Two Components of the Delayed Rectifier K+ Current in Ventricular Myocytes of the Guinea Pig Type
Theoretical Formulation and Their Role in Repolarization
Abstract Two distinct delayed rectifier K+ currents, IKr and IKs, 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 IKr and IKs. Our results show that IKs is the major outward current during the plateau repolarization. A specific block of either IKr or IKs 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 IKr does not result in early afterdepolarizations (EADs). In contrast, >80% block of IKs results in abnormal repolarization and EADs. This behavior reflects the high IKs-to-IKr density ratio (≈8:1) in this cell and can be reversed (ie, IKr block can cause EADs) by reducing the ratio of Iks to IKr. 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 IKr and IKs is the primary determinant of the APD restitution at shorter DIs, with Ca2+ current through L-type channels (ICa) playing a minor role. The rate of APD change depends on the relative densities of IKr and IKs; it increases when the IKr-to-IKs 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 ICa and different degrees of Ca2+-dependent conductance of IKs at very long DIs (>5 s) for different BCLs. This BCL dependence of ICa and IKs 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.
In ventricular myocytes, several types of K+ currents are involved in the repolarization of the action potential. These include the inward rectifier K+ current, IK1; the delayed rectifier K+ current, IK; the transient outward current, Ito; and perhaps the recently described plateau K+ current, IKp. IK1 is known to be important during fast phase-3 repolarization of the action potential.1 2 IK,3 4 5 Ito,6 and possibly IKp7 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, IK but not Ito is responsible for plateau repolarization.4 10 Recent studies show that IK, with its slow activation characteristics, is composed of two distinct K+ component currents, IKr and IKs.10 11 12 13 IKr (“rapid”) displays fast activation and is sensitive to the IKr channel blocker E-4031 or sotalol. IKs (“slow”) is similar to the classically described IK and is characterized by slower activation kinetics. The kinetics of these currents have been determined recently by Sanguinetti and Jurkiewicz10 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 IK activation plays a major role in APD restitution.
The objectives of the present study are (1) to formulate IKr and IKs and incorporate them into the recent Luo-Rudy (L-R) model1 3 19 of the ventricular action potential, (2) to update the L-R model by adjusting IKp on the basis of recent experimental data8 and by adding the T-type Ca2+ current, ICa(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 IKr and IKs 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 IKr and IKs 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
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 fCa 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, IKr and IKs. ICa(T) is added in the model. IKp 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⇓.
Two Components of IK: IKr and IKs
IK 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 IK, IKr and IKs, coexist in guinea pig ventricular cells. Two components of IK are also found in atrial cells of the same species.11 13
IKr 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 IKr incorporates both a time-dependent activation gate, Xr, 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 IKr. In the Hodgkin-Huxley–type formalism, IKr can be expressed as where V is the membrane potential, EKr is the reversal potential, and G̅Kr is the maximum conductance of IKr. Experimental studies show that lowering [K+]o decreases IKr.14 Considering the increase of the driving force when [K+]o is reduced, we would expect IKr to increase when [K+]o is lowered. From experimental results, we deduce that IKr conductance decreases at lower [K+]o. Following the formulation of IK in the L-R model3 and the experimental conductance measurements,10 we introduce a square root dependence of G̅Kr on [K+]o and express it as IKr is purely selective to K+ ions. Hence, EKr is set to be the equilibrium potential of K+ ions across the cell membrane. We use the formulation of Xr steady state activation, Xr∞, from Sanguinetti and Jurkiewicz10 and formulate the time constant of activation, τXr, 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 IKr.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.
IKs is the slow component of IK with characteristics similar to the classically described IK. This current shows no inward rectification.10 14 Upon depolarization, the activation of IKs 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 IKs as where G̅Ks is the maximum conductance, Xs is the activation gate, and EKs is the reversal potential. Neither inward rectification nor inactivation of IKs 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 IKs.10 This phenomenon may result from a decrease of EKs and an increase of the driving force. Dependence of G̅Ks on [K+]o has not been observed. We assume G̅Ks to be independent of [K+]o. We set G̅Ks to be in the range of measured values24 27 and verify that this value is consistent with the 24% increase of APD observed by Sanguinetti and Jurkiewicz10 when IKr 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 IK (IKr+IKs) was ≈11.4 times larger than the fully activated IKr upon repolarization to −40 mV. IKs 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 G̅Ks. G̅Ks is larger at higher [Ca2+]i. At [Ca2+]i of 0.12 μmol/L and [K+]o of 4 mmol/L, G̅Ks is 0.1737 millisiemens (mS)/μF, G̅Kr is 0.0225 mS/μF, and the ratio of G̅Ks to G̅Kr is 7.72:1. Formulation of G̅Ks is given in Appendix 2.
