Ca transients from Ca channel activity in rat cardiac myocytes reveal dynamics of dyad cleft and troponin C Ca binding

Sivan Vadakkadath Meethal, Katherine T. Potter, David Redon, Dennis M. Heisey, and Robert A. Haworth

Department of Surgery, University of Wisconsin, Madison, Wisconsin 53792

Submitted 12 May 2003 ; accepted in final form 22 September 2003


    ABSTRACT
 TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 
The properties of the dyad cleft can in principle significantly impact excitation-contraction coupling, but these properties are not easily amenable to experimental investigation. We simultaneously measured the time course of the rise in integrated Ca current (ICa) and the rise in concentration of fura 2 with Ca bound ([Ca-fura 2]) with high time resolution in rat myocytes for conditions under which Ca entry is only via L-type Ca channels and sarcoplasmic reticulum (SR) Ca release is blocked, and compared these measurements with predictions from a finite-element model of cellular Ca diffusion. We found that 1) the time course of the rise of [Ca-fura 2] follows the time course of integrated ICa plus a brief delay (1.36 ± 0.43 ms, n = 6 cells); 2) from the model, high-affinity Ca binding sites in the dyad cleft at the level previously envisioned would result in a much greater delay (>=3 ms) and are therefore unlikely to be present at that level; 3) including ATP in the model promoted Ca efflux from the dyad cleft by a factor of 1.57 when low-affinity cleft Ca binding sites were present; 4) the data could only be fit to the model if myofibrillar troponin C (TnC) Ca binding were low affinity (4.56 µM), like that of soluble troponin C, instead of the high-affinity value usually used (0.38 µM). In a "good model," the rate constants for Ca binding and dissociation were 0.375 times the values for soluble TnC; and 5) consequently, intracellular Ca buffering at the rise of the Ca transient is inferred to be low.

excitation-contraction coupling; adenosine triphosphate; fura 2; modeling; fuzzy space


EXCITATION-CONTRACTION COUPLING in heart occurs via the coupling of Ca influx via L-type Ca channels to the Ca-induced release of Ca (CICR) from the junctional sarcoplasmic reticulum (JSR) through ryanodine receptors (RyR) (15). Looseness in the coupling between Ca channels and RyR could contribute to the decline in function seen in some models of heart failure (16). The efficiency of CICR depends critically on the spatial relationships and kinetics of L-type Ca channels and RyR in the dyad cleft and also on cleft Ca concentration ([Ca]) (48) and sarcoplasmic reticulum (SR) Ca content (44). The time course of [Ca] changes within the cleft is potentially strongly influenced by Ca binding sites that may be present (26). Significant retardation of Ca diffusion from the cleft could strongly influence Ca efflux via Na/Ca exchange, which would also alter the Ca dynamics of excitation-contraction coupling (26, 35). However, the extent to which such binding sites retard Ca diffusion from the dyad cleft has not yet been experimentally determined. In one model of excitation-contraction coupling in which diffusion of Ca from the cleft space was included, a time constant for this process was specified arbitrarily (23); in others, cleft Ca binding sites were assumed but the effect of ATP was not considered (35, 46).

From modeling of Ca dynamics in skeletal muscle, a significant impact of ATP has been appreciated from its ability to bind Ca and facilitate its diffusion (3). ATP is thus likely to significantly impact the effect of Ca binding sites on Ca efflux from the dyad cleft. We have therefore developed a kinetic model for Ca efflux from the dyad cleft to the cytosol that incorporates the effect of ATP. Furthermore, we have incorporated fura 2 and have tested the model by comparing predictions of the kinetics of the rising phase of the Ca transient with high-resolution measurements of Ca transients using fura 2, for conditions under which all Ca enters via Ca channels. This has allowed us to constrain the model parameters to values that agree with experiment and to evaluate the impact of dyad cleft Ca binding sites on Ca kinetics. We find that even though ATP would attenuate the effect of the high-affinity binding sites in the cleft, the measured rise of the Ca transient is so fast that the presence of putative high-affinity Ca binding sites at the levels previously envisioned (26) can be excluded. Our data further suggest that the affinity for Ca binding to troponin C (TnC) is lower than that commonly used in modeling (1, 35) and more closely reflects that of isolated TnC. This also has the effect of substantially reducing the intracellular buffering of Ca seen by the rising phase of the Ca transient.


    METHODS
 TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 
Model

The model is based on the pioneering dyad cleft model formulated by Langer and Peskoff (26), which describes the time and position dependence of [Ca] in the dyad cleft after Ca entry through a Ca channel and SR Ca release, from the equation

(1)
where [c] is free [Ca]; Dc is the diffusion coefficient for Ca; r is the radial distance from the center of the cleft of height h; JSR, Jbnd, Jchnl, and JNCX are the fluxes of Ca per unit area into the cleft from the SR, from fixed sarcolemmal Ca binding sites, from the Ca channel, and from Na/Ca exchange, respectively; and t is time. For the modeling presented here, which focuses on the appearance in the cytosol of Ca that enters the cell via Ca channels, the Ca efflux from the SR and the Ca flux through the Na/Ca exchanger were set to zero. To include the effect of solutes that bind Ca such as fura 2 or ATP, two terms are added to the above equation for each solute f. Also, we did not use the equilibrium assumption of Langer and Peskoff for cleft Ca binding but included the association and dissociation of Ca from these sites explicitly

(2)
where kf+ is the rate constant for formation of the Ca complex fc from the solute f, kf is its rate of dissociation, and Jcbnd is the flux per unit volume from the cleft Ca binding

(3)
where kl+ and kl are the forward and reverse rate constants, respectively, for binding of Ca to the fixed cleft Ca binding sites l.

In the calculations of Peskoff et al. (36) and Langer and Peskoff (26), the Ca bound to these sites in the cleft was assumed to be in instantaneous equilibrium with the local [Ca], resulting in the flux Jbnd (Eq. 1). For all conditions reported here, even with the faster diffusion rates, results calculated with this assumption differ only very slightly from results calculated without this assumption but instead using the explicit binding and dissociation equation (Eq. 3) (data not shown). To avoid any uncertainty, we used Eq. 3 in the calculations presented here.

There are in addition similar flux equations describing the diffusion of each solute f

(4)

We make the approximation that the diffusion coefficients for solutes with Ca bound are the same as for free solute. With this approximation the total solute concentration ([f]t) is a constant for all r,t

(5)
Any gradient in fc will cause an equal and opposite gradient in free f, which will result in diffusion of free f into any element to replace fc that diffuses out of the element. Thus

(6)
Ca reuptake by the SR is not included for our purposes here, because the SR Ca pump is blocked. Equation 2 for the cytosol thus becomes

(7)
where r is now the radial distance from the T-tubule and Jvbnd is the flux of Ca per unit volume from cytosolic Ca binding sites, similar to Eq. 3.

Equations 4–6 for solutes still apply in the cytosol, with r now the radial distance from the T-tubule. To extend the model to the whole cytosol we defined an intermediate zone (zone 2) between the clefts (zone 1) and the cytosol (zone 3) (Fig. 1). Zone 2 has cylindrical symmetry like zone 3, coaxial with the T-tubule, and extends from one equivalent radius Req1 up to a second equivalent radius Req2. The equivalent radius Req1 is defined as the radius of the cylinder that has the same surface area as the surface area of dyad cleft edges, per unit length (µm) of T-tubule. Hence

(8)
where n is the number of clefts per axial micrometer of T-tubule and A is the area of one cleft edge

(9)
where Rd is the radius of the dyad cleft. Likewise, the equivalent radius Req2 is defined as the radius of the cylinder with surface area equivalent to that of the t-tubular surface not covered by dyad clefts. This is given by

(10)
where {rho}Ca is the density of Ca channels (15/µm2, from Refs. 6, 28) and Rt is the radius of the T-tubule. In this way, the interface area out of zone 1 matches that into zone 2 and the interface area out of zone 2 matches that into zone 3. Zone 2 provides a link between the geometry of zone 1 and that of zone 3, while retaining a simple geometry. No Ca binding sites were put into zone 2. Ca binding sites for calmodulin and TnC in zone 3 were initially those used by Balke et al. (1). The size of zone 3 is defined by

(11)
where Lt is the density of T-tubules (0.83 µm/µm3, from Ref. 32).



View larger version (29K):
[in this window]
[in a new window]
 
Fig. 1. Definition of model geometry. Zone 2 is an expansion zone that provides spatial continuity between the dyad cleft (zone 1) and the cytosol (zone 3). Values for parameters are given in Table 1. SR, sarcoplasmic reticulum. Diagram is not to scale, for clarity.

 


View this table:
[in this window]
[in a new window]
 
Table 1. Parameters used in model

 
To numerically find values for c(r,t) and fc(r,t), an explicit finite difference method was applied as described by Peskoff et al. (36). With this method, values of c(r,t) and fc(r,t) begin at time zero with resting values from r = 0 out to the edge of the cleft and through zones 2 and 3. Time is increased by a small step {delta}t, and values of c(r,t+{delta}t) and fc(r,t+{delta}t) are then calculated based on finite difference approximations of the differential equations and the values already calculated, as described in detail in Appendix B of Peskoff et al. (36). The flux of free Ca between zones 1 and 2 was determined by

(12)
where Jz12 is the flux per unit area at the interface, and ({delta}[c]/{delta}x)z12 is the concentration gradient between the last element of zone 1 and the first element of zone 2. Thus the flux of Ca between zones is controlled by Fick's law of diffusion across the equal-area interface, providing a continuous diffusion pathway with an expansion geometry that approximately emulates that seen in vivo. A similar relation was used to calculate flux between zones 2 and 3 and for other diffusible Ca-binding species, using their own diffusion coefficients. These fluxes were then included as source terms in the border elements. To otherwise prevent Ca loss from zone 1, [Ca] in the volume element just beyond the zone edge was, as a boundary condition, set equal to [Ca] in the last element of the zone (that is, in Eq. B6 of Ref. 36, where , [Ca] in the last element of the zone, was allowed to rise as Ca reached it). The effectiveness of this boundary condition was tested by setting Jz12 to zero: total Ca did not change after influx through the Ca channel, even as [Ca] at the cleft edge rose, as Ca equilibrated through the cleft. A similar boundary condition was also used for the outer edges of zones 2 and 3 and additionally at their inner edges: [Ca] in the volume element just before the inner edge of zone 2 was set equal to [Ca] in the first element of zone 2 and likewise for zone 3. The same procedure was used for other diffusible Ca-binding species. We used a step size of 2 nm in zone 1, 2 nm in zone 2, and 5 nm in zone 3; fixed time steps {delta}t of 2–10 ns were used, depending on how fast the diffusion coefficients of diffusible species were. Calculations were implemented in Microsoft Visual Basic, on a 450-MHz personal computer. The model parameters used are shown in Table 1.



