INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX I APPENDIX II REFERENCES

Understanding of the mechanisms by which Ca²⁺ influx via voltage-gated L-type Ca²⁺ channels (LCCs) triggers Ca²⁺ release from the junctional sarcoplasmic reticulum (SR) has advanced tremendously with the development of experimental techniques for simultaneous measurement of LCC currents and Ca²⁺ transients (Wier et al., 1994; Cannell et al., 1987; Nabauer et al., 1989), and detection of local Ca²⁺ transients (Cannell et al., 1984; Lopez-Lopez et al., 1994, 1995; Cheng et al., 1995). This has given rise to the local control theory of excitation-contraction (EC) coupling (Stern, 1992; Bers, 1993; Wier et al., 1994; Sham, 1997), which asserts that opening of an individual LCC located in the transverse (T) tubular membrane triggers Ca²⁺ release from a small cluster of SR Ca²⁺ release channels known as ryanodine receptors (RyRs) located in the closely apposed junctional SR membrane (Fabiato, 1985; Cheng et al., 1993; Cannell et al., 1995; Santana et al., 1996, Sham et al., 1995; Collier et al., 1999; Wang et al., 2001). Tight regulation of this Ca²⁺-induced Ca²⁺ release (CICR) is made possible by the fact that LCCs and RyRs are sensitive to local rather than global Ca²⁺ levels. The local control theory also asserts that graded control of SR Ca²⁺ release, in which Ca²⁺ release from junctional SR is a smooth, increasing function of Ca²⁺ influx, is achieved by statistical recruitment of elementary SR Ca²⁺ release events by trigger Ca²⁺ entering via single LCCs (Stern, 1992; Beuckelmann and Wier, 1988; Wier and Balke, 1999). In addition to triggering SR Ca²⁺ release, increases of local Ca²⁺ promote Ca²⁺-dependent inactivation of LCCs (Peterson et al., 1999; Bers and Perez-Reyes, 1999). Because L-type Ca²⁺ current (I_CaL) plays a primary role in determining action potential (AP) shape and duration, local control theory therefore implies that the microscopic properties of Ca²⁺ release are likely to contribute to macroscopic electrophysiological responses of the cardiac myocyte.

Several computational models have been developed to investigate properties of local Ca²⁺ release at the level of the cardiac dyad (Rice et al., 1999; Stern et al., 1999; Langer and Peskoff, 1996; Cannell and Soeller, 1997; Soeller and Cannell, 1997). Each of these model formulations incorporates 1) one or a few LCCs; 2) a cluster of RyRs; 3) the dyadic volume in which the events of CICR occur; and 4) anionic binding sites, which buffer Ca²⁺. In some of these models, detailed descriptions of diffusion and Ca²⁺ binding in the dyadic cleft are used to demonstrate the effects of geometry, LCC, and RyR properties and organization, and surface charge on the spatiotemporal profile of Ca²⁺ within the dyad, and hence on the efficiency of CICR (Langer and Peskoff, 1996; Cannell and Soeller, 1997; Soeller and Cannell, 1997). Stern et al. (1999) have simulated CICR stochastically using numerous RyR schemes to demonstrate conditions necessary for stability of EC coupling, and have suggested a possible role for allosteric interactions between RyRs. The functional release unit model of Rice et al. (1999) has demonstrated that local control of CICR (i.e., graded SR release and high EC coupling gain) can be obtained without including computationally intensive descriptions of Ca²⁺ gradients within the dyadic space. Isolated EC coupling models such as these, however, cannot elucidate the nature of the interaction between local events of CICR and integrative cellular behavior.

Existing models of the cardiac ventricular myocyte do not incorporate mechanisms of local control of SR Ca²⁺ release (Winslow et al., 1999; Jafri et al., 1998; Luo and Rudy, 1994; Priebe and Beuckelmann, 1998; Pandit et al., 2001; Noble et al., 1998; Fox et al., 2002). Rather, in these models all Ca²⁺ influx through sarcolemmal LCCs and Ca²⁺ release flux through RyRs is directed into a common Ca²⁺ compartment. As defined by Stern (1992), a "common pool" model is one in which trigger Ca²⁺ reaches the SR via the same cytosolic Ca²⁺ pool into which SR Ca²⁺ is released, where activation of the SR release mechanism is controlled by Ca²⁺ concentration in this cytosolic pool. The result of this physical arrangement is that once RyR Ca²⁺ release is initiated, the resulting increase of Ca²⁺ concentration in the common pool stimulates regenerative, all-or-none rather than graded Ca²⁺ release (Stern, 1992). This "latch up" of Ca²⁺ release can be avoided, and graded SR release can be achieved in a model of EC coupling by formulating Ca²⁺ release flux as an explicit function of sarcolemmal Ca²⁺ influx (Priebe and Beuckelmann, 1998; Luo and Rudy, 1994; Faber and Rudy, 2000; Fox et al., 2002). Models of this type are not common pool models based on the definition given by Stern (1992), and do not suffer an inability to stably exhibit both high gain and graded SR Ca²⁺ release. These phenomenological formulations, however, lack mechanistic descriptions of the processes that are the underlying basis of CICR. Both common pool models and models with phenomenological Ca²⁺ release mechanisms are therefore inadequate for the study of how detailed microscopic features of EC coupling have an impact on macroscopic electrophysiological properties of the myocyte, such as the whole-cell Ca²⁺ transient and AP morphology.

In this study, we develop a comprehensive model of the ventricular myocyte based on the theory of local control of SR Ca²⁺ release. This is accomplished by updating and extending the canine ventricular myocyte model of Winslow et al. (1999) to include a population of dyadic Ca²⁺ release units. Local interactions of individual sarcolemmal LCCs with nearby RyRs in the JSR membrane are simulated stochastically, with these local simulations embedded within the numerical integration of the differential equations describing ionic and membrane pump/exchanger currents, SR Ca²⁺ cycling, and time-varying cytosolic ion concentrations. We demonstrate that this model faithfully reproduces experimentally observed features of LCC voltage- and Ca²⁺-dependent gating (Linz and Meyer, 1998; Sipido et al., 1995; Hobai and O'Rourke, 2001; Sham et al., 1995; Sham, 1997; Rose et al., 1992; Herzig et al., 1993), microscopic EC coupling (Wier et al., 1994; Sham et al., 1998; Song et al., 2001), and macroscopic whole-cell AP and Ca²⁺ cycling properties (O'Rourke et al., 1999). Simulations demonstrate that local control is an essential property for stability of APs when the LCC inactivation process depends more strongly on local Ca²⁺ than on membrane potential, a scenario that is supported by experiments (Linz and Meyer, 1998; Peterson et al., 1999, 2000), but which cannot be implemented successfully using a common pool model where the inherent positive feedback of rising Ca²⁺ levels on RyR activation is intact. Modeling supports the hypothesis that the robust bidirectional interaction between Ca²⁺ dynamics and membrane potential in the local control environment plays a central role in establishing the integrative electrophysiological properties of the cardiac myocyte. Preliminary results from this study were presented previously in abstract form (Greenstein and Winslow, 2001a, 2001b).

