 |
GLOSSARY |
| x, y, z |
= |
space coordinates (µm) |
| t |
= |
time coordinate (ms) |
| C |
= |
free cytosolic Ca2+ concentration (µM) |
| FD |
= |
free fluo-3 concentration (µM) |
| GD |
= |
Ca-bound fluo-3 concentration |
| HD |
= |
total fluo-3 concentration (FD + GD) |
| FB |
= |
free endogenous buffer concentration (µM) |
| GB |
= |
Ca-bound endogenous buffer concentration (µM) |
| HB |
= |
total endogenous buffer concentration (µM) |
| DCs, DCy,
DCz |
= |
Ca2+ diffusion coefficients along x, y, and
z directions (µm2/ms) |
| DDx, DDy,
DDz |
= |
fluo-3 diffusion coefficients of both free and Ca-bound forms |
| ISR |
= |
current through Ca2+ release unit (pA) |
| Topen |
= |
open time of Ca2+ release unit (ms) |
| JSR |
= |
molar flux of Ca2+ through Ca2+ release unit
(pmol/ms) |
| JP |
= |
SR-ATPase pump rate (µM/ms) |
| VP |
= |
maximum SR pump rate (µM/ms) |
| KP |
= |
SR pump Michaelis constant (µM) |
| np |
= |
SR pump Hill coefficient |
| Jleak |
= |
Ca2+ leak rate through SR (µM/ms) |
kj+, kj |
= |
forward and reverse rate constants for buffer reactions (j = D for dye, j = B for endogenous buffer) |
 x, y, z |
= |
full-width at half-maximum (FWHM) of GD along
x, y, z (µm) |
| Co |
= |
resting Ca2+ concentration (µM) |
| |
INTRODUCTION |
Current mathematical models of cardiac Ca2+ sparks
(Smith et al., 1998
; Izu et al., 1998)
are deficient in several respects. First, most models replicate the
amplitude of the average confocally recorded Ca2+ spark
[typical peak fluorescence ratio (F/F0) of 2],
yet, such a spark most likely represents an "out-of-focus" event
whose peak amplitude may be several times less than that of an
"in-focus" Ca2+ spark. Recently, we recorded
Ca2+ sparks of peak F/F0 of up to
6.0, larger than any reported previously (Wier et al.,
2000). Currents required to generate such large Ca2+ sparks could be much larger than previously assumed,
with important implications for the molecular mechanisms of cardiac
Ca2+ sparks. Second, all mathematical models fail by a
large margin to reproduce the spatial characteristics of recorded
Ca2+ sparks, both cardiac (Smith et al.,
1998; Izu et al., 1998) and skeletal
(Jiang et al., 1999). The fluorescence distribution in model cardiac Ca2+ sparks is typically spherically
symmetrical, and Gaussian in profile with a full-width at half maximum
(FWHM) of about 1.0 µm. Although recorded Ca2+ sparks are
indeed Gaussian in profile, they are often not spherically symmetrical
(Parker et al., 1996; Cheng et al.,
1996b) and both cardiac and skeletal Ca2+ sparks
typically have an FWHM in the longitudinal direction (i.e., along the
long cell axis) of about 2.0 µm. The possible significance of this
deficiency of model cardiac Ca2+ sparks was made apparent
to us when we were unable to model satisfactorily the evolution of
cardiac Ca2+ waves from stochastically occurring
Ca2+ sparks (Izu et al., 1999). We
hypothesized that this attempt failed in part because the properties of
the model cardiac Ca2+ sparks were not correct. Previous,
deterministic, one-dimensional models of Ca2+ waves
(Backx et al., 1989) would not have encountered this
difficulty, but should now be regarded as unrealistic, because the
experimental evidence is that cardiac Ca2+ waves arise from
sequential activation of Ca2+ sparks (Cheng et al.,
1996a; Wier et al., 1997; Lukyanenko et al., 1999). The present study was undertaken to remedy these
deficiencies by verifying the existence of Ca2+ sparks of
such large amplitude in mammalian cardiac muscle and by producing a
model of cardiac Ca2+ sparks that matched the spatial
properties and their peak amplitude.
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METHODS |
Preparation of cells and recording of Ca2+ sparks
Two-month-old Sprague-Dawley rats (body weight, 180-280 g) were
heparinized (10 iu g1 i.p.) and anesthetized with sodium
pentobarbital (170 mg kg1 injected i.p.). The heart was
removed while still beating by means of a mid-line thoracotomy. Single
ventricular cells were obtained by an enzymatic dispersion technique
described previously (López-López et al.,
1995). Cells were loaded with the acetoxymethyl ester form of
fluo-3 or fluo-4 using a dye stock solution consisting of 50 µg of
dye in 40 µl of DMSO. Pluronic (25% w/v in DMSO), 1.5 µl, was
added (this solution was vortexed). Ten microliters of this solution
was added to 500 µl of the physiological salt solution (see below)
containing the cells. This solution was kept in the dark for 30 min and
mixed gently. Cells were placed in a rotatable chamber so that they
could be aligned for confocal scanning, parallel to their long axis
(x axis). Recordings were made at room temperature, with
cells bathed in a physiological salt solution containing (composition
in mM): NaCl, 140; dextrose, 10; HEPES, 10; KCl, 4.0;
MgCl2, 1; CaCl2, 1; pH adjusted to 7.3-7.4 with NaOH. The performance (spatial resolution and dynamic range) of
the confocal microscope used for recording Ca2+ sparks has
been described in detail recently (Wier et al., 2000). With the 63X 1.4 NA oil immersion objective used in the present study,
the resolution of the confocal system is 0.25 µm (laterally) and 0.52 µm (axially). Line-scan images were typically 256 pixels per line, at
0.1 µm per pixel, and 512 lines per frame, at 3.0 ms per line. The
relatively small pixel size ensures that the Nyquist criterion is met,
because the FWHM of the point spread function (PSF) of the microscope
is 0.25 µm. Pixels are typically 10.0 µs in duration, giving time
for linear scanning of 2.560 ms, with 440 µs for mirror
"flyback." The number of counts per pixel typically did not exceed
50, corresponding to a count rate of 5 × 106 cps,
with a pixel duration of 10.0 µs.
