MODELING IN PHYSIOLOGY
Predicted changes in concentrations of free and bound ATP and ADP
during intracellular Ca2+
signaling
M. E.
Kargacin and
G. J.
Kargacin
Department of Physiology and Biophysics, University of
Calgary, Calgary, Alberta, Canada T2N 4N1
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ABSTRACT |
High Ca2+ concentrations
can develop near Ca2+ sources
during intracellular signaling and might lead to localized regulation
of Ca2+-dependent processes. By
shifting the amount of Ca2+ and
other cations associated with ATP, local high
Ca2+ concentrations might also
alter the substrate available for membrane-associated and cytoplasmic
enzymes. To study this, simultaneous equations were solved over a range
of Ca2+ and
Mg2+ concentrations to determine
the general effects of Ca2+ on the
concentrations of free and Ca2+-
and Mg2+-bound forms of ATP. To
obtain a more specific picture of the changes that might occur in
smooth muscle cells, mathematical models of
Ca2+ diffusion and regulation were
used to predict the magnitude and time course of near-membrane
Ca2+ transients and their effects
on the free and bound forms of ATP near the membrane. The results of
this work indicate that changes in free
Ca2+ concentration over the range
of 50 nM-100 µM would result in significant changes in free ATP
concentration, MgATP concentration, and the CaATP-to-MgATP
concentration ratio.
smooth muscle; striated muscle; computer modeling; calcium
channels; restricted diffusion; adenosinetriphosphate-sensitive
channels
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INTRODUCTION |
IN CELLS, AN INCREASE in intracellular
Ca2+ concentration is often one of
the first events in signal transduction. For
Ca2+-dependent signaling to be
effective, intracellular Ca2+
concentration must be precisely controlled. In resting cells, average
free Ca2+ concentration is
generally maintained below 100-200 nM and, in muscle, may rise to
~1 µM during contractile signaling. However, theoretical
calculations (15, 23, 27-29, 33) and experimental measurements (8,
20, 25) indicate that localized
Ca2+ levels much higher than this
may develop near the plasma membrane of cells after cell stimulation as
a result of Ca2+ influx through
plasma membrane Ca2+ channels.
Predictions of maximum near-membrane free
Ca2+ concentrations during a
Ca2+ transient range from several
micromoles a few nanometers away from the membrane (15, 16, 23) to
>100 µM in the immediate vicinity of
Ca2+ channels or clusters of
channels (7, 28, 29, 31, 33). Measurements of maximum near-membrane
free Ca2+ concentration in cells
range from 2-10 µM in muscle and cultured cells (8, 20) to
200-300 µM in presynaptic terminals (25).
It is becoming increasingly clear that
Mg2+ is also regulated in cells
(reviewed in Refs. 22 and 24). Determinations of free
Mg2+ concentration in cardiac and
smooth muscle cells range from ~0.1 to ~3 mM (13, 22), with more
recent estimates falling in the lower part of this range (18, 22, 24).
Corkey et al. (5) measured a total cytosolic
Mg2+ concentration of 6.4 mM in
isolated rat liver hepatocytes and free
Mg2+ concentration of 0.38 mM; the
total Mg2+ content of rabbit
urinary bladder smooth muscle was determined by Kushmerick et al. (18)
to be 6.6 µmol/g wet wt. Although Mg2+ itself may be involved in the
regulation of cellular function, one of its primary roles in cells is
in enzymatic processes, where it is complexed to ATP as a substrate for
various ATPases. In muscle cells, estimates of the cellular content of
ATP range from ~0.3-2 µmol/g wet wt in smooth muscle cells to
~5 mmol/kg wet wt in skeletal muscle cells (12). The dissociation
constants (Kd)
for Mg2+ binding to
ATP4
reported in the
literature are in the micromolar range, and those for binding to
HATP3
are in the micromolar
to millimolar range (9, 18, 21, 30).
Mg2+ is also likely to bind to
intracellular Ca2+ buffers with a
lower affinity than Ca2+ (16, 22).
The fact that Ca2+ and
Mg2+ can bind to a number of
intracellular molecules suggests the possibility that in regions where
high Ca2+ concentration develops
in cells during signal transduction, the relative amounts of
Mg2+ and
Ca2+ associated with these
molecules might be altered. Although
Ca2+ binds to
ATP4
and
HATP3
with
Kd values higher
than those reported for Mg2+,
changes in Ca2+ concentration
might be expected to lead to changes in the amount of
Mg2+ bound to ATP or in the
relative amounts of Mg2+ and
Ca2+ bound to ATP. This could have
important physiological consequences, since MgATP is the preferred
substrate for most intracellular enzymes. Any effect of
Ca2+ on intracellular MgATP might
be especially significant in smooth muscle cells, where levels of
intracellular ATP (~1 mM) and free Mg2+ (
1 mM) are normally within
an order of magnitude of the free Ca2+ concentration that might be
reached near the plasma membrane during a
Ca2+ transient. It is also thought
that the MgATP-to-MgADP ratio may be an important factor in determining
the ability of smooth muscle cells to develop and maintain force (see
discussion in Ref. 13). Because ADP can also bind
Ca2+, the development of high
Ca2+ concentration in cells might
be predicted to alter the MgADP-to-MgATP ratio and, as a consequence,
influence contractility.
To examine the possible effects of high
Ca2+ concentration on the amount
of Ca2+ and
Mg2+ associated with ATP, ADP, and
intracellular Ca2+ buffers, we
solved a set of simultaneous equilibrium equations at various
Ca2+,
Mg2+, ATP, ADP, and intracellular
Ca2+ buffer concentrations. Our
results predict that MgATP concentration is likely to remain fairly
constant in the face of Ca2+
concentration changes of <1 µM but may then decrease significantly when higher concentrations are reached. The concentration of CaATP is
likely to rise and may approach that of MgATP. The simulations also
predict that the concentrations of unbound forms of ATP
(ATP4
,
HATP3
, and
H2ATP2
)
and the MgADP-to-MgATP ratio are likely to decrease. To further explore
these predictions, a mathematical model of
Ca2+ diffusion and regulation was
used to simulate the changes in Ca2+ and accompanying changes in
ATP that might occur in smooth muscle cells during contractile
signaling. Our results indicate that a local high
Ca2+ concentration may
significantly influence local ATP-dependent processes after
Ca2+ influx through the plasma
membrane.
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METHODS |
Equilibrium equations.
