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Regulation of Ca²⁺ and electrical alternans in cardiac myocytes: role of CAMKII and repolarizing currents

Cardiac Bioelectricity and Arrhythmia Center, Washington University in St. Louis, St. Louis, Missouri

Submitted 10 December 2006 ; accepted in final form 26 January 2007

	ABSTRACT

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX GRANTS REFERENCES

 
Alternans of cardiac repolarization is associated with arrhythmias and sudden death. At the cellular level, alternans involves beat-to-beat oscillation of the action potential (AP) and possibly Ca²⁺ transient (CaT). Because of experimental difficulty in independently controlling the Ca²⁺ and electrical subsystems, mathematical modeling provides additional insights into mechanisms and causality. Pacing protocols were conducted in a canine ventricular myocyte model with the following results: 1) CaT alternans results from refractoriness of the sarcoplasmic reticulum Ca²⁺ release system; alternation of the L-type calcium current has a negligible effect; 2) CaT-AP coupling during late AP occurs through the sodium-calcium exchanger and underlies AP duration (APD) alternans; 3) increased Ca²⁺/calmodulin-dependent protein kinase II (CaMKII) activity extends the range of CaT and APD alternans to slower frequencies and increases alternans magnitude; its decrease suppresses CaT and APD alternans, exerting an antiarrhythmic effect; and 4) increase of the rapid delayed rectifier current (I_Kr) also suppresses APD alternans but without suppressing CaT alternans. Thus CaMKII inhibition eliminates APD alternans by eliminating its cause (CaT alternans) while I_Kr enhancement does so by weakening CaT-APD coupling. The simulations identify combined CaMKII inhibition and I_Kr enhancement as a possible antiarrhythmic intervention.

arrhythmia; calcium; sudden death; electrophysiology; calcium/calmodulin-dependent protein kinase II

Due to tight coupling between the Ca²⁺ and electrical cellular subsystems, it is difficult to determine cause and effect experimentally, because the ability to independently control each subsystem is limited (23). Even more challenging is the study of interactions between specific SR processes and sarcolemmal currents (50). Several studies have shown that CaT alternans persists when the cell is voltage clamped, with either constant voltage (13, 48, 50) or constant duration APs (9), suggesting that Ca²⁺ oscillation plays the primary role in alternans generation at a moderately fast rate of pacing. Theoretical modeling can realize precise independent control of individual components and is, therefore, very useful for studying the highly interactive mechanism of alternans. While important insights have been obtained from simplified models (1, 2, 20, 55, 59), processes relevant to alternans formation, such as dynamic ion accumulation and regulation by Ca²⁺-dependent regulatory pathways, have not been considered.

Moreover, properties of CaT-AP coupling are species dependent (4, 15, 23, 50), each with remarkably different CaT and AP morphologies and durations. In addition, there is the well-documented transmural heterogeneities in AP and CaT cycling properties in the same species, which have been documented to affect the onset and amplitude of alternans (51, 68). It has been observed that the large CaT during beat-to-beat (large-small) CaT alternans is accompanied by a short APD in some species (or certain experimental conditions) (33, 46), while in other species by a prolonged APD (48, 50). It was suggested that the Na⁺/Ca²⁺ exchanger (I_NaCa) is responsible for prolongation of APD during large CaT, while Ca²⁺-dependent inactivation of I_Ca(L) is the mechanism of APD shortening (48, 59). However, the specific mechanism of CaT-APD coupling during alternans and its modulation by the whole cell environment require further exploration.

A delicate balance between repolarizing and depolarizing currents provides for precise control of the AP time course (54). Because this balance is modulated by [Ca²⁺]_i, it is important to use physiologically detailed models of the cardiac myocyte for studying the interaction between the Ca²⁺ and electrical subsystems in the study of alternans. Here, we investigate the cellular mechanism of alternans that involves both CaT and APD alternation. Specifically, we examine the following hypotheses. 1) Calcium alternans drives APD alternans via coupling of the Ca²⁺ and electrical subsystems through I_NaCa. 2) Calcium alternans is caused by refractory properties of the SR Ca²⁺ release process and steep dependence of Ca²⁺ release on SR Ca²⁺ load. 3) Repolarizing currents have a modulatory effect on alternans by influencing APD in a Ca²⁺-independent manner. 4) By modulating SR Ca²⁺ cycling, Ca²⁺/calmodulin-dependent protein kinase II (CaMKII) is a major determinant of alternans and its rate dependence.