Mathematical modeling of Ca2+ binding and diffusion
The reactions we consider are those of Ca2+
(C) with the free endogenous Ca2+-buffer
molecules (FB) and with the free fluorescent
indicator dye, fluo-3 (FD). We assume simple
one-to-one binding reactions, given as
The forward and reverse kinetic constants are
kj+ and kj
where j is either B or D.
Gj denotes the concentration of the
Ca2+-bound species and Hj, is the
total concentration (bound + unbound). Ca2+, free dye,
and Ca2+-bound dye (GD) are assumed
to be mobile, with the latter two having the same diffusion
coefficient. Free endogenous buffer and Ca2+-bound
endogenous buffer (GB) are assumed to be
immobile. Including terms for release and leak of Ca2+ from
the sarcoplasmic reticulum (SR) and active transport of Ca2+ into the SR (see below), the reaction-diffusion
equations are then
|
(1)
|
|
(2)
|
|
(3)
|
where
|
(4)
|
is the net rate of the bimolecular reaction and
|
(5)
|
where j = D or B.
Depending on our purpose, we will assume either that diffusion is
spherically symmetric or anisotropic. For spherically symmetric diffusion, 
· (DCC) is
DC(
2C/r2 + (2/r)C/r) (Crank, 1975), and for
anisotropic diffusion, it is given by
|
(6)
|
Note that we are assuming that the principal diffusion axes
coincide with the axes of the cell. The x-axis of the cell
is the long axis, the y-axis is transverse to the long axis,
and the z-axis coincides with the microscope's optical
axis. Since the transverse and axial directions appear similar in
cardiac cells, we will always assume that
DCy = DCz.
JSR
(r) is the point source of
Ca2+ release from the SR, located at the origin, and
(r) is the Dirac delta-function. Following Franzini-Armstrong et al. (1999), we call the
Ca2+ source the Ca2+ release unit (CRU).
JSR is related to the CRU current
ISR by JSR = ISR/(zF), where z = 2
and F is Faraday's constant. Jp
represents Ca2+ pumping by the SR-ATPase and is
|
(7)
|
where Vp = 200 µM/s,
Kp = 184 nM, and
np = 4 (Balke et al., 1994).
Jleak is the Ca2+ leak from the SR
and is adjusted so that, at the resting Ca2+ concentration
(100 nM), Jleak 
Jp = 0.
A substantial amount of fluorescent indicator can be bound to proteins
(Blatter and Wier, 1990; Harkins et al.,
1993) that immobilize the dye. In a previous paper (Izu
et al., 1998), we included the reaction of Ca2+
with the protein-bound indicator. This time, to reduce the
computational load, we take the approach of Smith et al.
(1998) and eliminate the protein-bound indicator reaction with
Ca2+. Smith et al. (1998) compensate for the
elimination of the dye immobilizing reaction by reducing the diffusion
coefficient of the indicator from the calculated value (from Stoke's
law) of 0.09 to 0.02 µm2/ms. The rate constants for the
reaction of Ca2+ with fluo-3,
kD+ = 80/µM/s and
kD = 90/s, are from Smith et al.
(1998). We will refer to the set of reactions given in Eqs.
1-5 and the set of rate constants and diffusivities just given as the
"Smith buffer model" to distinguish them from another set of
reactions, which we describe later. Simulation parameters for the Smith
buffer model are given in Table 1. The simulations always start with all chemical species in chemical equilibrium and no gradients. Thus a conservation relationship exists
between Gj, Fj, and
Hj, and there is no differential equation for
Gj. Zero-flux boundary conditions were imposed
in all cases.
Numerical solution of the reaction-diffusion equations
The model equations were solved numerically by Facsimile (AEA
Technologies, Harwell, UK), using the method of lines. The spatial derivatives were approximated by center differences. Accuracy of the
codes (spherically symmetric and anisotropic models) were checked by
eliminating all reactions and comparing simulation results with the
analytic solution (see Appendix C). In both cases, the simulation
agreed with the analytic solution to a few percent for distances >0.1
µm. To check the accuracy of the solution in the spherically
symmetric case when nonlinear reactions were present, we halved the
step size and found the solutions to be virtually identical to the
solution with the 0.01-µm step size. Such high resolution was not
possible (because of computer memory limitations) for the anisotropic
case. For this case, the spatial step size (equal in all three
directions) was either 0.05 or 0.1 µm. The domain length was 6 µm
for the spherically symmetric case and either 2 or 3 µm for the
anisotropic case. In all cases, we checked to ensure that C,
FB, and FD at the boundaries
were within 0.1% of their resting values.
Simulating blurring by the microscope
To simulate optical blurring of the spark by the microscope, we
convolved the 3-dimensional (3D) distribution of Ca2+-bound
indicator, GD, with a Gaussian kernel that
approximates the PSF of a microscope (Izu et al., 1998).
This was done by first multiplying the discrete Fourier transform (DFT)
of GD with the DFT of the PSF and then
performing the inverse transform. The size of the DFTs were 128 × 128 × 128, corresponding to a spatial length of 3 µm in all
directions. The standard PSF had an FWHM of (fwhmx,
fwhmy, fwhmz) = (0.4 µm, 0.4 µm,
0.8 µm).
| |
RESULTS |
Ca2+ sparks
A histogram of the peak amplitudes (F/F0)
of a large number of Ca2+ sparks from 21 cells from one
animal is presented in Fig.
1 A. The FWHM of these sparks
is presented in Fig. 1 B. All recordings were made with
scanning along the longitudinal or x axis of the cell.
