From the Department of Physiology, University of Pennsylvania School of Medicine, Philadelphia, Pennsylvania 19104-6085
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ABSTRACT |
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Cannell and Allen (1984. Biophys. J. 45:913-925) introduced the use of a multi-compartment model
to estimate the time course of spread of calcium ions (Ca2+) within a half sarcomere of a frog skeletal muscle fiber
activated by an action potential. Under the assumption that the sites of sarcoplasmic reticulum (SR) Ca2+ release
are located radially around each myofibril at the Z line, their model calculated the spread of released Ca2+ both
along and into the half sarcomere. During diffusion, Ca2+ was assumed to react with metal-binding sites on parvalbumin (a diffusible Ca2+- and Mg2+-binding protein) as well as with fixed sites on troponin. We have developed a
similar model, but with several modifications that reflect current knowledge of the myoplasmic environment and SR Ca2+ release. We use a myoplasmic diffusion constant for free Ca2+ that is twofold smaller and an SR Ca2+ release function in response to an action potential that is threefold briefer than used previously. Additionally, our
model includes the effects of Ca2+ and Mg2+ binding by adenosine 5'-triphosphate (ATP) and the diffusion of
Ca2+-bound ATP (CaATP). Under the assumption that the total myoplasmic concentration of ATP is 8 mM and
that the amplitude of SR Ca2+ release is sufficient to drive the peak change in free [Ca2+] ([Ca2+]) to 18 µM
(the approximate spatially averaged value that is observed experimentally), our model calculates that (a) the spatially averaged peak increase in [CaATP] is 64 µM; (b) the peak saturation of troponin with Ca2+ is high along the
entire thin filament; and (c) the half-width of
[Ca2+] is consistent with that observed experimentally. Without
ATP, the calculated half-width of spatially averaged
[Ca2+] is abnormally brief, and troponin saturation away
from the release sites is markedly reduced. We conclude that Ca2+ binding by ATP and diffusion of CaATP make
important contributions to the determination of the amplitude and the time course of
[Ca2+].
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INTRODUCTION |
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During normal activation of a skeletal muscle fiber, an
action potential in the transverse tubular membranes
triggers the opening of Ca2+ release channels in the sarcoplasmic reticulum (SR).1 The released Ca2+ produces
an increase in the myoplasmic free [Ca] ([Ca2+]),
which activates the fiber's contractile response.
The SR calcium release channels ("ryanodine receptors") are found primarily at triadic junctions, where
the transverse tubules and the terminal cisternae membranes of the SR are closely apposed. In frog fibers, the
triadic junctions are located primarily at the Z lines of
the sarcomeres and surround each myofibril with a geometry that approximates an annulus (Peachey, 1965).
With this anatomical arrangement, intra-sarcomeric
gradients in myoplasmic free [Ca2+] are expected
when Ca2+ release is active. An understanding of these
gradients and the associated movements of Ca2+ is important in the interpretation of spatially averaged Ca2+
measurements of the type that have been made with a
variety of Ca2+ indicators. They are also important in
the interpretation of local Ca2+ measurements of the
type that have been made recently with high-affinity indicators and confocal microscopy (Escobar et al., 1994
;
Tsugorka et al., 1995
; Klein et al., 1996
).
Cannell and Allen (1984) were the first to use a computer model of a half-sarcomere to estimate the binding and diffusion of Ca2+ after its release at the Z line
in response to an action potential. A principal motivation was to compare the model predictions about the
amplitude and time course of
[Ca2+] with measurements of
[Ca2+] that had been obtained from frog
single fibers injected with the indicator aequorin. In
this article, we describe a similar computer model developed from a similar motivation. In comparison with
Cannell and Allen (1984)
, our model incorporates
three significant differences about the myoplasmic environment and the SR Ca2+ release process.
First, we assume a twofold smaller diffusion constant
for myoplasmic free Ca2+ (3 × 106 cm2 s
1 at 16°C vs. 7 × 10
6 cm2 s
1 at 20°C). This difference is based on the
finding that the viscosity of myoplasm is approximately
twofold higher than that of a simple salt solution (Kushmerick and Podolsky, 1969
; Maylie et al., 1987a
,b,c).
Second, the temporal waveform that we assume for
SR Ca2+ release in response to an action potential
(half-width, 1.9 ms at 16°C) is approximately threefold
briefer than that assumed by Cannell and Allen (1984)
(half-width, 5.8 ms at 20°C). This difference derives
from measurements of spatially averaged
[Ca2+] in
frog fibers injected with lower-affinity Ca2+ indicators
such as purpurate-di-acetic acid (PDAA; Southwick and
Waggoner, 1989
) or furaptra (Raju et al., 1989
). These
indicators, which appear to track
[Ca2+] in skeletal
muscle with 1:1 stoichiometry and little or no kinetic delay (Hirota et al., 1989
; Konishi et al., 1991
; Zhao et al., 1996
), report Ca2+ signals that are substantially briefer
than estimated with aequorin (Cannell and Allen, 1984
).
Consequently, estimates of SR Ca2+ release with these
indicators (Maylie et al., 1987b
; Baylor and Hollingworth, 1988
; Hollingworth et al., 1992
, 1996
), which to date have been based on spatially averaged models
(e.g., Baylor et al., 1983
), are substantially briefer than
assumed by Cannell and Allen (1984)
.
Third, we include the reactions of Ca2+ and Mg2+
with ATP, which is present in the myoplasm of skeletal
muscle at millimolar concentration (probably 5-10 mM
in a rested fiber; Kushmerick, 1985; Godt and Maughan,
1988
; Thompson and Fitts, 1992
). Although the fraction of ATP in the Mg2+-bound form (MgATP) at rest is
expected to be large (~0.9) at the free [Mg2+] level of
myoplasm (see RESULTS), the ATP reaction kinetics
(Eigen and Wilkins, 1965
) are such that a significant
rise in the concentration of Ca2+ bound to ATP
(
[CaATP]) is predicted during activity. Furthermore, ATP is sufficiently small (mol wt, ~500), with an expected myoplasmic diffusion constant of ~1.4 × 10
6
cm2 s
1 at 16°C (Kushmerick and Podolsky, 1969
), that
a significant transport of Ca2+ along the sarcomere in
the CaATP form should occur. This transport of Ca2+
by CaATP appears to permit a more uniform and synchronous binding of Ca2+ to troponin along the thin
filament. These effects of ATP in skeletal muscle point
to a likely role of ATP in the shaping of local Ca2+ gradients in other cells (cf., Zhou and Neher, 1993
; Kargacin and Kargacin, 1997
).
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MATERIALS AND METHODS |
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Overview of the Multi-compartment Model
Our computational model is similar in principle to that of Cannell
and Allen (1984). We divide the myoplasmic space corresponding to a half-sarcomere of one myofibril into a number of compartments that have equal volume and radial symmetry (cf., Fig. 1,
where there are six longitudinal by three radial compartments).
Within each compartment, appropriate metal-binding sites for
Ca2+ and Mg2+ are included at the total concentrations and with
the diffusion constants listed in Table I, B and C (described below). Resting occupancies of the sites by Ca2+ and Mg2+ are based
on appropriately chosen values of dissociation constants (Kd,Ca for
Ca2+, Kd,Mg for Mg2+) and resting levels of free [Ca2+] and free
[Mg2+]. The time-dependent calculation is initiated by the introduction of a finite amount of total Ca2+, with an appropriate time
course, into the compartment comprising the outermost annulus
nearest the Z line (corresponding to the location of the SR release
sites; see Fig. 1, downward arrow). The calculation is advanced in
time by simultaneous integration of the first-order differential
equations for the concentration changes of the various species
(free Ca2+; metal-free and metal-bound sites) in all compartments.
For the integration, it is assumed that: (a) Ca2+ and Mg2+ react
with available binding sites according to the law of mass action;
and (b) the various species move by the laws of diffusion across any
immediately adjacent compartment boundary. Additionally, Mg2+
is assumed to be well buffered, so that possible changes in free [Mg2+] are neglected in all compartments.
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In each compartment, the binding steps are governed by a mass-action reaction of the type illustrated here for Ca2+:
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Site and CaSite denote the metal-free and Ca2+-bound forms
of the site, respectively, and k+1 and k1 denote the on- and off-rate constants, respectively, for the reaction. The corresponding
functional form used in the integration is:
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(1) |
where [Ca2+] denotes the free Ca2+ concentration ([Ca2+] + resting [Ca2+]). For the sites that bind Mg2+ (e.g., parvalbumin;
cf., Johnson et al., 1981
; Gillis et al., 1982
; Baylor et al., 1983
), an
analogous equation for Mg2+ is included in each compartment.
The diffusion of each species across each internal compartment boundary is calculated with an approximation from Fick's law, illustrated here for Ca2+:
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(2) |
A denotes the area of the boundary, and D denotes the relevant
diffusion constant; "CaSpecies" denotes either free Ca2+ or one of
the Ca2+-binding species listed in Table I; "[CaSpecies]" denotes
the difference in species concentration for the two compartments
on either side of the boundary being crossed; and
x denotes the center-to-center distance between the two compartments. The
number of boundaries varies from two to four per compartment,
according to the compartment's location (see Fig. 1). Ca2+'s
spread within the half-sarcomere thus occurs both as diffusion of
the free ion and as diffusion of Ca2+ bound to mobile sites (parvalbumin, ATP, and indicator but not troponin). For the integration,
the number of moles of each species that moves into or out of each
compartment per unit time is divided by the compartment volume
to determine the effect of diffusion on the change in concentration of that species in that compartment per unit time.
The removal of Ca2+ from the half-sarcomere is assumed to take
place only from the outermost compartments (see Fig. 1, upward arrows). This corresponds to the location of the longitudinal membranes of the SR, which extend from Z line to Z line at the periphery
of a myofibril (Peachey, 1965) and contain calcium ATPase molecules (Ca2+ pumps) at a high density (Franzini-Armstrong, 1975
).
