to those described under Numerical Modeling. The reduced diffusion coefficient D_F3,app for the SBA was obtained from the free diffusion coefficient D_F3 by reducing it in proportion to the mobile dye fraction (determined at resting [Ca²⁺] from the Harkins scheme). On and off rates of 0.8 · 10⁶ M¹s¹ and 80 s¹, respectively, were used in the SBA reconstruction, similar to values used in previous studies (Blatter et al., 1997; Smith et al., 1998).

The total Ca²⁺ current (I_spark) was computed from the reconstructed release flux q_Ca by volume integration,

(4)

where r_i and q_Ca,i are the radius and release flux at pixel number i, respectively. To partially remove the optical blurring due to the microscope line scan, images were enhanced by digital deconvolution using an iterative maximum-likelihood algorithm (Soeller and Cannell, 1999a) prior to flux reconstruction. The PSF used in the procedure had an FWHM of 0.4 µm, reflecting the experimentally observed in-plane resolution. All data-processing routines were implemented in PDL (http://pdl.perl.org).

Flux reconstruction by parameter fitting

An alternative approach to reconstructing the fluxes underlying Ca²⁺ sparks is to fit a given model to experimental data by adjustment of the parameters that define the model. If the fit of the model converges to the experimental data, then, within the constraints of the model, we obtain an estimate of the release flux. The primary advantage of the parameter fitting algorithm (or PFA) is its decreased sensitivity to noise in the input data (because no explicit numerical derivatives need to be computed). The fitting procedure was initiated by starting a model simulation run with an appropriately chosen initial SR Ca²⁺ release waveform (that reasonably reproduced the experimental spark in terms of amplitude and width). The simulation program compared the fluorescence time course of the computed spark to that of the experimental spark at each point of a regular sampling grid over a period of 35 ms and computed a residual sum of squares of differences. In subsequent runs, the parameters characterizing the release waveform were varied to minimize the residual sum of square differences until the best fit was obtained. This was indicated by a reduction of the residual square error to within a range that was expected based on the noise in the experimental data (using an estimated accuracy of ~5% of the average spark data). The whole fitting process was automated under control of the simulation package FACSIMILE (Chance et al., 1977) that was used to implement our spark model as described in Numerical Modeling above. The release waveform q_Ca was defined by seven adjustable parameters I₀, t₁, ₁, t₂, ₂, ₀ and _s,

(5)

where F is Faradays constant. The width of the release region  was a function of the two parameters ₀ and _s and could change linearly in time when t > t₁:

(6)

The temporal shape of the release waveform was a product of two Boltzmann equations:

(7)

By appropriate choice of time constants t₁, ₁, t₂, and ₂ this function can assume approximately rectangular shapes and exponentially decaying/rising time courses (see Results). The experimental input sparks used as a fitting target were digitally deconvolved prior to fitting (using an iterative maximum-likelihood algorithm as described above) to remove the blurring by the microscope. The dye binding model and parameter set (describing dye binding, Ca²⁺ buffering and diffusion) used in the fitting process were identical to those used with the NBA (see also Fig. 1).

Imaging of rat cardiac myocytes

Enzymatically isolated rat cardiac myocytes were prepared as described elsewhere (Evans and Cannell, 1997) in accordance with the Guide for the Care and Use of Laboratory Animals published by the National Institutes of Health (NIH publication No. 85-23 1985). Briefly, 250-g Wistar rats were killed by lethal injection of pentobarbital (100 mg/kg) and hearts quickly mounted on a Langendorff perfusion system. After initial perfusion with a nominally Ca²⁺ free solution (containing in mM: NaCl, 120; KCl, 5.4; HEPES, 10; pyruvate, 5; glucose, 20; taurine, 20; pH 7.05 with NaOH) for 5 min 4 U/ml protease (Sigma type XXIV), 1 mg/ml collagenase (Worthington type II) and 100 µM CaCl₂ were added. After 10-12 min of enzyme treatment, the ventricles were cut free and chopped into small pieces. A series of triturations was performed using a wide-bore pipette. The resulting cell suspension was filtered through gauze and diluted with solution containing 100 µM CaCl₂. The free Ca²⁺ concentration was gradually increased to 2 mM, and the final cell suspension was stored at room temperature.