Similar results were obtained in a total of 6 animals. In all cases,
large numbers of Ca2+ sparks were recorded with peak
amplitudes greater than 4.0, and ranging, in some cases, up to 9.0. The
distribution of amplitudes conforms roughly to the inverse hyperbolic
form predicted from the theory of confocal sampling (Izu et al.,
1998). Except in rat atrial cells (Blatter et al.,
1997), we are not aware of any previous studies in which
Ca2+ sparks greater than 4.0 have been recorded in cardiac
tissue. For example, the maximum F/F0 reported
in a recent study of a large number of cardiac Ca2+ sparks
was less than 3.0 (Cheng et al., 1999). The
Ca2+ sparks illustrated in Fig. 1 were detected "by
eye," rather than by an automatic detection system. Thus, the
amplitude distribution is distorted at the low end, due to the
inability of a human observer to distinguish small Ca2+
sparks from noise. Spark detection by the observer has no effect on the
upper end of the distribution. We attribute the detection of large
Ca2+ sparks primarily to the use of a confocal microscope
with high spatial resolution (Wier et al., 2000). The
modal value of the distribution of FWHM was ~2 µm. The largest
Ca2+ sparks, with which we are concerned here, typically
had FWHM of ~2 µm and were Gaussian in profile, as shown in Fig.
1 D.
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FIGURE 1
Large Ca2+ sparks in rat ventricular cell.
(A) Ca2+ spark amplitude histogram.
(B) Corresponding values of full-width-half-maximum (FWHM).
(C) Shade-surface representation of a typical large
Ca2+ spark, with peak F/F0 > 6.0. Calibration bars are: x, 2.5 µm; y,
100 ms; z, F/F0 from 1.0 to 5.0. (D)
Spatial profile of spark in (C). FWHM is 2.0 µm, solid
line is a Gaussian distribution fit to the data (open
circles).
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Modeling the cardiac Ca2+ spark
Equilibrium, Gaussian distribution model
Our objective is to calculate the current through the CRU that is
required to produce a spark of a given FWHM in the longitudinal (x), transverse (y), and axial
(z) directions. A schematic representation of a spark
with these dimensions is shown in Fig. 3, inset. The initial
calculations will be done with the assumption that calcium binds only
to the indicator dye, that C and FD
are always in chemical equilibrium, and that the spatial distribution of Ca2+-bound dye, GD, (hence
fluorescence) is Gaussian in all spatial dimensions, at all times. We
first present an analytic mathematical proof that, under these
conditions, the current found to generate a spark of a given FWHM is
the absolute minimum current; the addition of any other specific
conditions, such as the inclusion of other Ca2+ buffers,
and realistic (finite) reaction kinetics will only increase the current
required to produce the spark. This is true, as long as none of the
chosen conditions violates the assumption of a Gaussian distribution of
Ca2+-bound dye. The results of the equilibrium analysis
will show that even the absolute minimal currents required to produce
sparks with an FWHM of ~2.0 µm are substantially larger than
previously estimated.
Analytical proof of minimum current hypothesis
Let GD(x, y, z) be the
Ca-bound dye concentration and assume it has a Gaussian
distribution
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(8)
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The reason for choosing a Gaussian idealization of the Ca-bound
dye concentration is that experimentally measured sparks (Fig.
1 C) and simulated sparks (shown in Fig. 5 A)
can be well fit to a Gaussian distribution. The FWHM, , is related
to 
by = 
. Let
GD
be the distribution assuming infinite
kinetic rates (i.e., all reactions are in equilibrium). We make the
important assumption, the validity of which we will examine more
closely later, that
|
(9)
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We prove in Appendix A that GD has a
larger FWHM than GD. In other words, the FWHM of
the spark generated assuming infinite kinetic rates is larger than the
spark generated otherwise. This is important because the current needed
to produce a spark of a given amplitude and spatial size calculated
using the equilibrium assumption is a lower bound provided the spark
profile is Gaussian.
Now we examine the assumption that GD 
GD for all x, y, and z.
Certainly, at the spark origin GD
(0, 0, 0) GD(0, 0, 0), as long as
the channel remains open. By continuity, there is a region around the
origin where the inequality holds. Away from the origin, the inequality
need not hold for all times. Figure 2,
A-B, shows the spatial distribution of
GD from simulations in which the Ca-Dye
reaction rate was at standard values (dashed line) and when
the rates were multiplied by 100 (solid line), to
approximate infinite reaction rates. Figure 2, A and
B, shows GD at 1 and 5 ms after the
channel has opened, respectively. The difference between the two
(dotted line) is nonnegative everywhere. In this case, where
the dissociation constant was KD = 1.125 µM, GD using almost infinite kinetics,
GD(~
), is everywhere larger than the
GD obtained for moderate kinetic rates,
GD(moderate). When KD was
halved, there is a small region near the base of the spike (about 0.5 µm) where GD(~) is less than
GD(moderate) immediately after the channel
opens. For times >2 ms, however,
GD(~) > GD(moderate) everywhere. Because channel openings are on the order of 5-10 ms, the inequality in Eq. 9 is a
valid practical assumption, although it is not strictly true for all
times.
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FIGURE 2
Comparison of spatial distribution of Ca-bound dye for
very fast and standard kinetics. GD was
numerically computed using the Smith buffer model with the Ca-dye
reaction kinetics set to their standard values (dashed line)
or multiplied by 100 (solid line) to approximate an
infinitely fast reaction. Panels A and B show the
distribution 1 and 5 ms, respectively, after the channel opens. The
difference (dotted line) is everywhere positive at 5 ms,
but, just after the channel opens, (1 ms)-fast kinetics distribution is
about 0.5% lower than the standard kinetic distribution for
0.33 < r < 0.85. For times >2 ms, however, the
faster kinetic distribution is always greater than the slower
distribution. This observation supports the assumption that the
equilibrium distribution of GD is an upper bound
for GD for finite kinetic rates. Panel
C shows the FWHM increasing with increasing reaction rate.
In all cases,
kD/kD+ = 1.125 µM.
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Numerical simulations of equilibrium, Gaussian distribution
model
Numerical simulations provide a complementary way of showing that
the FWHM is largest when equilibrium dynamics is assumed. Simulations
have the advantage over the analytic proof in not requiring the
assumption of a Gaussian distribution of Ca-bound dye and in not
assuming the inequality in Eq. 9. (The shortcoming of numerical
simulations is that they sample only a tiny portion of parameter
space). We solved Eqs. 1-4 with the assumption of radial symmetry.
