Ca2+ currents recorded from Xenopus oocytes expressing only the α1C pore-forming subunit of the cardiac Ca2+ channel show Ca2+-dependent inactivation with a single exponential decay. This current-dependent inactivation is not detected for inward Ba2+ currents in external Ba2+. Facilitation of pore opening speeds up the Ca2+-dependent inactivation process and makes evident an initial fast rate of decay. Facilitation can be achieved by (a) coexpression of the β2a subunit with the α1C subunit, or (b) addition of saturating Bay K 8644 (−) concentration to α1C channels. The addition of Bay K 8644 (−) to α1Cβ2a channels makes both rates of inactivation faster. All these maneuvers do not induce inactivation in Ba2+ currents in our expression system. These results support the hypothesis of a mechanism for the Ca2+-dependent inactivation process that is sensitive to both Ca2+ flux (single channel amplitude) and open probability. We conclude that the Ca2+ site for inactivation is in the α1C pore-forming subunit and we propose a kinetic model to account for the main features of α1Cβ2a Ca2+ currents.

Introduction

Regulation of Ca2+ currents by Ca2+ influx includes a negative feedback mechanism that inactivates the current itself when Ca2+ is the charge carrier. Currents elicited by depolarizing steps show a fast activating phase followed by a Ca2+-dependent inactivating phase. This feature has been extensively studied in native channels (Eckert and Chad, 1984; Chad, 1989; Gutnick et al., 1989; Kostyuk, 1992; Shirokov at el., 1993) and has been recently demonstrated in cloned channels (Neely et al., 1994; Zong et al., 1996; de Leon et al., 1995). It has been shown that Ca2+ chelators can reduce the efficiency of the inactivation process (Imredy and Yue, 1992; Haack and Rosenberg, 1994) and that the Ca2+ influx through a channel can contribute to the inactivation of adjacent channels (cross talk) (Mazzanti et al., 1991; Imredy and Yue, 1992; Galli et al., 1994). These results suggested the presence of a specific Ca2+ site on the intracellular face of the channel protein (Huang et al., 1989). Two questions emerge: (a) Is the α1C pore-forming subunit alone capable of Ca2+-dependent inactivation, or, on the contrary, is the accessory β2a subunit required (Neely et al., 1994; Zong et al., 1996)? (b) Is this inactivating process related to intracellular Ca2+ build-up (Chad et al., 1984; Mazzanti et al., 1991), or does Ca2+ entry through a single channel inactivate the same channel by a Ca2+ regulatory site located deep inside the pore (Yue et al., 1990)?

To address these questions, we performed whole cell experiments with the cut-open oocyte Vaseline gap technique (Stefani et al., 1994) on Xenopus oocytes expressing the cloned α1C subunit of the rabbit cardiac Ca2+ channel, with and without the accessory β2a subunit (Neely et al., 1994). Ca2+ and Ba2+ currents were recorded in oocytes after the intracellular injection of the fast Ca2+ chelating agent Na4-BAPTA (1,2-bis(o-aminophenoxy)-ethane-N,N,N′,N′-tetraacetate, Kon = 6 × 108 M−1 s−1) to prevent contaminant Ca2+-activated Cl currents (Neely et al., 1994). The action of the dihydropyridine (DHP)1 agonist Bay K 8644 (−) on the inactivation rates was also investigated, since this agent increases the size of the macroscopic current by changing the channel open probability without significantly changing the single channel amplitude (Hess et al., 1984). In this respect, the effect of Bay K 8644 (−) would mimic the effect of the β2a subunit.

We found that Ca2+ currents from the α1C subunit expressed alone can inactivate in a Ca2+-dependent manner. The single exponential fits to these currents have time constants that decrease when the Ca2+ concentration is increased. The coexpression of the β2a subunit makes evident a double exponential decay with a faster time course and with rates that are Ca2+ dependent. Similarly, the addition of saturating concentrations of Bay K 8644 (−) to α1C channels induces the appearance of the fast rate of Ca2+-dependent inactivation. These results confirm the view that the Ca2+ binding site for the inactivation is part of the pore-forming α1C subunit (Neely et al., 1994; de Leon et al., 1995; Zhou et al., 1997) and is located in a region very close to the inner mouth pore, within a microdomain where the local Ca2+ concentration can reach its steady state in a few microseconds. Based on the fact that the rates of Ca2+-dependent inactivation are sensitive to both Ca2+ flux through the channel (single channel current) and open probability, we propose a kinetic model for the Ca2+-dependent inactivation process in α1Cβ2a channels.