For line-scan imaging, cells were incubated with 5 µM of the AM form of fluo-4 for 25 min at room temperature (20-22°C). The cells were then placed in the basic solution with 2 mM CaCl₂ and 10-20 µM of nifedipine. (The nifedipine concentration was adjusted to allow identification of well-separated spark sites.) The labeled cells in the nifedipine solution were placed in the chamber of a confocal microscope (Zeiss LSM 410) that had a glass coverslip as its bottom. To image repeatedly active spark sites, a protocol similar to that used by Bridge et al. (1999) was used. Trains of action potentials were invoked by field stimulation at a frequency of 0.33 Hz. Indicator fluorescence was excited using the 488-nm line of an argon laser and detected between 510 and 550 nm (HQ filter, Chroma Technology, Brattleboro, VT). Fifty confocal lines were acquired during each field stimulation. The field stimulus was triggered after the first 10 lines and the resulting Ca²⁺ transients were recorded during the remaining 40 line scans. Line scanning was performed at the maximum scan speed of 1.4 ms/line. The total exposure time was therefore 70 ms/stimulus, and trains of up to 100 stimuli could be recorded without detectable cell damage. Images were stored to hard disk for off-line analysis.

Data analysis

Line-scan images were analyzed using custom routines written in IDL (Research Systems Inc, Boulder, CO). The fluorescence during the prestimulus period was averaged and used to calculate self-ratio images by dividing the recorded fluorescence transient by the resting fluorescence intensity. To compute average sparks, all line scan images of a stimulation series were displayed and those spark sites manually selected that gave rise to sparks that were well separated from neighboring sites (to avoid contributions from Ca²⁺ release originating at adjacent sites). After analyzing 30 individual experiments, three spark sites were found that would be suitable for full analysis. Although the stimulus occurred at a fixed time with respect to the start of the scan, sparks were activated with a jitter of 1-2 lines (that was presumably due to the latency of L-type channel opening). This jitter was removed by aligning the rising phase of sparks before averaging.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B REFERENCES

Flux reconstruction from model sparks

To critically test the ability of our flux reconstruction method to reconstruct the local release flux, we estimated the flux underlying artificial model sparks produced by TPEFP. Example line-scan images of model sparks generated by TPEFP in the DM-Nitrophen solution are shown in Fig. 2 A. The left-hand panel shows images recorded in response to a Ca²⁺ step release generated by switching the photolysis spot on for 10 ms. The right-hand panel shows responses to an illumination time course that resulted in a release spike, which decayed approximately exponentially with a time constant of 4 ms. Using the SBA (because no immobile buffers were present in the test solution) the underlying [Ca²⁺] distributions and Ca²⁺ release fluxes were calculated, and Fig. 2 B shows that the [Ca²⁺] reaches a concentration of ~1 µM in the center of the photolysis spot, similar to the [Ca²⁺] concentrations achieved during a cardiac Ca²⁺ spark. The reconstructed Ca²⁺ flux peak amplitudes were ~30 mM/s (Fig. 2 C). The time course of photolysis was measured by applying the same illumination waveforms to a solution of fluorescein and recording the resulting fluorescence time course (see Methods). Comparison of the reconstructed release time course with that measured by fluorescein shows good agreement in both cases (Fig. 2 D). Similarly, the release time course reconstructed with a variant of the PFA (that used a SBA-type dye-binding model) agrees well with the expected time course (Fig. 2 E). The fluxes calculated with the PFA were slightly larger than those calculated with the SBA. This was largely explained by the fact that the release region reconstructed with the PFA was slightly narrower (data not shown).

View larger version (42K): [in this window] [in a new window]   FIGURE 2   Experimental tests of Ca2+ release flux reconstruction. Artificial model sparks were generated by two-photon spot photolysis of the Ca2+ cage DM-Nitrophen with brief flashes of light from a Ti:Sapphire laser. Ca2+ release was monitored by imaging the fluorescence changes of the indicator fluo-3 in confocal mode. (A) Two line-scan images of normalized fluorescence (F/F0) recorded in response to an uncaging flash with a step time course (left) and a flash with an exponentially decaying time course (right). (B) The Ca2+ distributions that were reconstructed. (C) The resulting flux estimates. A comparison of the reconstructed flux time courses (solid lines) with the squared time course of the flash intensities (dashed linesmeasured by two-photon fluorescence of fluorescein) shows good agreement (D). (D) The left panel also shows release time courses (dotted) reconstructed from fluorescence data that was recorded out of focus (600 nm) or had an ~4 times lower signal-to-noise ratio (dashed). (E) Release time courses reconstructed with the PFA. In all line-scan images, time increases from the bottom to the top. This convention was also adopted in the remaining figures.

These results show that our reconstruction methods can distinguish between the square step and a spiked release waveform. In the case of the step-shaped release waveform, the effects of a lower signal-to-noise ratio (decreased by a factor of ~4) and out-of-focus recording were also tested (Fig. 2 D). Although the reconstructed release waveform was clearly more noisy when using the lower signal-to-noise input data, the general square step shape was still reproduced. Using slightly out-of-focus (600 nm) input data, the waveform was similar to that of the in-focus reconstruction but reduced in amplitude.

