Kinetic Properties of the Cardiac L-Type Ca²⁺ Channel and Its Role in Myocyte Electrophysiology: A Theoretical Investigation

Abstract

The L-type Ca²⁺ channel (Ca_V1.2) plays an important role in action potential (AP) generation, morphology, and duration (APD) and is the primary source of triggering Ca²⁺ for the initiation of Ca²⁺-induced Ca²⁺-release in cardiac myocytes. In this article we present: 1), a detailed kinetic model of Ca_V1.2, which is incorporated into a model of the ventricular mycoyte where it interacts with a kinetic model of the ryanodine receptor in a restricted subcellular space; 2), evaluation of the contribution of voltage-dependent inactivation (VDI) and Ca²⁺-dependent inactivation (CDI) to total inactivation of Ca_V1.2; and 3), description of dynamic Ca_V1.2 and ryanodine receptor channel-state occupancy during the AP. Results are: 1), the Ca_V1.2 model reproduces experimental single-channel and macroscopic-current data; 2), the model reproduces rate dependence of APD, [Na⁺]_i, and the Ca²⁺-transient (CaT), and restitution of APD and CaT during premature stimuli; 3), CDI of Ca_V1.2 is sensitive to Ca²⁺ that enters the subspace through the channel and from SR release. The relative contributions of these Ca²⁺ sources to total CDI during the AP vary with time after depolarization, switching from early SR dominance to late Ca_V1.2 dominance. 4), The relative contribution of CDI to total inactivation of Ca_V1.2 is greater at negative potentials, when VDI is weak; and 5), loss of VDI due to the Ca_V1.2 mutation G406R (linked to the Timothy syndrome) results in APD prolongation and increased CaT.

INTRODUCTION

Most cellular action potential (AP) models compute the membrane potential starting from macroscopic transmembrane ionic currents through large ensembles of ion channels. The ionic currents are computed using the Hodgkin-Huxley (H-H) scheme (1), which represents voltage and time-dependent conductance changes in terms of “gating variables” (e.g., activation, inactivation) that are independent of each other. Current understanding of ion-channel gating clearly indicates strong coupling between kinetic gating transitions that cannot be reproduced within the H-H paradigm (2). Also, the H-H formalism does not represent kinetic states of the ion channel (such as open, closed, inactivated) and is therefore not suitable for simulating molecular interactions, ion-channel mutations that alter specific kinetic transitions, or effects of drugs that bind to the channel in a specific conformational state. To overcome these limitations we have developed and introduced into a model of the whole-cell, single-channel based Markov models of cardiac I_Na (3,4), I_Kr (5,6), and I_Ks (6), channels that are major determinants of the ventricular AP. Markov models represent discrete ion-channel states and their interactions; this property allowed us to examine kinetic state transitions of these channels during the AP (2), simulate the cellular arrhythmogenic manifestations of ion-channel mutations (2–5), and study the role of molecular subunit interactions in channel function (2,6). Recently, we used Markov models to study the cellular electrophysiological effects of state-specific drug binding (to open or inactivated state) in wild-type and mutant cardiac Na⁺ channels (7). Given the important role of Ca_v1.2, the L-type Ca²⁺ channel, in AP generation, rate-dependence (8,9), and conduction (10,11), and its participation in cellular calcium cycling and excitation-contraction (EC) coupling (12), it is important to develop a detailed kinetic model of this channel. Such model could be used to study various aspects of channel gating in the complex interactive cell environment, to predict the effects on the whole-cell AP of state-specific drug binding to the channel, and to simulate the cellular electrophysiological consequences of Ca_v1.2 mutations, such as a recently described missense mutation G406R that causes Timothy syndrome and is characterized by QT interval prolongation on the electrocardiogram and arrhythmia development, as well as an array of other organ dysfunction (13,14).

Ca_v1.2 gating is both voltage and Ca²⁺ dependent; the time course and magnitude of [Ca²⁺] near the channel is a major determinant of its inactivation kinetics. In ventricular myocytes, Ca_v1.2 interacts with ryanodine receptors (RyR) to trigger Ca²⁺ release from the sarcoplasmic reticulum (SR) through the calcium-induced-calcium-release (CICR) process. The interaction occurs in a restricted subcellular space in triad formations (15,16). Modeling the Ca_v1.2 channel kinetics requires representation of the dynamic Ca²⁺ changes in a restricted subspace for calcium. We introduce a subcellular restricted space where Ca_v1.2 and RyR interact to generate a Ca²⁺-transient. Because our interest is in the L-type Ca²⁺ current, its properties, and its role in the cellular AP and arrhythmia, we adopt a global cellular approach in the simulations; that is, the model subcellular space represents a subsarcolemmal space, along the length of the t-tubules, into which both L-type Ca²⁺ channels and RyR open. Clearly, this approach cannot be used to simulate local microscopic (molecular) processes of EC-coupling such as Ca²⁺ spark formation, as done by others (17,18). However, as shown in the Results section, the model recreates the relevant global properties of calcium cycling that influence the L-type Ca²⁺ current, I_Ca(L). This global approach results in major reduction of computing time, making the simulations of steady-state pacing protocols that are necessary for studying cell electrophysiology possible and practical.

Using this model, the following phenomena are studied: 1), kinetic state transitions of Ca_v1.2 and RyR during the AP at slow and fast rates; 2), the relative contributions of Ca²⁺-dependent and voltage-dependent inactivation to total inactivation of I_Ca(L); 3), sensitivity of Ca²⁺-dependent inactivation to Ca²⁺ entry through the channel and from SR release; 4), rate dependence of AP duration (APD adaptation) and APD restitution; 5), rate dependence of [Na⁺]_i and of the Ca²⁺-transient; restitution of the Ca²⁺-transient during premature stimuli; 6), I_Ca(L) and AP modification by the Ca_v1.2 missense mutation G406R that has been associated with the Timothy syndrome and its arrhythmic manifestations.

METHODS

A complete list of abbreviations, parameter definitions, model equations, and initial conditions can be found in Table 1 and the Appendix.

TABLE 1.