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX I APPENDIX II REFERENCES

The Ca²⁺ release unit model

We seek to define a model of local control of SR Ca²⁺ release that captures fundamental properties such as graded release, while at the same time is computationally tractable such that it may be incorporated into an integrative model of the ventricular myocyte. Models describing diffusion of Ca²⁺ within the dyadic space, detailed dyad geometry, and surface charge effects (Cannell and Soeller, 1997; Soeller and Cannell, 1997; Langer and Peskoff, 1996) are too computationally demanding for this application. As a compromise between structural and biophysical detail versus tractability, a "minimal model" of local control of Ca²⁺ release, referred to as the Ca²⁺ release unit (CaRU) model, is implemented. A full mathematical description of the stochastic state models, dynamical equations, parameters, and initial conditions defining the myocyte model are given in Appendix I.

Fig. 1 A shows a schematic of the CaRU model based in part on the previous model of Rice et al. (1999). The CaRU model is intended to mimic the properties of Ca²⁺ sparks in the T-tubule/SR (T-SR) junction. Ca²⁺ sparks are elementary SR Ca²⁺ release events arising from a cluster of RyRs (Cheng et al., 1993). Fig. 1 B shows a cross-section of the model T-SR cleft, which is divided into four individual dyadic subspace compartments arranged on a 2 × 2 grid. Each subspace (SS) compartment contains a single LCC and 5 RyRs in its JSR and sarcolemmal membranes, respectively. All 20 RyRs in the CaRU (5 RyRs/SS × 4 SSs/CaRU = 20 RyRs/CaRU) communicate with a single local JSR volume. The 5:1 RyR/LCC stoichiometry is chosen to be consistent with recent estimates indicating that a single LCC typically triggers the opening of four to six RyRs (Wang et al., 2001). Each subspace is treated as a single compartment in which Ca²⁺ concentration is uniform; however, Ca²⁺ may diffuse passively to neighboring subspaces within the same CaRU. The rate of Ca²⁺ transfer between two adjacent subspace compartments is assumed to be 10-fold slower than that from subspace to cytosol. This yields an inter-subspace transfer rate (r_iss) of 20 ms¹, which corresponds to a diffusion coefficient of ~3.3 × 10⁶ cm² s¹ when the assumed height of the model subspace is 12 nm. This value is similar to estimates for Ca²⁺ diffusion in the presence of RyR "feet" structures in the restrictive dyadic subspace volume (Soeller and Cannell, 1997). The division of the CaRU into four subunits allows for the possibility that an LCC may trigger Ca²⁺ release in adjacent subspaces (i.e., RyR recruitment) under conditions where unitary LCC currents are large. The existence of communication among adjacent subspace volumes is supported by the findings that Ca²⁺ release sites can be coherent over distances larger than that occupied by a single release site (Parker et al., 1996), and that the mean amplitude of Ca²⁺ spikes, local SR Ca²⁺ release events that consist of one or a few Ca²⁺ sparks (Song et al., 1998), exhibits a bell-shaped voltage dependence, indicating synchronization of multiple Ca²⁺ release events within a T-SR junction (Song et al., 2001). The choice of four subunits allows for semi-quantitative description of dyadic Ca²⁺ diffusion while retaining minimal model complexity.

View larger version (24K): [in this window] [in a new window]   FIGURE 1   Schematic representation of the Ca2+ release unit model (CaRU). (A) Trigger Ca2+ influx through the LCCs enters into the T-SR cleft (dyadic space). The rise in local Ca2+ level promotes the opening of RyRs and ClChs. The excess local Ca2+ passively diffuses out of the cleft into the cytosol, and JSR Ca2+ is refilled via passive diffusion from the NSR. (B) The T-SR cleft (shown in cross-section) is composed of four dyadic subspace volumes, arranged on a 2 × 2 grid, each containing 1 LCC, 1 ClCh, and 5 RyRs. Ca2+ in any subspace may diffuse to a neighboring subspace (Jiss) or to the cytosol (Jxfer). Jiss,i,j,l represents Ca2+ flux from the jth subspace to the lth subspace within the ith CaRU. Similarly, Jxfer,i,j represents Ca2+ flux from the jth subspace to the cytosol from the ith CaRU (see Appendix I).

One of the bases for local control of SR Ca²⁺ release is the structural separation of T-SR clefts at the ends of sarcomeres (i.e., RyR clusters are physically separated) (Franzini-Armstrong et al., 1999). Each CaRU is therefore simulated independently in accord with this observation. Upon activation of RyRs, subspace Ca²⁺ concentration will become elevated. This Ca²⁺ will freely diffuse to either adjacent subspace compartments (J_iss) or into the cytosol (J_xfer) along its concentration gradient. The local JSR compartment is refilled via passive diffusion of Ca²⁺ from the network SR (NSR) compartment (J_tr).

The model for the L-type Ca²⁺ channel is identical in structure to the mode-switching model developed previously by Jafri et al. (1998). The following modifications have been made to model parameters: 1) voltage-dependence of forward and reverse activation transition rates ( and , respectively) have been adjusted based on recent measurements of I_CaL obtained in canine midmyocardial cells (Hobai and O'Rourke, 2001); 2) voltage-independent transition rates into open states f and f' have been adjusted to yield peak open probability in the range of 5-15% in response to a maximally activating voltage clamp stimulus (Rose et al., 1992; Herzig et al., 1993; Handrock et al., 1998); 3) transition rates between the normal gating mode (Mode Normal) and the Ca²⁺-inactivation mode (Mode Ca)  and  are adjusted to enhance Ca²⁺-dependent inactivation, while the voltage-dependent steady-state inactivation function (y) is modified to have an asymptotic value of 0.6 for large positive membrane potentials. The latter is based on the observation that there is a small sustained component of Ca²⁺ current in response to voltage clamp stimuli in canine ventricular cells (Kääb et al., 1996; Tseng et al., 1987), and that Ca²⁺-dependent inactivation dominates the I_CaL inactivation process while voltage-dependent inactivation is relatively weak and incomplete (Linz and Meyer, 1998; Hadley and Hume, 1987; Peterson et al., 1999, 2000); 4) permeation through the LCC is described by the Goldman-Hodgkin-Katz current equation as originally presented by Luo and Rudy (Luo and Rudy, 1994) where Ca²⁺ concentration at the inner mouth of the channel is assumed to be equivalent to Ca²⁺ concentration of the adjacent subspace, rather than assuming it is constant (Jafri et al., 1998; Rice et al., 1999); 5) LCC permeability (P_CaL) is adjusted to a value of 9.13 × 10¹³ cm³ s¹, which yields a single channel slope conductance of 8.2 pS and a unitary current of ~0.12 pA at 0 mV (see Fig. 2 G) (Rose et al., 1992; Yue and Marban, 1990).

Whole-cell Ca²⁺ current can be described as a function of the total number of channels (N_LCC), the single channel current magnitude (i), the open probability (p_o), and the fraction of channels that are available for activation (f_active, i.e., in a phosphorylated mode), where I_CaL = N_LCC × i × p_o × f_active (Handrock et al., 1998). Under conditions where f_active remains constant, I_CaL = N_active × i × p_o, where N_active = N_LCC × f_active is constant. As described above, single channel parameters are based on experimentally obtained constraints on both i and p_o. N_active is therefore chosen such that the amplitude of the ensemble current summed over all LCCs corresponds to whole-cell measurements in canine myocytes (Hobai and O'Rourke, 2001). This approach yields a value of 50,000 for N_active, which agrees with experimental estimates of LCC density (Rose et al., 1992; McDonald et al., 1986), and which corresponds to 12,500 CaRUs (N_CaRU). The process of slow cycling between active and inactive states is not included in this model; rather only active LCCs are simulated.

