## Abstract

We wrote a program that runs as a Microsoft Excel spreadsheet to calculate the diffusion of Ca^{2+} in a spherical cell in the presence of a fixed Ca^{2+} buffer and two diffusible Ca^{2+} buffers, one of which is considered to be a fluorescent Ca^{2+} indicator. We modeled Ca^{2+} diffusion during and after Ca^{2+} influx across the plasma membrane with parameters chosen to approximate amphibian sympathetic neurons, mammalian adrenal chromaffin cells, and rat dorsal root ganglion neurons. In each of these cell types, the model predicts that spatially averaged intracellular Ca^{2+} activity ([Ca^{2+}]_{avg}) rises to a high peak and starts to decline promptly on the termination of Ca^{2+} influx. We compared [Ca^{2+}]_{avg} with predictions of ratiometric Ca^{2+} measurements analyzed in two ways. *Method 1* sums the fluorescence at each of the two excitation or emission wavelengths over the *N* compartments of the model, calculates the ratio of the summed signals, and converts this ratio to Ca^{2+} ([Ca^{2+}]_{avg,M1}). *Method 2* sums the measured number of moles of Ca^{2+} in each of the *N* compartments and divides by the volume of the cell ([Ca^{2+}]_{avg,M2}). [Ca^{2+}]_{avg,M1} peaks well after the termination of Ca^{2+} influx at a value substantially less than [Ca^{2+}]_{avg} because the summed signals do not reflect the averaged free Ca^{2+} if the signals come from compartments containing gradients in free Ca^{2+} spanning nonlinear regions of the relationship between free Ca^{2+} and the fluorescence signals. In contrast, [Ca^{2+}]_{avg,M2} follows [Ca^{2+}]_{avg} closely.

- intracellular calcium
- kinetic model
- diffusion coefficient
- fura 2ff
- furaptra

the importance of intracellular Ca^{2+} activity ([Ca^{2+}]_{i}) as a regulator of cellular activities has led to the development of methods to study the control and distribution of Ca^{2+} in the cytoplasm. Two approaches that have revealed aspects of [Ca^{2+}]_{i} signaling are the measurement of [Ca^{2+}]_{i} with fluorescent Ca^{2+} indicators and mathematical modeling of Ca^{2+} diffusion through the cytoplasm. Each of these approaches has strengths and weaknesses. Fluorescent Ca^{2+} indicators provide a relatively simple method for monitoring localized changes in [Ca^{2+}]_{i}, but they can have a limited signal-to-noise ratio and there are frequently difficulties in loading them into the cytoplasm and obtaining in vivo calibrations. Furthermore, Ca^{2+} indicators are diffusible Ca^{2+} buffers that can modify the amplitude and kinetics of the changes in [Ca^{2+}]_{i} they are intended to measure (cf. Refs. 17, 32). Mathematical modeling of Ca^{2+} diffusion enables one to examine quantitatively the effects on the temporal and spatial distribution of [Ca^{2+}]_{i} of buffer concentration, affinity, on rate, and mobility for multiple buffering species (reviewed in Ref. 21). However, application of these models is limited by the availability of information on the properties of endogenous Ca^{2+} buffering mechanisms and the difficulty of modeling complex geometries. One particularly productive application of models of Ca^{2+} diffusion has been to investigate the impact of Ca^{2+} indicators on [Ca^{2+}]_{i} in invertebrate neurons (3), amphibian neurons (22), and adrenal chromaffin cells (13, 19).