Definitions and abbreviations

AP	Action potential
APD	Action potential duration measured at 90% repolarization
BCL	Basic cycle length
CaV1.2	Cardiac L-type Ca2+ channel
RyR	Ryanodine receptor SR Ca2+ release channel
CaT	Calcium transient
CICR	Calcium induced calcium release
VDI	Voltage-dependent inactivation
CDI	Calcium-dependent inactivation
ModeV	L-type Ca2+ channel states in VDI gating mode
ModeCa	L-type Ca2+ channel states in CDI gating mode
INa	Fast Na+ current, μA/μF
m	Activation gate of INa
h	Fast inactivation gate of INa
j	Slow inactivation gate of INa
ICa(L)	Ca2+ Current through L-type Ca2+ channel, μA/μF
ICa,Na	Na+ Current through L-type Ca2+ channel, μA/μF
ICa,K	K+ Current through L-type Ca2+ channel, μA/μF
IKr	Rapid delayed rectifier K+ current, μA/μF
xr	Activation gate of IKr
rKr	Time-independent rectification gate of IKr
IKs	Slow delayed rectifier K+ current, μA/μF
xs1	Fast activation gate of IKs
xs2	Slow activation gate of IKs
IK1	Time-independent K+ current, μA/μF
K1	Inactivation gate of IK1
IKp	Plateau K+ current, μA/μF
ICa,b	Background Ca2+ current, μA/μF
ICa(T)	T-Type Ca2+ current, μA/μF
INa,b	Background Na+ current, μA/μF
INaCa	Na+-Ca2+ exchanger in myoplasm, μA/μF
INaCa,ss	Na+-Ca2+ exchanger in subspace, μA/μF
γINaCa	Position of energy barrier controlling voltage dependence of INaCa
INaK	Sodium-potassium pump, μA/μF
fNaK	Voltage-dependent parameter of INaK
σ	[Na+]o dependent factor of INaK
Ip,Ca	Sarcolemmal Ca2+ pump, μA/μF
	Maximum conductance of channel x, mS/μF
Km	Half-saturation concentration, mM/L
PS	Membrane permeability to ion S, cm/s
PS,A	Permeability ratio of ion S to ion A
γS	Activity coefficient of ion S
	Maximum current carried through channel x, μA/μF
Vm	Transmembrane potential, mV
zs	Valence of ion S
Cm	Total cellular membrane capacitance, 1 μF
ACap	Capacitive membrane area, cm2
AGeo	Geometric membrane area, cm2
RCG	Ratio of ACap/AGeo = 2
Vx	Volume of compartment x, μL
Δ[S]x	Change in concentration of ion S in compartment x, mM
CASQ2	Calsequestrin, Ca2+ buffer in JSR
TRPN	Troponin, Ca2+ buffer in myoplasm
CMDN	Calmodulin, Ca2+ buffer in myoplasm
BSR	Anionic SR binding sites for Ca2+ in the subspace
BSL	Anionic sarcolemmal binding sites for Ca2+ in the subspace
SR	Sarcoplasmic reticulum
JSR	Junctional SR
NSR	Network SR
ss	Subspace
myo	Myoplasm
Ex	Reversal potential of current x, mV
[S]o	Extracellular concentration of ion S, mM
[S]i	Intracellular concentration of ion S, mM
[S]ss	Subspace concentration of ion S, mM
[Ca2+]JSR	Ca2+ concentration in JSR, mM
[Ca2+]JSR,t	Total Ca2+ concentration in JSR ([Ca2+]JSR + [csqn]), mM
[Ca2+]NSR	Ca2+ concentration in NSR, mM
Irel	Ca2+ release from JSR to subspace, mM/ms
adap	Function describing RyR channel adaptation
gradedrel	ICa(L) dependent function for determining graded response of Irel
vgainofrel	Function describing voltage dependence of gain
Iup	Ca2+ uptake from myoplasm to SR, mM/ms
Ileak	Ca2+ leak from NSR to myoplasm, mM/ms
Itr	Ca2+ transfer from NSR to JSR, mM/ms
τtr	Time constant of Ca2+ transfer from NSR to JSR, ms
Idiff	Ca2+ transfer from subspace to myoplasm, mM/ms
τdiff	Time constant of Ca2+ transfer from subspace to myoplasm, ms
	Ca2+ flux from ICa(L)
	Ca2+ flux from Irel
F	Faraday constant, 96,487 C/mol
R	Gas constant, 8314 J/kmol/K
T	Temperature, 310°K
τ0	Rate constant of monoexponential decay for the probability density function fit to the open probability data of the L-type Ca2+ channel
ICa,t	Total transmembrane Ca2+ current
	ICa,t = ICa(L) + ICa,b + Ip,Ca − 2*INaCa − 2*INaCass
INa,t	Total transmembrane Na+ current
	INa,t = INa + INa,b+3*INaK + ICa,Na + 3*INaCa + 3*INaCass
IK,t	Total transmembrane K+ current
	IK,t = IKs + IKr + IK1 + ICa,K + IKp − 2*INaK
Itot	Total transmembrane current
	Itot = ICa,t + INa,t + IK,t
Istim	Stimulus current, μA/μF

L-type Ca²⁺ channel Markov model

A Markov representation of the L-type Ca²⁺ channel (Fig. 1) was developed and incorporated into a ventricular cell model. L-type Ca²⁺ channels are known to inactivate due to an increase in membrane potential or an increase in intracellular Ca²⁺. The former, known as voltage-dependent inactivation (VDI), is accompanied by a gating current whereas the latter, known as Ca²⁺-dependent inactivation (CDI), is not. This discovery by Hadley and Lederer (19) suggests that the mechanisms by which these inactivation processes occur are separate, implying that channels that have inactivated due to voltage can still be inactivated by Ca²⁺ and vice versa. This is further supported by experiments where VDI or CDI are altered or eliminated by point mutations to the Ca_V1.2 gene. This is perhaps best illustrated by the Ca_V1.2 mutation G406R, which is the underlying cause of the Timothy syndrome, which almost completely eliminates VDI while leaving CDI unaffected (13,14). VDI of Ca_V1.2 is also affected by mutations to the pore-encoding portion of the channel (20) or mutations within the I-II linker (13,21). CDI, which involves Ca²⁺ binding to calmodulin (CaM) that is constitutively tethered to a region of the Ca_V1.2 C-terminus known as the IQ motif, is eliminated by mutations to either the IQ motif (22–24) or to CaM (25). There has been recent evidence that VDI may also be dependent on the IQ-CaM complex (26), giving rise to the possibility that VDI and CDI are not entirely independent. We model the channel with two distinct kinetic modes (27,28): a voltage-gating mode (ModeV) with a single conducting state, and a mode where the channel is inactivated via a Ca²⁺-dependent process (ModeCa). We selected the minimum number of channel states necessary to reflect structural properties of the channel (four voltage sensitive transitions before channel opening reflecting movement of four voltage sensors, one for each of the four homologous domains making up the α₁ subunit of Ca_V1.2) and to describe the complex channel behavior (fast and slow voltage-dependent inactivation states, I_Vf and I_Vs, (29)). The two-tier structure (upper tier, ModeV; lower tier, ModeCa) implies that the channel can be inactivated by Ca²⁺ at any time, independent of the channel being closed, open, or inactivated by voltage. In addition, this means that channels can be simultaneously inactivated via both voltage-dependent and Ca²⁺-dependent mechanisms. Despite the relatively large number of channel states, the total number of transition rates that must be computed each time step is small (rates computed for state transitions within ModeV are the same for ModeCa and transition rates between ModeV and ModeCa are the same from any state).

FIGURE 1.

State diagram of the L-type Ca²⁺ channel Markov model. The upper tier (ModeV, gray) represents a voltage-dependent inactivation gating mode (VDI) and the lower tier (ModeCa, black) represents a Ca²⁺-dependent inactivation gating mode (CDI). See Appendix for transition rates.

Experimental protocols to study L-type Ca²⁺ channels often utilize charge carriers other than Ca²⁺ to separate voltage-dependent inactivation from Ca²⁺-dependent inactivation. To reproduce these experimental protocols with the model, we eliminate Ca²⁺-dependent inactivation by setting the transition rates from ModeV to ModeCa to zero. Simulations where this is done are noted in the text and in the figure caption.

The L-type Ca²⁺ channel Markov model was validated utilizing a wide range of experimental data. Our guide was to select experimental data that were recorded in guinea pig ventricular myocytes at 37°C. In addition, we preferentially selected experimental data that were recorded in the absence of β-agonists as these have a significant effect on current magnitude, voltage-dependent properties, and Ca²⁺-dependent properties of the channel.

The L-type Markov model (Fig. 1) includes four closed states (C₀, C₁, C₂, and C₃), a single open state (O), two states representing channels that have undergone fast or slow voltage-dependent inactivation (I_Vf and I_Vs), five states representing channels that have undergone Ca²⁺-dependent inactivation (C_0Ca, C_1Ca, C_2Ca, C_3Ca, and I_Ca), and two states that represent inactivation via both voltage and Ca²⁺-dependent mechanisms (I_VfCa and I_VsCa). The transition rates in each loop of the model (e.g., C₃-O-I_Vs) obey microscopic reversibility.