In each compartment, a mass conservation equation is used to
track the change in total Ca2+ concentration in that compartment (denoted [CaT]), equal to the change in compartment
Ca2+ concentration due to SR release (if any) minus that due to
SR pumping (if any) minus the net change in concentrations due
to diffusive movements out of the compartment of free Ca2+,
Ca2+ bound to parvalbumin, and Ca2+ bound to ATP. The
[Ca2+] level in each compartment (for use in Eq. 1) is calculated
as the
[CaT] of the compartment minus the change in compartment concentrations of Ca2+ bound to troponin, parvalbumin,
and ATP (denoted
[CaTrop],
[CaParv], and
[CaATP], respectively). If the maximum removal rate by the Ca2+ pump is set
to zero, the mass equations provide a check on the accuracy of
the calculation, since the values of
[CaT], if summed over all
compartments, should then equal the integral of the Ca2+ release
waveform (Eq. 3, described below) after referral of both quantities to the total myoplasmic volume. This check of the model was
satisfied at the level of a fraction of one percent.
Parameters of the Model
Table I gives general information about the model, including
the standard dimensions of the half-sarcomere and the most
common choice for the number of longitudinal and radial subdivisions. Part B lists the spatial locations of the different metal-binding species and their diffusion constants. In all cases, metal-free and metal-bound diffusion constants are assumed to be identical. The troponin sites are assumed to be fixed because of their
attachment to the thin filaments, which in a frog twitch fiber extend 1.0 µm away from the Z line (Page and Huxley, 1963). The
other values of the diffusion constants are half those estimated to
apply to free solution at 16°C, since the viscosity of myoplasm appears to be about twice that of free solution (Kushmerick and
Podolsky, 1969
; Maylie et al., 1987a
,b,c).
Table I lists the assumed concentrations and reaction rate
constants for the metal-binding sites on troponin, parvalbumin, and ATP. The values assumed for ATP are explained in the next section. The values for troponin are taken from "model 2" of Baylor et al. (1983), modified slightly as described in Baylor and
Hollingworth (1988)
. The values for parvalbumin are also taken
from "model 2" of Baylor et al. (1983)
, but with two changes. The
value for the total concentration of metal sites on parvalbumin is
1,500 rather than 1,000 µM, which reflects a more recent estimate for frog twitch fibers (Hou et al., 1991
). The value assumed
for the parvalbumin on-rate for Ca2+ is threefold smaller than
that assumed by Baylor et al. (1983)
. This latter change is related
to our assumption that resting free [Ca2+] is 0.1 µM (cf., Kurebayashi et al., 1993
; Harkins et al., 1993
; Westerblad and Allen,
1996
) rather than the fivefold smaller value assumed by Baylor et al.
(1983)
. There is uncertainty in the values of the parvalbumin reaction rates (Johnson et al., 1981
; Ogawa and Tanokura, 1986
),
and if the Ca2+-parvalbumin on-rate assumed by Baylor et al.
(1983)
is used, the fraction of the parvalbumin sites bound with
Ca2+ at a resting [Ca2+] of 0.1 µM is quite large (0.676). This
large fraction decreases somewhat the ability of parvalbumin to
accelerate the rate of decline of
[Ca2+] after the termination of
release. In any event, a threefold variation in the value assumed
for the Ca2+-parvalbumin on-rate had only minor effects on the
calculations (see RESULTS).
Table I also gives the values of Kd (dissociation constant, calculated as k1/k+1). Part D lists the fractional occupancies of the
metal sites in the resting state, as calculated from the values of Kd
and the values assumed for resting [Ca2+] and [Mg2+].
The Reactions of Ca2+ and Mg2+ with ATP
The competitive reaction of ATP with Ca2+ and Mg2+ is summarized as follows:
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For ATP4, Eigen and Wilkins (1965)
report values of k+1 and
k+2 of > 109 M
1 s
1 and 1.3 × 107 M
1 s
1, respectively (25°C;
ionic strength, 0.1-0.2 M), whereas under similar conditions the
values of Kd,Ca (= k
1/k+1) and Kd,Mg (= k
2/k+2) are approximately 60 µM and 30 µM, respectively (Botts et al., 1965
; Phillips
et al., 1966
). Thus, k
1 and k
2 are calculated to be >60,000 s
1
and 390 s
1, respectively. At 16°C and a viscosity of 2 cP (i.e., appropriate to the model conditions), reaction rates would be
smaller, with k
1 and k
2 values of perhaps 30,000 s
1 and 150 s
1,
respectively. Moreover, at the pH (~7) and K+ concentration
(~140 mM) of myoplasm, the effective values of Kd,Ca and Kd,Mg
are elevated because of partial binding of K+ and H+ to ATP4
.
Under these conditions, we estimate that Kd,Ca and Kd,Mg are ~200 and ~100 µM, respectively (Botts et al., 1965
; Phillips et al.,
1966
; Martell and Smith, 1974
). Thus, in the model, the values assumed for k+1 (= k
1/Kd) and k+2 (= k
2/Kd) are 1.5 × 108 M
1
s
1 and 1.5 × 106 M
1 s
1, respectively.
Given these reaction rates, single-compartment (i.e., spatially
homogeneous) calculations were carried out to estimate the kinetic response of the ATP reactions if driven by a substantial Ca2+
transient. The total concentration of ATP was assumed to be 8 mM,
a value near the middle of the range of values recently reported for fast-twitch fibers, 5-10 mM (Kushmerick, 1985; Godt and
Maughan, 1988
; Thompson and Fitts, 1992
). (Note: As for the
other species of this article, the ATP concentration is referred to
the myoplasmic water volume; see Baylor et al., 1983
; Godt and
Maughan, 1988
.) The free [Mg2+] was assumed to be 1 mM and
constant.
Fig. 2 shows the responses of Schemes B and C if driven simultaneously by a [Ca2+] of peak amplitude 18.0 µM, a time-to-peak of 2.90 ms, and half-width of 5.90 ms, i.e., similar to that expected for the spatially averaged
[Ca2+] of a single myofibril
(see RESULTS). The
[CaATP] response (upper trace) has a time-to-peak of 2.98 ms and a half-width of 6.09 ms; as a waveform, it is
virtually indistinguishable from that of
[Ca2+] (not shown). The
amplitude of
[CaATP], however, at 63.9 µM, is 3.6-fold larger
than that of
[Ca2+]. The factor 3.6 comes from the ratio of total
[ATP] (8 mM) to the effective value of Kd,Ca in the presence of 1 mM
free [Mg2+] (2.2 mM = the actual Kd,Ca of 200 µM times the factor {1 + [Mg2+]/Kd,Mg}; see Scheme B). Fig. 2 shows that, on a
millisecond time scale, ATP behaves as a rapid and linear Ca2+
buffer, with the concentration of Ca2+ bound to ATP being
nearly fourfold larger than that of free [Ca2+].
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The lower trace in Fig. 2 shows the [MgATP] response for
the same calculation; the peak change is
53.4 µM. With a time-to-peak of 3.73 ms and half-width of 6.96 ms, the
[MgATP]
waveform also closely tracks
[Ca2+], although not quite as faithfully as does
[CaATP]. Because ATP transiently releases ~53
µM total Mg2+, an increase in free [Mg2+] would occur if the solution were not well buffered for Mg2+. In the case of myoplasm,
any Mg2+ released by ATP would be buffered by phospho-creatine (primarily), which would limit the increase in free [Mg2+] to
about one-third the increase in total [Mg2+] (e.g., Baylor et al.,
1985
). Thus, in myoplasm, spatially averaged free [Mg2+] would
remain nearly constant, rising by only ~2% relative to the resting
level of 1 mM.
Since the [CaATP] response in Fig. 2 is fast and linear and
the implied increase in myoplasmic free [Mg2+] is small, the
[CaATP] response can be closely approximated by an equivalent reaction (termed here the "reduced" reaction), which omits
consideration of
[MgATP]:
For this reaction, it is assumed that k1 has the same value as does
Scheme B, but that k'+1 is 11-fold smaller than k+1, 1.36 × 107
M
1 s
1 (= 1.5 × 108 M
1 s
1/11). This decrease reflects the assumption that resting free [Mg2+] is 1 mM (10-fold higher than
Kd,Mg), which reduces by 11-fold the fraction of total ATP that is
immediately available to react with Ca2+. Thus, the 11-fold reduction in k+1 accounts for the 11-fold increase in effective value of
Kd,Ca due to 1 mM [Mg2+]. The response of Scheme D to the
same
[Ca2+] driving function used for Fig. 2 was also calculated
(not shown). As expected, this
[CaATP] response was virtually
identical to that of
[CaATP] shown in Fig. 2; it had a peak amplitude of 64.9 µM, a time-to-peak of 2.93 ms, and a half-width of
5.94 ms (vs. 63.9 µM, 2.98 ms and 6.09 ms, respectively, for
[CaATP] in Fig. 2). Thus, the reduced reaction (Scheme D),
which speeds and simplifies the calculations of
[CaATP] in the
multi-compartment model, closely approximates the complete
reaction system (Schemes B and C). Although it is possible that
other constituents of myoplasm might also bind significant concentrations of Ca2+, our examination of the list of constituents
for frog myoplasm (Godt and Maughan, 1988
) indicates that
ATP is the major (known) species that, to date, has not been included in kinetic models of Ca2+ binding in skeletal muscle.