In preparation for a series of experiments examining the control of [Ca^{2+}]_{i} in rat dorsal root ganglion neurons, we developed a program in the Visual Basic language feature of the Excel spreadsheet program (Microsoft, Bellevue, WA) to solve the buffer and diffusion equations described by others (3, 13, 19, 22). Although the resulting program is slow, it is easy to use and is portable to any computer that has Excel. A copy of the program and a manual to guide in its application can be downloaded at **http://www.physio.unr.edu/faculty/kenyon/diffusion.htm**. We used this program to study the temporal and spatial distribution of [Ca^{2+}]_{i} in the presence of ratiometric fluorescent Ca^{2+} indicators with different affinities for Ca^{2+}. In addition, we compared averaged [Ca^{2+}]_{i} ([Ca^{2+}]_{avg}) with predictions of fura 2 measurements analyzed in two ways. *Method 1* sums the fluorescence at each of the two excitation or emission wavelengths over the *N* compartments of the model, calculates the ratio of the summed signals, and converts this ratio to Ca^{2+} ([Ca^{2+}]_{avg,M1}). This approach corresponds to measurements made with photomultiplier tubes or the integration of the intensity of pixels forming a two-dimensional (2D) image. *Method 2* sums the measured number of moles of Ca^{2+} in each of the *N* compartments and divides by the volume of the cell ([Ca^{2+}]_{avg,M2}). In principle, this could be done experimentally by analyzing the fluorescence intensity of the voxels making up a three-dimensional (3D) image of the cell. We find that [Ca^{2+}]_{avg,M1} is not a reliable measure of [Ca^{2+}]_{avg} in the presence of a gradient of Ca^{2+} because the summed fluorescence ratio does not reflect the averaged free Ca^{2+} when the gradients span nonlinear regions of the relationship between free Ca^{2+} and the fluorescence ratio. In contrast, [Ca^{2+}]_{avg,M2} follows [Ca^{2+}]_{avg} closely.

## METHODS

We facilitated our programming by adopting unified units for all the parameters. Thus time is measured in milliseconds, distance in micrometers, area in square micrometers, volume in cubic micrometers, amount in femtomoles (i.e., 10^{–15} mol) and concentration in femtomoles per cubic micrometer (conveniently, 1 fmol/μm^{3} = 1 mol/l). Parameters were entered into the program in these units. The equations were described in detail previously (3, 13, 19, 22). Briefly, the model cells are spheres made up of *N* compartments including a central sphere and *N* – 1 concentric compartments identified by indexes *n* = 0 to *N* – 1, where *n* = 0 identifies the central sphere and *n* = *N* – 1 identifies the outermost compartment. Associated with each compartment are concentrations of free Ca^{2+} ([Ca^{2+}]* _{n}*), unbound diffusible (mobile) buffer ([MB]

*), Ca*

_{n}^{2+}-bound MB ([Ca-MB]

*), unbound fixed buffer ([FB]*

_{n}*), and bound FB ([Ca-FB]*

_{n}*). In addition, most simulations included a second diffusible buffer component with the properties of a fluorescent Ca*

_{n}^{2+}indicator, fura ([fur]

*, [Ca-fur]*

_{n}*).*

_{n}Each of the components except the fixed buffer can diffuse between compartments as described by one of four forms of a conventional diffusion equation. For 1 ≤ *n* ≥ *N* – 2 the change over time of the concentration of solute *S* by diffusion is given by (1) where *t* is time, [*S*]* _{n}* is the concentration of diffusible species (i.e., [Ca

^{2+}]

*, [MB]*

_{n}*, [Ca-MB]*

_{n}*, [fur]*

_{n}*, [Ca-fur]*

_{n}*) in compartment*

_{n}*n*(fmol/μm

^{3}),

*D*is the diffusion coefficient of that species (μm

_{S}^{2}/ms), vol

*is the volume of the*

_{n}*n*th compartment (μm

^{3}), δ is the thickness of the compartments (μm), and

*A*

_{n}_{–1}and

*A*are the surface areas of the inner and outer spheres defining compartment

_{n}*n*(μm

^{2}).

For the outermost compartment, *n* = *N* – 1, the equation is modified such that there is only diffusional exchange with compartment *n* = *N* – 2. For the buffers the equation is (2) For Ca^{2+}, terms are added describing a Ca^{2+} influx from the extracellular space [i.e., Ca^{2+} current (*I*_{Ca})] and Ca^{2+} efflux via a pump mechanism (3) where *I*_{Ca} is expressed in coulombs per milliseconds, *F* is the Faraday constant (9.65 × 10^{–11} C/fmol), *V*_{max} is the maximum velocity of Ca^{2+} pumping (fmol·μm^{–2}·ms^{–2}), and *K*_{m} is the equilibrium constant for the Ca^{2+} pump (fmol/μm^{3}). *I*_{Ca} is set to one of two values, an influx equal to the efflux generated by the pump at resting [Ca^{2+}] or a much larger influx (e.g., 5 nA) through voltage-gated Ca^{2+} channels.

