INTRODUCTION

Top Abstract Introduction Materials & Methods Results Discussion Appendix References

A detailed understanding of calcium (Ca²⁺) dynamics in excitable cells requires a quantification of different sources and sinks for Ca²⁺ ions. A generic scheme of Ca²⁺ signaling can be viewed as consisting of the following components (Clapham, 1995): influx of Ca²⁺ from some extracytosolic compartment into the cytosol, diffusion of Ca²⁺ as well as binding of Ca²⁺ to different buffers (which can be endogenously present in the cell or added exogenously by means of patch pipettes), and uptake into internal Ca²⁺ stores or extrusion across the plasma membrane. The influx of Ca²⁺ across the plasma membrane is well-characterized by means of patch-clamp recordings whereas the diffusional spread of Ca²⁺ ions while binding to different cellular buffers, the so-called "buffered diffusion problem," has received much less attention. Once the impact of these buffers as Ca²⁺ sinks is understood, one can quantitatively study the remaining determinants of Ca²⁺ signals, which are given by fluxes across different biological membranes, such as membranes of the endoplasmic reticulum or mitochondria. It is the study of the buffered Ca²⁺ diffusion in single bovine chromaffin cells that this paper is dedicated to.

Several imaging studies have revealed intracellular gradients of the free Ca²⁺ concentration ([Ca²⁺]) in different cell types (Williams et al., 1985; O'Sullivan et al., 1989; Kasai and Augustine, 1990; Huser et al., 1996). In bovine adrenal chromaffin cells, [Ca²⁺] gradients were seen to dissipate within a few hundred milliseconds (Neher and Augustine, 1992) and this time course was prolonged if the concentration of the Ca²⁺ indicator was increased. Since the dye competes with endogenous buffers in binding Ca²⁺, the dye-dependent changes in the recovery time course of the [Ca²⁺] signal were used to estimate the average Ca²⁺ binding ratio of the cytoplasm of chromaffin cells under whole-cell recording conditions. Later, using the perforated patch method, Zhou and Neher (1993) were also able to distinguish between the capacity of mobile and immobile endogenous Ca²⁺ buffers in chromaffin cells, but again as a cellular average. These studies have prompted a series of investigations that aimed at determining the cytosolic buffering power of various cell types (for a review see Neher, 1995).

Ca²⁺ signaling inherently involves three problem dimensions, namely time, space, and amplitude, and the range of action of a Ca²⁺ signal heavily depends on spatial buffering properties of the cytoplasm (Allbritton et al., 1992; Wagner and Keizer, 1994). This in turn raises the question of whether a given cell can be regarded as spatially homogeneous with respect to its buffering power or whether it constitutes a highly differentiated medium for spatial Ca²⁺ signals: an intriguing possibility, especially for polarized cells that use Ca²⁺ as a fast second messenger. Nevertheless, there is no quantitative report on spatially resolved measurements of Ca²⁺ binding properties of cells up to this day. Here, we attempt to establish methods to assess the distribution of Ca²⁺ binding sites in a cell under the whole-cell mode of the patch-clamp technique. We show that the effective spatial resolution is dictated by the acquisition rate of the imaging system and out-of-focus effects due to light diffraction. In the specific case of cultured adrenal chromaffin cells, we do not see signs of heterogeneity of fixed Ca²⁺ buffer distribution on a micrometer spatial scale, which, of course, does not exclude gradients of Ca²⁺ buffers on a submicrometer scale.

	MATERIALS AND METHODS

Top Abstract Introduction Materials & Methods Results Discussion Appendix References

Cell preparation and solutions

Chromaffin cells in primary culture from bovine adrenal glands were prepared and cultured as described previously (Smith and Neher, 1997). Cells were used for experiments 1-4 days after plating in culture dishes. The standard external bath solution for experiments contained (in mM): 140 NaCl, 2.8 KCl, 5 CaCl₂, 1 MgCl₂, 20 HEPES, and 2 mg/ml glucose (pH 7.2, 310 mOsm). The internal solutions were based on a 2× concentrated buffer containing (in mM): 290 cesium glutamate, 40 HEPES, and 12 NaCl (pH 7.2). After adding appropriate amounts of indicator dye or DM-nitrophen (DMN) or CaCl₂, this solution was diluted twofold to give the final internal solution with an osmolarity of 300-320 mOsm and pH 7.20.

DMN was purchased from Calbiochem (La Jolla, CA); Fura-2, Bis-Fura-2, and BAPTA were purchased from Molecular Probes (Eugene, OR). All other chemicals were from Sigma (St. Louis, MO).

Combining patch-clamp, digital Ca²⁺ imaging, and flash-photolysis of DMN

Our experiments required patching the chromaffin cells, loading them with the fluorescent dye, and activating the voltage-dependent Ca²⁺ channels by short depolarizing pulses as well as imaging the [Ca²⁺] distribution. In some experiments it was also necessary to photorelease Ca²⁺ from Ca²⁺-loaded DMN by short pulses of UV light. The apparatus to achieve this goal is schematically depicted in Fig. 1. It is centered around an inverted Zeiss microscope, Axiovert 135 TV. Two light sources are coupled into the epiillumination port of the microscope: a UV flash lamp (Rapp Optoelektronik, Hamburg, Germany) and a polychromatic light source (T.I.L.L. Photonics, Gräfeling, Germany), which is based on a xenon lamp. It chooses the appropriate wavelength by positioning a grid on a galvanometric scanner. The grid position is set by analog signals from a control unit, which receives commands from a master PC. The same PC is also used to trigger the flash lamp and the depolarizing voltage pulses of the patch-clamp amplifier (EPC-9, HEKA Electronic, Lambrecht, Germany) via a Macintosh Quadra 950 computer. The flash light passes a UG11 filter and an appropriate neutral density filter and is finally directed via a 50%/50% beam splitter to the back pupil of a 40× water immersion Zeiss objective (NA = 1.2, C-APOCHROMAT). The 50% transmission of the same beam splitter is also used to feed the fluorescence excitation light of the polychromatic source into the objective. By directing a fraction (8%) of the light to a photodiode, we also monitored the light intensities by sampling the output of a photodiode amplifier using the master PC. In addition, the objective was mounted on a piezoelectric element (Pifoc, Physik Instrumente, Germany), driven by a PC-controlled unit, which enabled us to acquire images at different focal planes with a precision of 10 nm, if needed.

