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Department of Cardiovascular Diseases, Medical Research Institute, Tokyo Medical and Dental University, 1-5-45, Yushima, Bunkyo-ku, Tokyo 113-8510, Japan
Correspondence: Address reprint requests to Yuji Hirano, MD, PhD, Dept. of Cardiovascular Diseases, Medical Research Institute, Tokyo Medical and Dental University, 1-5-45, Yushima, Bunkyo-ku, Tokyo 113-8510, Japan. Tel.: 81-3-5803-5830; Fax: 81-3-5684-6295; E-mail: hirano.card{at}mri.tmd.ac.jp.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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0.7 and fCa-entry to 0.1, whereas deactivation caused fading of ICa,L during final repolarization. These results support experimental findings that Ca2+ entering through ICa,L is essential for inactivation. After responses to standard voltage-clamp protocols were examined, the new model was applied to analyze the behavior of ICa,L when action potential was prolonged by several maneuvers. Our study provides a basis for theoretical analysis of ICa,L during action potentials, including the cases encountered in long QT syndromes. |
INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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). ICa,L also contributes to maintenance of the plateau or elongated depolarization of cardiac action potentials. Because ICa,L is important to our understanding of cardiac functions in physiological as well as pathological conditions, mechanisms of its modulation have been studied extensively (McDonald et al., 1994).
One outstanding feature of ICa,L is that it is inactivated not only by voltage but also by Ca2+, the channel's own charge carrier (Eckert and Chad, 1984; Kass and Sanguinetti, 1984). Although these two different inactivation mechanisms have been known for a long time, precise mechanisms that control ICa,L during action potentials have remained uncertain, including the relative contributions of the two inactivation mechanisms. Recently, however, several studies demonstrated that Ca2+ entering the cell through ICa,L played predominant roles during the inactivation process of cardiac action potential (Linz and Meyer, 1998; Alseikhan et al., 2002).
We now have various types of models for cardiac action potentials. In most models, inactivation is mainly the result of a voltage-dependent process, with only a minor part for [Ca2+]i-mediated modulation. One reason for this underestimation of the role of Ca2+ may result because experimental data utilized for the computational models did not always discriminate two different inactivation mechanisms explicitly. Furthermore, due to the complicated nature of intracellular Ca2+ dynamics, modeling of CICR has often been implemented in a highly schematized or simplified style.
Recent studies have increasingly demonstrated the importance of local Ca2+ dynamics in cardiac functions. For example, studies on Ca2+ sparks and the "local control theory" implies distinct local Ca2+ signals in restricted space (reviewed in Wier and Balke, 1999). Several studies indicate the existence of a subcellular "fuzzy" space where concentration of ions may reach high levels. Also, studies on Ca2+-dependent inactivation suggested direct interaction between Ca2+ ions entering through the channel and calmodulin that is constituently tethered to the channel protein (Zuhlke et al., 1999; Peterson et al., 1999). Thus, whereas modeling with detailed intracellular Ca2+ dynamics is necessary, there are many phenomena to be quantified before it can be implemented into a cardiac action potential model. Realistic modeling of the intracellular organization of Ca2+ and its effects on function should handle multiple cytosolic Ca2+ compartments, and may require development of higher resolution measurement techniques for imaging functional structures. Despite these obstacles, we do have several important experimental and theoretical insights into the dynamics of Ca2+ release. Clearly, incorporation of these insights into cardiac action potential models is an important way to advance our understandings of cardiac function.
In this study, we employed an alternative approach. We present a novel formulation of Ca2+-dependent inactivation, based on a simple assumption that does not require detailed knowledge of intracellular Ca2+ dynamics. Our results support experimental findings that Ca2+ entered through ICa,L largely determine the inactivation process, and provide a basis for further theoretical analysis of ICa,L during action potentials, including those encountered during long QT syndromes.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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). Those models were based on the model proposed by DiFrancesco and Noble (1985), with several modifications to represent guinea-pig ventricular action potentials (Nakajima et al., 1999).