View larger version (19K):
[in this window]
[in a new window]
 
Fig. 7. Effect of TnC Ca binding rates on the delay. Parameters used were as in Table 1, except that no Ca binding sites were in the dyad cleft, unless indicated. Both on-rates and off-rates were multiplied by the factors shown, keeping the affinity constant. A: effect of TnC Ca binding rates on the delay in the U(t). B: effect of TnC Ca binding rates on the delay in the W(t). The delay is the time for half-rise of W(t) minus the time for half-rise of the no-delay curve in Fig. 6C. {bullet}, No-cleft binding sites; {blacktriangleup}, with low-affinity cleft binding sites as in Table 1; dashed line, the delay observed for cell 1 in Fig. 4B.

 
Experimental

Krebs-Henseleit-HEPES buffer. Krebs-Henseleit (KH)-HEPES medium contained (mM) 118 NaCl, 4.8 KCl, 25 HEPES, 1.2 MgSO4, 2.0 CaCl2, 11 glucose, 1 probenecid (to inhibit fura 2 efflux from cells; Ref. 13), 5 Na pyruvate, and 0.05 diethylenetriaminepentaactic acid (to remove any heavy metals), with 1 µM insulin, adjusted to pH 7.4 with NaOH.

Cell isolation, labeling with fura 2, and SR inactivation. Cells were isolated from hearts of female retired breeder rats (19) excised after pentobarbital anesthesia in accordance with institutional guidelines. Cells were loaded with fura 2 by incubation with 2.5 µM fura 2-AM for 5 min at 25°C in KH-HEPES medium. Loaded cells were treated with 5 µM thapsigargin (TG) and 1 µM ryanodine (R) for 5 min at 37°C to inactivate the SR and then kept at 25°C with TG-R until being used. In the chamber, cells were superfused with KH-HEPES medium with or without 20 mM caffeine for 8 s before Ca transients were measured.

Field-stimulated Ca transients in KH-HEPES medium. Ca transients were measured on TG-R-treated cells in KH-HEPES medium at 37°C. Cells on the stage of a Nikon Diaphot microscope with a Nikon CF fluor x40 (oil, NA 1.3) objective were field stimulated with Pt electrodes and illuminated with a Till Photonics Polychrome IV xenon light source. The excitation wavelength was 360 nm, switched to 380 nm just before stimulation. Cell fluorescence was measured with a Till Photonics photodiode and custom LabView software. The Nikon filter cube contained a 400-nm dichroic mirror and a 460- to 625-nm long-pass filter (Omega Optical XF3091).

Simultaneous measurement of Ca transients and Ca currents. Ca currents were measured at 37°C under voltage clamp by using a Dagan 3100 patch-clamp amplifier with a 3911A whole cell expander unit and pCLAMP 6 software, according to the method of Puglisi et al. (38). Voltage clamp was via amphotericin B perforated patch. In brief, cells were superfused with Na- and K-free buffer containing (mM) 118 tetraethylammonium Cl, 4.7 CsCl, 25 HEPES, 1.2 MgSO4, 11 glucose, and 0.05 diethylenetriaminepentaacetic acid, with 1 µM insulin, adjusted to pH 7.4 with tetraethylammonium OH. After 10 min, perfusion was switched to the same buffer containing 2 mM CaCl2 and also 0.2 mM 4,4'-diisothiocyanostilbene-2,2'-disulfonic acid (DIDS) where indicated. Cells were patched with 2-M{Omega} electrodes with a pipette solution containing (mM) 80 Cs glutamate, 55 CsCl, 10 MgCl2, 10 HEPES, 0.1 EGTA, and 0.26 amphotericin B, pH adjusted to 7.4 with CsOH. Access resistance was 12.95 ± 4.4 M{Omega}. Capacitance and series resistance were compensated, and Ca currents were recorded with 20-ms voltage clamp steps to –13 mV (see RESULTS). Integrated Ca currents were calculated after subtraction of the mean current measured between 50–60 ms, where the current had become time invariant. Cell fluorescence was measured simultaneously with Ca current, as described above.

Fura 2 calibration. We previously (20) measured the Kd for intracellular fura 2 for Ca and found a value of 371 nM. Calibrating [Ca] also requires knowledge of Rmax and Rmin, where R is the ratio of the cytosolic 380-nm signal to the cytosolic 360-nm signal and Rmax and Rmin refer to the value of R under conditions of saturating [Ca] and zero [Ca], respectively. We measured Rmax and Rmin values for 3 µM fura 2 salt in the chamber at 37°C with and without Ca and found values of Rmax = 0.1381 and Rmin = 1.3118. These values were used to calculate cytosolic [Ca] at rest. The fluorescence from noncytosolic dye plus autofluorescence was measured by releasing cytosolic dye from cells in voltage clamp medium plus 25 µg/ml digitonin and 1 mM EGTA in place of 2 mM CaCl2. Because there was no Na in this medium, mitochondrial Ca was expected to be relatively stable during release of cytosolic dye under these conditions, and this was confirmed by the stability of the fluorescence ratio after dye release (Fig. 2). Cytosolic [Ca] in cells at rest ([cd] in Eq. 13) was then calculated after subtraction of the noncytosolic dye signal at each wavelength, as previously described (20). The fraction of cytosolic fura 2 with Ca bound in cells at rest is then given by the equilibrium relationship

(13)
where Kd is the dissociation constant of fura 2 (f in Eq. 13) for Ca (0.371 µM here, from Ref. 20) and [ft] is the total concentration of fura 2 in the cytosol (see below). These values were then used as the starting parameters in the model (Table 1).



View larger version (19K):
[in this window]
[in a new window]
 
Fig. 2. Calibration of fura 2 signal from fura 2-AM-loaded cell. Superfusion medium was the Na-free voltage clamp medium containing 2 mM Ca (see METHODS), which was switched to the same medium without Ca but containing 1 mM EGTA (pause –Ca +EGTA) and then to the same medium + 25 µg/ml digitonin (pause + digitonin) to release cytosolic dye. [Ca], Ca concentration.

 

Similarly, the fraction of cytosolic fura 2 with Ca bound can be calculated for cells during the Ca transient from the relationship

(14)
Note that this equation is valid even for situations in which dye and Ca ([c]) are not in equilibrium.

Estimation of magnitude of total Ca entry. The magnitude of total Ca entry is given by

(15)
where ICa is the unitary Ca current, {Delta}t is the time the channel is open, N is Avogadro's number (6.023 x 1023/mol), Z is the valence of Ca (2), e is the electronic charge (1.602 x 10–19 coulomb), and nt is the number of Ca channels per unit length of T-tubule, given by

(16)

Estimation of cellular fura 2 concentration. Cellular [fura 2] was estimated by comparison of the cell fluorescence per unit area at 360-nm excitation wavelength with the fluorescence per unit area of slides of fura 2 salt of known [fura 2] and known fura 2 film thickness. Such slides were made from known volumes of fura 2 salt solution in a KH-3-(N-morpholino)propanesulfonic acid (MOPS) medium (in mM: 100 KCl, 10 MOPS, pH brought to 7.2 with KOH) and sealed with Permount (Fisher Scientific); film thickness was calculated from (fura 2 volume)/(coverslip area). Fluorescence per unit area of fura 2 film was measured with a Till Photonics Imago cooled charge-coupled device camera from (mean pixel value)/(milliseconds of exposure) measured on different areas of the slide after subtraction of values from a slide without fura 2. To make these data independent of illumination intensity and hence usable over time, we normalized this mean pixel value/millisecond by comparison with the mean pixel value/millisecond from a fluorescent plastic slide, which we measured after the fura 2 salt measurements and also after every experiment with cells. We found that this normalized mean pixel value measured on fura 2 was proportional to [fura 2] at any film thickness and also to film thickness up to 20 µm at any [fura]. We therefore combined these data into a single linear relationship, with slope Slopefura, shown in Fig. 3. In an experiment with cells loaded with fura 2 (see above), the mean pixel value/millisecond was measured on a chosen area within a cell and the value from an adjacent equal area outside the cell was subtracted. This value was similarly normalized, to give the normalized mean pixel value NMPVcell. The cell thickness dcell in the area chosen was then measured by observing under bright-field illumination the focus change when the objective lens was z-axis translated, using a PI piezo objective driver. The cellular [fura 2] ([fura 2]cell) was then estimated from

(17)



View larger version (16K):
[in this window]
[in a new window]
 
Fig. 3. Fura 2 standard used for estimation of cellular fura 2 concentration ([fura 2]). Fluorescence excited at 360 nm was measured on slides with films of fura 2 of known [fura 2] and thickness (see METHODS). Each point is mean ± SD normalized fluorescence per unit area from 3 areas on a slide.

 

Statistics

The relationship between the normalized mean integrated Ca current and the normalized mean fluorescence change was determined as follows.