View larger version (41K): [in this window] [in a new window]   FIGURE 1   Scheme of the apparatus for Ca2+ imaging and flash photolysis. The setup is built around an Axiovert 135 TV. Two light sources are coupled into the microscope using a 50%/50% beam splitter: steady-state light (from a xenon lamp) for fluorescence excitation and flash light for photolysis. The power supply of the xenon lamp was pulsed for a few hundred milliseconds to increase the excitation power. The light intensity and the flash time course are monitored by a fast photodiode, which receives ~8% of the total light power. All the equipment is controlled and synchronized by a "master PC," which also reads in the image data from a 12 bit water-cooled frame transfer CCD camera.

For the acquisition of the fluorescent images, a water-cooled frame-transfer CCD camera was used. The images were stored with a dynamic range of 12 bits on the PC and later transferred to a SPARC-10 (SUN Microsystems, Cupertino, CA) UNIX workstation for analysis. Finally, in the diffusion experiments, it was crucial to maximize the signal-to-noise ratio (SNR) of the Ca²⁺ images. We achieved this goal by increasing the supply current of the xenon lamp and, thus, the excitation light intensity. A voltage between 1 and 2 V, provided to a controlled power supply, was converted to a current (with a gain of 1 V/5 A), which was added to the normal supply current (5.4 A) for the xenon lamp. This pulsing of the power supply enabled us to increase the excitation intensity by a factor of 3 for a few hundred milliseconds, which gave rise to a corresponding increase in the fluorescent counts during the pulsing. However, it was not possible to maintain the high excitation intensity for >500 ms because the lamp would otherwise become unstable (as was monitored by the photodiode).

Mathematical model for diffusion and method for calculating local diffusion coefficients

The idea in the first set of experiments was to observe the diffusive dissipation of [Ca²⁺] gradients and then to extract local apparent Ca²⁺ diffusion coefficients from a fit of the theoretically expected [Ca²⁺] time course to the experimentally observed one. Consequently, we need a mathematical model of the expected [Ca²⁺] time course in the presence of different mobile or immobile calcium buffers. Intuitively, the presence of an immobile endogenous Ca²⁺ buffer would slow down the diffusion and, thus, show up as a reduced apparent Ca²⁺ diffusion coefficient. A mobile Ca²⁺ buffer, however, would shift the apparent Ca²⁺ diffusion coefficient toward the mobile buffer's diffusion coefficient as an increasing fraction of Ca²⁺ is being carried by the buffer. A formalization of this idea was given by Wagner and Keizer (1994) within the framework of the rapid buffer approximation (rba) to the buffered diffusion problem. In the rba, one assumes that all buffers, whether mobile or immobile, have such fast Ca²⁺ binding kinetics that they are in chemical equilibrium with local [Ca²⁺] at every instant of time and at every point in the cell. In other words, the time scale for dissipation of the [Ca²⁺] gradients is supposed to be much slower than the time scale on which the Ca²⁺ binding reactions approach chemical equilibrium. If this is true, the dynamics of [Ca²⁺] can be described by a single nonlinear partial differential equation

(1)

where D_i and _i are the diffusion coefficient and the Ca²⁺ binding ratio of the ith buffer B_i. Note that the binding ratios _i = ([B_i]_TK_i/(K_i + [Ca²⁺])²) are nonlinear functions of [Ca²⁺] and the buffer's dissociation constant K_i. If, however, the affinity of the buffers is low, i.e., [Ca²⁺]  K_i, then _i = [B_i]_T/K_i is independent of [Ca²⁺]. Otherwise, in case of small [Ca²⁺] excursions, _i can be regarded as independent of [Ca²⁺] as well. Then, assuming that the free and Ca²⁺-bound form of mobile buffers have the same diffusion coefficient, Eq. 1 can be simplified to

(2)

which is the classical diffusion equation with an apparent Ca²⁺ diffusion coefficient given by D_app = (D_Ca + _iD_ii)/(1 + _ii). In whole-cell recordings, any endogenous mobile Ca²⁺ buffer washes out within minutes (see Zhou and Neher, 1993). Consequently, one is left with the fluorescent Ca²⁺ indicator, which is a mobile buffer, and immobile endogenous Ca²⁺ buffers, which we lump together into one species. This results in

(3)

with _endo representing the binding ratio of the immobile endogenous buffer, the quantity we want to determine. Note that a heterogeneous distribution of the fixed buffer shows up in a heterogeneous distribution of D_app, as expected intuitively. The problem can now be restated as follows: by using [Ca²⁺] imaging, we observe the solution to the system (Eq. 2) and ask for its structural parameters, namely the spatial distribution of D_app. This problem is mathematically referred to as a so-called "inverse problem"; the "direct problem" being the calculation of the [Ca²⁺] time course from a knowledge of all system parameters, which is the classical domain of simulation studies.

