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(Circulation. 2004;110:3168-3174.)
© 2004 American Heart Association, Inc.

Arrhythmia/Electrophysiology

Rate Dependence and Regulation of Action Potential and Calcium Transient in a Canine Cardiac Ventricular Cell Model

Thomas J. Hund, PhD; Yoram Rudy, PhD

From the Departments of Biomedical Engineering (T.J.H., Y.R.) and Pathology (T.J.H.), Washington University, St. Louis, Mo.

Correspondence to Yoram Rudy, Department of Biomedical Engineering, Washington University, Campus Box 1097, One Brookings Dr, St. Louis, MO 63130-4899. E-mail rudy{at}wustl.edu

Received January 29, 2004; de novo received April 19, 2004; accepted June 7, 2004.

	Abstract

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Background— Computational biology is a powerful tool for elucidating arrhythmogenic mechanisms at the cellular level, where complex interactions between ionic processes determine behavior. A novel theoretical model of the canine ventricular epicardial action potential and calcium cycling was developed and used to investigate ionic mechanisms underlying Ca²⁺ transient (CaT) and action potential duration (APD) rate dependence.

Methods and Results— The Ca²⁺/calmodulin-dependent protein kinase (CaMKII) regulatory pathway was integrated into the model, which included a novel Ca²⁺-release formulation, Ca²⁺ subspace, dynamic chloride handling, and formulations for major ion currents based on canine ventricular data. Decreasing pacing cycle length from 8000 to 300 ms shortened APD primarily because of I_Ca(L) reduction, with additional contributions from I_to1, I_NaK, and late I_Na. CaT amplitude increased as cycle length decreased from 8000 to 500 ms. This positive rate–dependent property depended on CaMKII activity.

Conclusions— CaMKII is an important determinant of the rate dependence of CaT but not of APD, which depends on ion-channel kinetics. The model of CaMKII regulation may serve as a paradigm for modeling effects of other regulatory pathways on cell function.

Key Words: electrophysiology • action potentials • calcium • ion channels

	Introduction

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The dependence of action potential duration (APD) and the Ca²⁺ transient (CaT) on pacing rate is a fundamental property of cardiac myocytes that, when altered, may promote life-threatening cardiac arrhythmias. We have developed a detailed and physiologically based mathematical canine ventricular cell model (Hund-Rudy dynamic [HRd] cell model) that simulates rate-dependent phenomena associated with ion-channel kinetics, AP properties, and Ca handling. The dog is commonly used to investigate cardiac electrophysiology, making it a logical choice for modeling. An epicardial myocyte was chosen rather than endocardial or midmyocardial myocytes because epicardial cells contain the highest density of I_to1 (transient outward K⁺ current), producing a unique and complex AP morphology.

Ca²⁺/calmodulin-dependent protein kinase¹ (CaMKII) mediates an important regulatory pathway in myocytes.² On activation by Ca²⁺/calmodulin, CaMKII phosphorylates neighboring subunits (autophosphorylation), which enables detection of Ca²⁺ spike frequency.³ In cardiomyocytes, CaMKII substrates include L-type Ca²⁺ channels (LTCCs), ryanodine receptor Ca²⁺-release channels (RyRs), sarcoplasmic reticulum Ca²⁺-ATPase (SR Ca²⁺-uptake pump), and phospholamban (PLB).^4–10 This suggests an important role for CaMKII in cardiac Ca²⁺-handling rate dependence and electrophysiology. We used the HRd model to gain new insights into ionic processes underlying AP and CaT rate dependence and how CaMKII regulates these processes.

	Methods

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Complete HRd equations, definitions, and detailed comments appear in the online-only Data Supplement. Important model properties (schematic in Figure 1A) are summarized here.

View larger version (45K): [in this window] [in a new window]   Figure 1. A, Canine ventricular cell model. Symbols are defined in text and in online-only Data Supplement Table I. B, Ratio of peak SR Ca2+ release flux to peak LTCC flux vs test potential (Vtest) in model (line in each panel) and experiment19 (circles in each panel). Model was clamped for 50 ms to –40 mV holding potential, followed by 50-ms test pulse. C, ICa(L) I-V relation compared with canine ventricular data.21 [Ca2+]o=2.0 mmol/L. D, Left, ICa(L) voltage-dependent inactivation compared with experiment.21 Variable 1-second prepulse (Vpre) was followed by 10-ms holding interval at –50 mV and +80 mV test pulse. Right, ICa(L) recovery from voltage-dependent inactivation compared with canine ventricular data.52 Prepulse of 350 ms to +20 mV was followed by varying interpulse interval at –40 mV and +20 mV test pulse. Model Ca2+-dependent inactivation gates were held constant to isolate voltage-dependent inactivation. E, Peak IKs and IKr tail currents on repolarization to –40 mV holding potential after 5-second test pulse, compared with canine epicardial data.24 F, Ito1 I-V relation compared with canine epicardial data.27