For *n* = 0 the equation is modified such that there is only diffusional exchange with compartment *n* = 1 (4)

In addition to the fluxes into and out of a compartment, the concentrations of Ca^{2+} and the buffers will change in response to the binding and unbinding of Ca^{2+} to each of the buffers as described by (5) where [CaB] and [B] are bound and unbound forms of the buffer present (MB, FB, and fura in fmol/μm^{3}), *k*_{off} is the off rate constant (ms^{–1}), and *k*_{on} is the on rate constant (μm^{3}/fmol·ms = M^{–1}·ms^{–1}). The equilibrium constant for Ca^{2+} binding is then *K*_{D} = *k*_{off}/*k*_{on} (fmol/μm^{3}). *Equations 1–5* were solved with a first-order Euler algorithm.

The program also calculated spatially averaged [Ca^{2+}]_{i} ([Ca^{2+}]_{avg}) defined as (6)

The fluorescence signal from the indicator was modeled from the equations of Grynkiewicz et al. (6) with an approach similar to that described by Marengo and Monck (13). The fluorescence signals from the *n*th compartment in response to excitation at 340 and 380 nm (F_{340}* _{n}* and F

_{380}

*) are given by (7) and (8) where*

_{n}*S*

_{f340}= 11,

*S*

_{f380}= 14.2,

*S*

_{b340}= 35, and

*S*

_{b380}= 1 are proportionality constants relating the concentration of free ([fur]) and bound ([Ca-fur]) indicators to the amplitude of the fluorescence signal resulting from excitation at 340 and 380 nm, respectively (6). The values were chosen to mimic fluorescence changes in response to excitation at 340 and 380 nm that are proportional to those seen in in vitro measurements of fura 2 [cf. Fig. 3 in Kao (10)] and to obtain minimum fluorescence ratio (R

_{min}) =

*S*

_{f340}/

*S*

_{f380}= 0.77 and maximum fluorescence ratio (R

_{max}) =

*S*

_{b340}/

*S*

_{b380}= 35 as found in the original report by Grynkiewicz et al. (6). The fluorescence ratio signal from the

*n*th compartment is then (9) and the measured [Ca

^{2+}] in that compartment ([Ca

^{2+}]

_{n}_{,M}) is given by the relationship developed by Grynkiewicz et al. (6) (10) Note that the fluorescence signals in the model do not include noise or background fluorescence. Thus the determination of [Ca

^{2+}]

_{n}_{,M}does not face practical limitations and at equilibrium or in a steady state [Ca

^{2+}]

_{n,M}will equal [Ca

^{2+}]

*. At other times, the finite rates of Ca*

_{n}^{2+}binding and unbinding can cause [Ca

^{2+}]

_{n}_{,M}to differ from [Ca

^{2+}]

*.*

_{n}We examined two methods for the determination of [Ca^{2+}]_{avg} from the fluorescence data. In *method 1* we took the sum of the individual fluorescence signals over the entire cell and calculated the fluorescence ratio of the summed signals (R_{sum}) as (11) and then obtained a measured value ([Ca^{2+}]_{avg,M1}) as (12) This approach corresponds to measurements made with photomultiplier tubes (cf. Refs. 23, 25, 29, 30, 32) or the integration of the intensity of pixels forming a 2D image.

In *method 2*, we used the modeled fluorescence signals to calculate the number of moles of free Ca^{2+} in each of the compartments, summed the measured Ca^{2+}, and divided by the volume of the cell (13) where [Ca^{2+}]_{n}_{,M} was determined with *Eqs. 7–10*. An experimental counterpart to this method would be the measurement of the Ca^{2+} content of spatially localized voxels of known volume. Although technical obstacles prevent implementation of this approach in the laboratory (see discussion), this calculation provides a convenient test of the ability of the Ca^{2+} indicator to follow changes in Ca^{2+} in limited volumes of the model.