As a first method, one is tempted to calculate D_app as the ratio ([Ca²⁺]/t)/([Ca²⁺]). This, however, corresponds to multiple high-pass filtering operations on [Ca²⁺] images and leads to an explosion of the noise level, which makes any meaningful interpretation of the results impossible. Consequently, we decided to take another approach for estimating D_app based on nonlinear regularization theory (Tarantola, 1987; Louis, 1989). Let us denote by D_theo an arbitrary distribution of apparent Ca²⁺ diffusion coefficients in the cell. Assume that at time zero we have a [Ca²⁺] distribution, which we denote by c_init. t ms later, diffusive spread of Ca²⁺ will cause a [Ca²⁺] profile c_theo, which of course depends on D_theo. We write this dependence as c_theo = c(D_theo). We can now compare the theoretically expected Ca²⁺ distribution, c_theo, with the experimentally observed one, c_obs, and adjust the diffusion coefficients in such a way that c_theo best matches c_obs. This is the regularization approach, which results in D_app as a solution to the multidimensional optimization problem of minimizing the error functional f(D_theo):

(4)

In other words, we look for a D_app distribution that optimally reproduces the observed spatiotemporal [Ca²⁺] profile (the first term in the above sum) and simultaneously has some degree of smoothness (the second term in the sum). The trade-off between these two aspects is controlled by the regularization parameter , which in general is selected empirically and causes the solution to the above problem to be unique. The next issue is then how to solve the minimization problem given in Eq. 4. For this end, we used many methods under which the Gauss-Newton and the conjugate-gradient method (Press et al., 1992) were the fastest algorithms. Here, we outline the conjugate-gradient method:

1.	Choose initial distribution Dtheo(0) (spatially homogeneous), set i = 0;
2.	Compute d(0) = r(0) = (f(Dtheo(0))/Dtheo). This calculation of the derivatives of the error functional f with respect to the parameters Dtheo involves solving a parabolic system of partial differential equations, which we performed using the Cranck-Nicholson algorithm. The dimension of this system is identical to the number of pixels in the cell;
3.	Find (i) that minimizes f(Dtheo(i) + (i)d(i)) using a line search algorithm;
4.	Set Dtheo(i+1) = Dtheo(i) + (i)d(i);
5.	Compute r(i+1) = (f(Dtheo(i+1)/Dtheo) [exactly like (2)];
6.	Set (i+1) = (r(i+1)2/r(i)2),  d(i+1) = r(i+1) + (i+1)d(i);
7.	Increment i to i + 1 and go to step (3).

The regularization parameter  was determined empirically by simulating the diffusion process and adding noise (before backcalculating the diffusion coefficients) to achieve the same SNR as in the [Ca²⁺] measurement. Then we applied the inverse algorithm with different values for  and compared the estimated diffusion coefficients with the ones we used for the simulation of the diffusion process. This procedure identified a range of applicable parameters, the values used were between 0.001 and 0.005.

Estimation of endogenous Ca²⁺ binding ratios with photolysis of DMN

Some inherent problems, which are detailed in the Discussion part of this paper, prompted us to reattack the estimation problem of the binding ratios with a different approach. The rationale here was the following: if we manage to photorelease Ca²⁺ from DMN quantitatively (Zucker, 1993) and rapidly compared with the mean diffusional equilibration time of Ca²⁺ gradients, the Ca²⁺ ions can either bind to some buffers or appear as free Ca²⁺ only at the pixel where they have been released. The more buffer one has at a given pixel, the smaller the increase in [Ca²⁺] upon photorelease will be. In other words, the increase in [Ca²⁺] is a measure of the total Ca²⁺ binding ratio of the cell at each and every pixel, as long as no significant spread of Ca²⁺ ions to neighboring pixels happens. The total Ca²⁺ binding ratio at every pixel is in turn the sum of binding ratios of exogenous and endogenous buffers. This consideration results in the following pixelwise identity:

(5)

where [Ca²⁺] is the difference in [Ca²⁺] before and after the flash. Thus, we need to know the total amount of Ca²⁺ released by a flash (from calibration measurements), [Ca²⁺]_total, the binding ratios _DMN and _ind, and measure the difference between free Ca²⁺ concentration after and before flash, [Ca²⁺], to calculate _endo. If the exogenous buffers are homogeneously distributed in the cell (by virtue of their mobility) and the Ca²⁺ release pattern is also homogeneous in the cell, every spatial heterogeneity in [Ca²⁺] can only result from a heterogeneous distribution of the fixed endogenous buffer. This again gives us quantitative information about binding ratios and their distribution without expensive and noise-sensitive diffusion measurements.

	RESULTS

Top Abstract Introduction Materials & Methods Results Discussion Appendix References

Diffusion experiments

As indicated above, our goal was to study the relaxation of depolarization-induced [Ca²⁺] gradients in the cytosol of chromaffin cells. For this end, we recorded in whole-cell mode with pipette solutions, which contained 100 µM Fura-2 but no ATP to exclude any further exogenous Ca²⁺ buffer. The choice of the Fura-2 concentration was a compromise between the following opposing constraints. a) One would like to aim at as small Fura-2 concentrations as possible, since the indicator is only a reporter of the Ca²⁺ signal and should minimally perturb the system. Increasing dye concentration would eventually lead to an exogenous binding ratio much bigger than the endogenous one. Then, the endogenous buffer will not affect the Ca²⁺ signal by modulating the apparent Ca²⁺ diffusion constant and, hence, will not be visible in the measurement. b) Alternatively, the SNR of the Ca²⁺ measurement is a critical factor for the inverse estimation procedures, which we outlined above. Since the SNR is itself determined by the number of photoelectrons generated at each pixel of the camera chip during acquisition of an image, the aim is to catch as many photons as possible per pixel and image integration time. But what parameters do we have at our disposal to increase the number of photons? These are threefold: higher dye concentrations, higher excitation intensities, and longer integration times per image. Observation of the gradient dissipation necessitates high acquisition rates; hence, we cannot afford to increase image integration time. Consequently, we chose to use 100 µM Fura-2, which gives rise to a binding ratio of 45 at [Ca²⁺] = 500 nM, the maximal acquisition rate of 40 Hz for single wavelength measurements, and to increase the excitation intensity for 200-300 ms by a factor of 2-3 within the acquisition time, by pulsing the power supply as outlined in Materials and Methods.