Phospho-creatine, although present in resting fibers at a concentration that is approximately four times larger than that of ATP,
has, in the presence of 1 mM free [Mg2+], an effective value of
Kd,Ca that is about 16-fold larger (36 mM vs. 2.2 mM) (cf., Smith
and Alberty, 1956
; O'Sullivan and Perrin, 1964
; Sillen and Martell, 1964
). Thus, the ability of phospho-creatine to act as a Ca2+
buffer is expected to be only ~25% of that of ATP. For other compounds that are present at millimolar or near millimolar
concentrations in myoplasm, e.g., inorganic phosphate and carnosine, the Ca2+ buffering effect is expected to be no more than
a few percent of that of ATP (Sillen and Martell, 1964
; Lenz and
Martell, 1964
; Godt and Maughan, 1988
).
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Ca2+ Release from the SR
The form of the equation assumed in the multi-compartment model for SR Ca2+ release in response to an action potential is
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(3) |
Release rate has units of micromoles of Ca2+ per liter of myoplasmic water per millisecond (µM/ms) and its time course, in the absence of SR Ca2+ depletion, reflects the open time of the SR
Ca2+-release channels. The choice of a product of exponentials,
as given on the right-hand side of Eq. 3, is empirical. The values selected for 1,
2, L, and M (1.5 ms, 1.9 ms, 5 and 3, respectively) give a waveform of SR Ca2+ release that is similar to the release
waveform estimated with our single-compartment model when
driven with experimental measurements of
[Ca2+] (see RESULTS). With these selections, the time-to-peak and half-width of
the release rate are 1.70 and 1.93 ms, respectively. The value chosen for R varied with the particular model being examined (see RESULTS) but was usually adjusted so that the peak of spatially averaged
[Ca2+] would be 18 µM, the value expected from the experimental measurements (cf., the first section of RESULTS). For
the standard multi-compartment calculation with ATP (cf., Fig.
4), the value of R corresponds to a peak release rate of 141 µM/
ms. The corresponding spatially averaged total concentration of
released Ca2+, which is given by the integral of release rate with
respect to time, is 296 µM.
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Ca2+ Uptake by the SR
The form of the equation assumed for Ca2+ uptake from the half-sarcomere by the SR Ca2+ pump is
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(4) |
The minus sign signifies that Ca2+ is removed from the myoplasm, and P gives the maximum removal rate (units of µM/ms).
The choice of functional form for the remaining terms reflects
the relatively short time scale of the calculations (30 ms) and is largely empirical. With
and N chosen to be 1 ms and 10, respectively, the exponential term gives a small delay (2-3 ms) for
pump activation after initiation of the calculation. The introduction of this delay, while somewhat arbitrary, permits the initial
binding of Ca2+ by troponin to precede the initial pumping of
Ca2+ by the SR Ca2+ pump. With the parameter P selected to be
1.5 µM/ms (concentration referred to the entire half-sarcomere)
and with Kd selected to be 1 µM, the return of spatially averaged
[Ca2+] towards baseline at later times in the calculation (10-30
ms) is similar to that observed experimentally (cf., Figs. 3 and 4 A
of RESULTS). Although it is possible in principle to include a reaction mechanism for the pump that explicitly calculates the concentration of Ca2+ bound by the pump (e.g., the 11-state cycle of
Fernandez-Belda et al., 1984
for example, as implemented by
Pape et al., 1990
, in their single-compartment model), this approach was deemed too complicated and very unlikely to change
the main conclusions of this article. As an additional simplification, the resting removal of Ca2+ by the SR Ca2+ pump and the
resting leak of Ca2+ through the efflux channels were assumed to
be zero.
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Implementation
Calculations and figure preparation were carried out on a DOS platform (100 MHz Pentium computer) with programs written in MLAB (Civilized Software, Bethesda, MD), a high-level language for differential equation solving, curve fitting, and graphics. In the 18-compartment model with ATP included, the total number of differential equations requiring simultaneous solution is ~100. This number is close to the maximum possible number of such equations that the 1997 DOS version of MLAB can handle. Because of this constraint, the "reduced" reaction of Ca2+ with ATP (Scheme D) was used in the multi-compartment calculations with ATP included.
Single Fiber Measurements
Intact single twitch fibers of semi-tendinosus or iliofibularis muscles of Rana temporaria were isolated and pressure injected with
furaptra. The indicator concentration in myoplasm was sufficiently small (<0.2 mM) that the fiber's [Ca2+] signal in response to action potential stimulation was not altered significantly by the indicator. The furaptra fluorescence signal was measured and calibrated as described previously (Konishi et al., 1991
;
Zhao et al., 1996
).
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RESULTS |
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Summary of Experimental Features of [Ca2+] in Response to
an Action Potential
Our previous experiments that measured spatially averaged [Ca2+] in response to an action potential (16°C)
provide an important constraint for the evaluation of
the multi-compartment model of this article. Most of
these experiments used intact frog twitch fibers of typical diameter (~90 µm) and used furaptra, a lower-affinity, rapidly reacting fluorescence indicator (Konishi et al., 1991
; Hollingworth et al., 1996
; Zhao et al.,
1996
). However, any attempt to relate the properties of
the furaptra fluorescence measurements to the
[Ca2+]
of a single myofibril involves several complications.
First, the myoplasmic value of furaptra's Kd,Ca is uncertain. Our calibration of the furaptra fluorescence
signal uses a Kd,Ca of 98 µM (16°C), which is the value
obtained from a comparison of the furaptra measurements with the [Ca2+] signal from PDAA (Konishi
and Baylor, 1991
; Konishi et al., 1991
). Because PDAA
is a rapidly reacting Ca2+ indicator of low-affinity (Kd,Ca
1 mM) and does not bind strongly to myoplasmic constituents, PDAA is thought to give the most reliable available estimate of
[Ca2+] (Hirota et al., 1989
). The
value of 98 µM for furaptra's myoplasmic Kd,Ca is about
twofold higher than the 49 µM value estimated for the
indicator in a salt solution (16°C, free [Mg2+] = 1 mM);
an increased value is expected in myoplasm because of
the binding of furaptra to myoplasmic constituents
(Konishi et al., 1991
). From the average experimental
value in frog fibers (0.144) observed for the peak of
furaptra's
fCaD signal (the change in the fraction of indicator in the Ca2+-bound form due to an action potential), the average value calibrated for the peak of
[Ca2+] is 16.5 µM (Hollingworth et al., 1996
; Zhao et al.,
1996
). From the same measurements, the average values
estimated for time-to-peak and half-width of
[Ca2+] are
5.0 and 9.6 ms, respectively.
Second, as noted by Konishi et al. (1991), who made
simultaneous measurements of
[Ca2+] with PDAA
and furaptra from the same region of the same fiber, the furaptra measurements may overestimate slightly
the actual values for the time-to-peak and half-width of
[Ca2+]. This follows because the time-to-peak and
half-width values measured with PDAA were slightly
briefer (by about 0.3 and 1.5 ms, respectively) than the
furaptra measurements.
Third, as noted by Hollingworth et al. (1996), a
slightly larger and briefer
[Ca2+] signal is found in experiments with smaller-diameter frog fibers. In four
such fibers (diameters 45-54 µm), the average furaptra
[Ca2+] values were 17.3 µM for peak, 4.4 ms for time-to-peak, and 8.2 ms for half-width (compared with 16.5 µM, 5.0 and 9.6 ms, respectively, for ordinary-sized fibers
mentioned above). These differences presumably arise because delays associated with radial propagation of the tubular action potential (Adrian and
Peachey, 1973
; Nakajima and Gilai, 1980
) are smaller
in smaller diameter fibers. Thus, the dispersive effects
on the spatially averaged
[Ca2+] signal due to nonsynchronous activation of individual myofibrils should be
smaller. We assume that if measurements could be
made in the absence of any radial delays,
[Ca2+]
would be slightly larger and briefer.
Based on these considerations, we expect that the following approximate values should apply to [Ca2+] of
a single myofibril at 16°C: peak amplitude, ~18 µM;
time-to-peak, ~4 ms; half-width, ~6 ms. In the absence
of longitudinal propagation delays (appropriate for the
multi-compartment model), the value for time-to-peak
is expected to be ~3 ms.
Summary of Estimates of SR Ca2+ Release Obtained with the Single-compartment Model
The furaptra [Ca2+] measurements can be used as input to the single compartment model of Baylor et al.
(1983)
to estimate the amplitude and time course of SR
Ca2+ release (e.g., Hollingworth et al., 1996
). With this
model, it is assumed that myoplasmic changes occur
uniformly in space and that the change in total myoplasmic Ca2+ concentration due to SR release (
[CaT])
can be estimated from the summed changes of Ca2+ in
four pools: (a)
[CaD] (the change in concentration
of Ca2+ bound to furaptra, which can be directly calibrated from the measured change in indicator fluorescence,
F), (b)
[Ca2+] itself (calibrated as described
in the previous section), (c)
[CaTrop], and (d)
[CaParv]. Given
[Ca2+] and the assumed resting [Ca2+]
of 0.1 µM, changes c and d can be calculated from Eq. 1
(described in MATERIALS AND METHODS) and the reaction parameters given in Table I.
Fig. 3 shows an example of this model applied to
measurements from a frog fiber of small diameter (45 µm). The four lower traces show the estimated changes
in Ca2+ concentration in the four pools described in
the preceding paragraph. The next trace ([CaATP])
shows the estimated concentration change in a fifth
pool, that of Ca2+ bound to ATP (cf., Fig. 2). Two
[CaT] traces were computed (not shown). The first,
which equaled the sum of the concentration changes in
the original four pools, had a peak value of 291 µM and
a time-to-peak of 6.5 ms; the second, which also included the contribution of
[CaATP], had a peak
value of 339 µM and a time-to-peak of 5.5 ms. The two
traces at the top of Fig. 3 show the time derivative
(d
[CaT]/dt) of the two
[CaT] signals; these traces
supply two estimates of the net flux of Ca2+ between SR
and myoplasm (i.e., release rate minus uptake rate). The large early positive deflections essentially reflect
the release process. The effect of Ca2+ uptake is apparent only at later times when, with the cessation of release, the traces go slightly negative. The smaller of the two d
[CaT]/dt signals (second from top) had a peak
value of 146 µM/ms, a time-to-peak of 2.5 ms, and a
half-width of 1.8 ms, whereas the larger signal (top),
which includes the contribution of
[CaATP], had a
peak value of 183 µM/ms, a time-to-peak of 2.5 ms, and
a half-width of 1.8 ms.