Cell model

The theoretical LRd model of a mammalian ventricular AP (30–32) (Fig. 2 A) provides the basis for the simulations of cellular behavior in this study. The model is based mostly on guinea pig ventricular myocyte experimental data; it includes membrane ionic channel currents, pumps, and exchangers. The model also accounts for processes that regulate intracellular concentration changes of Na⁺, K⁺, and Ca²⁺. Intracellular Ca²⁺ cycling processes represented in the model include Ca²⁺ uptake and release by the SR and its buffering. Buffers include calmodulin and troponin (in the myoplasm), sarcolemmal and SR Ca²⁺ binding sites (in the subspace), and calsequestrin (in the SR).

FIGURE 2.

(A) Schematic for the four compartment model (bulk myoplasm, JSR, NSR, and subspace) of the guinea pig ventricular myocyte. Symbols are defined in Table 1. (B) State diagram of the ryanodine receptor, RyR, (modified from Fill et al. (39)). See Appendix for transition rates.

Subspace compartment

Electron microscopic views of myocytes reveal invaginations in the cell membrane, known as transverse or t-tubules, that increase the total surface area of the myocyte and provide a pathway by which extracellular Ca²⁺ can be readily available for entry upon cell depolarization. The junctional sarcoplasmic reticular membrane makes close contact with sarcolemmal membrane all along the t-tubules, the distance between the two membranes being very small (15–20 nm). The confined volume between the two membranes is commonly referred to as the restricted space or the subsarcolemmal space. It is known that Ca²⁺ entry via L-type Ca²⁺ channels is the signal for the opening of RyRs and Ca²⁺ release from the SR. Immunogold-labeling techniques (33,34) have shown that L-type Ca²⁺ channels cluster in t-tubules and the areas of the plasma membrane that overlie the SR membrane, supporting their important role in the CICR process. Ca²⁺ entry into the restricted space via L-type Ca²⁺ channels (followed closely by Ca²⁺ release from the SR) generates Ca²⁺ concentrations near the inner membrane surface that are much greater than those observed in the bulk myoplasm. The magnitude of these concentrations and the rate at which the concentrations return to basal levels is dependent upon several factors including: 1), subspace volume; 2), Ca²⁺ diffusion rates from the subspace to the bulk myoplasmic volume; 3), the presence of buffers within the subspace; 4), the rate of Ca²⁺ entry into the subspace (i.e., via L-type and RyR channels); and 5), the rate of Ca²⁺ removal (i.e., via forward Na⁺/Ca²⁺ exchange). We model the subspace as a single compartment comprising 2% of the total volume of the myocyte into which both L-type Ca²⁺ channels and RyR open (see Appendix for derivation of subspace volume). Immunofluorescence of the Na⁺-Ca²⁺ exchange protein (35,36) has shown that these proteins are present with greater density within the t-tubules, hence 20% of the Na⁺-Ca²⁺ exchanger is included in the modeled subspace (37). Finally, we include sarcolemmal and SR membrane binding sites that buffer calcium (38).

Ryanodine receptor Markov model

A Markov model of the RyR, modified from that originally presented by Fill et al. (39) is utilized in the simulations (Fig. 2 B). The RyR Markov model is based upon experiments by Zahradniková et al. (40) where individual RyR channels were fused into a planar lipid bilayer. This preparation allows for direct control of Ca²⁺ concentrations to determine rates of channel activation and inactivation (an approach that shows that RyR channels both activate and inactivate due to a rise in subspace calcium ([Ca²⁺]_ss). In addition, the four transitions necessary to reach the open state reflect the homotetrameric structure of RyR (41). All of the channel transition rates are increased to adjust for the temperature differences between the experiments (23°C) and model (37°C). An addition to the original Fill (39) model is the inclusion of four additional inactivation states (I₁–I₄) connected to the closed states (C₁–C₄). This change allowed RyR channels to deactivate without having to pass through the open state, preventing reopening of the channel during repolarization and during rest. In our simulations, when channels could only recover from inactivation through the open state, Ca²⁺ leaked from the SR for the duration of the AP preventing SR refilling and normal decay of the calcium transient. This behavior could not be observed in the lipid bilayer experiments where RyRs were not stimulated by APs.

It has been suggested that calsequestrin (CASQ2) may also be an important modulator of RyR activity (42,43). CASQ2 is a high-capacity, low-affinity buffer of Ca²⁺ located in the SR. Kawasaki and Kasai (44) demonstrated that introduction of CASQ2 to the SR increased the RyR open probability and Hidalgo and Donoso (45) showed that interventions that led to conformational change of CASQ2 resulted in opening of RyRs. Studies by Györke et al. (46) showed that low SR Ca²⁺ led to CASQ2 inhibition of RyR opening and that upon elevation of SR Ca²⁺ this inhibition decreased. Communication between CASQ2 and RyR may be mediated by the auxiliary proteins triadin and junctin (47,48). Because RyR channels in the experiments of Zahradniková et al. (40) were studied in isolation, their dependence on CASQ2 was not apparent and hence not incorporated into their Markov model of RyR. We incorporate this dependence into the RyR Markov model presented here.

Simulation protocols

During pacing, a stimulus of −80 μA/μF is applied for a duration of 0.5 ms. The model is paced with a conservative current stimulus carried by K⁺ (49). A discrete time step of 0.0002 ms is used during computation of the AP (or during voltage clamp simulations) and 0.002 ms during the diastolic interval. APD is measured as the interval between the time of maximum upstroke velocity (dV/dt_max) and 90% repolarization (APD).

RESULTS

L-type Ca²⁺ channel: model validation and gating properties

Fig. 3 A compares the simulated I_Ca(L) current-voltage relationship to the experimentally measured current-voltage relationship (50). L-type Ca²⁺ channels typically begin to activate between −40 and −30 mV and peak current is reached between 0 and +10 mV. Fig. 3 C shows simulated voltage-dependent inactivation (VDI) properties of I_Ca(L) compared to the experimentally measured data reproduced in Fig. 3 B (51). To eliminate CDI, Findlay substituted extracellular Ca²⁺ with Mg²⁺. We eliminate CDI by not permitting transitions from ModeV to ModeCa (setting transition rates to zero). In both model and experiment, channel inactivation increases with increased prepulse voltage and with increased prepulse duration. Because the experiments were conducted at 23°C, the values for the prepulse duration were adjusted to 37°C utilizing a Q₁₀ = 2 in the simulation. Findlay observed two time constants of inactivation for the L-type Ca²⁺ channel. The model accounts for this observation by including two voltage-dependent inactivation states, one for which transitions into the state are fast (I_Vf) and one for which transitions are slow (I_Vs).

FIGURE 3.

L-type Ca²⁺ channel current-voltage relationship and voltage-dependent inactivation. (A) Simulated I_Ca(L) current-voltage relationship (normalized) compared to experimental data recorded in guinea pig ventricular myocytes (from Rose et al. (50), with permission) (•). (B and C) Voltage-dependent inactivation (VDI) of I_Ca(L) obtained from a double-pulse protocol (inset). The simulations are conducted in the absence of Ca²⁺-dependent inactivation (transitions from ModeV to ModeCa are prevented). Panels (B) show experimental time course (left) and voltage dependence (right) of VDI recorded in guinea pig ventricular myocytes at 23°C (Findlay (51), with permission) utilizing Mg²⁺ as the charge carrier. The voltage dependence curves (right panel), from upper to lower, correspond to prepulse durations of 10, 20, 50, 100, 200, 500, and 1000 ms. Panels (C) show the corresponding simulated time course and voltage dependence of VDI. The voltage dependence curves, from upper to lower, correspond to prepulse durations of 4, 8, 20, 40, 80, 200, and 400 ms (the simulations are conducted for body temperature, 37°C).