Materials And Methods

RNA Synthesis and Oocyte Injection

The plasmids containing cDNA fragments encoding the cardiac α1C and β2a subunits were digested with HindIII (Wei et al., 1991). The linearized templates were treated with 2 μg proteinase K and 0.5% SDS at 37°C for 30 min, and then twice extracted with phenol/chloroform, precipitated with ethanol, and resuspended in distilled water to a final concentration of 0.5 μg/μl. The cRNAs were transcribed from 0.5 μg of linearized DNA template at 37°C with 10 U of T7 RNA polymerase (Boehringer Mannheim Biochemicals, Indianapolis, IN), in a volume of 25 μl containing 40 mM Tris-HCl (pH 7.2), 6 mM MgCl2, 10 mM dithiothreitol, 0.4 mM each of ATP, GTP, CTP, and UTP, 0.8 mM 7-methyl-GTP. The transcription products were extracted with phenol and chloroform, twice precipitated with ethanol and resuspended in double distilled water to a final concentration of 0.2 μg/μl, and 50 nl were injected per oocyte. Before injection, oocytes were defolliculated by collagenase treatment (type I, 2 mg/ml for 40 min at room temperature; Sigma Chemical Co., St. Louis, MO). Oocytes were maintained at 19.0°C in Barth solution. Recordings were done 4–12 d after RNA injection.

Recording Technique and Solutions

Recording of macroscopic current was performed using the cut-open oocyte Vaseline gap technique (Stefani et al., 1994) on Xenopus laevis oocytes. The oocyte was placed in a triple compartment perspex chamber; voltage clamped currents were recorded from the top chamber, while the middle chamber (set at the same voltage as the recording chamber) isolated the top from the bottom chamber. The oocyte membrane exposed to the bottom chamber was permeabilized with 0.1% saponin in the internal saline (see bottom chamber solution). Microelectrodes probing voltage across the membrane (top chamber) had a resistance of ∼0.5–1 MΩ, and they were filled with (M): 2.7 NaMES (Na-methanesulfonic acid), 0.01 Na2-EGTA, 0.01 NaCl. Holding potential was −90 mV. All experiments were performed at room temperature. Subtraction of linear components was digitally obtained by scaled currents elicited by small control pulses of one-fourth the amplitude of the stimulating pulse (P/−4). Acquisition and data analysis were done on a personal computer. Signals were filtered at one-fifth the sampling frequency.

External solutions, used in the top and guard compartments were: NaMESBa-10 (10 mM BaOH; 96 mM NaMES; 10 mM HEPES, titrated to pH 7.0 with CH3SO3H), NaMESCa-2, -5, or -10 (2, 5, or 10 mM CaOH, 102 mM NaMES, 10 mM HEPES, titrated to pH 7.0 with CH3SO3H), or NMGMESCa-10 (10 mM Ca2+, 96 mM N-methyl-d-glucamine [NMG+], 10 mM HEPES, titrated to pH 7.0 with CH3SO3H). The solution in the bottom chamber in contact with the oocyte cytoplasm was 110 mM K-glutamate (10 mM HEPES, titrated to pH 7.0 with KOH). Na4-BAPTA (tetrasodium-1,2-bis (o-aminophenoxy)-ethane-N,N,N  ′,N  ′-tetraacetate) was loaded into a glass micropipette of ∼20-μm tip diameter, and ∼100 nl were injected with an automatic microinjector immediately before mounting the oocyte in the recording chamber. Na4-BAPTA stock solutions of 50 mM were made in distilled water and titrated at pH 7.0 with CH3SO3H. The BAPTA injections were performed before the experiments to prevent contamination of Ca2+ currents with Ca2+- and Ba2+-activated Cl currents (Miledi, 1982; Barish, 1983; Neely et al., 1994). This BAPTA concentration selectively eliminated Cl currents without any significant action on Ca2+-dependent current decay (Neely et al., 1994). To probe the accessibility of the internal Ca2+ site, we had to dramatically increase the internal BAPTA concentration (500 mM BAPTA and 10 mM NaCl) and to continuously perfuse the oocytes (1 ml/h) via a glass pipette inserted into the bottom side of the oocyte.