Fig. 2 shows two such series of [Ca²⁺] images from different cells with 25 ms integration per frame. In A and B the lowermost images were taken at rest with a holding potential of 60 mV. During the next frame, the cell was depolarized to 0 mV while all other images were taken again at 60 mV. In both cases, one can clearly identify initial [Ca²⁺] rises underneath the plasma membrane, whereas the pattern of spatial spread of these Ca²⁺ signals is quite distinct. In (A), the nucleus is located in the lower-right quarter of the cell (as identified in transmission images) while Ca²⁺ entry mostly occurs within a rim of the membrane on the opposite site of the cell. The incoming Ca²⁺ gives rise to cytosolic [Ca²⁺] gradients, which spread toward the nucleus within 100 ms. The nucleus seems to constitute a diffusion barrier since, even after 200 ms, there is a marked difference between the cytosolic and nuclear Ca²⁺ concentration. This is a type of cell which is not accessible to our inverse methodology for estimating apparent Ca²⁺ diffusion coefficients, because major parts of the cell only have very flat Ca²⁺ gradients and, consequently, hardly any information can be obtained. In (B), Ca²⁺ appears to spread into the cytosol in a radial fashion (in the first two or three images after the depolarizing pulse) while we still see signs of hindered diffusion around the nucleus. Nevertheless, when attempting to estimate Ca²⁺ diffusion coefficients in two-dimensional (2-D) images, we have to cope with some inherent problems.

View larger version (103K): [in this window] [in a new window]   FIGURE 2   Pseudocolored images of depolarization-induced [Ca2+] gradients in adrenal chromaffin cells. Depicted are two series of [Ca2+] images (from two different cells) acquired at 40 Hz with 25 ms exposure time per frame. In (A) and (B), the image at the bottom is acquired at a resting membrane potential of 60 mV. During the next frame, the cell was depolarized to 0 mV while all other images are again at Vm = 60 mV. In (A) the nucleus is located at the lower right quarter of the cell (visible in transmission images, not shown here) while the Ca2+ influx mostly happens at the opposite quarter. Furthermore, the nucleus seems to constitute a pronounced diffusion barrier for Ca2+. The Ca2+ influx in (B) occurs across a major part of the plasma membrane, and consequently a Ca2+ wave spreads toward the center of the cell.

1) The measured fluorescence is influenced by out-of-focus light, which causes blurring. Likewise, Ca²⁺ diffuses into and out of the focal plane (which should be rather called a focal slice) while [Ca²⁺] is measured at only one slice. Unfortunately, we are not able to reconstruct the spatiotemporal [Ca²⁺] profile in response to a depolarizing pulse in 3-D because of the acquisition time needed for a frame; and we cannot repeat the same experiment at different focal planes since channel statistics and current rundown imply that the [Ca²⁺] distribution is never guaranteed to be the same by repeating the same pulse many times at different focal planes.

2) A cross-section of a typical chromaffin cell is represented by ~1000 pixels in 2-D with a physical pixel size of ~500 nm × 500 nm. Thus, from the relaxation of the gradients, one needs to estimate 1000 unknowns, namely the diffusion coefficients at each pixel. This is algorithmically a very expensive task and requires high-end computing power in conjunction with a very high SNR.

We could derive a partial solution to these problems by using the observation that in some cells, the Ca²⁺ gradients appeared to be approximately radial. To qualify a cell for this radial approach, it had to fulfill two conditions: the integral of [Ca²⁺] had to be constant between consecutive images (to exclude release or uptake processes), and the [Ca²⁺]-profiles along neighboring lines through the center of the cell had to be almost identical (note that within the image acquisition time, the Ca²⁺ signal spreads ~1-2 µm, as discussed later). This reduced the problem to estimating the diffusion coefficients along a line. Two line profiles, 25 ms apart, together with the theoretically expected profile using the estimated diffusion coefficients, are plotted in the top panel of Fig. 3. The bottom panel shows the corresponding distribution of the diffusion coefficients where one can clearly discern spatial heterogeneities of Ca²⁺ mobility. The Ca²⁺ diffusivity is lowest under the plasma membrane. Moving further into the cell, one identifies two local minima of D_app. Comparison with transmission images reveals that these minima are located right at the positions where the nuclear membrane is, while the high D_app values correspond to the nucleoplasm. This would translate into the following pattern for the immobile endogenous buffer distribution: higher levels close to plasma or nuclear membrane and low expression levels within the nucleus.

View larger version (15K): [in this window] [in a new window]   FIGURE 3   Relaxation of [Ca2+] gradients and estimation of the apparent Ca2+ diffusion coefficients along a line through the center of the cell. The top panel shows two consecutive [Ca2+] line profiles, which are taken 25 ms apart. Superimposed is also the theoretically expected [Ca2+] profile (filled circles), which one would see as the solution to the diffusion equation 25 ms after the observed initial [Ca2+] distribution (t = 0 ms, observed), if the distribution of the apparent diffusion coefficients, Dapp, is as given in the bottom panel. The regularized estimates for the diffusion coefficients show local minima at the boundary of the cell, i.e., close to the plasma membrane, as well as close to the nuclear membrane.

