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(Circulation Research. 1995;77:140-152.)
© 1995 American Heart Association, Inc.

Articles

Two Components of the Delayed Rectifier K⁺ Current in Ventricular Myocytes of the Guinea Pig Type

Theoretical Formulation and Their Role in Repolarization

Presented in part in abstract form at the 39th meeting of the Biophysical Society, San Francisco, Calif, February 12-16, 1995.

Jinglin Zeng, Kenneth R. Laurita, David S. Rosenbaum, Yoram Rudy

From the Cardiac Bioelectricity Research and Training Center, Department of Biomedical Engineering, Case Western Reserve University, Cleveland, Ohio.

	Abstract

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 Appendix 2 References

 
Abstract Two distinct delayed rectifier K⁺ currents, I_Kr and I_Ks, were found recently in ventricular cells. We formulated these currents theoretically and investigated their roles in action potential repolarization and the restitution of action potential duration (APD). The Luo-Rudy (L-R) model of the ventricular action potential was used in the simulations. The single delayed rectifier K⁺ current in the model was replaced by I_Kr and I_Ks. Our results show that I_Ks is the major outward current during the plateau repolarization. A specific block of either I_Kr or I_Ks can effectively prolong APD to the same degree. Therefore, either channel provides a target for class III antiarrhythmic drugs. In the simulated guinea pig ventricular cell, complete block of I_Kr does not result in early afterdepolarizations (EADs). In contrast, >80% block of I_Ks results in abnormal repolarization and EADs. This behavior reflects the high I_Ks-to-I_Kr density ratio (8:1) in this cell and can be reversed (ie, I_Kr block can cause EADs) by reducing the ratio of I_ks to I_Kr. The computed APD restitution curve is consistent with the experimental behavior, displaying fast APD variation at short diastolic intervals (DIs) and downward shift at longer DIs with the decrease of basic drive cycle length (BCL). Examining the ionic currents and their underlying kinetic processes, we found that activation of both I_Kr and I_Ks is the primary determinant of the APD restitution at shorter DIs, with Ca²⁺ current through L-type channels (I_Ca) playing a minor role. The rate of APD change depends on the relative densities of I_Kr and I_Ks; it increases when the I_Kr-to-I_Ks density ratio is large. The BCL-dependent shift of restitution at longer DIs is primarily attributed to long-lasting changes in [Ca²⁺]_i. This in turn causes different degrees of Ca²⁺-dependent inactivation of I_Ca and different degrees of Ca²⁺-dependent conductance of I_Ks at very long DIs (>5 s) for different BCLs. This BCL dependence of I_Ca and I_Ks that is secondary to long-lasting changes in [Ca²⁺]_i is responsible for APD changes at long DIs and can be viewed as a "memory property" of cardiac cells.

Key Words: K⁺ current • action potential duration • repolarization • simulation • optical mapping

	Introduction

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 Appendix 2 References

 
In ventricular myocytes, several types of K⁺ currents are involved in the repolarization of the action potential. These include the inward rectifier K⁺ current, I_K1; the delayed rectifier K⁺ current, I_K; the transient outward current, I_to; and perhaps the recently described plateau K⁺ current, I_Kp. I_K1 is known to be important during fast phase-3 repolarization of the action potential.^{1 2} I_K,^{3 4 5} I_to,⁶ and possibly I_Kp^{7 8} are thought to play an important role in late phase-2 (plateau) repolarization and in determining the shape and duration of the action potential.^{7 9} In guinea pig–type ventricular cells, I_K but not I_to is responsible for plateau repolarization.^{4 10} Recent studies show that I_K, with its slow activation characteristics, is composed of two distinct K⁺ component currents, I_Kr and I_Ks.^{10 11 12 13} I_Kr ("rapid") displays fast activation and is sensitive to the I_Kr channel blocker E-4031 or sotalol. I_Ks ("slow") is similar to the classically described I_K and is characterized by slower activation kinetics. The kinetics of these currents have been determined recently by Sanguinetti and Jurkiewicz^{10 13 14} and by Chinn¹² and Horie et al.¹¹ However, their respective roles and relative importance in plateau repolarization and in determining action potential duration (APD) and its restitution are not well understood.

Abrupt shortening of pacing cycle length may affect the shape and duration of the action potential.^{15 16 17 18} In guinea pig ventricular myocytes, APD is shortened¹⁵ when the pacing coupling interval is reduced. Upon increase of the coupling interval, APD is restored, a phenomenon known as APD restitution. APD restitution is thought to contribute to the development of arrhythmias by affecting the recovery of excitability and creating conditions that favor induction of reentry.¹⁶ The sensitivity of APD to pacing rate is believed to be related to slow time-dependent activation and inactivation of membrane ionic currents.^{15 16} A widely accepted hypothesis is that I_K activation plays a major role in APD restitution.

The objectives of the present study are (1) to formulate I_Kr and I_Ks and incorporate them into the recent Luo-Rudy (L-R) model^{1 3 19} of the ventricular action potential, (2) to update the L-R model by adjusting I_Kp on the basis of recent experimental data⁸ and by adding the T-type Ca²⁺ current, I_Ca(T),^{20 21} (3) to construct a theoretical APD restitution curve based on the updated L-R model and to compare the theoretical behavior to that obtained experimentally by use of optical (voltage-sensitive dye) recordings, and (4) to investigate the contributions of I_Kr and I_Ks to the repolarization of the action potential and to APD restitution. A clear understanding of the repolarization process and of APD dependence on rate is important to the understanding of arrhythmogenic phenomena and their mechanisms. This is true not only for single-cell behavior but also for propagation of premature action potentials where excitation and repolarization interact (eg, head-tail interaction during reentry). In addition to investigating the relative importance of I_Kr and I_Ks to action potential repolarization and to rate-dependent changes in APD, we provide a theoretical model of these single-cell processes that can be incorporated into models of propagation and arrhythmias in cardiac tissue.

	Materials and Methods

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 Appendix 2 References

 
Simulations
The theoretical model of the ventricular action potential developed by Luo and Rudy (the L-R model)^{1 3 19} provides the basis for the theoretical simulations in this article. In this model, the guinea pig–type ventricular action potential is numerically constructed on the basis of experimental data. Included in the model are the membrane ionic channel currents, represented mathematically by Hodgkin-Huxley–type formalism, as well as ionic pumps and exchangers. In addition, the model accounts for processes that regulate ionic concentration changes, especially dynamic changes of [Ca²⁺]_i. Intracellular processes include Ca²⁺ uptake and Ca²⁺ release by the sarcoplasmic reticulum (SR) as well as Ca²⁺ buffering. The SR release can be triggered in two ways: externally (by a sufficiently fast increase in total myoplasmic Ca²⁺, a process known as Ca²⁺-induced Ca²⁺ release [CICR]) or internally (by overloading the SR with calcium above a threshold level [spontaneous release]). In the model, CICR occurs if total [Ca²⁺]_i (ie, free plus buffered) increases by >0.18 µmol/L in the first 2 ms of the action potential. A property of the L-R model that is important to the present study is that the L-type Ca²⁺ current is inactivated by both voltage-dependent and Ca²⁺-dependent processes (f gate and f_Ca gate, respectively). The reader is referred to Reference 3³ for a detailed description of the model and a list of equations governing the model behavior.