Ca²⁺/Calmodulin-Dependent Protein Kinase II
The CaMKII formulation was adapted from Hanson et al³ and responds dynamically to [Ca²⁺]_ss (subspace Ca²⁺ concentration) elevation during the CaT. Kinase subunits can be inactive, in a Ca²⁺/calmodulin-bound active state (CaMK_bound), or in a "trapped" state (CaMK_trap), wherein the subunit remains active for some time after [Ca²⁺]_ss returns to diastolic values. Autophosphorylation of 1 subunit by another promotes transition from CaMK_bound to CaMK_trap. Trapped subunits are dephosphorylated at a constant rate, ß_CaMK, of 0.00068 ms^–1, a moderate value compared with the cycle-length (CL) range investigated here.

Subspace Compartment
The junctional SR membrane abuts the sarcolemma along t-tubules, where LTCC and RyR clusters are localized,¹¹ creating a subspace in which the Ca²⁺ concentration ([Ca²⁺]_ss) rises faster and reaches larger values compared with that of the bulk myoplasm. We modeled the subspace as a compartment into which LTCCs and RyRs open, generating a local Ca²⁺ concentration, [Ca²⁺]_ss. Anionic sarcolemmal and SR membrane binding sites act as calcium buffers.¹² The Ca²⁺-dependent transient outward current (I_to2) is a ligand-gated Cl^–-selective channel. Its low Ca²⁺ sensitivity (K_0.5=0.1502 mmol/L)¹³ supports its incorporation into the Ca²⁺ subspace. CaMKII forms a complex with RyR¹⁴ and is also assumed to be in the subspace.

RyR Ca²⁺-Release Channel I_rel
The I_rel formulation includes activation by the L-type Ca²⁺ current, Ca²⁺-dependent inactivation,^15,16 and open-probability modulation by junctional SR [Ca²⁺] and [Ca²⁺]_ss¹⁷. Although it is generally accepted that the RyR is regulated by SR Ca²⁺ content¹⁷ and inactivated by cytosolic Ca,^15,16 the relative contribution of each process to SR Ca²⁺-release termination is unknown. I_rel in our formulation terminates via both inactivation and SR regulation of the activation gate.¹⁸ Graded release is achieved by making steady-state activation a continuous function of I_Ca(L). Voltage-dependent SR release gain¹⁹ (variable gain, Figure 1B) is introduced through a multiplicative factor dependent on the I_Ca(L) driving force.

Though controversial (online-only Data Supplement section J), CaMKII phosphorylation is thought to promote RyR channel opening.^5,14,20 Accordingly, the I_rel inactivation time constant (_ri) depends on CaMKII activity. A 10-ms maximal CaMKII-dependent increase in _ri yields a steady-state CaT amplitude (CaT_amp) 95%²⁰ greater for control than with CaMKII suppressed at rapid pacing (CL=300 ms).

SR Ca²⁺-ATPase and PLB
CaMKII phosphorylates the SR,⁶ targeting SERCA2a (SR Ca²⁺-ATPase)⁷ and PLB,^8,10 which associates with SERCA2a to inhibit uptake. PLB phosphorylation shifts the Ca²⁺-binding K_0.5 and relieves inhibition,⁹ whereas direct SERCA2a phosphorylation increases the maximum uptake rate,⁷ although this is controversial⁹ (online-only Data Supplement section K). and K_0.5 depend on CaMKII activity to represent this behavior. The maximal CaMKII-dependent increase is 75%,⁷ whereas the maximal K_0.5 decrease is 0.17 µmol/L⁹.

L-Type Ca²⁺ Channel
I_Ca(L) steady-state activation and current density yield a current-voltage (I-V) relation consistent with canine ventricular data²¹ (Figure 1C). The activation variable, d, is raised to a time-dependent and voltage-dependent power (see the online-only Data Supplement). We assume 2 voltage-dependent inactivation gates with steady-state values (Figure 1D) and time constants fitted to canine ventricular data^21,21a (online-only Data Supplement Figure I). Ca²⁺-dependent inactivation has a fast process²² dependent on [Ca²⁺]_ss and LTCC Ca²⁺ entry (approximated as I_Ca(L)²³) and a slow process dependent on I_Ca(L).²²

Ca²⁺-dependent facilitation occurs via CaMKII phosphorylation.⁴ The rapid Ca²⁺-dependent inactivation time constant (_fca) is CaMKII dependent, producing a maximal 40%²⁰ increase in peak I_Ca(L) relative to the model, with CaMKII suppressed at rapid pacing (CL=500 ms).