Numerical tests of flux reconstruction algorithms

A number of numerical tests of the performance of the flux reconstruction algorithms were carried out. It is known that dye binding to proteins occurs in muscle cells (Harkins et al., 1993) and that it must occur reversibly with finite on and off rates (for thermodynamic reasons). Therefore, the reversible dye binding model underlying the NBA was used to generate the simulated sparks because it reflects these dye properties most accurately (among the models used in this study, cf. Fig. 1). Simulated fluorescence records were calculated using a square step-release waveform corresponding to a release current of 11 pA. For ideal conditions, the fluorescence records were not blurred and represented an in-focus recording through the center of the spark site. The results generated by the NBA were, as expected, in good agreement with the simulated [Ca²⁺] release fluxes, as shown in Fig. 3 B. The FBA (that used only the experimentally observable concentration of TF3Ca) also produced reasonable time courses but with slightly increased amplitudes (Fig. 3 C). In contrast, estimates produced by the SBA showed large discrepancies with regard to time course, spatial distribution and amplitude of both [Ca²⁺] (not shown) and q_Ca (Fig. 3 D). These major errors occurred despite the fact that the effective diffusion coefficient for fluo-3 in the reconstruction was reduced in direct proportion to the level of dye binding to proteins (i.e., D_F3 was reduced by a factor of 0.13 because 87% of the dye was bound to protein). Because all previous studies have used a modification of the diffusion coefficient (to mimic the effects of dye binding) we suggest that, in general, the release flux will have been underestimated. To mimic the PSF of our confocal microscope the simulated fluorescence transient was blurred using a Gaussian-shaped kernel (FWHM 0.4 × 0.4 × 0.9 µm in x, y, and z, respectively). In the presence of optical blurring and recording from a line that passed through the center of the spark (in-focus recording, Fig. 3 E), the fast-binding algorithm produced a release flux estimate that was spatially wider than the underlying release region (~0.7 µm versus 0.4 µm). As might be expected, the calculated peak flux amplitude was reduced (~8% of the correct amplitude when using the FBA). However, the integral of the flux was only slightly reduced when compared to the actual amplitude (~8 pA versus 11 pA). The effects of blurring could be partly restored by deconvolving the line-scan data prior to reconstruction. Using the deconvolved data, the amplitude was restored to ~20% of the correct one and the integrated current was nearly fully recovered (~10 pA). The reconstructions generated by the SBA underestimated peak flux and integrated flux more severely. When the input data was further degraded by recording out-of-focus (600 nm from the center of the spark site, Fig. 3 F) a further reduction in flux was observed (~4 pA versus 11 pA for the FBA), which was only partially recovered by deconvolution.

View larger version (38K): [in this window] [in a new window]   FIGURE 3   Numerical tests of flux reconstruction algorithms. (A) The Harkins scheme of fluo-3 binding that was used in numerical calculations to generate simulated spark records. F3, freely diffusing fluo-3; F3Ca, freely diffusing fluo-3-Ca complex; PF3, protein bound fluo-3; PF3Ca, protein bound fluo-3-Ca complex. The values of the various binding constants are listed in Table 1. (B-D) Ca2+ and Ca2+ release flux distributions reconstructed from a simulated spark record with an underlying square release waveform that had an amplitude of 11 pA. The simulated spark data was not blurred and the recording line was placed through the center of the spark. The graphs show the reconstructed Ca2+ release flux time course at the center of the spark (solid lines) compared to the actual time course extracted from the model (dashed). The reconstructions in (B), (C), and (D) were calculated with the NBA, FBA, and SBA, respectively. (E) Reconstructions of Ca2+ release flux calculated with the FBA and the SBA using blurred input data. The graphs of release flux and integrated current (Ispark) show that blurring results in an underestimation of the "true" quantities. This effect is increased when recording from an out-of-focus location (600 nm from the center of the spark site, Panel F). The graphs show the results of reconstructing from the raw (solid lines) and deconvolved data (dashed lines).

When model sparks with exponentially decaying release time courses were modeled, the FBA reported that the release time course was sensitive to the amplitude of the release function (Fig. 4 A). In Fig. 4 B, this artifact of the FBA is examined for release fluxes between 0.5 and 16.5 pA. Over this range, the calculated release time constant varied between ~120% and ~80% of the actual release function time constant (7 ms). When the SBA was applied to the same data, the distortion of the calculated release time course was more severe (compare Fig. 4 A and C with B and D). For the smallest release flux (0.5 pA), the error was small (~2%) whereas, for the largest release flux (16.5 pA), the error increased so that the reconstructed time constant was only 37% of the actual time constant (Fig. 4 D). These results suggest that caution must be applied to the interpretation of release waveforms calculated from Ca²⁺ sparks of different amplitudes, a problem compounded by the use of the SBA. In connection with this point, it has been suggested that large amplitude sparks are associated with a faster decaying release flux (Lukyanenko et al., 1998).