IKs, unlike IKr, is not purely selective to K+ ions. The formulation of its reversal potential, EKs, follows that of IK in the L-R model.3
It should be mentioned that IKr and IKs, as formulated here, duplicate the experimental current-voltage relations in the presence or absence of IKr block (ie, before and after exposure to E-4031; see Fig 6⇓ of Reference 1010 ).
IKp, the Plateau K+ Current
Formulation of IKp 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 IKp by using the whole-cell patch-clamp protocol. On the basis of their measurements, the maximum conductance of IKp, G̅Kp, 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.
ICa(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 ICa(T) to the model to formulate a more complete theoretical model of the ventricular action potential. ICa(T) is formulated as where G̅Ca(T) is the maximum conductance; ECa 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 ICa(T) during the action potential (not shown). ICa(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 ICa(T) shows that the addition of ICa(T) to the model has a minimal effect on the shape and duration of the action potential.
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.
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 IK1, IKr, IKs, and IKp. In the following simulations, we investigate the roles of IKr and IKs during the different phases of the action potential and compare these currents with IK1 and IKp.
Fig 2⇓ depicts the behavior of IKr and IKs during an action potential. The fast component, IKr, increases faster than IKs 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, IKr increases in magnitude because of the decreased inward rectification at less positive potentials. Compared with IKr, IKs attains a much larger magnitude during the plateau, reflecting its larger conductance. The dotted curve in Fig 2B⇓ is the sum of IKr and IKs. Its behavior is similar to the behavior of IK in the original L-R model (Fig 13 in Reference 33 ). Another K+ current that activates during the plateau is IKp. The simulations (Fig 2C⇓) show that IKp is larger than IKs only at the early phase of the plateau but is much smaller than IKs during most of the plateau. Therefore, we conclude that IKs is the major repolarizing current during the plateau phase.
During most of phase-3 repolarization of the action potential, the IKr and IKs curves cross over and IKr is larger than IKs, especially near the end of repolarization (Fig 2B⇑). This reflects the different driving forces of these currents. IKr is purely selective to K+ ions. Its reversal potential is the EKr (−95.86 mV at a [K+]o of 4 mmol/L), which is more negative than the resting potential. IKs is carried by K+ ions as well as Na+ ions. At a [K+]o of 4 mmol/L, EKs is ≈−83.26 mV. Therefore, during late repolarization the larger driving force of IKr results in a larger outward magnitude than IKs. The important role of IK1 in fast repolarization during late phase 3 is well established. In Fig 2D⇑, IKr is compared with IK1. Although IKr is larger than IKs during the late repolarization phase, its magnitude is much smaller than that of IK1. We conclude that even when IKr is considered, IK1 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 IKp 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 IK1 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 IK1 results in an increase of only 4.3% in APD at 50% repolarization but in a 23% increase in APD100. A 100% increase of IK1 maximum conductance accelerates phase-3 repolarization and shortens APD100 by 3.7%.
As mentioned previously in “Materials and Methods,” the value of the maximum IKs 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 IKr channel blocker E-4031. Fig 4A⇓ shows how the action potential shape and duration are affected when the maximum conductance of IKr is either reduced or increased. APD100 at 100% block of IKr increases by 24%, and at 100% increase of G̅Kr, it decreases by 14%. Fig 4B⇓ shows the results of similar protocols for modulation of IKs. With <80% block of IKs, repolarization of the action potential is similar to that under control conditions, but APD is prolonged. At 50% decrease of G̅Ks, APD100 is increased by 23.4%. When G̅Ks is decreased by >80%, the action potential cannot repolarize normally, and early afterdepolarizations (EADs) are observed. A 100% increase of G̅Ks results in an 18% decrease of APD100. As shown in Fig 4C⇓, if both G̅Kr and G̅Ks 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 IKr and IKs, 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.
