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(Circulation Research. 1999;84:571-586.)
© 1999 American Heart Association, Inc.

Original Contribution

Mechanisms of Altered Excitation-Contraction Coupling in Canine Tachycardia-Induced Heart Failure, II

Model Studies

Raimond L. Winslow, Jeremy Rice, Saleet Jafri, Eduardo Marbán, Brian O'Rourke

From the Departments of Biomedical Engineering (R.L.W., J.R., S.J.) and Computer Science (R.L.W.) and Center for Computational Medicine and Biology (R.L.W., J.R., S.J.), The Johns Hopkins University School of Medicine and Whiting School of Engineering, and Section of Molecular and Cellular Cardiology (E.M., B.O.), Division of Cardiology, Department of Medicine, The Johns Hopkins University School of Medicine, Baltimore, Md.

Correspondence to Raimond L. Winslow, PhD, The Johns Hopkins University School of Medicine, Department of Biomedical Engineering, 411 Traylor Research Bldg, 720 Rutland Ave, Baltimore, MD 21205. E-mail rwinslow{at}bme.jhu.edu

	Abstract

Top Abstract Introduction Materials and Methods Results Discussion Appendix References

 
Abstract—Ca²⁺ transients measured in failing human ventricular myocytes exhibit reduced amplitude, slowed relaxation, and blunted frequency dependence. In the companion article (O'Rourke B, Kass DA, Tomaselli GF, Kääb S, Tunin R, Marbán E. Mechanisms of altered excitation-contraction coupling in canine tachycardia-induced heart, I: experimental studies. Circ Res. 1999;84:562–570), O'Rourke et al show that Ca²⁺ transients recorded in myocytes isolated from canine hearts subjected to the tachycardia pacing protocol exhibit similar responses. Analyses of protein levels in these failing hearts reveal that both SR Ca²⁺ ATPase and phospholamban are decreased on average by 28% and that Na⁺/Ca²⁺ exchanger (NCX) protein is increased on average by 104%. In this article, we present a model of the canine midmyocardial ventricular action potential and Ca²⁺ transient. The model is used to estimate the degree of functional upregulation and downregulation of NCX and SR Ca²⁺ ATPase in heart failure using data obtained from 2 different experimental protocols. Model estimates of average SR Ca²⁺ ATPase functional downregulation obtained using these experimental protocols are 49% and 62%. Model estimates of average NCX functional upregulation range are 38% and 75%. Simulation of voltage-clamp Ca²⁺ transients indicates that such changes are sufficient to account for the reduced amplitude, altered shape, and slowed relaxation of Ca²⁺ transients in the failing canine heart. Model analyses also suggest that altered expression of Ca²⁺ handling proteins plays a significant role in prolongation of action potential duration in failing canine myocytes.

Key Words: excitation-contraction coupling • heart failure • midmyocardial ventricular action potential • Ca²⁺ transient

	Introduction

Top Abstract Introduction Materials and Methods Results Discussion Appendix References

 
Recent studies using the canine tachycardia pacing-induced model of heart failure^{1 2 3 4 5 6 7 8} demonstrate that changes in cellular electrophysiological and excitation-contraction (E-C) coupling processes are qualitatively similar to those observed in cells isolated from failing human heart. In human heart failure, I_K1 current density measured at hyperpolarized membrane potentials is reduced by 50%,^{9 10} and density of the transient outward current I_to1 is reduced by 75% in subepicardial¹¹ and 40% in midmyocardial ventricular cells⁹ and is unchanged in subendocardial ventricular cells.¹¹ The magnitude of I_K1 is reduced by 40%, and that of I_to1 by 70% in failing canine midmyocardial cells.⁵ Expression of proteins involved in E-C coupling is also altered in human heart failure. Sarcoplasmic reticulum (SR) Ca²⁺ ATPase mRNA level,^{12 13 14 15 16} protein level,^{12 17 18} and uptake rate¹⁹ are reduced by 50% in end-stage heart failure. Na⁺/Ca²⁺ exchanger (NCX) mRNA levels are increased by 55% to 79%,^{12 20} and NCX protein levels increase 36% to 160%.^{12 20 21 22} Less information is available with regard to NCX function in heart failure. However, Reinecke et al²² reported an 89% increase in sodium-gradient–stimulated ⁴⁵Ca²⁺ uptake in human heart sarcolemmal vesicles.

As described in the preceding article by O'Rourke et al,²³ alterations of intracellular Ca²⁺ handling in failing canine midmyocardial ventricular myocytes parallel those observed in human. In particular, the time constant of Ca²⁺ uptake in the absence of Na⁺/Ca²⁺ exchange is prolonged in failing cells (576±83 versus 282±30 ms in controls), suggesting a functional downregulation of the SERCA2a. This observation is consistent with Western blot analyses indicating that SR Ca²⁺ ATPase protein levels are reduced in failing heart by 28%. Additionally, in the presence of cyclopiazonic acid (CPA, a blocker of the SR Ca²⁺ ATPase pump), the time constant of Ca²⁺ extrusion is larger in normal than failing cells (813±269 versus 599±48 ms). This observation is consistent with Western blot analyses indicating a 104% increase in the level of expression of the NCX in failing cells. Taken together, these results suggest that SR Ca²⁺ uptake is impaired and that Ca²⁺ extrusion via the NCX is enhanced in myocytes isolated from the failing canine heart in a way that is similar qualitatively to that seen in human patients.

In this article, we use the data of O'Rourke et al²³ to develop a computational model of the action potential and of intracellular Ca²⁺ handling in normal and failing canine ventricular myocytes using biophysically detailed descriptions of both sarcolemmal currents and key components of E-C coupling. With the limits of individual alterations fixed using experimentally derived values, the model is used to quantify the extent to which each parameter (I_to1, I_K1, SR Ca²⁺ ATPase, and NCX) contributes to the overall change in electrical and Ca²⁺ dynamics in heart failure. The results support the hypothesis that differences in expression of sarcolemmal ion channels and Ca²⁺ handling proteins measured experimentally are sufficient to account for the altered action potential waveform and Ca²⁺ transient of the failing canine cardiomyocyte.