A computer program using a matrix inversion algorithm was written to
solve a set of simultaneous equilibrium equations for the binding of
various ionic species to ATP, ADP, and intracellular Ca2+ and
Mg2+ buffers. The binding
constants (Table 1) were from Martell and Smith (21, 30) and Fabiato (9). The equilibrium equations are linear if
total Mg2+ concentration is
calculated from a known free Mg2+
concentration. Thus it was necessary to use an iterative method of
solution (17) for equations in which total
Mg2+ concentration remained
constant while free Mg2+
concentration varied as free Ca2+
concentration was changed in a given simulation. Before a simulation was started, a total Mg2+
concentration was determined for a desired starting free
Mg2+ concentration at 50 nM free
Ca2+. The total
Mg2+ concentration corresponding
to this free Mg2+ concentration
was then used throughout the simulation as an end point for the
iterative calculations. Briefly, as
Ca2+ concentration was changed, an
estimate of free Mg2+
concentration based on its previous value was entered into the program
and used to calculate a new total
Mg2+ concentration; if the
calculated total Mg2+
concentration was not within a preset margin of error (0.01%) of the
starting total Mg2+ concentration,
the difference between the calculated and starting total
Mg2+ concentration was used to
make a corrected estimate of free
Mg2+ concentration. For each free
Ca2+ concentration, this iterative
process was continued until the calculated total
Mg2+ concentration was within the
allowed margin of error of the starting total
Mg2+ concentration. The
concentrations of the other ionic species included in the simulations
are given in Table 2 and in
RESULTS. The equilibrium equations
were solved using a program written in C language; simulations were run
on a 486 personal computer, and results were plotted using commercially
available graphics software (Sigma Plot, Jandel Scientific, San Rafael,
CA).
Unless otherwise noted, [MgATP] is used to refer to the
total concentration of all species of ATP bound to
Mg2+ (i.e.,
[MgATP] = [MgATP2
] + [MgHATP
]).
Similarly, [MgADP] = [MgADP2
] + [MgHADP
],
[CaATP] = [CaATP2
] + [CaHATP
],
[CaADP] = [CaADP2
] + [CaHADP
], and
free ATP concentration
([ATP]free) = [ATP4
] + [HATP3
] + [H2ATP2
].
Smooth muscle cell model.
To study the changes in
[Ca2+] that are likely
to develop in smooth muscle cells during contractile signaling and the
effects of these changes on ATP concentration, a mathematical model of Ca2+ diffusion and regulation was
used. The model was a higher-resolution version of the models used by
Kargacin and Fay (15, 16). Details of the model and its method of
solution are given in RESULTS and the
APPENDIX. Simulations were run on a
486 personal computer or a personal computer with a Pentium processor.
 |
RESULTS |
Predicted effects of
Ca2+
concentration on free and bound ATP at different starting
Mg2+ and ATP
concentrations.
As discussed above (also see Ref. 12), measurements of the ATP content
of smooth muscle cells range from 0.3 to 2 µmol/g wet tissue wt, and
free Mg2+ concentration is
generally thought to be <1 mM. To examine the effect of changes in
Ca2+ concentration on the
concentration of bound and free ATP in such cells, free
Ca2+ concentration was varied in
the equilibrium equations from 50 nM to 100 µM at various starting
free Mg2+ concentrations ranging
from 0.1 to 1 mM. For these simulations, total ATP concentration was 1 mM, the
Kd Ca for
the binding of Ca2+ to an
intracellular Ca2+ buffer was
assumed to be 1 µM (15, 16), and the
Kd Mg for binding to the same buffer was 1 mM (the effects of using different values for
Kd Mg are
discussed below). To achieve the starting free
Mg2+ concentrations listed above,
starting total Mg2+ concentrations
ranged from 0.77 to 2.07 mM. The equilibrium equations (Fig.
1A)
predict that, under the conditions of the simulations, intracellular
[MgATP] would change very little over free
Ca2+ concentrations ranging from
50 nM to ~10 µM. For a starting free Mg2+ concentration of 0.5 mM,
[MgATP] was predicted to increase slightly from 904 to 907 µM as free Ca2+ concentration
increased from 50 nM to 1.5 µM; at 10 µM free
Ca2+, [MgATP] was 898 µM. Between 10 and 100 µM free
Ca2+, [MgATP]
decreased significantly, and at 100 µM free
Ca2+, [MgATP] had
dropped to 797 µM. The ability of the model cell to maintain a fairly
constant [MgATP] between 50 nM and 10 µM free
Ca2+ was aided by the displacement
of Mg2+ from the intracellular
Ca2+ buffer as
Ca2+ concentration increased (Fig.
1B).
[ATP]free was also
predicted to decrease significantly as free
Ca2+ concentration increased (Fig.
1C). As expected, the most dramatic change occurred for the lowest starting free
Mg2+ concentration (0.1 mM). In
this case, [ATP]free
declined by ~60% from ~260 to ~100 µM. The
[CaATP]-to-[MgATP] ratio increased to 0.1 for
the simulation with a starting free
Mg2+ concentration of 1 mM and to
0.6 for the simulation with a starting free
Mg2+ concentration of 0.1 mM.

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Fig. 1.
Effects of changes in Ca2+
concentration on MgATP ([MgATP]) and free ATP
([ATP]free)
concentrations and CaATP-to-MgATP concentration ratio
([CaATP]/[MgATP]) on cells with 1 mM total ATP.
A: [MgATP] vs. free
Ca2+ concentration
([Ca2+]free)
for starting free Mg2+
concentration
([Mg2+]free) = 0.1, 0.25, 0.5, and 1 mM. B:
displacement of Mg2+ from
intracellular Ca buffer as Ca2+
concentration increased (starting
[Mg2+]free = 0.5 mM).
C: [ATP]free
vs.
[Ca2+]free
for starting
[Mg2+]free = 0.1, 0.25, 0.5, and 1 mM. D:
[CaATP]/[MgATP] vs.
[Ca2+]free
for starting
[Mg2+]free = 0.1, 0.25, 0.5, and 1 mM. Numbers to
right of traces in A,
C, and D are starting
[Mg2+]free.
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Predicted effects of
Ca2+
concentration on various species of ATP at different
Kd Ca-to-Kd Mg
ratios for an intracellular
Ca2+ buffer.
The total Ca2+ buffer capacity of
cells may be as high as 250 µM and the buffer
Kd Ca
1 µM (1, 26, 28, 29, 34, 35; also see discussions in Refs. 15 and 16).
Although these intracellular buffers are thought of primarily as
Ca2+ buffers, they are likely to
bind Mg2+.
Kd Ca for
striated muscle troponin range from
10
8 to
10
6 M, and
Kd Ca for
parvalbumin is ~10
8 M. Kd Mg for
these proteins is ~10
4 M
(26). In general, however,
Kd Mg for
various Ca2+-binding proteins have
not been extensively studied; therefore, the equilibrium equations were
solved for various
Kd Mg
values (1 µM-10 mM) for an intracellular
Ca2+ buffer. To simplify the
calculations, a single
Ca2+/Mg2+
buffer was included, and the total buffer concentration and
Kd Ca were
kept constant at 250 and 1 µM, respectively; the starting free
Mg2+ concentration for the
simulations was 0.5 mM;
[ATP]total was 1 mM
(Fig.