CaMKII is a regulatory pathway that modulates its activity in response to frequency, amplitude, and duration of Ca²⁺ pulses (10, 26). It plays an essential role in frequency-dependent augmentation of cardiac contractility (69) and acceleration of relaxation (12), particularly during stress or exercise. CaMKII hyperactivity can lead to structural heart disease and arrhythmias (3, 32, 75).

For the purpose of this study, an updated mathematical formulation of SR Ca²⁺ release (I_Rel) was developed. It includes activation of RyR by I_Ca(L) and its regulation by junctional SR (JSR) Ca²⁺ concentration ([Ca²⁺]_JSR) and CaMKII. This formulation was incorporated into theoretical models of ventricular epicardial myocytes of two species, guinea pig (16, 43) and canine (28). The article outline is as follows. First, reformulated I_Rel is validated by reproducing experimental protocols that reveal properties of the Ca²⁺-induced Ca²⁺ release (CICR) process. Second, the dependence of CaMKII on CaT and APD and its inotropic effect are simulated and compared with experiment. Third, the roles of I_Ca(L), SR Ca²⁺ fluxes, and CaMKII in alternans onset and termination at moderately fast rate are studied. Fourth, the nature of bidirectional CaT-APD coupling during alternans is investigated, particularly the role of I_Ca(L), I_NaCa, and rapid delayed rectifier K⁺ current (I_Kr). Aspects of this work have been presented in abstract form (41).

	METHODS

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX GRANTS REFERENCES

 
Myocyte Models

Table 1 in the APPENDIX contains parameter definitions. The theoretical Luo-Rudy dynamic (LRd) (16, 43) and Hund-Rudy dynamic (HRd) (28) models of mammalian ventricular AP provide the basis for the simulations. The LRd model is based on guinea pig data; it includes membrane ion-channel currents, pumps, and exchangers, and accounts for dynamic concentration changes of Na⁺, K⁺, and Ca²⁺. The HRd model (Fig. 1A) is based on epicardial canine data (28) and adds to LRd processes of chloride (Cl^–) homeostasis and the CaMKII regulatory pathway. The model includes the following phosphorylation targets of CaMKII: Ca²⁺ uptake flux by SERCA pump (I_up), I_Ca(L), and I_Rel. I_up includes effects of CaMKII on both the SERCA pump maximal turnover rate and its affinity to Ca²⁺. I_Ca(L) and I_Rel interact in a subsarcolemmal restricted subspace for Ca²⁺ distribution. Models equations and computer codes can be found in the research section of http://rudylab.wustl.edu.

View larger version (38K): [in this window] [in a new window]   Fig. 1. A: ventricular myocyte model. Symbols are defined in APPENDIX Table 1. Model equations are provided (including computer code) in the research section of http://rudylab.wustl.edu. B: SR Ca2+ release model. C: variable gain. Measured (shaded) (63, reproduced with permission) and simulated (solid) excitation-contraction (EC) coupling gain are shown as function of membrane voltage. D: graded release. Ca2+ released by RyR (IRel dt, CL = 1,000 ms) is shown as function of peak ICa(L). E: fractional SR Ca2+ release. Ca2+ released by RyR (IRel dt, CL = 1,000 ms) is shown as function of JSR end-diastolic loading ([Ca2+]JSR,t) in percentage of [Ca2+]JSR,t.

Simulation protocols. The 0.5 ms or 1 ms of –80 µA/µF current stimuli were used to pace LRd or HRd, respectively. The stimulus current was assigned Cl^– and/or K⁺ ions as charge carrier to ensure charge conservation and model stability (29). Numerical integration was performed using Matlab (Mathworks, Natick, MA) (56), with error tolerance of 10^–6. Steady state was defined when all state variables showed <0.1% variability over 100 beats (1 min). The models were tested for convergence and long-term stability over the entire frequency range and parameter values considered. Steady-state APD (90% repolarization) and peak CaT [or Ca²⁺ = max(CaT) – min(CaT), where max(CaT) and min(CaT) are maximum and minimum CaT, respectively] were used to create rate-adaptation curves. Results are shown for HRd simulations, except for Fig. 4, where alternans are also shown for LRd to demonstrate model (species) independence of the alternans phenomenon.