View larger version (27K): [in this window] [in a new window]   FIGURE 4   Amplitude dependence of reconstructed release flux time course. The FBA and SBA were used to reconstruct the Ca2+ release fluxes underlying simulated sparks that had an exponentially decaying time course (time constant (sim) 7 ms) and varying release current amplitude. (A) Time courses of Ca2+ release fluxes reconstructed with the FBA for simulated release current amplitudes (Ispark) of 0.5, 4.5, 10.5, and 16.5 pA. (B) The reconstructed time constants (rec) obtained from fits to reconstructed release flux traces for current magnitudes varying between 0.5 and 16.5 pA. (C) and (D) Corresponding graphs of reconstructions that were calculated with the SBA.

To test for linearity in the relationship between the reconstructed release current amplitude and the actual current amplitude, release currents were calculated from simulated Ca²⁺ sparks with different release flux amplitudes. Blurring and a defocus of 600 nm was applied to the simulation data. The results are summarized in Fig. 5 A, which shows that the amplitude of the reconstructed release current is essentially proportional to the actual release current. However, the reconstructed release amplitudes systematically underestimate the true current by a factor of ~2.5 (without deconvolution) or ~2.2 (with deconvolution). This effect was due to the out-of-focus recording (see also Fig. 3 F). The effect of defocus is explored in Fig. 5 B. For no defocus, the FBA reports a peak release flux of 77% of the actual, and this figure decreased with increasing defocus to 15% at 1 µm when the raw blurred data was used as input. With deconvolution prior to reconstruction the peak release flux is restored to ~95% for an in-focus spark and decreases to ~16% at 1 µm.

View larger version (12K): [in this window] [in a new window]   FIGURE 5   Summary graphs of algorithm performance. The reconstructions were performed with the FBA. (A) The relationship between the magnitude of the release current in the simulation (Ispark,sim) and the reconstructed release current (Ispark,rec) suggesting that the relationship is linear. (B) A graph of the dependence of the reconstructed release current on the degree of defocus for a fixed simulation current of 11 pA. Circles, deconvolved input data; squares, raw input data.

The novel PFA was also tested in a numerical experiment. To demonstrate that the PFA can distinguish between abruptly terminating and more gradually decaying release waveforms, it was used to reconstruct the fluxes underlying two noisy simulated sparks. To simulate the photon noise in experimental spark records, Poisson noise corresponding to a signal of 150 photon counts at the peak of the fluorescent transient was added to the simulated spark data. Figure 6, A and B, show results of using the PFA to reconstruct the release flux underlying a spark that resulted from a fast rising and exponentially decaying release waveform. The reconstructed release flux (that resulted in the best fit to the noisy data) approximated the actual waveform well (Fig. 6 B). Note that the release flux reconstructed from the same data using the FBA was more noisy. A second simulated spark that was generated using a nearly rectangular-shaped release waveform was also well fit with the PFA (Fig. 6 C). Figure 6 D shows that the reconstructed release flux reproduced the abrupt onset and termination of release (albeit with slightly increased rise and fall times). Notably, the reconstructed fluxes calculated with the PFA were largely unaffected by the noise in the simulated spark records.

View larger version (36K): [in this window] [in a new window]   FIGURE 6   Reconstruction of SR Ca2+ release flux by parameter fitting. The PFA was used to reconstruct the Ca2+ release flux underlying two simulated spark records. Poisson noise corresponding to 150 photons at the peak of the fluorescence transient was added to the simulated records. (A) and (B) The simulated spark used as fitting target was generated with a release function that had an exponentially decaying release time course. (A) The fluorescence time course of the best fit spark (solid lines) overlaid on data points extracted from the simulated noisy input spark that were used as a fitting target (connected by dashed lines). (B) The release flux at the center of the simulated spark record is shown (dashed) overlaid by the reconstructed release flux waveform that resulted in the best fit of the fluorescence data (solid line). (C) and (D) Corresponding plots of the fitted time course and best-fit release flux using a simulated spark record generated with an approximately rectangular release flux waveform. (B) and (D) The release flux reconstructed with the FBA is also plotted for comparison (grey solid lines).