	Materials and Methods

Top Abstract Introduction Materials and Methods Results Discussion Appendix References

 
Normal Canine Ventricular Cell Model
Jafri et al²⁴ have presented a model of Ca²⁺ handling in the guinea pig ventricular myocyte that incorporates the following: (1) sarcolemmal ion currents of the Luo-Rudy phase II ventricular cell model,²⁵ (2) a state model of the L-type Ca²⁺ current in which Ca²⁺-mediated inactivation occurs via the mechanism of mode switching,²⁶ (3) calcium-induced calcium release from SR via ryanodine-sensitive calcium release (RyR) channels using a model adapted from that of Keizer and Levine,²⁷ and (4) a restricted subspace located between the junctional SR (JSR) and T tubules into which both L-type Ca²⁺ and RyR channels empty. The model of the canine midmyocardial ventricular cell used in this study is derived from this guinea pig ventricular cell model. All dynamic equations, parameters, and initial conditions for this new model are given in the Appendix. The following modifications to the model of Jafri et al²⁴ have been made to better represent properties of canine midmyocardial ventricular cells.

I_to1
Canine epicardial and midmyocardial ventricular cell action potentials exhibit a prominent notch in phase 1 of the action potential that results from the presence of 2 transient outward currents: a Ca²⁺-independent 4-aminopyridine (4-AP)–sensitive current (I_to1)^{5 28 29} and a Ca²⁺-dependent current (I_to2).^{29 30} The Ca²⁺-independent component I_to1 is modeled on the basis of the formulation of Campbell et al³¹ for ferret ventricular cells. Peak I_to1 conductance (G_to1) was adjusted to yield a linear plot of peak current density in response to 500-ms–duration voltage-clamp stimuli from a holding potential of –80 mV, with slope 0.3 pA/pF-mV and y-intercept 4.6 pA/pF. This agrees well with experimental measurements reported for canine I_to1 at 37°C by Liu et al²⁸ (see their Figure 10B: slope, 0.28 pA/pF-mV, and y-intercept, 5 pA/pF). Activation rate constants were scaled to yield a time to peak of 8 ms at a clamp potential of +10 mV (see Figure 5B of Tseng and Hoffman).²⁹ Inactivation rate constants were adjusted to yield a decay time constant of 20 ms.²⁹ The Ca²⁺-dependent chloride (Cl–) current I_to2 was not incorporated in this model.

View larger version (21K): [in this window] [in a new window]   Figure 5. A, Cytosolic Ca2+ concentration (µmol/L) as a function of time in response to a 10-second–duration periodic sequence of voltage-clamp stimuli with frequency of 1 Hz. Holding potential is –97 mV, and clamp potential is +3 mV, with duration of 200 ms. Only response to the final stimulus is shown. Results of 4 simulations are shown. Solid line indicates normal model Ca2+ transient (KNaCa=0.30; KSR=1.0). Dotted line indicates the average failing model Ca2+ transient computed using values of KSR and KNaCa estimated in the presence/absence of CPA, as described in the text (KSR=0.38; KNaCa=0.53). Short-dashed lines show failing model Ca2+ transients computed using KSR=0.26 and KNaCa=0.60. Long-dashed lines show failing model Ca2+ transients computed using KSR=0.51 and KNaCa=0.48. B, Experimentally measured cytosolic Ca2+ concentration (µmol/L) vs time in response to voltage-clamp stimuli (as described in panel A) in normal (solid line) and failing (dotted line) canine ventricular myocytes. C, L-type Ca2+ current as a function of time for the stimuli and model parameters described in panel A. D, Subspace Ca2+ concentration (µmol/L) as a function of time for the stimuli and model parameters described in panel A.

I_Kr
The delayed rectifier current I_K in both canine and guinea pig ventricular myocytes consists of rapid- and slow-activating components known as I_Kr and I_Ks, respectively. Models of I_Kr and I_Ks in guinea pig ventricular cells have been developed.³² These models have been modified to approximate properties of corresponding currents measured in isolated canine midmyocardial ventricular cells. I_Kr is described using a closed-open–state model in which forward (K₁₂) and backward (K₂₁) rate constants are exponential functions of voltage (V) with the following form:

(1)

Parameters of this model are fully constrained by knowledge of the time constant (V), defined as

(2)

at 2 voltages and by knowledge of the steady-state activation function. Activation was fit using a Boltzmann function determined by Liu and Antzelevitch³³ (see their Figure 11). The time constant of activation at +5 mV was set to 100 ms,³³ and the time constant of deactivation at –60 mV was set to 3000 ms,³⁴ thereby constraining the rate constants K₁₂(V) and K₂₁(V). A fixed increment of 27 ms was added to Equation 2 to bound the time constant away from 0 at depolarized potentials. The maximum conductance _Kr was adjusted to yield a tail current density of 0.2 pA/pF in response to a voltage-clamp step to +25 mV for 3.0 seconds, followed by a step to –35 mV for 1.0 seconds, as described by Gintant.³⁵

I_Ks
The slow-activating delayed rectifier current I_Ks is present in epicardial, midmyocardial, and endocardial canine ventricular cells. I_Ks is modeled as described in Zeng et al,³² with the exception that the steady-state activation function is fit using a Boltzmann function determined by Liu and Antzelevitch.³³ The voltage-dependent time constant is also shifted by +40 mV in the depolarizing direction to fit the experimental data of Liu and Antzelevitch³³ (see their Figure 13). Maximum conductance (_Ks) is adjusted to yield a tail current density of 0.4 pA/pF in response to 3.0-second–duration voltage-clamp steps from the holding potential of –35 to +25 mV, followed by a return to the holding potential³⁴ (see Figure 5). The Ca²⁺ dependence of I_Ks described in the Luo-Rudy phase II guinea pig model is not included, as there are no experimental data constraining this dependence in canine ventricular cells.

I_K1
I_K1 is fit using data measured at 22°C in isolated canine midmyocardial ventricular myocytes measured by Kääb et al⁵ and scaled to 37°C. These data indicate that maximum outward I_K1 density is 2.5 pA/pF at –60 mV⁵ (see Reference ⁵ , Figure 4B). These data also show that I_K1 density is nonnegligible at voltages within the plateau range of the canine action potential. For example, I_K1 density is 0.3 pA/pF at 0 mV, a value comparable with the density of I_Kr during the plateau phase of the action potential. The functional representation of I_K1 in the Luo-Rudy phase II model can therefore not be used, as it approaches 0 at plateau membrane potentials. An alternative formulation better approximating the canine data is presented in the Appendix.