In this paper, we use the Luo-Rudy Phase II model as the control action potential model, based on the C++ source code downloaded from the research section of http://www.cwru.edu/med/CBRTC (Clancy and Rudy, 2001). This version of the Luo-Rudy dynamic model (referred to as the LRd model) has been revised several times since the original version (Luo and Rudy, 1994) was published, including the formulation of IKr and IKs current (Zeng et al., 1995; Viswanathan et al., 1999) and the on-off dynamics of Ca2+-induced Ca2+-release (Faber and Rudy, 2000). The program was modified as described below, keeping the essential part for the computation untouched. These included an adaptive time step method for integration. When the model was in voltage clamp mode (Fig. 6–8), however, time step interval was assigned with a fixed value of 0.002 ms. The C++ source code was compiled using Microsoft Visual C++ (ver. 6) and executed on a Pentium-III based personal computer. Numerical results were written in ASCII files, and were visualized using Microcal Origin (ver. 7.0J).
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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; Zuhlke et al., 1999). The most simple formulation of this effect might be that transition of channels into the Ca2+-inactivated state is proportional to the amount of Ca2+ (charges) entering through the channel openings.
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We define a Ca2+entry-dependent inactivation property with a parameter fCa-entry as the proportion of unbound Ca channels (Unbound / (Unbound + Ca-inactivated)). Changes in fCa-entry (
fCa-entry) were determined by a term proportional to Ca2+ influx through the channel (-
x ICa,L) and time-dependent recovery which takes place in the proportion of Ca2+-inactivated channels (ß x (1 - fCa-entry)):
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 | (1) |
ICa,L is now computed by the Goldman-Hodgkin-Katz current equation:
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Eqs. 1 and 2 are the central scheme of this study. We investigate the outcome of this formulation in the framework of a standard action potential model.
Reevaluation of voltage-dependent inactivation of ICa,L
In the LRd model, and also in many available cardiac action potential models (DiFrancesco and Noble, 1985; Jafri et al., 1998), the voltage-dependent inactivation property was set as a major contributor of inactivation of ICa,L. Simple addition of a Ca2+ entry-dependent inactivation mechanism to the LRd model, then, would pose too much inactivation of ICa,L during action potentials. Therefore, as a first step, it was necessary to reassess the voltage-dependent inactivation properties of ICa,L.
As shown in Fig. 1, settings for voltage-dependent inactivation properties showed large variations among different computer models. One reason for this discrepancy is that basal experimental data adopted for modeling did not necessarily discriminate Ca2+-dependent inactivation from voltage-dependent inactivation properties. During experiments, Ca2+-dependent inactivation is thought to be minimized or removed by using Ba2+, Sr2+, or monovalent cation (such as Na+) as the charge carrier. Although Ba2+ is often used for this purpose, it has been recognized that Ba2+ allows some current-dependent inactivation (Markwardt and Nilius, 1988; Ferreira et al., 1997). There are several reports of steady-state voltage-inactivation curve for cardiac ICa,L, where the effect of Ca2+-dependent inactivation was minimized using Na+ as the charge carrier (often denoted as Ins). These studies described inactivation curves that decrease monotonically to a nonzero value (0.4 by Hadley and Hume, 1987; 0.55 by Sun et al., 1997; and Linz and Meyer, 1998). We followed this type of formulation (Fig. 1 A, thick line):
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). In the heart, Hadley and Hume (1987) reported that decay of Na+ current through the L-type Ca2+ channel (onset of inactivation) could often be described by two exponentials. However, data showed such large variations among cells that they had to characterize the decay kinetics of Ins by the half-time of inactivation to obtain consistent results. Later, Sun et al. (1997) and Mitarai et al. (2000) reported that double exponentials were required to describe the inactivation time course when test voltage was raised above 0 mV. In guinea-pig ventricular myocytes at 35°C (Mitarai et al., 2000), the time constant of the slow component was 500 ms without apparent voltage dependence. The time constant of the fast component was shortened to 20 ms as the test potential was elevated. Based on their Fig. 2, the amplitude of the fast component was more than two times larger at test potentials between 10 and 30 mV. They also found that the slow component dominated the inactivation when channel protein was phosphorylated. Then, slow decay observed by Linz and Meyer (1998) might be explained by the use of isoproterenol in their Ins measurements.