We used the parameters describing the properties of endogenous Ca^{2+} buffers, diffusion, and Ca^{2+} influx identified by Sala and Hernández-Cruz (22) to model an amphibian sympathetic neuron and parameters identified by Nowycky and Pinter (19) to model a mammalian adrenal chromaffin cell, and we identified parameters to model a rat dorsal root ganglion neuron (see below). The standard parameters used in our calculations are listed in Table 1. The properties of the second diffusible Ca^{2+} buffer were selected to represent three indicators that differ primarily in their rates for Ca^{2+} unbinding: fura 2, fura 2ff, and furaptra (parameters listed in Table 2).

## RESULTS

We verified our program by using the parameters in Table 1 to reproduce previously published simulations (19, 22). A typical result is shown in Fig. 1*A*, where we plot the calculated time courses of [Ca^{2+}]* _{n}* in response to a 5-nA, 50-ms Ca

^{2+}influx in our model amphibian sympathetic neuron for

*n*= 39 (the outermost compartment), 29 (5 μm in from the plasma membrane), 19 (10 μm in from the plasma membrane), and 0 (the central sphere). We found no differences between our results and results presented in the original publications [compare Fig. 1

*A*with Fig. 1

*B*of Sala and Hernández-Cruz (22)], and we conclude that our model is valid for these calculations. We then examined the effects of a second diffusible buffer (75 μM) with the properties of fura 2 added to the model amphibian neuron with the other parameters as listed in Table 1. Although this concentration was chosen as being at the low end of those typically used to study [Ca

^{2+}]

_{i}in neuronal cell bodies, it adds a Ca

^{2+}binding ratio of ∼296 at 50 nM free Ca

^{2+}and has a marked effect on [Ca

^{2+}]

*as illustrated in Fig. 1*

_{n}*B*. We found that the additional Ca

^{2+}buffering provided by the indicator reduces the peak amplitude of [Ca

^{2+}]

*in each compartment, in agreement with previous results (22). However, in our model, fura 2 slowed the distribution of Ca*

_{n}^{2+}through the cytoplasm, in contrast to the increased rate of distribution expected from the addition of a diffusible buffer (19, 22, 24). Our result is due to the relatively low diffusion coefficient for fura 2 (7, 26), making it a relatively immobile buffer in our simulations (see discussion).

Figure 1*C* plots [Ca^{2+}]_{avg}, [Ca^{2+}]_{avg,M1}, and [Ca^{2+}]_{avg,M2} from the simulation shown in Fig. 1*B*. [Ca^{2+}]_{avg} rises to >1 μM during the influx of Ca^{2+} and starts to decline when the Ca^{2+} influx ends. Interestingly, the amplitudes and time courses of [Ca^{2+}]_{avg,M1} and [Ca^{2+}]_{avg,M2} are very different. [Ca^{2+}]_{avg,M2} follows [Ca^{2+}]_{avg} closely such that the two lines are superimposed after the peak response. The slight delay in the [Ca^{2+}]_{avg,M2} trace is largely due to the finite binding kinetics of the fura 2 (results not shown). In contrast, [Ca^{2+}]_{avg,M1} rises relatively slowly, misses the peak of [Ca^{2+}]_{avg}, and continues to increase after the end of the Ca^{2+} influx. Thus the time course and amplitude of [Ca^{2+}]_{avg,M1} resemble experimental observations using photomultiplier tubes to detect fluorescence from high-affinity Ca^{2+} indicators in mammalian neuronal cell bodies (23, 25) and in mammalian adrenal chromaffin cells (18). Specifically, each of these reports noted that [Ca^{2+}]_{avg,M1} increased after the end of the Ca^{2+} influx and attributed this to the kinetics of the distribution of Ca^{2+} through the cytoplasm. As described below, our model clarifies this issue.