View larger version (26K): [in this window] [in a new window]   Figure 4. A, Cytosolic Ca2+ concentration (µmol/L) as a function of time in response to a 10-second–duration periodic sequence of voltage-clamp stimuli with frequency of 1 Hz. Holding potential is –97 mV, and clamp potential is +3 mV, with duration of 200 ms. Only response to the final stimulus is shown. A family of responses is shown in which KNaCa is varied from 0.5 to 2.5 in steps of 0.5 (response for KNaCa=0.25 is also shown), while KSR is held constant at the value accounting for the Ca2+ relaxation rate measured in normal myocytes under 0-Na conditions. B, Model NaCa exchange current INaCa as a function of time in response to the stimuli described in Figure 3A.

I_Ca,L
The model of L-type Ca²⁺ current used is identical to the mode-switching model presented in Jafri et al,²⁴ with 3 exceptions. First, the voltage dependence of the activation transition rates (V) and ß(V) and the inactivation variable y(V) are shifted by +10 mV in the depolarizing direction to position the peak L-type Ca²⁺ current in response to voltage-clamp stimuli at +5 mV, as measured experimentally.⁵ Second, the monotonic decreasing steady-state (voltage-dependent) inactivation function y is modified to have an asymptotic value of 0.2 for large positive membrane potentials V. This modification reproduces the slow component of Ca²⁺ current observed under voltage-clamp stimuli in canine ventricular cells.^{5 37} Finally, peak L-type Ca²⁺ current density is adjusted to a value of 2.5 pA/pF at a clamp voltage of +5 mV.

J_up
In the model of Jafri et al,²⁴ Ca²⁺ uptake into network SR (NSR) is modeled using a Hill function with coefficient of 2. Reverse pump rate is assumed to be 0, and Ca²⁺ leak from NSR to cytoplasm is assumed to be proportional to the gradient of NSR and cytosolic Ca²⁺ concentrations. Recently, Shannon et al³⁸ have proposed the hypothesis that SR Ca²⁺ accumulation at rest is not limited by leak of Ca²⁺ from SR but rather is limited by a reverse component of SR Ca²⁺ ATPase pump current. They have proposed a new model of the SR Ca²⁺ ATPase pump that includes forward- and reverse-current components, each with its own binding constant and peak forward and reverse rates (denoted V_maxf and V_maxr, respectively).³⁹ The forward mode exhibits slight cooperativity, whereas the reverse mode is noncooperative. The relative magnitudes of forward- and reverse-current components determine whether SR load increases, is constant, or decreases during diastole. The model is presented in the Appendix.

Failing Canine Ventricular Cell Model
Kääb et al⁵ have shown that in the canine tachycardia pacing-induced model of heart failure, I_to1 and I_K1 are downregulated on average by 66% and 32%, respectively, in terminal heart failure. Only the number of expressed channels is changed; the kinetic properties of I_to1 and gating behavior of I_K1 are unaltered. On the basis of these data, the effects of terminal heart failure are modeled by reducing the peak conductance of I_to1 and I_K1 by the factors indicated above. Downregulation of the SR Ca²⁺ ATPase is modeled by simultaneous scaling of both the forward and reverse maximum pump rates V_maxf and V_maxr by a scale factor, K_SR. Upregulation of the NCX is modeled by increasing a scale factor, K_NaCa.

Numerical Methods
The dynamical equations in the Appendix are solved on a Silicon Graphics workstation using the Merson modified Runge-Kutta fourth-order adaptive step algorithm (No. 25, Reference 52⁵² ), with a maximum step size of 100 microseconds and maximum error tolerance of 10^–6. The error from all variables is normalized to ensure that each contributes equally to the calculation of global error, as described in Jafri et al.²⁴ Initial conditions listed in the Appendix are used in all calculations, unless noted otherwise. These initial conditions were computed in response to a periodic pulse train of frequency 1 Hz and were determined immediately before the 11th pulse. Action potentials are initiated using 0.1 µAµF^–1 current injection for 500 microseconds.

The canine ventricular cell model is used to derive quantitative estimates of the NCX scale factor K_NaCa and the SR Ca²⁺ ATPase scale factor K_SR from experimental data by fitting model Ca²⁺ transient decay rates to those measured experimentally. To do this, a series of 10 voltage-clamp stimuli (–97-mV holding potential, 3-mV step potential, and 200-ms duration) are applied at a frequency of 1 Hz. Ca²⁺ transient decay rate is estimated from response to the final voltage-clamp stimulus to assure that model SR Ca²⁺ concentrations have reached equilibrium values.

	Results

Top Abstract Introduction Materials and Methods Results Discussion Appendix References

 
Action Potentials and Ca²⁺ Transients: Model Versus Experimental Results
Figure 1 demonstrates the ability of the model to reconstruct action potentials and Ca²⁺ transients of both normal and failing canine midmyocardial ventricular myocytes. The solid and dotted lines in Figure 1A show experimental measurements of normal and failing action potentials, respectively. Model action potentials are shown in Figure 1C. In this figure, the solid line shows a normal action potential. The dashed line shows an action potential when I_to1 is reduced by 66% of the normal values and I_K1 by 32% of the normal values (the average percentage reductions observed in terminal heart failure.)⁵ The dotted line corresponds to these same reductions of I_to1 and I_K1, in addition to a 62% reduction of the SR Ca²⁺ ATPase pump and a 75% increase of the NCX. These values are model-based estimates of the average percentage change in activity of these proteins determined using experimentally derived limits on their function, as described in the following sections.

View larger version (19K): [in this window] [in a new window]   Figure 1. Model vs experimental action potentials and Ca2+ transients. Each action potential and Ca2+ transient is in response to a 1-Hz pulse train, with responses measured in the steady state. A, Experimentally measured membrane potential as a function of time in normal (solid line) and failing (dotted line) canine myocytes. B, Experimentally measured cytosolic Ca2+ concentration (µmol/L) as a function of time for normal (solid line) and failing (dotted line) canine ventricular myocytes. C, Membrane potential as a function of time simulated using the normal canine myocyte model (solid line), the myocyte model with Ito1 and IK1 downregulation (dashed line; downregulation by 66% and 32%, respectively), and the heart failure model (dotted line; downregulation of Ito1 and IK1 as described previously, KSR=0.38 corresponding to 62% downregulation and KNaCa=0.53 corresponding to 75% upregulation). D, Cytosolic Ca2+ concentration (µmol/L) as a function of time simulated using the normal (solid line) and heart failure (dotted line) model, with parameters as described in panel A. E, L-type Ca2+ current as a function of time for the normal (solid line) and failing (dotted line) cell models. F, Na+/Ca2+ exchange current as a function of time for the normal (solid line) and failing (dotted line) cell models.