But there are alternative ways to explain these results. The nuclear membrane can constitute a diffusion barrier and, thus, its effect can show up as a reduced Ca²⁺ diffusion coefficient. This in combination with optical blurring of the fluorescence can then give rise to the broad minima of D_app, which we observe in Fig. 3, without an underlying increase in the immobile Ca²⁺ binding ratio. Likewise, at the boundary of the cell, restricted diffusion and blurring effects can cause reduced D_app values. We could not exclude any of these possibilities, since we cannot observe the [Ca²⁺] time course in 3-D, and both hindered diffusion and high immobile binding ratios will show up as low D_app within our framework. To distinguish between these alternative interpretations, we used flash photolysis as another means to estimate _endo.

Photolysis experiments

In this set of experiments, it was crucial to measure the [Ca²⁺] distribution immediately after photolytic release of Ca²⁺. If the release pattern is homogeneous and the exogenous buffers are uniformly distributed in the cell, every heterogeneity in [Ca²⁺] can only result from a nonhomogeneous pattern of fixed endogenous buffer distribution.

As a reporter dye, we decided to use Bis-Fura-2 for several reasons. It has the same Ca²⁺ binding group as Fura-2 but two fluorophores (of Fura-2 type) are attached via linkers to the chelating group. This should give rise to a higher fluorescence yield and, thus, enable us to use smaller dye concentrations to achieve a desired SNR. The top panel in Fig. 4 shows the absorption spectra of 50 µM Bis-Fura-2 at different [Ca²⁺] levels. Comparison with Fura-2 data (not shown here) reveals that the absorption is indeed twice that of Fura-2. A plot of the fraction of Ca²⁺-bound dye over [Ca²⁺] in the bottom panel shows that the dissociation constant of Bis-Fura-2 for Ca²⁺ is 500 nM, i.e., two-fold higher than Fura-2. This implies that, at a given concentration, its binding ratio is smaller than that of the high-affinity Fura-2 and, hence, its distortion of Ca²⁺ signals less pronounced. Another relevant issue for our measurements is the binding kinetics of the dye. It determines two factors: 1) conventional [Ca²⁺] measurements with fluorescent indicators use the equilibrium form of the law of mass action to deduce [Ca²⁺] from fluorescence data, using some modification of the Grynkiewicz et al. (1985) formalism (see the Appendix in our case). Consequently, we must allow the dye to get in chemical equilibrium with free Ca²⁺. This imposes a lower limit on the minimal integration time per frame. The kinetics of Bis-Fura-2 was investigated in a recent report (Naraghi, 1997), which resulted in an on-rate of 5.5 × 10⁸ M s¹ and an off-rate of 260 s¹. Thus, the equilibration time constant is <4 ms. This matches our integration time of 25 ms/frame.

View larger version (21K): [in this window] [in a new window]   FIGURE 4   Absorption spectra of 50 µM Bis-Fura-2 at different [Ca2+] levels and the titration curve for Ca2+ binding to Bis-Fura-2. The top panel shows the molar extinction coefficients of Bis-Fura-2 at different Ca2+ concentrations, ranging from <1 nM (top curve at 380 nm) to >5 mM (bottom curve at 380 nm), demonstrating that it undergoes a shift of its absorption upon Ca2+ binding just like Fura-2. From these data the ratio of the Ca2+-bound Bis-Fura-2 over total Bis-Fura-2 was calculated and plotted as a function of [Ca2+] in the bottom panel. Fitting these data with a binding curve (superimposed line) reveals a KD value of 500 nM. (Note: The [Ca2+] for the curves in the top panel can be seen as abscissa values in the bottom panel.)

2) The flash lamp generates short pulses of UV light with a total duration of 2-3 ms. Within this time, Ca²⁺ is released into the cell and is subject to binding and unbinding to free DMN, Bis-Fura-2, and endogenous buffers. Dependent on the Ca²⁺ binding kinetics of these buffers, there might be a transient Ca²⁺ spike, which is more pronounced if the Ca²⁺ release rate is higher than the binding rates of the different buffers (Heinemann et al., 1994). The Grynkiewicz formalism is not applicable during this transient spike if the dye is not in equilibrium with Ca²⁺. These considerations imply that we need to estimate the amplitude and duration of such a probable spike. With this information, one can decide on the appropriate timing for the [Ca²⁺] measurement after the UV flash. Xu et al. (1997) have measured the in vivo Ca²⁺ binding kinetics of DMN and Fura-2 in chromaffin cells. They found the kinetics of Fura-2 to be little affected by the cytosolic medium, which gives us the justification to assume the same for Bis-Fura-2. Furthermore, they also showed that the chromaffin cells contain 4 mM of an immobile endogenous buffer with a K_D of 100 µM and an on-rate of 1.0 × 10⁸ M s¹. By using this well-defined set of parameters and the rates of release of Ca²⁺ from DMN (Ellis-Davies et al., 1996), we performed a 4th order Runge-Kutta simulation of the temporal evolution of the concentrations in response to a flash of light. The time course of the flash as a perturbation (which shifts the system from one equilibrium state to another) was sampled by a fast photodiode and used in the simulation. Fig. 5 shows the outcome of such a simulation for a typical experimental condition. Clearly, there is a transient Ca²⁺ spike of a few hundred nM amplitude within the first 3-4 ms after the flash. Nevertheless, it is not seen by the indicator, which is acting as a low-pass filter and achieves equilibrium within 5 ms. The fast endogenous buffer with its low affinity, however, does follow the [Ca²⁺] time course faithfully. The conclusion from these simulations is if we start the post-flash [Ca²⁺] measurement 3 ms after the onset of the flash, we can be sure that there is no significant contamination of the recorded fluorescence by nonequilibrium conditions. This is exactly what we did.