These results suggest that the FBA and the PFA (but not the SBA) can provide a good estimation of actual release currents. As noted above, a problem in reconstructing the fluxes underlying Ca²⁺ sparks is the generally poor signal-to-noise ratio associated with their recording. Reconstruction tests using records of individual sparks suggested that the reconstruction algorithms are so sensitive to noise as to render the reconstructed flux distribution almost useless (data not shown). This was the case for previously published (SBA) and algorithms developed in this study (NBA, FBA, PFA). To ameliorate this problem, we can apply signal averaging to Ca²⁺ sparks recorded from well-separated, individual Ca²⁺ spark sites (Bridge et al., 1999). This protocol allows reconstructing the average release waveform at a fixed spark site at steady state. Although it has been suggested that individual Ca²⁺ spark sites respond in a stereotypical way (Bridge et al., 1999), it is possible the release flux time course could vary between individual Ca²⁺ sparks. Although Fig. 5 A suggests that the FBA reconstructed flux amplitude scales in proportion to the underlying flux, we also tested whether the release flux computed from the averaged Ca²⁺ spark was the same as the average of fluxes from the individual Ca²⁺ sparks. Figure 7 A shows ten simulated sparks where the release waveform deviated randomly from a mean waveform. The time of the upstroke of the underlying step release was exponentially distributed with a mean of 1.5 ms, and the duration of the release step varied exponentially with a mean of 3 ms. Additionally, the amplitude of the release current during the plateau deviated from a mean of 11 pA according to Poisson statistics with a standard deviation of 1.1 pA. The individual release time courses and the average release waveform are shown in Fig. 7 B. The Ca²⁺ distribution (Fig. 7 C) and release flux (Fig. 7, D and E) reconstructed from the average spark agree very closely with the averaged Ca²⁺ distribution and the averaged release flux, respectively. This suggests that the flux reconstruction procedures behave approximately linearly and the release flux reconstructed from average sparks at a fixed site will correspond to the average release flux at that site.

View larger version (41K): [in this window] [in a new window]   FIGURE 7   Linearity of flux reconstruction. (A) The row of images at the top shows ten simulated spark records where the release wave form deviated randomly from a mean release waveform and the resulting average spark. (B) The time courses of the release waveforms underlying the simulation of individual sparks and the mean waveform (red). (C)-(E) Compare the time courses of the average Ca2+ distribution and the average Ca2+ release flux (red) to the average of the reconstructed Ca2+ and Ca2+ release flux time courses. (D) The release flux was reconstructed from unblurred in-focus data. (E) Optical blurring and a defocus of 600 nm were also included, and the amplitude of the average release flux (red) was rescaled to match the reduced average reconstructed release flux when blurring and out-of-focus recording were included.

Experimental average sparks at fixed sites and flux reconstruction

The above tests suggest that the FBA and the PFA can provide an estimate of the release waveform associated with Ca²⁺ sparks, and that averaging to improve signal-to-noise does not seriously distort the average release flux. Figure 8 A shows ten line-scan images (from a sequence of 100) and evoked Ca²⁺ sparks in response to the electrical stimulation are clearly visible. Several repeatedly firing sites can be seen, and individual line-scan images from the sequence were selected on the basis that no adjacent sites appeared to be active (Fig. 8 B). Note the stereotypical appearance of these sparks with regard to amplitude and spatial extent in these images, supporting the idea (Bridge et al., 1999) that individual spark sites appear to respond in an all-or-none fashion. Figure 8 C shows the spatio-temporal properties of the resulting Ca²⁺ spark obtained by averaging the individual Ca²⁺ spark records. The averaged Ca²⁺ spark was associated with a peak fluorescence ratio of 3.5 and a spatial extent of ~1.7 µm (FWHM) at the peak. These properties are within the range reported elsewhere (Cheng et al., 1996a; Song et al., 1997).

View larger version (68K): [in this window] [in a new window]   FIGURE 8   Example of sparks recorded in the presence of nifedipine. (A) Nine from a sequence of 100 line-scan images recorded in a field stimulated rat cardiac myocyte. A frequently firing site is indicated (arrow). (B) Several individual sparks from that site and the resulting average spark. (C) Graphs showing the fluorescence time course at the center, the spatial fluorescence profile at the peak, and the time course of increase in spark width, summarizing the spatio-temporal characteristics of the average spark.