The model data of Figure 1C show that downregulation of I_to1 and I_K1 reduces the depth of the phase 1 notch. However, notch depth is larger in the experimental measurements from the failing myocyte (Figure 1A, dotted line) than is predicted by the model (Figure 1C, dashed line). This greater notch depth is due to the presence of the Ca²⁺-dependent transient outward current I_to2, which is not included in the model. The most significant change in model action potential duration (APD) occurs with upregulation of the NCX and downregulation of the SR Ca²⁺ ATPase (Figure 1C, dotted line). These 2 changes alone increase APDs at 90% repolarization (APD₉₀) by 200 ms.

Figure 1D illustrates model normal (solid line) and failing (dotted line) Ca²⁺ transients. Amplitude of the Ca²⁺ transient is reduced significantly in the heart failure model. Ca²⁺ transient shape is flattened, duration is prolonged, and relaxation is slowed. These changes are similar qualitatively to those seen in the experimental data of Figure 1B.

Figures 1E and 1F show L-type Ca²⁺ and Na⁺/Ca²⁺ exchange currents for normal (solid lines) and failing (dotted lines) model cells. The reduction in peak magnitude of the L-type Ca²⁺ current seen in Figure 1E for the failing model cell results from downregulation of I_to1, which reduces depth of the phase 1 notch and therefore driving force during onset of the L-type Ca²⁺ current. Figure 1E also shows that L-type Ca²⁺ current is increased during the later plateau phase of the action potential in failing model cells. The mechanism of this increase will be considered in subsequent sections. Figure 1F shows that Na⁺/Ca²⁺ exchange operates in reverse mode, generating a net outward current during most of the plateau phase of the action potential. The magnitude of this outward current decreases during the plateau phase, and in the failing cell model the current becomes significantly smaller than the inward L-type Ca²⁺ current.

These simulations demonstrate the ability of the model to reproduce both normal and failing canine myocyte action potentials and Ca²⁺ transients. The following sections describe application of the model to estimation of the degree of functional change in the NCX and SR Ca²⁺ ATPase in control and failing myocytes. The approach is as follows: (1) the time constant of Ca²⁺ decay (_Ca) measured with SR function blocked using CPA data is used to estimate the model Na⁺/Ca²⁺ exchange scale factor K_NaCa; (2) with K_NaCa fixed at this value, the model SR Ca²⁺ ATPase scale factor K_SR required to reproduce the _Ca measured in physiological solutions is determined; (3) the SR Ca²⁺ ATPase reduction in heart failure is cross-checked independently by determining the model SR Ca²⁺ ATPase scale factor required to reproduce the _Ca measured under Na⁺-free conditions (0-Na data) with the model Na⁺/Ca²⁺ exchange set to 0; and (4) the model Na⁺/Ca²⁺ exchange scale factor is estimated independently from _Ca in physiological solutions using the estimate of SR function determined in step 3.

Estimation of NCX and SR Ca²⁺ ATPase Activity in Normal and Failing Myocytes: CPA Experiments
In the preceding article by O'Rourke et al, Ca²⁺ transients in response to voltage-clamp stimuli were measured in the presence and absence of CPA, a blocker of the SR Ca²⁺ ATPase pump. In the presence of CPA, Ca²⁺ transient decay rate (_Ca) following termination of a depolarizing voltage step reflects the rate of extrusion of Ca²⁺ from the cytosol by the NCX (extrusion by the sarcolemmal Ca²⁺ ATPase is small). Estimates of the NCX pump current scale factor K_NaCa may therefore be obtained by setting the model value of K_SR to 0 and varying K_NaCa until model Ca²⁺ transient decay rates match those measured experimentally in the presence of CPA. K_NaCa may then be fixed at this value and K_SR varied until model Ca²⁺ transient decay rate matches that measured experimentally using physiological solutions. This procedure can be applied to data obtained from both normal and failing cells to assess the extent of functional upregulation and downregulation of the NCX and SR Ca²⁺ ATPase in heart failure.

To estimate K_NaCa, model K_SR was set to 0, 10 voltage-clamp steps (holding potential –97 mV, step potential 3 mV, and duration 200 ms) were applied at a frequency of 1 Hz to assure that Ca²⁺ levels in each model Ca²⁺ pool were equilibrated, and model _Ca was measured by fitting an exponential function to the decay phase of the final Ca²⁺ transient. Figure 2A plots model _Ca (ordinate, ms) as a function of K_NaCa (abscissa) with K_SR=0.0 (open triangles). K_NaCa=0.30 yields a _Ca equal to the average value measured experimentally in normal myocytes in the presence of CPA (813±269 ms). One SD of experimental variability is accounted for by K_NaCa values in the interval (0.21, 0.48). This same curve shows that K_NaCa=0.53 produces a _Ca matching that measured in failing myocytes in the presence of CPA (599±48 ms). One SD experimental variability is encompassed by K_NaCa values in the interval (0.48, 0.60). Assuming the normal value of K_NaCa to be 0.30, these data suggest a functional upregulation of the NCX in heart failure in the range of 60% to 100%, with average value 75%.

View larger version (11K): [in this window] [in a new window]   Figure 2. A, Model voltage-clamp Ca2+ transient relaxation time constant Ca as a function of NCX scale factor KNaCa at SR Ca2+ ATPase pump scale factors of KSR=0.0 (corresponding to CPA block), 0.51 (value estimated in 0-Na experiments for failing myocytes), and 1.0 (normal value). indicates KSR=0.0; , KSR=0.51; and , KSR=1.0. B, Model Ca2+ transient relaxation time constant Ca as a function of SR Ca2+ ATPase pump scale factor KSR for KNaCa values of 0.0 (0-Na conditions), 0.30 (value estimated in normal myocytes in presence of CPA), and 0.53 (value estimated for failing myocytes during CPA block). indicates KNaCa=0.0; , KNaCa=0.53; and , KNaCa=0.30.