2A).
[MgATP] remained relatively constant between 50 nM and 1 µM free Ca2+ for all values of
Kd Mg;
however, it remained constant over the greatest range of
Ca2+ concentrations for
Kd Mg
values of 100 µM or 1 mM. When
Kd Mg was
10 mM, little Mg2+ was bound to
the buffer, and at
Kd Mg of 1 or 10 µM, Mg2+ was displaced
from the buffer only at high free
Ca2+ concentration (Fig.
2B). Free ATP was least affected by
the change in free Ca2+
concentration for the
Kd Mg
values in which little Mg2+ was
displaced from the buffer (Fig. 2C).

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Fig. 2.
Effects of ratio of dissociation constant for
Ca2+
(Kd Ca)
to dissociation constant for Mg2+
(Kd Mg)
for an intracellular Ca2+ buffer
on [MgATP] and
[ATP]free.
A: [MgATP] vs.
[Ca2+]free
for Kd Mg ranging from 1 µM to 10 mM. B: release of
Mg2+ from buffer by
Ca2+.
C:
[ATP]free vs.
[Ca2+]free
for Kd Mg ranging from 1 µM to 10 mM. For simulations in A-C, starting
[Mg2+]free
was 0.5 mM and
[ATP]total was 1 mM.
Kd Ca for buffer was 1 µM;
total buffer concentration was 250 µM.
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Effects of free
Ca2+
concentration on ADP-to-ATP ratio.
ADP reduces the rate of relaxation of tonic smooth muscle. This is
presumably the result of the high affinity of the actomyosin cross
bridge for MgADP, which competes with MgATP for dephosphorylated cross
bridges (reviewed in Ref. 13). The most dramatic effects of ADP on
muscle contraction are likely to occur under conditions of hypoxia or
ischemia, when ATP declines and ADP increases. During hypoxia and
inhibition or glycolysis, Allen et al. (2) noted a 44% decrease in ATP
concentration and an increase in ADP concentration from 5 to 180 µM.
Ischemia has also been shown experimentally (reviewed in Ref. 22) to be
accompanied, at least initially, by an increase in free
Mg2+ concentration, presumably as
the result of the release of Mg2+
from ATP.
To examine possible effects of high
Ca2+ concentration on the
MgADP-to-MgATP ratio, equations were included in the equilibrium model
for the binding of Mg2+,
Ca2+,
H+, and
K+ to ADP. For the nonischemic
condition (total ATP concentration = 1 mM, total ADP concentration = 5 µM, starting free Mg2+
concentration = 0.5 mM, total
Mg2+ concentration = 1.49 mM), the
MgADP-to-MgATP concentration ratio was 0.005 and decreased by 6% over
the range of free Ca2+
concentrations of 50 nM-100 µM. With total
Mg2+ concentration constant at
1.49 mM, the starting free Mg2+
concentration in the model increased to 0.82 mM when total ATP concentration was decreased to 0.5 mM and total ADP concentration increased to 0.1 mM. Under these conditions, the MgADP-to-MgATP concentration ratio was ~0.2 and decreased by only 5% when free Ca2+ concentration increased from
50 nM to 100 µM. Thus it is unlikely that changes in
Ca2+ would have a significant
physiological effect on the high-energy phosphate ratio.
Changes in
Ca2+ and ATP in
restricted diffusion spaces.
A restricted diffusion space had been proposed to exist in smooth
muscle cells, where the sarcoplasmic reticulum comes into close
apposition to the plasma membrane (see discussion in Ref. 15). Free
diffusion of Ca2+ into the central
cytoplasm of the cell would be inhibited by the physical presence of
the sarcoplasmic reticulum membrane, and one might expect high
Ca2+ concentrations to develop in
such spaces after Ca2+ influx
through the plasma membrane. To study the magnitude and time course of
Ca2+ signals that are likely to
develop in restricted diffusion spaces and the effects of these signals
on local ATP concentrations, the one-dimensional diffusion model
described previously (15) was used. A diagram illustrating the model is
shown in Fig.
3A. The
equations describing radial diffusion into a cylindrical cell and the
Ca2+ regulatory processes included
in the model are given in the
APPENDIX. The parameters used in the
model are summarized in Table 3. The model
was a higher-resolution version of the model described in detail by
Kargacin and Fay (15, 16). Briefly, the model describes radial
diffusion through a restricted diffusion space and into the central
cytoplasm of a smooth muscle cell (3-µm radius). At the beginning of
a simulation, Ca2+ moved into the
cell through the plasma membrane. Plasma membrane Ca2+ influx was described by two
exponentials (see APPENDIX and Table 3) and matched the time course of the
Ca2+ current measured
experimentally by Becker et al. (3) (see also Refs. 15 and 16). The
maximum influx rate was adjusted so that the maximum average
cytoplasmic free Ca2+
concentration reached in a simulation (<1 µM) was typical of the
free Ca2+ level measured in smooth
muscle cells during contractile signaling. A single immobile
intracellular Ca2+ buffer (total
concentration = 250 µM, rate constant for
Ca2+ binding = 108 · M
1 · s
1,
off rate for bound Ca2+ = 102/s,
Kd = 1 µM; see
also APPENDIX) was included in the
model, as were kinetic equations (see
APPENDIX) describing
Ca2+ uptake by the sarcoplasmic
reticulum Ca2+ pump and extrusion
of Ca2+ through the plasma
membrane. As discussed above, the total
Ca2+ buffer concentration used in
the simulations is consistent with estimates reported in the
literature. The rate constants for
Ca2+ binding to the buffer are
typical of those reported for the binding of
Ca2+ to various
Ca2+-binding proteins (26, 29).
The terms describing the velocities of the sarcoplasmic reticulum and
plasma membrane Ca2+ pumps were
adjusted to provide a rate of Ca2+
removal from the cytoplasm (~70 nM/s), in agreement with experimental estimates of the rate at which
Ca2+ is removed from the smooth
muscle cell cytoplasm (60-100 nM/s) (3) and with the percent
contribution of the sarcoplasmic reticulum Ca2+ pump to this removal (75%)
(17). The
Na+/Ca2+
exchanger was not modeled explicitly, but its contribution to Ca2+ removal was assumed to be
equal to that of the plasma membrane Ca2+ pump (see discussion in Ref.