The L-type Ca²⁺ channel Markov model recreates single channel properties as shown in Fig. 4. The simulated open time histogram for a voltage step to +10 mV (Fig. 4 A) can be fit by a monoexponential probability density function with a τ_o = 0.8 ms, in agreement with the experiments of Cavalié et al. (52). The latency to first opening is shown in Fig. 4 B, with the simulation showing strong correlation to experimental data by Cavalié et al. (52) who observed >90% channel openings within 6 ms of depolarization. The simulations were conducted without Ca²⁺-dependent inactivation.

FIGURE 4.

Single channel properties of the L-type Ca²⁺ channel. The experimental data (from Cavalié et al. (52), with permission) are from single channel recordings in guinea pig ventricular myocytes at 32°C, and were measured utilizing Ba²⁺ as the charge carrier. To model Ba²⁺ as the charge carrier, all simulations were conducted without transitions to ModeCa. (A) Simulated (left) and experimental (right) (52) open time histograms measured for a voltage step to +10 mV from a holding potential of −65 mV (B, inset). The open probability data were fit by a monoexponential probability density function with a time constant of τ₀ = 0.8 for both measured and simulated data. (B) Simulated latency to first opening. Channels are simulated with the same square pulse protocol as panel A.

Following SR Ca²⁺ release, Ca²⁺ concentrations within the subspace can reach a value 50 times higher than myoplasmic concentrations. Therefore, the L-type Ca²⁺ channel Markov model must be responsive to large changes in Ca²⁺. Fig. 5 A shows the sensitivity of I_Ca(L) to [Ca²⁺]_ss, both simulated and experimentally measured (53). It should be noted that the experimental data are a measure of single channel activity, while the simulated data are a measure of peak I_Ca(L). There is a strong correlation between the simulated and experimental data, both exhibiting a K_D = 3 μM. In addition, for both experiment and model, even for very high concentrations of subspace Ca²⁺, the channel does not inactivate completely (to zero). In the simulation (and experiment), the tested [Ca²⁺]_ss was applied and the model was allowed to reach steady state at that concentration before application of the square pulse. Hence, this study provides the steady-state dependence of I_Ca(L) on [Ca²⁺]_ss and gives no indication of the time course of Ca²⁺-dependent inactivation. This time course was determined by optimizing the morphology of I_Ca(L) during the AP clamp protocol shown in Fig. 8, where SR release was kept intact. The time course of I_Ca(L) recovery from Ca²⁺-dependent inactivation is shown in Fig. 5 B. The experimental data (solid circles) (54) were measured in guinea pig ventricular myocytes at 35°C with SR Ca²⁺ release intact. The experimentally measured time constant of recovery is τ = 92 ms, and the model time constant of recovery is τ = 101 ms. When we eliminate CDI in the model, the time constant of recovery shows only a slight decrease (τ = 88 ms). This is in agreement with experimental results that show no significant difference in the rate of I_Ca(L) recovery when utilizing either Ca²⁺ or Ba²⁺ as the charge carrier (Ian Findlay, PhD, Université de Tours, France personal communication, 2004).

FIGURE 5.

(A) Dependence of peak I_Ca(L) on intracellular subspace Ca²⁺. Values are normalized to control conditions of [Ca²⁺]_ss= 0.02 μM. For each value of [Ca²⁺]_ss, the channel is allowed to reach steady state before application of a square voltage pulse to 0 mV. The experimental data (from Höfer et al. (53), with permission) (•) provide normalized channel activity recorded utilizing a cell attached patch in guinea pig ventricular myocytes at 37°C. (B) I_Ca(L) recovery from inactivation. The experimental data (from Vornanen and Shepherd (54), with permission) (•) were recorded in guinea pig ventricular myocytes at 35°C utilizing Ca²⁺ as the charge carrier with a time constant of recovery τ = 92 ms. The solid curve shows simulated I_Ca(L) recovery from VDI (no transitions to ModeCa; non-Ca²⁺ charge carrier) with a time constant of recovery τ = 88 ms. The gray dashed curve shows simulated I_Ca(L) recovery from both VDI and CDI (Ca²⁺ as the charge carrier) with a time-constant of recovery τ = 101 ms.

FIGURE 8.

I_Ca(L) during the action potential in the presence of SR Ca²⁺ release. The upper panels show the AP clamp waveform from Grantham and Cannell (59) used to generate the corresponding I_Ca(L) (lower panels). The experimental I_Ca(L) (nifedipine-sensitive current; lower left panel) is measured in guinea pig ventricular myocyte at 36°C. Experimental traces are reproduced from Grantham and Cannell (59), with permission.

The relative contribution of CDI to total I_Ca(L) inactivation is shown in Fig. 6 for two different times (t_early and t_late) after application of a voltage step. CDI's contribution to inactivation is computed as the ratio of inactivation with Ca²⁺ as the charge carrier relative to the inactivation when Ba²⁺ is the charge carrier. The experimental data (solid squares) (55) were recorded at room temperature (23°C) so the time at which the data are sampled is adjusted to 37°C in the simulation (Experiment: t_early = 20 ms, t_late = 200 ms; Simulation: t_early = 8 ms, t_late = 80 ms). The relative contribution of CDI to total inactivation is both voltage dependent and time dependent. CDI is responsible for a greater percentage of inactivation at negative potentials when VDI is weak and in the period shortly after depolarization. With increased time and voltage, the contribution of CDI to total inactivation decreases and VDI becomes dominant.

FIGURE 6.

(A and B) Voltage dependence of the contribution of CDI to total inactivation for two different times (early, t_early; late, t_late) after application of a voltage step to varying potentials from a holding potential of −80 mV. The plotted values are the ratio of current decay recorded with Ca²⁺ as the charge carrier relative to that recorded with Ba²⁺ as the charge carrier. Experiments (from Findlay (55), with permission) (▪) were conducted at room temperature (23°C) and simulations (solid line) at 37°C; values of t_early and t_late were adjusted for 37°C in the simulations. (Experiment: t_early = 20 ms, t_late = 200 ms; Simulation: t_early = 8 ms, t_late = 80 ms). (C–F) Relative contributions of Ca²⁺ through the channel and Ca²⁺ released from the SR to total CDI during a voltage step. (C) Trace 3 (thin curve) shows simulated I_Ca(L) in the absence of Ca²⁺-dependent inactivation, corresponding to current carried by monovalent cations or Ba²⁺, for a voltage step from −80 to −10 mV. Trace 2 (medium curve) shows I_Ca(L) with Ca²⁺ as the charge carrier (hence with Ca²⁺-dependent inactivation) but with no Ca²⁺ release from the SR (I_rel = 0) for a voltage step from −80 to 0 mV (the 10-mV difference in test potential compared to trace 3 is to reproduce the experimental protocol in panel D for comparison). Trace 1 (thick curve) shows I_Ca(L) for the same square pulse with Ca²⁺ as the charge carrier and SR calcium release intact. (D) Experimental recordings of I_Ca(L) in human atrial myocytes measured at room temperature. Trace 3 is the current through the channel utilizing monovalent cations as the charge carrier tested with a voltage step from −80 to −10 mV. Trace 2 shows I_Ca(L) with Ca²⁺ as the charge carrier but with SR release blocked with ryanodine during a voltage step from −80 to 0 mV. Trace 3 shows I_Ca(L) with Ca²⁺ as the charge carrier and SR release intact for the same voltage step. (E) Simulated fraction of Ca²⁺-induced inactivation that results from Ca²⁺ released from the SR (gray) or from Ca²⁺ entering via I_Ca(L) (black). The formula for determining the relative contribution is from Sun et al. (8) where the I_Ca(L) dependent fraction = (trace3 − trace2)/(trace3 − trace1) and the SR dependent fraction = (trace2 − trace1)/(trace3 − trace1). (F) Experimental fractional contribution of I_Ca(L)-dependent (black) and SR dependent (white) Ca²⁺-induced inactivation of I_Ca(L). Experimental curves in panels D and F are reproduced from Sun et al. (8), with permission.