Specific block of IKr is found to result in EADs in various preparations.29 30 31 However, EADs are not generated by IKr block in guinea pig ventricular cells10 or in our simulations for an IKr-to-IKs ratio and IKr and IKs conductances that are typical of the guinea pig ventricular cell. However, if we increase IKr by 50% and decrease IKs by 30%, EADs develop when IKr 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.
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 (ICa) at DIs of 20 and 100 ms (Fig 6⇓). Just before the application of S2 stimuli, both voltage-dependent (f) and Ca2+-dependent (fCa) 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 ICa 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, ICa 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 ICa 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 IK (Fig 7⇓), we observe that both IKr and IKs 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 IKr and IKs act to shorten the APD at smaller DIs. Our previous simulations (Fig 2⇑) show that IKr and IKs 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.
Both IKr and IKs are characterized by relatively long time constants. As shown in Fig 7D⇑, 7E⇑, and 7F⇑, at both DIs the activation gates of both IKr (Xr gate) and IKs (Xs 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 IKr and IKs 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 Xr compared with that of Xs. As shown in Fig 7D⇑, at a DI of 100 ms Xr is already close to complete deactivation just before the S2 stimulus. Hence, IKr has a smaller effect on APD restitution in the range of longer DIs. For the short DI (20 ms), both Xr and Xs are partially activated so that both IKr and IKs affect the APD and the restitution curve is steeper. It is known that the conductance of IKs depends on free [Ca2+]i. The conductance is higher for a larger [Ca2+]i. In the model, G̅Ks 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 ICa 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 G̅Ks depends only on [Ca2+]i, its value is also similar at the two DI values (Fig 7F⇑). We conclude that the Ca2+ dependence of G̅Ks 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 ICa. In Fig 8⇓, membrane potential, intracellular Ca2+ transient, ICa, and fCa 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 fCa; Fig 8D⇓) and, as a result, a smaller ICa during the plateau (Fig 8C⇓). The smaller ICa results in a smaller APD at a DI of 132 ms, overcoming the prolongation effect of the decreases in IKr and IKs 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.
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 IK 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⇓).
Several components in the model depend on the intracellular Ca2+ transient. These include the Na+-Ca2+ exchanger, ICa, and IKs. 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 ICa depends on [Ca2+]i. As shown in Fig 9D⇑, the fCa 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 ICa 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 ICa is a factor that contributes to the downward shift of the single-cell APD restitution curves at fast pacing rates.
In the model, G̅Ks is dependent on [Ca2+]i. G̅Ks is larger at larger [Ca2+]i. Because the intracellular Ca2+ transient is larger at a BCL of 300 ms, G̅Ks is larger at a BCL of 300 ms than at a BCL of 600 ms (Fig 10B⇑). However, IKs is larger at a BCL of 600 ms, not at a BCL of 300 ms (Fig 10A⇑). Accompanying the smaller IKs is a reduced activation (smaller Xs) 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 ICa (Fig 9⇑). This causes the reduced activation of IKs at a BCL of 300 ms through the voltage-dependent characteristics of Xs. Therefore, despite the Ca2+-dependent increase of G̅Ks at fast pacing rates, IKs is smaller at a BCL of 300 ms. A smaller IKs 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 IKs—increased G̅Ks and reduced Vm—the Ca2+-dependent increase of G̅Ks 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 IKr and IKs. It is important to know how the current densities of IKr and IKs affect the restitution of APD because the relative densities of IKr and IKs may vary considerably between species and within the same species.
Scaling G̅Kr and G̅Ks 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, τ, of the APD restitution curve, fitted by a single exponential function, APD=1−A · exp (−DI/τ), changes minimally from 46.4 ms under control to 44 ms under 20% block of both IKr and IKs. However, if the relative densities of IKr and IKs 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 IKr and IKs are given. When IKr is fully blocked, the time constant of the APD restitution curve is 55.8 ms. However, when IKr is made larger relative to IKs, the time constant decreases. It is only 38.4 ms when IKr is increased fourfold and IKs is reduced by 50%. In other words, the rate of APD change increases when the density ratio of IKr to IKs 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 IKr and IKs 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 IKr-to-IKs density ratio is high.