Two Components of the Delayed Rectifier K⁺ Current
The canine delayed rectifier K⁺ current has a rapidly activating component (I_Kr) and a slowly activating component (I_Ks).²⁴ The model I_Ks has fast (x_s1, with time constant _xs1) and slow (x_s2) activation gates. Voltage dependence of _xs1 fits canine data.²⁴ The slow activation gate is 10 times slower than x_s1²⁴. I_Kr has 1 activation gate, x_r, based on experimental data.²⁴ I_Ks and I_Kr conductances were chosen to agree with experimental data²⁴ (Figure 1E).

Transient Outward K⁺ Current
The 4AP-sensitive transient outward K⁺ current, I_to1, incorporates the activation and inactivation kinetics of Dumaine et al.²⁵ We add a second inactivation gate with a slower time constant²⁶ than in Dumaine’s formulation. The model I_to1 I-V curve agrees with experimental data²⁷ (Figure 1F).

Other Formulations
Cl Homeostasis
I_to2 inclusion requires modeling intracellular Cl^– regulation by the Na⁺-dependent Cl^– cotransporter²⁸ CT_NaCl, the K⁺-Cl^– cotransporter²⁹ CT_KCl, and the background Cl^– current I_Cl,b.

Na⁺-Ca²⁺ Exchanger
The Na⁺-Ca²⁺ exchanger (I_NaCa), from Weber et al,³⁰ includes an allosteric interaction between intracellular Ca²⁺ and the exchanger.

Late Na⁺ Current
Our slowly inactivating late sodium current I_Na,L³¹ formulation uses activation from the Luo-Rudy dynamic (LRd) fast sodium current I_Na.³² Steady-state inactivation and a 600-ms, voltage-independent inactivation time constant were taken from Maltsev et al.³³

Pacing Studies
The model was paced with a conservative current stimulus³⁴ (carried by KCl) for 2000 seconds from rest (initial conditions in online-only Data Supplement Table II) at a constant CL. Steady-state APD (at 90% repolarization) and CaT_amp (peak systolic [Ca²⁺]_i–minimal diastolic [Ca²⁺]_i) were used to create the APD adaptation curves and the CaT_amp-frequency curves, respectively.

	Results

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APD Rate Dependence (Adaptation)
The model AP morphology (Figure 2A) and APD (Figure 2B) agree with canine epicardial recordings²⁶ over a CL range from 8000 to 300 ms. Most important for canine APD rate adaptation is I_Ca(L), supported by the fact that eliminating I_Ca(L) from the model reduced adaptation (CL range of 8000 to 300 ms) from 99 to 30 ms (71% decrease, Figure 2B). I_to1 also plays a role: its elimination reduced adaptation by 38% (Figure 2B). Of secondary importance are I_NaK (clamping [Na⁺]_i during pacing reduces adaptation by 15%) and I_Na,L (eliminating I_N,aL reduces adaptation by only 9%). CaMKII inhibition had very little effect on APD adaptation (Figure 2B).

View larger version (22K): [in this window] [in a new window]   Figure 2. A, Steady-state AP simulated (top) and measured in canine epicardial myocyte24 (bottom) for CLs of 8000, 4000, 2000, 1000, 500, and 300 ms. B, Steady-state APD vs CL (adaptation curve) in canine epicardial myocyte26 (circles) and in model under control conditions (bold line), in presence of CaMKII inhibition (thin line), without Ito1 (dashed line), and without ICa(L) (long dashed line). Abbreviations are as defined in text.

Interestingly, a decrease in the repolarizing current I_to1 facilitates APD shortening at a fast rate. Comparing steady-state AP with (=0.19 mS/µF) and without (=0 mS/µF) I_to1 reveals the effect of I_to1 on APD (Figure 3). At a slow rate, a large I_to1 produced a rapid phase 1 repolarization (Figure 3A, arrow) which increased the I_Ca(L) driving force and enhanced its voltage-dependent activation during the AP plateau (Figure 3B, arrow). Phase 1 repolarization also increased I_Kr activation and decreased the driving force for reverse-mode I_NaCa, thus reducing these repolarizing currents (Figure 3C and 3D, respectively). By increasing I_Ca(L) and decreasing I_Kr and I_NaCa, I_to1 indirectly prolonged APD. During rapid pacing, I_to1 decreased owing to slow recovery from inactivation, and phase 1 repolarization slowed (Figure 2A). Consequently, indirect APD prolongation by I_to1 was suppressed at fast pacing rates.