View larger version (41K): [in this window] [in a new window]   Fig. 2. Simulated (A) and experimental (B) (10) time course of CaMKII activity at stimulation rates of 1, 2.5, and 4 Hz; Ca2+ transient (CaT) duration and amplitude are held constant at 200 ms and 20 µmol/l, respectively. Different time scales reflect different isoforms in model and experiment (see text). Simulated (C) and measured (D) [from De Koninck and Schulman (10), reproduced with permission] frequency dependence of CaMKII activity are shown for indicated CaT durations (CaTd). E: time course of AP, CaT, CaMKII activity, and [Ca2+]JSR at 1-Hz stimulation rate. Solid lines: control; shaded lines: AP-clamp pacing with twice APD. F: CaMKII activity as function of pacing rate for different AP-clamp APDs; control (solid lines), 1.5x APD (dashed-dotted gray line), 2x APD (dashed gray line).

View larger version (38K): [in this window] [in a new window]   Fig. 3. A: simulated (top) and measured (bottom) [Sipido et al. (60) with permission] CaT during pacing at 0.25, 0.5, 1, and 2 Hz. Descending limb of CaT is fit by a single exponential with time constant . Simulated (B) and measured (mice, force) (C) [from De Santiago et al. (12) with permission from Elsevier] effect of CaMKII inhibition (by KN-93) on rate of CaT decline and mechanical relaxation are shown. D: simulated effect of CaMKII inhibition on force-frequency (CaT-frequency) relation. E: corresponding experimental data [Wehrers XHT et al. (69) with permission] [rabbit, time-derivative of ventricular pressure (dP/dt)].

View larger version (35K): [in this window] [in a new window]   Fig. 4. APD (A and C) and CaT (B and D) rate-adaptation curves. Insets show bifurcation portions on enlarged scale. A and B: guinea pig. C and D: canine. Inset in C shows experimental data (23, reproduced with permission).

	RESULTS

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX GRANTS REFERENCES

Because we hypothesize that SR Ca²⁺ cycling plays a key role in CaT alternans, it is essential to verify that the models of SR Ca²⁺ release and CaMKII activity reproduce the experimentally observed behaviors that are relevant to alternans generation. The following sections and Figs. 1–3 provide such validation. Tables 4 and 5 in the APPENDIX contain documentation of the electrophysiological data used for the canine model validation and CaMKII data, respectively.

Sensitivity of CaMKII to CaT and APD. In the model-experiment comparison, we use in vitro experimental results from neuronal CaMKII isoform due to lack of direct data regarding the rate of CaMKII phosphorylation in cardiac myocytes. However, as shown by Gaertner et al. (22), all CaMKII isoforms have very similar catalytic and regulatory properties (5). It should be noted that CaMKII isoforms with very similar biochemical characteristics can demonstrate remarkably different targeting properties (e.g., anchoring to target proteins) (25, 72). Simulated CaMKII activity and steepness of frequency dependence increase at a fast rate. The CaMKII model reproduces qualitatively the experimentally (in vitro) observed dependence of CaMKII activity [fraction of active CaMKII binding sites (CaMK_active)] on CaT frequency, amplitude, and duration (CaTd) (10). Figure 2 shows simulated (A) and measured (10) (B) time course of CaMK_active (% of maximum) at rates of 1, 2.5, and 4 Hz (CaTd and amplitude held constant at 200 ms and 20 µmol/l). Both experiment (10) and simulation show increased CaMK_active and steepness at fast rate (slope at 4 Hz is twice steeper than at 2.5 Hz). This rate dependence is due to the autocatalytic nature of the autophosphorylation reaction (27). The different time course in model and experiment is due to faster kinetic parameters used in the simulations (28), reflecting higher activity of cardiac isoform- compared with isoforms- and - in the experiments (22). Figure 2 also shows simulated (C) and measured (10) (D) frequency dependence of CaMKII activity for different CaTd: 500, 200, 80, and 40 ms (40-ms data are not available in the experiment, but are included in simulation, because it is comparable to CaTd in the restricted subspace). As CaTd increases, the curve shifts to lower frequencies. Note that the same CaMKII activity can be reached at slower rates for longer CaTd. For example, CaMKII activity of 20% (horizontal line) is achieved at 0.4, 1, 3, and 6 Hz for CaTd of 500, 200, 80, and 40 ms, respectively (arrows). Both experiment (10) and model show a threshold phenomenon, i.e., no CaMKII activity occurs for specific CaTd until minimal frequency is reached. Ability of CaMII to respond to CaT frequency and morphology is important, because CaMKII senses different CaT transients when tethered (26) to different targets; for example, RyR and L-type calcium channel in the restricted tubular subspace experience a different CaT and consequently CaMKII activity than SERCA/PLB in the bulk myoplasm (Fig. 1A). Both experiment (10) and model show a threshold phenomenon, i.e., no CaMKII activity occurs for specific CaTd until minimal frequency is reached. In addition, model simulations show time-dependent saturation of CaMK_active (Fig. 2A).