HB and HD were fixed, as
were kB+, kB,
and ISR. We multiplied
kD+ and kD by
from their standard values. The reaction of dye with
Ca2+ is speeded up or slowed down relative to the standard,
depending on whether is greater than or less than unity. Figure
2 C shows that the FWHM of the simulated spark increases as
the reaction speeds up. The numerical simulations and the analytic
proof both show that the spatial extent of the spark grows larger as
the reaction rates increase. Thus, the current that is required to produce a spark of a given spatial extent is smallest when equilibrium conditions prevail. Next, we calculate the actual value of that current, given the FWHM of the spark.
The minimum requisite current for a spark of a given FWHM
Assuming all reactions involving Ca2+ and buffers
(B and F) are in equilibrium, we can calculate
C and GB from the assumed
distribution of GD. C is
|
(10)
|
from which we calculate
|
(11)
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In these equations, Ki = ki/ki+,
i = B or D. The total concentration of
C at any point is 
= C + GB + GD and the total
number of moles of C is the integral of
over the entire volume
|
(12)
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GDp is the concentration of Ca-bound
dye at the peak of the spark, t = T. Accordingly,
GDp = GD0 × F/F0, where F/F0 is the
usual measure of spark amplitude. GD0 is the
concentration at rest and is given by
|
(13)
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Numerical quadrature of Eq. 12 was done using Monte Carlo
integration (Kahaner et al., 1988). We now want to find
the absolute minimal current to produce a Gaussian-shaped spark of a
given spatial size. We do this by setting HB to
zero so all released Ca2+ is available for binding to
fluorescent indicator and not taken up by endogenous buffers. Figure
3 shows the minimum current required to
produce sparks of various spatial sizes and amplitude when initial
FD = 50 µM. The lower
(circles) and upper (triangles) curves show the
currents for spherically symmetric sparks having FWHM of 1 and 2 µm,
respectively. The F/F0 values that we calculate are for an unblurred spark. The spark amplitudes of actual sparks are
expected to be larger if blurring were eliminated. To check if the
calculated minimum current is valid, we simulated a spherically symmetric spark using a current of 2.4 pA, setting
HB to zero, and using values of
kD+ and kD that
were 100 times larger than standard, to approximate an infinitely fast
reaction. The FWHM of this simulated spark was 0.92 µm, which is
close to the predicted value of 1.0 µm. The amplitude
(F/F0) of the unblurred spark was 12.1.
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FIGURE 3
Minimum current required to produce a spark of a given
spatial size. For these calculations, the fluorescent indicator is the
only buffer (HB = 0, FD = 50 µM). Spark dimensions are given
by the FWHM (in µm) along the x, y, and z axes,
(x, y, z). Inset shows a
schematic of an ellipsoidal spark. Note for a spherical spark
x = y = z.
Circles show the current required to produce a spherical spark whose
FWHM is 1 µm spark; squares are for an elliptical (2, 1.5, 1.5)
spark and triangles are for a spherical (2, 2, 2) spark. The current
varies linearly with the spark amplitude F/F0,
and the slopes are proportional to the spark volume. Even without
endogenous buffers, about 20 pA of current is needed to produce a
spherically symmetric spark having FWHM of 2 µm.
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Effects of spark symmetry
In cardiac cells, sparks are often spatially asymmetric, being
larger along the longitudinal, x, direction than in the
transverse, y and presumably z, direction
(Parker et al., 1996). The center curve (Fig. 3,
squares) has been calculated for an asymmetric spark that
has longitudinal FWHM of 2 µm and transverse (along y and
z) FWHM of 1.5 µm. To produce such a spark with amplitude of 12 requires 10.7 pA. The top curve (triangles) gives the
current for a spherical spark with FWHM of 2 µm. These plots show
that the requisite current increases linearly with
F/F0, and the slope depends on the spark
dimensions. In fact, the ratio of the slopes equals the ratio of the
volumes of the sparks, where the volume is V = (4
/3)(yz/2)2(x/2).
The volume of the asymmetric spark (x = 2.0 µm,
yz = 1.5 µm) is 4.5 times the volume of the small
spherically symmetric spark (x = 1.0 µm,
yz = 1.0 µm) and the large spherical spark (x = 2.0 µm, yz = 2.0 µm)
has 8 times the volume. To produce a spherical spark that has an FWHM
of 2.0 µm, instead of 1.0 µm, requires eight times as much current,
18.96 pA. It is, therefore, not surprising that previous spark
simulations (Smith et al., 1998; Izu et al.,
1998; Jiang et al., 1999) that used
Ca2+ release channel currents of ~2 pA produced sparks
that had a spatial FWHM of only ~1 µm.
The requisite current in the presence of endogenous
Ca2+ buffers
The current required to produce a spark of a given FWHM in the
presence of endogenous Ca2+ buffers can be found by setting
HB (total concentration of free and bound
buffer) to the desired value. Figure
4 A shows the required current to produce a spherical spark having FWHM = 1.0 µm
(circles) and an ellipsoidal spark with dimensions
(x = 2.0 µm, yz = 1.5 µm)
(squares) when the endogenous buffer concentration was fixed to 123 µM (Berlin et al., 1994) and the total dye
concentration varied. The current varies linearly with the free dye
concentration, FD, and, as before, the ratio of
the slope of the lines equals the ratio of the volumes of the two
sparks. As expected, much larger currents are required when endogenous
buffers are present. For example, without endogenous buffers, 10.7 pA
is needed to produce the asymmetric spark (marked with asterisk in
Figure 3) but 37.7 pA is required when the endogenous buffer
concentration is 123 µM (asterisk in Figure 4).
ISR is a rather weak function of the total dye
concentration, however. Despite a 15-fold increase in
HD, the current had to only double to produce a
spark of similar size and amplitude. Thus small, inevitable,
differences in dye loading should not significantly affect the spark
amplitude or FWHM. In contrast, ISR is a fairly
strong function of the dye's dissociation constant
KD, as seen in Fig. 4 B. A change in
KD from 500 nM to 1 µM, typical values used
for calculating Ca2+ concentrations, changes
ISR by about 10 pA, a 36% increase.