Let C(t) be the integrated Ca current as a function of time. Let F(t) be the mean fluorescence as a function of time. We fit the model F(ti) = C(ti{theta}) + {epsilon}i, where ti is the time of the ith data point, {theta} is a time-delay parameter, and {epsilon}i is random noise and measurement error. The least squares estimate of {theta} is the value that minimizes the sum of [F(ti) – C(ti {theta})]2 across all data points i = 1,..., N, a problem in nonlinear least squares. Note that because {theta} is a real-valued parameter, C(ti{theta}) will not generally correspond to an observed value. However, this is easily handled by first finding an approximating function to describe C(t). It was discovered that by carefully positioning three internal knots, a quadratic polynomial spline (also known as a grafted polynomial) resulted in an excellent approximation over the time period 0–30 ms (adjusted R2 >= 0.9999). (Because the plateau region had little influence in determining the delay parameter, time was truncated at 30 ms.) This allowed C(t) to be expressed as the linear equation C(t) = {alpha}1B1(t) +... + {alpha}6B6(t), where Bk(t) is a truncated quadratic power basis function and {alpha}k is a coefficient obtained from a linear regression least squares fit. An estimate of {theta} ({theta}*) was then obtained by using nonlinear least squares to obtain the best fit to the model F(ti) = C(ti{theta}) +{epsilon}i, representing C(u) with the previously estimated quadratic spline. An initial estimate of {theta} was first obtained without restricting the data. This estimate was then used to truncate the data so that t < {theta} would not occur, i.e., to avoid extrapolating the spline back before the Ca current was observed. After restriction, the model was then refitted. The change from the initial to the restricted estimate was small. SAS statistical software (PROC NLIN, SAS Institute, Cary, NC) was used for all calculations. To check these results, we found the value of the integer k that minimized the sum of [F(ti) – C*(tik)]2, where C* is the calcium current actually observed at time ti k. We observed that there was always very close agreement between titik and {theta}. The primary advantage of the nonlinear least squares approach is that standard asymptotic theory can be applied to obtain standard errors for the estimates.


    RESULTS
 TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 
Experimental

Cells loaded with fura 2 by the AM ester method contained 9.25 ± 2.90 µM fura 2 (mean ± SD, n = 14 preparations; see METHODS). Using these cells, we sought to measure fura 2 fluorescence changes and Ca current simultaneously, under conditions in which Ca influx was only via Ca current, the measured current was only Ca current, and that Ca current was the only source of the Ca transient.

To create a condition where Ca current is the only source of the Ca transient, cells were pretreated with thapsigargin and ryanodine, to inactivate the SR. We first determined how complete the elimination of SR Ca stores was. This was especially important because superfusion of cells in the bath in the absence of inhibitors could potentially reverse inhibition. To do this we measured the magnitude of Ca transients induced by field stimulation before and during (8 s after rapid switching) exposure of cells to KH-HEPES medium containing 20 mM caffeine. After a brief delay the Ca transient rose quickly to a plateau, as measured by the decrease in fura 2 fluorescence excited at 380 nm (supplemental Fig. S1A).1 These Ca transients appear to result almost entirely from Ca influx via L-type Ca channels, because inclusion of 5 µM KB-R7943 along with the caffeine did not affect the time course or magnitude of the Ca transient (data not shown). This level of KB-R7943 is reported to block reverse-mode Na/Ca exchange in rat heart cells (41), which would be the other possible Ca influx pathway under these conditions because rat heart cells contain no T-type Ca channels (9). Caffeine did however, decrease the magnitude of the Ca transients, from 11.8 ± 0.3% of total fluorescence down to 7.1 ± 1.6% of total fluorescence, a decrease to 60.7% of the transient magnitude without caffeine (n = 3 cells). To see whether this reduction was caused by release of residual SR Ca or by a direct effect of caffeine on the Ca current, the effect of a similar level of caffeine was measured on the Ca current measured under voltage-clamp conditions (see METHODS). A caffeine-induced decrease in the magnitude of the integrated Ca current to 61.6% of its original level was observed (supplemental Fig. S1B). We therefore conclude that most to all of the caffeine-induced decrease in Ca transient is the result of a direct inhibition of the Ca current and that the thapsigargin-ryanodine treatment alone is adequate to eliminate any significant contribution of SR Ca to the Ca transient measured under these conditions. This conclusion was further reinforced by the observation that inclusion of thapsigargin and ryanodine (up to 10 µM) in the superfusate had no further effect than the pretreatment on the Ca transients of thapsigargin- and ryanodine-pretreated cells (data not shown). Consequently, we did not routinely include these inhibitors in the superfusate. We also conclude that it is not necessary to perform the simultaneous measurement of Ca current and fluorescence after a rapid exposure to caffeine to ensure depletion of SR Ca. This is advantageous because the simultaneous measurement is itself technically challenging, and needing to incorporate a brief caffeine exposure would have made the measurement even more difficult.

To measure the Ca current we used the perforated patch method of Puglisi et al. (38), in which Na currents are eliminated by using a Na-free medium. This also ensures that there is no Ca entry by reverse-mode Na/Ca exchange, so that Ca current is the only route for Ca entry. To ensure that the measured current is only Ca current, it is important to eliminate Cl currents. One difficulty is that DIDS, which is used in the superfusate to block Cl currents under the conditions of Ca channel measurement (38), interfered with the fura 2 fluorescence measurement. To overcome this we used a voltage-clamp step to –13 mV, the calculated reversal potential of Cl based on the Cl concentrations in the bath and pipette, as a strategy that would allow us to omit DIDS from the bath. With this clamp step, the measured current was only slightly affected by the presence of DIDS (supplemental Fig. S2A).

Figure 4 shows the raw Ca transients and the simultaneously measured raw Ca currents from a train of five stimulii (Fig. 4A) measured on a fura 2-loaded cell under the Ca current measuring condition (see METHODS) without DIDS. We next removed the stimulus transient from the current trace by substituting a linear interpolation for the first 2 ms and last 2 ms of the voltage step (not shown). Integration of the Ca current after this adjustment gave a smooth curve; the mean normalized integrated Ca current from all five beats is shown in Fig. 4B along with the mean normalized fluorescence change. The change in the Ca transient lagged behind the integrated Ca current (Fig. 4B). The solid curve superimposed on the fluorescence data (Fig. 4B, the "good model") is the curve predicted from the model using Table 1 parameters, which is the end result of our modeling (see below).



View larger version (19K):
[in this window]
[in a new window]
 
Fig. 4. Simultaneously recorded Ca transients and Ca currents. A: superimposed train of 5 Ca transients measured at 380-nm excitation (0.5 Hz; negative deflection indicates [Ca] rise) along with their superimposed Ca currents measured simultaneously. B: mean normalized integrated Ca current compared with mean normalized Ca transient change, from the 5 pairs in A, recorded on cell 1. Integrated Ca current was 0.244 ± 0.003 pC/pF (mean ± SD); intracellular [Ca] rose from 254 ± 18 nM to 401 ± 11 nM (mean ± SD). Good model, normalized Ca transient change predicted from the model with Table 1 values.

 

Similar measurements were made on five other cells. To obtain the best estimate of the delay between the measured integrated Ca current and the measured Ca transient rise, we emulated the integrated Ca current by using a quadratic polynomial spline and used nonlinear least squares analysis to find the best fit of the time delay parameter (see METHODS). This gave a very accurate measure of the delay for each cell (Table 2). The standard deviation of the best fit values between cells suggests that there was some cell-to-cell variability. However, the delay was still very short for any cell (1.358 ± 0.43 ms, mean ± SD).


View this table:
[in this window]
[in a new window]
 
Table 2. Best-fit values of delay between integrated Ca current and fluorescence change

 

Because the difference in kinetics of the integrated Ca current and the Ca transient was so small, we examined some sources of error that may contribute to that difference. First, we investigated the impact of the simple linear adjustment to the Ca current data on the time course of the integrated Ca current trace. This was evaluated by comparison with a more complex adjustment, in which the measured Ca current trace between 1 and 20 ms was fit to a later-rising multiexponential function. This function caused a slight early delay in the rise of the integrated Ca current (supplemental Fig. S3), and the time to 50% rise of the integrated Ca current increased from 8.875 to 9.037 ms, an increase of 0.162 ms. Next, we investigated the contribution of residual Cl current to the time course of the integrated Ca current. The effect of the Cl current inhibitor DIDS (0.2 mM) on the integrated normalized current was to speed up the later part of the rise by ~0.3 ms (supplemental Fig. S2B). We also measured the response time of the measuring photodiode to light emitted from a light-emitting diode. This had a time to 50% rise of 0.28 ms (supplemental Fig. S3).

Any realistic model of Ca and fura 2 diffusion, including the dyad clefts, must account for the shortness of the measured delay (Table 2).

Modeling

Comparison of the measured data with the predictions of our model at first appears complex. The Ca current measured at a particular time reflects not only the number of channels open but also the unitary current defined by the voltage of the clamp potential at that time. The Ca transient is thus impacted not only by the delay introduced by the efflux of Ca from the cleft but also by the different times of Ca channel opening and by two different unitary Ca currents, corresponding to that at –13 mV and that on repolarization to –70 mV. The unitary currents for these voltages with 2 mM extracellular Ca are calculated to be 0.152 and 0.315 pA, respectively, based on a reversal potential of 40 mV, a maximum conductance of 5.3 pS, and a Kd of 1.7 mM (18). The complexity introduced by multiple unitary currents can be simplified, because the predicted rise in concentration of fura 2 with Ca bound [Ca-fura 2] resulting from time zero opening of Ca channels is almost exactly proportional to the unitary current over its entire time course (supplemental Fig. S4). This implies that the rise in [Ca-fura 2] can be defined per unit of total current, regardless of the unitary current: the total current at any time, rather than the unitary current, is the key quantity in defining the subsequent rise in [Ca-fura 2]. The measured Ca current describes this total current at different times. The measured [Ca-fura 2] transient can thus be predicted from the time profile of [Ca-fura 2] predicted for time zero Ca channel opening simply by taking into consideration the time dependence of the magnitude of the total current. The measured [Ca-fura 2] transient at any time is the sum of the [Ca-fura 2] transients resulting from the Ca current entering at each time up until that time. The measured whole cell [Ca-fura 2] transient is then given mathematically by the predicted whole cell [Ca-fura 2] profile for time zero Ca channel opening convoluted with the normalized Ca current, after adjusting current magnitude to ensure that the total Ca entry in the model is the same as the total Ca entry measured. This is described in detail below.