Results similar to those in Fig. 3 were observed in a total of four small-diameter frog experiments. Without inclusion of ATP, the average values (±SEM) estimated
for the [CaT] signal were 298 ± 4 µM for peak amplitude and 6.5 ± 0.1 ms for time-to-peak; with inclusion of
ATP, the values were 351 ± 9 µM and 5.6 ± 0.1 ms, respectively. For the d
[CaT]/dt signal, the average values without inclusion of ATP were 142 ± 4 µM/ms for peak
amplitude, 2.9 ± 0.2 ms for time-to-peak, and 1.9 ± 0.1 ms for half-width; with ATP, the values were 176 ± 7 µM/ms, 2.9 ± 0.2 ms, and 1.9 ± 0.1 ms, respectively. All
values for time-to-peak likely include a small delay, ~1
ms, because of action potential propagation.
These calculations indicate that the inclusion of ATP,
with properties as specified in Table I, in the single-compartment model of Baylor et al. (1983) increases
the estimated peak value of
[CaT] by about 53 µM
(18%) and that of d
[CaT]/dt by about 34 µM/ms
(24%). Interestingly, these changes occur with very little change in the main time course of the d
[CaT]/dt
signal, as the estimates for time-to-peak and half-width
of release were unaltered. This finding supports the use
of the SR Ca2+ release function described in MATERIALS
AND METHODS (Eq. 3) as the starting point for the calculations with the multi-compartment model.
Results of the Multi-compartment Model without Inclusion of ATP
At the outset, it is useful to note two important conceptual differences between single- and multi-compartment modeling. First, with a single-compartment model,
calculations can be applied in either of two logical directions: (a) backward, from [Ca2+] to a release waveform (e.g., as in Fig. 3) or (b) forward, from the release
waveform to
[Ca2+] (not shown). In contrast, with the
multi-compartment approach, only calculations in the
forward direction are practical because spatially averaged
[Ca2+] results from the summed changes in a
number of different compartments (e.g., 18 as in Fig.
1). The procedure adopted for the multi-compartment calculations was thus to assume an SR Ca2+ release
waveform as driving function and evaluate its success by
a comparison of calculated spatially averaged
[Ca2+]
with expectations from the measurements of
[Ca2+]
(cf., first section of RESULTS). This evaluation compared
values for peak amplitude, time-to-peak, and half-width
of
[Ca2+]. Secondly, only the multi-compartment
model calculates concentrations as a function of spatial
location. Thus, single-compartment calculations are expected to have errors associated with an inability to estimate local gradients in
[Ca2+] and the associated gradients in Ca2+ bound to nonlinear (saturable) binding
sites. In consequence, inconsistencies are expected to
arise between single- and multi-compartment calculations with otherwise identical parameters.
The first calculations with the multi-compartment
model did not include ATP and provide a useful baseline for assessment of the effect of the inclusion of ATP
(next section). The amplitude initially selected for the
parameter R in the release waveform driving function
(Eq. 3) corresponds to a spatially averaged release rate
of 142 µM/ms, the value estimated from the single-compartment model without ATP (preceding section).
A striking result of this calculation (not shown) is that
spatially averaged [Ca2+] is very different from the expectations outlined in the first section of RESULTS. Its
peak amplitude, 58 µM, is about threefold larger than
expected (~18 µM), and its half-width, 3.6 ms, is markedly briefer than expected (~6 ms). The time-to-peak
(3.2 ms), however, is close to expected (~3 ms). This
large discrepancy between the single- and multi-compartment results has two possible sources. First, there
might be a significant error in the d
[CaT]/dt signal
used to drive the multi-compartment model, in which
case the effect of other parameter selections (including
the omission of ATP) becomes difficult to evaluate. Alternatively, the d
[CaT]/dt signal may be approximately correct, in which case the omission of ATP and/
or the choice of the other model parameters must be quite significant.
Although it is possible that the d[CaT]/dt signal,
which is based on the single-compartment model, may
have errors in both amplitude and time course, other
experimental evidence supports the conclusion that
the time course of the d
[CaT]/dt signal is approximately correct. This evidence comes from action potential experiments on fibers that contained millimolar
concentrations of a high-affinity Ca2+ buffer such as
fura-2 (Baylor and Hollingworth, 1988
; Hollingworth et al., 1992
; Pape et al., 1993
) or EGTA (Jong et al.,
1995
). At millimolar concentrations, these buffers rapidly bind most of the Ca2+ that is released from the SR,
and thus their optical signal, which is proportional to
the amount of bound Ca2+, closely tracks
[CaT]. The
time derivative of this signal had a half-width of ~3 ms.
Although this value is ~1 ms larger than that of the
d
[CaT]/dt waveform defined by Eq. 3, a larger experimental half-width is expected for two reasons. First, the
fibers of these experiments were of typical diameter
(~90 µm) rather than small diameter. Second, because
the myoplasmic
[Ca2+] signal in these fibers was reduced and abbreviated (due to the presence of millimolar Ca2+ buffer), there was likely relief from the process of Ca2+-inactivation of SR Ca2+ release (Baylor et al.,
1983
; Schneider and Simon, 1988
). This process normally serves to abbreviate the time course of SR release.
Given this support for the time-dependent part of
Eq. 3, it was of interest to redo the multi-compartment
calculation described above with the amplitude factor
R reduced so that the peak value of spatially averaged
[Ca2+] would be 18 µM, the value expected from the
experimental measurements (cf., first section of RESULTS). To achieve this result, an R value of 89 µM/ms
is required (instead of 142 µM/ms). In this case, however, the half-width of
[Ca2+] is only 2.6 ms, which is
even briefer than calculated initially (3.6 ms) and less
than half the expected value (~6 ms). In summary, because these calculations failed to produce a spatially averaged
[Ca2+] that is acceptable in both amplitude
and time course, the multi-compartment model appears to have some important error or omission unrelated to the use of Eq. 3 as driving function.
Results of the Multi-compartment Model with Inclusion of ATP
The next calculations included ATP, with the value of R
set initially to 176 µM/ms (the value estimated from
the single-compartment model with ATP; see second
section of RESULTS). In this case, spatially averaged
[Ca2+] (not shown) has a peak value of 27.7 µM and a
half-width of 10.6 ms. Both values are substantially
larger than expected from the measurements (~18 µM
peak and ~6 ms half-width) and again imply some significant error or omission.
As in the preceding section, the multi-compartment
calculation with ATP was then repeated but with the
value of R lowered (to 141 µM/ms) so as to yield an
amplitude of 18 µM for spatially averaged [Ca2+].
The results of this calculation are shown in Fig. 4. Interestingly, spatially averaged
[Ca2+] (Fig. 4 A) has values for time-to-peak and half-width of 3.2 and 5.2 ms,
respectively, which are quite close to the expected values (~3 and ~6 ms, respectively).
Fig. 4 B shows the associated calculations of [CaTrop],
which involve nine troponin-containing compartments.
For
[CaTrop], a value of 446 µM on the ordinate corresponds to 100% occupancy of the troponin sites with
Ca2+. (The 446 µM value is calculated from the 240 µM
value given in Table I times a factor of two [since the
troponin sites are located in only half of the compartments in Fig. 1] minus the resting occupancy of troponin with Ca2+, 34 µM [= 0.071 × 480 µM; cf., Table I].) In all nine compartments, the occupancy of troponin with Ca2+ reached a peak level that is close to saturation (>85%). Thus, the underlying Ca2+ transients in
the troponin-containing compartments are of sufficient amplitude and duration to give nearly complete
activation of troponin along the entire thin filament, as
expected from fiber mechanical measurements (e.g.,
Gordon et al., 1964
).
Fig. 4, C and D, shows the calculations of [CaATP]
and
[CaParv], respectively, in the 18 compartments.
The calculations of
[Ca2+] for the individual compartments are not shown, but the time course and relative
amplitude of these changes are closely similar to those
shown in Fig. 4 C for
[CaATP]. This follows because (a) as mentioned in MATERIALS AND METHODS, on the
time scale shown, the Ca2+-ATP reaction is virtually in
kinetic equilibrium with
[Ca2+], and (b) since the effective value of ATP's Kd,Ca is large (2.2 mM; Table I),
the Ca2+-ATP reaction deviates by <10% from linearity
even for Ca2+ transients as large as 100 µM (the amplitude of
[Ca2+] in the outer-most compartment nearest the Z line in the calculation of Fig. 4; not shown).
Hence, the
[Ca2+] changes for all compartments can
be closely approximated from the
[CaATP] changes
in Fig. 4 C if the latter are scaled by the factor 1/3.6
(see MATERIALS AND METHODS). Similarly, spatially averaged
[CaATP] can be closely approximated from the
spatially averaged
[Ca2+] waveform shown in Fig. 4 A
if scaled by the factor 3.6. For spatially averaged
[CaATP], the actual values of peak amplitude, time-to-peak, and half-width are 63.7 µM, 3.2 ms, and 5.3 ms, respectively.
The principal conclusion from the calculation of Fig.