The rate of transition between ModeV and ModeCa is dependent on the Ca²⁺ concentration within the subspace ([Ca²⁺]_ss). The relative contribution to CDI of Ca²⁺ that enters the subspace via I_Ca(L) versus that which enters the subspace via SR release is explored in Fig. 6, C–F. In the simulation and experiment (8) (Fig. 6, C and D, respectively), cells are clamped from holding potential to −10 mV for the trace where Ca²⁺ is not the charge carrier (trace 3) and from holding potential to 0 mV for the two traces where Ca²⁺ is the charge carrier (traces 1 and 2). In the experiment (Fig. 6 D, trace 2), SR Ca²⁺ release is blocked with the application of ryanodine that we simulate by setting I_rel = 0 (Fig. 6 C, trace 2). In summary, trace 1 represents total I_Ca(L) inactivation (VDI + CDI from I_Ca(L) and I_rel), trace 2 represents I_Ca(L) inactivation in the absence of SR release (VDI + CDI from I_Ca(L) only), and trace 3 represents pure VDI (no CDI). Fig. 6, E and F show the percent contribution to CDI from SR release ((trace 2 − trace 1)/(trace 3 − trace 1)) compared to the percent contribution to CDI from Ca²⁺ entering via I_Ca(L) ((trace 3 − trace 2)/(trace 3 − trace 1)). It is clear that Ca²⁺ released from the SR dominates CDI initially, then, as Ca²⁺ decreases due to SR reuptake of Ca²⁺, the SR dependent contribution declines with participation from Ca²⁺ entry via I_Ca(L) dominating. For the simulated time period shown in Fig. 6 E RyRs open quickly, inactivate, and then remain almost fully inactivated; thus, there is very little steady-state SR-dependent contribution to CDI, as appears to be the case in the SR-dependent experimental curve of Fig. 6 F. Reasons for this observed difference can be attributed to the fact that the experimental data were recorded in human atrial cells at room temperature (23°C). Despite these differences, the behavior of the model follows that of the experiment, especially during the initial phase following depolarization when a sharp decline in current is observed, correlating with the large increase in [Ca²⁺]_ss during the period of SR Ca²⁺ release and dominance of SR-dependent inactivation (Fig. 6 C, trace 1, and E, SR-dependent curve). This initial fast decline is followed by a much slower decrease in current (Fig. 6 C, trace 1), a combination of VDI and CDI from Ca²⁺ entry via I_Ca(L). The crossover of the SR-dependent and I_Ca(L)-dependent CDI curves occurs 36 ms after depolarization in the simulation and 55 ms after depolarization in the experiment.

RyR model in the subspace: graded release, fractional release, and variable gain

Communication between L-type Ca²⁺ channels and RyRs occurs in local diadic spaces and the stochastic opening of a single L-type Ca²⁺ channel can trigger the opening of several local RyRs. The magnitude of Ca²⁺ entry via I_Ca(L) determines the number of RyR openings, thus small I_Ca(L) results in small SR release compared to large I_Ca(L) resulting in large SR release. This phenomenon is known as graded release or graded response and is reproduced by the model as shown in Fig. 7 A. Note that in our macroscopic formulation the spatial distribution of RyRs is not represented and the ratio of RyR channels that open in response to a given I_Ca(L) is introduced to recreate the macroscopic properties of global release.

FIGURE 7.

(A) Graded release of SR Ca²⁺. In the simulation, voltage is clamped from −80 mV to the test potential. The solid trace shows normalized peak I_Ca(L) and the dashed trace shows normalized peak [Ca²⁺]_i. (B) Fractional SR release as a function of SR content. [Ca²⁺]_JSR is clamped at each value for 20 s. to allow RyR states to reach a new equilibrium, then the clamp is removed and a stimulus is applied. [Ca²⁺]_JSR,t is total JSR Ca²⁺ (free + Ca²⁺ bound CASQ2) and is integrated over the interval from the time of stimulus to the peak of the CaT. (C) Voltage-dependent variable gain of SR Ca²⁺ release. The right panel shows experimentally measured dependence of Δ[Ca²⁺]_i (difference between the peak [Ca²⁺]_i and resting [Ca²⁺]_i) on I_Ca(L) for different V_m values (indicated on curve) in guinea pig ventricular mocytes (from Beuckelmann and Wier (57), with permission). The corresponding simulated data are shown in the left panel. Note that for the same amplitude of I_Ca(L), a larger release occurs at negative potentials.

The model exhibits a steep nonlinear dependence of fractional SR release as a function of JSR Ca²⁺ content as shown in Fig. 7 B, in agreement with experiments (56). In the model, the rate of transition between inactivated and closed states is modulated by Ca²⁺-bound CASQ2 and when SR content is low, channels are unavailable for opening.

Another property that myocytes exhibit is variable gain. Gain is the ratio between the amount of Ca²⁺ released from the SR and the amount of Ca²⁺ entry into the myocyte that triggers SR release. In myocytes, it has been observed that for the same amount of triggering Ca²⁺, the magnitude of release can vary as a function of the transmembrane potential. For example, in Fig. 7 A, the magnitude of I_Ca(L) at V_m = −10 mV and at V_m = +30 mV is approximately the same, but there is a large difference in the magnitude of the Ca²⁺ transient at these two potentials, with the negative potential exhibiting a larger SR Ca²⁺ release. In Fig. 7 C we show the simulated relationship between [Ca²⁺]_i and I_Ca(L) over a range of V_m values (indicated on the curve) and compare the model results to experimentally measured data in guinea pig ventricular myocytes (57). From the shape of the curve, it is clear that for the same magnitude of I_Ca(L) the amount of SR release is greater at negative potentials. At 0 mV, the gain of SR Ca²⁺ release is 15.4 as computed with the following formula , where is the Ca²⁺ flux from the SR and is the Ca²⁺ flux through I_Ca(L), integrated over the period from the beginning of the voltage clamp to the peak of the CaT. This value for gain is comparable to values measured in rabbit (56) and canine (58).

I_Ca(L) during the action potential

An essential property of the L-type Markov model is the ability to recreate measured I_Ca(L) amplitude and morphology during the AP. Fig. 8 shows an experimentally measured I_Ca(L) (nifedipine-sensitive current) (59) compared to a simulated I_Ca(L) during the application of an identical AP clamp waveform (10 APs at 1 Hz). The experiment was conducted in a guinea pig ventricular myocyte at 37°C with SR Ca²⁺ release intact. Note that both the simulated and measured I_Ca(L) exhibit a peak value of ∼6 pA/pF, a period of rapid inactivation, and a current magnitude during the AP plateau of ∼3 pA/pF.

Rate dependence of APD, [Na⁺]_i, and [Ca²⁺]_i

Fig. 9 A compares simulated AP and [Ca²⁺]_i transient during the AP to their experimental counterparts (60). For this comparison, the amplitudes of the AP and [Ca²⁺]_i are not provided in the experimental tracings because the measured AP and [Ca²⁺]_i are expressed as fluorescence ratios of the voltage-sensitive dye RH237 and Ca²⁺-sensitive dye Rhod-2. However, important temporal relationships between the AP and the [Ca²⁺]_i transient are observed in both the simulation and experiment, including a 5-ms delay between the AP upstroke and the initiation of release, a 25-ms delay between the AP upstroke and the time of peak [Ca²⁺]_i transient, and a time constant of decay of the [Ca²⁺]_i transient of ∼150 ms.

FIGURE 9.