To get the [Ca-fura 2] profile for time zero Ca channel opening we first calculate the spatiotemporal distribution of all Ca-containing species for a current of 0.152 pA flowing for 0.1 ms from time zero. We chose this time for convenience because 0.1 ms is the time resolution of the data points for Ca current measurement and for [Ca-fura 2] measurement. This is not meant to imply that 0.1 ms is the average open time of single channels. The predicted dyad cleft [Ca], cytosolic [Ca], and cytosolic [Ca-fura 2] for the conditions in Table 1 are shown in Fig. 5, A–C, respectively. The diastolic [Ca] and total [fura 2] values used in the model, shown in Table 1, were the values measured for cell 1 in Fig. 4 (see METHODS). We adjusted the other parameters in Table 1 to fit the model to the data. As described below, this mainly required the use of low-affinity Ca binding parameters for TnC, like those measured for soluble TnC (14) rather than the value commonly used for myofibrillar TnC (39). The affinity of calmodulin for Ca was reduced to values that are more like those of unbound calmodulin (29); for simplicity, they were set the same as those chosen for TnC (see below and DISCUSSION for justification). Also, SR Ca pump binding of Ca was not included, because in cells treated with thapsigargin not only is the Ca pump inhibited but also Ca binding to the pump is inhibited (40). The predicted whole cell [Ca-fura 2] profile was then given by the integration over space of the [Ca-fura 2] at each time.



View larger version (38K):
[in this window]
[in a new window]
 
Fig. 5. Calculated dyad cleft and cytosolic free [Ca] for good model conditions. Parameters used were as in Table 1. A: [Ca] in the dyad cleft for 0.152-pA unitary current. B: [Ca] in the cytosol. C: concentration of fura 2 with Ca bound ([Ca-fura 2]) in the cytosol. Note that the time axis scale in A is different from that in B and C.

 

Next, we adjusted the total Ca entry in the model to be equal to the measured Ca entry. To do this, we expressed the modeled Ca entry in picocoulombs per picofarad

(18)
where CA is the capacitance per unit area, taken as 1 µF/cm2. The [Ca-fura 2] profile for time zero channel opening was then adjusted for magnitude by multiplying the modeled increase in [Ca-fura 2] by a factor that is the measured Ca entry (from the integral of the measured Ca current, 0.244 pC/pF; Fig. 4) divided by 0.0228 pC/pF (Eq. 18). This gives a [Ca-fura 2] profile U(t) predicted by the model for the measured amount of Ca entry had the Ca entry all occurred between time zero and 0.1 ms. The relative magnitude of these numbers also indicates that each Ca channel was open for an average total time of 0.1 x 0.244/0.0228 = 1.07 ms. Some predicted U(t) are shown in Fig. 6A for the parameters given in Table 1 except that Ca binding parameters were varied for each curve as shown.



View larger version (23K):
[in this window]
[in a new window]
 
Fig. 6. Comparison of predicted [Ca-fura 2] change with measured change. A: [Ca-fura 2] profile [U(t)] predicted for 0.152-pA Ca current and then scaled to the measured total Ca entry (0.244 pC/pF); parameters used were all from Table 1 except for the following comparisons: low-aff TnC: soluble TnC parameters (Kd = 4.565 µM, kl+ = 200 µM–1 · s–1, kl = 913 s–1) from Ref. 14, adjusted for temperature as described in the text; high-aff TnC: TnC parameters from Ref. 39 (Kd = 0.38 µM, kl+ = 39 µM–1s–1, kl = 13 s–1); cleft sites: parameters from Table 1. U(t) is the [Ca-fura 2] profile that would result if all the Ca channels opened simultaneously, from time zero to 0.1 ms. B: time dependence of normalized Ca entry [Pr(t)] from measured Ca current. C: convolution of A and B to give predicted whole cell [Ca-fura 2] for actual Ca entry (lines as in A) compared with measured [Ca-fura 2] (dots). The no-delay curve used a U(t) function that rose from 0 immediately (0.1 ms) to the plateau value of the low-aff TnC no-cleft sites curve shown in A.

 

Actual predicted whole cell signals W(t) were then calculated by convolution (*) of U(t) with the probability function Pr(t) (the inverse of the normalized mean Ca current; Fig. 6B), which describes the fraction of Ca entering at any time

(19)
This predicted profile for one of the beats in Fig. 4, along with the measured [Ca-fura 2] profile expressed as the fraction of cytosolic [Ca-fura 2] (from Eq. 14), is shown in Fig. 6C. These simulations are quite informative, as described below.

First, the presence of both low- and high-affinity Ca binding sites within the dyad cleft as envisioned by Langer and Peskoff (26) strongly delayed the 50% rise time of U(t), up to 4.2 ms, and also had some impact on the cell buffering capacity (Fig. 6A). This delay translated into a predicted delay in the rise of W(t) that was not observed in the experimental data (Fig. 6C). This was the case whether TnC had low affinity ("low-aff TnC low-and high-aff cleft sites") or high affinity ("high-aff TnC low- and high-aff cleft sites"). Without cleft Ca binding sites, a fast, early rise in U(t) was seen (0.7 ms was needed for 50% rise), followed by a slower rise to the plateau value ("low-aff TnC no-cleft sites," Fig. 6A).

Second, and perhaps the most striking finding, was the impact of TnC Ca binding parameters on U(t). Although removal of both low- and high-affinity cleft sites speeded up the rise time of U(t) and W(t), only the low-affinity TnC (low-aff TnC no-cleft sites) reached a steady plateau like the experimental data. If TnC Ca binding was high affinity ("high-aff TnC no-cleft sites"), and even if there were also low- and high-affinity cleft sites (high-aff TnC low- and high-aff cleft sites), a rapid transient peak was predicted in W(t) that was not observed experimentally (Fig. 6C). The parameters used here for low-affinity Ca binding by TnC were those measured on soluble TnC (14), whereas those used for high-affinity Ca binding were those measured on myofilaments (39). The latter values, or similar, are commonly used in modeling (1, 7, 35, 45, 49).

Third, adding back the low-affinity Ca binding sites in the cleft made the early rise slightly slower, requiring 1.3 ms for 50% rise in U(t) ("low-aff TnC low-aff cleft sites," Fig. 6A). This suggests that most of the delay seen when both sites were present was the result of the high-affinity sites. The later, slower rise was less affected by adding back the low-affinity Ca binding sites (Fig. 6A).

In the absence of ATP the rise times for U(t) were changed to 0.3, 1.6, and 6.5 ms for no-, low-, and low- plus high-affinity binding sites, respectively (not shown). Thus, in the absence of Ca binding sites, the rise in [Ca-fura 2] was actually slightly retarded by ATP, whereas in the presence of Ca binding sites, and especially high-affinity sites, the rise in [Ca-fura 2] was strongly promoted by ATP.

Inspection of the curves in Fig. 6A shows that although the Ca binding properties of the cleft determined the early part of U(t), the TnC Ca binding parameters determined the later part. Moreover, the low-affinity TnC values actually appeared to slow this later rate of rise of U(t) (compare high-aff TnC no-cleft sites with low-aff TnC no-cleft sites, Fig. 6A). But was the effect of low-affinity TnC a consequence more of the higher on-rate for Ca than of the low affinity? We tested this by changing both the on- and off-rates for TnC Ca binding by the same factor, so that the equilibrium binding constant kept the same value. This showed that indeed the rate of rise of the later phase of U(t) was accelerated by reducing these rates, until a point where a peak began to emerge (Fig. 7A). Furthermore, inspection of Fig. 6C shows that the slower rate of rise of the later phase of U(t) had a significant impact even on the early rate of rise of the convolved function W(t): the curve for low-aff TnC no-cleft sites rose much more slowly than the curve for high-aff TnC no-cleft sites (Fig. 6C), even though both had the same very fast early rise in U(t) resulting from little delay in the cleft (Fig. 6A). In fact, this delay in W(t) caused by low-affinity TnC was considerably greater than the delay caused by low-affinity, but not high-affinity, sites in the cleft (Fig. 6C). This was confirmed by using the U(t) functions in Fig. 7A for convolution. Soluble TnC Ca binding and dissociation rates (Fig. 6) gave a U(t) that resulted in a delay of 1.7 ms in W(t) (Fig. 7B), much greater than was observed experimentally (Fig. 4B), whereas using TnC rates that were a factor of 0.375 of these values [which gave the fastest U(t) without emergence of a peak (Fig. 7A)] resulted in a delay of only 0.65 ms in W(t), in the absence of dyad cleft Ca binding sites (Fig. 7B). Adding the low-affinity binding sites to the cleft simply added a constant amount to the delay in W(t) (Fig. 7B), an amount similar to the time to half-rise of U(t). This shows that TnC Ca binding kinetics (Fig. 7B) as well as dyad cleft Ca binding properties (Fig. 6C) both strongly influence the measured delay between integrated Ca current and the Ca transient. Moreover, TnC Ca binding kinetics had almost no impact on Ca efflux from the cleft: time to half-efflux from the cleft was only 0.3% faster with 2x TnC rates than with 0.375x TnC rates. A model based on these lower (0.375x) TnC rates, plus low-affinity Ca binding sites within the dyad cleft (parameters in Table 1), can thus nicely account for the delay observed between the integrated Ca current and the Ca transient (Figs. 4C and 7B, good model).

Because the experimental data caused us to favor the faster U(t) functions, we became concerned that the spatial approximations made by the model could have a significant impact on these functions. We tested this by altering the model in two ways. First, decreasing the radius of the dyad cleft to 40 nm speeded up U(t), as expected, and increasing it to 120 nm slowed it down (Fig. 8A, Table 3). However, the average of U(t) for 40- and 120-nm clefts was similar to U(t) for 80-nm clefts (Fig. 8A). In reality, there will be a distribution of cleft radii with some mean. Running a convolution in which one-third of the clefts had each (40, 80, and 120 nm) radius gave a function W(t) that closely overlaid W(t) for all clefts with 80-nm radius (not shown). Second, we removed zone 2 so that Ca leaving the dyad cleft went directly into zone 3, the cytosol, which was also altered to begin (more realistically) at the T-tubule surface. This change had no effect on the time course of U(t) (Fig. 8B). This alteration did, however, increase the inaccuracy of the calculation: total Ca was 5.4% in error, compared with 2.1% using all three zones, under the conditions of Fig. 8B.