4 is that, with ATP included as a diffusible Ca2+-binding
species, spatially averaged [Ca2+] is close to expectation if the value of R in Eq. 3 is ~140 µM/ms. Based on
(a) the fact that ATP is present in myoplasm at millimolar concentrations and presumably reacts with Ca2+
with reaction rate constants close to those listed in Table I, and (b) the finding of a great improvement in the
agreement between calculated and measured
[Ca2+]
with inclusion of ATP in the multi-compartment model,
two conclusions appear to be warranted. First, ATP
likely plays an important role in the binding and transport of myoplasmic Ca2+. Second, apart from a small
time shift due to action potential propagation, the SR
Ca2+ release function used in Fig. 4 is probably quite
close to the actual SR Ca2+ release function of a small-diameter frog fiber.
As discussed in a later section of RESULTS, the need in Fig. 4 for an SR Ca2+ release function with an amplitude ~20% smaller than that estimated from the single-compartment model with ATP included reflects errors in the single-compartment model due to its inability to calculate effects of local saturation of Ca2+-binding sites. The somewhat fortuitous result that the amplitude of the release function used in Fig. 4 (141 µM/ ms) is very close to that estimated in the single-compartment calculation without ATP included (142 µM/ ms; second section of RESULTS) is a related point that is also considered in a later section of RESULTS.
Role of ATP in Transporting Ca2+ within the Sarcomere
An additional feature of the calculation in Fig. 4 is that
the diffusion of Ca2+ in the CaATP form is responsible
for the spread of more total Ca2+ throughout the sarcomere than is the diffusion of free Ca2+. This follows from
the observation that, at any myoplasmic location, [CaATP] is ~3.6-fold greater than
[Ca2+], whereas
the diffusion constant of free Ca2+ is only 2.1-fold greater
than that of ATP (Table I). Thus, the flux of Ca2+ across
compartment boundaries will be ~1.7-fold (= 3.6/2.1)
greater for CaATP than for free Ca2+ (cf., Eq. 2).
To explore the importance of CaATP diffusion, it was
of interest to repeat the multi-compartment calculation
of Fig. 4 with the value of DATP reduced from 1.4 × 106
cm2 s
1 to 0. In this circumstance, the spread of Ca2+
depends primarily on the diffusion of free Ca2+. Fig. 5
shows the result, which reveals two significant points. First, a comparison of Figs. 5 B and 4 B shows that, with
DATP reduced to 0, there is an increased occupancy of
troponin with Ca2+ in the compartments nearest the Z
line but a reduced occupancy in the compartments
nearest the m-line, as well as a reduced rate of rise in
the latter compartments. Thus, the transport of Ca2+ in
the CaATP form that occurs if DATP = 1.4 × 10
6 cm2
s
1 results in a Ca2+-troponin occupancy that is more
uniform and more synchronous. This presumably enables a more uniform and synchronous activation of fiber force.
|
Second, spatially averaged [Ca2+] in Fig. 5 A has a
peak amplitude of 24.7 µM, a time-to-peak of 3.4 ms,
and a half-width of 6.2 ms. Although these values are
not markedly different from those in Fig. 4 A (18.0 µM,
3.2 ms, and 5.2 ms, respectively), they are substantially
different from the values mentioned in the first multi-compartment calculations of RESULTS. In those calculations, with ATP omitted entirely,
[Ca2+] had a peak
amplitude of 58.0 µM, a time-to-peak of 3.2 ms, and a
half-width of 3.6 ms. Because the value of R in Eq. 3 was
essentially identical for that calculation and the calculation of Fig. 5 (142 vs. 141 µM/ms, respectively), it follows that ATP produces a much smaller and broader
Ca2+ transient simply through its ability to bind Ca2+
during the rising phase of
[Ca2+] and release it during the falling phase. Thus, independent of its ability to
transport Ca2+, ATP acts as an important "temporal filter" of
[Ca2+].
Conclusions Based on an Examination of Changes to Other Parameters Listed in Table I
As described in the preceding sections, a significant
binding and diffusive role for ATP is supported by the
finding that inclusion of millimolar ATP in the model
results in good agreement between the properties of
calculated [Ca2+] and those extrapolated from the
measurements of
[Ca2+]. A further test of the significance of this result is to examine whether, without
ATP, adjustment of one or several of the many other
parameters of the model listed in Table I might produce a comparable improvement in the properties of
calculated
[Ca2+]. Although it was not possible to
make an exhaustive exploration of all such model adjustments, several changes were investigated that, in the
absence of ATP, were designed specifically to improve the
agreement between calculated and measured
[Ca2+].
None of the changes was found to make the substantial
qualitative difference that resulted from the inclusion
of ATP. These other changes included (a) a threefold
reduction in the peak rate of SR Ca2+ pumping (the parameter P in Eq. 4), (b) a twofold increase in the value
of the diffusion constant of free Ca2+ (DCa in Table I),
(c) a threefold increase in the Ca2+-parvalbumin on-rate constant (k+1 for parvalbumin in Table I), and (d)
use of a smaller and broader SR Ca2+-release function.
With changes a-c, whether implemented individually or simultaneously, there was no major improvement in
the agreement between modeled and measured
[Ca2+].
With changes of type d, if sufficiently large, it was possible to produce a
[Ca2+] with a peak amplitude of ~18
µM and a half-width of 5-6 ms, but these improvements
were achieved only at the expense of the appearance of
a slow foot on the rising phase of
[Ca2+] and a delayed time-to-peak of
[Ca2+] (~5 ms). In sum, the inability of these changes to produce an acceptable spatially averaged
[Ca2+] further supports the idea that
ATP does indeed contribute importantly to the determination of
[Ca2+].
The Possible Importance of other Myoplasmic Ca2+-binding Species
A related question is whether inclusion of other types of
Ca2+-binding species in the multi-compartment model
can produce improvements similar to that produced by
ATP. For example, some neuronal cells appear to contain substantial concentrations of a nondiffusible, low-
affinity Ca2+ buffer(s), which may strongly influence
[Ca2+] (Helmchen et al., 1996
). This possibility was examined in our multi-compartment model by a comparison of the effects of such a hypothetical fixed buffer
(HFB) with those of ATP. For these comparisons, HFB
was assumed to be distributed in all myoplasmic compartments and have values of k+1 and k
1 identical to those
listed in Table I for ATP. A further constraint for these
calculations was that, for each concentration of HFB considered, the value of R (Eq. 3) was always adjusted so that
the peak amplitude of
[Ca2+] would be 18 µM.
The first calculation assumed that ATP was absent
but that HFB was present at a concentration of 8 mM.
This situation is similar to that shown in Fig. 5, except
that a smaller value of R is used (118 µM/ms) so as to
yield an 18 µM [Ca2+] transient. In this case, the values for time-to-peak and half-width of
[Ca2+] are 3.4 and 5.4 ms, respectively, which are essentially identical to those in Fig. 4 A (3.2 and 5.2 ms, respectively). Thus,
in terms of the ability to generate a satisfactory
[Ca2+]
response, the presence of HFB in the multi-compartment is very comparable to that of ATP. However, this
calculation also reveals that, because of the inability of
HFB to diffuse, there is substantially less occupancy of
troponin with Ca2+ in the three troponin-containing
compartments most distant from the Z line
on average, only 64% with HFB (vs. 86% with ATP; Fig. 4 B).
As in Fig. 5, this calculation provides another demonstration of the importance of the diffusibility of a low-affinity buffer for achieving a high Ca2+-occupancy of
troponin all along the thin filament and indicates that
the calculations with HFB alone are not as satisfactory
as those with ATP alone.
The second calculations with HFB assumed that ATP
was present in the usual amount (8 mM) and examined
how the presence of different concentrations of HFB
affected the time course of [Ca2+]. The first such calculation assumed a concentration of HFB equal to that
of ATP, 8 mM. In this case, the required value of R for an 18 µM
[Ca2+] was 171 µM/ms, and the time-to-peak and half-width of
[Ca2+] were 3.6 and 9.9 ms, respectively. Since the value for half-width is substantially
longer than expected (~6 ms), it seems unlikely, given
that skeletal muscle contains ~8 mM ATP, that it also
contains a similar or larger concentration of HFB.
The next step was to reduce the concentration of
HFB to identify the value that would give a half-width
for [Ca2+] of 6 ms, i.e., essentially that expected from
the experimental measurements. This concentration
was 1.8 mM (with associated value of R = 148 µM/ms),
and the value for time-to-peak of
[Ca2+] was 3.3 ms.
Since the occupancy of troponin with Ca2+ in this calculation was also high in all of the troponin-containing compartments (>85%), the presence of this concentration of HFB in muscle seems plausible. Indeed, with 1.8 mM HFB, the time-to-peak and half-width of
[Ca2+]
are in better overall agreement with the values expected from the experimental measurements than is
the
[Ca2+] of Fig. 4 (time-to-peak, 3.2 ms; half-width,
5.2 ms).
In summary, these calculations indicate that it is unlikely that skeletal muscle contains a concentration of
low-affinity fixed buffer (in ATP-equivalent units) as large
as 10% of that postulated for nerve (Helmchen et al.,
1996). However, the possibility that muscle contains a few
percent of that postulated for nerve cannot be ruled out
and, in fact, may be supported by the calculations.
A final calculation in this general category was to
omit HFB entirely and identify what concentration of
ATP alone would give values for peak and half-width of
[Ca2+] that were essentially the same as noted in the
preceding paragraphs with the inclusion of 1.8 mM
HFB. (Again, a constraint for these calculations was
that, for each concentration of ATP considered, the
value of R was readjusted to give a peak amplitude of
18 µM for
[Ca2+].) With 9.2 mM ATP and with an R
of 148 µM/ms, the time-to-peak and half-width values
of
[Ca2+] are 3.3 and 6.0 ms, respectively. Thus, inclusion of ATP alone at 9.2 mM (rather than 8 mM) gives
a calculated
[Ca2+] that is virtually identical to that
obtained with inclusion of 8 mM ATP and 1.8 mM
HFB. As noted in MATERIALS AND METHODS, the concentration of phospho-creatine found in muscle, ~40
mM, approximates 2 mM of ATP-equivalent (diffusible) low-affinity Ca2+ buffer. Phospho-creatine thus
provides a basis for a modest increase in the ATP-equivalent concentration used in the model.