Each RyR channel is represented by the model developed by Keizer and Smith (1998) and later modified by Rice et al. (1999). This model was originally designed to describe the property of RyR adaptation, a slow spontaneous decrease in open probability that has been observed to occur after activation by a step increase in Ca²⁺ when measured in single channels reconstituted in lipid bilayers (Gyorke and Fill, 1993). A channel in the adapted state can be reactivated by an additional increase in Ca²⁺. In contrast, the findings of Sham et al. (1998) suggest that RyR inactivation into an absolute refractory state occurs in vivo during EC coupling. It is difficult to incorporate single channel RyR data obtained in vitro into models of EC coupling due to the lack of quantitative information regarding in vivo regulation by accessory proteins and other ligands. Therefore, our approach has been to constrain RyR model parameters based on experimentally obtained properties of EC coupling (Song et al., 2001; Wier et al., 1994), without an explicit attempt to retain the property of adaptation. The Ca²⁺ dependence of the RyR model state transition rates have been adjusted based on the assumption that four Ca²⁺ ions must bind to the channel before it can enter the open state (Zahradnikova et al., 1999).

Ca²⁺ buffering in each CaRU is implemented as described previously (Rice et al., 1999) using the rapid buffer approximation (Wagner and Keizer, 1994). It is assumed that Ca²⁺ is buffered in the subspace by the phospholipid anion groups on both the SR and sarcolemmal membranes, and that these buffers are immobile (Smith et al., 1998). Buffering parameters for JSR Ca²⁺ are based on measures of Ca²⁺-calsequestrin binding (Shannon et al., 2000).

The Ca²⁺-dependent transient outward chloride (Cl) current (I_to2) is included as part of the CaRU. Experimental evidence indicates that the Ca²⁺ binding affinity of this Cl channel (ClCh) is low (K_d,ClCh ~ 150 µM) relative to normal cytosolic Ca²⁺ concentrations (Collier et al., 1996), and that I_to2 is abolished in the presence of caffeine and exhibits rate-dependent properties that correlate with those of SR Ca²⁺ load (Zygmunt, 1994), suggesting that ClChs are activated by subspace Ca²⁺. Estimates of ClCh density are in the range of 3-4 µm² (Collier et al., 1996), a value similar to the density of active LCCs (2-5 µm²) (Rose et al., 1992; McDonald et al., 1986). Each CaRU therefore includes an equal number of ClChs as LCCs, i.e., one Cl channel per subspace (Fig. 1 B). I_to2 is modeled as a voltage- and time-independent ligand-gated channel, using a simple two-state, closed-open model (see Fig. 12 C), based on the gating and permeation properties of the unitary current as measured by Collier et al. (1996). For simplicity, intracellular Cl concentration is assumed to be constant.

Whole myocyte model

Definitions for global state variables (e.g., K⁺ currents, cytosolic Ca²⁺ concentration) of the local control myocyte model are based on those of the Winslow et al. (1999) canine ventricular cell model with the following modifications: 1) the voltage-dependent Ca²⁺-independent transient outward potassium (K⁺) current (I_to1) is modeled as described by Greenstein et al. (2000); 2) the rapid delayed rectifier K⁺ current (I_Kr) is modeled as described by Mazhari et al. (2001); 3) the SR Ca²⁺ ATPase (SERCA2a pump) has been updated based on the model and parameter set of Shannon et al. (2000), which accounts for both a forward and a backward Ca²⁺ pump flux; and 4) some membrane currents and ionic fluxes are scaled to preserve cytosolic ionic concentrations and AP shape (see Appendix I).

A detailed description of the local control simulation algorithm is given in Appendix II. Within the whole-cell simulation, the time progression of states within each CaRU is calculated individually. The simulation of each CaRU requires both numerical integration of local (subspace and JSR) Ca²⁺ balance equations and Monte Carlo simulation of channel gating (LCC, RyR, and ClCh). The state of each channel is described by a set of discrete valued random variables that evolve in time as described by Markov processes. Time steps for CaRU simulations are adaptive and are chosen to be sufficiently small based on channel transition rates. The CaRU simulations occur within the (larger) time step used for the numerical integration of global state variables (e.g., V_m). The majority of computational time is spent in stochastic simulation of the large number of independent CaRUs. This simulation is therefore highly parallelizable, and is implemented on an SGI Power Challenge symmetric multiprocessing computer configured with 12 R10,000 processors and 1 Gbyte memory.

As a result of the embedded Monte Carlo simulation, all state variables (e.g., V_m) and ionic currents/fluxes (e.g., I_CaL) will contain a component of stochastic noise (e.g., Fig. 3 A). These fluctuations introduce a degree of variability to simulation output. Therefore, where appropriate, simulation data are presented as mean ± SE, where the specified value for n refers to the number of simulation runs.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX I APPENDIX II REFERENCES

L-type Ca²⁺ current

Because local control of Ca²⁺ release is the central feature of the myocyte model presented here, it is necessary to verify that the newly parametrized L-type Ca²⁺ channel model behavior is consistent with experimental data for both single LCCs and whole-cell currents. Fig. 2 demonstrates single LCC properties of the model under normal physiological conditions (i.e., with EC coupling intact and 2 mM extracellular Ca²⁺). Sample LCC unitary currents in response to 200-ms membrane depolarizations to test potentials of 20 mV and 0 mV from a holding potential of 100 mV are shown in Fig. 2, A and B, respectively. At test pulses 0 mV, 200-ms sweeps with no LCC openings seldom occur because 1) no silent mode behavior (Herzig et al., 1993; Handrock et al., 1998) is implemented in this model; 2) voltage-dependent inactivation is relatively slow and incomplete with respect to activation kinetics; and 3) the likelihood that Ca²⁺-mediated inactivation occurs is very low before the first opening of an LCC. Multiple openings within the same record are common because the steady-state inactivation probability of LCCs at depolarized potentials is substantially less than unity in canine myocytes (i.e., inactivation is incomplete) (Kääb et al., 1996; Tseng et al., 1987; Rose et al., 1992). Fig. 2, C and D show open time histograms and Fig. 2, E and F show cumulative first latency distributions, determined at test potentials of 20 mV and 0 mV, respectively, based on a random sampling of 500 LCCs. Open time histograms are well-described by a single exponential (_open = 0.481 ms and 0.492 ms at 20 mV and 0 mV, respectively), indicating that mean open time does not vary with test potential. Open durations, and first latency distributions are consistent with previous measurements after accounting for differences in experimental conditions (e.g., temperature) (Herzig et al., 1993; Handrock et al., 1998; Schroder et al., 1998; Rose et al., 1992). The fraction of sweeps exhibiting no openings is lower in the model than found in experiments due to the exclusion of LCC silent mode behavior in the model (Handrock et al., 1998). The sharpening of the first latency distribution at depolarized potentials indicates that channel openings become less temporally dispersed with increasing depolarization, in agreement with experimental findings (see Fig. 5 of Rose et al., 1992). Fig. 2 G shows unitary currents as a function of membrane potential. Single channel slope conductance, as measured in the range between 80 mV and 20 mV, is 8.2 pS and agrees with measurements made in near physiological solutions (6.9-9.1 pS, see Fig. 3 B of Rose et al., 1992, and Fig. 4 B of Yue and Marban, 1990).