(A) Simulated AP and Ca²⁺ transient (left) compared to experimentally recorded AP and Ca²⁺ transient (right) in guinea pig ventricular myocyte at 37°C. The experiment is a simultaneous optical recording using the voltage-sensitive dye RH237 and Ca²⁺ sensitive indicator Rhod-2 (from Choi and Salama (60), with permission). (B) APD adaptation, [Na⁺]_i rate dependence, and [Ca²⁺]_i rate dependence. The figure shows the steady-state relationship (achieved after 5 min of pacing) between cycle length and APD (▪). Corresponding changes in [Ca²⁺]_i,max (▴), [Ca²⁺]_i,diastolic (△), and [Na⁺]_i (•) are also shown.

Simulated changes to APD, [Na⁺]_i, peak [Ca²⁺]_i, and diastolic [Ca²⁺]_i as a function of rate are summarized in Fig. 9 B. APD adaptation (61), [Na⁺]_i accumulation (62), and the positive force-frequency relationship (increase in peak [Ca²⁺]_i with increasing rate) (62) are consistent with experimental comparisons made in a previous publication (32).

APD and [Ca²⁺]_i restitution

Fig. 10, A and B, show restitution properties of the myocyte model. The myocyte is paced at a constant cycle length until steady state. Then a premature stimulus is applied with a coupling interval DI (diastolic interval, DI = 0 is APD₉₀ of the last paced AP). Fig. 10 A shows the last paced beat (CL = 500) and five APs with the associated I_Ca(L) and [Ca²⁺]_i for five DI values (50 ms, 150 ms, 250 ms, 350 ms, and 450 ms). Note that at DI = 50 ms a normal [Ca²⁺]_i transient fails to occur despite a large I_Ca(L) (i.e., a large triggering source). Also, note that the CaT at DI = 450 ms is larger then the CaT generated at steady state at CL = 500 ms. This is because the DI + APD₉₀ is longer than the initial CL = 500 ms and the SR has had a longer time to fill and the RyR has had more time to recover from inactivation resulting in a larger release and CaT.

FIGURE 10.

Restitution properties. The myocyte is paced at a constant cycle length for 5 min followed by a premature stimulus applied at varying diastolic intervals (DI). (A) Representative changes to AP, I_Ca(L), and [Ca²⁺]_i for five different DIs (50 ms, 150 ms, 250 ms, 350 ms, and 450 ms) after pacing at CL = 500 ms. (B) Summary of restitution behavior for a myocyte paced at three different cycle lengths (CL = 300, 500, and 1000 ms). Δ[Ca²⁺]_i is the difference between peak [Ca²⁺]_i and the [Ca²⁺]_i just before application of the premature stimulus.

Restitution data for three different pacing cycle lengths (CL = 1000 ms, 500 ms, and 300 ms) are shown in Fig. 10 B. At very short DI, dV/dt_max is slow due to incomplete recovery of I_Na. This results in decreased V_max and a larger driving force for I_Ca(L), increasing the current at these short DIs. Although the triggering source is large (Peak I_Ca(L)), RyR channels remain inactivated preventing release. Only when RyR channels have recovered from inactivation does release occur. The duration of this interval where no SR release can occur is dependent on the pacing frequency (Peak RyR Open Probability and Peak I_rel). This phenomenon is related to recovery of SR Ca²⁺ content (63) and the relationship between bound CASQ2 and RyR recovery (46,63). At fast rates, the increased amount of Ca²⁺ in the cytosol (Δ[Ca²⁺]_i) leads to faster recovery of SR calcium content (Peak [Ca²⁺]_JSR) and faster recovery of RyR and I_rel (Peak I_rel). At very long DI, it can be observed that the magnitude of I_rel is strongly dependent on SR content when Peak RyR Open Probability and Peak I_Ca(L) are similar in magnitude for all three CLs.

Currents during the AP at fast and slow pacing

Steady-state AP, [Ca²⁺]_i, [Ca²⁺]_ss, and the significant currents that determine APD and AP morphology are shown in Fig. 11 for two different pacing frequencies. At rapid rate (CL = 300 ms, gray curve) I_Ca(L) exhibits a larger peak and plateau compared to slow rate (CL = 1000 ms, black curve) (B). Although there is a greater CDI at CL = 300 ms due to the larger Ca²⁺ transient, the lower AP plateau increases the driving force for I_Ca(L), resulting in a larger current. The lower plateau and shortened APD (A) are mostly due to the large I_Ks current that accumulates at fast rate (F).

FIGURE 11.

Action potential (AP), calcium transient ([Ca²⁺]_i), I_Ca(L), subspace calcium transient ([Ca²⁺]_ss), I_NaCa, I_NaK, I_NaCa,ss, I_K1, I_Kr, and I_Ks for two different pacing frequencies, CL = 1000 ms (black curve) and CL = 300 ms (gray curve).

The Na⁺-Ca²⁺ exchanger within the subspace (I_NaCa,ss; D) provides little triggering Ca²⁺ compared to that provided by I_Ca(L) and upon release of Ca²⁺ from the SR (and the subsequent increase in subspace Ca²⁺) quickly shifts to the forward mode to remove Ca²⁺. At fast rates when [Ca²⁺]_ss (H) is elevated, I_NaCa,ss remains in the forward mode during most of the AP plateau.

L-type Ca²⁺ channel state residencies during the AP

The time course of L-type Ca²⁺ channel state residencies during the AP at fast and slow rates is shown in Fig. 12 (see Fig. 1 for the state diagram of the channel). At the slow rate (CL = 1000 ms, black curve), almost all the channels are able to fully recover to the closed states in ModeV (Fig. 12 B) before the pacing stimulus, whereas at the rapid rate (CL = 300 ms, gray curve) ∼15% of the channels are still in ModeCa at the time of stimulus (Fig. 12, B and G, arrows). Despite a greater number of channels locked in ModeCa, the magnitude of I_Ca(L) is larger at rapid rate due to the reduced V_m and increase in driving force. Upon depolarization, channels quickly move to the open state (Fig. 12 C). From the open state, channels inactivate via either fast VDI (I_Vf, Fig. 12 D) or slow VDI (I_Vs, Fig. 12 E). Upon the initiation of SR Ca²⁺ release, channels transition from ModeV to ModeCa and inactivate by CDI (Fig. 12, G–J); I_VfCa and I_VsCa contain channels that have inactivated via both VDI and CDI.

FIGURE 12.

State residencies of the L-type Ca²⁺ channel during the time course of the action potential for two different pacing frequencies, CL = 1000 ms (black curve) and CL = 300 ms (gray curve). For identification of the channel states, refer to Fig. 1.

At rapid rate, a greater percentage of channels become inactivated by CDI due to the elevated [Ca²⁺]_i. For the rapid rate, at t =100 ms after the time of stimulus, ∼10% of the channels are inactivated solely by CDI, 40% solely by VDI, and 40% via both CDI and VDI. This is compared to 5%, 55%, and 30%, respectively, at the slow rate. The remaining 10% of channels are in either the closed or open states.

RyR channel state residencies during the AP

The time course of RyR channel state residencies during the AP at fast (CL = 300 ms, gray curve) and slow rate (CL = 1000 ms, black curve) is shown in Fig. 13 (see Fig. 2 B for the state diagram of RyR). Upon depolarization and elevation of [Ca²⁺]_ss by Ca²⁺ entry via I_Ca(L), channels move rapidly from the closed state (Fig. 13 D) to the open state (Fig. 13 C) (Ca²⁺-dependent activation). Channels then transition into the near inactivated states (I₄ and I₅, Fig. 13 F) due to the high concentration of Ca²⁺ in the subspace following SR release (Ca²⁺-dependent inactivation). After AP repolarization, channels move from inactivated states I₄ and I₅ to I₁, I₂, and I₃ (Fig. 13, F and E) as subspace Ca²⁺ decreases. From I₁, I₂, and I₃, channels then recover to the closed states (Fig. 13 D) as the SR stores recover. This recovery from inactivated to closed states is dependent on the concentration of Ca²⁺ bound CASQ2. As can be seen in Fig. 13 D, the delay of recovery from inactivation is longer at slow rates than at fast rates. This is due to increased Ca²⁺ loading of the myocyte at rapid rate, which allows faster recovery of Ca²⁺-bound CASQ2 and, in turn, faster recovery of RyR from inactivation after AP repolarization.