The L-R model is updated here to include the slow and fast components of the delayed rectifier K⁺ current, I_Kr and I_Ks. I_Ca(T) is added in the model. I_Kp is adjusted on the basis of recent experimental data. Relevant equations are in Appendix 1 and Appendix 2, and details are provided below. A schematic diagram of the updated model is provided in Fig 1.

View larger version (45K): [in this window] [in a new window]   Figure 1. Schematic diagram shows the updated Luo-Rudy (L-R) single ventricular cell model. INa indicates fast Na+ current; ICa(L), Ca2+ current through L-type Ca2+ channels; ICa(T), Ca2+ current through T-type Ca2+ channels; IKr, fast component of the delayed rectifier K+ current; IKs, slow component of the delayed rectifier K+ current; IK1, inward rectifier K+ current; IKp, plateau K+ current; INaK, Na+-K+ pump current; INaCa, Na+-Ca2+ exchange current; Ip(Ca), Ca2+ pump in the sarcolemma; INa,b, Na+ background current; ICa,b, Ca2+ background current; Ins(Ca), nonspecific Ca2+-activated current; Iup, Ca2+ uptake from myoplasm to network sarcoplasmic reticulum (NSR); Irel, Ca2+ release from junctional sarcoplasmic reticulum (JSR); Ileak, Ca2+ leakage from NSR to myoplasm; and Itr, Ca2+ translocation from NSR to JSR. For details, see Reference 3.

Two Components of I_K: I_Kr and I_Ks
I_K is one of the major currents in ventricular myocytes of certain species such as the guinea pig. It has been traditionally characterized as a single type of channel current.^{4 22 23 24} Recently, Sanguinetti and Jurkiewicz¹⁰ found and Chinn¹² further showed that two components of I_K, I_Kr and I_Ks, coexist in guinea pig ventricular cells. Two components of I_K are also found in atrial cells of the same species.^{11 13}

I_Kr exhibits rapid activation and prominent inward rectification.^{10 11 13} It is blocked by E-4031 or sotalol.^{10 11 12 13} The formulation of I_Kr incorporates both a time-dependent activation gate, X_r, and a time-independent inactivation gate, R, to approximate the very fast inactivation process of this channel. The inclusion of R in the formulation introduces the inward-rectification property of I_Kr. In the Hodgkin-Huxley–type formalism, I_Kr can be expressed as

(1)

where V is the membrane potential, E_Kr is the reversal potential, and _Kr is the maximum conductance of I_Kr. Experimental studies show that lowering [K⁺]_o decreases I_Kr.¹⁴ Considering the increase of the driving force when [K⁺]_o is reduced, we would expect I_Kr to increase when [K⁺]_o is lowered. From experimental results, we deduce that I_Kr conductance decreases at lower [K⁺]_o. Following the formulation of I_K in the L-R model³ and the experimental conductance measurements,¹⁰ we introduce a square root dependence of _Kr on [K⁺]_o and express it as

(2)

I_Kr is purely selective to K⁺ ions. Hence, E_Kr is set to be the equilibrium potential of K⁺ ions across the cell membrane. We use the formulation of X_r steady state activation, X_r, from Sanguinetti and Jurkiewicz¹⁰ and formulate the time constant of activation, _Xr, to fit their measurements. We also adopt the formulation of R from the same article. Introduction of R in the model accounts for the "hook" phenomenon that is observed in deactivating tail currents of I_Kr.^{25 26} In the model, the very fast inactivation is approximated by a time-independent process (R gate). Therefore, the increase in tail current upon repolarization that generates the hook appearance is instantaneous in our simulations. In the experiments, a very short time delay is observed.^{25 26} Equations are provided in Appendix 2.

I_Ks is the slow component of I_K with characteristics similar to the classically described I_K. This current shows no inward rectification.^{10 14} Upon depolarization, the activation of I_Ks may follow a sigmoidal time course,^{4 10} not a single exponential function. The second power of activation in the Hodgkin-Huxley formalism provides an adequate fit to the measured traces.^{3 4 10} Therefore, we express I_Ks as

(3)

where _Ks is the maximum conductance, X_s is the activation gate, and E_Ks is the reversal potential. Neither inward rectification nor inactivation of I_Ks is observed.¹⁰ Hence, there is no time-independent inactivation gate in Equation 3.

It has been found that lowering [K⁺]_o from 4 mmol/L to 0 mmol/L increases I_Ks.¹⁰ This phenomenon may result from a decrease of E_Ks and an increase of the driving force. Dependence of _Ks on [K⁺]_o has not been observed. We assume _Ks to be independent of [K⁺]_o. We set _Ks to be in the range of measured values^{24 27} and verify that this value is consistent with the 24% increase of APD observed by Sanguinetti and Jurkiewicz¹⁰ when I_Kr is blocked by 3 µmol/L E-4031. We also verify this value on the basis of the experimental observation that the fully activated tail current of I_K (I_Kr+I_Ks) was 11.4 times larger than the fully activated I_Kr upon repolarization to -40 mV. I_Ks has been found to be sensitive to [Ca²⁺]_i.¹⁰ On the basis of the results of studies by Tohse,²⁷ we introduce [Ca²⁺]_i dependence of _Ks. _Ks is larger at higher [Ca²⁺]_i. At [Ca²⁺]_i of 0.12 µmol/L and [K⁺]_o of 4 mmol/L, _Ks is 0.1737 millisiemens (mS)/µF, _Kr is 0.0225 mS/µF, and the ratio of _Ks to _Kr is 7.72:1. Formulation of _Ks is given in Appendix 2.

I_Ks, unlike I_Kr, is not purely selective to K⁺ ions. The formulation of its reversal potential, E_Ks, follows that of I_K in the L-R model.³

Formulation of X_s is given in Appendix 2. Steady state X_s is based on the data provided by Sanguinetti and Jurkiewicz,¹⁰ and the time constant of X_s is based on data from Matsuura et al.⁴

It should be mentioned that I_Kr and I_Ks, as formulated here, duplicate the experimental current-voltage relations in the presence or absence of I_Kr block (ie, before and after exposure to E-4031; see Fig 6 of Reference 10¹⁰ ).

View larger version (22K): [in this window] [in a new window]   Figure 6. Graphs show comparisons of the S2 action potential and underlying processes at two diastolic intervals (DIs) of 20 and 100 ms. A, Action potential. B, L-type Ca2+ current (ICa). C, Voltage-dependent inactivation (f) and Ca2+-dependent inactivation (fCa). Protocol is the same as in Fig 5A.