View larger version (18K): [in this window] [in a new window]   FIGURE 2   Single L-type Ca2+ channel model properties. (A) and (B), sample unitary Ca2+ currents in response to a 200-ms test pulse to 20 mV and 0 mV, respectively. The small positive deflection in each current record indicates the time at which voltage is switched to the test potential. (C) and (D), open duration histograms measured from 500 sweeps at 20 mV and 0 mV, respectively. Exponential fits (black line) yield mean open duration of 0.481 ms and 0.492 ms at 20 mV and 0 mV, respectively. (E) and (F), first latency distribution measured at 20 mV and 0 mV, respectively. (G) Single channel current-voltage relation of model LCCs with 2 mM extracellular Ca2+. Unitary conductance is 8.2 pS when measured in the range of 80 mV to 20 mV.

The summation of all unitary Ca²⁺ currents within the myocyte, such as those shown in Fig. 2, A and B, yields macroscopic I_CaL. Simulated whole-cell currents elicited by a family of depolarizing voltage steps from 30 mV to 40 mV in 10-mV increments are shown in Fig. 3 A. Currents activate rapidly (<6 ms) and decay over ~100 ms. In Fig. 3 B, peak I_CaL amplitude is plotted as function of test potential (where data points represent the mean of five runs at each potential). Maximum inward Ca²⁺ current is produced at a test potential of 10 mV (n = 5). The bell-shaped peak current profile is in close agreement with peak currents measured in canine (compare to Fig. 1 E of Hobai and O'Rourke, 2001) (Kääb et al., 1996; O'Rourke et al., 1999), guinea pig (Rose et al., 1992), and human (He et al., 2001) ventricular myocytes. Fig. 3 C demonstrates the underlying processes that govern the time course of I_CaL during the voltage clamp to 0 mV. The quantities shown are LCC open probability (p_o, black solid line), the probability of occupancy of Mode Normal (Prob{Norm} = 1  Prob{Ca²⁺-mediated inactivation has occurred}, gray solid line), and the fraction of channels available for voltage-dependent inactivation (y = 1  Prob{voltage-dependent inactivation has occurred}, dashed line). LCC p_o reaches a peak value of ~0.1, consistent with studies that indicate that peak p_o with Ca²⁺ as the charge carrier (Rose et al., 1992) and time-averaged p_o with Ba²⁺ as the charge carrier (minimizing Ca²⁺-mediated inactivation) (Herzig et al., 1993; Handrock et al., 1998) is in the range of 0.05-0.15. A comparison of the time progression of Prob{Norm} and y clearly demonstrates that at 0 mV, Ca²⁺-mediated inactivation of I_CaL develops more rapidly and progresses more completely than voltage-dependent inactivation, in agreement with recent experiments (Linz and Meyer, 1998; Sipido et al., 1995; Hadley and Hume, 1987; Peterson et al., 1999, 2000). In addition, Ca²⁺-mediated inactivation is partially relieved in the latter portion of the pulse (Fig. 3 C), indicative of decaying local Ca²⁺ levels.

View larger version (26K): [in this window] [in a new window]   FIGURE 3   Properties of macroscopic ICaL. (A) Simulated whole-cell currents as a function of time in response to a family of depolarizing voltage steps from 30 mV to 40 mV in 10 mV increments. (B) Mean peak current-voltage relation based on five simulations at each potential (i.e., n = 5). (C) LCC open probability (black solid line), probability of occupancy of Mode Normal (Prob{Norm}, gray solid line), and probability that voltage-dependent inactivation has not occurred (y, dashed line) in response to a voltage clamp to 0 mV. (D) Steady-state inactivation curve obtained using a double-pulse protocol (see text) with (filled circles) and without (open circles) Ca2+ as the charge carrier.

The feedback of local Ca²⁺ signals on LCC gating plays an important role in the determination of properties of both CICR and APs, and is a key feature of the local control myocyte model, which is explored in more detail below. Steady-state inactivation properties of I_CaL are measured using a double-pulse protocol where membrane potential is first stepped from 100 mV to various pre-pulse potentials for 400 ms and then to a 0-mV test potential. Peak currents, normalized by current obtained in the absence of a pre-pulse, are shown as a function of the pre-pulse potential in Fig. 3 D. Simulations were performed under normal conditions (filled circles) and with subspace Ca²⁺ clamped to diastolic levels to mimic an alternative charge carrier such as Ba²⁺ that would not significantly promote Ca²⁺-mediated inactivation (open circles). Under normal conditions, the I_CaL steady-state inactivation curve is U-shaped. Disabling of Ca²⁺-mediated inactivation yields an inactivation curve that decreases monotonically with depolarization. Inactivation is incomplete, and asymptotically approaches ~50% for highly depolarized pre-pulses. These features agree well with inactivation curves obtained experimentally in native myocytes (compare to Fig. 10 of Hadley and Hume, 1987, Fig. 1 E of Linz and Meyer, 1998, and Fig. 7 B of Li et al., 1999). Ca²⁺-mediated inactivation makes a dominant contribution to the total inactivation of I_CaL in the range of potentials of 10 mV to +30 mV, consistent with the range of potentials where LCC Ca²⁺ influx is maximal (Fig. 3 B). In addition, this is within the range of plateau potentials where inactivation would normally occur during an AP. The U-shape is therefore a consequence of the variation in local Ca²⁺ transients that arise due to the voltage dependence of LCC Ca²⁺ influx, and the subsequent graded CICR.

Excitation-contraction coupling

Fig. 4, A and B, demonstrate the most elementary model release event, as triggered by a single LCC. A typical response to a 200-ms 0 mV voltage clamp pulse is shown. Ca²⁺ flux through an LCC (gray line) and the net SR Ca²⁺ release flux through the five adjacent RyRs (black line) are shown in Fig. 4 A. At the onset of the voltage pulse, the LCC first opens after ~5 ms, and then exhibits only one additional opening. Local JSR release flux is triggered by the first LCC opening and lasts ~20 ms, far longer than the LCC open duration. The amplitude of the release flux varies with the number of open RyRs and the local Ca²⁺ gradient across the JSR membrane. Individual RyR channel gating events can be discerned as step-like changes in local JSR release flux (Fig. 4 A, arrow 1), while changes due to the time varying local Ca²⁺ gradient across the RyRs (i.e., effects of JSR depletion) occur more gradually over time (Fig. 4 A, arrow 2). Mean RyR open time varies from ~7 ms, when subspace Ca²⁺ is high early in the voltage clamp pulse to <1 ms after the subspace Ca²⁺ has subsided later in the pulse (data not shown), similar to values reported previously (Rice et al., 1999; Lukyanenko et al., 1996). Fig. 4 B shows the corresponding dyadic subspace Ca²⁺ concentration, which reaches a peak value of ~40 µM. The small amplitude deflections in subspace Ca²⁺ level that continue to occur following termination of the transient are the result of inter-subspace Ca²⁺ diffusion (J_iss), and are indicative of trigger events (LCC openings) occurring in neighboring subspace compartments within the CaRU.