Figure 2, E and F, shows modulation of CaMKII activity by APD. Figure 2E shows (clockwise) time course of AP, CaT, [Ca²⁺]_JSR, and CaMK_active at 1-Hz stimulation rate. Control AP waveform (solid lines) is prolonged to double APD at CL = 1,000 ms from 215 to 430 ms (shaded lines). The prolonged AP is used as command waveform to pace the cell to steady state (shaded lines). APD prolongation leads to dramatic increase of CaT and [Ca²⁺]_JSR, >200% of initial values; consequently, CaMK_active is increased by 75%. These simulations show that APD can modulate CaMKII activity by increasing intracellular Ca²⁺ loading and CaT. Figure 2F shows CaMK_active at rates from 0.5 to 3 Hz for control APD and APD increased by 50 or 100%. The same level of CaMKII activity is achieved at lower frequency for longer APD. For example, CaMKII activity of 20% (horizontal line) is achieved at 1.8, 2.4, and 2.8 Hz for 2.0, 1.5, and 1.0 control APD, respectively (arrows). The sensitivity of CaMKII activity level to APD is important when considering that CaMKII serves as a frequency sensor in different species (mouse, guinea pig, dog, human), which have remarkably different AP morphologies and durations (4, 21, 40).

CaMKII underlies frequency-dependent acceleration of relaxation and positive force-frequency relation. When the heart rate increases, greater force (implying greater CaT) (6) is generated. The increase of force (or CaT) with frequency is termed positive force-frequency relation (PFFR). At a fast rate, less time is available for cardiac relaxation or pumping Ca²⁺ back into the SR. In this subsection, we study the effects of CaMKII on the frequency dependence of CaT amplitude and decay. These properties affect CaT in a rate-dependent manner and are therefore relevant to formation of alternans. Experimental data on CaMKII regulation in ventricular myocytes are limited. For experiment-model comparison, we use the measured time derivative of left ventricular pressure (69) or twitch relaxation (12) as surrogates of CaT when data for myocyte CaT are not available. Figure 3A shows experimental (bottom) (60) and simulated (top) CaT at different rates. Both experiment and simulation show that CaT amplitude and rate of decay increase at a fast rate. Note that simulated and measured (60) peak CaT increase monotonically with pacing frequency (PFFR). The descending limb of CaT is fit by a single exponential, with time constant of relaxation . Both experiments and simulations show that, with increasing frequency, decreases monotonically. Tenfold increase in frequency from 0.25 to 2 Hz results in about twofold increase of relaxation rate, with decreasing from 450 to 200 ms. This phenomenon is called frequency-dependent acceleration of relaxation (FDAR) and is essential for normal diastolic function. Complete suppression of CaMKII effect on all targets in the model (Fig. 3B, shaded line) slows FDAR and blunts its frequency dependence compared with control (panel B, solid line). Similar behavior is seen experimentally (panel C) (12).

Figure 3, D and E, shows the effect of CaMKII inhibition on force-frequency relation. For control with CaMKII active (solid curves), experiment (E) (69) and simulation (D) show similar 40% increase of contractility or CaT, respectively, as rate increases over the range shown. Total CaMKII inhibition greatly suppresses this rate dependence.

Figures 3 was generated by setting CaMKII activity to zero for all of its targets [i.e., I_Ca(L), I_up, and I_Rel], which mimics the acute effect of KN-93 application to the whole cell. The agreement between model and experiment is qualitative. The onset of increased contractility (or CaT) is shifted to lower frequencies in simulations relative to experiments, reflecting the slower heart rate of canine (simulation) compared with rabbit (experiment) (69).