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reported that recovery was a single exponential process with steep voltage-dependence. At -50 mV, recovery time constant was 313 ± 26 ms with Na+ and 275 ± 26 ms with Ca2+ as the charge carrier. Kass and Sanguinetti (1984) used divalent cations (Ca2+, Sr2+, or Ba2+) to study inactivation properties. In their Purkinje preparations, however, recovery time course required double exponentials when duration of the inactivating prepulse exceeded 250 ms. Recovery time constants obtained by brief prepulses, or those for fast phase of inactivation with 250-ms prepulse (which is dominant at -48 mV; see their Fig. 7) were 50 ms at -50 mV. When combined with the time constant of onset of inactivation (for fast phase with Sr2+ or Ba2+ as the charge carrier), inactivation time constant showed bell-shaped voltage dependence with the peak at -25 mV.
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The slower phase of the onset of inactivation was assumed to have 30% of the total amplitude, and was assigned a time constant of 500 ms (Mitarai et al., 2000) at potential levels above -20 mV. This is illustrated in Fig. 1 B (thick dotted line).
The Ca2+ entry-dependent inactivation model
We can now go back to the central issue—the new formulation of ICa,L incorporating Ca2+ entry-dependent inactivation defined by Eqs. 1 and 2. In this formulation, Ca2+-dependent inactivation parameter (fCa) is determined by two terms: onset of inactivation proportional to the influx of Ca2+ (association of Ca2+) and time-dependent recovery (dissociation). This proposal was implemented into the LRd model under the revised settings of voltage-dependent inactivation as described above. Other factors to determine ICa,L, including voltage activation (d), the value of PCa (a permeability of Ca2+), and the dynamics of intracellular Ca2+ were unchanged.
One of the most distinctive features of the new formulation was that changes in association and dissociation constants ( and ß) greatly influenced action potential durations (APDs). Increase in -accelerated Ca2+-dependent inactivation and shortened APDs. On the other hand, increase in ß decreased the extent of Ca2+-inactivation and prolonged APDs. For the present, we cannot directly determine values of and ß in Eq. 1. We therefore tried to find a set of values which reproduce the experimental data of Ca2+-dependent inactivation of ICa,L during action potentials. It was reported that Ca2+-dependent inactivation proceeded to 90–95% during the plateau phase. Later, during repolarization, ICa,L recovered up to 25% (Linz and Meyer, 1998; see their Fig. 11).
Fig. 2 A shows a case when is set to 0.0051 µA-1 x µF x ms-1 and ß to 0.02 ms-1. With this setting, APD was almost identical with the original LRd model shown in B. As the plateau phase proceeds, fCa-entry continues to decline to 0.1, and then recovers. In the original LRd model (Fig. 2 B), decline in ICa,L during action potential was produced by the voltage-dependent inactivation variable. Here, Ca2+-dependent inactivation rapidly increases at the upstroke and then declines. Thus, the model produced similar temporal profile of ICa,L, but with different gating mechanisms.
In Fig. 3, we examined the effects of association and dissociation parameters ( and ß) on the shape and durations of action potentials. It should be noted that changes in membrane permeability or maximum conductance of ICa,L had little effect on APDs. Increased ICa,L amplitude produced higher plateau levels initially, but was counterbalanced by entry-dependent inactivation, leaving minor and variable effects on APDs (Fig. 4 B). This observation was in contrast with the LRd model (Fig. 4 A), where the plateau height as well as APDs were modified concordantly as the maximum conductance of ICa,L was altered.
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). The most likely source of increased Ca2+ in this case was the sarcoplasmic reticulum (SR). It was also reported that block of Ca2+ release from SR by ryanodine/thapsigargin could reduce ICa,L inactivation during initial 50 ms of action potentials (Linz and Meyer, 1998). These findings indicate a need to include additional Ca2+-mediated inactivation mechanisms. The comprehensive modeling of intracellular Ca2+ handling, however, is not the purpose of the present report (see Discussion). As our primary concern was to evaluate the new formulation of Ca2+ entry-dependent inactivation in the framework of standard action potential model, we tried to preserve the features of the Luo-Rudy model as much as possible, and followed their intracellular Ca2+ dynamics. Then, to implement the effect of Ca2+ released from SR, we adopted the same strategy as in the case of Ca2+ entered through sarcolemma; Ca2+ released from the SR inactivates Ca channels but with different effectiveness.