Why does [Ca^{2+}]_{avg,M1} differ from [Ca^{2+}]_{avg}? Although kinetic and saturation limitations on the ability of fura 2 to report free Ca^{2+} are well known (see discussion and Refs. 2, 8), the observation that [Ca^{2+}]_{avg,M2} follows [Ca^{2+}]_{avg} quite well implies that the large discrepancy between [Ca^{2+}]_{avg,M1} and [Ca^{2+}]_{avg} is not due to the properties of fura 2. Rather, the fault must lie in the analysis that produces [Ca^{2+}]_{avg,M1}. In particular, *Eq. 11* is not valid if the fluorescence signals come from regions that include a Ca^{2+} gradient that spans a nonlinear portion of the curve relating free Ca^{2+} to the fluorescence ratio. This can be illustrated by a simple example (see also Ref. 11). Consider two compartments (*n* = 1 and 2) with volumes such that vol_{1} makes up 25% of the total and vol_{2} the remainder. If [Ca^{2+}]_{1} = 3.8 μM and [Ca^{2+}]_{2} = 0.19 μM, then [Ca^{2+}]_{avg} = (3.8 + 3 × 0.19)/4 = 1.09 μM. The fluorescence ratio from the two compartments (from *Eqs. 7, 8*, and *11*) is R_{sum} = 5.22, which gives [Ca^{2+}]_{avg,M1} = 0.29 μM, ∼27% of [Ca^{2+}]_{avg}.

This mechanism suggests that low-affinity indicators might perform better in the measurement of [Ca^{2+}]_{avg,M1} because a given change in Ca^{2+} will span a shorter (more linear) range of the relationship between Ca^{2+} and fluorescence ratio. We tested this by modeling the responses of the lower-affinity Ca^{2+} indicators fura 2ff and furaptra (Table 2). As shown in Fig. 2, [Ca^{2+}]_{avg,M1} obtained from either of these indicators is closer to [Ca^{2+}]_{avg} than that obtained from fura 2.

The magnitude of the discrepancy between [Ca^{2+}]_{avg} and [Ca^{2+}]_{avg,M1} depends on the nonlinear relationship between free Ca^{2+} and the fluorescence signal, the magnitude of the Ca^{2+} gradient between the compartments, and the relative volumes of the compartments. The values chosen for the simple example above were selected to approximate the situation near the peak in [Ca^{2+}]_{avg} in Fig. 1*C*, where the Ca^{2+} in the outer four compartments (∼25% of the total volume) peaks near 3.8 μM whereas the remainder of the cell is near 0.19 μM. The effects of varying these values systematically are shown in Fig. 3 for fura 2, fura 2ff, and furaptra. For these calculations [Ca^{2+}]_{2} was set to 135 nM and [Ca^{2+}]_{1} was varied between 1.35 nM and 13.5 μM. In addition, we varied the relative volume of *compartment 1*, setting vol_{1}/(vol_{1} + vol_{2}) to 0.25, 0.1, and 0.01 in Fig. 3, *A*–*C*, respectively. These calculations show that, for fura 2, the decline in the ratio [Ca^{2+}]_{avg,M1}/[Ca^{2+}]_{avg} is more severe when [Ca^{2+}]_{1} is greater than [Ca^{2+}]_{2} and that [Ca^{2+}]_{avg,M1}/[Ca^{2+}]_{avg} falls below 0.8 for physiological gradients between [Ca^{2+}]_{1} and [Ca^{2+}]_{2} even when vol_{1} is only 10% of the total cell volume. Figure 3 also shows that [Ca^{2+}]_{avg,M1}/[Ca^{2+}]_{avg} is near 1 for a range of Ca^{2+} gradients for low-affinity Ca^{2+} indicators. This is because free Ca^{2+} ranging from 135 nM to 1–2 μM spans only short parts of the relationship between free Ca^{2+} and fluorescence ratio for these indicators.