View larger version (18K):
[in this window]
[in a new window]
 
Fig. 8. Effect of model geometry on U(t). Parameters used were as in Table 1. A: effect of cleft radius. B: effect of zone 2. No Ca binding sites were in the dyad cleft for B.

 

View this table:
[in this window]
[in a new window]
 
Table 3. Times to half-efflux of Ca from clefts and to half-rise of U(t)

 

A notable difference between the U(t) curves based on the TnC Ca binding parameters in Fig. 6 is that the extent of cytosolic Ca buffering is much greater for the high-affinity TnC than for the low-affinity TnC (supplemental Fig. S5). The high-affinity parameters predict buffering that is similar to or greater than that calculated in previous studies in ferret heart cells, where Ca efflux after caffeine was measured by Na/Ca exchange current and Ca transients were measured with fluo 3 (50). On the other hand, the low-affinity parameters (which fit our data) predict cytosolic Ca buffering closer to or less than that measured in rat heart cells, where Ca transients measured with indo 1 were compared with Ca entry measured from a Ca current (5). This curve ("modified Berlin et al.," supplemental Fig. S5) has been modified to account for a too-low Kd value used for intracellular indo 1 in Ref. 5, as found subsequently (2). Even though the measured level of total fura 2 in the cell is low, it still has a significant impact on the cytosolic buffering, because the buffering is so low (supplemental Fig. S5). Removing the fura 2 from the model shows, however, that this low level of fura 2 does not much impact the dyad cleft Ca kinetics or the cytosolic [Ca] transient (data not shown).

Finally, the other major determinants of Ca kinetics in the dyad cleft are the diffusion coefficient chosen for Ca and the unitary current. Peak [Ca] increased linearly with unitary Ca current, as was found by others (46), but was affected little by diffusible carriers (supplemental Fig. S6). Peak [Ca] increased markedly as the diffusion coefficient was reduced. The cytosolic Ca diffusion coefficient had little impact on peak [Ca] in the cleft when the cleft Ca diffusion coefficient was held constant. Thus the use of the Peskoff and Langer (26) diffusion coefficient for cleft Ca (1 x 10–6 cm2/s) had a strong impact, increasing peak [Ca] to 430 µM for 0.3-pA unitary current. Removing ATP had little impact on this, increasing peak [Ca] to 432 µM for 0.3-pA unitary current. The cleft Ca diffusion coefficient also had a significant impact on times to half-efflux of Ca from the cleft (supplemental Fig. S6; Table 3). In summary, peak [Ca] is strongly influenced by the choice of diffusion coefficient for cleft Ca and by unitary Ca current but not by diffusible carriers, even though the latter may strongly influence Ca efflux from the cleft.


    DISCUSSION
 TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 
Effect of Cleft Ca Binding Sites

Previous dyad cleft modeling showed that cleft Ca binding sites, if present, could strongly retard radial diffusion of Ca (26, 46), and our modeling also supports that conclusion (Table 3). A significant finding of our study, however, is that the existence of putative high-affinity dyad cleft Ca binding sites as envisioned in previous work (26) is unlikely, because of the brevity of the delay observed between the integrated Ca current and the Ca transient (Fig. 4B). If high-affinity sites were present to that extent, we would expect a delay about threefold greater than that seen (Table 3). In the modeling of Langer and Peskoff (26), both low- and high-affinity Ca binding sites were presumed to be present in the dyad cleft on the basis of analysis of Ca binding to isolated sarcolemmal preparations by Post and Langer (37). The low-affinity sites appeared to be phospholipids, whereas the high-affinity sites were thought to be proteins (37). Although these binding sites were measured on gasdissected membranes, their presence within dyad clefts was never specifically demonstrated.

The above conclusion does not mean that no high-affinity Ca binding sites at all are present in the clefts. Specifically, Ca binding sites on the RyR will be present. The density of high-affinity sites (0.016 nmol/cm2) from the Post and Langer (37) measurements translates to 1,938 sites for an 80-nmradius cleft. A tetrameric RyR is ~30 nm square (49), so that even when closely packed only ~16 could completely fit into an 80-nm cleft. If each monomer binds 1 Ca at the activator site and 1 on each calmodulin, this would correspond to 128 Ca binding sites, <7% of the high-affinity sites measured by Post and Langer. The effect of this low level of high-affinity sites would not be resolvable from our measurements.

Effect of ATP

ATP has been found to have a small but significant impact on rates of Ca diffusion in models of skeletal muscle (3, 21). The magnitude of its effect was considerable in our model when Ca binding sites were included in the cleft (Table 3). The geometry of the cardiac dyad cleft, in combination with the Ca binding sites, could retard Ca like an ion exchange column. ATP will compete with these sites for Ca binding and will carry Ca out of the cleft by diffusion. This could account for the size of the ATP effect, and it underscores the importance of including ATP in cardiac modeling. Previous dyad cleft modeling did not include ATP (26, 46).

Effect of Ca Diffusion Coefficients

From our measurements, the Ca diffusion coefficient within the dyad cleft cannot be as low as that used by Langer and Peskoff (26), unless there are no Ca binding sites in the cleft at all, which is unlikely. Low DCa values have been used extensively (26, 46) because of obstruction of Ca by dyad cleft "feet" that restrict radial diffusion. Our results suggest that radial diffusion may not be very restricted.

Effect of TnC

The data presented provide strong evidence that TnC Ca binding affinity is low at the onset of the Ca transient: first, from the absence of a peak in the transient (Fig. 6C) and second, from the change in plateau [Ca] resulting from the measured amount of Ca entry (Fig. 6C). The latter depends on the validity of the calibration (see below), but the former does not. The predicted peak in the transient does, however, result from the presumed much greater on-rate of fura 2 for Ca than of TnC for Ca. Because the off-rate we used (Table 1) was measured with fura 2 in protein-free solution, rather than with intracellular fura 2, the predicted peak height could be too high, if the intracellular on- and off-rates of fura 2 for Ca are a lot lower than their values in protein-free solution. Indeed, there is evidence that at 16°C the Ca-fura 2 off-rate in skeletal muscle fibers is a factor of 4 slower than its value in free solution (4). We therefore investigated the impact of reducing the fura 2 on-rate and off-rate for Ca by a factor of 4. We kept the same Kd (Table 1), because this is an intracellular value (20). Reducing these rates did decrease the peak in U(t) (supplemental Fig. S7A). However, a peak was still evident, and after convolution it is clear that the predicted W(t) for this condition still has a large predicted decline after the peak (supplemental Fig. S7B), which is not observed experimentally. Moreover, when the low-affinity TnC Ca binding parameters are used (Table 1), U(t) is slowed dramatically and so is W(t) (supplement Fig. S7). This clearly also could not fit the observed data. Thus our data are consistent only with low-affinity TnC and fast fura 2 Ca binding.

There is some uncertainty about the level of calmodulin in the heart, depending on which assay is used to measure it (12, 24). Because of this, the value we used (Table 1), which is based on phosphodiesterase activity (15, 24) could be an underestimate rather than an overestimate (12). Such an error would result in a too-high value being used for the Ca binding affinity in Table 1, rather than a too-low value. Reducing the content of TnC and/or calmodulin in the model reduces Ca buffering and hence raises the plateau [Ca-fura 2] in the same way as lowering the affinity. It does not, however, affect the shape of the Ca transient in the same way as lowering the affinity. This was tested by reducing the TnC content twofold. This change increased the plateau, but the peak height above the plateau was retained undiminished, resulting in a predicted W(t) with a large peak, which was not seen experimentally (supplemental Fig. S8). Errors in the values used for the content of these Ca binding proteins therefore cannot account for our observations.

Because of the absence of a peak in the measured transient, as well as the change in plateau [Ca] resulting from the measured amount of Ca entry (Fig. 6C), we are encouraged to believe that the low measured Ca buffering under these conditions is not an artifact of calibration but rather reflects the true affinity of TnC under these conditions. Even though cardiac modeling has extensively used the high-affinity Ca binding parameters of Robertson et al. (39), evidence has long existed that myofibrillar TnC in the absence of actomyosin interaction binds Ca with low affinity: Ca binds to isolated TnC with low affinity (~105 M–1), which increases by a factor of 10 on incorporation into troponin (~106 M–1), but then the affinity drops back down again (to ~105 M–1) when troponin is incorporated into the thin filament (53).

A further improvement of fit was achieved by reducing the rate constants for TnC Ca binding and release while keeping the affinity low and constant (Fig. 7A). This actually speeded up the predicted [Ca-fura 2] transient (Fig. 7B) by decreasing the competition between TnC and fura 2 for Ca. Conversely, the TnC rate constants had essentially no impact on dyad cleft Ca kinetics (Fig. 7B). Thus, although we conclude that TnC affinity is low like that of soluble TnC, to fit the data while allowing low-affinity cleft sites requires that its Ca exchange rate could well be lower by a factor of ~0.375 (Fig. 7B).

Accounting for Observed Delay: the Good Model

What, then, best accounts for the delay between the integrated Ca current and the Ca-fura 2 transient (Fig. 4B)? About 0.3 ms of this comes from the response time of the photodiode (supplemental Fig. S3C); however, this may be balanced by the effect of residual Cl current, which could have delayed the measured integrated Ca current by ~0.3 ms (supplemental Fig. S2B). Using the simple linear interpolation to remove the stimulus artifact may have advanced the integrated Ca current by up to ~0.16 ms, based on the effect of using alternative interpolation methods (supplemental Fig. S3B). We think, however, that the linear interpolation probably is the most accurate.