We define a SR release-dependent inactivation property with a parameter fCa-SR, as the proportion of SR-release unaffected Ca channels (Unaffected/(Unaffected + SR-Ca inactivated)). Changes in fCa-SR were determined by a term proportional to Ca2+-release current from SR (-SR x Irel) and time-dependent recovery which takes place in the proportion of SR-release modified channels (ßSR x (1 - fCa-SR)):
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Ca2+-dependent inactivation parameter in Eq. 2 now should be further reduced for the SR-release modified channels. We assumed that Ca2+-dependent inactivation by the influx through ICa,L and by the release from SR could be treated independently:
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With the values of SR and ßSR set to 0.05 µA-1xµF x ms-1 and 0.02 ms-1, respectively, released Ca2+-dependent inactivation rapidly increased at the upstroke of action potential and then declined slowly (Fig. 5 A).
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1/1000) permeabilities for K+ and Na+. However, due to high intracellular concentration of K+, its outward component is very large, especially following the upstroke of the action potential. This is not consistent with experimental observations that there should be no monovalent permeation while the channel is passing significant amounts of Ca2+ (Matsuda and Noma, 1984; Sun et al., 2000; see also Jafri et al., 1998). Therefore, we tentatively deleted the K outward current and the Na inward current component of L-type Ca channels (Fig. 5 B). We then asked how these changes affect maintenance of ionic gradients across the cell membrane. The LRd model showed gradual changes in [Na+]i and [K+]i during long courses of stimulation. This also occurred with our monovalent cation-deleted version, with very similar gradual changes in ionic concentrations. After 200 stimulations with BCL of 300 ms, [Na+]i increased from 9 mM to 13.0 mM in the Ca2+-inactivation model without Na+ and K+ permeation, and to 12.2 mM in the LRd model. On the other hand, [K+]i decreased from 141.2 mM to 136.3 mM in our Ca2+-inactivation model and to 137.3 mM in the LRd model.
Voltage-clamping of Ca2+ entry inactivation model
We applied several voltage-clamp protocols on the model shown in Fig. 5 B, to further characterize and illustrate the properties of the new formulation when compared with the original LRd model.
In Fig. 6, the model was depolarized to various potential levels from the holding potential of -60 mV. Depolarization to 0 mV elicited a large inward Ca current (5 x the amplitude that was observed during an action potential), as shown in A. The current initially showed rapid decay due to Ca release from SR (fCa-SR), followed by a slower decay mainly determined by fCa-entry. In the LRd model shown in B, this role was played by voltage-dependent inactivation, fv. In C, we compared the current decay at -30, 0, and 30 mV. The speed of current decay was fastest at 0 mV where current amplitude was the largest. Fig. 6 D shows the peak current-voltage relationships of the new model and the LRd model. The IV curves were very similar between two models equipped with different inactivation properties and identical voltage-dependent activation properties of ICa,L.
In Fig. 7, we measured the voltage-dependent inactivation curve using a standard double-pulse protocol. Duration of conditioning was set to 1000 ms. Upper panels show current traces with the prepulse potential of -30 mV and -60 mV (holding potential). When normalized to the current amplitude obtained with prepulse potential of -80 mV (B), data points clearly deviated from the voltage-dependent inactivation curve defined by equation 3 (finf(
)). Records of gating parameters shown in A indicated that decrease in current amplitude at -30 mV was mainly brought by fCa-entry. These results are completely different from the LRd model (C), where observed points closely followed the "pure" voltage-inactivation curve finf().