In summary, the calculations of this section do not exclude, but also do not necessarily support, the presence of a small concentration of HFB in myoplasm. They do, however, argue against the likelihood of a concentration of HFB as large as 10% of that found in nerve.
Comparison of the Single-compartment Model without ATP and the Multi-compartment Model with ATP
In Fig. 4, the multi-compartment model with ATP was driven by an SR Ca2+ release function of amplitude 141 µM/ms, which is essentially identical to the 142 µM/ms value estimated from the single-compartment model without ATP (cf., Fig. 3). This similarity implies that the error in the single-compartment estimates of SR release associated with the omission of ATP are offset by other errors. Several factors appear to contribute to these other errors.
First, a single-compartment model does not consider
separate myoplasmic regions with differing degrees of
local saturation of binding sites. Thus, a single-compartment model will, for a given spatially averaged [Ca2+],
maximize
and thus over-estimate
the amount of Ca2+
captured by the intrinsic buffer sites included in the
model (which are assumed to react with Ca2+ with a 1:1
stoichiometry). Moreover, the erroneous extra Ca2+
that the single-compartment model assigns to binding
by the intrinsic buffers occurs early in time, when the
myoplasmic gradients in [Ca2+] (as estimated by the
multi-compartment model) are large. For example,
the multi-compartment model estimates that
[Ca2+]
in the compartments nearest the Z line rises rapidly to
~100 µM, and as a result, there is rapid, local saturation of the troponin sites in these regions (Fig. 4 B). In
contrast, in the other compartments, significant diffusional delays affect the rise of
[CaTrop]. Thus, in the
single-compartment model, both kinetic and steady-state errors arise from the spatially homogeneous estimation of Ca2+ binding to the intrinsic buffers.
Second, calculations with the multi-compartment
model show that some local saturation of furaptra with
Ca2+ also occurs at early times near the release sites
(see next section). This local saturation results in an estimate of [Ca2+] from furaptra that has a later time-to-peak and broader half-width than does the actual
[Ca2+]. By itself, use of a delayed
[Ca2+] to drive the
single-compartment model will result in an estimate of
SR Ca2+ release that is delayed with respect to the actual release waveform.
Third, because of the same early local saturation of
furaptra, the amplitude of spatially averaged [Ca2+],
if calibrated with the actual myoplasmic Kd,Ca of the indicator, will be underestimated. As discussed earlier, a
value of 98 µM was assumed for furaptra's Kd,Ca so that
the amplitude of
[Ca2+] calibrated from the indicator's
fCaD would agree with
[Ca2+] measured with
PDAA. The next section shows that the 98 µM value is
probably larger than the actual myoplasmic value, and
its use in the single-compartment model partially compensates for the other errors that arise because of the
local saturation of sites with Ca2+.
Characterization of Probable Error in the Previous Estimate of Furaptra's Kd,Ca and in the Single-compartment Estimates of SR Ca2+ Release
Fig. 6 shows several additional calculations associated
with the multi-compartment model of Fig. 4. For these
calculations, a nonperturbing concentration of furaptra (1 µM) was included as a separate Ca2+-binding species in all compartments, and the diffusion constant of
furaptra was assumed to be 0.68 × 106 cm2 s
1 (Konishi et al., 1991
). In Fig. 6 A, the continuous trace is identical to the spatially averaged
[Ca2+] shown in
Fig. 4 A (called here "true" spatially averaged
[Ca2+],
i.e., as calculated under the assumptions of the model).
In Fig. 6 B, the spatially averaged
fCaD signal for furaptra was simulated by the multi-compartment model under two different assumptions about indicator properties. For both simulations, furaptra was assumed to have
a value of k
1 (Scheme A) of 5,000 s
1 (Zhao et al.,
1997). For the first calculation (Fig. 6 B, dotted trace), a
value of 5.1 × 107 M
1 s
1 was assumed for k+1 (thus
Kd,Ca = 98 µM, as assumed by Konishi et al. [1991] and
Zhao et al. [1996]); for the second calculation (Fig. 6 B,
dashed trace), the k+1 value was 7.1 × 107 M
1 s
1 (Kd,Ca = 70 µM). Fig. 6 B shows that, with a Kd,Ca of 70 µM, the
amplitude of
fCaD is significantly larger, 0.151 (vs.
0.120 if Kd,Ca is 98 µM). The 0.151 value is essentially
identical to the average value of 0.150 observed for
fCaD in the experiments on small-diameter frog fibers
(described in the first section of RESULTS). Thus, the
dotted trace in Fig. 6 B indicates that the peak of ~0.15
for furaptra's spatially averaged
fCaD signal cannot be
explained under the assumptions that Kd,Ca is 98 µM and
that the peak of spatially averaged
[Ca2+] is 18 µM.
Since the peak of
[Ca2+] is thought to be close to 18 µM, we conclude that furaptra's myoplasmic Kd,Ca is
likely to be closer to 70 µM than to 98 µM.
|
Considered as temporal waveforms, the two fCaD responses in Fig. 6 B are essentially identical (times-to-peak, 3.8-3.9 ms; half-widths, 8.1-8.2 ms). Both times-to-peak are noticeably slower than the 3.2 ms time-to-peak
of true spatially averaged
[Ca2+] (Fig. 4 A, continuous
trace). The delay in time-to-peak of
fCaD is due to local
saturation of furaptra with Ca2+ because no such delay
is found if
fCaD is driven by
[Ca2+] in a single-compartment simulation (not shown).
The dashed trace in Fig. 6 A is a single-compartment
calculation of spatially averaged [Ca2+] based on the
dashed
fCaD response in Fig. 6 B; for this conversion, a
furaptra Kd,Ca of 98 µM was used, and the steady-state
form of the 1:1 binding equation was assumed. This
trace thus simulates previous experimental estimates of
[Ca2+] based on a furaptra
fCaD signal of typical amplitude and the previously assumed value of Kd,Ca. In
Fig. 6 A, the peak amplitudes of
[Ca2+] are nearly
identical (18 µM for the continuous trace, 17.5 µM for
the dashed trace); this is expected since Kd,Ca for furaptra was chosen previously to make the amplitude of
furaptra's
[Ca2+] agree with that of PDAA's (cf., first
section of RESULTS). The time courses of the two
changes in Fig. 6 A, however, are obviously different
(time-to-peak of 3.2 ms and half-width of 5.2 ms for the continuous trace vs. 3.9 and 8.1 ms, respectively, for
the dashed trace). As mentioned above, the fact that the
time course of the simulated furaptra
[Ca2+] is slower
than that of true
[Ca2+] reflects the effects of local saturation of the indicator with Ca2+, which the single-compartment calculation cannot take into account. This error in time course would be smaller for an indicator of lower affinity, which would undergo less local
saturation. Indeed, if calculations analogous to those of
Fig. 6 are carried out with PDAA (Kd,Ca
1 mM), the
simulated peak amplitude of
fCaD is only 0.017, and
the single-compartment conversion of
fCaD to spatially
averaged
[Ca2+] yields a peak amplitude of 17.3 µM, a
time-to-peak of 3.3 ms, and a half-width of 5.4 ms (calculations not shown). As expected, these values are very
close to those of true
[Ca2+].
Since a number of previous publications, from this
and other laboratories, have used a single-compartment model without ATP to estimate SR Ca2+ release
parameters, it was of interest to use the multi-compartment model with ATP to characterize the likely errors
in these estimates. Table II gives this information for estimates obtained with PDAA and furaptra. Column 1 (part A for [CaT]; part B for d
[CaT]/dt) gives the information related to the release function used to drive
the standard multi-compartment calculation with ATP
(Fig. 4) and thus provides the "true" reference point for
the comparisons in Table II. For the estimates in column
2, the dashed trace in Fig. 6 A was used as the
[Ca2+]
to drive the single-compartment model. (As mentioned
above, this trace simulates a furaptra
[Ca2+] signal,
calibrated as in Fig. 3.) A comparison of column 2 with column 1 shows that, somewhat fortuitously, the single-compartment model without ATP provides generally
accurate estimates of the true release parameters; the
main error is a modest overestimation of the time-to-peak of release. Column 3 shows analogous release parameters based on use of the simulated PDAA
[Ca2+]
signal mentioned above. Again, the release parameters
in column 3 are in reasonable agreement with those in
column 1. Overall, the SR release parameters estimated
from furaptra are in slightly better agreement with the
true release parameters than are those from PDAA, even
though there is more error in the single-compartment estimate of spatially averaged
[Ca2+] with furaptra than
with PDAA (see above and next section). This result,
which is again somewhat fortuitous, indicates that the totality of errors inherent in the difference between the
single-compartment model without ATP and the multi-compartment model with ATP (see preceding section) is
offset slightly better with furaptra and its previous
method of calibration (Kd,Ca = 98 µM) than with PDAA.
|
General Analysis of Errors in [Ca2+] Associated with
Single-compartment Calculations
The preceding section compared single- and multi-compartment estimations of spatially averaged [Ca2+]
from furaptra and PDAA and noted several sources of
error inherent in the single-compartment estimates.
This section further characterizes these errors by means
of analogous calculations applied to a hypothetical family of indicators. For this analysis, all indicators are assumed to react with Ca2+ with an identical value of k+1
but with different values of k
1 and hence different
values of Kd,Ca. The value selected for k+1, 5 × 107 M
1
s
1, lies in the range considered for furaptra in the previous section (5-7 × 107 M
1 s
1) and is probably also
similar to that which applies to many members of the
family of tetra-carboxylate Ca2+ indicators (cf., Tsien,
1980
) when in the myoplasmic environment, e.g., indo-1,
fura-2, fluo-3, calcium-orange-5N, etc. (Zhao et al., 1996
).