15) and was included in the equation describing the plasma membrane
pump. Release of Ca2+ from the
sarcoplasmic reticulum was not included in the simulations described
below. The diffusion coefficient for
Ca2+ in the cell cytoplasm was 2.2 × 10
6
cm2/s on the basis of the
measurements of Allbritton et al. (1). A barrier to the free diffusion
of Ca2+ located 12.5 nm from the
plasma membrane (Fig. 3A) was
represented in the model as a region (125 nm thick) with a lower
diffusion coefficient (0.22 × 10
6 or 0.022 × 10
6
cm2/s; see below). The presence of
this barrier resulted in the formation of a restricted diffusion space
between the plasma membrane and the barrier. The width of this space in
the model (12.5 nm) is consistent with the spacing between the
sarcoplasmic reticulum and the plasma membrane seen in electron
micrographs of smooth muscle cells (10-20 nm) (11, 13). The
magnitudes and time courses of
Ca2+ transients predicted by the
model [with a diffusion coefficient in the barrier region
(Dbarrier) of
0.22 × 10
6
cm2/s] are shown in Fig.
3B. The free
Ca2+ concentration reached a
maximum of ~10 µM in the restricted diffusion space 15 ms after
influx started (Fig. 3B, trace a).
The free Ca2+ concentration
transients at three locations in the central cytoplasm of the model
cell are also shown in Fig. 3B. The
initial rapid Ca2+ rise seen in
the restricted space was not present in the central cytoplasm (Fig.
3B, traces b-d), but rather a
more gradual increase in free Ca2+
concentration occurred over the time course of the simulation. The
average central cytoplasmic free
Ca2+ concentration in the cell
reached a maximum of ~590 nM for this simulation (result not shown).

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Fig. 3.
A: smooth muscle cell model. Section
of a portion of a smooth muscle cell is shown by light shading. Long
axis of cell is in vertical direction. Areas of restricted diffusion
(darker shading, not drawn to scale) are shown near plasma membrane on
either side of cell. Cell was assumed to be cylindrical, and
Ca2+ influx through plasma
membrane from extracellular space and radial diffusion through cell
cytoplasm are modeled. Regulatory processes incorporated into model are
described in RESULTS.
Ca2+ release by sarcoplasmic
reticulum is not included in model; however, Ca2+ uptake into sarcoplasmic
reticulum is included. Ca2+ pumps
were located at surfaces of restricted diffusion space facing central
cytoplasm and plasma membrane. To calculate central cytoplasmic
[Ca2+]free
in model,
[Ca2+]free
was averaged over volume included in 4.8-µm-diameter cylinder at
center of cell (diameter of central cytoplasm = 80% of cell diameter).
This approximates experimental measurements of
Ca2+ concentration made with a
Ca2+-sensitive dye through a mask
centered over a cell. a-d,
Approximate locations of Ca2+
transients in B.
B:
Ca2+ transients obtained with
model in restricted diffusion space (a) and at 3 cytoplasmic locations
(b-d). For simulation,
diffusion coefficient in barrier region
(Dbarrier) was
0.22 × 10 6
cm2/s. Trace
b was obtained 2.4 µm from center of cell,
trace c 1.5 µm from center of cell,
and trace d at center of cell.
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Additional results from the simulation in Fig.
3B are shown in Fig.
4, along with those obtained for a second
simulation with a lower
Dbarrier. The
permeability of the plasma membrane to
Ca2+ was increased to the same
extent at the start of each simulation. As noted above,
Dbarrier for the
first simulation was 0.22 × 10
6
cm2/s; for the second simulation
Dbarrier was
0.022 × 10
6
cm2/s. Figure
4A compares the free
Ca2+ concentration in the
restricted diffusion space for the two simulations. With
Dbarrier of 0.022 × 10
6
cm2/s, local free
Ca2+ concentration reached ~80
µM in the space. The average cytoplasmic free
Ca2+ concentration for the
simulation with
Dbarrier of 0.022 × 10
6
cm2/s (shown over a 3-s time
interval in Fig. 4D) reached a
maximum of ~600 nM. On the basis of previous work (15, 16), it
appears unlikely that Ca2+
extrusion mechanisms would have a major influence on the time course or
magnitude of the Ca2+ transient
over an initial 100-ms time interval. The time course of the
Ca2+ transients in the restricted
space for simulations without Ca2+
uptake by the sarcoplasmic reticulum included in the model and the
transients with Ca2+ removal
processes included are shown in Fig. 4A. Figure
4B shows the amount of
Ca2+ bound to an immobile
intracellular Ca2+ buffer (see
APPENDIX and Table 3) over the 100-ms
time interval. With the rate constants used in the simulation,
Ca2+ bound very rapidly to the
buffer and was released relatively slowly. Thus neither
Ca2+ removal processes nor
Ca2+ binding to the buffer can
account for the decline in free
Ca2+ concentration with time seen
in Fig. 4A. Instead, this decline is
due primarily to the decrease in plasma membrane permeability and
diffusion of Ca2+ away from the
site of influx. The change in near-membrane
[ATP]free predicted by
the equilibrium equations for the
Ca2+ transients shown in Fig.
4A are shown in Fig.
4C. In these simulations, starting
free Mg2+ concentration was 0.5 mM
and total ATP concentration was 1 mM. During the
Ca2+ transient,
[ATP]free dropped by
~10% when
Dbarrier was 0.22 × 10
6
cm2/s and by ~30% when
Dbarrier was
0.022 × 10
6
cm2/s. [MgATP]
decreased by 7.6 and 9.6%, respectively, and the CaATP-to-MgATP ratio
rose to 0.02 and 0.14, respectively, for the two diffusion coefficients
(results not shown). Figure 4D shows
the average free Ca2+
concentration in the central cytoplasm of the model cell for Dbarrier of 0.022 × 10
6
cm2/s and the corresponding change
in [ATP]free in the
same area. There was only a minimal decrease in
[ATP]free from an
initial level of ~70 µM in the central cytoplasm.

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Fig. 4.
Predicted near-membrane Ca2+
transients and accompanying changes in
[ATP]free in smooth
muscle cells. A: time course of
[Ca2+]free
(solid lines and ) in a restricted diffusion space near plasma
membrane.
Dbarrier values
in model were 0.022 × 10 6
cm2/s (top
trace) and 0.22 × 10 6
cm2/s (bottom
trace). Near-membrane
Ca2+ transient is shown with
(solid lines) and without ( ) sarcoplasmic reticulum
Ca2+ uptake included in model.
B:
Ca2+ bound to intracellular
Ca2+ buffer. Solid line,
Dbarrier = 0.022 × 10 6
cm2/s; dashed line,
Dbarrier = 0.22 × 10 6
cm2/s.
C: changes in near-membrane
[ATP]free calculated
from equilibrium equations for
Dbarrier = 0.022 × 10 6
cm2/s (solid line) and
Dbarrier = 0.22 × 10 6
cm2/s (dashed line).
D: average central cytoplasmic
[Ca2+]free
(left, solid line) and
[ATP]free
(right, dashed line) changes for a
simulation with
Dbarrier = 0.022 × 10 6
cm2/s (average
[Ca2+]free
was calculated as described in
METHODS;
[ATP]free was
calculated from average
[Ca2+]free
curve using equilibrium equations). Time scale in
D is different from time scale in
A-C.
