Theoretical analysis of the Ca²⁺ spark amplitude distribution

A difficulty of using confocal microscopy to study Ca²⁺ sparks is the uncertainty of the linescan position with respect to the source of Ca²⁺ release. Random placement of the linescan is expected to result in a broad distribution of measured Ca²⁺ spark amplitudes (a) even if all Ca²⁺ sparks were generated identically. Thus variations in Ca²⁺ spark amplitude due to positional differences between confocal linescans and Ca²⁺ release site are intertwined with variations due to intrinsic differences in Ca²⁺ release properties. To separate these two sources of variations on the Ca²⁺ spark amplitude, we determined the effect changes of channel current or channel open time collectively called the source strength, α - had on the measured Ca²⁺ spark amplitude histogram, N(a). This was done by 1) simulating Ca²⁺ release, Ca²⁺ and fluo-3 diffusion, and Ca²⁺ binding reactions; 2) simulation of image formation of the Ca²⁺ spark by a confocal microscope; and 3) using a novel automatic Ca²⁺ spark detector. From these results we derived an integral equation relating the probability density function of source strengths, f(α)(α), to N(a), which takes into account random positional variations between the source and linescan. In the special, but important, case that the spatial distribution of Ca²⁺-bound fluo-3 is Gaussian, we show the following: 1) variations of Ca²⁺ spark amplitude due to positional or intrinsic differences can be separated, and 2) f(α)(α) can, in principle, be calculated from the Ca²⁺ spark amplitude histogram since N(a) is the sum of shifted hyperbolas, where the magnitudes of the shifts and weights depend on f(α)(α). In particular, if all Ca²⁺ sparks were generated identically, then the plot of 1/N(a) against a will be a straight line. Multiple populations of channels carrying distinct currents are revealed by discontinuities in the 1/N(a) plot. 3) Although the inverse relationship between Ca²⁺ spark amplitude and decay time might be used to distinguish Ca²⁺ sparks from different channel populations, noise can render the measured decay times meaningless for small amplitude Ca²⁺ sparks.