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FIGURE 4
Current required to produce sparks of given spatial
size. (A) Required current varies only slowly with the total
dye concentration. A 10-fold increase in dye concentration only doubles
the required current. Thus, small differences in dye loading should not
materially affect FWHM measurements. Circles show the current for a
spherical (1, 1, 1) spark and squares for an ellipsoidal
(2, 1.5, 1.5) spark. (B) The current required to produce a
(2, 1.5, 1.5) spark is strongly dependent on the dye's
KD. Other parameters used here are the standard
values of the Smith buffer model.
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Nonequilibrium models
Effect of dye saturation. The main result from the
analysis above is that the currents required to produce Gaussian sparks that have the spatial extent of ~2 µm are much larger (~20-30 pA) than previously thought. In fact, however, large Ca2+
currents (e.g., 20 pA), will saturate a dye like fluo-3, so that GD is no longer Gaussian, but platykurtic
(flat-topped). This will make the measured FWHM larger than predicted
from the analysis above and reduce the requisite current. Figure
5 A shows unblurred spark
profiles in a spherically symmetrical spark for various currents (5 ms
in duration) using the Smith buffer model. The currents used were 1, 2, 5, 10, 20, 30, 40, and 50 pA. The amplitudes increase roughly
proportionally to small currents, but reach an asymptotic value of
1 + KD/Co (= 12.25)
(Fig. 5 B) at large currents. Figure 5 C shows
that sparks do not attain an FWHM of 2 µm until ISR is 50 pA (circles). Also plotted
in this figure are the FWHM derived from backcalculation assuming
chemical equilibrium and a Gaussian Ca2+-bound dye profile
(squares). Note that, for ISR < 20 pA (where saturation is less severe) the backcalculated FWHM is
larger than that from simulation, supporting our earlier conclusion
that the equilibrium solution provides a lower bound for the current
needed to produce a spark of a given spatial size. When dye saturation becomes severe (ISR > 20 pA) and the
Gaussian distribution no longer accurately describes the actual
Ca2+-bound dye distribution, the equilibrium estimate of
the FWHM is lower than the simulation values. It should be noted that, in a linear system (approximating the case of no dye saturation), the
FWHM is independent of the current because as the spark broadens the
peak rises proportionally.
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FIGURE 5
Spark properties for a range of release currents.
(A) The unblurred Ca2+-bound dye spatial profile
along the x-axis at 5 ms, just before channel closing.
Currents are 1, 2, 5, 10, 20, 30, 40, 50 pA. Spark amplitude for each
current is given in Panel B. The amplitude increases
approximately in proportion to the current when the currents are small
but approaches the asymptote of 12.125 for currents above 10 pA.
(C) The FWHM of the spherically symmetric sparks for
standard dye kinetic parameters (circles) and infinite
reaction rates (i.e., equilibrium distribution, squares).
(Da) The normalized profiles of the unblurred (solid
line) spark generated by 50-pA current and sparks blurred with
either the homebrew PSF (dashed) or standard PSF
(dot-dashed). The profiles for the blurred sparks were
displaced downward slightly for clarity. After adding noise to the
image blurred with the homebrew PSF, the characteristic flat-top
profile is difficult to discern and the profile could be fit reasonably
well with a Gaussian function (Db). Computation parameters
are the standard Smith buffer model values.
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Difficulty in detecting platykurtic sparks. We mentioned
earlier that the profiles of measured sparks could be well fit to a
Gaussian distribution (Fig. 1 C). The spark profiles in
Fig. 5 A, which have not been subjected to blurring by the
microscope, are platykurtic when the currents are 30 pA so we might
expect to see flat-topped sparks if saturation were occurring. To test whether we could detect such flat-topped sparks, we simulated optical
blurring using the standard PSF (see Methods) and the PSF obtained from
our "homebrew" confocal (Wier et al., 2000). The
parameters of the homebrew PSF were (0.25, 0.25, 0.52) (in µm), which
are better than the standard PSF. The standard PSF was chosen to
account for the greater optical distortion likely to occur in the cell.
The solid curve in Fig. 5 Da is the unblurred profile of
the spark generated by ISR = 50 pA. The
spark profiles for blurring by the homebrew PSF is shown with the
dashed curve and the profile using the standard PSF is given by the
dot-dashed curve. Curves for the blurred sparks were displaced downward
slightly to enhance clarity. The flat-top appearance of the spark is
preserved with the homebrew PSF but becomes much less distinct with the standard PSF. At this time, we cannot deconvolve the effect of blurring
for actual sparks because x-y-z scans of sparks cannot be
collected within the time scale of sparks. But noise, not blurring, is
probably more serious in masking dye saturation. Figure
5 Db shows that, after introducing noise (Izu et
al., 1998) into the image, it becomes difficult to detect the
signature flat-top of a saturated spark even with the superior PSF and
we could fit the blurred, noisy spark profile with a Gaussian function.
Averaging multiple sparks would decrease the noise and might reveal dye
saturation. To check this possibility, we averaged sparks (after
normalization to the peak) collected with linescans at different
positions (y, z) relative to the CRU at the origin. We used
the homebrew blurring PSF to maximize the chance of detecting platykurtic sparks. Using linescans with coordinates (y = 0.25 µm × k, z = 0.25 µm × k; k = 0, ... , 4) yielded an average spark that
was almost perfectly Gaussian. These results indicate that the lack of
observed platykurtic sparks does not indicate the absence of dye saturation.
Thus optical blurring eliminates one avenue for testing whether the dye
is saturated. Other effects of microscope blurring on spark properties
are considered later.
Asymmetric nonequilibrium cardiac Ca sparks. The last
simulations showed that, when we take dye saturation into account, a 50-pA current is needed to generate a spherical 2-µm spark. Cardiac sparks are less than 2 µm in their transverse and axial dimensions, however (Parker et al., 1996). An asymmetric spark
having dimensions (x, y, z) = (2, 1.5, 1.5) would have about the same spatial profile (that is the same
FWHM) as a spherically symmetric (2, 2, 2) spark, when viewed in a
longitudinal linescan. Because the volume of the asymmetric spark is
only 0.58 times that of the spherical spark, we predict that less
current would be required to generate the asymmetric spark. To test
this prediction, we generated spatially asymmetric sparks by reducing
the diffusion coefficients of dye, bound dye, and Ca2+ to
half standard values along the y and z directions
from the x direction values (Parker et al.,
1996). Figure 6 shows the
variation in the longitudinal and transverse FWHM of the simulated
sparks with current. A 20-pA current generated a spark with dimensions (2, 1.4, 1.4). A slightly smaller spark having dimensions
(1.6, 1.2, 1.2) is generated with only 10 pA.