We also used our model to investigate the distribution and measurement of Ca^{2+} in cells of different sizes and containing different endogenous Ca^{2+} buffers. We first investigated a model of a mammalian adrenal chromaffin cell (19). Although the model buffers for this cell give a Ca^{2+} binding ratio similar to that in the model amphibian neuron (Table 1), the relative contributions of fixed and diffusible buffering are reversed such that most of the Ca^{2+} buffering is provided by the diffusible buffer in the adrenal chromaffin cell model. In addition, the adrenal chromaffin cell has a smaller radius and *I*_{Ca} than the amphibian neuron. Combined, these properties speed the distribution of Ca^{2+} through the cytoplasm, thereby reducing the discrepancy between [Ca^{2+}]_{avg} and [Ca^{2+}]_{avg,M1} compared with the amphibian neuron. Nevertheless, as shown in Fig. 4*A*, [Ca^{2+}]_{avg,M1} misses the peak of [Ca^{2+}]_{avg} and continues to increase after the end of the Ca^{2+} influx. As in the model amphibian sympathetic neuron, [Ca^{2+}]_{avg,M2} is much closer to [Ca^{2+}]_{avg} than [Ca^{2+}]_{avg,M1} is, but it is clear in Fig. 4*A* that [Ca^{2+}]_{avg,M2} does not follow [Ca^{2+}]_{avg} in the adrenal chromaffin cell as well as it does in the amphibian sympathetic neuron or dorsal root ganglion neuron. The reasons for this are considered in discussion.

The models of the amphibian neuron and mammalian adrenal chromaffin cell are characterized by Ca^{2+} binding ratios that are higher than those reported in many cell types [reviewed by Neher (17)], including rat dorsal root ganglion neurons, in which Zeilhofer et al. (31) reported a value of 370. To examine the distribution of Ca^{2+} and the ability of spatially averaged fura 2 fluorescence to measure [Ca^{2+}]_{avg} in these cells, we chose parameters that approximated the geometry and endogenous Ca^{2+} buffers of small-diameter dorsal root ganglion neurons from adult rats (Table 1). Briefly, the Ca^{2+} binding ratio of the diffusible endogenous buffer was set to 57 based on results from chick dorsal root ganglion neurons (28) with the remaining Ca^{2+} binding attributed to the fixed buffer. As shown in Fig. 4*B*, the model predicts a large discrepancy between [Ca^{2+}]_{avg} and [Ca^{2+}]_{avg,M1} during and after a 3-nA Ca^{2+} influx. As in the amphibian sympathetic neuron, [Ca^{2+}]_{avg,M2} follows [Ca^{2+}]_{avg} quite closely.

## DISCUSSION

The model of Ca^{2+} diffusion used here was considered in detail in the original publications (3, 13, 19, 22), and we will not discuss the details of the model here. However, a brief consideration of the value we chose for the diffusion coefficient of fura 2 illustrates several points arising from our work. In previous studies, values spanning an order of magnitude from 0.2 × 10^{–6} to 2 × 10^{–6} cm^{2}/s have been used, with a cluster of relatively high values (1 × 10^{–6}-2 × 10^{–6} cm^{2}/s; Refs. 5, 13–16, 20). For our modeling we considered measurements of the diffusion of fura 2 and fluo 3 (7, 26) and modeling studies using low values (1, 3, 4, 27) and used a diffusion coefficient of 0.2 × 10^{–6} cm^{2}/s (0.02 μm^{2}/ms).

How important is the value of the diffusion coefficient for the Ca^{2+} indicator? Qualitatively, a larger value will enhance the ability of fura 2 to buffer the Ca^{2+} near the plasma membrane and increase the rate of distribution of Ca^{2+} through the cytoplasm. These effects will shorten the lifetime of the gradient in Ca^{2+} that leads to the discrepancy that we found between [Ca^{2+}]_{avg,M1} and [Ca^{2+}]_{avg}. Indeed, setting the diffusion coefficient to 2.5 × 10^{–6} cm^{2}/s (0.25 μm^{2}/ms), the value chosen by Sala and Hernández-Cruz (22), reduces the initial spike of [Ca^{2+}]_{avg} to ∼500 nM and shortens the lifetime of the gradient of Ca^{2+} between the periphery and the center of the cell to ∼0.5 s. In addition, the discrepancy between [Ca^{2+}]_{avg} and [Ca^{2+}]_{avg,M1} was much smaller than that seen when using the lower diffusion coefficient (data not shown).