CaT and APD Alternans

Frequency dependence of alternans. Figure 4, A and B, shows steady-state APD and Ca²⁺ rate dependence (adaptation curves) generated by the guinea pig (LRd) model. Figure 4, C and D, shows similar curves for canine (HRd). As pacing rate is increased, APD shortens, until it reaches a point of bifurcation at which for the same pacing rate APD oscillates between long and short values. Figure 4, B and D, shows corresponding CaT adaptation curves; CaT amplitude increases at a fast rate until, exactly at the same frequency as APD, bifurcation occurs. The bifurcation portions of APD and CaT curves are shown in insets on the expanded scales. The guinea pig model alternates at CL from 150 to 250 ms, consistent with experimental data (49). Maximal APD and CaT differences between two consecutive beats occur at CL = 200 ms with magnitudes of 12 ms and 0.75 µmol/l, respectively. The canine model alternates at CL = 155–275 ms, also consistent with experimental data (23). Maximal APD and CaT differences between two consecutive beats occur at CL = 250 ms, with magnitudes of 35 ms and 0.5 µmol/l, respectively. APD and CaT curves bifurcate also when plotted against the preceding diastolic interval (DI) (not shown), as frequently presented (23, 35). Note that the bifurcation portions of APD adaptation curves are smooth functions of CL; as CL decreases, alternans amplitude increases to a maximum and then decreases (23) (Fig. 4C, inset). Both canine and guinea pig model simulations are shown in Fig. 4, demonstrating model (species) independence of the alternans phenomenon. The simulated frequency ranges and amplitudes of alternans are consistent with corresponding experimental data (24, 68).

CaT-AP coupling during alternans and its mechanism. To elucidate the link between APD (electrical) and CaT (mechanical) alternans, we pace the cell under conditions of AP clamp or CaT clamp (Fig. 5). Figure 5A shows AP (top) and CaT (bottom) during alternans at 4 Hz; note that large CaT is accompanied by long APD. In Fig. 5B, steady-state behavior is shown during pacing at CL = 250 ms with AP (top) clamped to either its short APD = 133 ms (shaded line) or long APD = 165 ms (solid line). Despite elimination of AP alternans by the clamp protocol, CaT alternans persists (Fig. 5B, bottom). In Fig. 5C, CaT (bottom) is clamped to either its small (shaded line) or large (solid line) morphology. In either case, AP alternans is eliminated (Fig. 5C, top). The SR Ca²⁺ subsystem continues to oscillate during clamping with large CaT morphology, and the SR Ca²⁺ release rate is higher during large depletion than during small depletion (not shown). The results reveal that, at this pacing rate, CaT alternans is causing AP alternans; in other words, oscillation of the Ca²⁺ subsystem is driving the APD oscillations. Simulations over the entire bifurcation range (170 < CL < 270) show the same Ca²⁺-driven mechanism of AP alternans.

View larger version (48K): [in this window] [in a new window]   Fig. 5. AP and CaT clamp protocols. A: AP (top), CaT (middle), and ICa(L) (bottom), during alternans at 4 Hz. B: AP clamp with short (shaded lines) or long (solid lines) AP. Despite AP clamping (top), calcium subsystem oscillates (bottom). C: clamping CaT (bottom) to its small (shaded lines) or large (solid lines) waveform eliminates AP alternans (top). D: clamping ICa(L) (bottom) does not eliminate AP (top) or CaT (middle) alternans.

Figure 6A shows (clockwise) superimposed AP, I_Kr, CaT, and I_NaCa during alternans for two consecutive beats, with long (solid line) and short (shaded line) APD. The higher early plateau of the short AP (70 ms, arrow) is mainly due to enhanced I_Ca(L) caused by less Ca²⁺-dependent inactivation (Fig. 6A, bottom) during the small CaT (shaded line). Early plateau Ca²⁺-dependent transient outward Cl^– current (I_to2) is also Ca²⁺ dependent, but is a small current, and its effect on AP morphology changes during alternans is small. During the large CaT (solid line), I_NaCa is more inward than during the small CaT (shaded line), slowing AP repolarization to cause crossover of the APs and prolongation of APD. The higher plateau of the short AP and the APs crossover are in agreement with experimental data (24) (Fig. 6B) from canine ventricular myocytes. The simulations identify I_NaCa as the major coupling link between CaT alternans and APD alternans, due to its major role late in the AP, when repolarization and APD depend on a delicate balance of currents and are easily modulated.

View larger version (22K): [in this window] [in a new window]   Fig. 6. A: superimposed AP, CaT, INaCa, and IKr of consecutive beats during alternans at CL = 250 ms. B: measured (canine) (24) AP and IKr during alternans.