I_Kp, the Plateau K⁺ Current
Formulation of I_Kp in the L-R model³ was based on measurements from single channel recordings⁷ and fitting of the total time-independent current. Recently Backx and Marban⁸ studied I_Kp by using the whole-cell patch-clamp protocol. On the basis of their measurements, the maximum conductance of I_Kp, _Kp, is adjusted from 0.0183 mS/µF in the L-R model³ to 0.00552 mS/µF in the updated version of the L-R model presented here.

I_Ca(T), the T-Type Ca²⁺ Current
The T-type Ca²⁺ channel, also called the low-threshold Ca²⁺ channel, activates at potentials ranging from -50 mV to -30 mV and displays fast inactivation.^{20 21} Its role in the cardiac action potential is still unclear. In the L-R model,³ this channel was not included. We add I_Ca(T) to the model to formulate a more complete theoretical model of the ventricular action potential. I_Ca(T) is formulated as

(4)

where _Ca(T) is the maximum conductance; E_Ca is the reversal potential and equals the equilibrium potential of Ca²⁺ ions across the cell membrane; and b and g are the activation and inactivation gates, respectively. Their formulation is based on the experimental data of Droogmans and Nilius²⁰ and is provided in Appendix 2. We simulate the action potential at 4 mmol/L [K⁺]_o and the behavior of I_Ca(T) during the action potential (not shown). I_Ca(T) displays fast activation and inactivation. It attains a maximum inward magnitude of 1.05 µA/µF 8.7 ms from the time of [dV/dt]_max. A comparison of simulated action potentials with and without I_Ca(T) shows that the addition of I_Ca(T) to the model has a minimal effect on the shape and duration of the action potential.

Experimental Methods
The theoretical APD restitution curve, computed by use of the updated version of the L-R model, is compared with an experimental restitution curve obtained by use of optical recordings of cardiac action potentials. A short description of the experimental methodology follows (see Reference 28²⁸ for details).

Guinea pig hearts were perfused in Tyrode's solution containing (mmol/L) NaCl 130, NaHCO₃ 12.5, MgSO₄ 1.2, KCl 4.75, dextrose 5.0, and CaCl₂ 1.25 (pH 7.40). The right atrium was excised to avoid competitive stimulation from the sinoatrial node. The heart was immersed in coronary effluent draining into the chamber and maintained at a constant temperature (31°C to 32°C). Action potentials were measured by an optical action potential mapping system with high spatial resolution (0.1 mm to 1.0 mm between each recording site) and high temporal resolution (0.5 ms) and a high signal-to-noise ratio. In this system, light fluoresced from membrane-bound voltage-sensitive dye (di-4ANEPP) was recorded to measure the membrane potential variation. Action potential recordings were limited to ventricular epicardial cells.

The ventricular epicardial surface was stimulated at a baseline cycle length of 400 ms until stable recordings were observed. Restitution measurements were made from 128 ventricular recording sites simultaneously by introducing an extrastimulus following a 50-beat drive train at the basic drive cycle length (BCL) of 400 ms. Representative action potentials recorded from sites having similar APDs during BCL pacing were analyzed in these studies. In our experiments and simulations, APD was defined as the interval between the point of the maximum positive derivative of membrane potential during the upstroke, [dV/dt]_max, and the point of the maximum positive curvature during repolarization. Diastolic interval (DI) was defined as the interval between the point of the maximum positive curvature during repolarization and [dV/dt]_max of the following action potential.

	Results

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 Appendix 2 References

 
Roles of the Different K⁺ Currents in Repolarization
The important outward currents responsible for repolarization of the cardiac action potential are carried by K⁺ ions. Unlike the other ionic currents, K⁺ currents are diverse. Under normal (nonischemic, no Ca²⁺ overload) conditions, the updated L-R model includes I_K1, I_Kr, I_Ks, and I_Kp. In the following simulations, we investigate the roles of I_Kr and I_Ks during the different phases of the action potential and compare these currents with I_K1 and I_Kp.

Fig 2 depicts the behavior of I_Kr and I_Ks during an action potential. The fast component, I_Kr, increases faster than I_Ks at the very beginning of the action potential. However, it does not attain a large magnitude because of its instantaneous inward rectification. During the slow repolarization of the plateau, I_Kr increases in magnitude because of the decreased inward rectification at less positive potentials. Compared with I_Kr, I_Ks attains a much larger magnitude during the plateau, reflecting its larger conductance. The dotted curve in Fig 2B is the sum of I_Kr and I_Ks. Its behavior is similar to the behavior of I_K in the original L-R model (Fig 13 in Reference 3³ ). Another K⁺ current that activates during the plateau is I_Kp. The simulations (Fig 2C) show that I_Kp is larger than I_Ks only at the early phase of the plateau but is much smaller than I_Ks during most of the plateau. Therefore, we conclude that I_Ks is the major repolarizing current during the plateau phase.

View larger version (19K): [in this window] [in a new window]   Figure 2. Graphs show results seen when the cell is paced 40 times at 0.2 Hz. A, Action potential. B, Fast component of delayed rectifier K+ current (IKr), slow component of delayed rectifier K+ current (IKs), and IKr+IKs. C, Comparison of IKs and plateau K+ current (IKp). D, Comparison of IKr and inward rectifier K+ current (IK1). Traces are for the 40th action potential time course. Note crossover of IKr and IKs curves at beginning and end of the action potential in panel B.

During most of phase-3 repolarization of the action potential, the I_Kr and I_Ks curves cross over and I_Kr is larger than I_Ks, especially near the end of repolarization (Fig 2B). This reflects the different driving forces of these currents. I_Kr is purely selective to K⁺ ions. Its reversal potential is the E_Kr (-95.86 mV at a [K⁺]_o of 4 mmol/L), which is more negative than the resting potential. I_Ks is carried by K⁺ ions as well as Na⁺ ions. At a [K⁺]_o of 4 mmol/L, E_Ks is -83.26 mV. Therefore, during late repolarization the larger driving force of I_Kr results in a larger outward magnitude than I_Ks. The important role of I_K1 in fast repolarization during late phase 3 is well established. In Fig 2D, I_Kr is compared with I_K1. Although I_Kr is larger than I_Ks during the late repolarization phase, its magnitude is much smaller than that of I_K1. We conclude that even when I_Kr is considered, I_K1 is still the major contributor to fast repolarization during late phase 3.

Spatial nonuniformities of APD are known to exist in cardiac tissue. Because the K⁺ currents are the major determinants of repolarization, it is interesting to examine how changes in their conductances affect APD. Fig 3A shows that complete block of I_Kp or 100% increase of its maximum conductance has little effect on APD. APD at full repolarization (APD₁₀₀) increases by only 1% or decreases by only 0.03%, respectively, as a result of these conductance changes. Block of I_K1 has little effect on early repolarization but reduces significantly the rate of repolarization at the end of the action potential (Fig 3B). An 80% block of I_K1 results in an increase of only 4.3% in APD at 50% repolarization but in a 23% increase in APD₁₀₀. A 100% increase of I_K1 maximum conductance accelerates phase-3 repolarization and shortens APD₁₀₀ by 3.7%.