FIGURE 13.

State residencies of the RyR channel during the time course of the action potential for two different pacing frequencies, CL = 1000 ms (black curve) and CL = 300 ms (gray curve). For identification of the channel states, refer to Fig. 2 B.

It can be observed that the RyR open probability (Fig. 13 C) at rapid rate is smaller than that at slow rate, but the maximum SR Ca²⁺ flux is greater (Fig. 13 B). The reason for the decreased open probability is that 55% of the channels remain inactivated at rapid rate (Fig. 13 E, arrow), primarily due to the decreased time for recovery between beats. Despite the decreased open probability, I_rel is larger due to the larger SR Ca²⁺ stores ([Ca²⁺]_JSR) that result from rapid pacing (see Fig. 11 J). Increased [Ca²⁺]_ss (Fig. 11 H) at rapid rate (due to increased I_rel, Fig. 13 B) increases the percentage of RyR channels that are activated by Ca²⁺ (rightward transition on the state diagram) and the percentage of channels that are inactivated by Ca²⁺ (upward transition on the state diagram). Hence, at rapid rate more channels reside in I₄ and I₅ compared to slow rate (Fig. 13 F).

Timothy syndrome and loss of Ca_V1.2 VDI

The Ca_V1.2 mutation G406R, linked to the Timonthy syndrome, almost completely removes VDI of I_Ca(L). In Fig. 14 A we show steady-state inactivation curves of I_Ca(L) for the control model (black trace) and for G406R (gray trace) compared to experimentally measured values (symbols). Analysis of mRNA in heart and brain tissue by Splawski et al. (13) showed that ∼23% of Ca_V1.2 channels contain the exon 8A in which the mutation G406R occurs. For this reason we show AP (Fig. 14 B), I_Ca(L) (C), and CaT (D) for three scenarios—wild-type (WT, black traces), heterozygous expression (11.5% G406R Ca_V1.2 channels, dashed gray traces), and homozygous (23% G406R Ca_V1.2 channels, gray traces). The G406R mutation results in I_Ca(L) current with a larger peak (Fig. 14 C, arrow) due to the loss of fast VDI. Recovery of current during the late phase of the AP corresponds with the decline of the CaT. This recovery from CDI results in an increased depolarizing I_Ca(L) current that prolongs APD (clinically observed as prolongation of QT interval on the ECG) and increases myocyte Ca²⁺. The increase of [Ca²⁺]_i results from both increased Ca²⁺ entry via I_Ca(L) and reduced diastolic interval, a time during which excess Ca²⁺ is removed by the Na⁺-Ca²⁺ exchanger.

FIGURE 14.

(A) Steady-state voltage-dependent inactivation curves of Ca_V1.2 for wild-type (WT, black) and mutation G406R (gray). The experimental data (symbols) (from Splawski et al. (13), with permission) were recorded in the absence of Ca²⁺. (B) Steady-state APs at CL = 1000 ms for WT myocyte (black trace), a myocyte with 11.5% G406R channels (dashed trace), and with 23% G406R channels (gray trace). (C) Corresponding I_Ca(L) (arrow indicates peak of I_Ca(L) for WT) and (D) calcium transient, CaT.

DISCUSSION

In this study we present detailed kinetic models of Ca_V1.2 and RyR that interact within a subcellular restricted space. These models were incorporated into the LRd model of the mammalian ventricular cell with the following results: 1), The model of Ca_V1.2 reproduces experimental single channel data (mean open time, latency to first opening) and macroscopic current data (VDI, CDI, recovery from inactivation, I_Ca(L) during the AP). 2), The Ca_V1.2-RyR interaction in the subspace reproduces graded SR Ca²⁺ release and voltage-dependent variable gain of release. 3), The model accurately reproduces adaptation of APD to changes in rate, rate dependence of [Na⁺]_i and the Ca²⁺ transient (force-frequency relationship), and restitution properties of APD and the Ca²⁺ transient during premature stimuli. 4), CDI of Ca_V1.2 is sensitive to Ca²⁺ that enters the subspace from L-type Ca²⁺ channels and from SR release. The relative contributions of these Ca²⁺ sources to total CDI vary with time after depolarization, with transition from early SR Ca²⁺ dominance to late L-type Ca²⁺ dominance, as seen experimentally. 5), The relative contribution of CDI to total inactivation of I_Ca(L) is greater at negative potentials when VDI is weak. 6), CDI is greater at rapid rates due to elevated subspace Ca²⁺. 7), Suppression of VDI due to the Ca_V1.2 mutation G406R results in APD prolongation and increased [Ca²⁺]_i.

Divalent cations and I_Ca(L) inactivation

In the I_Ca(L) model development we utilized extensive experimental data obtained with divalent cations other than Ca²⁺ (i.e., Ba²⁺, Mg²⁺, Sr²⁺, Cd²⁺) as the charge carrier to eliminate CDI. There has been some evidence to indicate that these substitutions (specifically Ba²⁺) for Ca²⁺ may also cause some inactivation of the channel (64). The mechanism by which other divalent cations cause inactivation is not known. Making the assumption that divalent cations other than Ca²⁺ can be used to indicate pure VDI can lead to an overestimation of the magnitude of VDI compared to CDI. To avoid this potential pitfall, studies of VDI have been conducted utilizing monovalent cations as the charge carrier (65,66). However, in these studies isoproternol (ISO) was used to amplify the current for facilitation of measurement. In experiments conducted by Findlay in the absence of ISO (67), the current through I_Ca(L) carried by Ba²⁺, Sr²⁺, and Na⁺ all exhibited similar inactivation kinetics, implying that Ba²⁺ and other divalent cations are suitable substitutes for determining VDI when ISO is absent. All simulations in this study were conducted under such basal conditions, in the absence of β-adrenergic effects.

I_Ca(L): Ca²⁺ versus voltage dependent inactivation

There has been some debate as to which mechanism dominates inactivation of L-type Ca²⁺ channels, VDI, or CDI. In our simulations, we show that in the absence of CDI, VDI can cause >80% inactivation of the current (Fig. 3 B) for voltage clamps to positive potentials. Experimental data of steady-state inactivation from Linz and Meyer (66) show that in the absence of CDI only 50% of the channels are inactivated due to VDI for voltage clamps to positive potentials. The primary reason for differences between our simulation and the Linz and Meyer experiments is their use of ISO in the solution. In a series of studies (51,67–69), Findlay demonstrated that when β-adrenergic stimulation effects were induced by the addition of ISO to the bath solution, the contribution of VDI to total inactivation was greatly reduced and CDI became dominant. Findlay proposed a “switch” whereby phosphorylation of the channel turns off rapid VDI and CDI becomes dominant (55). Our simulations are conducted in the absence of β-adrenergic stimulation and are in agreement with Findlay experiments conducted in the absence of ISO. In Fig. 6, A and B, we show that the relative contributions of CDI and VDI to total inactivation are both voltage and time dependent. We also show that the relative contribution of CDI and VDI to total inactivation is rate dependent (Fig. 12); specifically the contribution of CDI to total inactivation increases with increased rate due to the elevation of total [Ca²⁺] within the myocyte. Loss of VDI, as shown in simulations of the Ca_V1.2 mutation G406R (Fig. 14), leads to APD prolongation and an increase in [Ca²⁺]_i. It is clear that both VDI and CDI are important mechanisms of I_Ca(L) inactivation. The loss of either of these mechanisms of inactivation can lead to APD prolongation and calcium overload, resulting in increased susceptibility to arrhythmia.