View larger version (29K): [in this window] [in a new window]   Fig. 7. Steady-state SR Ca2+ flux balance during alternans. A: dependence of Ca2+ = max(CaT) – min (CaT) on JSR Ca2+ content during alternans at 5 Hz. Simulation (top) is compared with experiment (bottom) (13, reproduced with permission). Only 40% change in [Ca2+]JSR leads to fourfold change in Ca2+. [Ca2+]JSR,t = free and buffered [Ca2+]JSR before release; solid and shaded bars correspond to larger and smaller [Ca2+]JSR,t, respectively. Data are normalized to small [Ca2+]JSR,t. B: total end-diastolic SR Ca2+ content ([Ca2+]JSR,t and [Ca2+]NSR) as function of rate. C: free Ca2+ in JSR ([Ca2+]JSR) as function of frequency. D: total Ca2+ released by RyR (IRel dt, solid thick line), reloaded into NSR [(Iup – Ileak) dt, shaded lines] and translocated from NSR to JSR (Itr dt, solid thin line) over one cycle, as function of frequency.

View larger version (24K): [in this window] [in a new window]   Fig. 8. A: APD (top) and Ca2+ (bottom) adaptation curves for three different levels of CaMKII activity: control (100%), 25% elevation (125%), or complete block (0%). B: effect of IKr on APD and CaT alternans. APD (top) and CaT (bottom) adaptation curves for three different levels of IKr conductance are shown: control (100%), 300% elevation, and 50% reduction. Inset: superimposed consecutive APs at 5 Hz for IKr increase of 300%.

	DISCUSSION

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX GRANTS REFERENCES

 
This study shows that, at a moderately fast rate (between 3.5 and 5.5 Hz), the SR Ca²⁺ subsystem, strongly modulated by CaMKII, can initiate CaT alternation that induces APD alternans.

CaT Alternans

At a moderately fast rate, the guinea pig (4–6.5 Hz) and canine (3.5–5.5 Hz) models produce sustained alternans of both APD and CaT. Simulated AP and CaT clamp protocols confirm (38) that oscillation of the Ca²⁺ subsystem is driving the APD alternans in both species. The mechanism underlying CaT alternans is explored by evaluating the roles of the trigger for SR Ca²⁺ release I_Ca(L), SR load, SR Ca²⁺ fluxes, and CaMKII activity during alternans. Model simulations show that refractoriness of the SR Ca²⁺ release process is the main mechanism of CaT alternans. Specifically, two rate-limiting processes, I_up and I_tr (Fig. 7D), in conjunction with steep dependence of SR Ca²⁺ release on SR Ca²⁺ load (Fig. 1E), determine the onset and offset of sustained alternans at moderately fast rates.

I_tr in the model represents both (local) RyRs intrinsic recovery from refractoriness and (global) Ca²⁺ diffusion (8) through the SR. While the steep dependence of release and rate of uptake are sufficient to induce alternans in the model (see also Ref. 70), I_tr also contributes to alternans formation (Fig. 7D). In addition, the model predicts that, during Ca²⁺ overload, the SR Ca²⁺ cycling subsystem can oscillate even without corresponding beat-to-beat oscillations of CaT (not shown). In contrast to previous modeling reports (20, 55), we find that alternation of I_Ca(L) is not necessary to evoke steady-state CaT alternans; such alternans are not eliminated under I_Ca(L) clamp, only reduced in amplitude (Fig. 5D). This observation is consistent with experimental data (13, 48, 50) showing that contraction or CaT alternans can occur without I_Ca(L) fluctuations.