Figure 2B plots model _Ca (ordinate, ms) as a function of K_SR (abscissa). The curve marked with open circles plots this dependence when K_NaCa is constant at the normal value estimated above (K_NaCa=0.30). The experimental value of _Ca measured in normal myocytes using physiological solutions is 219±36 ms. The maximum forward and reverse SR Ca²⁺ ATPase pump rates V_maxf and V_maxr given in Table 4 of the Appendix have been selected to yield a similar time constant when K_SR=1.0. Measured variation about this value is accounted for by K_SR values in the interval (0.85, 1.15).

The experimental value of _Ca measured in failing myocytes using physiological solutions is 292±23 ms. Dependence of model _Ca on K_SR when K_NaCa is fixed at the value estimated for failing canine myocytes (0.53) is shown by the curve labeled with open squares in Figure 2B. K_SR=0.38 yields a model _Ca equal to that observed experimentally. The experimental deviation of _Ca is accounted for by K_SR values in the interval (0.26, 0.51). Assuming the average value of K_SR in normal cells to be 1.0, these data suggest a functional downregulation of the SR Ca²⁺ ATPase pump in heart failure in the range of 49% to 74%, with average value 62%.

Estimation of NCX and SR Ca²⁺ ATPase Activity in Normal and Failing Myocytes: 0-Na Experiments
To provide a second, independent measure of altered Ca²⁺ handling protein expression in heart failure, O'Rourke et al²³ have measured _Ca in the presence and absence of Na⁺/Ca²⁺ exchange by removing Na⁺ ions from both intracellular and extracellular solutions. In the absence of Na⁺/Ca²⁺ exchange, _Ca reflects primarily the rate of Ca²⁺ uptake from the cytosol by the SR Ca²⁺ ATPase pump. Estimates of the SR Ca²⁺ ATPase pump rate scale factor K_SR under 0-Na conditions may therefore be obtained by setting the model value of K_NaCa to 0 and varying K_SR until simulated voltage-clamp Ca²⁺ transient decay rates match those measured experimentally. Once the model value of K_SR is constrained, K_NaCa can then be determined by changing its value until model Ca²⁺ transient decay rates match those measured experimentally using physiological solutions. This procedure can be applied to data obtained from both normal and failing cells to assess the extent of functional upregulation and downregulation of the NCX and SR Ca²⁺ ATPase in heart failure.

To mimic 0-Na conditions, K_NaCa was set equal to 0. K_SR was then varied, and the time constant for Ca²⁺ reuptake into SR was computed. Model _Ca values are plotted as a function of K_SR in Figure 2B (open triangles). Experimentally measured values of this time constant are 282±30 ms in normal and 576±83 ms in failing canine ventricular cells studied under 0-Na conditions. A K_SR value of 1.0 accounts for _Ca measured experimentally in normal cells (282±30 ms), and values in the interval (0.92, 1.07) account for the observed SD in these measurements. This estimate of the average K_SR value in normal myocytes based on block of the NCX agrees with that estimated using the CPA data. A K_SR value of 0.51 accounts for the average _Ca measured experimentally in failing cells (576±83 ms), and K_SR values in the interval (0.46, 0.59) account for the SD. Assuming the normal K_SR value to be 1.0, these data suggest a functional downregulation of the SR Ca²⁺ ATPase pump in failing myocytes in the range of 41% to 54%, with average value 49%. This estimate of SR Ca²⁺ ATPase downregulation is qualitatively similar to that obtained using CPA.

Dependence of model _Ca on K_NaCa when K_SR is fixed at the value estimated for normal canine myocytes (1.0) is shown by the curve labeled with open circles in Figure 2A. K_NaCa=0.22 yields a model _Ca equal to that observed experimentally using physiological solutions (219±36 ms). Experimental deviation of _Ca is accounted for by K_NaCa values in the interval (0.13, 0.43). Dependence of model _Ca on K_NaCa when K_SR is fixed at the value estimated for failing canine myocytes under 0-Na conditions (0.51) is shown by the curve labeled with open squares in Figure 2A. K_NaCa=0.35 yields a model _Ca equal to the average value observed experimentally using physiological solutions (292±23 ms). Experimental deviation of _Ca is accounted for by K_NaCa values in the interval (0.26, 0.46). Assuming the normal value of K_NaCa to be 0.22, these data suggest a functional upregulation of the NCX in heart failure in the range of 18% to 109%, with average value 38%. This estimate of altered expression of NCX in heart failure has greater variability than that obtained previously using the CPA data but is consistent in that it also indicates increased expression.

Parametric Dependence of Voltage-Clamp Ca²⁺ Transients on SR Ca²⁺ ATPase and NCX Levels
The above analyses provide estimates of K_SR and K_NaCa in normal and failing myocytes. Results indicate functional downregulation of the SR Ca²⁺ ATPase pump and upregulation of the NCX in heart failure. The parametric dependence of model cytosolic Ca²⁺ transients on K_SR and K_NaCa is examined next.

Model cytosolic Ca²⁺ concentration (ordinate, µmol/L) versus time (abscissa, seconds) is shown in Figure 3A as K_SR is varied. In these simulations, K_NaCa is constant at the value estimated using CPA data from normal cells (K_NaCa=0.30). K_SR is varied from 1.0 to 0.0 in steps of 0.1. Ca²⁺ transients are in response to a 1-Hz voltage-clamp stimulus (holding potential –97 mV, step potential 3 mV, and duration 200 ms). Response to the final stimulus of 10 stimulus cycles is shown, with the time origin translated to 0 seconds. These data show that reduction of the model SR Ca²⁺ ATPase pump, simulating the effects of downregulation of this pump in heart failure, reduces the amplitude of the early peak of the Ca²⁺ transient (marked by the arrow). This early peak disappears as K_SR approaches 0. Figure 3B shows JSR Ca²⁺ levels for each of the responses in Figure 3A. Reduction of the early peak in the data of Figure 3A coincides with depletion of JSR Ca²⁺ at small values of K_SR. Thus, the early peak in the model Ca²⁺ transient is generated by Ca²⁺ release from JSR, and the slow second peak, which is present even when JSR is depleted, results from influx of Ca²⁺ through sarcolemmal L-type Ca²⁺ channels and reverse-mode Na⁺/Ca²⁺ exchange. As K_SR decreases, Ca²⁺ levels in JSR decrease, and the Ca²⁺ transient becomes reduced in peak amplitude. The Ca²⁺ transient exhibits a decrease, no change, or an increase of amplitude during the course of the voltage-clamp stimulus, depending on the value of K_SR. Decay rate of the Ca²⁺ transient decreases with decreasing K_SR values, as shown in the data of Figure 3A, as well as Figure 2B (open circles).