View larger version (24K): [in this window] [in a new window]   Figure 3. A, AP (arrow identifies rapid phase 1 repolarization); B, ICa(L) (arrow indicates augmentation); C, IKr; and D, INaCa computed for steady-state pacing at CL=8000 ms. Values were computed with (= 0.19 mS/µF, thick line) and without (= 0 mS/µF, thin line) Ito1. Abbreviations are as defined in text.

Figure 4 compares rate-dependent changes in AP (Figure 4A) and major ionic currents between the HRd canine model and the LRd guinea pig model.³² Reduction in I_Ca(L) at fast rates was observed in the dog but not in the guinea pig (Figure 4B). Guinea pig adaptation instead is primarily caused by an accumulation of slow deactivating I_Ks at fast rates (Figure 4C, arrow).³⁵ Canine I_Ks is small and does not accumulate between beats because of its faster deactivation (Figure 4C). Whereas guinea pig I_Ks during the AP was larger than I_Kr, canine I_Kr is much larger than I_Ks (Figure 4C and 4D).

View larger version (33K): [in this window] [in a new window]   Figure 4. Currents during the AP in HRd canine (left panels) and LRd guinea pig32 (right panels) cell models. Steady-state values are shown at CLs of 300 ms (thin line) and 2000 ms (thick line). A, AP; B, ICa(L); C, IKs (arrow indicates IKs accumulation); D, IKr. Abbreviations are as defined in text.

CaT_amp-Frequency Relation
Steady-state CaT_amp and morphology (Figure 5A), measured³⁶ and simulated, agreed over a wide pacing range. Consistent with experiment,³⁶ the model diastolic [Ca²⁺]_i and CaT_amp increased as pacing frequency increased from 0.25 to 2.0 Hz (positive CaT_amp-frequency relation, Figure 5B), associated with an increase in simulated CaMKII activity, excitation-contraction coupling (ECC) gain (Figure 5C), and PLB phosphorylation by CaMKII (Figure 5D). CaMKII inhibition produced a negative CaT_amp-frequency relation for frequencies >1 Hz (Figure 5B) and flattened the gain-frequency relation (Figure 5C). CaMKII inhibition during pacing (2.0 Hz) (Figure 6) reduced CaT_amp (Figure 6B) by decreasing I_up (Figure 6C), which reduced SR Ca²⁺ load (Figure 6D) by decreasing peak I_Ca(L) (Figure 6E), which reduced the trigger for SR release, and by reducing I_rel directly (Figure 6F).

View larger version (33K): [in this window] [in a new window]   Figure 5. A, Simulated (bottom) and measured36 (top) steady-state CaT for 0.25-, 0.5-, 1-, and 2-Hz pacing. Adapted from Sipido et al.36 B, CaTamp-frequency relation for experiment (circles), model under control conditions (line), and in presence of CaMKII inhibition (dashed line). C, Minimal diastolic CaMKII activity (normalized to 3.3 Hz) and ECC gain. ECC gain=, where Frel and FCa(L) are fluxes through RyRs and LTCCs, respectively, and integration interval, A, equals 1 cycle. Gain is shown for control model (thin line) and in presence of CaMKII inhibition (dashed line). D, PLB phosphorylation vs pacing frequency compared with experimental data.10 Abbreviations are as defined in text.

View larger version (23K): [in this window] [in a new window]   Figure 6. Effect of CaMKII inhibition on CaT. A, CaMKII activity; B, [Ca2+]I; C, Iup; D, [Ca2+]JSR (arrows indicate loading); E, ICa(L) (arrow indicates peak); and F, Irel for steady-state pacing at CL=500 ms (2.0 Hz) in model under control conditions (thick line) and in presence of CaMKII inhibition (thin line). Quantities are shown during AP. Abbreviations are as defined in text.

	Discussion

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We present a dynamic model of the canine ventricular epicardial cell that reproduces experimentally measured AP and CaT over a wide pacing frequency range. Given the broad frequency range during cardiac arrhythmias and the interplay between Ca cycling and cellular electrophysiology in arrhythmogenesis, this model serves as a valuable research tool.

Summary of Important Mechanistic Findings
The major findings of this study are that (1) canine APD adaptation is determined primarily by I_Ca(L) reduction at fast rates; (2) I_to1 contributes to APD adaptation indirectly by augmenting the phase 1 notch at slow rate; (3) ECC gain increases with frequency owing to increased CaMKII activity, producing a positive CaT_amp-frequency relation; and (4) CaMKII is important for rate-dependent changes in CaT but does not significantly effect APD adaptation.