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The results shown in Fig. 4C were
derived from the Ca2+ transients
in Fig. 4A using equilibrium
equations. They indicate that the most significant changes in the
restricted diffusion space were in
[ATP]free and
CaATP-to-MgATP ratio. Because of the dynamic nature of the transient,
however, it might be argued that equilibrium conditions are unlikely to
be met in a cell; therefore, in the results presented below, rate
equations for the binding of Ca2+
to ATP were incorporated into the diffusion model itself (using Eqs. 5 and 6 in the
APPENDIX) to better approximate the
situation in a living cell. To simplify the calculations, only the
binding and release of Ca2+ from
free ATP were modeled, and diffusion of CaATP out of the restricted
space was not included. On the basis of the results obtained with the
equilibrium equations, starting
[ATP]free was set at
70 µM. Two different sets of on and off rates for the binding of
Ca2+ to ATP were used (on rate = 108 · M
1 · s
1,
off rate = 1.7 × 104/s; on
rate = 106 · M
1 · s
1,
off rate = 170/s), both giving a
Kd equal to that
for Ca2+ binding to
ATP4
(log
Kd =
3.77;
Table 1). Figure 5 shows that the magnitude of the change in near-membrane
[ATP]free
predicted using either of the two sets of rate constants was quite
similar to that obtained with the equilibrium calculation. The time
courses of the equilibrium trace and that determined with the faster on
and off rate constants were also similar, although the rate equations
predicted a slightly faster return of
[ATP]free toward its
starting level. The decrease in
[ATP]free seen with
the slower on and off rates was delayed in onset and was slightly lower
in magnitude. One possible reason for the differences between the
results obtained with the equilibrium equations and those obtained with
the faster rate constants is that the equilibrium equations included
binding of Ca2+ to other ionic
species (e.g., HATP3
) and
the release of Mg2+ from the
Ca2+ buffer as
Ca2+ concentration increased,
which were not included in the simplified calculation with the rate
constants. This indeed appeared to be the case. When the equilibrium
equations were modified to include only the binding of
Ca2+ to the intracellular
Ca2+ buffer and to
ATP4
(log
Kd =
3.77,
starting [ATP]free = 70 µM), a trace virtually identical to that calculated with the rate
equations was obtained (Fig. 5).

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|
Fig. 5.
Comparison of predicted changes in near-membrane
[ATP]free calculated
with equilibrium equations or determined from rate constants incorporated into diffusion model. Changes in
[ATP]free were
calculated from near-membrane
[Ca2+]free
transient shown by top trace in Fig.
4A using equilibrium equations (solid
line: starting
[Mg2+]free = 0.5 mM, [ATP]total = 1 mM) or rate constants for Ca2+
binding to ATP (dotted line: on rate = 106 · M 1 · s 1,
off rate = 170/s; dashed line: on rate = 108 · M 1 · s 1,
off rate = 1.7 × 104/s).
Trace shown by was obtained from near-membrane
[Ca2+]free
transient shown by top trace in Fig.
4A and equilibrium equations that
included only Ca2+ binding to
cellular Ca2+ buffer and to a
single species of ATP (starting
[ATP]free = 70 µM,
log Kd = 3.77). For simulations with diffusion model,
Dbarrier = 0.022 × 10 6
cm2/s.
|
|
 |
DISCUSSION |
Our results predict that the changes in
Ca2+ concentration that occur
during signal transduction may significantly alter the free and bound
concentrations of ATP and the CaATP-to-MgATP concentration ratio in
cells. Although during Ca2+
signaling inhomogeneities in Ca2+
concentration are likely to be present throughout cells (15, 16), the
steepest gradients and highest
Ca2+ concentrations occur near
plasma membrane Ca2+ channels or
clusters of channels and near
Ca2+-release sites on the
sarcoplasmic reticulum. In smooth muscle cells, high
Ca2+ concentrations would be
especially prominent in the restricted diffusion spaces that have been
postulated to exist where the sarcoplasmic reticulum is in close
apposition to the plasma membrane. In the simulations shown in Fig. 4,
two diffusion coefficients for
Ca2+ diffusion through a barrier
region were used to predict the
Ca2+ concentrations that might
develop in the restricted diffusion space between this barrier and the
plasma membrane. The higher Dbarrier (0.22 × 10
6
cm2/s, 1/10th of that in the
absence of the barrier) predicted a maximum free
Ca2+ concentration of ~10 µM
in the restricted diffusion space and models diffusion into a
restricted space that has fairly good communication with the central
cytoplasm of the cell. A lower Dbarrier (0.022 × 10
6
cm2/s, 1/100th of that in the
absence of a barrier) predicted a maximum free
Ca2+ concentration of ~80 µM.
A region such as this might be found where influx occurs into a space
where access to the central cytoplasm is limited. Electron micrographs
of smooth muscle cells obtained from longitudinal or transverse
sections can show regions of close contact between the plasma membrane
and the sarcoplasmic reticulum that extend for
1 µm (11).
Ca2+ moving into such a region
would be expected to have very limited access to the rest of the cell
and could reach very high levels. The magnitude of the
Ca2+ transient predicted in our
simulations may indeed be conservative on the basis of the measurements
and predictions made by others for the free
Ca2+ concentrations near single
channels or clusters of Ca2+
channels. Models developed by a number of investigators (23, 28, 29,
31, 33, 34) predict free Ca2+
concentrations of this magnitude or greater near the pore of single
Ca2+ channels, near clusters of
channels, in presynaptic terminals, and in the diadic region of cardiac
muscle cells. Llinás et al. (20) measured 200-300 µM
Ca2+ in the presynaptic terminal
of the squid giant axon.
ATP-dependent reactions near the plasma membrane and/or the
sarcoplasmic reticulum would be the most affected by alterations in the
relative amount of ATP bound to different cations. In addition to
effects on ATP-dependent enzymes in this region, changes in Ca2+ concentration could also
alter the permeability of ATP-sensitive channels. In ventricular
muscle, ATP-sensitive potassium
(KATP) channel permeability is
decreased by an increase in [MgATP] or ATP4
concentration (10,
19). KATP channels are also
activated by nucleotide diphosphates (19). Thus local changes in
[ATP]free, [MgATP], and/or the ADP-to-ATP concentration ratio
induced by changes in Ca2+
concentration could locally alter
KATP channel activity.