View larger version (21K): [in this window] [in a new window]   Figure 3. A, Cytosolic Ca2+ concentration (µmol/L) as a function of time in response to a 10-second–duration periodic sequence of voltage-clamp stimuli with frequency of 1 Hz. Holding potential is –97 mV, and clamp potential is +3 mV, with duration of 200 ms. Only response to the final stimulus is shown. A family of responses is shown in which KNaCa is constant at 0.30 (value estimated in CPA experiments with normal canine myocytes), and KSR is varied from 1.0 to 0.0 in steps of 0.1. B, Model JSR Ca2+ concentration in response to the stimuli in panel A.

Figure 4A shows model cytosolic Ca²⁺ concentration (ordinate, µmol/L) versus time (abscissa, seconds) as K_NaCa is varied in steps of 0.5 from 0.5 to 2.5. A plot for K_NaCa=0.25 is also shown. K_SR is constant at the value estimated using CPA data from normal myocytes (K_SR=1.0). Voltage clamp steps from –97 mV to +3 mV with 200 ms duration are applied at a rate of 1.0 Hz. The final Ca²⁺ transient in a sequence of 10 is displayed, with the time origin translated to 0 seconds. There are 3 effects of increased K_NaCa. These are (1) increased rate of Ca²⁺ extrusion and lower diastolic Ca²⁺ at the holding potential, (2) reduction in Ca²⁺ transient amplitude in response to the +3 mV voltage-clamp step, and (3) "flattening" of the Ca²⁺ transient during the voltage-clamp step. The increased Ca²⁺ extrusion at the holding potential is a direct consequence of increased NCX activity when the exchanger is operating in the forward mode at the –97-mV holding potential, as shown in Figure 4B. This figure also shows that the NCX operates in reverse mode at the +3 mV clamp potential, thus generating Ca²⁺ influx. The reduction in Ca²⁺ transient amplitude in response to the voltage step is a consequence of the fact that total Ca²⁺ extrusion at the holding potential is greater than total Ca²⁺ influx at the step potential. This produces a smaller Ca²⁺ transient through reductions in SR Ca²⁺ loading and therefore a smaller Ca²⁺ release. The flattening of the Ca²⁺ transient with increased K_NaCa is a direct consequence of increased Ca²⁺ influx during the voltage step, as shown in Figure 4B. Decreased K_NaCa values also produce smaller Ca²⁺ transient decay rates, as seen by the data of Figure 4A, as well as Figure 2A (open circles).

Ca²⁺ Transients in Response to Voltage-Clamp Stimuli: Model Versus Experimental Results
Figure 5A shows model Ca²⁺ transients in response to a 1-Hz voltage-clamp pulse train. These transients were computed using K_SR and K_NaCa parameter values determined from the experimental series in the presence and absence of CPA. The solid line is the normal model Ca²⁺ transient (K_NaCa=0.30 and K_SR=1.0). The peak Ca²⁺ level (480 nmol/L) agrees well with the value measured experimentally in normal myocytes (450±75 nmol/L).²³ The dotted line is the model Ca²⁺ transient computed using the average K_NaCa (0.53) and K_SR (0.38) values for failing myocytes. The remaining 2 Ca²⁺ transients (dashed lines) correspond to K_Naca and K_SR values selected at ±1 SD from the average for failing myocytes. The short dashed line represents parameter choices producing a high degree of SR unloading (large NCX activity, K_NaCa=0.60; small SR Ca²⁺ ATPase activity, K_SR=0.26). The long-dashed line represents parameter choices that minimize SR unloading (small NCX activity, K_NaCa=0.48; large SR Ca²⁺ ATPase activity, K_SR=0.51). These data show that as K_NaCa is increased from a normal value of 0.30 (taking on values of 0.48, 0.53, and 0.60) and K_SR is decreased from the normal value of 1.0 (taking on values 0.51, 0.38, and 0.26), Ca²⁺ transient peak decreases monotonically from the normal value of 480 nmol/L, taking on values of 300, 266, and 230 nmol/L. These values agree well with the average experimental values measured in failing cells of 230±40 nmol/L.²³

Figure 5B shows a Ca²⁺ transient measured experimentally. The amplitude and waveform of the model predictions in Figure 5A are in close agreement with these experimental data.

Figure 5C shows a plot of the L-type Ca²⁺ current during the Ca²⁺ transients of Figure 5A. The parameter changes have relatively little effect on peak current, but increases in K_NaCa or decreases in K_SR produce a monotonic increase in the late component of the L-type Ca²⁺ current. As shown in Figure 5D, these same parameter changes also produce monotonic decreases of the subspace Ca²⁺ transient peak. Thus, the increase in the late component of L-type Ca²⁺ current seen in Figure 5C results from a decrease in Ca²⁺-mediated inactivation of this current due to reductions in magnitude of the subspace Ca²⁺ transient, which is in turn a consequence of reduced SR Ca²⁺ load. As can be appreciated by examining the magnitude of the change in L-type Ca²⁺ current density with alterations in Ca²⁺ handling, this late component of the L-type Ca²⁺ current would be expected to play an important role in determining the action potential plateau. This suggests that in heart failure, alterations in the expression of Ca²⁺ handling proteins that decrease SR Ca²⁺ load and reduce the amplitude of the Ca²⁺ transient may contribute substantially to prolongation of APD by reducing Ca²⁺-mediated inactivation of the L-type Ca²⁺ current.

	Discussion

Top Abstract Introduction Materials and Methods Results Discussion Appendix References

 
In this article, we present a model of the canine midmyocardial action potential and Ca²⁺ transient. The model is used to estimate the magnitude of SR Ca²⁺ ATPase pump rates and NCX current in normal and failing myocytes²³ using 2 methods. In the first method, model SR Ca²⁺ ATPase current is set to 0, and the NCX current is scaled to yield Ca²⁺ relaxation time constants in response to voltage-clamp stimuli matching those measured experimentally in normal and failing myocytes in the presence of CPA, a blocker of the SR Ca²⁺ ATPase. The extent of functional upregulation of the NCX in heart failure estimated using this approach is in the range of 60% to 100%, with average value 75%. Having constrained the model NCX current, model SR Ca²⁺ ATPase pump current is then estimated by matching the model Ca²⁺ relaxation rate to experimental data obtained in the absence of CPA. Comparison of model SR Ca²⁺ ATPase pump currents estimated for normal and failing myocytes suggests a functional downregulation in heart failure in the range of 49% to 74%, with average value 62%.