CaT-AP Coupling

While the magnitude of CaT alternans is comparable in guinea pig and canine (100% relative to minimum CaT, Fig. 4, B and D), that of APD alternans is twice as large in canine (20% canine, 10% guinea pig, of maximum APD, Fig. 4, A and C), indicating stronger CaT-AP coupling in this species. These values are comparable with experimental data (23, 36, 50) that reflect a modest level of CaT-APD coupling during alternans. This is in contrast to recently published simulations (55) where 50% alternation of CaT caused greatly exaggerated (>100%) alternation in APD. Such strong dependence of APD on CaT during alternans has never been observed experimentally (24, 30, 36, 49, 50). While the roles of I_NaCa and I_Ca(L) in CaT-AP coupling during alternans were discussed previously in general terms (70), the precise nature of these interactions in detailed myocyte models was not addressed. The stronger CaT-AP coupling in the canine compared with guinea pig is due to differences in ion channel expression levels and kinetic properties. On the background of smaller I_Kr and slow delayed rectifier K⁺ current (I_Ks) in the canine (27), in conjunction with a much smaller I_Ca(L) during the late AP plateau (4), CaT-induced changes in I_NaCa have a much greater modulatory effect on AP repolarization and APD. This makes the canine myocyte more susceptible to Ca²⁺-induced AP alternans and suggests that similar sensitivity to arrhythmia is characteristic of the human heart, the cell electrophysiology and AP morphology of which resemble those of the canine (21). The results show that prolongation of APD secondary to a large CaT is mainly due to large inward I_NaCa at the late AP plateau and repolarization phase, identifying I_NaCa as the major CaT-APD coupler during alternans. The other Ca²⁺-dependent currents, I_Ca(L) and I_to2, play a role in shaping the AP during its initial plateau phase, causing crossover between consecutive APs during alternans (Fig. 8A), but have a minimal effect on APD. Transient outward K⁺ current (I_to1) that contributes to APD rate adaptation (28) has little effect on AP morphology during alternans. The situation can be different, with I_Ca(L) playing a role in APD alternans, in species, where I_Ca(L) persists into the late phase of the AP (e.g., the guinea pig) and Ca²⁺-dependent I_Ks has a large conductance (17, 40). Under such conditions, a large CaT can lead to APD shortening during alternans, due to increased Ca²⁺-dependent inactivation of I_Ca(L). Heart failure shifts the onset of APD alternans to slower frequencies and causes a remarkable increase in its amplitude (71). Upregulation of I_NaCa has been reported in human and animal models of heart failure (6). This observation supports the role of I_NaCa as the major CaT-APD coupler during alternans. It should be commented that exploration of such mechanistic details requires detailed species-specific and ionic-based cell models. It cannot be accomplished with simplified models (55, 58), where the levels of Ca²⁺-dependent and voltage-dependent inactivation of I_Ca(L) are treated as model parameters, not based on experimental data.

Modulation of CaT and APD Alternans by CaMKII and Repolarizing Currents

Elevated CaMKII activity, as occurs in hypertrophy and heart failure (3), extends the range of CaT alternans and consequently APD alternans to slower frequencies and increases alternans magnitude, suggesting its role in arrhythmia and sudden death in these pathologies. Decrease of CaMKII activity suppresses both CaT and APD alternans, thereby exerting an antiarrhythmic effect. Unfortunately, the decrease blunts the PFFR and FDAR (Fig. 3, E and D), thereby compromising cardiac mechanical function. Modification of I_Kr has been suggested as a possible intervention for reducing APD alternans (20, 23). Here we describe the first study of the role of Ca²⁺-independent currents during Ca²⁺-driven APD alternans. The simulations show (Fig. 8, A and B) that only a large threefold increase of I_Kr can completely suppress APD alternans, which limits its potential use as antiarrhythmic intervention. Moreover, increase of I_Kr has no effect on the onset and magnitude of CaT alternans (Fig. 8D). Thus, unlike CaMKII inhibition that suppresses APD alternans by eliminating its cause, CaT alternans, increased I_Kr weakens CaT-AP coupling, thereby suppressing APD alternans by disrupting its link to persistent alternans of CaT. Similar results were obtained by modulating other repolarizing currents, namely the I_Ks, the inward rectifier K⁺ current (I_K1), and the ATP-dependent K⁺ current (I_K,ATP) (not shown). However, results are shown only for I_Kr, because the conductance of I_Ks is Ca²⁺ dependent, increase of I_K1 markedly decreases excitability and conduction velocity (44), and I_K,ATP is activated only during pathological conditions of ischemia (acidosis) (54). These results suggest two possible antiarrhythmic strategies for alternans suppression: 1) prevention of CaT alternans by partial CaMKII inhibition; or 2) modification of the coupling between the Ca²⁺ and electrical subsystems by modulating repolarizing currents such as I_Kr or I_K,ATP. A combined approach of 1 and 2 above seems reasonable, providing more flexibility for alternans suppression with minimization of deleterious effects on contractility and mechanical performance. At a very fast rate (>7 Hz), APD alternans is primarily an electrical phenomenon. This electrical alternans (not shown) has been attributed to slow recovery from inactivation of either I_Na (44, 52, 66) or I_Ca(L) (70). Several studies have shown (73) that cells in the heart can be exposed to such fast and even faster rates (e.g., 11 Hz) (73) during fast ventricular tachycardia and fibrillation. APD alternans at these rates can lead to propagation failure and transition from ventricular tachycardia to fibrillation via wavebreak mechanisms (73).