There therefore likely still remains ~1–1.2 ms in the delay between the integrated Ca current and the Ca transient to account for. From our calculations, half of this could come from TnC Ca binding with 0.375x the rates of soluble TnC and half could arise from low-affinity sites within the dyad cleft (Table 3). A curve calculated with these parameters is shown in Fig. 4B, labeled as the good model, and is seen to fit the data quite well. This, of course, is not a unique solution, because the total absence of cleft Ca binding sites combined with rather faster (0.5x) TnC Ca binding rates could easily give a similar delay (Fig. 7B). It would, however, seem unreasonable not to include low-affinity sites in the clefts, because phospholipids are very likely to be present. If low-affinity binding sites are present in the cleft, then the effect of all other parameters can only add up to the remaining half of the delay observed. This means that the TnC Ca binding kinetics that give the shortest delay without introducing a peak are to be preferred, hence our choice of rates for soluble TnC, 0.375x (Table 1). Also, the mean distance from Ca channels to the cleft edge could be closer to 40 nm than to 80 nm, and this would save 0.97 – 0.72 = 0.25 ms (Table 3). This falls far short of allowing enough time to accommodate the delay of milliseconds predicted for when high-affinity sites are present. Indeed, the entire observed delay is much shorter than the delay just from high-affinity sites would predict (Table 3). This does allow us to exclude the presence of high-affinity Ca binding sites, at the level previously envisioned (26), within the construct of the model.

Ca Buffering

A corollary to the low affinity inferred for TnC is that the extent of cytosolic Ca buffering is low, because TnC makes up the major part of the Ca buffer (supplement Fig. S5). Ca buffering measured in our study is similar to that concluded from a previous study with indo 1 (5) after reconsideration of the intracellular Kd for indo 1 (2). That study (5) was also based on changes in [Ca] resulting from Ca entry via Ca channels. Equilibrium Ca binding studies on ventricular homogenates, on the other hand, give rather higher values for Ca buffering (2). A higher value for cytosolic Ca buffering was also measured for ferret myocytes by using Na/Ca exchange current after caffeine-induced SR Ca release as a measure of total Ca change while measuring the Ca transient with fluo 3 (Ref. 50; supplemental Fig. S5). We think that there is no conflict here, but rather the intracellular buffering capacity of the myocyte during a rapid Ca transient is a dynamic rather than a fixed quantity. This was also clearly evident in the study of Berlin et al. (5). Each buffering measurement accurately reflects Ca buffering under the measurement conditions used.

Methods like the caffeine method (50) that measure total Ca coming down from elevated [Ca] levels could indicate more Ca binding and slower Ca release (higher-affinity Ca binding) than we see in our measurements for two reasons. First, TnC has Ca/Mg binding sites with high affinity for Ca, in addition to the Ca-specific regulatory site (14). The high-affinity sites exchange slowly on the time scale of our measurements, but they will be significantly more occupied during a caffeine-induced Ca transient of several seconds' duration. Second, Ca binding at the regulatory site of TnC is very state dependent. In the absence of actomyosin interaction, myofibrillar TnC binding of Ca is low affinity (53). A low-affinity value for TnC Ca binding in cells at rest (Table 1) therefore makes physiological sense. Cross-bridge interaction with the thin filament during force development then reversibly increases the affinity of TnC for Ca (17, 31, 51).

The value we used for soluble TnC off-rate in (Fig. 6) is from isolated TnC, adjusted to 37°C with a Q10 of 1.813 (14). In that study, a value of 140 µM–1s–1 was reported for the on-rate at 4°C at [Ca] <1 µM. This rate has a low Q10 (14), resulting in the value (200 µM–1s–1) for 37°C we used in Fig. 6. The high-affinity value used for cardiac troponin C (Fig. 6 and Ref. 39) has been widely used in modeling studies (1, 7, 35, 49). In that study (39), a very slow off-rate from myofilaments was measured experimentally from Ca dissociation from myofilaments in high [Ca]; the slow on-rate (39 µM–1s–1) was then inferred from the measured off-rate and the equilibrium binding constant (22). This on-rate may actually be close to correct: our modeling supports a value of 75 µM–1s–1 (Table 1). The real variable, controlled by actomyosin interaction, is in the off-rate.

In summary, these considerations imply that the intracellular buffering of Ca is considerably lower when rapid changes in [Ca] from resting levels are measured than in studies where equilibrium or slow Ca binding is measured. This lower Ca buffering, therefore, rather than the higher buffering, is more appropriate for use in modeling conditions during excitation. This phenomenon may well have a physiological benefit: low Ca buffering during excitation will promote even myofilament activation, by allowing Ca to diffuse more extensively before thin filament activation (and affinity increase) occurs.

Sources of Error

Several sources of error could have impacted our measurements. It is possible that a small amount of SR Ca remained after the treatment of cells with thapsigargin and ryanodine because of reversal of pump inhibition during superfusion in the bath. At first, we thought that such reversal was indeed occurring significantly, because the size of Ca transients was reduced when measured during a brief exposure to caffeine (supplemental Fig. S1A, Data). This level of caffeine did, however, inhibit the integrated Ca current to a similar extent (supplemental Fig. S1B), suggesting that most if not all of the decrease in the Ca transient resulted from a direct inhibition of the Ca current. Such a direct effect of caffeine on the Ca current has also been observed by others (52). We therefore consider the contribution of residual SR Ca to the Ca transient to be minimal under the conditions used without caffeine.

The estimate of cellular [fura 2] is only an approximation, because cell autofluorescence is not subtracted from the fluorescence value used in the estimate (see METHODS), and also it is assumed that the fluorescence of intracellular fura 2 per mole is the same as that of fura 2 salt in KH-MOPS buffer. Cell autofluorescence for cells loaded under these conditions is ~25% of the total fluorescence (20). A significant amount of intracellular fura 2 is bound to protein (8). This, and high viscosity, could both increase fura 2 fluorescence by as much as 50% (10, 25). These factors will tend to result in an overestimate of [fura 2]. Also, dye distribution within the cell is not homogeneous. Dye loading by the AM ester method results in dye loading into noncytosolic (probably mitochondrial) compartments (20, 30). Because the fraction of noncytosolic dye is similar to the fraction of the cell volume that is not cytosolic, we have simply used the estimate of mean cell dye concentration as our estimate of cytosolic dye concentration without attempts at correction.

Inaccuracy in the estimate of [fura 2] is acceptable because the dye concentration (at this level) is not a critical determinant of either the modeled dyad cleft Ca kinetics or the measured [Ca]. It has little impact on the Ca kinetics because there is not enough dye in the dyad cleft to significantly compete with the phospholipids and ATP for binding Ca. It does impact [Ca] by buffering Ca (supplemental Fig. S5), but the measurement of [Ca] is otherwise independent of dye concentration because the dye is ratiometric.

There are approximations associated with the calibration of the fura 2 fluorescence signals. First, a potential error comes from the assumption that mitochondrial [Ca] does not change between beats. The first beat should be accurately calibrated, because the calibration was done on cells at rest under identical conditions (Fig. 2). On the subsequent beats we assumed that the fluorescence change between beats was all associated with cytosolic [Ca] changes, which may not have been the case, because changes in mitochondrial [Ca] could also have occurred. The impact of this assumption was, however, minimal, because the variance of the starting [Ca] calculated with our assumption was only 18 nM (Fig. 4). Moreover, the difference between starting and plateau [Ca] for each beat was 147 ± 7 nM, an even smaller variance that reflects the fact that both starting and plateau cytosolic [Ca] gradually increased with beat number (Fig. 4). If this gradual increase actually reflected mitochondrial and not cytosolic [Ca] changes, the difference per beat would be even more constant between beats.

The second type of error, arising from dye compartmentation, is the possibility that a part of the fluorescence change during the rapid phase of the Ca transient could arise from noncytosolic (mitochondrial) dye. Generally, mitochondrial changes are thought to be slower than cytosolic, although some evidence for mitochondrial Ca transients during cytosolic Ca transients has been reported (11). If we suppose, for the sake of argument, that mitochondrial dye was just as responsive to incoming Ca as cytosolic dye, how much difference would it make? We can estimate this by recalculating our calibated [Ca] with the assumption that all of the 360-nm signal and all of the 380-nm signal comes from an equally accessible (cytosolic + mitochondrial) dye pool. Again referring to cell 1, the starting [Ca] would change to 0.224 ± 0.017 µM and the plateau [Ca] would be 0.301 ± 0.014 µM, for a difference of 0.077 ± 0.003 µM. This is a factor of 2 less, suggesting Ca buffering of a factor of 2 greater. We could fit this to our model if we assumed a Kd for TnC and calmodulin binding of Ca that was 1.5 µM. This is still low affinity. However, it is hard to justify the assumption that mitochondrial Ca uptake occurs within 1 ms of Ca entry at such low levels of cytosolic [Ca]. More important, even if this did occur, we would still conclude that the TnC Ca binding affinity at the rise of the Ca transient was low, a factor of 4.5 lower than the Robertson et al. value (39). Finally, the inference about low-affinity buffering is supported by the absence of a peak in the Ca transient (Fig. 6C), an observation that does not depend directly on fura 2 calibration.

In a later development of the model of Peskoff et al. (36) in which the dyad clefts were connected to the cytosolic space, the dyad clefts were modeled as flat disks with a 200-nm radius, such that the Ca efflux from the clefts was already spread out over a large area (35). A novel feature of our model is the use of radial symmetry around the T-tubule axis to emulate the efflux of Ca from a more realistic compact source on that axis. We used an imaginary expansion zone (zone 2) to connect the dyad cleft with the cytosol. The actual geometry of this connection is likely to be rather irregular and different from one junction to the next. Theoretically, no zone 2 is needed for the modeling. The simulation will work simply by transferring the efflux from zone 1 directly into zone 3. However, we find greater errors in total Ca associated with transfer between zones when we eliminate zone 2. Calculation errors without zone 2 could be larger because the solute gradients at the zone edge are much larger. Zone 2 appears to minimize these errors by creation of a space that by virtue of its geometry allows a natural attenuation of the high Ca levels as the zone is traversed. Using cylindrical geometry coaxial with the T-tubule thus conserves the geometry of the origin of cytosolic Ca, while keeping the modeling geometry simple. The fact that zone 2 is geometrically situated "outside" the cell ("inside" the T-tubule) does not impact the validity of the model: zone 2 is simply a tool to facilitate the connection between the dyad cleft and the cytosol for modeling purposes. Although this approach is not strictly correct geometrically, the impact of zone 2 on the modeling result is in practice found to be negligible (Fig. 8B). This is because Ca crosses zone 2 extremely quickly, because it contains no Ca binding sites, and the total amount of Ca in this zone at any time is negligible (supplement Fig. S9). This result thus establishes the validity of using this construct.