In the second method, model NCX current is set to 0, and the SR Ca²⁺ ATPase current is scaled to yield Ca²⁺ relaxation time constants matching those measured experimentally under 0-Na conditions. Functional downregulation of the SR Ca²⁺ ATPase current in heart failure estimated using this approach is in the range of 41% to 54%, with average value 49%. Having constrained the model SR Ca²⁺ ATPase current, NCX current is estimated by matching the model Ca²⁺ relaxation rate to experimental data obtained in control intracellular and extracellular sodium concentrations. Functional NCX upregulation in heart failure estimated using this approach is in the range of 18% to 109%, with average value 38%.

Analysis of protein levels in canine hearts subjected to the tachycardia pacing protocol reveal that both SR Ca²⁺ ATPase and phospholamban proteins are reduced on average by 28%²³ and that NCX protein is increased on average by 104%.²³ Both steady-state mRNA and expressed levels of E-C coupling proteins in failing human ventricular cells have been measured. The majority of reports agree that there is a 50% reduction of: (1) mRNA encoding the SR Ca²⁺ ATPase pump,^{12 13 14 15 16} (2) expressed SR Ca²⁺ ATPase protein level,^{12 17 18} and (3) direct SR Ca²⁺ ATPase uptake rate during heart failure.¹⁹ There is a 55% to 79% increase in Na-Ca exchanger mRNA levels,^{12 20} a 36% to 160% increase in expressed protein levels,^{12 20 21 22} and an approximate factor of 2 increase in Na⁺/Ca²⁺ exchange activity in human heart failure.²²

The model-based estimates of functional upregulation and downregulation of the NCX and SR Ca²⁺ ATPase pump reported here are consistent with these reports. Model estimates of average SR Ca²⁺ ATPase functional downregulation are 49% and 62%, depending on the estimation methods used. These values agree well with estimates of mRNA level, protein level, and SR Ca²⁺ ATPase uptake rate measured in human heart failure, but suggest a slightly larger degree of downregulation than indicated by measurements of protein level in canine tachycardia pacing-induced heart failure²³ (28%). Model estimates of average NCX upregulation are 38% and 75%. These estimates agree well with measured increases in mRNA levels in human heart failure and are within the range of variability of measured NCX protein levels in human heart failure. However, the model estimates are slightly lower than is suggested by the increased protein levels measured in the failing canine heart.²³

Ca²⁺ transients measured in failing human and canine ventricular myocytes exhibit reduced amplitude and slowed relaxation.^{5 40 41 42 43} Model simulations of Ca²⁺ transients in response to voltage-clamp stimuli reported here demonstrate that the altered expression of the NCX and SR Ca²⁺ ATPase pump measured in failing canine myocytes is sufficient to account for these properties. Both changes contribute to reduced SR Ca²⁺ load and release and therefore reduced amplitude of the early Ca²⁺ transient peak (Figures 3A and 4A). The shape of the Ca²⁺ transient is also controlled by both NCX and SR Ca²⁺ ATPase levels. As the Ca²⁺ ATPase pump is downregulated (Figure 3A), the shape of the plateau portion of the voltage-clamp Ca²⁺ transient changes from negative to 0, then to positive slope. This change in slope is produced by a decrease in early Ca²⁺ release from JSR, which in turn increases the dependence of Ca²⁺ transient shape on Ca²⁺ entry through the L-type Ca²⁺ channel. Upregulation of NCX also influences Ca²⁺ transient shape, tending to flatten the Ca²⁺ transient plateau by increasing reverse-mode Ca²⁺ entry at depolarized potentials (Figure 4A). The interplay between both of these factors accounts for the flattened Ca²⁺ transient shape seen in failing myocytes (Figure 1D, model; Figure 1B, experimental data).

Model Ca²⁺ transients in response to voltage-clamp stimuli exhibit a "knob" at the early peak of the transient (see Figure 3A, for example) that does not appear to be present in the experimental data. This knob disappears as the SR Ca²⁺ level becomes small (Figure 3A), indicating that the knob is dependent on SR Ca²⁺ release. The knob is likely an artifact of model construction. All SR Ca²⁺ release in this model occurs from a single functional unit, defined as a set of L-type Ca²⁺ channels, RyR channels, and the subspace within which they interact. Stern has referred to such models as common pool models.⁴⁴ The knob reflects a large, single Ca²⁺ release event from this single functional unit. In contrast, real cardiac cells have a large number of functional units in which there is local control of calcium-induced calcium release. We have recently implemented a local control model of Ca²⁺ release consisting of an ensemble of functional units, in which each functional unit is defined as an L-type Ca²⁺ channel interacting with a small set of RyR channels through a diadic space. Both L-type Ca²⁺ channels and RyR channels are modeled stochastically using the channel models presented in Jafri et al.²⁴ In such a model, the stochastic nature of RyR channel openings produces a variable latency of Ca²⁺ release in each functional unit. The Ca²⁺ transients computed using this model exhibit the property of gradedness and do not exhibit the knob seen in Figure 3A due to temporal smearing of Ca²⁺ release times.

A recent study has put forth the hypothesis that coupling between L-type Ca²⁺ channels and RyR channels may be altered in heart failure and that this altered coupling leads to a reduction in amplitude of the Ca²⁺ transient.⁴⁵ The results presented here cannot refute this hypothesis. Indeed, structurally detailed models of RyR channel and L-type Ca²⁺ channel interactions in the diadic space predict a strong dependence of these interactions on geometric factors.^{46 47 48} However, the results reported here indicate that such an assumption is not necessary to account for reduced amplitude of Ca²⁺ transients in failing myocytes. Rather, these simulations indicate that the altered expression of Ca²⁺ handling proteins reported by several different groups in both failing human and canine myocytes could account for changes in Ca²⁺ transient amplitude and shape.

The data of Figure 1 demonstrate that downregulation of the outward repolarizing currents I_K1 and I_to1, together with altered expression of the NCX and SR Ca²⁺ ATPase pump, can account for differences in both action potential and Ca²⁺ transient shape in heart failure. However, the data of Figure 1C also indicate that downregulation of I_K1 and I_to1, at least to the extent measured on average in failing cells, has a small effect on APD. Instead, altered expression of Ca²⁺ handling proteins plays a significant role in APD prolongation.