View larger version (17K): [in this window] [in a new window]   FIGURE 5   Time course of the concentrations of Ca2+ and Ca2+-bound buffers after a flash. The measured time course of the flash is used in this simulation to perturb the kinetic system from one equilibrium state to another one. Here, we assume to have 1 mM DMN, 0.2 mM Bis-Fura-2, and 4 mM of an endogenous buffer with a KD of 100 µM according to Xu et al. (1997). The kinetic parameters for the exogenous buffers are taken from Naraghi (1997) or Ellis-Davies et al. (1996). Clearly, there is a transient overshoot of [Ca2+], which lasts ~2 ms and is seen by the endogenous buffer by virtue of its fast kinetics. Nevertheless, this is invisible to the dye (acting as a low-pass filter of the [Ca2+] time course), which attains equilibrium after 3 ms. Thus, we can start the [Ca2+] measurement 3 ms after the onset of the flash without any transient contaminations.

The next important issue was to check whether the photolysis is spatially homogeneous. First, using a mirror in the object plane, we imaged the intensity distribution of the widefield excitation pattern as well as that of the flash light in the focal plane. Both appeared to overlap quite well and were homogeneous, but this could not exclude the possibility that the Ca²⁺ source, i.e., CaDMN, was still compartmentalized. If this was the case, one would see spatial gradients of [Ca²⁺] after the flash, which were the result of compartmentalized CaDMN rather than different endogenous buffer concentrations. To exclude this possibility, we designed a control experiment to prove that the Ca²⁺ source strength was uniform throughout the cell. This was a simple "buffer overload" experiment: with 2 mM of Bis-Fura-2 and 1-2 mM fully loaded DMN in the pipette, we outcompeted the endogenous buffers in binding Ca²⁺. Thus, a homogeneous [Ca²⁺] distribution after the flash could only be the result of a homogeneous source and sink, i.e., CaDMN and indicator distribution. Fig. 6 demonstrates the outcome of such a control experiment. We have plotted the fluorescence ratios as a function of pixel number before and after the flash. Clearly, the ratios are homogeneous. This means that we can now proceed with our experiments designed to assess the endogenous buffer distribution.

View larger version (40K): [in this window] [in a new window]   FIGURE 6   Homogeneity of the photolysis pattern. A cell was loaded with 2 mM Bis-Fura-2 (to overcome the endogenous buffers) and 1 mM DMN. Depicted are the fluorescence ratios (R) before and after a flash. We see that the photolysis efficiency is spatially uniform since the same is true for the ratio distribution.

Here, we used the same timing protocol for photolysis and imaging as above but with an internal solution, which contained 1 mM fully loaded DMN, 200 µM Bis-Fura-2, and different amounts of CaCl₂ such that [Ca²⁺] was adjusted to values between 500 and 1100 nM. Note that under these conditions the binding ratio of the dye is between 100 and 39, i.e., of the same order of magnitude as the endogenous buffer according to Xu et al. (1997), and between 20 and 4 for DMN. The fraction of DMN, which we wanted to cleave upon a flash, i.e., the photolysis efficiency, was adjusted to 2-7%. This choice was dictated by many considerations: a) calibration parameters of the dye are not changed for small flash intensities, b) the identity in Eq. 5 is only valid for small values of [Ca²⁺] since it is based on the linear approximation and c) only small increments in [Ca²⁺] guarantee that the binding ratios of the dye and the DMN are not changed significantly upon flash. This is not an issue for the endogenous buffer since it is expected to have a K_D around 100 µM. Fig. 7 (top) shows the result of such an experiment where we have plotted the [Ca²⁺] distribution before and after flashes. Clear postflash gradients of [Ca²⁺] are not discernible. The bottom panel shows the calculated distribution of the endogenous binding ratio according to Eq. 5. The values for _endo scatter between 30 and 55. Similar experiments at different [Ca²⁺] levels with 17 cells never revealed a pronounced heterogeneity in the distribution of _endo; and in accordance with the low affinity of the endogenous buffer, we did not see any signs of a decrease in the average cellular _endo value by increasing [Ca²⁺] up to 1.1 µM. These results must be contrasted with the results of the diffusion experiments, which is what we do in the next section.

View larger version (30K): [in this window] [in a new window]   FIGURE 7   Distribution of pre and postflash [Ca2+] as well as the calculated endogenous binding ratios. The top panel shows the [Ca2+] profile before and in response to a UV flash. From these two images, the distribution of endo was calculated according to Eq. 11 and plotted in the bottom panel.

	DISCUSSION

Top Abstract Introduction Materials & Methods Results Discussion Appendix References

In light of the increasing evidence for functional but not necessarily morphological Ca²⁺ compartments (Chad and Eckert, 1984; Imredy and Yue, 1992; Llinas et al., 1995), it has become clear that Ca²⁺ signaling must rely heavily on a balanced, local, and quantitative interplay between different sources and sinks of Ca²⁺. The generic objective of a cell, using Ca²⁺ as a highly controlled and ubiquitous second messenger, is to achieve a specific pattern of [Ca²⁺] distribution in response to a specific stimulus, which fulfills three conditions: a) the strength of the [Ca²⁺] signal must be within a well-defined amplitude window, which is matched to the affinity of the desired target of the Ca²⁺ signal; b) the duration of the [Ca²⁺] signal must be within a well-defined time window to account for the activation kinetics of the desired target, which in turn is governed by the binding and unbinding kinetics of Ca²⁺ to the sensor responsible for triggering the signaling cascade; and c) the transient [Ca²⁺] elevation must happen at the location where the corresponding Ca²⁺ sensor is located, and maybe only there.