View larger version (25K): [in this window] [in a new window]   Figure 3. A, Graph shows action potentials under control conditions, when plateau K+ current (IKp) is completely blocked (100% block), and when the maximum conductance of IKp is doubled (100% increase). B, Graph shows action potentials under control conditions, under 80% block of inward rectifier K+ current (IK1), and when IK1 is doubled.

As mentioned previously in "Materials and Methods," the value of the maximum I_Ks conductance in the model is consistent with the observation of Sanguinetti and Jurkiewicz¹⁰ that APD increased by 24% when the cell was exposed to 3 µmol/L of the I_Kr channel blocker E-4031. Fig 4A shows how the action potential shape and duration are affected when the maximum conductance of I_Kr is either reduced or increased. APD₁₀₀ at 100% block of I_Kr increases by 24%, and at 100% increase of _Kr, it decreases by 14%. Fig 4B shows the results of similar protocols for modulation of I_Ks. With <80% block of I_Ks, repolarization of the action potential is similar to that under control conditions, but APD is prolonged. At 50% decrease of _Ks, APD₁₀₀ is increased by 23.4%. When _Ks is decreased by >80%, the action potential cannot repolarize normally, and early afterdepolarizations (EADs) are observed. A 100% increase of _Ks results in an 18% decrease of APD₁₀₀. As shown in Fig 4C, if both _Kr and _Ks are reduced by 20%, 40%, and 50%, APD₁₀₀ is increased by 11.2%, 30.9%, and 50.3%, respectively. The action potential repolarizes adequately under these partial blocks of I_Kr and I_Ks, and effective prolongation of the plateau is obtained. However, if the degree of block of both currents exceeds 55%, eg, 60% as in Fig 4C, EAD develops and the membrane cannot repolarize normally.

View larger version (30K): [in this window] [in a new window]   Figure 4. A, Graph shows action potentials under control conditions (bold line) and when the maximum conductance of the fast component of the delayed rectifier K+ current (IKr) is reduced by 25%, 50%, 75%, or 100% or increased by 100%. B, Graph shows action potentials under control conditions (bold line) and when the maximum conductance of the slow component of the delayed rectifier K+ current (IKs) is reduced by 25%, 50%, or 90% or increased by 100%. C, Graph shows action potentials under control conditions (bold line) and when the maximum conductances of both IKs and IKr are simultaneously reduced by 20%, 40%, 50%, or 60%.

Specific block of I_Kr is found to result in EADs in various preparations.^{29 30 31} However, EADs are not generated by I_Kr block in guinea pig ventricular cells¹⁰ or in our simulations for an I_Kr-to-I_Ks ratio and I_Kr and I_Ks conductances that are typical of the guinea pig ventricular cell. However, if we increase I_Kr by 50% and decrease I_Ks by 30%, EADs develop when I_Kr is completely blocked.

Restitution of APD
It is well established that APD of guinea pig–type ventricular cells is shortened when the pacing rate is increased or when the coupling interval of a premature beat is shortened.¹⁵ A prolongation of the coupling interval results in an increase in APD (APD restitution). Several processes have been proposed as being responsible for the observed APD shortening. These include K⁺ accumulation in extracellular clefts, activation of K⁺ currents, Na⁺ window current inactivation, and Ca²⁺ current inactivation.¹⁵ In this article we use the single ventricular cell model to study this phenomenon and its underlying mechanism. This approach offers an opportunity to exclude a priori a factor such as K⁺ accumulation in the extracellular space because no interstitial clefts are associated with a single cell. We can therefore focus on the effects of transmembrane currents and their activation and inactivation kinetics on APD restitution.

To study the mechanism of APD restitution, we stimulate the cell 39 times at a constant pacing rate and then apply an additional stimulus (S2) at various DIs. Fig 5A depicts the 39th paced action potential (S1) and the S2 action potential when the cell is paced at a BCL of 400 ms. Numbers in Fig 5A indicate DIs. APD increases when DI is prolonged. APD restitution curves for BCLs of 300, 400, 600, and 1000 ms are shown in Fig 5B. APD increases sharply until DI reaches 100 ms (inset of Fig 5B). It then continues to increase slowly and saturates to a steady state APD. A measurable notch was observed in the APD restitution curve for a BCL of 300 ms (bold arrow in Fig 5B). The APD values at very long DIs (>5000 ms) are still different at different BCLs. A downward displacement of the restitution curves at fast BCLs was observed. In the following simulations, we first investigated the mechanism of the sharp variation of APD at short DIs of <100 ms and then the mechanism underlying the differences of APD at very long DIs.

View larger version (28K): [in this window] [in a new window]   Figure 5. Graphs show action potential duration (APD) restitution. The cell is initially paced 39 times at a constant basic drive cycle length (BCL), and then an additional stimulus (S2) is applied after a variable pause (diastolic interval [DI], defined as the interval between the end of repolarization of the last paced action potential and the S2 stimulus). A, Last paced action potential (AP) and the S2 APs for four different DIs (4.5, 30, 130, and 230 ms). The cell is initially paced at a BCL of 400 ms. B, APD at full repolarization plotted as a function of DI to create APD restitution curves. Four restitution curves are shown for four different initial pacing cycle lengths of 300, 400, 600, and 1000 ms. The inset expands the range of shorter DIs (<200 ms). C, Myocytes are paced at a BCL of 400 ms in both experiments and simulations. Normalized APD restitutions at DIs of <200 ms are compared. indicates experimental recordings; , simulation results.

In Fig 5C, we compare the theoretical APD restitution in Fig 5B with an experimental restitution curve that we obtained by using optical action potential recordings (see "Materials and Methods"). Both curves were obtained for a basic pacing cycle length of 400 ms and are shown for the DI range of fast APD changes (DI <200 ms). The theoretical and experimental restitution curves correspond very closely. If APD restitution is fitted by a single exponential function, the time constant of the theoretical APD restitution curve is 46.4 ms, and the time constant of the experimental APD restitution is 41.4 ms in Fig 5C. Restitution time constants in our experiments fall in the range of 15 to 42 ms (data from five animals). Time constants measured by others,¹⁵ also at 37°C, are in the range of 35 to 65 ms.

To investigate the mechanism of the fast variation of APD at DIs of <100 ms, we compare selected transmembrane currents and their channel kinetics during the S2 action potential at DIs of 20 and 100 ms. The cell is initially paced at a BCL of 400 ms.

First, we compare the L-type Ca²⁺ current (I_Ca) at DIs of 20 and 100 ms (Fig 6). Just before the application of S2 stimuli, both voltage-dependent (f) and Ca²⁺-dependent (f_Ca) inactivation show less recovery from the inactivation caused by the previous action potential at a DI of 20 ms (bold arrow in Fig 6C). This would favor the shortening of the S2 APD at a DI of 20 ms. However, as shown in Fig 6A, the plateau potential at a DI of 20 ms is less positive, which results in a larger driving force of I_Ca at a DI of 20 ms than at a DI of 100 ms. The larger driving force compensates for the greater inactivation of the current. As a result, I_Ca attains a similar magnitude at both DIs (Fig 6B). Hence, the inactivation of the L-type Ca²⁺ channel alone cannot account for the fast shortening of APD at smaller DIs. The similar I_Ca magnitude at both DIs also suggests that the inactivation of this channel cannot play a dominant role in the shortenings of APD at smaller DIs.