Figure 9 shows flux reconstructions from three individual Ca²⁺ spark sites. The upper panel shows the averaged Ca²⁺ sparks (Fig. 9 A), and the calculated release wave forms using the FBA are shown below (Fig. 9 B). Despite averaging, noise in the reconstruction is large, and, without averaging, the computed waveform would probably be useless. Because the deconvolution increased noise in the reconstructed release waveform (data not shown), the FBA calculations were performed on the raw data. Figure 9, C and D, shows the deconvolved average sparks and the best-fit sparks determined with the PFA, respectively. The release currents (integrated within a radius of 0.7 µm) that were calculated with the FBA and the PFA are compared in Fig. 9 E, and peak amplitudes ranged from 7 to 12 pA. The time courses calculated with the FBA and the PFA are in reasonable agreement with respect to rise time and peak amplitude, although the traces calculated with the FBA are generally more noisy. The release currents calculated with the PFA decay approximately exponentially. This point is explored further in Fig. 10, which shows PFA reconstructions of the release flux at the center of the three averaged sparks. Although the amplitudes of these sparks were variable, the time constants of decay of the release flux were all within the range ~7-8 ms.

View larger version (71K): [in this window] [in a new window]   FIGURE 9   Reconstruction of the Ca2+ release flux underlying average sparks. (A) Averaged sparks from three different sites. (B) The corresponding Ca2+ release flux distributions calculated with the FBA. (C) The deconvolved average sparks are shown above (D) corresponding fitted sparks generated with the PFA. (E) The integrated release currents calculated using the FBA (black) and the PFA (grey).

View larger version (23K): [in this window] [in a new window]   FIGURE 10   Comparison of the reconstructed Ca2+ release flux waveforms. An overlay of the release fluxes (calculated with the PFA) at the center of the three average sparks plotted in Fig. 9 shows that the decay times fall within a narrow range of 7-8 ms. Also shown is the normalized reconstructed release time course computed with the FBA from one of the sparks (noisy light gray curve).

Because the calculated Ca²⁺ release flux also depends on the values of the various parameters (describing binding and movement of the various species), we performed a sensitivity analysis. The release flux from one of the sparks shown in Fig. 9 was reconstructed using a number of parameter sets in which individual parameters deviated by a factor of 1.5 from our standard parameter set (see Methods). The resulting time courses of Ca²⁺ release flux and integrated current were generally unaffected, but the calculated current amplitudes varied depending on the parameter set. When the fluo-3 affinity was reduced by 50%, the calculated current was increased to 17.2 pA or ~160% of the value calculated with the standard parameter set. A 50% increase in fluo-3 Ca²⁺ affinity resulted in a reduction of the estimated amplitude to ~70%. Changes in the resting Ca²⁺ concentration caused similar changes in the estimated current amplitude (to 170 and 62% when the concentration was increased or reduced by 50%, respectively). We additionally tested the effect of similar changes in D_F3, D_Ca, total troponin and fluo-3 concentration, and kinetics of fluo-3 binding. The resulting changes in current amplitude were less than 20%.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B REFERENCES

In this study, we have tested the ability of four flux reconstruction algorithms to reconstruct local Ca²⁺ fluxes from confocal line-scan images and identified two algorithms that can provide useful measurements of the Ca²⁺ release fluxes underlying Ca²⁺ sparks in rat cardiac myocytes. By using artificial sparks, we provide evidence that our flux reconstruction algorithms produce reliable estimates of the underlying Ca²⁺ fluxes. We have also shown that detailed modeling of dye binding to immobile structures is required for accurate reconstruction, in accord with the conclusions of Hollingworth et al., (2000). Failure to account for dye movement can lead to serious artifacts in flux reconstruction, the most serious of which is a strongly amplitude-dependent time course. Because such a phenomenon has been reported, we suggest that such data need to be re-evaluated using our FBA or PFA. Using these reconstruction algorithms, we have estimated that the release fluxes underlying Ca²⁺ average sparks in rat cardiac myocytes are 7-12 pA in amplitude and last ~10 ms.

Experimental tests of the flux reconstruction algorithm

Although flux reconstruction algorithms have been derived from theoretical models of spark formation (Blatter et al., 1997), the present study provides the first direct test of such methods in an experimental model. To generate artificial Ca²⁺ sparks with known release waveforms, we used the unique ability of two-photon excited flash photolysis to create a local source of Ca²⁺ by photolysis of DM-Nitrophen. Although the region of photorelease was not spherical (but rather the ellipsoidal shape of the microscope PSF) the resulting fluorescence distribution should rapidly become spherical as diffusion takes place. The Ca²⁺ release time course underlying these artificial sparks is known because the rate of photolysis is proportional to the time course of the squared flash intensity (provided the intensity is well below saturating levels, as was the case heredata not shown). It was therefore possible to directly compare the calculated release waveforms to the known release waveform to test the reliability of the algorithms. To our knowledge, this is the first time that any such test of flux reconstruction algorithms has been performed.