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FIGURE 6
Dimensions of an ellipsoidal spark. Ellipsoidal sparks
were generated using the Smith buffer model by reducing the transverse
and axial diffusion coefficients of Ca2+,
FD, and GD by half from
their longitudinal values of 0.3, 0.02, and 0.02 µm2/ms,
respectively. Circles indicate the longitudinal FWHM, and squares, the
transverse FWHM. Note that only 20 pA of current is needed to produce a
spark that has a longitudinal FWHM of 2 µm when the axial and
transverse FWHM are 1.4 µm, but 50 pA is required to produce a
spherical spark with FWHM = 2 µm (see Fig. 4 C).
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From these results, we conclude that cardiac sparks having a
longitudinal FWHM of ~2 µm require about 20 pA of current for generation, assuming that the transverse and axial FWHM are ~1.5 µm. Because the volume of the spark scales with the current,
relatively small changes in spark dimensions might reflect large
changes in the underlying current.
Other factors that might increase the FWHM of cardiac
Ca2+ sparks
We considered a number of schemes that might lead to spark sizes
x ~ 2 µm without invoking large currents. These
schemes are superclusters of CRUs, a different buffer model, and
assuming the existence of Ca2+-releasing channels carrying
small currents that surround the central large current CRU(s).
Superclusters of RyRs
In cardiac muscle, sparks might sometimes arise from the
near-simultaneous release of calcium from a number of CRUs
(Parker et al., 1996). It seemed possible that
simultaneous release from multiple clusters of ryanodine receptors, or
superclusters, could produce sparks that appear broader. First, sparks
arising off the confocal linescan have larger FWHM, slowed kinetics,
and decreased amplitude (Izu et al., 1998; Smith
et al., 1998; Jiang et al., 1999). If the
linescan passed perpendicularly through a planar lattice of release
sites that fired simultaneously, then the linescan would pass, on
average, between release sites. The resulting spark would be broad
(because of its distance from the release site) but the amplitude would
not be greatly decreased (because of summation from multiple release
sites), producing a large FWHM. Additionally, when the CRUs are close
together, buffer saturation can become prominent even when the currents
from individual CRUs are relatively small. We simulated four release
sites on the corners of a vertically oriented square having edges that
measured either 0.4 or 0.8 µm, in rough approximation to 4 CRU
surrounding a single myofibril at the z line. The smaller
length is about the mean minimum distance between CRUs reported by
Franzini-Armstrong et al. (1999). The larger length is
about the mean distance between sparks measured in confocal linescans
oriented transverse to the long axis of the cardiac cell (Parker
et al., 1996). Table 2 shows
x and y for sparks generated by 1 or 4 CRUs. The first three entries (labeled 1 × ISR) are for a single CRU carrying current
ISR. The linescan goes directly through the CRU.
The next four entries (labeled 4 × ISR)
are for 4 CRUs on the corners of a square whose length is L, where each
CRU carries current ISR. The linescan for these
simulations goes through the center of the square. When each CRU on the
square carried 2 pA of current, the resulting spark had a longitudinal
FWHM of just 1.2 µm. When the current through each CRU increased to 5 pA, the spark had a longitudinal FWHM of 2 µm, the same as a spark
generated by a single CRU carrying 20 pA. Less dye saturation accounts
for the fact that the 4 × 2-pA spark is considerably smaller than
a spark generated by a single CRU carrying 10 pA.
The results show that, when clusters of closely packed CRUs (spacing
~0.4-0.8 µm) fire simultaneously, the resulting spark would have a
longitudinal FWHM ~ 2 µm although the current through each CRU
is a modest 5-10 pA. However, a cluster of CRUs carrying smaller (~2
pA) currents still cannot generate sparks that have FWHM close to the
observed values.
Ultrasmall channels
Lipp and Niggli (1999) observed highly localized
(0.4-µm FWHM) tiny increases in fluorescence, which they suggested
might be "quarks" (Lipp and Niggli, 1996). They
estimated that these putative quarks are produced by currents only
-
of those that produce a typical spark.
Moreover, they occur away from the site of origin of the spark. In a
preliminary model of Ca2+ waves, we found it necessary to
intersperse small Ca2+ release channels carrying ~0.1 pA
between channels carrying 2 pA to get Ca2+ wave propagation
that matched experimental observations (Izu et al.,
1999). Here we examined whether Ca2+ release from
ultrasmall current channels surrounding a central channel carrying 2 pA
would generate a spark that had an FWHM of ~2 µm. To test this, we
placed the 2-pA channel at the center of a 2-dimensional square lattice
having lattice period of 0.25 µm. (See Izu et al.,
2001 for conversion of current to a flux appropriate for
2-dimensional systems.). Ultrasmall current channels carrying 0.1 pA
were placed at each lattice site. To simulate Ca2+-induced
Ca2+-release, these ultrasmall channels began to release
Ca2+ when the ambient Ca2+ concentration
reached about 500 nM. Within 1 ms after the central channel opened the
adjacent 0.1 pA channels started releasing Ca2+. However,
the contribution of the 0.1 pA channels to elevating the fluorescence
(GF) was almost completely masked by the
fluorescence increase due to the much larger central channel. Thus, it
is unlikely that if quarks exist that Ca2+ release from
them is responsible for generating sparks with FWHM ~ 2 µm.