Comparing the two components of I_K (Fig 7), we observe that both I_Kr and I_Ks are larger at the smaller DI of 20 ms. The further increase due to the longer APD at the larger DI should not be considered in the comparison. The larger outward I_Kr and I_Ks act to shorten the APD at smaller DIs. Our previous simulations (Fig 2) show that I_Kr and I_Ks are the major repolarizing currents during the plateau of an action potential. Their different magnitudes at DIs of 20 and 100 ms during the early repolarization phase of the action potential indicate that they play an important role in APD restitution at the range of smaller DIs.

View larger version (30K): [in this window] [in a new window]   Figure 7. Graphs show comparisons of the S2 action potential and underlying processes at two diastolic intervals (DIs), 20 (dotted line) and 100 ms (solid line). A, Action potential. B, Fast component of delayed rectifier K+ current (IKr). C, Slow component of delayed rectifier K+ current (IKs). D, Activation gate of IKr (Xr). E, Activation gate of IKs (Xs). F, [Ca2+]i-dependent maximum conductance of IKs (Ks). Protocol is the same as in Fig 5A.

Both I_Kr and I_Ks are characterized by relatively long time constants. As shown in Fig 7D, 7E, and 7F, at both DIs the activation gates of both I_Kr (X_r gate) and I_Ks (X_s gate) are still partially activated just before the S2 stimulus. There is a smaller degree of deactivation at the shorter DI (20 ms), which results in the larger I_Kr and I_Ks at this DI and in greater shortening of APD. As shown in Fig 5B, the rate of APD change is greater at smaller DIs. This is caused by the relatively fast time constant of X_r compared with that of X_s. As shown in Fig 7D, at a DI of 100 ms X_r is already close to complete deactivation just before the S2 stimulus. Hence, I_Kr has a smaller effect on APD restitution in the range of longer DIs. For the short DI (20 ms), both X_r and X_s are partially activated so that both I_Kr and I_Ks affect the APD and the restitution curve is steeper. It is known that the conductance of I_Ks depends on free [Ca²⁺]_i. The conductance is higher for a larger [Ca²⁺]_i. In the model, _Ks is dependent on [Ca²⁺]_i. At both values of DI, [Ca²⁺]_i before the S2 stimulus is high, which results in a greater degree of Ca²⁺-dependent inactivation of I_Ca and decreased Ca²⁺ entry into the cell at the initial phase of the S2 action potential. Reduced Ca²⁺ entry implies little or no release of Ca²⁺ by the SR through the CICR process. In the simulations, SR Ca²⁺release is not observed during the S2 action potential for either a DI of 20 ms or a DI of 100 ms (not shown). This is consistent with the observation³² that cell contraction is absent or smaller for early premature stimuli. In the absence of SR release, an intracellular Ca²⁺ transient is not generated and [Ca²⁺]_i is similar for both DI values. Because _Ks depends only on [Ca²⁺]_i, its value is also similar at the two DI values (Fig 7F). We conclude that the Ca²⁺ dependence of _Ks plays a negligible role in APD restitution at the range of fast APD change (DI <200 ms).

As demonstrated in Fig 5B (bold arrow), a notch is present in the APD restitution curve for a BCL of 300 ms. A similar biphasic behavior of restitution has also been observed experimentally.^{18 33 34 35 36 37} However, the mechanism underlying this phenomenon remains unclear. Using the model, we attempted to identify the processes that cause the biphasic behavior (notch) of the restitution curve. Inspection of the restitution curve for a BCL of 300 ms (Fig 5B) reveals that APD increases monotonically with DI until a DI of 102 ms is reached. Further increase of DI causes APD to decrease, reaching a minimum (the notch) at a DI of 132 ms. Our simulations show that Ca²⁺ release from the SR begins at a DI of 102 ms. For a shorter DI, Ca²⁺ entry is not sufficient to cause CICR, mostly because of incomplete recovery of I_Ca. In Fig 8, membrane potential, intracellular Ca²⁺ transient, I_Ca, and f_Ca are compared at a DI of 102 ms (solid line) and a DI of 132 ms (dashed line). APD is shorter at the larger DI of 132 ms (Fig 8A). The intracellular Ca²⁺ transient is larger at a DI of 132 ms (Fig 8B), causing a greater degree of Ca²⁺-dependent inactivation (a smaller f_Ca; Fig 8D) and, as a result, a smaller I_Ca during the plateau (Fig 8C). The smaller I_Ca results in a smaller APD at a DI of 132 ms, overcoming the prolongation effect of the decreases in I_Kr and I_Ks at the longer DI (Fig 7). With continued decrease of the K⁺ currents as DI is further increased beyond 132 ms, APD increases again and the notch is formed.

View larger version (21K): [in this window] [in a new window]   Figure 8. Graphs show results seen when the cell is paced for 39 times at a cycle length of 300 ms and a premature stimulus is applied. Comparisons during the extra premature stimuli at diastolic intervals (DIs) of 102 and 132 ms are shown for action potential (A), [Ca2+]i (B), L-type Ca2+ current (ICa) (C), and Ca2+-dependent inactivation gate (fCa) (D).

As shown in Fig 5B, restitution curves are different for different BCLs. There is a downward displacement of the restitution for faster basic pacing rates. Even at a DI of 5000 ms, where APD has reached a flat portion of the curve for quite some time, APDs are different for different pacing rates. This theoretical result is consistent with the experimental results observed by Bjornstad et al.¹⁵ The underlying mechanism of this behavior has not yet been clearly elucidated. One hypothesis is that extracellular accumulation of K⁺ at faster pacing rates and the [K⁺]_o dependence of I_K provide the mechanism. However, this phenomenon has also been observed in isolated cell preparations³⁷ and in our simulations using the single-cell model. For both situations, significant extracellular accumulation of K⁺ cannot occur. In isolated cell preparations, extracellular cleft space is very small. In our simulations, an extracellular cleft space is not present. Therefore, accumulation of K⁺ cannot explain the phenomenon exclusively.