View larger version (22K): [in this window] [in a new window]   FIGURE 4   Sample results for a single CaRU in response to a 200-ms voltage clamp to 0 mV. (A) Ca2+ flux through a single LCC (gray line) and through the set of five RyRs (black line) within a single dyadic subspace compartment. Arrows 1 and 2 highlight changes in SR Ca2+ release flux due to changes in the number of open RyRs and due to the changing local Ca2+ gradient arising from JSR Ca2+ depletion, respectively. (B) Subspace Ca2+ concentration associated with the events of panel A. (C) Ca2+ flux through the set of four LCCs (gray line) and the set of 20 RyRs (black line) within a single CaRU. (D). Mean subspace Ca2+ concentration in the four subspace compartments associated with the events in the CaRU described in panel C.

A local Ca²⁺ spike (a localized Ca²⁺ release event within a single T-SR junction) is modeled by a single CaRU in Fig. 4, C and D, for the same voltage clamp stimulus. Total Ca²⁺ influx through the set of four LCCs (gray line) and the net SR Ca²⁺ release flux through the set of 20 RyRs (black line) are shown in Fig. 4 C. LCC Ca²⁺ influx rises to a level consistent with two open channels within a short time following the initiation of the pulse, indicating some degree of temporal synchronization in the onset of trigger Ca²⁺ influx at 0 mV. Net JSR Ca²⁺ release flux follows a similar time course to that observed in an individual subspace. The spatial average of Ca²⁺ concentration in the four subspace compartments of the CaRU is intended to represent a Ca²⁺ spike (Fig. 4 D). The peak amplitude of this signal is less than that seen in a single subspace (Fig. 4 B) due to temporal dispersion of Ca²⁺ release events. The Ca²⁺ spike duration is ~25 ms (at half-maximal amplitude), similar to that measured in myocytes using confocal imaging techniques (20-50 ms) (Cheng et al., 1993; Song et al., 2001; Sham et al., 1998).

Whole cell Ca²⁺ signals, which can be explained as the spatial and temporal summation of local elementary Ca²⁺ release events, are shown in Fig. 5. Total LCC Ca²⁺ influx (gray line) and RyR Ca²⁺ release flux (black line) in response to a 0 mV voltage clamp are plotted as a function of time in Fig. 5 A. EC coupling gain, measured as the ratio of peak RyR Ca²⁺ flux to peak LCC Ca²⁺ flux, is ~12 at 0 mV. Although peak Ca²⁺ flux through both RyRs and LCCs occurs within a few milliseconds following the onset of the voltage pulse, there is a relatively small sustained component of flux that lasts throughout the duration of the voltage clamp, indicative of a small number of release events associated with LCC reopenings and/or late openings. Fig. 5 B demonstrates mean subspace free Ca²⁺ concentration (solid line) averaged over all CaRUs, and bulk cytosolic Ca²⁺ concentration (dashed line). The peak amplitude of mean subspace Ca²⁺ concentration is ~18 µM, substantially greater than the cytosolic Ca²⁺ level, which peaks at <1 µM. The mean subspace Ca²⁺ concentration is, however, less than that observed for individual simulated Ca²⁺ release events due to temporal dispersion in the occurrence of Ca²⁺ release events, and because Ca²⁺ release fails to occur in some CaRUs. On average, the local Ca²⁺ transient displays fast kinetics. It rises and decays within ~70 ms at 0 mV, while the cytosolic Ca²⁺ transient lasts 200 ms. The late sustained Ca²⁺ fluxes shown in Fig. 5 A give rise to a similar sustained component of the subspace Ca²⁺ signal, which lasts for the duration of the voltage clamp pulse. Fig. 5 C shows corresponding free Ca²⁺ levels in the JSR (solid black line, average over all JSR volume compartments) and NSR (solid gray line), and total SR Ca²⁺ load (dashed line), which includes both free and buffer-bound Ca²⁺ in all SR compartments. JSR and NSR Ca²⁺ pools are at similar levels at all times during the pulse, indicative of the fast Ca²⁺ diffusion rate between these compartments (_tr = 3 ms). Preceding the pulse, SR free Ca²⁺ concentration is ~730 µM, which corresponds to total SR Ca²⁺ content of ~118 µmol L-cytosol¹, in agreement with measurements of SR load in canine myocytes (Hobai and O'Rourke, 2001). Upon CICR, total SR Ca²⁺ is reduced to ~80 µmol L-cytosol¹, resulting in ~32% decrease in total SR Ca²⁺ content. Similar values of fractional SR Ca²⁺ release have been obtained in experiments (~35%) (Bassani et al., 1995; DelBridge et al., 1996).

View larger version (17K): [in this window] [in a new window]   FIGURE 5   Whole-cell Ca2+ fluxes and concentrations in response to a 200-ms voltage clamp to 0 mV. (A) Ca2+ fluxes through the entire population of LCCs (gray line) and the entire population of RyRs (black line) are shown as a function of time. For consistency, fluxes are reported in units of mmol L-cytosol1 s1 (B) Average subspace Ca2+ transient (black line, left axis) and bulk cytosolic Ca2+ transient (dashed line, right axis). (C) Free Ca2+ concentration in the NSR (gray solid line, left axis), mean free JSR Ca2+ concentration (averaged over all CaRUs, black solid line, left axis), and total SR Ca2+ concentration (includes both free and bound Ca2+ in the NSR and JSR, dashed line, right axis).