Bezprozvanny and Erlich (4) found that MgATP and
Na2ATP, in the presence of
inositol trisphosphate (IP3),
increased the open probability of
IP3-sensitive
Ca2+ channels in planar lipid
bilayers. The effect was maximum at ~1 mM ATP but was apparent at
concentrations as low as 10 µM. At higher concentrations, ATP
inhibited the channels. ATP has also been shown to increase the open
probability of ryanodine receptor channels (32). Hofer et al. (14)
recently reported an effect of ATP concentration on the leak of
Ca2+ through the membrane of the
endoplasmic reticulum of fibroblasts. Thus the presence of high
Ca2+ near the plasma membrane and
the sarcoplasmic or endoplasmic reticulum could alter the permeability
of plasma membrane K+ channels
and/or IP3 receptor,
ryanodine receptor, and Ca2+ leak
channels in the sarcoplasmic reticulum. This could result in changes in
plasma membrane permeability, membrane potential, or the efficiency of
Ca2+ uptake into intracellular
storage sites.
The effects of a decrease in [MgATP] in the presence of a
local high Ca2+ concentration
would depend on the concentration requirements of the enzymes near the
high-Ca2+ site. The most important
effect of Ca2+, however, might be
on the CaATP-to-MgATP ratio. Our results indicate that a significant
increase in this ratio is likely to occur in cells as
Ca2+ concentration increases.
Because MgATP appears to be the preferred substrate for most
ATP-dependent processes in cells, an increased availability of CaATP
could have an important inhibitory influence on enzymes that can bind
but cannot use CaATP efficiently as a substrate.
The results of this study also suggest that the ability of
intracellular Ca2+ buffers to also
bind Mg2+, but with a lower
affinity than they bind Ca2+,
could stabilize [MgATP] in the presence of changes of free
Ca2+ concentration from 50 nM to
~10 µM. Given a
Kd Ca of 1 µM for the buffer,
Kd Mg-to-Kd Ca
ratios of 100-1,000 would provide the cell with relatively stable
[MgATP] over the greatest range of free
Ca2+ concentrations. These
Kd ratios are
generally consistent with those reported by Robertson et al. (26) for
the binding of Ca2+ and
Mg2+ to intracellular proteins.
Because the precise nature of the Ca2+ buffers in cells and their
capacities remains the subject of ongoing research, the role of these
buffers as regulators of Mg2+ and
MgATP levels remains to be determined. In the present work, all the
cellular buffers were considered as a single immobile buffer. It is
more realistic, however, to assume that different proteins would have
different relative affinities for
Ca2+ and
Mg2+ and that mobile and immobile
buffers are present (see discussions in Refs. 28, 29, 34, and 35)
Different buffers are also likely to be localized to different cellular
regions. Relatively high concentrations of
Mg2+ associated with buffers on or
near the plasma membrane of cells might tend to better stabilize
[MgATP] in these regions, provided Ca2+ is able to displace
Mg2+ from these buffers.
In considering the implications of the present study, it is important
to keep in mind that changes in
Ca2+ concentration may be highly
localized in cells. Ca2+ signaling
is also a dynamic process, and the high
Ca2+ concentrations and steep
gradients achieved near sites of
Ca2+ influx and/or release
may be present only for short periods of time before they are
dissipated by diffusion, extrusion, and sequestration. Comparison of
Figs. 3-5, however, indicates that high
Ca2+ concentrations of short
duration, such as those predicted by our models, could significantly
alter local ATP concentrations. The results obtained with the
equilibrium equations and those obtained with the two sets of rate
constants were qualitatively and quantitatively quite similar. The
equilibrium equations, which can provide more detailed information
about specific ionic species, appear able to provide a good starting
point for evaluating questions such as those posed in this study. The
simplifying assumption made in the present simulations that the
cytoplasmic Ca2+ buffers and ATP
were immobile did not allow us to explore the effects of the diffusion
of Ca2+-bound species on local
Ca2+ or ATP concentrations. On the
basis of the work of others (29, 35), it is, nevertheless, possible to
make some qualitative predictions. One would expect the diffusion of
CaATP away from the restricted space to increase the concentration of
this species and Ca2+ at more
central sites. Because of this, free
Ca2+ concentration in the
restricted space would also be expected to decline somewhat faster than
predicted by our model. The diffusion of unbound ATP back into the
restricted space would tend to bring [ATP]free toward its
resting level. This effect would be partially offset, however, by
additional Ca2+ binding to ATP.
Using Ca2+ and buffer
concentrations and other parameters quite similar to those used in our
model, Smith et al. (29) predicted that a mobile
Ca2+ buffer [diffusion
coefficient for the buffer
(Dbuffer) = 0.75 × 10
6
cm2/s] near a
Ca2+ pore would be ~15% less
saturated than immobile buffers near the pore. We would, therefore,
expect that the magnitude of the decrease in
[ATP]free would be
slightly less than that shown in Fig. 5. Zhou and Neher (35) and Smith
et al. (29) estimated that the presence of a mobile buffer
(mobile-to-immobile buffer concentration ratio
1:4;
Dbuffer
0.6 and 0.7 × 10
6
cm2/s) can increase the effective
diffusion coefficient for Ca2+ by
a factor of 2-3. On the basis of these results, the rate of return
of [ATP]free toward
baseline levels obtained with the rate equations (dashed line in Fig.
5) might be a good approximation of the rate when diffusion of CaATP
away from the restricted space is considered.
Although further experimental and theoretical work is required to
better determine the precise magnitudes and time courses of the changes
in Ca2+ concentration,
[MgATP],
[ATP]free,
CaATP-to-MgATP concentration ratio, and ADP-to-ATP ratio, the present
study raises a number of interesting possibilities concerning ways in
which these changes could fine tune regulatory processes in cells. This
regulation would be especially significant near the plasma membrane and
near the sarcoplasmic reticulum in muscle cells. The results obtained in this study, however, are quite general and should be applicable to a
number of other cell types that rely on
Ca2+-dependent signal transduction
events.
 |
APPENDIX |
In Eqs. 1-10, Ca denotes free
Ca2+ concentration.
The time rate of change of Ca2+
due to radial diffusion and the uptake and release of
Ca2+ by the
Ca2+ regulatory processes in a
cylindrical cell at radial position r
can be described by the following equation
|
(1)
|
The
first term on the right in Eq. 1 is
derived from Fick's laws of diffusion for radial diffusion in a medium
with cylindrical symmetry. The second term on the right,
S(Ca, r,
t), includes the
Ca2+ concentration-, position-,
and time-dependent processes (described below) that act as sources or
sinks of Ca2+ in the cell. In the
work described here, only radial diffusion was modeled. This assumes
that there was no dependence of
Ca2+ concentration on the axial
(z) or angular (
) dimensions in the region modeled. A numerical solution for the diffusion part of
Eq. 1 was obtained using the explicit
finite differences method described by Crank (6). For this solution,
the cell was divided into concentric annuli of thickness
r. The rate of change of Ca2+ due to diffusion (diffusion
coefficient, D) into and out of the ith annulus is
|
(2)
|
|
(3)
|
Equations 2 and 3 were applied for diffusion within
the restricted diffusion space near the plasma membrane, the barrier region (where the diffusion coefficient was lower), and the central cytoplasm of the model cell diagramed in Fig. 3. As discussed by Crank,
finite differences solutions agree with analytic solutions of the
diffusion equation, provided
D
t/(
r)2 < 0.5. This condition was met by adjusting the time interval (
t) for the calculations once
D and
r were chosen for the cytoplasmic and barrier spaces. As noted, diffusion within each of the spaces shown
in Fig. 3 could be calculated using Eqs.