To make our tests realistic, we produced artificial sparks, which resembled experimentally recorded Ca²⁺ sparks in terms of size, duration, fluorescence amplitude, and signal-to-noise ratio. The latter factor is particularly important because all inverse reconstruction algorithms (i.e., those that use differentiation) are extremely sensitive to noise. In fact, even with signal averaging, the resulting computed spark release waveform was quite noisy, and, without averaging, would have been useless. Although the PFA is less noise sensitive than other inverse reconstruction algorithms, signal averaging is still important to constrain the fitted release flux distribution. An additional complicating factor resides in the problem that the exact locus of the origin of the spark is unknown. It was therefore of interest to examine the sensitivity of the reconstructed waveform to small amounts of microscope defocus (Izu et al., 1998). We found that, as defocus increased, the reconstructed flux decreased in amplitude with minimal changes of the time course (Fig. 2). It follows that our reconstructions of evoked Ca²⁺ sparks must be a lower limit of the actual release flux, although the release waveforms should be reasonably accurate.

Reversible dye binding and flux reconstruction

Experimental observations suggest that a significant proportion of fluo-3 (>80%) is bound to immobile structures in the cytoplasm (Harkins et al., 1993). In models of Ca²⁺ spark formation, the effect of dye binding is generally approximated by reducing the effective fluo-3 diffusion coefficient in proportion to the estimated bound fluo-3 ratio (e.g., Pratusevich and Balke, 1996; Smith et al., 1998). Such an approach is justified if the binding kinetics are instantaneous and there are no differences in affinity between bound and free forms, and previous studies have used this simplifying approximation (Blatter et al., 1997; Lukyanenko et al., 1998; Rios et al., 1999). We found that a simplified dye binding model (SBA) does not correctly reconstruct the release time course, although optical blurring by the microscope helped reduce the reconstruction error of the SBA (albeit at the expense of a reduced reconstructed flux amplitude, Fig. 3). Prompted by the generally poor performance of the SBA, we developed novel algorithms (the FBA and the PFA) that are based on improved approximations to the Harkins (1993) scheme. The idea that SBA models are inadequate is supported by the observation of intrasarcomeric Ca²⁺ movements in skeletal muscle, which require a more detailed model of dye binding to explain them (Hollingworth et al., 2000). Even in the presence of optical blurring and out-of-focus recording, the FBA reconstructed time course matched the actual time course reasonably accurately, although the peak amplitude was reduced by de-focus and blurring.

Dye binding schemes

We examined three different approaches to account for dye binding. The simplest class of model (SBA) assumes that dye binding can be modeled by a reduction in dye diffusion coefficient. As described above, such models do not adequately describe either the time course or amplitude of the release flux. Even more serious is the fact that the models produce a release flux time course that is strongly amplitude dependent. Clearly, such an artifact would be unacceptable for classes of experiment which seek to relate the time course of RyR inactivation to the amplitude of the Ca²⁺ release flux (Lukyanenko et al., 1998). The next class of model would explicitly deal with protein-bound and free forms of the dye and assume instantaneous equilibration between bound and free forms. We examined this FBA model and showed that it was able to reproduce known release waveforms with a reasonable degree of accuracy. A more complete model also includes the kinetics of the dye-protein interaction. This NBA model was also good at flux reconstruction, but was not markedly better than our FBA method. However, the NBA cannot be used to reconstruct fluxes underlying experimental sparks because it requires the (nonobservable) calcium-free fluo-3 as input. We have overcome this limitation with the novel PFA, a fitting algorithm that uses the full binding model (including dye-protein binding kinetics) and is also less sensitive to noise in the input data.

Flux reconstruction from average sparks

Ca²⁺ flux reconstruction from individual experimental sparks records is problematic due to the generally low signal-to-noise ratio. Averaging spark records provides, in principle, a useful way to improve the signal-to-noise ratio. Because the FBA contains nonlinear steps (e.g., Eq. A13) it was necessary to show that the FBA behaved linearly with respect to averaging, and that this was the case was demonstrated in Fig. 7. We electrically evoked sparks and selected well-separated, repeatedly firing spark sites. To obtain such well-defined sites, the probability of Ca²⁺ spark occurrence is reduced by a Ca²⁺ channel blocker, such as nifedipine, which does not alter the properties of Ca²⁺ sparks (Collier et al., 1999). In addition, sites chosen for averaging must not be contaminated by release events from nearby sites; this strict criterion resulted in <10% of experiments yielding sites for further analysis. Furthermore, we viewed it to be important to select sparks that were at least as large (F/F₀ > 2.5) as the average spark amplitude to ensure that the site was close to the line scanned by the confocal microscope (Pratusevich and Balke, 1996; Izu et al., 1998; Cheng et al., 1999). From consideration of our selection criteria (selecting from the brightest 25% of sparks), we estimate that the sparks analyzed here should have been, on average, ~0.25 µm from the scan line (Izu et al., 1998). We therefore predict that we underestimate the true current amplitude by a factor of ~1.4 (see Fig. 5 B). In contrast, sparks have been shown to be larger in the longitudinal direction by ~18% (Cheng et al., 1996b). Because we recorded sparks in the longitudinal direction and assumed a spherical fluorescence distribution, we would have overestimated the current by a volume factor of 1.18² or ~1.4. This analysis suggests that these two effects, taken together, compensate each other, and our calculated current amplitudes should be a reasonable estimate of the underlying release fluxes.