Recently, Song et al. (2001) examined voltage-dependent recruitment and amplitude of Ca²⁺ spikes. Under control conditions, they found a bell-shaped voltage dependence for the likelihood of Ca²⁺ spike occurrence and a shallower bell-shaped dependence for the amplitude of Ca²⁺ spikes. These data demonstrated that the gradation of SR Ca²⁺ release is predominantly attributable to graded recruitment of T-SR junctions, with a smaller contribution due to variations in the amplitude of local Ca²⁺ release flux. Fig. 6 demonstrates the ability of the local control myocyte model to reproduce these findings. Fig. 6 A shows the fraction of CaRUs that fire at least one Ca²⁺ spike during a 200-ms depolarizing test pulse as a function of test pulse potential. Ca²⁺ spikes are detected by monitoring the mean subspace Ca²⁺ concentration within each CaRU, and are considered to have occurred if Ca²⁺ concentration rises above a threshold value of 4 µM for a time greater than 5 ms. Values reported in Fig. 6 were determined by analyzing a minimum of 250 CaRUs at each potential (n  250). Under control conditions (filled circles), the model voltage dependence of the probability of firing a Ca²⁺ spike is bell-shaped and saturates at ~1.0 in the range of 20 mV to +20 mV, indicating that it is extremely rare for SR release not to occur in this voltage range. The same simulations were performed using an altered version of the model in which there is a single dyadic subspace per unit (total number of LCCs, RyRs, and subspace compartments per myocyte being conserved) and therefore lacks inter-subspace diffusion (open circles). These results represent the probability of elementary release events within the model, and peak at ~0.9 in the range of 10 mV to +10 mV. In either case, the bell-shaped voltage dependence indicates that gradation of SR Ca²⁺ release arises, in part, from voltage-dependent recruitment of T-SR junction (i.e., CaRU) activation. If it is assumed that the results for the modified model are similar to the properties of a single subunit within a control CaRU and that CaRU activation (i.e., a Ca²⁺ spike) occurs as long as at least one elementary release event occurs within a CaRU, then the probability of CaRU activation in the presence of LCC silent mode behavior can be estimated. The estimate assumes that only 40% of LCCs are active at any given time (f_active = 0.4) (Handrock et al., 1998) and produces peak fractional T-SR activation similar to experiments (Song et al., 2001), as shown in Fig. 6 A (triangles). Fig. 6 B shows the model voltage dependence of local Ca²⁺ spike amplitude (as CaRU Ca²⁺ release flux) averaged over all CaRUs for the same two simulations. For the control case (closed circles), Ca²⁺ spike amplitude has a shallow bell-shaped voltage dependence and peaks at ~170 nmol L-cytosol¹ s¹ in the range of 10 mV to +10 mV. The rise in mean Ca²⁺ spike amplitude in the central voltage range occurs as a result of enhanced synchronization of RyR release events within CaRUs contributing to gradation of SR Ca²⁺ release, and agrees well with the experiments of Song et al. (2001). In the modified model lacking inter-subspace Ca²⁺ diffusion (open circles), subspace Ca²⁺ release flux is smaller at all potentials. This is the case because these signals represent SR Ca²⁺ release flux from a set of five RyRs, whereas the control simulations represent flux from all 20 RyRs in the CaRU. More interesting however, is that the shape of the voltage dependence of the Ca²⁺ transient amplitude in the absence of inter-subspace coupling is inverted compared to the control simulations. In the absence of inter-subspace coupling, synchronization within a single unit is not possible because only a single Ca²⁺ release event occurs. There is therefore no enhancement of local Ca²⁺ release signals in the central range of potentials. The depressed release at central voltages in the modified model is due to a reduction in the Ca²⁺ gradient across the SR membrane as a result of SR depletion. In the central voltage range, where SR release is maximal (Fig. 6 A, open circles), reduction in global SR Ca²⁺ load over the duration of the pulse will lead to a reduction in Ca²⁺ transient amplitude for events that occur late within the pulse (data not shown). In the control case, this effect is masked by the enhancement of Ca²⁺ spike amplitude that occurs within the same range of potentials due to synchronization of release events within CaRUs.

View larger version (17K): [in this window] [in a new window]   FIGURE 6   Microscopic properties of model Ca2+ spikes. (A) The fraction of CaRUs that fire a Ca2+ spike as a function of voltage under control conditions is shown (filled circles). For the modified model lacking intersubspace diffusion, the fraction of subspaces that exhibit a Ca2+ release event is shown (open circles). The fraction of CaRUs that fire a Ca2+ spike in the presence of LCC slow cycling between active and silent modes is estimated (see text, triangles), n  250 simulations at each potential. (B) Ca2+ spike amplitude as a function of voltage clamp potential under control conditions (filled circles) and for the modified model lacking intersubspace diffusion (open circles).

While Fig. 6 shows results regarding voltage-dependent gradation of SR Ca²⁺ release at the level of the CaRU, Fig. 7 demonstrates the macroscopic properties of SR Ca²⁺ release. Previous experimental studies (Wier et al., 1994; Santana et al., 1996; Janczewski et al., 1995; Cannell et al., 1995) and mathematical models (Stern, 1992; Stern et al., 1999) have shown that there can be significant differences between the voltage dependence of LCC Ca²⁺ influx (F_LCC) and RyR Ca²⁺ release flux (F_RyR) even though SR Ca²⁺ release is controlled by Ca²⁺ entry via I_CaL. These differences underlie the phenomenon of "variable" EC coupling gain. We use the definition of gain given by Wier et al. (1994) as the ratio F_RyR(max)/F_LCC(max). Fig. 7 A shows the voltage dependence of F_LCC(max) (filled circles) and F_RyR(max) (open circles) obtained using the same voltage protocols as in Fig. 3 B. In Fig. 7 B, the peak fluxes of Fig. 7 A have been normalized based on their respective maxima. Although both F_LCC(max) and F_RyR(max) are bell-shaped functions of membrane potential, they do not share identical voltage dependence. Maximal LCC Ca²⁺ influx occurs at 10 mV, whereas maximal RyR Ca²⁺ release flux occurs at 0 mV. EC coupling gain as defined above is plotted as a function of voltage in Fig. 7 C (triangles), and is monotonically decreasing with increasing membrane potential. The similarity in shape of the EC coupling gain curve and the unitary LCC current-voltage relation (dashed line, scaled for comparison) suggests that EC coupling gain may be more closely related to unitary LCC current, rather than macroscopic I_CaL. The simulated data of Fig. 7 agrees well with experimentally obtained measurements of whole-cell Ca²⁺ fluxes (Wier et al., 1994; Santana et al., 1996; Song et al., 2001; Cannell et al., 1995; Janczewski et al., 1995). The value of r_iss (inter-subspace Ca²⁺ transfer rate) was chosen to match the model gain function with experiments and to be consistent with estimates of Ca²⁺ diffusion within the dyad (Soeller and Cannell, 1997). The role of inter-subspace coupling on gain is demonstrated in Fig. 7 C, by comparison of control simulations (triangles) to those in the absence of inter-subspace coupling (i.e., r_iss = 0, squares). With inter-subspace coupling intact, EC coupling gain is greater at all potentials, but the increase in gain is most dramatic at more negative potentials. In this negative voltage range LCC p_o is submaximal, leading to sparse LCC openings. However, unitary current magnitude is relatively high, such that the opening of an LCC efficiently triggers adjacent RyRs. An increased r_iss value allows the rise in local Ca²⁺ due to the triggering action a single LCC to recruit and activate RyRs in adjacent subspace compartments within the same T-SR junction (where the LCCs have not opened, as openings are sparse), effectively raising the functional RyR/LCC ratio. The net effect of inter-subspace coupling is to increase the magnitude and slope of the gain function preferentially in the negative voltage range. The local control myocyte model predicts that Ca²⁺ diffusion in the T-SR junction (across the subspace compartments) is an important mechanism underlying the rate at which gain decreases with increasing voltage. Previous models of EC coupling have similarly achieved a steeply decreasing gain function with high amplitude at negative potentials by incorporating details of spatial Ca²⁺ gradients in the dyadic space (Stern et al., 1999).

View larger version (14K): [in this window] [in a new window]   FIGURE 7   Voltage dependence of macroscopic LCC Ca2+ influx, SR Ca2+ release, and EC coupling gain. (A) Mean peak Ca2+ flux amplitudes, FLCC(max) (filled circles) and FRyR(max) (open circles) as a function of membrane voltage, n = 5 simulations at each voltage. (B) Peak Ca2+ fluxes (data of panel A) normalized by their respective maxima. (C) EC coupling gain as a function of membrane potential defined as FRyR(max)/FLCC(max) under control conditions (triangles) and in the absence of intersubspace coupling within the CaRUs (squares), as well as L-type unitary current (dashed line, scaled to match the gain function at 40 mV).