Several issues are raised here. First, we find that intuition is frequently misleading with regard to Ca^{2+} diffusion in the presence of fixed and diffusible buffers and the behavior of this system is best understood by calculations. Second, the state of knowledge with regard to important parameters of the model is poor. Third, the identification of the relative importance of the various parameters generally requires calculation. For example, our experience with the model is that the diffusion coefficient for fura 2 has a large effect on its ability to buffer and to measure free Ca^{2+} in our simulations. Finally, given the number of issues to be investigated, no individual can be expected to make all of the important simulations. Others will be interested in different endogenous buffers, or different Ca^{2+} indicators, or different geometries, etc., and they will need to run their own simulations to answer their questions. To meet this need, we have developed a program to model Ca^{2+} diffusion in the presence of fixed and diffusible buffering that will run in a widely available spreadsheet.

One calculation we could not find in the literature is the predicted time course of spatially averaged Ca^{2+}, i.e., [Ca^{2+}]_{avg}. Accordingly, we set up our program to calculate this parameter and we also examined two approaches to obtain it from the modeled fluorescence signal of a ratiometric Ca^{2+} indicator. *Method 1* sums the fluorescence at each of the two excitation or emission wavelengths over the *N* compartments of the model, calculates the ratio of the summed signals, and converts this ratio to Ca^{2+} ([Ca^{2+}]_{avg,M1}). Experimentally this is done by recording the fluorescence from the whole cell with one or two photomultiplier tubes or by summing the intensity of pixels in fluorescence images. Our calculations show that, relative to [Ca^{2+}]_{avg}, [Ca^{2+}]_{avg,M1} increases more slowly to a lower peak value and continues to increase after the end of the Ca^{2+} influx. These characteristics resemble measurements of spatially averaged Ca^{2+} from a number of laboratories (cf. Refs. 18, 23, 25) and have been attributed to limited fura 2 binding kinetics and the distribution of Ca^{2+} through the volume of the cell. Our modeling demonstrates that the observed waveforms are expected as a consequence of the gradients of Ca^{2+} within the cell and a flawed analysis. Specifically, summing the fluorescence signals, calculating the ratio (*Eq. 11*), and converting this ratio to Ca^{2+} is not valid if the signals come from regions that include a Ca^{2+} gradient that spans a nonlinear portion of the curve relating free Ca^{2+} to the fluorescence ratio.

*Method 2* sums the measured number of moles of Ca^{2+} in each of the *N* compartments and divides by the volume of the cell. Experimentally, this could be accomplished by determining [Ca^{2+}]_{n}_{,M} (as described in *Eq. 10*) and the volume for each voxel in a 3D image of the cell. Although this information could be obtained in principle by a high-time-resolution *z*-scan with a confocal microscope, we are not aware of hardware suitable for the task. In addition, if a high-affinity indicator is used, our simulations show that this approach will face the problem of measuring Ca^{2+} in volumes in which nearly all the indicator is bound with Ca^{2+} (results not shown). In this case, the denominator of the fluorescence ratio will be very small and subject to errors caused by background fluorescence and noise and consequent uncertainty in the estimation of the free Ca^{2+}. Thus the analysis of voxels is problematic for a number of reasons. Similarly, quantitative measurement of [Ca^{2+}] from the analysis of pixels from a 2D image is difficult because interpretation of these images requires that the fluorescence be collected from all parts of the cell and the contribution of signal coming from outside the focal plane be corrected in the image. A complete understanding of these issues is beyond the scope of the present work, but we do not see how voxel or pixel information can be used to determine [Ca^{2+}]_{avg} in the presence of gradients in Ca^{2+}.