In Fig. 8 A, we compared the responses to train of depolarizing pulses in voltage-clamp mode (upper panel) and to repetitive stimulation in current-clamp mode (lower panel). In each panel, the model was initially depolarized (stimulated) at 2 Hz for 12 s. The frequency was then transiently increased to 3 Hz, before subsequent return to 2 Hz. In voltage-clamp mode (upper panel), when pacing frequency was increased, a transient suppression of peak Ca2+ transient was followed by an augmentation toward a new steady level. Upon a reduction of pacing frequency, peak Ca2+ transient declined after a transient increase. This behavior was essentially the same with trains of action potentials (lower panel), although recovery of peak Ca2+ transient was slow following the increase upon reduction of pacing frequency to 2 Hz. The same protocol was applied to the LRd model (Fig. 8 B). The two models equipped with identical properties of CICR responded with almost identical behavior, except that the amplitude of Ca2+ transient was higher in our new model. Also, the level of Ca2+ concentration in network sarcoplasmic reticulum (NSR) reached higher level during trains of stimulations.
Behavior during prolonged APDs
Prolongation of APD is an important substrate for clinical arrhythmias (Chiang and Roden, 2000). In long QT syndromes, this is brought about by a decreased K current amplitude or increased Na current during the plateau, due to the mutation of channel proteins. Using the model shown in Fig. 5 B, we next examined how the reduction of IKr and IKs current modulate the action potentials and the profiles of ICa,L.
As shown in Fig. 9 A, b and c, reduction of K currents in the new model prolonged APDs and induced early after-depolarization (EADs). During the plateau phase where EADs took place, not only voltage inactivation but also Ca2+-dependent inactivation was incomplete. As determined by the balance of inward and outward current, membrane potentials oscillate with prominent changes in activation parameters (activation and deactivation). EADs were also accompanied by changes in Ca2+ entry-dependent inactivation parameter (fCa-entry).
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), reduction of IKr alone failed to produce EADs. However, reduction of IKs below 30% produced EADs or failure of repolarization. In Fig. 9 B, c where IKs was reduced to 25% of the control, EADs appeared without appreciable changes in Ca2+-dependent inactivation parameter (fCa).
Prolongation of APDs can be produced also by changes in ICa,L parameters. As shown in Fig. 3, changes in association and dissociation constants of Ca2+ and the channel protein ( and ß) were the most efficient way in Ca2+-inactivation models. When EADs were produced by reducing by 30% (Fig. 9 A, d), profiles of gating parameters underlying the EADs were surprisingly similar with the cases produced by reduced K-current amplitudes.
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DISCUSSION |
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0.1 during the plateau phase and recovered up to 0.3 at final repolarization. Our results support the view that Ca2+ entry-dependent inactivation can play a significant role to determine the time course of L-type Ca current during cardiac action potentials, consistent with recent experimental findings (Sun et al., 1997; Linz and Meyer, 1998; Alseikhan et al., 2002).
Comparison with the LRd model
In this study, we replaced the ICa subsystem in the LRd model. The new gating mechanisms resulted in several important properties not simulated in the LRd model, while leaving some basic properties untouched.
In our formulation (Fig. 3), APDs were greatly affected by the alteration of binding efficacy between Ca2+ ion and the Ca channel (actually through calmodulin constitutively tethered to channel protein; see below). Recently, prominence of Ca2+-dependent inactivation of L-type Ca channels in controlling APDs was experimentally demonstrated by the expression of a Ca2+-insensitive mutant calmodulin, which virtually eliminated Ca2+-dependent inactivation and strikingly prolonged APDs (Alseikhan et al., 2002). On the other hand, changes in maximum conductance or PCa had small and variable effects on APDs in a standard condition (Fig. 4). This is in contrast to the original LRd model.
During voltage clamp simulations, peak IV relationships were kept almost identical (Fig. 6). However, mechanisms to determine the voltage-dependent inactivation curve were completely different between the two models (Fig. 7). In the present model, inactivation curve with a minimum 0 mV was not built into model equations, but emerged from Ca2+ entry-dependent inactivation.
Intracellular Ca2+ dynamics and the modulation of Ca current
In the present study, CICR or dynamics of intracellular Ca2+ affects ICa,L through the parameter fCa-SR, as determined by the amplitude of Ca2+-release from SR (Irel). Experimental findings indicate that the effect of SR release is particularly important during the initial 50 ms of action potentials (Linz and Meyer, 1998) or during depolarizing pulses in voltage clamp experiments using conventional square pulses (Sun et al., 1997). Although our new model is consistent with these studies, significant issues remain to be examined concerning the effects evoked by intracellular Ca2+ dynamics.