Values for the rates of Ca2+-dependent inactivation were obtained by fitting to a double exponential (α1Cβ2a; α1C in the presence of 500 nM Bay K 8644 (−)) or to a single exponential (α1C) the decay of the Ca2+ current (from the peak of the current up to 800 ms). The exponential functions were

 
\begin{equation*}P+{ \,\substack{ ^{n} \\ {\sum} \\ _{i=1} }\, }A_{i}e^{-\frac{t}{{\tau}_{i}}}\end{equation*}

with n = 1 and 2. P was the offset factor, and Ai the amplitude of each exponential component.

The model-fitting procedure was implemented with SCoP (Simulation Resources, Inc., Barren Springs, MI). The rates in the transitions were exponential functions of the voltage, as predicted by the Eyring theory. Simultaneous fitting of current traces at different potentials was performed to evaluate the kinetic parameters in a non–steady state model. The source file, containing a system of differential equations, was compiled and the resulting executable file was fed with ensembles of current recordings under different conditions, such as subunit expression, Ca2+ concentration, and presence of the DHP agonist Bay K 8644 (−).

Results

Ca2+-dependent Inactivation in α1C-expressing Oocytes

Fig. 1 shows Ba2+ and Ca2+ currents (left and right, respectively) recorded from an oocyte expressing the α1C subunit alone. The currents were elicited by depolarizing pulses from a holding potential of −90 mV. The figure shows that, during large depolarizing pulses, Ba2+ currents had a very slow decay, while Ca2+ currents showed a much faster decay that could be attributed to the Ca2+-dependent inactivation process. The decay phase of the currents was fitted to a single exponential function of the form Ae−tr + C, where A is the amplitude factor, t the time, r the rate, and C the offset. The fit was for Ba2+ (Fig. 1,A): A = −4.1 nA, r = 0.001 ms−1 and C = −17.9 nA at 0 mV and for Ca2+ (Fig. 1,B): A = −4.5 nA, r = 0.003 ms−1, and C = −4.8 nA at 0 mV. The relative amplitude factor, A/(A + C), was larger and the rate of decay was much faster in external Ca2+. This difference in the decay phase between Ca2+ and Ba2+ currents became more evident after potentiating the Ca2+ current by adding a submaximal concentration (50 nM) of the DHP agonist Bay K 8644 (−). The potentiated Ba2+ currents had a similar time course to the control (compare Fig. 1, C and A). Thus, Ba2+ does not substitute Ca2+ for the inactivation in the time scale used (0.5–1 s). During the time course of our experiments, the main effect of Bay K 8644 was to potentiate Ba2+ currents. On the other hand, in external Ca2+, Bay K 8644 induced an increase in the size of the peak current as well as an increase in the decay (Fig. 1, B and D). Thus, facilitation of pore opening by Bay K 8644 (−) makes more evident Ca2+-dependent inactivation, which is already present in the absence of the agonist.

Effect of the β2a Subunit on α1C Currents

Fig. 2 shows the effect of coexpression of the β2a subunit with the α1C subunit on the rates of Ca2+-dependent inactivation. Ba2+ and Ca2+ currents (Fig. 2, left and right, respectively) in oocytes coexpressing the pore-forming α1C subunit together with the regulatory β2a subunit had faster activation rates and larger amplitudes than in oocytes expressing the α1C subunit alone. This is expected from the facilitation of the pore opening by the coexpression of the β2a subunit. By comparing the recordings in external Ca2+ (Figs. 1 and 2, right), it becomes evident that the β2a subunit speeds up the Ca2+-dependent inactivation process (Fig. 2, B and D). As was the case for α1C currents, in α1Cβ2a, the addition of a submaximal concentration of Bay K 8644 (−) (50 nM) increased both Ba2+ and Ca2+ current amplitudes. From these results we can conclude that maneuvers that increase the open probability of the Ca2+ channel without affecting the single channel amplitude (addition of Bay K 8644 (−) and the coexpression of the β2a subunit; Costantin et al., 1995) speed up the Ca2+-dependent inactivation process.