It has been well established that CDI of I_Ca(L) occurs by Ca²⁺ binding to a constitutively bound calmodulin molecule on the C-terminus of the Ca_V1.2 α₁ subunit (22,23,70). But there has been some evidence that Ca²⁺ in the pore of the channel or Ca²⁺ very near the channel pore can also cause CDI (71). We formulated CDI as a function of both I_Ca(L) (representing Ca²⁺ passing through the channel pore) and [Ca²⁺]_ss (Ca²⁺ that can bind to the C-terminus inactivation site). However, extensive experimental data were fitted by the model with CDI dependence on [Ca²⁺]_ss alone, suggesting a minor role for pore Ca²⁺ compared to subspace Ca²⁺ in I_Ca(L) inactivation.

I_Ca(L) selectivity filter

The L-type Ca²⁺ channel is permeable to both monovalent and divalent cations, but there has been some evidence to suggest that when Ca²⁺ is present, the channel is selective almost exclusively for Ca²⁺ and does not allow passage of other ions (72,73). In our model of I_Ca(L) the channel is permeable to Ca²⁺, K⁺, and Na⁺ with membrane permeabilities of 1.215 × 10⁻³ cm/s, 4.34 × 10⁻⁷ cm/s, and 1.515 × 10⁻⁶ cm/s, respectively. We conducted simulations where the permeability of Na⁺ and K⁺ through I_Ca(L) was eliminated but discovered that the resultant AP morphology did not fit experimentally recorded guinea pig APs.

I_Ca(L) facilitation

L-type Ca²⁺ channels exhibit a property whereby the channel current magnitude increases with successive depolarizations. This property is known as facilitation and is thought to play a role in generating positive force frequency. I_Ca(L) facilitation is not reproduced by the model presented here, but the model still exhibits a positive force frequency relationship due to [Ca²⁺]_i loading at fast rate. Recent experimental evidence has shown that I_Ca(L) facilitation is a result of CaMKII activation (23,24,74–76). CaMKII is sensitive to frequency changes in [Ca²⁺]_i and when activated, can phosphorylate several targets including Ca_V1.2, SERCA, and RyRs. A recent model of the canine ventricular AP incorporated the CaMKII signaling pathways and its effects on the AP and Ca²⁺ transient (9). Modification of this signaling pathway for inclusion in the guinea pig myocyte model could be done in future development to incorporate I_Ca(L) facilitation. I_Ca(L) is also augmented by β-adrenergic stimulation (77). The simulations in this study are conducted under basal conditions in the absence of β-adrenergic effects. Models of the β-adrenergic signaling pathway (78) could also be adapted and incorporated in the cell model for studies of cellular behavior under various levels of β-adrenergic tone.

Gain and local control of SR Ca²⁺ release

Local control theory of SR Ca²⁺ release says that the individual discrete openings of single L-type Ca²⁺ channels signal the opening of nearby RyRs, the Ca²⁺ from which can recruit adjacent RyRs leading to SR release. This concept is important for understanding variable gain of SR release, where it is observed that for the same magnitude of I_Ca(L), a larger release is achieved at negative potentials than at positive potentials. The idea is that at negative potentials fewer Ca_V1.2 channels open but because the driving force for Ca²⁺ is large, a large amount of Ca²⁺ enters through the open channels creating sufficient local Ca²⁺ levels to trigger local RyR openings. At positive potentials, many more Ca_V1.2 channels will open, but the average current per channel will be low because of a small driving force. Consequently, at some sites [Ca²⁺]_ss is not large enough to trigger opening of local RyRs. There have been important modeling efforts to recreate local control of SR release at the level of these local interactions (17,18,79–81). The whole-cell model presented here uses a global formulation of I_rel to achieve graded response and variable gain. Using this macroscopic formulation, the model recreates behavior of I_Ca(L), I_rel, and [Ca²⁺]_i that corresponds well to experimentally measured data at a variety of test potentials and pacing rates. This approach is computationally efficient and permits simulating cellular behavior over many beats of pacing and during AP propagation in the multicellular tissue, a property that is necessary for studying cardiac arrhythmias. However, microscopic details of local control dynamics are not represented explicitly in the model.

RyR Ca²⁺-dependent inactivation and adaptation

The mechanism that terminates SR Ca²⁺ release is not fully understood and several possible control mechanisms have been proposed (16). There has been evidence that RyR channels do not undergo Ca²⁺-dependent inactivation (82), contrary to the proposed RyR kinetic model presented here. Suggested mechanisms of SR release termination, such as stochastic attrition (83) (closure of RyR channels and their neighbors due to reduction of local Ca²⁺ as a result of spontaneous stochastic closure of local RyRs) and RyR adaptation (a decrease of RyR sensitivity to local Ca²⁺ following a Ca²⁺ stimulus, a process distinct from inactivation in that the channels can exhibit further opening following an additional, larger Ca²⁺ stimulus) cannot be replicated in this study without including a more detailed model of local control (17,18,84) and/or RyR kinetics (41,85). Another proposed mechanism of SR Ca²⁺ release termination is due to depletion of luminal Ca²⁺ (46,63,86). This mechanism is similar to the CASQ2-mediated recovery from inactivation in the model presented here. In our formulation, both cytosolic and luminal SR Ca²⁺ (via CASQ2) play an important role in RyR inactivation and RyR recovery from inactivation. Despite the lack of clarity concerning RyR inactivation and SR Ca²⁺ release termination, the RyR model presented here accurately recreates the time course and magnitude of SR Ca²⁺ release for a multitude of voltage clamp protocols and pacing frequencies necessary for validating the Ca_V1.2 model and its use in the simulations presented.

I_rel

Rates of RyR activation and deactivation were increased from those based on lipid-bilayer studies in the original publication of Fill et al. (39) to account for differences in temperature and to create an SR flux that generates a time course and magnitude of the CaT that closely fit experimentally recorded values. For this reason, the rates were not increased uniformly. The activation rate was increased 3.5 times, and , , and were increased seven times. The deactivation rates , , and were increased five times and the deactivation rate was increased 25 times.

I_Ca(T)

If ,

else

I_Ks

I_Kr

I_NaCa & I_NaCa,ss

I_Na

If V_m < −40 mV

Time-independent currents

I_Ca,b

I_Na,b

I_Kp

I_NaK

I_K1

I_p,Ca

SR fluxes

Ca²⁺ buffers in myoplasm

Ca²⁺ buffer in JSR

Intracellular ion concentrations

SR calcium concentrations

Subspace concentrations

Conservative current stimulus

For the duration of the stimulation

V_m

Cell geometry

Derivation of subspace volume

We assume a myocyte with a radius of 11 μm and length of 100 μm with Z-lines spaced by 1.85 μm. T-tubules protrude into the myocyte along the Z-lines with spacing of 1.5 μm. With these values, we compute 54 Z-lines per myocyte with 46 t-tubules per Z-line for a total of 2484 t-tubules per myocyte. The diameter of each t-tubule is 0.3 μm and the distance between the SR membrane and the t-tubule is 0.02 μm (15). The volume of subspace along the length of a single t-tubule to the center of the myocyte is:

which is of the whole cell volume. In the model, we round to 2%.

Initial ionic concentrations and markov model state residencies

End diastolic steady-state values for the model paced at CL = 1000 ms.

L-type Ca²⁺ channel

Kinetics for G406R CA_V1.2

APPENDIX: MODEL EQUATIONS

L-type Ca²⁺ channel I_Ca(L)