Further approximations are associated with the dyad cleft geometry. We chose a cleft radius of 80 nm rather than the 200-nm radius used in Refs. 26 and 35 because at a Ca channel density of ~15/µm2 (6, 28) and one channel per cleft, the cleft area would exceed the membrane area by a factor of 1.885. With an 80-nm cleft, 0.3 of the membrane is covered by clefts, which seems more reasonable. An 80-nm radius also appears to us more compatible with T-tubules of only 57-nm radius (32) and their appearance by electron microscopy. Dyad clefts are not, however, circular: the junctional SR can interface with the T-tubule continuously over micrometer distances along the T-tubule (33), whereas the lateral distance to the edge of the cleft is closer to or less than the 80 nm distance we used. The shape of the zone of interaction is thus more like a strip with Ca channels scattered along it than a series of circles each with a Ca channel. There will in fact be a distribution of distances from the Ca channel to the dyad cleft edge. Our simulation in Fig. 8A suggests, however, that even though a distribution may exist, a distance parameter specified by a single average value is adequate to emulate this for our purposes here.

Errors in the estimate of T-tubule density will impact the estimate of cytosolic buffering but not the delay predicted between the integrated Ca current and the rise in [Ca-fura 2]. The value used, a length of 0.83 µm/µm3, equivalent to an area of 0.299 µm2/µm3 (32), could, however, be the most accurate of the values available in the literature. Earlier studies by Page and Surdyk-Droske (34), using morphometry of electron micrographs of thin sections to measure T-tubules, reported only 0.145 µm2/µm3. This measurement is somewhat subjective and open to underestimation because of the T-tubule membrane being obscured by overlying structures. In Ref. 32, on the other hand, silver-impregnated samples were imaged by high-voltage electron microscopy, which allowed the full extent of T-tubules to be beautifully visualized without interference from cytosolic tissue. An even larger T-tubule area of 0.44 µm2/µm3 was estimated from light microscopic measurements (47), a value that may be overestimated because of the lower resolution of optical microscopy.

Mg equilibria with ATP were not included in the model as dynamic variables, following the example of Baylor and Hollingworth (3). The effect of this approximation will be to underestimate the impact of ATP, because MgATP in the cleft will by dissociation serve as a reservoir of free ATP to replenish the free ATP bound by Ca (3). The peak [CaATP] for the model (Table 1) conditions is <12 µM, achieved right over the Ca channel just before it closes. This will cause a peak depletion of [CaATP] of 0.02%, which will cause the rate of CaATP formation to be underestimated at most by a similar factor and, consequently, also the impact of ATP on Ca diffusion.

In our model the concentration of Ca across the h dimension of the dyad cleft was uniform at all times and places. We did not consider surface charge effects, which will in reality result in a steep Ca gradient across the h dimension, near the membrane surface (46). Including surface charge effects explicitly would have greatly increased computation times and would have complicated boundary conditions between zones. Surface charge tends to buffer changes in cleft [Ca] by sequestering Ca into the diffuse double layer, in addition to binding Ca. Changes in cleft [Ca] away from the membrane are therefore smaller than when surface charge is not considered, for a given amount of Ca entry, an effect equivalent to a "volume expansion" of the cleft (46). The impact of including surface charge on predicted [Ca] at the cleft edge is complex, resulting in slowing of the initial rate of rise of [Ca] while accelerating the approach to equilibrium at later times (46). Including surface charge effects has much less impact on the timing of [Ca] changes within the cleft than including the cleft Ca binding sites (46). Also, surface charge effects are less evident at the cleft edge than they are opposite the Ca channel at the cleft center (46): a current of 0.2 pA for 0.3 ms resulted in a [Ca] of ~9 µM at 47 nm from the cleft center at 0.3 ms, which decayed to ~1 µM by 1 ms, whether or not surface charge was considered, although surface charge slowed the early decline and speeded up the later decline (46). [Ca] changes in the center of the dyad cleft (Fig. 5A), on the other hand, which can be thought of as an average in the h dimension, are likely to be overestimated in our model because we did not consider surface charge. On the other hand, we did not apply any correction to cleft volume for feet, which would result in an underestimate of [Ca] changes. Either way, because cleft Ca efflux depends on [Ca] at the edge of the cleft, the modeled rate of Ca efflux from the cleft should not be affected much by surface charge, compared with the effect of the other variables that we did consider.

Predictions of the model do not depend strongly on the accuracy of the figure used for Ca channel density. Two channels each passing 0.1 pA simultaneously look very much like one channel passing 0.2 pA, in terms of the total cytosolic [Ca-fura 2] transient (supplemental Fig. S4). If there were 30 channels/µm2 instead of 15, the average total open time per channel would be half of the value of 1.07 ms calculated from our measurements using 15 channels/µm2.

Finally, our modeling assumes that all Ca channels are situated in dyad clefts, and this may not be the case. Ca channels and RyR do show a high degree of colocalization (42), however, and are functionally coupled (43), suggesting that most are indeed situated in the dyad cleft. Any channels outside of clefts would cause the [Ca-fura 2] rise to be faster than if they were in clefts. However, because U(t) is already fast and strongly influenced by TnC Ca binding rates (Fig. 8A) as well as by dyad cleft properties (Fig. 6A), the presence of any Ca channels outside of clefts would be difficult to discern from the measured transients.

In conclusion, we have modeled Ca fluxes from the dyad cleft to the cytosol, and we have for the first time tested the model using high-resolution measurements of fura 2 fluorescence. We find that the Ca transient is too fast to allow for high-affinity dyad cleft Ca binding sites, even though ATP would strongly facilitate dyad cleft Ca efflux if such sites were present. Dyad cleft Ca dynamics are much faster than were previously speculated. Additionally, the experimental data can only be fit when Ca binding to TnC is modeled with low affinity, more like isolated TnC than like TnC in activated myofilaments: high-affinity TnC would give a peaked transient, which was not observed. If just low-affinity cleft Ca binding sites are present, TnC Ca binding rates also had to be reduced (by a factor 0.375x of the soluble TnC rates) if the data were to be fit well. Thus the high-resolution fura 2 measurements, combined with modeling, allowed the modeling parameters to be more realistically constrained. This combination of experiment and theory holds promise for the evaluation of dyad cleft Ca kinetics not only in normal cells but also in disease.

A user-friendly version of the model described in this paper, DyToCy.exe, is available for download at http://www.surgery.wisc.edu/transplant/research/DyToCy.shtml.


    ACKNOWLEDGMENTS
 
We thank Dr. Richard Moss and Dr. Timothy Kamp for comments on the manuscript.

GRANTS

This work was supported by National Heart, Lung, and Blood Institute Grants HL-33652 and HL-61534 (to R. A. Haworth).


    FOOTNOTES
 

Address for reprint requests and other correspondence: R. A. Haworth, Dept. of Surgery, Univ. of Wisconsin Clinical Sciences Center, 600 Highland Ave., Madison WI 53792-3236 (E-mail: haworth{at}surgery.wisc.edu).

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

1 Supplemental Figs. S1–S9 for this article may be found at http://ajpcell.physiology.org/cgi/content/full/00193.2003/DC1. Back


    REFERENCES
 TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 
1. Balke CW, Egan TM, and Wier WG. Processes that remove calcium from the cytoplasm during excitation-contraction coupling in intact rat heart cells. J Physiol 474: 447–462, 1994.[Abstract]

2. Bassani RA, Shannon TR, and Bers DM. Passive Ca binding in ventricular myocardium of neonatal and adult rats. Cell Calcium 23: 433–442, 1998.[ISI][Medline]

3. Baylor SM and Hollingworth S. Model of sarcomeric Ca2+ movements, including ATP Ca2+ binding and diffusion, during activation of frog skeletal muscle. J Gen Physiol 112: 297–316, 1998.[Abstract/Free Full Text]

4. Baylor SM and Hollingworth S. Fura-2 calcium transients in frog skeletal muscle fibres. J Physiol 403: 151–192, 1988.[Abstract]

5. Berlin JR, Bassani JWM, and Bers DM. Intrinsic cytosolic calcium buffering properties of single rat cardiac myocytes. Biophys J 67: 1775–1787, 1994.[Abstract]

5. Bers DM, Patton CW, and Nuccitelli R. A practical guide to the preparation of Ca2+ buffers. Methods Cell Biol 40: 3–29, 1994.[ISI][Medline]

6. Bers DM and Stiffel VM. Ratio of ryanodine to dihydropyridine receptors in cardiac and skeletal muscle and implications for E-C coupling. Am J Physiol Cell Physiol 264: C1587–C1593, 1993.[Abstract/Free Full Text]

7. Bianchi CP and Narayan S. Possible role of the transverse tubules in accumulating calcium released from the terminal cisternae by stimulation and drugs. Can J Physiol Pharmacol 60: 503–507, 1981.[ISI]

8. Blatter LA and Wier WG. Intracellular diffusion, binding, and compartmentalization of the fluorescent calcium indicators Indo-1 and Fura-2. Biophys J 58: 4191–1499, 1990.