In addition to the Ca2+-dependent inactivation phase, both α1C and α1Cβ2a currents show a slow smaller component of inactivation that is weakly voltage dependent. This slow inactivation is more clearly detected in Ba2+ currents (Figs. 1, A and C, and 2, A and C) since, in external Ca2+, the Ca2+ inactivation process predominates. This slower component can be attributed to a slow voltage-dependent inactivation (Lee et al., 1985; Hadley and Hume, 1987; Campbell et al., 1988; Gutnick et al., 1989; Hadley and Lederer, 1991; Giannattasio et al., 1991), or to a less efficient Ba2+-dependent inactivation process (Ferreira et al., 1997). This voltage-dependent component has not been analyzed here; since its time course is much slower than Ca2+-dependent inactivation, it should not interfere with the main conclusions of this paper.

Voltage and Ca2+ dependence of the Inactivation Rates in α1C and α1C β2a Currents

The currents in Fig. 3 were recorded in an oocyte expressing the α1Cβ2a Ca2+ channel at three different external Ca2+ concentrations: 2 (A), 5 (B), and 10 (C) mM Ca2+. Each panel shows currents elicited by three different voltages (holding potential −90 mV), together with superimposed fits to the decay phase of the currents. The inactivating currents were fitted with a double exponential function, yielding to a slow and a fast rate of inactivation. The fast rate depended on external Ca2+ concentration, while the slow rate was much less affected by external Ca2+ (see also Fig. 6). The peak of the ionic current occurred at +10 mV in 2 mM Ca2+, at +20 mV in 5 mM Ca2+, and at +25 mV in 10 mM Ca2+. The fast rate of inactivation, compared at equivalent voltages corrected for surface charge effect, increased from rf= 0.0126 ms−1 (0 mV, 2 mM Ca2+) to rf = 0.0224 ms−1 (+20 mV, 10 mM Ca2+).

The voltage dependencies of current (I-V) and fast rate of inactivation (r-V) for the experiment in Fig. 3 are shown in Fig. 4,A (normalized rates and peak current values). The graph shows a negative voltage shift between the peak of the r-V (open symbols) and the peak of the I-V (filled symbols). The fact that the peak of the r-V always occurred at more negative voltages than the peak of the I-V suggests a complex dependence of the Ca2+ inactivation mechanism on the parameters of channel activation. Possibly, open probability, single channel amplitude, Ca2+ influx and accumulation, buffer capacity, and diffusion could be involved. A similar voltage shift between the peak of the r-V and I-V curves was observed in α1C alone (Fig. 4 B).

Effect of DHP Agonist Bay K 8644 (−) on α1C and α1Cβ2a Ca2+ Currents

The ability of the DHP agonist Bay K 8644 (−) to increase the size of the current by enhancing the open probability, without affecting the single channel amplitude, was used to further investigate the role of the open probability in the Ca2+-dependent inactivation mechanism. We compared the effect of Bay K 8644 (−) on α1C and α1Cβ2a Ca2+ currents at three different Ca2+ concentrations. Fig. 5 shows α1Cβ2a Ca2+ currents elicited by three different voltages, both in the absence (A, 2 mM Ca2+) and presence (2, 5, and 10 mM Ca2+, B–D, respectively) of Bay K 8644 (−) 500 nM, with the corresponding superimposed fits. The addition of saturating concentration of Bay K 8644 (−) (500 nM) produced a small negative shift of the activation–voltage curve (5 mV), an approximately twofold increase in the size of the ionic current and a twofold increase in both rates.

Fig. 6 summarizes the effect of external Ca2+ and Bay K 8644 (−) on α1Cβ2a Ca2+-dependent rates of inactivation. The maximum value of the fast rates of inactivation in 2 mM Ca2+ and in the absence of the DHP agonist was 0.017 ms−1 (diamonds), and it became 0.038 ms−1 in 10 mM Ca2+ and in the presence of 500 nM Bay K 8644 (−), thus undergoing a more than threefold overall increase. An equivalent pattern as in Fig. 6 was observed in α1C channels. The slow time constant of the double-exponential inactivation in α1Cβ2a is of the same order of magnitude of the single exponential time constant in α1C alone, and it is Ca2+ dependent, as it can be seen in Fig. 6,A. Fig. 7 shows the relative position in the voltage axis of the r-V and I-V curves for α1Cβ2a in 5 mM Ca2+ and in the presence of 500 nM Bay K 8644 (−).