Possible protein-dye interactions
We showed above that the spark FWHM increases only slowly with
increasing current because the broadening of the
GD distribution is largely offset by the
increase in peak. In skeletal muscle, a similar problem has been noted;
model Ca2+ transients rise more rapidly at the release site
and more slowly away from the release site than do recorded ones,
making the model Ca2+ transients smaller in spatial spread
(Hollingworth et al., 1999). This problem was
ameliorated by considering possible protein-dye interactions, as
described earlier (Harkins et al., 1993). The Harkins
buffer model allows binding of dye (D) to protein, forming PD, the binding of Ca-bound fluo-3 (CaD) to
protein, forming CaPD, as well as the binding of
Ca2+ to PD, also forming CaPD. PD has
a lower affinity for Ca than free dye (D) (1.92 and 0.51 µM, respectively). Ca-bound fluo-3 (CaD) has a lower
affinity for protein than free fluo-3 (D) (1378 and 366 µM, respectively). Thus, when Ca2+ is released,
CaD rises and the loss of D is compensated by the unbinding of dye from its protein-bound form. The dynamic increase of
D and the greater diffusivity of CaD over
CaPD should decrease the amplitude and broaden the spark.
Numerical simulations of the Harkins model required a mesh spacing
finer than 0.1 µm used in the 3D simulations of the Smith model. This
requirement made the computations prohibitively long (>24 hrs per ms
of simulation time), so all results were obtained assuming spherical
symmetry. Ca and CaD were assumed to be mobile with diffusion coefficients of 0.3 and 0.09 µm2/ms,
respectively. Figure 7 A
shows the FWHM of the sparks generated from the Harkins
(squares) and Smith (circles) buffer models. The
FWHM without and with blurring (see below) are given by the open and
solid symbols, respectively. Sparks from the Harkins buffer model have
a larger FWHM than those from the Smith buffer model, which is in
accord with our intuitive predictions. However, the relatively large
currents (~40 pA) are still required to generate sparks having
FWHM ~ 2 µm in the Harkins buffer model.
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FIGURE 7
Comparison of sparks generated by the Smith and Harkins
buffer models. (A) The FWHM of spherical sparks for the
Smith (circles) and Harkins (squares) buffer
models. The empty symbols indicate no blurring, and filled symbols show
the values obtained with optical blurring. For the same current, the
Harkins model produces slightly larger sparks. The chief difference
between the two models lies in the spark amplitudes shown in Panel
B. The spark amplitudes for the Harkins model are about half
of those in the Smith model. See text for explanation for the
differences in blurred and unblurred values for the FWHM and amplitude.
Simulation parameters for Harkins buffer model (Hollingworth et
al., 1999): on- and off-rate constants of Ca2+ with
fluo-2 were, 3.5 × 108 per µM/s and 179/s for
protein-free fluo-3 and 2.25 × 107 per µM/s and
43/s for protein-bound fluo-3. On- and off-rate constants for the
reaction of protein with fluo-3 were 1 × 107 per
µM/s and 3.67 × 103 per s for Ca2+-free
fluo-3 and 1 × 107 per µM/s and 1.38 × 103 per s for Ca2+-bound-fluo-3.
DCx = 0.3 µm2/ms,
DDx = 0.09 µm2/ms, total
protein = 123 µM, Ca2+-free dye = 50 µM (at
rest). SR-pump parameters are the same as in Smith buffer model.
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The most striking difference between the Smith and Harkins models is
the spark amplitude shown in Fig. 7 B. The amplitude of
sparks from the Smith model is about twice those from the Harkins model. The difference arises from the different dye dissociation constants in the two models. For any dye, the maximum achievable amplitude is 1 + KD/Co. For the protein-free dye
in the Harkins model, KD1 = 0.51 µM, so
the maximum amplitude is 6.1. The protein-bound dye has
KD2 of 1.92 µM so the maximum amplitude is
20.2. However, because most of the Ca-bound dye is in the protein-free
form (CaD), the peak amplitude is close to 6.1. In the Smith
model, KD = 1.125 µM, and the peak
amplitude for large currents is about the maximum achievable of 12.25.
Blurring by the confocal microscope
Earlier, we have seen that optical blurring by the confocal
microscope made the detection of platykurtic sparks difficult. We now
consider other effects that blurring has on spark properties, particularly on the possible importance of this factor in producing sparks with large FWHM.
From Eq. B3, we see that the FWHM of the blurred spark is
FWHMi = 
, where the
subscripts i, m, and o refer to the image, microscope, and object; this
equation holds exactly when the object and blurring kernel are
described by Gaussian functions. Thus, when the spark is spatially
small, the FWHM of the blurred spark is determined to a large extend by
the microscope's FWHM. Accordingly, when the currents are small, the
unblurred sparks are narrower than the blurred sparks. This is seen in
Fig. 7 A, where the spark FWHM for the blurred image
(solid symbols) lie above those for the unblurred image
(open symbols) for currents <10 pA. The ratio of the spark
amplitude of the blurred to unblurred spark image scales as
FWHMo/FWHMm when this ratio is small (Eq. B3),
and approaches 1 when the object is large. This behavior is illustrated
in Fig. 7 B where the amplitude of the blurred spark
(closed symbols) lies below the unblurred amplitude. Thus,
blurring by the microscope will distort sparks generated by small
currents, making them appear wider and dimmer. However, as the sparks
become larger, the PSF of the microscope figures less prominently in
determining the measured FWHM of the spark. When the current is 10 pA,
the FWHM of the blurred and unblurred sparks are almost identical.
However, when the currents are large and dye saturation significant,
the unblurred spark profile is platykurtic. As seen earlier, blurring
rounds out the flat top giving the spark a more Gaussian profile.
Consequently, the FWHM of the blurred spark is narrower than the
unblurred spark (Fig. 5 Da). Thus, in Fig. 7 A
the FWHM of the unblurred sparks exceed those of the blurred sparks for currents >10 pA. The difference is fairly small, however, amounting to
<5% for the Smith model and <10% for the Harkins model. Thus, blurring by the microscope should have minimal effect on the measured FWHM and amplitude of sparks with FWHM ~ 1 µm. Nevertheless,
blurring is not benign because it makes detection of saturated sparks difficult.
Temporal properties of sparks for long channel openings
Cardiac Ca2+ sparks typically reach their peak in
5-10 ms (Cheng et al., 1993) then begin to decay
immediately. The fluorescence of some sparks, however, remain elevated
for long times (tens of ms) in the presence of ryanodine (Cheng
et al., 1993), or spontaneously (Parker and Wier,
1997), or when inactivation is blocked (Xiao et al.,
1997). These long-lived sparks presumably reflect long openings
of the RyRs. The fluorescence of these long-lived sparks rises abruptly
to a peak, and then stays there or falls rapidly to a plateau. We
compared the temporal features of sparks generated by 2-pA and 20-pA
currents that flowed for 25 ms, much longer than the standard 5-ms open
time. For the 2-pA channel, the fluorescence rises gradually and
continuously throughout the time the CRU stays open (Fig.