9. Bogdanov KY, Ziman BD, Spurgeon HA, and Lakatta EG. L- and T-type calcium currents differ in finch and rat ventricular cardiomyocytes. J Mol Cell Cardiol 27: 2581–2593, 1995.[CrossRef][ISI][Medline]

10. Busa WB. Spectral characterization of the effect of viscosity on Fura-2 fluorescence: excitation wavelength optimization abolishes the viscosity artifact. Cell Calcium 13: 313–319, 1992.[ISI][Medline]

11. Chacon E, Ohata H, Harper IS, Trollinger DR, Herman B, and Lemasters JJ. Mitochondrial free calcium transients during excitation-contraction coupling in rabbit cardiac myocytes. FEBS Lett 382: 31–36, 1996.[CrossRef][ISI][Medline]

12. Chafouleas JG, Dedman JR, Munjaal RP, and Means AR. Calmodulin development and application of a sensitive radioimmunoassay. J Biol Chem 254: 10262–10267, 1979.[Medline]

13. Di Virgilio F, Steinberg TH, Swanson JA, and Silverstein SC. Fura-2 secretion and sequestration in macrophages. A blocker of organic anion transport reveals that these processes occur via a membrane transport system for organic anions. J Immunol 140: 915–920, 1988.[Abstract/Free Full Text]

14. Dong W, Rosenfeld SS, Wang CK, Gordon AM, and Cheung HC. Kinetic studies of calcium binding to the regulatory site of troponin C from cardiac muscle. J Biol Chem 271: 688–694, 1996.[Abstract/Free Full Text]

15. Fabiato A. Calcium-induced release of calcium from the cardiac sarcoplasmic reticulum. Am J Physiol Cell Physiol 245: C1–C14, 1983.[Abstract/Free Full Text]

16. Gomez AM, Valdivia HH, Cheng H, Lederer MR, Santana LF, Cannell MB, McCune SA, Altschuld RA, and Lederer WJ. Defective excitation-contraction coupling in experimental cardiac hypertrophy and heart failure. Science 276: 800–806, 1997.[Abstract/Free Full Text]

17. Gordon AM, Homsher E, and Regnier M. Regulation of contraction in striated muscle. Physiol Rev 80: 853–924, 2000.[Abstract/Free Full Text]

18. Guia A, Stern MD, Lakatta EG, and Josephson IR. Ion concentration-dependence of rat cardiac unitary l-type calcium channel conductance. Biophys J 80: 2742–2750, 2001.[Abstract/Free Full Text]

19. Haworth RA, Goknur AB, Warner TF, and Berkoff HA. Some determinants of quality and yield in the isolation of adult heart cells from rat. Cell Calcium 10: 57–62, 1989.[ISI][Medline]

20. Haworth RA and Redon D. Calibration of intracellular Ca transients of isolated adult heart cells labelled with fura-2 by acetoxymethyl ester loading. Cell Calcium 24: 263–273, 1998.[ISI][Medline]

21. Hollingworth S, Soeller C, Baylor SM, and Cannell MB. Sarcomeric Ca2+ gradients during activation of frog skeletal muscle fibres imaged with confocal and two-photon microscopy. J Physiol 526: 551–560, 2000.[Abstract/Free Full Text]

22. Holroyde MJ, Robertson SP, Johnson JD, Solaro RJ, and Potter JD. The calcium and magnesium binding sites on cardiac troponin and their role in the regulation of myofibrillar adenosine triphosphatase. J Biol Chem 255: 11688–11693, 1980.[Abstract/Free Full Text]

23. Jafri MS, Rice JJ, and Winslow RL. Cardiac Ca dynamics: the roles of ryanodine receptor adaptation and sarcoplasmic reticulum load. Biophys J 74: 1149–1168, 2000.

24. Klee CB and Vanaman TC. Calmodulin. Adv Protein Chem 35: 213–321, 1982.[ISI][Medline]

25. Konishi M, Olson A, Hollingworth S, and Baylor SM. Myoplasmic binding of fura-2 investigated by steady-state fluorescence and absorbance measurements. Biophys J 54: 1089–1104, 1988.[Abstract]

26. Langer GA and Peskoff A. Calcium concentration and movement in the diadic cleft space of the cardiac ventricular cell. Biophys J 70: 1169–1182, 1996.[Abstract]

27. Lattanzio FA Jr and Bartschat DK. The effect of pH on rate constants, ion selectivity and thermodynamic properties of fluorescent calcium and magnesium indicators. Biochem Biophys Res Commun 177: 184–191, 1991.[ISI][Medline]

28. Lew WYW, Hryshko LV, and Bers DM. Dihydropyridine receptors are primarily functional L-type calcium channels in rabbit ventricular myocytes. Circ Res 69: 1139–1145, 1991.[Abstract]

29. Malmendal A, Linse S, Evenas J, Forsen S, and Drakenberg T. Battle for the EF-hands: magnesium-calcium interference in calmodulin. Biochemistry 38: 11844–11850, 1999.[CrossRef][ISI][Medline]

30. Miyata H, Silverman HS, Sollott SJ, Lakatta EG, Stern MD, and Hansford RG. Measurement of mitochondrial free Ca2+ concentration in living single rat cardiac myocytes. Am J Physiol Heart Circ Physiol 261: H1123–H1134, 1991.[Abstract/Free Full Text]

31. Moss RL. Plasticity in the dynamics of myocardial contraction: calcium, cross-bridge kinetics or molecular cooperation? Circ Res 84: 862–865, 1999.[Free Full Text]

32. Nakamura S, Asai J, and Hama K. The transverse tubular system of rat myocardium: its morphology and morphometry in the developing and adult animal. Anat Embryol (Berl) 173: 307–315, 1986.[ISI][Medline]

33. Ogata T and Yamasaki Y. High-resolution scanning electron microscopic studies on the three-dimensional structure of the transverse-axial tubular system, sarcoplasmic reticulum and intercalated disc of the rat myocardium. Anat Rec 228: 277–287, 1990.[ISI][Medline]

34. Page E and Surdyk-Droske M. Distribution, surface density, and membrane area of diadic junctional contacts between plasma membrane and terminal cisterns in mammalian ventricle. Circ Res 45: 260–267, 1979.[Abstract]

35. Peskoff A and Langer GA. Calcium concentration and movement in the ventricular cardiac cell during an excitation-contraction cycle. Biophys J 74: 153–174, 1998.[Abstract/Free Full Text]

36. Peskoff A, Post JA, and Langer GA. Sarcolemmal calcium binding 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.[ISI][Medline]

37. Post JA and Langer GA. Sarcolemmal calcium binding sites in heart. I. Molecular origin in "gas-dissected" sarcolemma. J Membr Biol 129: 49–57, 1992.[ISI][Medline]

38. Puglisi JL, Yuan W, Bassani JWM, and Bers DM. Ca2+ influx through Ca2+ channels in rabbit ventricular myocytes during action potential clamp. Circ Res 85: e7–e16, 1999.[ISI][Medline]

39. Robertson SP, Johnson JD, and Potter JD. The time-course of Ca2+ exchange with calmodulin, troponin, parvalbumin, and myosin in response to transient increases in Ca2+. Biochem J 34: 559–569, 1981.

40. Sagara Y and Inesi G. Inhibition of the sarcoplasmic reticulum Ca2+ transport ATPase by thapsigargin at subnanomolar concentrations. J Biol Chem 266: 13503–13506, 1991.[Abstract/Free Full Text]

41. Satoh H, Ginsburg KS, Qing K, Terada H, Hayashi H, and Bers DM. KB-R7943 block of Ca2+ influx via Na+/Ca2+ exchange does not alter twitches or glycoside inotropy but prevents Ca2+ overload in rat ventricular myocytes. Circulation 101: 1441–1446, 2000.[Abstract/Free Full Text]

42. Scriven DRL, Dan P, and Moore EDW. Distribution of proteins implicated in excitation-contraction coupling in rat ventricular myocytes. Biophys J 79: 2682–2691, 2000.[Abstract/Free Full Text]

43. Sham JS, Cleemann L, and Morad M. Functional coupling of Ca2+ channels and ryanodine receptors in cardiac myocytes. Proc Natl Acad Sci USA 92: 121–125, 1995.[Abstract]

44. Shannon TR, Ginsburg KS, and Bers DM. Potentiation of fractional sarcoplasmic reticulum calcium release by total and free intra-sarcoplasmic reticulum calcium concentration. Biophys J 78: 334–343, 2000.[Abstract/Free Full Text]

45. Sipido KR and Wier WG. Flux of Ca2+ across the sarcoplasmic reticulum of guinea-pig cardiac cells during excitation-contraction coupling. J Physiol 435: 605–630, 1991.[Abstract]

46. Soeller C and Cannell MB. Numerical simulation of local calcium movements during L-type calcium channel gating in the cardiac diad. Biophys J 73: 97–111, 1997.[Abstract]

47. Soeller C and Cannell MB. Examination of the transverse tubular system in living cardiac rat myocytes by 2-photon microscopy and digital image-processing techniques. Circ Res 84: 266–275, 1999.[Abstract/Free Full Text]

48. Stern MD. Theory of excitation-contraction coupling in cardiac muscle. Biophys J 63: 497–517, 1992.[Abstract]

49. Stern MD, Song LS, Cheng H, Sham JSK, Yang HT, Boheler KR, and Ríos E. Local control models of cardiac excitation-contraction coupling: a possible role for allosteric interactions between ryanodine receptors. J Gen Physiol 113: 469–489, 1999.[Abstract/Free Full Text]

50. Trafford AA, Díaz ME, and Eisner DA. A novel, rapid and reversible method to measure Ca buffering and time-course of total sarcoplasmic reticulum Ca content in cardiac ventricular myocytes. Pflügers Arch 437: 501–503, 1999.[CrossRef][ISI][Medline]

51. Wang YP and Fuchs F. Length, force, and Ca2+-troponin C affinity in cardiac and slow skeletal muscle. Am J Physiol Cell Physiol 266: C1077–C1082, 1994.[Abstract/Free Full Text]

52. Zahradnik I and Palade P. Multiple effects of caffeine on calcium current in rat ventricular myocytes. Pflügers Arch 424: 129–136, 1993.[ISI][Medline]

53. Zot HG, Iida S, and Potter JD. Thin filament interactions and Ca binding to Tn. Chemica Scripta 21: 133–136, 1983.[ISI]





This Article
Abstract
Full Text (PDF)
Data Supplement
All Versions of this Article:
286/2/C302    most recent
00193.2003v1
Alert me when this article is cited
Alert me if a correction is posted
Citation Map
Services
Email this article to a friend
Similar articles in this journal
Similar articles in PubMed
Alert me to new issues of the journal
Download to citation manager
Google Scholar
Articles by Vadakkadath Meethal, S.
Articles by Haworth, R. A.
Articles citing this Article
PubMed
PubMed Citation
Articles by Vadakkadath Meethal, S.
Articles by Haworth, R. A.


HOME HELP FEEDBACK SUBSCRIPTIONS ARCHIVE SEARCH TABLE OF CONTENTS
Visit Other APS Journals Online
Copyright © 2004 by the American Physiological Society.