The same protocol as shown in Fig. 5 for α1Cβ2a currents was applied to α1C alone. In α1C channels, the Ca2+ current decay in 2 mM external Ca2+ (Fig. 8,A) could be fitted to a single exponential function. In α1C alone, the Ca2+-dependent inactivation might be contaminated by the presence of a voltage-dependent inactivation rate recorded in external Ba2+ (Fig. 1). However, the addition of Bay K 8644 (−) 500 nM produced the expected negative shift of the activation–voltage curve of ∼5 mV together with a significant increase in the size of the current. This current potentiation was associated with a double exponential time course of decay. As expected for a Ca2+-dependent process, the fast component became faster as the Ca2+ concentration increased. Taking altogether the results in α1C and α1Cβ2a channels and the action of Bay K 8644 (−) on both channels, we can conclude that both single channel amplitude and open probability participate in the process of Ca2+-dependent inactivation. The effect of the single channel amplitude is reflected by the Ca2+ dependence of the inactivation rates and the left shift of the r-V vs. I-V curves, while the role of the open probability is manifested by the faster inactivation rates after facilitating pore opening by the addition of Bay K 8644 (−) and the coexpression of the β2a subunit.

The Effect of BAPTA on Ca2+-dependent Inactivation in α1Cβ2a Currents

To test the accessibility of the site to internal Ca2+ buffer, we investigated the effect of perfusing high BAPTA concentration on the Ca2+-dependent inactivation rates (Fig. 9). The six traces in Fig. 9,A have been recorded at different times after the oocyte was mounted and the internal perfusion started (BAPTA 500 mM at the speed of 1 ml/h). Ca2+ currents (5 mM Ca2+) were initially contaminated by outward Ca2+-activated Cl currents shown as inward slow component and slow tail currents (trace a, t = 1 min 15 s). As the perfusion progressed, the Cl currents were removed showing the Ca2+-dependent inactivation process. The initial phase of decay in trace c (t = 4 min 26 s) could be fitted with a single exponential function (r = 0.0099 ms−1). After 20 min 5 s (trace f), the rate of inactivation had become much slower (r = 0.0025 ms−1). The time course of the inactivation rates for the whole experiment is shown in Fig. 9,B. During the perfusion, peak current amplitude decreased due to run down. In equivalent prolonged recordings (20 min), when BAPTA was not perfused in the oocyte, the decrease in the size of the current due to run down did not slow down the inactivation rate (Fig. 9, right). Fig. 9,C shows the progressive run down of the current, while there were no changes in their time course. Fig. 9,D shows that the voltage dependence of the fast rate of inactivation at the beginning and after 23 min remained unmodified. Fig. 9 E shows small changes in the rates of inactivation during the whole experiment. These results confirm that the internal Ca2+ site is accessible to fast Ca2+ chelators (Imredy and Yue, 1992; Haack and Rosenberg, 1994), thus ruling out the possibility that this site is located within the conduction pathway.