It is not surprising that downregulation of model I_K1 has only a modest impact on APD, as I_K1 is primarily responsible for the terminal phase of repolarization. However, the finding that reduction of model I_to1 has only a small effect on APD differs from the experimental results of Kääb et al⁵ in dog myocytes and of Beuckelmann et al⁹ in human cells. These experiments were performed using EGTA as an intracellular Ca²⁺ buffer. This buffering minimizes the modulatory effects of Ca²⁺ and thus enhances the relative influence of outward K currents on action potential characteristics. When effects of EGTA buffering are simulated in the model described in this article, block of I_to1 has a greater influence on APD. An example is shown in Figure 6. The Ca²⁺ buffering effects of EGTA were modeled using the fast buffering approximation developed by Wagner and Keizer,⁴⁹ with EGTA= 10 mmol/L and the dissociation constant K_m=0.15 µmol/L. Block of I_to1 by 95% increases APD₉₀ by 73 ms, or 25% of the control value. These results again emphasize the important modulatory role of Ca²⁺ on action potential characteristics in the canine myocyte.

View larger version (15K): [in this window] [in a new window]   Figure 6. Membrane potential as a function of time for the normal model (solid line) and for the model with a 95% reduction in magnitude of Ito1 (dotted line). Simulations are done in the presence of EGTA (10 mmol/L; Km=0.15 µmol/L), using the fast buffering approximation of Wagner and Keizer.49 Response to fifth stimulus at a cycle length of 1200 ms is shown.

It is also possible that 4-AP block of K currents other than I_to1 occurred in the Kääb et al⁵ experiments, but that such effects were not resolvable. Steady-state current-voltage relations were measured in the presence and absence of 4-AP to assess whether or not this was the case. Data are shown in Figure 10C of Kääb et al⁵ and indicate that experimental variability in steady-state current at 0 mV (a potential near that of the action potential plateau) is roughly ±1.0 pA/pF. The sum of model outward currents I_to1, I_K1, I_Kr, and I_Ks during the plateau is comparable with the magnitude of this variability in the experimental measurements (1.0 pA/pF). Genetic approaches for selective suppression of I_to1^50,51 may turn out to be more useful than pharmacological approaches in determining the influence of this current on APD.

The model predicts that one important mechanism of APD prolongation in heart failure is that shown in Figures 1 and 5. Under conditions of reduced SR Ca²⁺ release, there is less Ca²⁺-mediated inactivation of the L-type Ca²⁺ current. The resulting increase of inward current, as shown for voltage-clamp stimuli in Figure 5C and for action potentials in Figure 1E, helps to maintain and prolong the plateau phase of the action potential. Investigation into the relative contribution of the various Ca²⁺-regulatory mechanisms and Ca²⁺-dependent membrane currents in determining the action potential shape and duration is an important area for future experimental and modeling studies.

	Appendix

Top Abstract Introduction Materials and Methods Results Discussion Appendix References

 

View this table: [in this window] [in a new window]   Standard Units (Unless Otherwise Noted) Used in the Following Set of Equations

Membrane Currents
Na⁺ Current I_Na

(A1)

(A2)

(A3)

(A4)

(A5)

(A6)

(A7)

For V-40 mV,

(A8)

(A9)

(A10)

(A11)

For V<-40 mV,

(A12)

(A13)

(A14)

(A15)

Rapid-Activating Delayed Rectifier K⁺ Current I_Kr

(A16)

(A17)

(A18)

(A19)

(A20)

(A21)

(A22)

Slow-Activating Delayed Rectifier K⁺ Current I_Ks

(A23)

(A24)

(A25)

(A26)

(A27)

Transient Outward K⁺ Current I_to1

(A28)

(A29)

(A30)

(A31)

(A32)

(A33)

(A34)

Time-Independent K⁺ Current I_K1

(A35)

(A36)

Plateau K⁺ Current I_Kp

(A37)

(A38)

NCX Current I_NaCa

(A39)

Na⁺-K⁺ Pump Current I_NaK

(A40)

(A41)

(A42)

Sarcolemmal Ca²⁺ Pump Current I_p(Ca)

(A43)

Ca²⁺ Background Current I_Ca,b

(A44)

(A45)

Na⁺ Background Current I_Na,b

(A46)

Membrane Potential

(A47)

Ca²⁺ Handling Mechanisms
L-Type Ca²⁺ Current I_Ca

(A48)

(A49)

(A50)

(A51)

(A52)

(A53)

(A54)

(A55)

(A56)

(A57)

(A58)

(A59)

(A60)

(A61)

(A62)

(A63)

(A64)

(A65)

(A66)

(A67)

(A68)

(A69)

(A70)

(A71)

RyR Channel (Keizer and Levine)²⁷

(A72)

(A73)

(A74)

(A75)

(A76)

SERCA2a Pump (Shannon et al)³⁸

(A77)

(A78)

(A79)

Intracellular Ca²⁺ Fluxes

(A80)

(A81)

(A82)

(A83)

(A84)

Intracellular Ion Concentrations

(A85)

(A86)

(A87)

(A88)

(A89)

(A90)

(A91)

(A92)

(A93)

View this table: [in this window] [in a new window]   Table 1. Cell Geometry Parameters

View this table: [in this window] [in a new window]   Table 2. Standard Ionic Concentrations

View this table: [in this window] [in a new window]   Table 3. Membrane Current Parameters

View this table: [in this window] [in a new window]   Table 4. SR Parameters

View this table: [in this window] [in a new window]   Table 5. L-Type Ca2+ Channel Parameters

View this table: [in this window] [in a new window]   Table 6. Buffering Parameters

View this table: [in this window] [in a new window]   Table 7. State Variable Initial Conditions

	Acknowledgments

 
This research was supported by National Science Foundation grant BIR91-17874, National Institutes of Health grant HL60133, NIH Specialized Center of Research on Sudden Cardiac Death grant P50 HL52307, Silicon Graphics Inc, and the Whitaker Foundation.

	Footnotes

 
This manuscript was sent to Harry A. Fozzard, Consulting Editor, for review by expert referees, editorial decision, and final disposition.

Received April 27, 1998; accepted December 18, 1998.

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