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, IKr and IKs, and (2) to examine the roles of IKr and IKs in the repolarization of the action potential and in the rate dependence and restitution of the APD. IKr 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. IKs 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 IKr deactivating tail currents can be fitted by either a monoexponential function10 23 38 or a biexponential function,11 12 26 depending on the preparation. In guinea pig ventricular myocytes, Sanguinetti and Jurkiewicz10 write, “IKr 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 IKs 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 ICa 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 IK39 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 IKr 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 IKr 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, IKr decreases because of deactivation. The slow component of IKr deactivation may reduce the rate of IKr decrease and accelerate phase-3 repolarization. However, the magnitude of IKr is much smaller than that of IK1 during this phase of the action potential (Fig 2D⇑). In addition, phase-3 repolarization lasts for <50 ms, whereas the slow component of IKr deactivation is rarely measurable during a holding period of 750 ms at the return potential from depolarizing pulses.10 Therefore, the slow component of IKr deactivation should have little effect on the action potential configuration and APD. This implies that even if a slow deactivation component of IKr 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 IKr 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 IKs 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 IKs conductance (density) in this cell type. Fig 4⇑ shows that a reduction of either IKs conductance or IKr 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 IKr 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 IKr was found to induce EADs in other preparations.29 30 31 In our simulations, EADs can be induced by a complete block of IKr if we decrease the conductance of IKs and increase the conductance of IKr while keeping APD similar to control (defined as APD for an IKr-to-IKs ratio typical of the guinea pig ventricular cell). This implies that our observation that IKr 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 IKr-to-IKs 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, IKr block induced EADs but IKs 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 ICa).
In contrast to the monotonic repolarization when IKr is blocked, a >80% block of IKs results in abnormal repolarization and the development of EADs. This result suggests that in guinea pig–type cells, in which the IKs-to-IKr density ratio is large, IKr is safer than IKs as a target for class III agents. This conclusion cannot be generalized, however, to cells with very different IKs-to-IKr 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 IKr and IKs curves cross over at this phase, with IKr obtaining a larger magnitude than IKs. However, IK1 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 IKr and IKs, and the restitution time constant decreases when the IKr-to-IKs density ratio is high (it is only 38.4 ms when IKr is increased fourfold and IKs 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 ICa 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 ICa is completely blocked by dihydropyridines.44 The steep portion of the restitution curve at the range of short DIs is determined by the IKr and IKs 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 IKr-to-IKs density ratio is high. This result is supported by the experimental observation of Todt et al45 that specific block of IKr 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 ICa through its Ca2+-dependent inactivation. It also acts to enhance IKs through its Ca2+-dependent conductance, G̅Ks. Both effects contribute to the downward shift of the restitution curves, with ICa 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 humans18 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 ICa 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 Ito plays a major role in repolarization23 38 47 or in which only a single type of IK is present.47 48
In addition to IKr and IKs, 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 IKr and IKs 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 IKr and IKs, for the antiarrhythmic effects of class III agents.
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; [C̅S̅Q̅N̅], maximum concentration of Ca2+ buffered by calsequestrin in JSR; [CMDN], concentration of Ca2+ buffered by calmodulin in cytosol; [C̅M̅D̅N̅], maximum concentration of Ca2+ buffered by calmodulin in cytosol; [TRPN], concentration of Ca2+ buffered by troponin in cytosol; [T̅R̅P̅N̅], maximum concentration of Ca2+ buffered by troponin in cytosol; Km,CSQN, Km,TRPN, and Km,CMDN, equilibrium constants of buffering by calsequestrin, troponin, and calmodulin, respectively; and Δ[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.
Relevant Model Equations
IKr, the Fast Component of the Delayed Rectifier K+ Current
IKs, the Slow Component of the Delayed Rectifier K+ Current
with [Ca2+]i in millimoles per liter. where PNa,K is 0.01833.
Ca2+ Current Through T-Type Ca2+ Channels
Ca2+ Current Through L-Type Ca2+ Channels
The introduction of IKr, IKs, and ICa(T) and the modification of IKp required an adjustment of the Ca2+-dependent inactivation gate (fCa). The adjustment is where Km,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.
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.
Reprint requests to Prof Yoram Rudy, Department of Biomedical Engineering, Case Western Reserve University, Cleveland, OH 44106-7207.
Presented in part in abstract form at the 39th meeting of the Biophysical Society, San Francisco, Calif, February 12-16, 1995.
- Received November 14, 1994.
- Accepted March 2, 1995.
- © 1995 American Heart Association, Inc.
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