Discussion

We have shown that raising the external Ca2+ concentration increased the Ca2+-dependent inactivation rates and left-shifted their voltage dependence: enhancing the Ca2+ influx through the channels determined faster decaying currents under conditions in which the open probability should remain unaffected. We also have shown that the rates of Ca2+-dependent inactivation are open probability dependent. Thus, both the single channel amplitude and the open probability contribute to the Ca2+-dependent inactivation mechanism. One hypothesis that would explain the dependence of the rates on external Ca2+ concentration would be a Ca2+ binding site facing the external medium. This possibility is ruled out by the role of internal Ca2+ buffers (Lee et al., 1985; Imredy and Yue, 1992; Haack and Rosenberg, 1994) and by recent molecular biology experiments. de Leon et al. (1995) and Zhou et al. (1997) showed that the required region for Ca2+-dependent inactivation in α1C Ca2+ channels is located within the COOH terminus of the protein, facing the cytoplasm. The Ca2+ binding site could be sensitive to the internal accumulation of Ca2+ in a shell underneath the plasma membrane. In this case, the build up of Ca2+ would depend on the time integral of the current, the open probability, and the efflux from the shell into the cytoplasm. This mechanism has been proposed and extensively studied for native Ca2+ channels (Chad et al., 1984; Lee et al., 1985); it was supported by the dependence of the rates of inactivation on the internal buffer concentration (Lee et al., 1985; Imredy and Yue, 1992; Haack and Rosenberg, 1994) and by observed cross talk among channels (Yue et al., 1990; Mazzanti et al., 1991; Imredy and Yue, 1992). In our case, the shell mechanism becomes unlikely for two reasons: (a) only extreme conditions of Ca2+ buffering capacity were able to slow down the inactivation process, and (b) the lack of observed cross talk among channels. In the oocyte expression, the rates of Ca2+ inactivation are independent of the level of expression as shown before by our group (Neely et al., 1994). We have confirmed this finding. In this new set of experiments in α1Cβ2a, changes of expression level measured as the peak currents in the I-V curve (from 100–1,200 nA; 2 mM Ca2+) did not affect the predominant fast rate of inactivation, which remained between 0.015 and 0.02 ms−1 (data not shown). Our results favor a domain mechanism in which the local Ca2+ concentration equilibrates in microseconds (Sherman et al., 1990; Shirokov et al., 1993). However, we cannot rule out the possibility that the absence of cross talk was due to the expression level. Under our experimental conditions, it is possible that the expression level was not high enough to reach the critical channel density necessary to the interaction among adjacent channels.

A concern that must be addressed refers to the accessibility of BAPTA to the Ca2+ binding site. It is unlikely that the lack of effect on the inactivation rates by the injected BAPTA is due to a slow diffusion in the oocyte cytoplasm. In fact, injections of small quantities of BAPTA or EGTA (Neely et al., 1994) was effective to eliminate the activation of the Ca2+-activated Cl channels.

A Minimum Model for Ca2+-dependent Inactivation: Role of the Single Channel Amplitude and Open Probability

Several models have been proposed for Ca2+-dependent inactivation. All these models agree on identifying the Ca2+ dependence of the inactivation in one (or more) state-to-state transitions where the rate is dependent on the internal Ca2+ concentration. Two main conditions are considered: the “shell” model and the “local domain” model.

In the shell model, the Ca2+ flowing through the channels accumulates into a shell underneath the plasma membrane (Standen and Stanfield, 1982; Chad et al., 1984) or it is thought of as charge accumulating on a leaky capacitor (Mazzanti et al., 1991). This assumption leads to second order rates of Ca2+-dependent inactivation where the parameters of channel opening (i.e., single channel conductance and open probability) usually have to be integrated over the variable time.

In the local domain model, the Ca2+-sensing site is located very close to the channel mouth, making it less accessible to chelators, as well as to Ca2+ ions coming from adjacent channels. The calculation would thus be restricted to a very small domain surrounding the mouth of the channel, where the Ca2+ concentration would reach its steady state value in a few microseconds. This assumption justifies the use of the steady state diffusion equation and results in a linear dependence of the second order rates of inactivation on the single channel amplitude (Sherman et al., 1990; Shirokov et al., 1993).

Both classes of models can account for the main features of the Ca2+-dependent inactivation in L-type Ca2+ channels. The domain models also include the possibility of an “extended local domain” in which the volume where the Ca2+ concentration is calculated is larger, and a channel can sense the flux of ions that is entering a neighboring channel (cross talk). The fact that Yue et al. (1990) reported that Ca2+ entry inactivates the channel it goes through, together with our finding that high concentrations of fast chelators are necessary to reduce the inactivation rates, leads us to test the local domain hypothesis proposed by Sherman et al. (1990). The local domain model has been used with an expanded kinetic scheme to account for the voltage shift between the r-V and I-V curves.