In summary, we find that for the measurement of [Ca^{2+}]_{avg}, *method 1* is invalid and that *method 2* is impractical. Thus we cannot suggest an approach that can use high-affinity indicators to reliably measure the rapid rise, high peak, and sharp decline of [Ca^{2+}]_{avg} predicted by our model of Ca^{2+} diffusion. Rather, our results suggest that the use of low-affinity indicators and *method 1* has potential for this measurement. Other workers have also pointed out advantages of low-affinity indicators, including their lower Ca^{2+} binding ratio and improved resolution of micromolar free Ca^{2+} due to a lower degree of saturation (8, 12). However, these issues are independent of our main finding that the spatially averaged fluorescence ratio from a low-affinity indicator more accurately reflects [Ca^{2+}]_{avg} in the presence of Ca^{2+} gradients because those gradients span shorter (more nearly linear) segments of the curve relating free Ca^{2+} to fluorescence ratio. Along with these advantages one must also consider disadvantages including difficulty in measuring resting Ca^{2+} and small signals in response to an increase in Ca^{2+} that may limit their usefulness.

We expected that our calculations would illustrate the inability of fura 2 to report the large and fast changes in Ca^{2+} near the plasma membrane because of a limited on rate for binding and saturation (cf. Refs. 2, 8). Instead, the agreement between [Ca^{2+}]_{avg,M2} and [Ca^{2+}]_{avg} and the ability of fura 2 to follow [Ca^{2+}]_{n}_{,M} in all of the compartments of the amphibian neuron (results not shown) suggest that fura 2 can follow physiological changes in Ca^{2+} quite well. However, this is not a general result and a sufficiently large flux density (fmol·s^{–1}·μm^{–3}) will deplete the unbound fura 2 in a compartment, thereby slowing the formation of Ca^{2+}-bound fura 2. This was documented by Nowycky and Pinter (19), who found that fura 2 underestimated the free Ca^{2+} in the outermost compartment of the model adrenal chromaffin cell. We confirmed this by using parameters from Table 1 for the cell and parameters from Table 2 for fura 2, finding that at the end of a 50-ms, 0.5-nA Ca^{2+} influx, [Ca^{2+}]_{74} peaked near 3.5 μM while [Ca^{2+}]_{74,M} peaked at ∼2.5 μM. This shortfall is why [Ca^{2+}]_{avg,M2} is clearly slower than [Ca^{2+}]_{avg} in the model chromaffin cell compared with the model neurons (compare Fig. 4*A* with Figs. 1*C* and 4*B*). This poor performance relative to that in the amphibian neuron is due to the 3.5 times higher flux density into the outermost compartment of the adrenal chromaffin model (0.074 fmol·s^{–1}·μm^{–3}) compared with that in the amphibian neuron (0.021 fmol·s^{–1}·μm^{–3}). The greater depletion of the unbound fura 2 also contributes to the discrepancy between [Ca^{2+}]_{avg} and [Ca^{2+}]_{avg,M1} in Fig. 4*A* but does not account for the major portion of the discrepancy that we attribute to the flawed nature of the [Ca^{2+}]_{avg,M1} measurement.

In summary, we developed an Excel-based model of Ca^{2+} diffusion that can be readily adapted to investigate the effects of fixed and diffusible Ca^{2+} buffering in spherical cells. We also investigated the amplitude and time course of [Ca^{2+}]_{avg} and the measurement of this parameter. Our results indicate that available measurements of [Ca^{2+}]_{avg} are not valid during times in which there are gradients of Ca^{2+} and that [Ca^{2+}]_{avg} rises faster to higher levels than the measurements indicate.

## Acknowledgments

We thank Dr. Martha Nowycky for pointing out the typographical error in the parameters listed by Sala and Hernández-Cruz (see Table 1) and Dr. Grant Nicol for finding a similar error of ours in an early draft and for comments on the manuscript. We thank Dr. Greg Dick for timely advice at several stages of this work.

GRANTS

This work was supported by National Institute of Neurological Disorders and Stroke Grant NS-41037.

## Footnotes

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- Copyright © 2004 the American Physiological Society