Limitations

A limitation of the study is that ECC spatial heterogeneity is not considered. A phenomenon associated with this heterogeneity is Ca²⁺ waves, which are known to be arrhythmogenic (13). However, Ca²⁺ waves are rarely observed in nonfailing myocytes during fast pacing (30, 50), the subject of our investigation. For the same reasons (the models are based on data from nonfailing myocytes), only acute up/downregulation of CaMKII was considered. The model does not include the -adrenergic/PKA regulatory pathway that, while sharing common targets [i.e., I_up, I_Ca(L), and I_Rel] with CaMKII, by itself plays an important role in ECC and cardiac repolarization. Incorporating in the model the -adrenergic/PKA regulatory pathway, together with effects of chronic upregulation of CaMKII as occurs in heart failure (75), will be an important step in future model development and simulation studies. As was recently shown in a transgenic mice model, chronic inhibition of CaMKII activity leads to upregulation of repolarizing (39) and I_Ca(L) (75) currents and can compensate for mechanical function impaired due to calcineurin overexpression (32).

The time constant of Ca²⁺ translocation flux from NSR to JSR (I_tr) in the model represents both (local) RyRs intrinsic recovery from refractoriness and (global) Ca²⁺ diffusion (8) through the SR. Separation of these time-limiting processes requires additional experimental data that are not yet available and development of a detailed kinetic model of RyR gating. This can be important future work, considering that some studies (50) stress the importance of intrinsic RyR refractoriness in CaT alternans development. The simulations also indicate the need for detailed experimental studies of CaMKII properties in ventricular myocytes and its interactions with RyRs and SERCA/PLB during alternans.

	APPENDIX

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX GRANTS REFERENCES

 
Equations and parameters of the guinea pig (LRd) and canine (HRd) models used in this study are as in previously published papers (16, 28, 74) and in the research section of http://rudylab.wustl.edu. Definitions are provided in Table 1 of this APPENDIX. Changes made for the purpose of this study are summarized below.

Formulation of I_Rel

See Table 2. The differential equation that describes I_Rel is of the form

(1)

where

(2)

and

(3)

_Rel is an amplitude coeffient, K_Rel, is a half-saturation coefficient, and is a maximal CAMKII-independent value of _IRel.

We make _IRel in Eq. 3 dependent on CaMKII to incorporate CaMKII-dependent facilitation into the model with a maximal change that produces a 100% facilitation of peak _,CaMK:

(4)

In addition, we make _IRel in Eq. 3 a function of [Ca²⁺]_JSR to prevent an unphysiological draining of JSR. Sensitivity of the release flux I_Rel to luminal [Ca²⁺]_JSR is modeled by Hill equation with a coefficient h_Rel (14, 57) and half-saturation constant K_Rel, (37).

Gating Variables of I_Ca(L) See (Table 3)

Fast Ca²⁺-dependent inactivation (f_Ca gate) formulation:

(5)

(6)

where _fca,CaMK is the maximal CaMKII-dependent change of _Ca (time constant of f_Ca gate) and set to 5 ms, K_m,CaMK is a half-saturation coefficient, f_Ca, is the steady-state value of f_Ca. In addition, to reflect higher [Ca²⁺] in the subspace, activity coefficient _Cai = 1 was replaced by _Cai = 0.341 in the constant field equation for _Ca(L).

Steady state formulation of activation d gate was modified as follows

(7)

where V is membrane voltage.

SR Fluxes

CaMKII dependence of I_up was set to up _up,CaMK = 0.9.

Model of SR Ca²⁺ Release and SR Fluxes for LRd Model

Numerical values for I_Rel, and _IRel are provided in Table 2. I_tr time constant _tr was set to 120 ms.

Table 4 provides documentation for the electrophysiological data used for the canine model validation. Table 5 contains CaMKII data used in the simulations.

	GRANTS

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX GRANTS REFERENCES

 
This research was funded by National Heart, Lung, and Blood Institute Merit Award R37-HL 33343 and RO1-HL 49054 (to Y. Rudy).

	ACKNOWLEDGMENTS

 
We thank members of the Rudy Lab: Keith Decker, Dr. Greg Faber, Namit Gaur, Tom O'Hara, Dr. Ali Nekouzadeh, and Jonathan Silva for helpful discussions. Special thanks go to Dr. Valentin Krinski for insightful comments on various aspects of the paper.

	FOOTNOTES

 

Address for reprint requests and other correspondence: Y. Rudy, Campus Box 1097 Whitaker Hall, Rm 290 Washington Univ. in St. Louis One Brookings Drive St. Louis, MO 63130–4899 (e-mail: rudy{at}wustl.edu)

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

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