The steady state equation for diffusion in a sphere is, when the diffusion happens only on the radial dimension:

 
\begin{equation*}C=B+\frac{{\phi}}{r},\end{equation*}
1

where C is the ion concentration, B is the boundary concentration (in our case, B is the cytoplasmic Ca2+ concentration), φ is the influx, and r is the radius. Eq. 1 means that, if [Ca2+]i is the internal Ca2+ concentration at rest, then the concentration near the pore will be

 
\begin{equation*}[Ca^{2+}]=[Ca^{2+}]_{i}+Ai,\end{equation*}
2

where i is the single channel amplitude and A is a constant. The constant A takes into account the diffusion coefficient for Ca2+ and the effect of chelators. In a voltage-independent transition from an open state to a Ca2+-inactivated state, the rate will be linearly dependent on the internal Ca2+ concentration: α = α0[Ca2+]. Thus, with the assumptions that Ca2+ binds instantaneously to a single site and at a fixed distance from the site, the rate of Ca2+-dependent inactivation depends linearly on the flux; i.e., on the single channel amplitude i. The inactivation process will also be open probability dependent since the inactivated state is sequentially connected to the open state.

Then we assumed that the transition rates between all the states in the kinetic model follow the Eyring rate theory; i.e., they are exponential functions of the form:

 
\begin{equation*}{\alpha}_{i}={\alpha}_{0i}e^{z_{i}{\delta}_{i}\frac{e^{-\;}V}{kT}}\end{equation*}
 
\begin{equation*}{\beta}_{i}={\beta}_{0i}{\mathit{e}}^{-z_{i}(1-{\delta}_{i})\frac{e^{-\;}V}{kT}},\end{equation*}

where αi are the forward rates and βi are the backward rates. α0i and β0i are the voltage-independent rates:

 
\begin{equation*}{\alpha}_{0i}=\frac{kT}{h}e^{-\frac{{\Delta}W_{if}}{kT}}\;and\;\end{equation*}
 
\begin{equation*}{\beta}_{0i}=\frac{kT}{h}e^{-\frac{{\Delta}W_{ib}}{kT}},\end{equation*}

where k is the Boltzmann constant (1.38 × 10−23  J/°K, T is the absolute temperature (K), h is the Planck constant (6.63 × 10−34 J s), e is the electronic charge (1.602 × 10−19 C), and ΔWif and ΔWib are the energies required for the transition to occur in the two directions (forward and backward). zi is the gating charge (e), δi is the fraction of the electric field sensed by zi, and V is the membrane voltage. The rates to the Ca2+-inactivated states are voltage independent:

 
\begin{equation*}{\alpha}_{Ca^{2+}}={\alpha}_{0Ca^{2+}}[Ca^{2+}]\end{equation*}
 
\begin{equation*}{\beta}_{Ca^{2+}}={\beta}_{0Ca^{2+}}.\end{equation*}

Within the above theoretical premises, we started building a kinetic scheme of closed, open, and inactivated states. The computer routine (SCoP) numerically solves the system of differential equations that describes the kinetic scheme, and assigns numbers to the parameters α0i, β0i, zi, δi, α0Ca2+, β0Ca2+. The internal Ca2+ concentration, [Ca2+], is calculated as a linear function of the single channel amplitude at each potential.

The kinetic scheme has to satisfy our experimental results, which are summarized as: (a) The inactivation is Ca2+ dependent: increasing the external Ca2+ concentration produces an increase in the absolute values of the rates of inactivation. (b) The process is open probability dependent: increasing the open probability with DHP agonist Bay K 8644 (−) or by coexpressing the β2a subunit together with the α1C subunit produces an increase in the rates of inactivation. (c) The time course of the decay can be fitted with a double exponential function. The initial fast rate of inactivation is not present (or not detectable) if the α1C subunit is expressed alone and when α1C currents are measured in the absence of Bay K 8644 (−). (d) The r-V curve peaks at more negative voltages than the I-V curve.

A sequential model of the form

graphic

cannot predict all of these features, specifically it cannot reproduce the observed shift between I-V and r-V. This model predicts that the r-V and the I-V peak at the same potentials under conditions of a fast activation and a slow inactivation (Sherman et al., 1990). If we were to relax these constraints, the decay phase of the current would be less evident due to the slow activation, and the peak of the r-V would tend to be more positive than the peak of the I-V curve.