Comparison With Existing Models
Canine ventricular AP models have been previously developed to study electrophysiologic remodeling after heart failure³⁷ and myocardial infarction³⁸ and APD alternans during rapid pacing.³⁹ The model presented here distinguishes itself by incorporating (1) dynamic CaMKII activity and regulation of intracellular Ca²⁺ handling; (2) the late Na⁺ current, I_Na,L, and the Ca²⁺-dependent transient outward current, I_to2; (3) dynamic intracellular Cl^– concentration changes; and (4) a novel I_rel formulation. Our model represents an important advance in the physiologic representation of rate-dependent cell processes through its inclusion of the CaMKII regulatory pathway, shown experimentally to play a role in the force-frequency relation and rate-dependent CaT abbreviation.^14,20,40

"Local-control"⁴¹ Ca²⁺ release has been integrated into a canine AP model,⁴² wherein SR Ca²⁺ release involves statistical recruitment of individual Ca²⁺ release units. Although this model reproduces macroscopic release based on individual diadic events, computational demands discourage its use in modeling cardiac arrhythmias. Therefore, we reproduced local-control features (variable gain and graded release) by using a macroscopic approach with reduced computational demand.

Effect of CaMKII on Ca²⁺ Handling
We have shown that increased CaMKII activity during rapid pacing augments SR Ca²⁺ release and promotes a positive CaT_amp-frequency relation. Our findings are supported by recent experiments measuring increased CaMKII activity and CaMKII-dependent SR Ca²⁺ release after pacing.¹⁴ It is important to note that additional factors determine the CaT_amp-frequency relation. Even in the presence of CaMKII inhibition, a positive relation exists over a limited frequency range from 0.125 to 1 Hz (see Figure 5B). Intracellular Na⁺ and Ca²⁺ accumulation during rapid pacing produces CaMKII-independent SR loading. Intracellular Ca²⁺ buffer saturation at fast rates also may contribute to a positive force-frequency relation.⁴³

APD Adaptation
In the guinea pig, I_Ks activates and deactivates more slowly than does I_Kr⁴⁴. In the dog, I_Kr and I_Ks deactivation kinetics are reversed, with I_Ks deactivating faster than I_Kr.²⁴ Our simulations suggest that I_Ks does not contribute significantly to canine APD adaptation owing to its small amplitude and fast deactivation, consistent with recent canine experiments.^45,46 However, ß-adrenergic stimulation enhances I_Ks,⁴⁷ possibly increasing its importance in AP repolarization and adaptation in vivo.

Our results also suggest a role for I_to1 in determining APD and APD adaptation. Consistent with previous theoretical⁴⁸ and experimental⁴⁹ studies, we found that I_to1 creates a phase 1 notch that increases the driving force for I_Ca(L) and facilitates activation of a sustained component. In addition, the phase 1 notch decreases the repolarizing currents I_Kr and reverse-mode I_NaCa. Together, these processes prolong APD. New insight is obtained into the role of I_to1 in APD adaptation: slow recovery promotes I_to1 reduction, notch suppression, and less associated APD prolongation at fast rates. Our findings are consistent with greater adaptation in epicardial than in endocardial cells (85 and 65 ms, respectively²⁶), which have a greatly diminished I_to1 density. We also found (not shown) that the notch accelerates the time to peak CaT (62 ms in control vs 83 ms without I_to1), consistent with experimental observations.⁵⁰

Limitations
The model formulation was based, wherever possible, on recent experimental data and current understanding of cardiac electrophysiology and Ca²⁺ handling. However, there are controversies regarding CaMKII and its regulatory effects. Disparate findings exist on whether or not CaMKII phosphorylates SERCA2a directly.^7,9 Similarly, conflicting reports exist on CaMKII regulation of RyR activity.^5,20,51,52 These issues remain unresolved (see online-only Data Supplement, sections J and K). We hope that this study will motivate detailed experimental characterization of CaMKII effects on cellular function.

	Acknowledgments

 
This work was funded by grants R01-HL49054 and R37-HL33343 (to Y.R.) from the National Institutes of Health—National Heart, Lung, and Blood Institute, and a Whitaker Foundation Development Award. We thank Greg Faber, Jonathan Silva, and Keith Decker for helpful discussions and Marlene Siegal for administrative assistance.

	Footnotes

 
An online-only Data Supplement is available at http://www.circulationaha.org

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