In general, these indicators bind heavily to myoplasmic constituents, and as a consequence, their rate constants
for reaction with Ca2+ appear to be substantially reduced in comparison with those of the indicator in free
solution. Six values of k
1 were selected for these calculations: 101 s
1, 102 s
1,..., 106 s
1, with the corresponding values of Kd,Ca being 0.2 µM, 2 µM,..., 20 mM.
Fig. 7 (described in detail beginning with the next
paragraph) summarizes the results of this analysis. As
in the preceding section (cf., Fig. 6 and Table II), the
multi-compartment model with ATP included is assumed to give the "true" results (Fig. 7, B and D, horizontal dotted lines) against which the simulated [Ca2+]
from each of the indicators can be compared. To calculate an indicator's
[Ca2+], the fraction of the indicator bound with Ca2+ (spatially averaged
fCaD plus the
resting fraction, fCaD; Fig. 7 A) was calculated by the
multi-compartment model, based on a nonperturbing
concentration of indicator (1 µM) included in all compartments. For simplicity, the myoplasmic diffusion
constant of all indicators was fixed in the calculations at
0.25 × 10
6 cm2 s
1 (cf., Zhao et al., 1996
). For the conversion of an indicator's fCaD +
fCaD response to
[Ca2+], two different single-compartment methods
were used. In the first method (Fig. 7, B and D, dashed
curves), fCaD +
fCaD was converted to
[Ca2+] by the
steady-state form of the 1:1 binding equation, i.e., as
was done for the conversion of the dashed
fCaD curve
in Fig. 6 B to the dashed
[Ca2+] curve in Fig. 6 A. In
the second method (Fig. 7, B and D, continuous curves;
also traces in Fig. 7 C), the kinetic form of the 1:1 binding equation was used (see for example Baylor and
Hollingworth, 1988
; Klein et al., 1988
; Hollingworth et al.,
1992
). Necessarily, the second method gives a
[Ca2+]
with a larger peak amplitude and a briefer half-width
than does the first method. Even though the second approach partially compensates for the kinetic lag between
[Ca2+] and
fCaD that arises when k
1 is small,
this method cannot be expected to correct for errors
related to gradients in indicator saturation.
|
In Fig. 7 A, six time-dependent calculations of fCaD + fCaD are plotted, corresponding to the six different
choices of k
1. As k
1 increases from 101 s
1 to 106 s
1,
fCaD becomes progressively smaller and
fCaD becomes
both briefer and smaller. At the largest value of k
1,
both fCaD and
fCaD are too small to be resolved above
baseline at the gain shown. With the first (steady-state)
method of conversion of the traces in Fig. 7 A to
[Ca2+] (
[Ca2+] traces not shown), values of k
1 approaching 105 s
1 or greater are required if both the
peak amplitude and half-width of
[Ca2+] (Fig. 7, B
and D, respectively, circle points connected by dashed curves)
are to agree well with those of true
[Ca2+] (Fig. 7, B
and D, horizontal dotted lines). With values of k
1
104
s
1, a progressively larger disparity is observed between
the parameters of calculated and true
[Ca2+].
The traces in Fig. 7 C show the results of the second
(kinetic) method of conversion of the traces in Fig. 7 A
to [Ca2+] and were used to calculate the second set of
points in Fig. 7, B and D (cross points connected by continuous curves). As mentioned above,
[Ca2+] parameters
are necessarily larger and briefer with this method and
thus the continuous curves in Fig. 7, B and D, lie closer to the horizontal dotted lines than do the dashed
curves. Interestingly, with this method of conversion,
[Ca2+] in Fig. 7 C becomes obviously biphasic at the
two smallest values of k
1 (101 s
1 and 102 s
1), and at
the next larger value of k
1 (103 s
1), a hump can be
seen on the rising phase of
[Ca2+]. The appearance
of two phases in
[Ca2+] is an artifact of local indicator
saturation in combination with the use of a single-compartment kinetic correction to convert the fCaD +
fCaD
response to
[Ca2+]. The earlier phase, which rises to a
plateau during the time of SR Ca2+ release, reflects effects of indicator saturation at sarcomeric regions close to
the release sites. The later phase, which involves a delay
in the rise of
[Ca2+] and
fCaD at sarcomeric locations
more distant from the release sites, reflects the time required for Ca2+ to diffuse and bind to indicator in these
locations. (Note: The biphasic response does not depend
on the diffusion of indicator since it is also seen if the diffusion constant of the indicator is set to zero in the multi-compartment part of the calculation; not shown.)
Overall, the simulations in Fig. 7 indicate that, at values of k1 < 104 s
1, effects due to Ca2+ gradients and
local saturation of indicator introduce significant error
in single-compartment methods for estimation of true
[Ca2+]. Moreover, at k
1
103 s
1, these errors are
quite severe. Thus, this analysis supports the conclusion
that, to achieve an accurate estimate of true
[Ca2+] in
spatially averaged measurements, it is highly desirable
to use a low-affinity, rapidly responding indicator (Hirota et al., 1989
).
Application of the Multi-compartment Model to Reestimate Myoplasmic Values of the Ca2+-fluo-3 Reaction Rates
Previous publications from this and other laboratories
(e.g., Baylor and Hollingworth, 1985, 1988; Klein et al.,
1988; Harkins et al., 1993
; Kurebayashi et al., 1993
;
Pape et al., 1993
; Westerblad and Allen, 1996
; Zhao et al.,
1996
) have described single-compartment methods for
estimation of Ca2+ reaction rates (k+1 and k
1) of a
number of different indicators when in the myoplasmic
environment. In the most common type of experiment,
the same region of the same fiber was exposed to two
indicators
usually a lower-affinity indicator (e.g., antipyrylazo III, furaptra, or PDAA) and a higher-affinity
indicator (e.g., fura-2, fura-red, or fluo-3)
and optical
measurements were made simultaneously from both indicators. The optical responses from the lower-affinity
indicators (whether recorded in the same or different
fibers) were usually very similar in time course, whereas the responses from the higher-affinity indicators, while
somewhat variable in time course, always had significantly later times-to-peak and broader half-widths than
did the lower-affinity responses. The slower responses
of the higher-affinity indicators were assumed to reflect
the smaller values of k
1 that are inherent in these indicators being of higher-affinity, and the timing of these responses relative to the lower-affinity responses was
used in single-compartment fits to estimate the values
of k
1 and k+1 of the higher-affinity indicators. The results presented in the previous two sections, however,
indicate that because of the effects of local indicator
saturation, use of the single-compartment method will
probably introduce significant error in the estimates of
k
1 and k+1. It was therefore of interest to use the multi-compartment model to reestimate k
1 and k+1 values
for one of the higher-affinity indicators, as a means of assessing the direction and magnitude of possible errors in the previous estimates.
The indicator selected for this analysis was fluo-3,
which has been used in a number of recent measurements of local Ca2+ signals in muscle (e.g., Cheng et al.,
1993; Tsugorka et al., 1995
; Klein et al., 1996
; Hollingworth et al., 1998
). Results with the multi-compartment
model were compared with the traces and single-compartment analysis of Harkins et al. (1993; cf., their Fig.
8). These authors reported average values for k+1 and
k
1 of fluo-3 of 1.31 × 107 M
1 s
1 and 33.5 s
1, respectively (16°C), with the corresponding value of Kd,Ca being 2.56 µM. If these values are used in a multi-compartment calculation of the type shown in Fig. 6, a poor
fit of the simulated data of Harkins et al. (1993)
is obtained (not shown). Additional calculations were therefore carried out with the multi-compartment model to find values of k+1 and k
1 that gave a better fit to these
data. A good fit was obtained with k+1 and k
1 values
of 3.5 × 107 M
1 s
1 and 55 s
1, respectively (Kd,Ca of 1.57 µM). We conclude that the new estimates of k+1 and k
1
are probably closer to the actual rates that apply to fluo-3
in myoplasm and that the use of a single-compartment method probably underestimates the actual rates of
higher-affinity indicators. Although the new estimates for
fluo-3 are significantly larger than the previous estimates,
they are still markedly smaller than the rates reported for
the indicator in a simple salt solution (~8 × 108 M
1 s
1
and ~400 s
1 at 22°C and a viscosity of 1 cP; Eberhard
and Erne, 1989
; Lattanzio and Bartschat, 1991
).
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DISCUSSION |
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---|
A Multi-compartment Model That Includes Ca2+-binding to ATP and Diffusion of CaATP
In this article, we describe a multi-compartment model
of a half-sarcomere that calculates the spread of Ca2+
within a single myofibril of a frog twitch fiber. This
model is similar to that first developed by Cannell and
Allen (1994) but incorporates several important changes
that reflect current knowledge of the Ca2+ release process and the myoplasmic environment. These changes include a smaller value for the diffusion constant of
free Ca2+, a larger and briefer SR Ca2+ release event in
response to an action potential, and the inclusion of
ATP as a diffusible species that binds both Ca2+ and
Mg2+. As shown in MATERIALS AND METHODS, at the
free Mg2+ concentration of myoplasm (~1 mM), ATP
behaves as a low-affinity, rapidly reacting Ca2+ buffer.
Additionally, ATP (mol wt 507) has a diffusion constant in myoplasm (~1.4 × 106 cm2 s
1; 16°C) that is about
half that of Ca2+ (Kushmerick and Podolsky, 1969
).