By averaging >9 individual Ca²⁺ sparks, the improved signal-to-noise ratio allowed the FBA and the PFA to produce an estimate of the amplitude and time course of the Ca²⁺ release flux underlying these sparks. The FBA and PFA reconstructed release current time courses were similar with regard to rise time (4-5 ms) and peak amplitude (7-12 pA). Although the currents calculated with the PFA decayed approximately exponentially, the traces calculated with the FBA seemed to terminate more abruptly. We suggest that this can be largely explained by the noise sensitivity of the FBA because we observed a similar effect when the FBA was used to reconstruct a noisy simulated spark arising from an exponentially decaying release waveform (Fig. 6 B). Comparison of the release time course of the three average sparks that were analyzed suggested that the decay times were very similar, although the amplitudes of these three sparks were quite different (Fig. 10 A), in contrast to the results by Lukyanenko et al. (1998).

It is also important to note that our spark model reproduces the spatial dimension of Ca²⁺ sparks, which has been a deficiency in all previous spark models. This improvement was achieved by allowing the region of release to grow with time during the period of release. Such an effect could be reasonably explained by recruitment of RyRs from the site of initiation during the early phase of a spark evolution. Indeed, electron micrographs of couplons show them to be quite extended (Franzini-Armstrong et al., 1999).

It is notable that the SR release flux of 7-12 pA is more than twice as large as that estimated by Cheng et al. (1993) and Blatter et al. (1997). When comparing local flux densities, our estimate of ~50 mM/s is five times larger than that reported by Blatter et al. (1997) and nearly an order of magnitude larger than that estimated by Lukyanenko et al. (1998). We suggest that the previous studies underestimate the fluxes underlying the spark for several reasons: 1) the use of extensive filtering to suppress noise (Lukyanenko et al., 1998); 2) the use of SBA-type approaches (Blatter et al., 1997; Lukyanenko et al., 1998); and 3) low values for the fluo-3 binding constants (Cheng et al., 1993; Blatter et al., 1997; Lukyanenko et al., 1998). In a recent study that sought to explain the spatial dimensions of Ca²⁺ sparks, a lower estimate of ~11 pA was obtained (Izu et al., 2001), which agrees reasonably with our estimates. However, the same study suggested that the current underlying a Ca²⁺ spark could be as large as ~40 pA, which would be associated with abnormal spark amplitudes (F/F₀  12). Our estimate is approximately five times larger than that estimated by calibrating spark amplitude with local fluorescence changes produced by dihydropyridine receptor (DHPR) gating (Wang et al., 2001). A part of this difference resides in the spark amplitudes being larger in our experiments. Additionally, it is possible that the site from which the DHPR current was recorded was some distance from the spark site in the experiments of Wang et al. (2001). This idea is supported by their observation of an abnormally long delay between DHPR opening and spark activation and a relatively low coupling efficiency, despite the (drug-induced) long DHPR open times in their study.

Given an SR release flux of >7.5 pA, we can estimate that at least ~15 RyRs must be activated during a Ca²⁺ spark (for a single RyR current of ~0.5 pA, Mejia-Alvarez et al., 1999). Because this cluster of RyRs opens within ~4 ms, we can estimate that the opening rate constant must be ~500/s (if 4 ms equals two time constants) which, if Ca²⁺ gated, implies a local trigger level of ~10 µM for a diffusion-limited on rate of 2 × 10⁸ M¹s¹. This estimate of the local trigger Ca²⁺ is in good agreement with previous detailed mathematical modeling of the dyadic cleft (Cannell and Soeller, 1997; Soeller and Cannell, 1997) and an analysis of the increase in spark rate during excitation-contraction coupling (Cannell et al., 1994). By examining ruthenium red inhibition of RyRs it was estimated that >10 RyRs should be involved in a Ca²⁺ spark (Lukyanenko et al., 2000). Using a different approach, Bridge et al. (1999) estimated that at least 19 RyRs were activated during a Ca²⁺ spark, in reasonable agreement with our results. To achieve the activation