2 and 3; however, because the diffusion coefficient was lower in the barrier region, it
was necessary to include expressions describing the movement of
Ca2+ across the boundaries between
the restricted space and the barrier region and between the barrier
region and the central cytoplasm, where the diffusion coefficient
changed. To do this, the method described by Crank for determining the
diffusion of a substance through the boundary separating regions with
different diffusion coefficients was used. For radial diffusion in a
cylinder, the change in Ca2+ at
the boundary between two such regions is
|
(4)
|
where
Dm and
rm are the
diffusion coefficient and thickness of the annuli on the central side
of the boundary, respectively, and
Dn and
rn are the
equivalent parameters on the other side of the boundary. The number
m is the number of annuli of thickness
rm between
the center of the cell and the boundary, and
n is the number of annuli of thickness
rn between
the center of the cell and the boundary (i.e., if
b is the radial distance between the
center of the cell and the boundary, b = m
rm = n
rn).
[The method for deriving this equation for radial diffusion in a
cylinder of composite media is given by Crank (6); details of the
derivation are provided in the first edition of the same volume.]
As was done previously (15, 16), intracellular
Ca2+ buffers were treated as a
single immobile buffer distributed uniformly throughout the cell.
Buffering of Ca2+ by the
intracellular buffer was described by the following equations
|
(5)
|
|
(6)
|
where
Kon and
Koff are the rate
constants for the binding of Ca2+
to free buffer and the release of bound
Ca2+ from the buffer,
respectively. Similar equations were used to describe the binding of
Ca2+ to free ATP for the
simulations shown in Fig. 5. Values for the rate constants used in the
simulations are given in Table 3.
Ca2+ influx into the cell through
the plasma membrane was described by the following equations
|
(7)
|
|
(8)
|
The
time constants
ton and
toff (Table 3)
were adjusted so that the time course of
Ca2+ influx in the model cell
matched the time course of the
Ca2+ current measured by Becker et
al. (3) in voltage-clamped smooth muscle cells. The constant
K0 was adjusted
so that the average central cytoplasmic free
Ca2+ concentration reached in the
model cell was <1 µM (in agreement with the free
Ca2+ levels measured
experimentally in smooth muscle cells during Ca2+ transients).
Extrusion of Ca2+ out of the model
cell was described by the Hill equation
|
(9)
|
where
Vmax is the
maximum velocity of extrusion,
Km is the
Ca2+ concentration at half-maximal
velocity, and n is the Hill
coefficient. The values for
Km are given in
Table 3 and are consistent with the values reported in the literature
(see discussion in Ref. 16).
Vmax was adjusted
as described in RESULTS. To balance
the Ca2+ efflux across the plasma
membrane when the cell was at rest, an inward
Ca2+ leak was included in the
simulations so that no net removal of Ca2+ from the cytoplasm occurred.
The leak was described by the following equation
|
(10)
|
and
the leak constant
(Kleak)
was adjusted so that the Ca2+
influx resulting from the leak was equal to the resting extrusion of
Ca2+ through the plasma membrane
(determined from Eq. 9 with 150 nM cytoplasmic Ca2+). A similar
pair of equations was used to describe
Ca2+ uptake by the sarcoplasmic
reticulum and a leak out of the sarcoplasmic reticulum membrane when
these elements were included in the model. For the sarcoplasmic
reticulum uptake and leak, extracellular Ca2+
(Caextracellular) in
Eq. 10 was replaced by the free
Ca2+ concentration in the
sarcoplasmic reticulum (1.5 mM) and
Kleak was
adjusted (as described above) to balance resting sarcoplasmic reticulum
Ca2+ uptake. The values used for
Km and
n for sarcoplasmic reticulum Ca2+ uptake are given in Table 3.
Vmax was adjusted
as described in RESULTS. The equations
describing Ca2+ influx through the
plasma membrane, Ca2+ buffering,
extrusion out of the cell, and the inward leak of Ca2+ through the plasma membrane
were included in all simulations. As described in
RESULTS, uptake of
Ca2+ by the sarcoplasmic reticulum
was included in the simulations except when the contribution of
sarcoplasmic reticulum Ca2+ uptake
to the time course of the near-membrane
Ca2+ transient was examined (Fig.
4A,
). Sarcoplasmic reticulum
uptake sites were located on both surfaces of the barrier region.
 |
ACKNOWLEDGEMENTS |
This work was supported by grants from the Heart and Stroke
Foundation of Alberta and the Medical Research Council of Canada to G. J. Kargacin and a grant from the Heart and Stroke Foundation of Alberta
to M. E. Kargacin. G. J. Kargacin is an Alberta Heritage Foundation for
Medical Research Scholar.
 |
FOOTNOTES |
Address for reprint requests: M. E. Kargacin, Dept. of Physiology and
Biophysics, University of Calgary, 3330 Hospital Dr. NW, Calgary, AB,
Canada T2N 4N1.
Received 9 January 1997; accepted in final form 2 June 1997.
 |
REFERENCES |
1.
Allbritton, N.,
L. T. Meyer,
and
L. Stryer.
Range of messenger action of calcium ion and inositol 1,4,5-trisphosphate.
Science
258:
1812-1815,
1992[Medline].
2.
Allen, D. G.,
P. G. Morris,
C. H. Orchard,
and
J. S. Pirolo.
A nuclear magnetic resonance study of metabolism in the ferret heart during hypoxia and inhibition of glycolysis.
J. Physiol. (Lond.)
361:
185-204,
1985[Abstract].
3.
Becker, P. L.,
J. J. Singer,
J. V. Walsh, Jr.,
and
F. S. Fay.
Regulation of calcium concentration in voltage-clamped smooth muscle cells.
Science
244:
211-214,
1989[Medline].
4.
Bezprozvanny, I.,
and
B. E. Erlich.
ATP modulates the function of inositol 1,4,5-trisphosphate-gated channels at two sites.
Neuron
10:
1175-1184,
1993[Medline].
5.
Corkey, B. E.,
J. Duszynski,
T. L. Rich,
B. Matschinsky,
and
J. R. Williamson.
Regulation of free and bound magnesium in rat hepatocytes and isolated mitochondria.