8 A). The spark generated by
20-pA current behaves differently (Fig. 8 B). The
fluorescence does not rise continuously during the CRU opening but
instead rises rapidly, "overshoots" slightly, then plateaus. The
absence of a continuously rising fluorescence signal in experimentally
measured sparks with long channel openings may indicate dye saturation
and large underlying currents. We note that the overshoot and plateau
in the simulated sparks is caused by the unloading of the
Ca2+ by the protein-bound dye, CaPD, to
regenerate PD. In all simulations, the differences between
the peak and plateau level were small, but, in some experimentally
measured sparks, there can be substantial differences between peak and
plateau fluorescence levels (Xiao et al., 1997). In
actual sparks, the overshoot and drop to a lower plateau may reflect a
more complicated mechanism such as the closing of a few channels in the
CRU.
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FIGURE 8
Comparison of temporal properties of sparks generated
by small and large currents for long channel opening. The temporal
profiles of sparks generated by (A) small (2 pA) or
(B) large (20 pA) differ significantly. Both channels were
open for 25 ms. For small currents, the fluorescence rises continuously
throughout the time the channel is open. By contrast, when the current
is large the fluorescence rises rapidly, overshoots, then plateaus. The
overshoot occurs in the Harkins buffer model but not in the Smith
buffer model.
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DISCUSSION |
In summary, we have presented a series of estimates of the total
current through a CRU needed to produce a spark of spatial dimensions
(x, y, z), where the are the FWHM (in µm) along the ventricular myocyte's longitudinal
axis, transverse axis, and the microscope's z-axis. The
first estimate assumed all buffer reactions were in equilibrium and
that the spatial distribution of the Ca2+-bound fluo-3 was
Gaussian. This estimate established a lower bound for the required
current in the absence of fluo-3 saturation. A spherical spark with
FWHM of 2 µm requires ~20 pA, even without endogenous buffers. The
required current scales linearly with the spark volume, so only 10.7 pA
is required to produce an ellipsoidal spark with dimensions
(2, 1.5, 1.5). To produce the same sized spark in the presence of 123 µM endogenous buffer however, requires ~38 pA.
Large currents however, cause significant dye saturation, so the
computed Ca2+-bound fluo-3 spatial profile is not Gaussian,
but platykurtic. The FWHM for the platykurtic distribution is
significantly larger, given the same current. By taking possible dye
saturation into account, we found that 20 pA was needed to produce a
(2, 1.4, 1.4) spark. A 10-pA current will generate a spark with
dimensions (1.6, 1.2, 1.2). This is about half the current estimated
when the spark had a Gaussian profile.
Currents of this magnitude (~10-20 pA) are close to the 16-20 pA
estimate that Rios et al. (1999) made for currents
underlying sparks in skeletal muscle. These large currents produce
Ca-dye distributions that are platykurtic. The fact that the largest Ca2+ sparks were not platykurtic seems to argue against the
notion that sparks arise from large currents from a small source
region. However, optical blurring and noise makes detecting the
flat-top saturation signature of even the widest spark (generated by 50 pA) difficult (Fig. 5, Da and Db). Hence, the
absence of flat-topped sparks cannot be used to rule out the
possibility that large currents underlie sparks. This is important,
because an alternate way of generating spatially broad sparks without
using large currents is to lengthen the size of the source along the
long axis of the cell (DiGregorio et al., 1999;
Gonzalez et al., 2000). This possibility is favored by
Gonzalez et al. (2000) for generating sparks in skeletal
muscle because it appears that dye saturation does not occur there. As
Gonzalez et al. point out, however, such an extended source requires
RyRs to lie outside the plane of the z-line.
For currents 5 pA the computed spark amplitudes
(F/F0) are large, ~6 for Harkins buffer model,
or ~12 for Smith buffer model. We and others (Cheng et al.,
1993, 1999;
López-López et al., 1995) measured sparks
with amplitudes of F/F0 of ~2-3, considerably smaller than predicted from the model. Recently, however,
Shirokova et al. (1999) have measured large amplitude
(~9) sparks in developing mouse skeletal muscle. We now also measure
sparks with large amplitude (~6) occasionally (Fig. 1 and Wier
et al., 2000).
We attribute the measurement of large amplitude sparks using our custom
confocal microscope (Wier et al., 2000), in part, to the
better optical resolution and differences in photon detection compared
to our BioRad 600. We measured a population of sparks in cells on the
same coverslip using the homebrew confocal and the BioRad within a few
minutes of each other. In this way, differences in spark
characteristics due to different animals, loading conditions, or cell
isolation are minimized or eliminated (C. Lamont, J. Mauban, and
W. G. Wier, unpublished observations). The mean spark amplitude was larger in the population measured with the custom confocal microscope than in the BioRad (3.1 versus 2.0, p < 0.001 using Student's t-test). The amplitudes of the
largest sparks were also larger when measured with the custom confocal
microscope than with the BioRad (8.5 versus 4.2). From the optical
standpoint, any reasonably aligned confocal microscope should record
about the same spark amplitude and FWHM for the largest sparks, because the FWHM of a large spark (~2 µm) is much larger than the lateral FWHM of the PSF of a typical microscope objective used for measuring sparks (<0.5 µm). For example, the spark amplitude and FWHM for the
blurred and unblurred spark are similar when the underlying current is
large, ~40 pA (Fig. 7). Thus, differences in photon detection between
the homebrew confocal and the BioRad is likely to be important in
accounting for the differences in spark amplitudes measured in
these two systems.
The magnitude of ISR is not the sole determinant
of the spark amplitude. The Kd of the dye
strongly affects amplitude. The spark amplitude is smaller in the
Harkins than in the Smith buffer model for the same current because, in
the former, most of the Ca-bound dye is in the free (not pr
TOP
ABSTRACT
GLOSSARY