These are conditions imposed on the source. Analogously, requirements can be formulated for the sinks, which can be quite distinct and heterogeneous in nature: they can be energy-consuming pumps and exchangers, intracellular organelles, or single chelating molecules. It is well known that the first two categories of sinks are often equipped with rather slow activation kinetics (in the range of hundreds of milliseconds and longer; see Neher and Augustine, 1992, or Markram and Sakmann, 1994) while the temporal window on which the buffers operate is dictated by their Ca²⁺ binding rate. Recent studies on Ca²⁺ binding kinetics of the endogenous buffer in adrenal chromaffin cells (Xu et al., 1997) show that they can constitute Ca²⁺ sinks, which act in a submillisecond time domain. Within a millisecond or so, Ca²⁺ is expected to diffuse <1 µm (Allbritton et al., 1992). Thus, the fast buffer effects are local effects. Consequently, in polarized neuroendocrine cells or in neurons, the notions of fast and local Ca²⁺ signals must be regarded as synonymous. And if this is true, one can expect to have a differential Ca²⁺ buffering power at different locations of the cell with different functional fingerprints. But how can we assess this possible differential buffer distribution?

An answer to this question can only be given operationally from a functional, i.e., physiologist's, point of view. Since we do not know how many different molecular species are involved in fast Ca²⁺ buffering and have no direct evidence for their chemical identity, we must look for a common property that defines their action regardless of their identity. In other words, this property must reflect the buffers' action in reducing and localizing the Ca²⁺ signal. Following Mathias et al. (1990) and Neher and Augustine (1992), this important property of a cellular buffer is the extent to which it is capable of binding Ca²⁺ ions at a given free Ca²⁺ concentration. That is, what is the buffers' effect on relating a change in total Ca²⁺ to a change in free Ca²⁺? This question readily suggests studying the so-called "Ca²⁺ binding ratio (_B)" of a buffer B, defined as _B = [CaB]/[Ca²⁺]. Several studies have investigated the cellular binding ratio by lumping together the combined action of all cellular buffers in terms of a hypothetical Ca²⁺ binding species (for review see Neher, 1995).

Zhou and Neher (1993) were able to dissect two different types of endogenous Ca²⁺ buffers in bovine adrenal chromaffin cells: an immobile species (because it was detectable after prolonged periods of whole-cell recording without significant washout) with a binding ratio of 40, and a slowly mobile species (because, after transition to whole-cell mode, it washed out slowly) with a binding ratio of 10. Furthermore, from the time course of the decline of this binding ratio after breaking into the cell, they could estimate the corresponding buffer's molecular weight to be ~10,000. Unexpectedly, they did not find any signs of highly mobile endogenous Ca²⁺ buffers, such as nucleotides, but these numbers are all cellular averages and do not reflect any spatial differentiation of the cell with respect to the distribution of its molecular Ca²⁺ sinks. Likewise, to our knowledge there is no report to date in any system which has systematically and quantitatively investigated the endogenous Ca²⁺ buffer distribution. Thus, we made an effort to shine some light on the quantitative distribution of the endogenous Ca²⁺ binding ratio as a determinant of local Ca²⁺ signals.

In this paper we present two complementary methods to study the endogenous Ca²⁺ buffer distribution: one aims at observing Ca²⁺ diffusion and the other avoids it. In both cases, we make the assumption that the reaction kinetics of the involved buffers is fast compared with the mean diffusional times, the so-called "rapid buffer approximation (rba)" (Wagner and Keizer, 1994). This is justified by virtue of recent studies (Naraghi, 1997; Xu et al., 1997) where the on- and off-rates for Ca²⁺ binding of some exogenous buffers as well as the endogenous buffer in adrenal chromaffin cells were investigated. There it was shown that the on-rate of the endogenous buffer is ~10⁸ M s¹ and the off-rate 10⁴ s¹. This implies that the endogenous buffer reaches chemical equilibrium within <1 ms. Similarly, the exogenous buffers, which we used here, achieve equilibrium within <5 ms. Together, these justify the use of rba in our system.

In the first approach, we explore the extent to which the apparent Ca²⁺ diffusion is influenced, i.e., slowed by the presence of an immobile endogenous buffer using Eq. 3. Neher and Augustine (1992) have shown that there are no signs of Ca²⁺ release or uptake within 100 ms after a short depolarizing pulse. Thus, we explicitly aim at observing the dissipation of Ca²⁺ gradients from which we extract local Ca²⁺ diffusion coefficients. This approach can be characterized by the following criteria:

1.	The source of the Ca2+ for inducing the gradients, and particularly its strength, is not a primarily relevant issue. The temporal spread of the Ca2+ signal and well-defined gradients of [Ca2+] throughout the cell are important. Our sources here are voltage-gated Ca2+ channels.
2.	We should be aware of the influence of the Ca2+ indicator in shaping the gradients. It expresses itself in two quantities: the indicator's binding ratio ind and its diffusion coefficient Dind.
3.	The deduction of diffusion coefficients, and thus binding ratios, from observations of Ca2+ diffusion is inverse to the classical simulation problem. There, knowing the parameters of the diffusion equation, one asks for the temporal evolution of its solution. Here, observing the solution to the diffusion equation by imaging techniques, one asks for the parameters of the system. This corresponds to a nonlinear high-pass filtering of the Ca2+ images. Consequently, this process amplifies noise, which in turn imposes high requirements on the SNR of the imaging system.
4.	For the applicability of the inverse method, it is essential to have moderate Ca2+ gradients such that the apparent Ca2+ diffusion coefficient in Eq. 3 can be regarded as independent of [Ca2+].
5.	Ca2+ diffusion in the cell is an inherently 3-D problem, whereas given the temporal resolution of presently available imaging systems, one is only able to observe the Ca2+ signals in one focal slice. Then, one needs to simplify assumptions about the underlying cellular geometry. Here, we assume radial Ca2+ diffusion.
6.	There are two possible sources for a locally reduced apparent Ca2+ diffusion coefficient: either a locally increased immobile Ca2+ binding ratio or deviations from the concept of diffusion in an isotropic medium, for instance because of local diffusion barriers. There is no direct way of distinguishing between these two possibilities.