One defining difference between a macroscopic model and a local control model of EC coupling is that it would be impossible for a macroscopic model to exhibit different values of gain for macroscopic L-type Ca²⁺ currents of the same amplitude (Stern, 1992). Wier et al. (1994) have explicitly demonstrated that Ca²⁺ currents of similar shape and amplitude can evoke very different responses of SR Ca²⁺ release. The local control myocyte model can reproduce the findings of this experiment, as shown in Fig. 8. Fig. 8, A and B show F_LCC, and Fig. 8, C and D show F_RyR in response to 200-ms depolarizing pulses to 20 mV and +50 mV, respectively. Although the amplitude and time course of macroscopic LCC Ca²⁺ influx is similar for the two test pulses, the SR Ca²⁺ release is triggered effectively in response to the 20 mV pulse (where gain is high), but only minimal Ca²⁺ release occurs at +50 mV (where gain is low). In addition, upon repolarization to 100 mV from +50 mV, the brief Ca²⁺ tail current triggers substantial SR Ca²⁺ release (Fig. 8 D).

View larger version (23K): [in this window] [in a new window]   FIGURE 8   Variable EC coupling gain. (A) and (B), whole-cell LCC Ca2+ influx (FLCC) in response to voltage clamp stimuli of 20 mV and +50 mV, respectively. (C) and (D), whole-cell SR Ca2+ release flux (FRyR) in response to voltage clamp stimuli of 20 mV and +50 mV, respectively. Note that while FLCC is similar in response to each of the two voltage clamp stimuli, the magnitude of FRyR is very different, indicating that EC coupling gain is not determined by properties of macroscopic ICaL. In addition, note that the large, but brief, tail of Ca2+ influx that occurs following repolarization from +50 mV triggers substantial SR Ca2+ release flux.

Action potentials

Fig. 9 demonstrates the ability of the model to reconstruct action potentials and Ca²⁺ transients of normal canine midmyocardial ventricular myocytes. In Fig. 9 A, a normal 1-Hz steady-state AP is shown. AP properties are similar to those measured in experiments (O'Rourke et al., 1999), with action potential duration (APD) of ~300 ms. Fig 9 B shows cytosolic (black line) and mean subspace (gray line) Ca²⁺ transients. While the cytosolic Ca²⁺ transient peaks at ~0.75 µM, and lasts longer than the duration of the AP, Ca²⁺ in the subspace reaches ~11 µM on average, and equilibrates to near-cytosolic levels rapidly, within ~100-150 ms. Fig. 9 C demonstrates the two model currents that communicate directly with the local subspaces within the CaRUs, I_CaL (black line) and I_to2 (gray line). I_CaL peaks at ~4.7 pA pF¹ and has a sustained component of ~0.7 pA pF¹, which lasts for nearly the entire duration of the AP. I_to2 peaks at ~0.6 pA pF¹ and also displays a minimal sustained current component. The sustained current appears because subspace Ca²⁺ remains moderately elevated on average throughout the AP due to LCC reopenings, and because I_to2 does not inactivate.

View larger version (14K): [in this window] [in a new window]   FIGURE 9   The action potential, Ca2+ transients, and membrane currents. Signals shown are in response to a 1-Hz pulse train, with responses shown in steady state. (A) Membrane potential as a function of time simulated using the local control myocyte model under normal conditions. (B) Cytosolic (black line, left axis) and mean subspace (gray line, right axis) Ca2+ concentrations corresponding to the AP simulated in panel A. (C) L-type Ca2+ current (ICaL, black line) and the Ca2+-dependent transient outward Cl current (Ito2, gray line) corresponding to the AP simulated in panel A.

While macroscopic I_CaL shown in Fig. 9 C has a similar shape as that of the common pool Winslow et al. (1999) canine myocyte model, the underlying LCC inactivation process in the local control model has been altered to depend more strongly on local Ca²⁺ than on membrane potential. This adjustment is based on experimental findings obtained from both isolated myocytes (Linz and Meyer, 1998; Sipido et al., 1995) and recombinant channels expressed in HEK 293 cells (Peterson et al., 1999, 2000), which show that LCC voltage-dependent inactivation is slow and incomplete while Ca²⁺-mediated inactivation is strong and dominates the inactivation process (see also Fig. 3 D). Strong Ca²⁺-dependent inactivation (in the absence of strong voltage-dependent inactivation) is a key mechanism in determining how graded SR Ca²⁺ release influences AP properties and whole-cell Ca²⁺ dynamics. Fig. 10 demonstrates the differences in I_CaL inactivation properties between the Winslow et al. (1999) common pool model and the local control myocyte model, and their consequences. Fig. 10, A and B, show steady-state APs (solid line), Prob{Norm} (dashed line), and y (dotted line), for the common pool and local control models, respectively. Prob{Norm} and y are the model quantities indicating the time progression of the Ca²⁺- and voltage-dependent inactivation processes, respectively, as previously described in Fig. 3 C. During the plateau of the AP, ~70% of LCCs become unavailable due to voltage-dependent inactivation while ~60% become unavailable due to Ca²⁺-dependent inactivation in the common pool model (Fig. 10 A). The balance between voltage- and Ca²⁺-dependent inactivation processes in the Winslow et al. (1999) common pool model are therefore in contrast to experimental findings. The guinea pig models of Jafri et al. (1998) and Luo and Rudy (1994) also exhibit very strong voltage-dependent inactivation of I_CaL and relatively weak Ca²⁺-mediated inactivation (data not shown, see Winslow et al., 2001). The roles of these processes are reversed (as they should be) in the local control myocyte model, with only ~35% of LCCs succumbing to voltage-dependent inactivation, while ~75% are shut down by Ca²⁺-dependent inactivation (Fig. 10 B, compare to Fig. 11 of Linz and Meyer, 1998). Understanding the fundamental differences in the CICR processes in the common pool versus the local control models provides the reason why the balance between each of the inactivation processes is incorrectly assigned in the common pool model. In a model where the release of SR Ca²⁺ is controlled by sensing Ca²⁺ levels in the same pool into which SR Ca²⁺ is released, Ca²⁺ release will be an all-or-none response (Stern, 1992). If Ca²⁺-dependent inactivation of LCCs were the dominant inactivation process in this type of model, then it follows that I_CaL inactivation would also exhibit all-or-none behavior, switching on in response to SR Ca²⁺ release. The single regenerative SR release event would rapidly and strongly promote Ca²⁺-dependent inactivation of I_CaL, and would therefore destabilize the plateau phase of the AP. An attempt at simulating APs using the Winslow et al. (1999) model modified to have strongly Ca²⁺-dependent and weakly voltage-dependent inactivation of I_CaL (with equations governing y identical to that of the local control model) is illustrated in Fig. 10 C. APs alternate between those with short duration (~150-250 ms) and those with very long duration (>1000 ms) with unstable oscillatory plateau potentials. The alternans indicate the presence of a bifurcation in the AP profile as a function of JSR Ca²⁺ load. Short-duration APs occur when the all-or-none SR Ca²⁺ release event strongly inactivates I_CaL, and hence terminates the AP. SR Ca²⁺ load will be gradually diminished following successive short APs due to the imbalance between cellular Ca²⁺ influx (via LCCs) and Ca²⁺ efflux (via Na⁺/Ca²⁺ exchangers and sarcolemmal Ca²⁺ pumps). When the SR becomes sufficiently depleted, the weak SR Ca²⁺ release flux produces only slight inactivation of I_CaL. In addition, the population of RyRs fails to adequately inactivate/adapt, leading to additional spontaneous release events and a long-lasting unstable oscillatory plateau. This unstable behavior occurs over a wide range of LCC inactivation parameters as long as voltage-dependent inactivation of I_CaL is relatively slow and incomplete (data not shown). Strong voltage-dependent inactivation of I_CaL, although contrary to experimental observations, is therefore necessary to enforce stability of those common pool models that incorporate the regenerative SR release mechanisms of CICR. As seen in Fig. 10 B, the local control myocyte model does not suffer from the consequences of all-or-none SR Ca²⁺ release (at the whole-cell level) and therefore successfully generates stable APs with LCCs whose inactivation process is dominated by local Ca²⁺-mediated inactivation.