The simulations of Figs 9 and 10 are aimed at elucidating the underlying mechanism of the downward shift of APD restitution in single cells. The cell is paced 39 times at a BCL of 300 or 600 ms, and an S2 stimulus is applied after a 5-s pause. The S2 action potentials at the two cycle lengths are compared in Fig 9A. APD is shorter at a DI of 300 ms. After the cell remains at diastolic potential for 5 s, all time- and voltage-dependent gates resume their steady state values that are not dependent on the basic pacing rate. The only processes that are sufficiently slow and do not reach a steady state after such a long time involve regulation of intracellular ionic concentrations. Ca²⁺ may accumulate intracellularly during fast pacing. Therefore, we examined the diastolic [Ca²⁺]_i 5 s after the last paced beat at the two different pacing rates. At a fast pacing rate (BCL of 300 ms), diastolic free [Ca²⁺]_i after the 5-s pause is larger than that at the slow pacing rate (BCL of 600 ms) (Fig 9B, arrow). This suggests that the total amount of Ca²⁺ stored in the cell is greater at the fast pacing rate even after 5 s. This explains the larger intracellular Ca²⁺ transient at a BCL of 300 ms than at a BCL of 600 ms (Fig 9B).

View larger version (20K): [in this window] [in a new window]   Figure 9. Graphs show results seen when the cell is initially paced for 39 times at basic drive cycle lengths (CLs) of 300 and 600 ms. After a 5-s pause at diastolic potential, a stimulus (S2) is applied. A, Comparison of S2 action potentials at basic CLs of 300 and 600 ms. B, Comparison of [Ca2+]i at basic CLs of 300 and 600 ms. Before the S2 stimulus, diastolic [Ca2+]i is higher at the basic CL of 300 ms (arrow). C, Comparison of L-type Ca2+ current (ICa) at basic CLs of 300 and 600 ms. ICa plateau magnitude is smaller for the protocol with the basic CL of 300 ms. D, Comparison of Ca2+-dependent inactivation gate of ICa (fCa) at basic CLs of 300 and 600 ms. There is greater Ca2+-dependent inactivation for the protocol with the basic CL of 300 ms.

View larger version (17K): [in this window] [in a new window]   Figure 10. Graphs show comparisons, with basic drive cycle lengths (CLs) of 300 and 600 ms for slow delayed rectifier K+ current (IKs) (A), maximum conductance of IKs (Ks) (B), and activation gate of IKs (Xs) (C). Simulation protocol is the same as in Fig 9.

Several components in the model depend on the intracellular Ca²⁺ transient. These include the Na⁺-Ca²⁺ exchanger, I_Ca, and I_Ks. We found that the Na⁺-Ca²⁺ exchange current is shifted inward at shorter pacing cycle lengths (not shown), which should act to prolong, rather than shorten, APD. Therefore, the Na⁺-Ca²⁺ exchange current cannot contribute to APD shortening at the shorter BCL.

The inactivation of I_Ca depends on [Ca²⁺]_i. As shown in Fig 9D, the f_Ca gate, which represents the Ca²⁺-dependent inactivation, is smaller (more inactivated) at the shorter BCL of 300 ms than at the longer BCL of 600 ms. In other words, there is more Ca²⁺-dependent inactivation at a BCL of 300 ms. This higher degree of Ca²⁺-dependent inactivation results in the smaller I_Ca at fast pacing rates (Fig 9C), which in turn causes the shortening of APD at fast pacing rates. We conclude that a greater degree of Ca²⁺ inactivation of I_Ca is a factor that contributes to the downward shift of the single-cell APD restitution curves at fast pacing rates.

In the model, _Ks is dependent on [Ca²⁺]_i. _Ks is larger at larger [Ca²⁺]_i. Because the intracellular Ca²⁺ transient is larger at a BCL of 300 ms, _Ks is larger at a BCL of 300 ms than at a BCL of 600 ms (Fig 10B). However, I_Ks is larger at a BCL of 600 ms, not at a BCL of 300 ms (Fig 10A). Accompanying the smaller I_Ks is a reduced activation (smaller X_s) at a BCL of 300 ms (Fig 10C). Because at both values of BCL the degree of deactivation is the same just before the S2 stimuli (Fig 10C), greater deactivation before the S2 stimulus cannot account for the reduced activation during the S2 action potential at a BCL of 300 ms. As shown in Fig 9A, the plateau potential of S2 is less positive at a BCL of 300 ms than at a BCL of 600 ms because of the reduced I_Ca (Fig 9). This causes the reduced activation of I_Ks at a BCL of 300 ms through the voltage-dependent characteristics of X_s. Therefore, despite the Ca²⁺-dependent increase of _Ks at fast pacing rates, I_Ks is smaller at a BCL of 300 ms. A smaller I_Ks acts to prolong (rather than shorten) APD at fast pacing rates. Therefore, our simulations suggest that as a result of two effects of elevated [Ca²⁺]_i on I_Ks—increased _Ks and reduced Vm—the Ca²⁺-dependent increase of _Ks only plays a minor role in the downward shift of restitution curves at fast pacing rates.

APDs vary over a wide range of values among species, within the same species, and even within a small tissue preparation. As shown in our simulations (Fig 4), APD depends strongly on the densities of I_Kr and I_Ks. It is important to know how the current densities of I_Kr and I_Ks affect the restitution of APD because the relative densities of I_Kr and I_Ks may vary considerably between species and within the same species.

Scaling _Kr and _Ks by the same factor (0.8, ie, 20% block), we found that APD₁₀₀ is increased from 179.7 to 220.1 ms (the cell is paced at 0.2 Hz). However, the time course of APD restitution is minimally affected. The time constant, , of the APD restitution curve, fitted by a single exponential function, APD=1-A · exp (-DI/), changes minimally from 46.4 ms under control to 44 ms under 20% block of both I_Kr and I_Ks. However, if the relative densities of I_Kr and I_Ks are changed, the time constant of the APD restitution curve is affected significantly. In Fig 11, normalized APD restitution curves for different degrees of block or enhancement of I_Kr and I_Ks are given. When I_Kr is fully blocked, the time constant of the APD restitution curve is 55.8 ms. However, when I_Kr is made larger relative to I_Ks, the time constant decreases. It is only 38.4 ms when I_Kr is increased fourfold and I_Ks is reduced by 50%. In other words, the rate of APD change increases when the density ratio of I_Kr to I_Ks is high. When the density ratio is kept constant but the magnitude of both currents is reduced, the steady state APD is prolonged, but the time constant of APD restitution is minimally affected. We conclude that both I_Kr and I_Ks are important to the fast rate of APD change at short DIs. The rate of APD change depends on the relative densities of these channels and increases when the I_Kr-to-I_Ks density ratio is high.

View larger version (23K): [in this window] [in a new window]   Figure 11. Graph shows normalized action potential duration (APD) restitution curves at a pacing cycle length of 400 ms for different combinations of the ratio of the fast component of the delayed rectifier K+ current (IKr) to the slow component of the delayed rectifier K+ current (IKs) (the maximum conductance ratio). The different conditions are indicated in the inset. Note increase of slope as the ratio of IKr to IKs increases. indicates the time constant of the restitution curve as fitted by a single exponential function: APD=1-A · exp(-DI/), where DI indicates diastolic interval.