Our diffusion measurements indicated local gradients of the apparent Ca²⁺ mobility at the boundary of the cell and at the nuclear envelope. One could attribute this to increased binding ratios close to biological membranes because of Ca²⁺ binding to membrane phospholipids or Ca²⁺ binding to specific buffers as an indication of functional organization of the cell, but this is only one possibility. These effects also may have been due to non-radial Ca²⁺ diffusion in 3-D and/or the nuclear envelope as a diffusion hindrance in conjunction with optical blurring effects. To separate these possibilities, we decided to apply flash photolysis of Ca²⁺-loaded DMN as a light-dependent Ca²⁺ source (Kaplan and Ellis-Davies, 1988) together with Ca²⁺ imaging using Eq. 5. The flash photolysis technique was also used in a study by Al-Baldawi and Abercrombie (1995) together with Ca²⁺-sensitive electrodes to measure the average Ca²⁺ binding ratio of the giant axon of the marine invertebrate Myxicola. The main characteristics of this approach can be summarized as 1) The strength of the source must be defined in order to extract the endogenous binding ratios from the comparison of preflash [Ca²⁺] with postflash [Ca²⁺]. This requires careful calibration of the flash system in terms of its photolysis efficiency. 2) It is important to acquire the images as fast as possible to prevent inhomogeneities of the postflash [Ca²⁺] distribution to disappear by means of Ca²⁺ diffusion during the acquisition time of the image. 3) The indicator's diffusion coefficient, D_ind, effectively determines the spatial resolution by determining the mean diffusional path length of Ca²⁺ during an image acquisition. Its binding ratio, _ind, determines the fraction of the released Ca²⁺, which is bound to the indicator. It should be of the same order of magnitude as _endo. 4) To attribute any heterogeneity of postflash [Ca²⁺] to the endogenous buffer distribution, it is critical to guarantee a spatially homogeneous photolysis efficiency.

With these constraints in mind, the performed photolysis experiments revealed no significant signs of heterogeneous buffer distribution. This attributes the slowing of Ca²⁺ diffusion near the nuclear membrane to its action as a diffusion barrier rather than to a high Ca²⁺ binding power. Furthermore, the near membrane D_app distribution must be interpreted as the result of deviations from radially symmetric diffusion in conjunction with blurring effects. Indeed, by looking at the Ca²⁺ independent fluorescence at different focal planes, the cell appears to be flat rather than spherical. We can conclude the photolysis experiments allow us to distinguish between different interpretations of the diffusion experiments and establish a more direct means of assessing the endogenous buffer distribution, but what is the spatial resolution of our measurement? Our pixel size (in the lateral direction) in the object plane is 580 nm. The effective resolution, nevertheless, is determined by the temporal resolution of the imaging system as well as the optical blurring effects in the axial direction. As mentioned earlier, we need 25 ms for the acquisition of an image.

Fig. 8 depicts the expected apparent Ca²⁺ diffusion coefficient and the mean displacement of Ca²⁺ ions (during this 25 ms) as a function of the exogenous binding ratio, assuming an endogenous ratio of 40 and a diffusion coefficient of 120 µm²/s for the dye. With an exogenous binding ratio of 50 to 100, we expect the mean Ca²⁺ displacement to be ~2 µm. Thus we have to restate our result in light of this temporal blurring: on a spatial grid with a lateral resolution of ~2 µm, there are no signs of heterogeneous buffer distribution in the bovine adrenal chromaffin cell. But there is also a spatial blurring because of the wide field imaging, which is applied in this paper. By using fluorescent beads as point sources of light, we experimentally measured the point spread function (PSF) of our optical system. From the measured PSF, we expect an axial resolution of 2-3 µm, which is of the same order of magnitude as the effective lateral resolution. In effect, we are dealing with cubic voxels with ~2 µm side length. To improve the spatial resolution, one would optimally need the axial resolution of a two-photon excitation system (reducing axial blurring) in conjunction with a significantly faster imaging system (reducing temporal blurring). For instance, to have a lateral resolution of ~500 nm, we would need image acquisition times of ~2 ms, which is far beyond generally available technology and at the cutting edge of the mean reaction times of the high-affinity indicators (see Fig. 5). In addition, since we cannot afford an exogenous binding ratio much more than 100, there is an upper limit to the tolerable dye concentration. To get a meaningful fluorescence signal within 2-ms integration, the fluorescence yield of the Ca²⁺ indicators must be very high. Simultaneously, we must maintain our ability to measure [Ca²⁺], but there is no two-photon system up to date that allows measurement of [Ca²⁺] with such high rates. To conclude, imaging systems with 1) significantly higher acquisition rates, 2) the axial resolution properties of a two-photon system, and 3) improved detector quantum yields in conjunction with much brighter Ca²⁺-sensitive dyes, are necessary to improve the effective spatial resolution of the Ca²⁺ buffer measurements. Our resolution of 2 µm, of course, does not exclude buffer gradients on a finer spatial grid.

View larger version (45K): [in this window] [in a new window]   FIGURE 8   Dependence of the apparent Ca2+ diffusion coefficient on the exogenous binding ratio. We have plotted the apparent Ca2+ diffusion coefficient according to Eq. 9 as a function of the exogenous binding ratio, assuming an immobile buffer with a binding ratio of 40 and a dye of Bis-Fura-2 type with Dind = 120 µm2/s. Even at the concentration range where the exogenous buffer has similar binding ratios like the endogenous one (50-100), Dapp is ~70-90 µm2/s. Within the 25-ms frame integration time, this gives rise to a mean Ca2+ displacement of ~1.8-2.2 µm. Consequently, although the pixel size in the object plane is ~580 nm, the effective spatial resolution is ~2 µm and is dictated by the acquisition time for a frame.