J. Biol. Chem.
261:
2567-2574,
1986[Abstract/Free Full Text].
6.
Crank, J.
The Mathematics of Diffusion. New York: Oxford University Press, 1975.
7.
DeFelice, L. J.
Molecular and biophysical view of the Ca channel: a hypothesis regarding oligomeric structure, channel clustering and macroscopic current.
J. Membr. Biol.
133:
191-202,
1993[Medline].
8.
Etter, E. F.,
A. Minta,
M. Poenie,
and
F. S. Fay.
Near-membrane [Ca2+] transients resolved using the Ca2+ indicator FFP18.
Proc. Natl. Acad. Sci. USA
93:
5368-5373,
1996[Abstract/Free Full Text].
9.
Fabiato, A.
Myoplasmic free Ca2+ in mammalian cardiac cells.
J. Gen. Physiol.
78:
457-497,
1981[Abstract].
10.
Findlay, I.
ATP4
and ATP-Mg inhibit the ATP-sensitive K+ channel of rat ventricular myocytes.
Pflügers Arch.
412:
37-41,
1988[Medline].
11.
Gabella, G.
Structure of smooth muscles.
In: Smooth Muscle: An Assessment of Current Knowledge, edited by E. Burbring,
A. F. Brading,
A. W. Jones,
and T. Tomoita. Austin, TX: University of Texas Press, 1983, p. 1-46.
12.
Hardin, C. D.,
and
R. J. Paul.
Metabolism and energetics of vascular smooth muscle.
In: Physiology and Pathophysiology of the Heart (3rd ed.), edited by N. Sperelakis. Boston, MA: Kluwer Academic, 1995, p. 1069-1086.
13.
Hartshorne, D. J.
Biochemistry of the contractile process in smooth muscle.
In: Physiology of the Gastrointestinal Tract (2nd ed.), edited by L. R. Johnson. New York: Raven, 1987, p. 423-482.
14.
Hofer, A. M.,
S. Curci,
T. E. Machen,
and
I. Schulz.
ATP regulates calcium leak from agonist-sensitive internal calcium stores.
FASEB J.
10:
302-308,
1986[Abstract/Free Full Text].
15.
Kargacin, G. J.
Calcium signaling in restricted diffusion spaces.
Biophys. J.
67:
262-272,
1994[Abstract].
16.
Kargacin, G. J.,
and
F. S. Fay.
Ca2+ movement in smooth muscle cells studied with one- and two-dimensional diffusion models.
Biophys. J.
60:
1088-1100,
1991[Abstract].
17.
Kargacin, M. E.,
and
G. J. Kargacin.
Direct measurement of Ca2+ uptake and release by the sarcoplasmic reticulum of saponin permeabilized isolated smooth muscle cells.
J. Gen. Physiol.
106:
467-484,
1995[Abstract].
18.
Kushmerick, M. J.,
P. F. Dillon,
R. A. Meyer,
T. R. Brown,
J. M. Krisanda,
and
H. L. Sweeney.
31P NMR spectroscopy, chemical analysis and free Mg2+ of rabbit bladder and uterine smooth muscle.
J. Biol. Chem.
261:
14420-14429,
1986[Abstract/Free Full Text].
19.
Lederer, W. J.,
and
C. G. Nichols.
Nucleotide modulation of the activity of rat heart at sensitive K+ channels in isolated membrane patches.
J. Physiol. (Lond.)
419:
193-211,
1989[Abstract].
20.
Llinás, R.,
M. Sugimori,
and
R. B. Silver.
Microdomains of high calcium concentration in a presynaptic terminal.
Science
256:
677-679,
1992[Medline].
21.
Martell, A. E.,
and
R. M. Smith.
Critical Stability Constants. New York: Plenum, 1982, vol. 5, suppl. 1.
22.
Murphy, E.
Cellular magnesium and Na/Mg exchange in heart cells.
Annu. Rev. Physiol.
53:
273-287,
1991[Medline].
23.
Peskoff, A.,
J. A. Post,
and
G. A. Langer.
Sarcolemmal calcium binding sites in heart. II. Mathematical model for diffusion of calcium released from the sarcoplasmic reticulum into the diadic region.
J. Membr. Biol.
129:
59-69,
1992[Medline].
24.
Quamme, G. A.,
D. Long-Jun,
and
S. W. Rabkin.
Dynamics of intracellular free Mg2+ changes in a vascular smooth muscle cell line.
Am. J. Physiol.
265 (Heart Circ. Physiol. 34):
H281-H288,
1993[Abstract/Free Full Text].
25.
Rizzuto, R.,
M. Brini,
M. Murgia,
and
T. Pozzan.
Microdomains with high Ca2+ close to IP3-sensitive channels that are sensed by neighboring mitochondria.
Science
262:
744-747,
1993[Medline].
26.
Robertson, S. P.,
J. D. Johnson,
and
J. D. Potter.
The time-course of Ca2+ exchange with calmodulin, troponin, parvalbumin and myosin in response to transient increases in Ca2+.
Biophys. J.
34:
559-569,
1981[Abstract].
27.
Sala, F.,
and
A. Hernández-Cruz.
Calcium diffusion modeling in a spherical neuron.
Biophys. J.
57:
313-324,
1990[Abstract].
28.
Smith, G. D.
Analytical steady-state solution to the rapid buffering approximation near an open Ca2+ channel.
Biophys. J.
71:
3064-3072,
1996[Abstract].
29.
Smith, G. D.,
J. Wagner,
and
J. Keizer.
Validity of the rapid buffering approximation near a point source of calcium ions.
Biophys. J.
70:
2527-2539,
1996[Abstract].
30.
Smith, R. M.,
and
A. E. Martell.
Critical Stability Constants. Amines. New York: Plenum, 1975, vol. 2.
31.
Smith, S. J.,
and
G. J. Augustine.
Calcium ions, active zones and synaptic transmitter release.
Trends Neurol. Sci.
11:
458-464,
1988. [Medline]
32.
Smith, J. S.,
T. Imagawa,
J. Ma,
M. Fill,
K. P. Campbell,
and
R. Coronado.
Purified ryanodine receptor from rabbit skeletal muscle is the calcium-release channel of sarcoplasmic reticulum.
J. Gen. Physiol.
92:
1-26,
1988[Abstract].
33.
Stern, M. D.
Buffering of calcium in the vicinity of a channel pore.
Cell Calcium
13:
183-192,
1992[Medline].
34.
Wagner, J.,
and
J. Keizer.
Effects of rapid buffers on Ca2+ diffusion and Ca2+ oscillations.
Biophys. J.
67:
447-456,
1994[Abstract].
35.
Zhou, Z.,
and
E. Neher.
Mobile and immobile calcium buffers in bovine adrenal chromaffin cells.
J. Physiol. (Lond.)
469:
245-273,
1993[Abstract].
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