Thus, there should be a significant transport of Ca2+
along the sarcomere in the CaATP form. Probably the
most important result of our modeling is that, without
ATP (or an analogous compound), the measurements
of spatially averaged
[Ca2+] in frog fibers could not
be adequately simulated with the SR Ca2+ release function that is thought to result from a single action potential (peak amplitude of ~140 µM/ms, half-width of ~2
ms). Thus, our calculations (a) implicate the presence
of a significant concentration of a low-affinity, rapidly
reacting Ca2+-binding species in myoplasm, and (b)
show that ATP, with properties as reported in the biochemical and physiological literature, satisfies the requirements of this species. Since ATP is present in myoplasm at millimolar concentrations, is mobile, and is capable of binding Ca2+, it seems hard to avoid the
conclusion that ATP plays a role similar to that inferred
with our model. It is possible, of course, that other
compounds, both mobile (e.g., phospho-creatine) and immobile (cf., Helmchen et al., 1996
), also contribute
to the effects that our model assigns to ATP. However,
the contribution of these other compounds appears to
be minor in comparison with that of ATP.
Other Implications of the Presence of ATP in the Model
The calculation in Fig. 2 shows that [CaATP] is expected to be three to four times larger than, and nearly
in kinetic equilibrium with,
[Ca2+]. Since the diffusion constant of ATP is about half that of free Ca2+ (Table I), it follows that more Ca2+ diffuses within myoplasm as CaATP than as free Ca2+. As shown by the
comparison between Figs. 4 B and 5 B, the diffusion of
CaATP helps synchronize the Ca2+-occupancy of troponin and thus activation of the myofilaments.
Given that the peak value of spatially averaged
[CaATP] is ~65 µM, the question arises whether the
presence of ATP requires that SR Ca2+ release be ~65
µM larger than it would otherwise be if ATP were not
present (or did not bind Ca2+). However, calculations
(not shown) indicate that, to achieve a given high-level
occupancy of the troponin regulatory sites, the amount
of extra SR Ca2+ that must be released due to the presence of 8 mM ATP is only about 30 µM (~270 µM in
the absence of ATP vs. ~300 µM in the presence of
ATP). The difference between 65 and 30 µM reflects the fact that Ca2+ quickly dissociates from ATP when
[Ca2+] declines (cf., Figs. 2 and 4). Thus, Ca2+ from
ATP is made available to bind to the troponin (and
other) sites. Since the multi-compartment model indicates that slightly more than half of the Ca2+ that is
bound by ATP is subsequently bound by troponin, the
extra load that ATP adds to the SR release requirement
is only ~10%. For comparison, the increase in load due
to the presence of parvalbumin is ~20%. This follows
from the observations that spatially averaged
[CaParv] also rises rapidly to about 65 µM but that the rate of Ca2+ dissociation from parvalbumin is very slow (cf.,
Table I). Thus, Ca2+ cannot dissociate from parvalbumin and be bound by troponin on the time scale of
twitch activation.
Other Comparisons between the Single- and Multi-compartment Models
We were initially surprised that the estimate of SR Ca2+
release from the single-compartment model without
ATP included supplied a satisfactory driving function
for the multi-compartment model with ATP included.
As considered in RESULTS, there appear to be several
sources of offsetting error in the single-compartment model that account for this situation. As shown by the
comparisons in Table II, the overall effect of these
errors is offset slightly better in the case of [Ca2+]
measurements with furaptra (if calibrated with a Kd,Ca
of 98 µM) than with PDAA, even though PDAA is
thought to give a more reliable estimate of the actual
time course of spatially averaged
[Ca2+].
Based on our multi-compartment model, we have reanalyzed the data reported by Harkins et al. (1993),
who estimated the values of k+1 and k
1 for the Ca2+-
fluo-3 reaction in myoplasm. While their reaction rates
provide a useful empirical way to relate, in a single-compartment calculation, the
fCaD signal of fluo-3 to the
[Ca2+] signal of furaptra (calibrated by the method of
Konishi et al., 1991
), these rates may not accurately reflect fluo-3's actual myoplasmic values of k+1 and k
1.
Indeed, our multi-compartment model indicates that
because of local saturation of the indicator with Ca2+
near the SR release sites, there is likely to be error in
the previous single-compartment analysis (as well as in
comparable analyses reported elsewhere in the literature for other indicators
see RESULTS). The new values that we estimate for fluo-3's k+1 and k
1 are 2.7- and
1.6-fold higher, respectively, than the rates reported by
Harkins et al. (1993)
. It should be noted, however, that
in our multi-compartment model of the half-sarcomere, Ca2+ enters the myoplasm only at the outermost
compartment nearest the Z line, whereas in reality,
some Ca2+ may enter a myofibril from adjacent myofibrils and/or SR release sites that are not in registration with the Z line. Thus, the local
[Ca2+] gradients
calculated by our model should probably be regarded as upper limits of the actual gradients. In support of
this conclusion, our previous experiments with PDAA
and furaptra (e.g., Fig. 6 B of Konishi et al., 1991
) indicate that the difference between the time courses of spatially averaged
[Ca2+] measured with these two indicators is slightly less extreme than calculated by our multi-compartment model (Figs. 6 A and 7 D). This, in turn,
suggests that the k+1 and k
1 values estimated for fluo-3
by the multi-compartment model may slightly overestimate the actual Ca2+-fluo-3 reaction rates in myoplasm.
Local indicator saturation probably also explains a
finding reported by Pape et al. (1993), who used a single-compartment approach to estimate the value of k+1
for fura-2 in the myoplasm of cut fibers from spatially
averaged PDAA and fura-2 measurements. They found
that the estimate of k+1 varied with indicator concentration, increasing about twofold, from 3.5 × 107 M
1 s
1
to 7 × 107 M
1 s
1, as myoplasmic [fura-2] rose from
0.5 to 2 mM. Since the spatially averaged
[Ca2+] signal in these experiments was relatively large at a [fura-2] of 0.5 mM but nearly eliminated at a [fura-2] of 2 mM,
it is possible that the estimate of k+1 increased because
local saturation of fura-2 with Ca2+ was substantially reduced at the higher fura-2 concentrations. This interpretation is supported by calculations (not shown) carried out with our multi-compartment model, which
were designed to simulate the experiment of Pape et al.
(1993)
. In these simulations, the estimate of k+1 also increased twofold as the fura-2 concentration was increased from 0.5 to 2 mM. (Note: The value of k+1 estimated for fura-2 in cut fibers is severalfold larger than
the value of k+1 estimated for fluo-3 in intact fibers [cf.,
last section of RESULTS]. Two effects probably underlie
this difference. First, the binding of indicator to myoplasmic constituents has been associated with reductions
in the values of indicator reaction rates, and fluo-3 appears to be more heavily bound in myoplasm than fura-2 [intact fiber measurements, summarized in Zhao et
al., 1996
]. Second, cut fibers appear to be ~40% more
hydrated than intact fibers [as judged by measurements
of intrinsic birefringence; Irving et al., 1987
]. Therefore, reaction rates, may, in general, be higher in cut
compared with intact fibers.)
A related point concerns previous estimations of the
resting level of myoplasmic [Ca2+] ([Ca2+]R) with high-affinity indicators such as fluo-3 or fura-2. Often, the
value estimated for [Ca2+]R depends directly on an associated estimate of the myoplasmic value of the indicator's Kd,Ca. If Kd,Ca is obtained from kinetic fits of spatially averaged measurements with two Ca2+ indicators,
the multi-compartment calculations indicate that the
value of Kd,Ca is probably overestimated. For example, with fluo-3, our multi-compartment analysis implies that Kd,Ca
for fluo-3 lies closer to 1.6 µM than to the 2.6 µM value estimated by Harkins et al. (1993). Corrected for this error,
the fluo-3 data of Harkins et al. imply a narrower range of
estimates for [Ca2+]R, 0.10-0.14 µM rather than 0.10-0.24
µM. This correction helps slightly to reconcile the ~10-fold difference for [Ca2+]R (0.03-0.3 µM) reported in
skeletal muscle with different techniques (cf., Baylor et al.,
1994
; Westerblad and Allen, 1994
).
Generalizations and Speculations
Since ATP is present in most cells at millimolar concentrations, it seems likely that the significant effects of
ATP deduced here on the shaping of local Ca2+ gradients also applies to the many other cell types that use [Ca2+] to control their activity. For example, in secretory cells, the binding of Ca2+ by ATP and the diffusion
of CaATP likely modify the amplitude and time course
of the local
[Ca2+] signals that control vesicle release.
A final speculation relates to the possibility that
[CaATP] might itself serve as an intracellular signal.
Since the value of Kd,Ca of ATP is large relative to
[Ca2+],
[CaATP] provides a rapid, local monitor of
the product of
[Ca2+] and total [ATP]. Thus, a large
[CaATP] indicates both a substantial
[Ca2+] and a
substantial total [ATP]. This signal might be used by the cell to activate novel regulatory pathways.
![]() |
FOOTNOTES |
---|
Address correspondence to Dr. S.M. Baylor, Department of Physiology, University of Pennsylvania School of Medicine, Philadelphia, PA 19104-6085. Fax: (215) 573-5851; E-mail: baylor{at}mail.med.upenn.edu
Original version received 8 May 1998 and accepted version received 29 June 1998.
A preliminary account of some of these results was previously published in abstract form (Baylor, S.M., and S. Hollingworth. 1998. Biophys. J. 74:A235).We thank Dr. W.K. Chandler for helpful comments on the manuscript.
This work was supported by a grant from the U.S. National Institutes of Health (NS 17620) and the Muscular Dystrophy Association.
![]() |
Abbreviations used in this paper |
---|
[Ca2+], free [Ca];
[CaT], the
change in total Ca concentration;
HFB, hypothetical fixed buffer;
PDAA, purpurate-di-acetic acid;
SR, sarcoplasmic reticulum.
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