View larger version (27K): [in this window] [in a new window]   FIGURE 10   Inactivation properties of ICaL in common pool and local control models of the action potential. (A) and (B), membrane potential (solid line, left axis), Prob{Norm} (dashed line, right axis), and y (dotted line, right axis) for the Winslow et al. (1999) common pool myocyte model and the local control myocyte model, respectively. Prob{Norm} and y are the model quantities indicating the time progression of the Ca2+- and voltage-dependent inactivation processes of ICaL (see text for details). (C) Membrane potential as a function of time for a 10-s simulation of a modified version of the Winslow et al. (1999) model with ICaL parametrized with strongly Ca2+-dependent and weakly voltage-dependent inactivation (similar to that of the local control model). (D) Membrane potential (solid line, left axis), Prob{Norm} (dashed line, right axis), and y (dotted line, right axis) for the local control myocyte model where SR Ca2+ load has been reduced to 33% of its normal level.

Altered EC coupling in both human (Lindner et al., 1998) and canine (Hobai and O'Rourke, 2001) heart failure is associated with decreased SR Ca²⁺ content. Both modeling (Winslow et al., 1999) and experimental studies (Ahmmed et al., 2000; O'Rourke et al., 1999) support the idea that altered Ca²⁺ handling plays a key role in heart failure-associated AP prolongation. Fig. 10 D shows an AP (solid line), Prob{Norm} (dashed line), and y (dotted line) for the local control model under conditions where SR Ca²⁺ load has been reduced to 33% of its normal level. This is a non-steady-state simulation in which the pre-stimulus initial condition for SR Ca²⁺ load (in NSR and all JSR compartments) has been reduced three-fold from control without any alteration in model parameters, thereby isolating the impact of depressed SR Ca²⁺ load on the AP. The voltage-dependent inactivation mechanism proceeds in a manner similar to that of the control case (dotted line). However, under conditions of reduced SR Ca²⁺ release, Ca²⁺-mediated inactivation of I_CaL occurs at a far slower rate and to a lesser extent (dashed line) than in the control case (compare to Fig. 10 B). The resulting increased magnitude of the late sustained component of I_CaL (not shown) maintains the plateau and dramatically prolongs the AP (solid line). This supports the hypothesis that in heart failure, alterations in Ca²⁺ handling proteins that decrease SR Ca²⁺ load and reduce the amplitude of local Ca²⁺ transients may contribute substantially to prolongation of APD by reducing Ca²⁺-mediated inactivation of the L-type current.

In a previous study (Greenstein et al., 2000), we examined the role of I_to1 in shaping AP morphology and duration using the common pool canine myocyte model (Winslow et al., 1999). These simulations predicted that reduction of I_to1 density from normal levels leads to modest shortening of APD. The reduction in I_to1 reduces the depth of the phase 1 AP notch, reducing the driving force for, and hence the peak level of, I_CaL. Because this study was performed in a model displaying all-or-none rather than graded SR Ca²⁺ release properties, the altered I_CaL had no ability to modulate F_RyR. In Fig. 11 the role of I_to1 is revisited, with particular attention to how an alteration of AP shape influences EC coupling. Under normal 1-Hz steady-state conditions (solid black line) the AP has a duration of ~315 ms (Fig. 11 A), peak I_CaL is ~4.8 pA pF¹ (Fig. 11 B), peak F_RyR is ~0.8 mmol L-cytosol¹ s¹ (Fig. 11 C), and peak cytosolic Ca²⁺ concentration is ~0.8 µM (Fig. 11 D). (Note that slight differences in control simulations in Fig. 11 compared to Fig. 9 arise due to stochastic noise inherent in the model.) The density of I_to1 was then reduced by 67%, an amount similar to that observed in failing canine myocytes (Kääb et al., 1996), and simulations were repeated using normal initial conditions to demonstrate the role of graded SR Ca²⁺ release under conditions of identical SR Ca²⁺ load. The phase 1 notch of the AP becomes less pronounced, and APD is shortened modestly to ~255 ms (Fig. 11 A, gray line). The corresponding peak I_CaL is reduced by 40% to ~2.9 pA pF¹ (Fig. 11 B). The property of graded SR Ca²⁺ release is evident by observing F_RyR (Fig. 11 C, gray line). The reduction in trigger Ca²⁺ reduces peak SR Ca²⁺ release flux by nearly 50% under conditions where SR load is unchanged. The resulting cytosolic Ca²⁺ transient is consequently reduced to ~0.65 µM (Fig. 11 D). Although this is a good demonstration of the effects of graded release during the AP, a sudden decrease in I_to1 is not a physiologically relevant event. Upon pacing to steady state with a 67% reduction of I_to1 (dotted line), both graded SR Ca²⁺ release and the new steady-state SR Ca²⁺ load affect Ca²⁺ cycling properties. The shortening of the APs results in a decreased steady-state SR Ca²⁺ load compared to control (data not shown), which in turn leads to a further decrease in SR Ca²⁺ release, reducing the amplitude of the cytosolic Ca²⁺ transient to ~0.6 µM, 25% less than control (Fig. 11 D). The model therefore predicts that reduction in I_to1, similar to that observed in heart failure (Kääb et al., 1996), may contribute to reduced force generation. This model prediction has been verified by recent experiments showing that slowed phase 1 repolarization during the AP reduces temporal synchrony and recruitment of Ca²⁺ release events, in conjunction with a reduced amplitude of I_CaL (Sah et al., 2002).

View larger version (29K): [in this window] [in a new window]   FIGURE 11   Role of Ito1 on the events of EC coupling. (A) Membrane potential, (B) L-type Ca2+ current (ICaL), (C) SR Ca2+ release flux (FRyR), and (D) cytosolic Ca2+ concentration as a function of time. Each of the signals is shown under normal 1-Hz steady-state conditions (solid black line), with 67% reduction in Ito1 density using the normal initial conditions (gray line), and with 67% reduction in Ito1 density under 1-Hz steady-state conditions (see text for details). The altered shape of the AP resulting from reduction in Ito1 reduces trigger influx Ca2+ via LCCs leading to decreased SR Ca2+ release flux, and therefore a depressed cytosolic Ca2+ transient.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX I APPENDIX II REFERENCES

In this study we present a biophysically detailed model of the normal canine ventricular myocyte that conforms to the theory of local control of EC coupling in cardiac muscle. Local control theory asserts that L-type Ca²⁺ curren