	Discussion

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 Appendix 2 References

 
The objectives of the present study are (1) to incorporate, in a quantitative model of the guinea pig–type ventricular action potential, the recently described components of the delayed rectifier K⁺ current, I_Kr and I_Ks, and (2) to examine the roles of I_Kr and I_Ks in the repolarization of the action potential and in the rate dependence and restitution of the APD. I_Kr in the updated L-R model incorporates the following properties: (1) fast activation (time constant of 175 ms at -30 mV), (2) a relatively negative activation threshold (-33 mV), (3) [K⁺]_o-dependent maximum conductance, (4) pure selectivity to K⁺ ions, and (5) inward rectification. I_Ks is characterized by (1) slow activation (time constant of 417 ms at -30 mV), (2) a relatively positive activation threshold (-9 mV), (3) [Ca²⁺]_i-dependent maximum conductance, and (4) selectivity to both K⁺ and Na⁺ with an Na⁺-to-K⁺ permeability ratio of 0.01833.

Measurements of the time course of I_Kr deactivating tail currents can be fitted by either a monoexponential function^{10 23 38} or a biexponential function,^{11 12 26} depending on the preparation. In guinea pig ventricular myocytes, Sanguinetti and Jurkiewicz¹⁰ write, "I_Kr tail current was adequately fit with a single exponential function in the majority of experiments, although some currents had a measurable slower component." They suggested that the slow component "could result from a slight decline in I_Ks between the time control and drug-exposed currents were recorded, or could represent a genuine second component." In contrast, Chinn¹² observed that deactivating tails may consist of both fast and slow components. Considering the discrepancy in these published experiments, we compared the differences between their experimental protocols. Sanguinetti and Jurkiewicz used nisoldipine to block I_Ca and Chinn used cadmium. Nisoldipine is a specific Ca²⁺ channel blocker and has no effects on K⁺ currents. In contrast, cadmium has complex effects on I_K³⁹ and might have affected Chinn's results. In addition, Chinn provided only four data points for each fast and slow time constant of deactivation. These values were limited to the potential range between -50 and -20 mV. In our model, we base the formulation of I_Kr on the data of Sanguinetti and Jurkiewicz. It should be recognized that Sanguinetti and Jurkiewicz's protocol involved holding the return potential from depolarizing pulses for a period of 750 ms. It is possible that a slow component of the current was not detected in most of their experiments because of the limited duration of their protocol. The slow component of I_Kr deactivation is found to be prominent between -20 and -50 mV.^{12 26} Phase-3 repolarization of the action potential occurs in this potential range. During phase-3 repolarization, I_Kr decreases because of deactivation. The slow component of I_Kr deactivation may reduce the rate of I_Kr decrease and accelerate phase-3 repolarization. However, the magnitude of I_Kr is much smaller than that of I_K1 during this phase of the action potential (Fig 2D). In addition, phase-3 repolarization lasts for <50 ms, whereas the slow component of I_Kr deactivation is rarely measurable during a holding period of 750 ms at the return potential from depolarizing pulses.¹⁰ Therefore, the slow component of I_Kr deactivation should have little effect on the action potential configuration and APD. This implies that even if a slow deactivation component of I_Kr existed, the monoexponential representation adopted in our model is adequate for simulating the action potential. As is clear from the above discussion, a complete characterization of I_Kr deactivation in guinea pig ventricular myocytes, including the existence of a slowly deactivating component, requires additional experiments.

Regulation of plateau currents is important in determining the APD.^{1 3 16} Our simulations show that I_Ks is the dominant outward current during the plateau of the action potential. This is the case for the guinea pig ventricular cell simulated here, reflecting the large I_Ks conductance (density) in this cell type. Fig 4 shows that a reduction of either I_Ks conductance or I_Kr conductance can effectively prolong the APD. The desired effect of class III antiarrhythmic agents is to prolong the refractory period by delaying repolarization of the action potential.^{16 40} However, delay of repolarization creates conditions that favor the development of arrhythmogenic EADs.^{19 41} Class III antiarrhythmic compounds are thought to be associated with EAD-related arrhythmogenic phenomena such as the long QT syndrome and torsade de pointes.¹⁶ The simulations of Fig 4 show that I_Kr can be completely blocked without producing EADs. Similar behavior was observed experimentally in guinea pig ventricular cells.¹⁰ However, it should be noted that specific block of I_Kr was found to induce EADs in other preparations.^{29 30 31} In our simulations, EADs can be induced by a complete block of I_Kr if we decrease the conductance of I_Ks and increase the conductance of I_Kr while keeping APD similar to control (defined as APD for an I_Kr-to-I_Ks ratio typical of the guinea pig ventricular cell). This implies that our observation that I_Kr block does not result in EADs cannot be generalized to cell types other than the guinea pig ventricular myocyte but might apply to other cells with a similar I_Kr-to-I_Ks density ratio. It should be added that in a report of an earlier theoretical study, Courtney et al⁴² predicted a behavior opposite to that of our simulations. In their study with a guinea pig ventricular cell model, I_Kr block induced EADs but I_Ks block did not. This is not consistent with experimental observations.¹⁰ However, their model was based on the simple Beeler-Reuter⁴³ representation of the action potential, which does not accurately represent processes that are crucial to EAD formation (eg, the kinetics of I_Ca).

In contrast to the monotonic repolarization when I_Kr is blocked, a >80% block of I_Ks results in abnormal repolarization and the development of EADs. This result suggests that in guinea pig–type cells, in which the I_Ks-to-I_Kr density ratio is large, I_Kr is safer than I_Ks as a target for class III agents. This conclusion cannot be generalized, however, to cells with very different I_Ks-to-I_Kr density ratios because the relative density, rather than the different kinetics of these channels, has the dominant effect on EAD formation.⁴¹ In the context of class III agents and prolongation of the refractory period, it should be emphasized that late phase-3 repolarization plays an important role in determining the refractory period and the recovery of excitability. Our simulations (Fig 2B) demonstrate that the I_Kr and I_Ks curves cross over at this phase, with I_Kr obtaining a larger magnitude than I_Ks. However, I_K1 becomes dominant at this phase (Fig 2D) and controls the final repolarization phase of the action potential.

APD is an important factor in arrhythmogenesis. Nonuniformities of APD create conditions that favor the induction of reentry. Rate dependence of APD can influence the degree of head-tail interaction during reentry and the stability of the reentrant activity. We study the dependence of APD on the degree of prematurity of the action potential by constructing the restitution curve and investigating the processes that determine its shape. The simulated restitution fits a measured restitution curve that we have constructed for the guinea pig ventricular epicardium by using an optical recording (voltage-sensitive dye) approach. It should be reiterated that restitution time constants in our experiments fall in the range of 15 to 42 ms. Time constants measured by others,¹⁵ also at 37°C, range from 35 to 65 ms. The time constant of the simulated restitution curve in Fig 5C is 46.4 ms. As stated in "Results," the rate of APD change depends on the relative densities of I_Kr and I_Ks, and the restitution time constant decreases when the I_Kr-to-I_Ks density ratio is high (it is only 38.4 ms when I_Kr is increased fourfold and I_Ks is reduced by 50%).