The LRd model generates action potentials through detailed description of current systems. However, its Ca2+ handling subsystem is very simple and phenomenological. As in the DiFrancesco-Noble model (1985), the LRd model has a single cytosolic Ca2+ compartment (common pool) to which both Ca2+ influx through ICa,L and SR Ca2+ release are directed. In cardiac myocytes, SR Ca2+ release is proportional to the influx of trigger Ca2+ (graded response) whereas released Ca2+ from SR is significantly larger than the trigger influx (high gain). It is difficult to accommodate "graded response" and "high gain" simultaneously in common pool models, because a high-gain system generally leads to all-or-none–type regenerative Ca2+ releases (Stern, 1992). In the LRd model, Ca2+ release is a function of Ca2+ influx through the sarcolemma, but independent of Ca2+ release from the SR. In this setting, released Ca2+ may have a suppressive effect on CICR because it reduces the concentration gradient of Ca2+ between the common pool and the junctional SR. The formulation of CICR employed here does not include regenerative Ca2+ release, a mechanism that admittedly is not biophysically accurate.
Recent local control theory established that CICR is determined by local Ca2+ dynamics (such as Ca2+ sparks), not by the average [Ca2+]i within the myocytes (Wier and Balke, 1999; Wang et al., 2001). For example, Wier et al. (1994) observed that gain of CICR (the ratio of released Ca2+ to influx of Ca2+ through the channel) is dependent on the voltage of clamp pulses, with the highest gain at -20 mV. This observation arises from the fact that the amplitude of unitary Ca channel current is high near threshold levels (located far from the reversal potential of Ca2+), and therefore is more effective to trigger Ca2+ sparks or release of Ca2+ from SR. These important properties of CICR could not be reproduced in the LRd model and accordingly, in our present model which faithfully followed the formulation presented in the LRd model. During rapid stimulations (Fig. 8), the new formulation of ICa,L inactivation increased the amplitude of Ca transient and slightly raised the level of NSR Ca2+ concentration. It did not, however, affect the basic properties of intracellular Ca2+ dynamics.
There are now several models of intracellular Ca2+ handling with more complex structures and detailed description of Ca2+-induced Ca2+ release. For example, Rice, et al. (1999) introduced stochastic methods to make a robust model with high gain and graded Ca2+ release in the functional unit of the cardiac dyadic space, whereas Stern et al. (1999) stressed the importance of allosteric interactions between ryanodine receptors. To extend the range of phenomena to be evaluated, it is necessary to incorporate a more detailed intracellular Ca2+ handling system into cardiac action models.
In the heart, Ca channel is not the sole current system modulated by [Ca2+]i. Additional Ca2+-modified channels or ion-transporters include not only Na+–Ca2+ exchanger and IKs potassium channel, but also Na channel (Tan et al., 2002). If the localization or distribution of channel proteins is not uniform with respect to local Ca2+ distribution in the myocyte, modeling of [Ca2+]i effect on channel function should be highly complicated.
Further revisions required
Several important modulatory functions are not implemented in this version of Ca current formulation, in addition to the problems related to handling of [Ca2+]i. Ca channel open probability was potentiated following large depolarization. In contrast to the case in skeletal muscle (Hulme et al., 2002), prepulse facilitation in cardiac myocytes may involve a mechanism related to voltage-dependent conformational change (Hirano et al., 1999). Modulation of ICa,L by Ca2+ include potentiation when increase in [Ca2+]i was modest (Hirano and Hiraoka, 1994). The latter effect is linked to phosphorylation of the channel protein by Ca2+ calmodulin-dependent kinase II, and is an example of pivotal roles of Ca2+ as the modulator of various enzyme activities.
Computational modeling is an important companion to experimental work, systematizing existing experimental observations and serving as a predictable tool for further experimental investigation. However, the model is only as accurate as the quality of the experimental data used in its development. As new experimental insight becomes available, models must be reassessed and revised.
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ACKNOWLEDGEMENTS |
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,b). Their voltage-inactivation curve was similar to that used in this study, when myocytes were under ß-adrenergic stimulation. Submitted on April 11, 2002; accepted for publication August 29, 2002.
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