We know from single channel data that the α1C and α1Cβ2a Ca2+ channels undergo “silent” transitions (i.e., they gate without opening), and they are able to open with different open probability patterns (Costantin et al., 1995). Thus, we have tested the model proposed by Bean (1989) with the modifications shown in Fig. 10. The kinetic scheme has three parallel lines of states that develop as branches from an initial closed state. The parallel transitions carry the same amount of charge. In the top line, the channel never opens, although it displaces all the charge of the voltage sensor. The middle (unwilling) and bottom (willing) lines have final open states connected to final inactivated states. The vertical transitions between the lines are voltage independent and can be Bay K 8644 (−) and β subunit sensitive.

Fig. 11 shows experimental data from an oocyte injected with the α1Cβ2a Ca2+ channel together with superimposed fitted traces to the model of Fig. 10. In this model, the rates connecting the open states to the Ca2+-inactivated states consist of a constant coefficient times the local Ca2+ concentration, according to Eq. 2 (Table I). The simulated traces in Fig. 11,B are shown together with their I-V and r-V curves (fast rate, Fig. 11 C). This model predicts the shift between the peaks of the two curves.

Thus, we can conclude that the kinetic scheme in Fig. 10 reproduces well the experimental data for α1Cβ2a channels. In the case of α1C alone, though it is possible to reproduce the main kinetic features of the data, such as the slower activation phase of the currents and the u-shaped voltage dependence of the inactivation rates, the possible presence of a contaminating voltage-dependent inactivation does not allow a clear analysis of the time course of decay. This intrinsic limitation in the modeling of α1C currents would not produce a reliable set of parameters for the kinetic scheme in the case of α1C alone.

In testing the effect of the β2a subunit on α1C Ca2+- dependent inactivation, a possible concern is whether the endogenous β3xo subunit changed the properties of the α1 subunits when injected alone. The coexpression β3xo with the α1E and α1C (data not published for α1C) clones produced the generic effect of β subunit of shifting to more negative potential G-V curves (Tareilus et al., 1997). We concluded that the endogenous β3xo has a role in channel expression, but the quantities present are not sufficient to induce kinetic changes.

A further question concerns the molecular mechanism for the Ca2+-dependent inactivation process. Since the Ca2+-sensitive region has been identified within the COOH terminus of the α subunit (de Leon et al., 1995; Zhou et al., 1997), we can speculate that the Ca2+ binding to the COOH terminus produces an allosteric change in the conformation of the protein, such as a collapse of the pore. Another possibility is that the COOH terminus, once Ca2+ has bound, folds backward and blocks the pore, with a mechanism analogous to the N-type inactivation in Shaker K+ channels (Hoshi et al., 1990). However, if the Ca2+ binding site is located in the COOH terminus of the α1C subunit, the COOH terminus could be folded in order for the Ca2+ site to be in close proximity to the pore. Differences in the tertiary structure between native and cloned channels that could arise from differences in the folding of the COOH terminus may modify the accessibility of the Ca2+ binding site to intracellular Ca2+ buffers.

Acknowledgments

We thank Dr. Ramon Latorre and Dr. Ligia Toro for kindly reading the manuscript and Ms. Jin for injection of Xenopus laevis oocytes.

This work was supported by National Institutes of Health (NIH) grant AR-38970 to E. Stefani. N. Qin is the recipient of NIH National Research Service Award GM17120-02 and of the American Heart Association (AHA) Scientist Development Grant 9630053N. This work was done during the tenure of a Grant-in-Aid 113-GI1 award to R. Olcese from the American Heart Association, Greater Los Angeles Affiliate (Los Angeles, CA).

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1

Abbreviations used in this paper: DHP, dihydropyridine; HP, holding potential; I-V, voltage dependencies of current; r-V, voltage dependencies of rate of inactivation; SHP, subtracting holding potential.

J. Zhou's current address is Zeneca Pharmaceuticals, Wilmington, DE 19897.

Author notes

Address correspondence to Dr. Enrico Stefani, UCLA, Dept. of Anesthesiology, BH-612 CHS, Box 951778, Los Angeles, CA 90095-1778. Fax: 310-825-6649; E-mail: estefani@ucla.edu