Extracellular K⁺ accumulation and incomplete recovery of the delayed K⁺ channel activation and of Ca²⁺ channel inactivation are suggested to be the mechanisms that determine the restitution behavior.^{15 18 32} Our simulations suggest that I_Ca is not the dominant factor in APD restitution, despite its incomplete recovery when premature stimuli are applied. A similar conclusion was reached on the basis of experimental observations that APD restitution of guinea pig ventricular myocytes occurs normally even when I_Ca is completely blocked by dihydropyridines.⁴⁴ The steep portion of the restitution curve at the range of short DIs is determined by the I_Kr and I_Ks that are partially activated when the premature stimuli are applied. The slope of the restitution curve at this range depends on the relative densities of these channels. It is steeper when the I_Kr-to-I_Ks density ratio is high. This result is supported by the experimental observation of Todt et al⁴⁵ that specific block of I_Kr results in a slow time course of APD restitution.

The downward shift of the simulated APD restitution curves due to an increase of the basic pacing rate (Fig 5) has been observed experimentally as well.^{15 18 37 46} Because the simulations are conducted in a model of a single cell, K⁺ accumulation in extracellular clefts, a mechanism proposed by Boyett and Jewell,³² cannot exclusively explain this phenomenon. The simulations show that, after pacing, a very long time (DI >5 s) is required for the intracellularly stored Ca²⁺ to reach its normal diastolic level. At a fast basic pacing rate, more Ca²⁺ accumulates intracellularly. This results in a larger intracellular Ca²⁺ transient during an S2 action potential that follows a period of fast pacing. The larger intracellular Ca²⁺ transient acts to reduce I_Ca through its Ca²⁺-dependent inactivation. It also acts to enhance I_Ks through its Ca²⁺-dependent conductance, _Ks. Both effects contribute to the downward shift of the restitution curves, with I_Ca playing a dominant role. It is interesting to note that the slow change of intracellular calcium has a long-lasting influence (>5 s) that affects the action potential over many cycles. This introduces a "memory" property into the process of cellular excitation in cardiac myocytes.

Another interesting observation is that the 300-ms restitution curve (Fig 5) displays an initial biphasic behavior (notch marked by bold arrow in Fig 5B). For this particular curve, APD increases rapidly with DI (steep portion), then decreases (notch), and then increases again more slowly. Similar biphasic behavior has been observed experimentally in humans^{18 34} and in various animal species.^{29 46} The potential importance of this behavior to arrhythmogenesis has been discussed in a recent publication.³³ The mechanism that underlies this behavior requires further investigation. Our simulations show that the beginning of the notch coincides with the resumption of SR Ca²⁺ release (for a shorter DI, CICR does not occur). The simulations (Fig 8) suggest that the increase in the intracellular Ca²⁺ transient results in a reduced I_Ca through the Ca²⁺-dependent inactivation process. Under certain conditions, this reduction of an inward current more than offsets the reduction in the outward potassium currents to cause APD shortening. As DI is further increased, the K⁺ currents continue to decrease, APD increases again, and the notch is formed. The fact that a notch is formed at a BCL of 300 ms but not at longer BCLs (Fig 5B) is consistent with the concept that intracellular calcium plays an important role in this phenomenon, because at shorter BCLs more Ca²⁺ accumulates intracellularly during pacing. Further experimental work is needed to validate this theoretical observation and to fully elucidate the mechanism that underlies biphasic restitution.

It is important to remember that the results presented here regarding APD, its prolongation by blocking agents, or its rate dependence cannot be extrapolated to cells other than the guinea pig type. In particular, extrapolation should not be made to species in which I_to plays a major role in repolarization^{23 38 47} or in which only a single type of I_K is present.^{47 48}

In addition to I_Kr and I_Ks, other delayed rectifying potassium currents have been observed. A Kv1.5 channel current has been found in rat and human myocardium.^{49 50 51} This current is distinguished from I_Kr and I_Ks by its rapidity of activation and limited slow inactivation. The results reported here cannot be extrapolated to the role of Kv1.5 or of other K⁺ currents in action potential repolarization or APD restitution. In view of its kinetics, Kv1.5 may constitute another target, in addition to I_Kr and I_Ks, for the antiarrhythmic effects of class III agents.

	Acknowledgments

 
This study was supported by grant HL-49054 from the National Heart, Lung, and Blood Institute of the National Institutes of Health (Dr Rudy) and by the Department of Veterans Affairs (Dr Rosenbaum). We thank Xiaoqin Zou and Robin Shaw for deriving the analytical buffering equations shown in Appendix 1 and for helpful discussions.

	Footnotes

 
Reprint requests to Prof Yoram Rudy, Department of Biomedical Engineering, Case Western Reserve University, Cleveland, OH 44106-7207.

	Appendix 1

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 Appendix 2 References

 
Analytical Computation of Ca²⁺ Buffering
In the early version of the L-R model used by Luo and Rudy,³ Steffensen's iterative method was used to compute, numerically, Ca²⁺ buffering in the junctional sarcoplasmic reticulum (JSR) and in the cytosol under steady state conditions. Recently, we derived an analytical expression for the same purpose. In the present study, the analytical formulation was used. Equations are provided below.

In JSR,

where

and

In cytosol,

where

where

Definitions are as follows (concentrations given in mmol/L): [Ca²⁺]_i indicates free [Ca²⁺]_i in cytosol; [CSQN], concentration of Ca²⁺ buffered by calsequestrin in JSR; [], maximum concentration of Ca²⁺ buffered by calsequestrin in JSR; [CMDN], concentration of Ca²⁺ buffered by calmodulin in cytosol; [], maximum concentration of Ca²⁺ buffered by calmodulin in cytosol; [TRPN], concentration of Ca²⁺ buffered by troponin in cytosol; [], maximum concentration of Ca²⁺ buffered by troponin in cytosol; K_m,CSQN, K_m,TRPN, and K_m,CMDN, equilibrium constants of buffering by calsequestrin, troponin, and calmodulin, respectively; and [Ca²⁺]_i, change in total Ca²⁺ amount during one time step. The subscripts i, new, and old indicate intracellular, present time step, and previous time step, respectively.

	Appendix 2

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 Appendix 2 References

 
Relevant Model Equations
I_Kr, the Fast Component of the Delayed Rectifier K⁺ Current

I_Ks, the Slow Component of the Delayed Rectifier K⁺ Current

with [Ca²⁺]_i in millimoles per liter.

where P_Na,K is 0.01833.

Ca²⁺ Current Through T-Type Ca²⁺ Channels

Ca²⁺ Current Through L-Type Ca²⁺ Channels
The introduction of I_Kr, I_Ks, and I_Ca(T) and the modification of I_Kp required an adjustment of the Ca²⁺-dependent inactivation gate (f_Ca). The adjustment is

where K_m,Ca=0.6 µmol/L. Note that a Hill coefficient of 1 is used, instead of 2 as in the original L-R model.³

A detailed table of all model equations is in Reference 3.

Received November 14, 1994; accepted March 2, 1995.

	References

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 Appendix 2 References

 
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