Modeling the Effect of Kv1.5 Block on the Canine Action Potential

Abstract

A wide range of ion channels have been considered as potential targets for pharmacological treatment of atrial fibrillation. The Kv1.5 channel, carrying the I_Kur current, has received special attention because it contributes to repolarization in the atria but is absent or weakly expressed in ventricular tissue. The dog serves as an important animal model for electrophysiological studies of the heart and mathematical models of the canine atrial action potential (CAAP) have been developed to study the interplay between ionic currents. To enable more-realistic studies on the effects of Kv1.5 blockers on the CAAP in silico, two continuous-time Markov models of the guarded receptor type were formulated for Kv1.5 and subsequently inserted into the Ramirez-Nattel-Courtemanche model of the CAAP. The main findings were: 1), time- and state-dependent Markov models of open-channel Kv1.5 block gave significantly different results compared to a time- and state-independent model with a downscaled conductance; 2), the outcome of Kv1.5 block on the macroscopic system variable APD₉₀ was dependent on the precise mechanism of block; and 3), open-channel block produced a reverse use-dependent prolongation of APD₉₀. This study suggests that more-complex ion-channel models are a prerequisite for quantitative modeling of drug effects.

Introduction

The ultrarapidly activating delayed rectifier K⁺ current, commonly denoted I_Kur, conducted by the Kv1.5 channel, has emerged as a target for pharmacological treatment of atrial fibrillation (AF; see Tamargo et al. (1) and references within). I_Kur contributes to both early and late repolarization of the action potential (AP) of the human atrial myocyte and because a similar current largely seems to be lacking in ventricular tissue, a blocker of the Kv1.5 channel would have potential to selectively increase the duration of the atrial AP (APD) and, hence, the refractory period (AERP). I_Kur blockers have been shown to affect human atrial AP repolarization in vitro, and selectively increase atrial refractoriness and terminate atrial arrhythmias in several animal species in vivo (2–8). Human in vitro data as well as computer modeling of human atrial cells using Hodgkin-Huxley representations of I_Kur have also shown that the effect of a block of I_Kur on AP repolarization depend on the relative densities of all involved ion channels and may be more effective in diseased tissue (3,9). In both animal disease models and humans, ion channel densities changes during progression of AF (see Tamargo et al. (1) and references within), thus changing the relative contribution of each involved ion channel in the repolarization process. This remodeling process also facilitates the progression of the disease. In a complex biological system with interwoven dependencies like this, good knowledge of the behavior of individual components in isolation does not automatically provide good knowledge of the behavior of the system as a whole. Mathematical modeling may be especially helpful in elucidating important system properties by combining mathematical formulations of the knowledge of single components. For example, complete models of the human atrial myocyte AP have been developed (10,11) and extensions of these models have been used to evaluate the effects of I_Kur block (3,9).

Animal models are used extensively throughout the drug development process from early discovery to late development, to improve understanding of biological mechanisms, and to allow for predictions of human responses to drug exposure. For this reason, mathematical models of animal models are of interest. Different animal models are used in different disease areas and for cardiovascular diseases, experimental dog models have been widely used to study, for example, atrial arrhythmia mechanisms in vivo. The Ramirez-Nattel-Courtemanche (RNC) model (12) is a mathematical model of the canine atrial AP. It is based on the Hodgkin-Huxley formalism, contains all major ionic currents, and has been validated using experimental measurements in canine atrial myocytes. In this model, the I_Kur current carried by Kv1.5 is known as I_Kurd.

However, most Kv1.5 blockers are selectively blocking only the activated or open state of Kv1.5 channels (2,13–18) and important features of the drug-ion channel interaction such as use- and voltage-dependence are not easily implemented in the Hodgkin-Huxley formalism. Rather than using the Hodgkin-Huxley formulation, state-dependent blocking mechanisms are usually modeled with Markov models. Therefore, a continuous-time Markov model of the guarded receptor type was set up for the canine Kv1.5 to enable studies on the effects of Kv1.5 open-channel blockers on the canine atrial AP in silico. Conceptually similar Markov models have previously been successfully used to describe the kinetics of block of human Kv1.5 by quinidine (13), quinine, clofilium, and tetrapentylammonium (14), loratadine (15), and bupi-, ropi-, and mepi-vacaine (16). The Markov model was subsequently inserted into the RNC model, replacing the original Hodgkin-Huxley expression. A subset of the parameters in our model reflects properties of a drug and drug-channel interactions such as the net charge of a drug, and the rates of binding to, and dissociation from, the receptor site on the channel. By performing simulations with the modified version of the RNC model, we have been able to examine the influence of these drug-defining parameters on the AP. The model has also been used to make predictions about the morphological changes of the AP for two particular I_Kur blockers, whose kinetic parameters have been determined experimentally elsewhere (2). In connection to this, our model of open-channel Kv1.5 block was extended to also account for drugs that, like the above-exemplified compounds, exhibit a voltage-dependent dissociation rate distinguished from the type of voltage dependence caused by drugs carrying a net charge. To our knowledge, this phenomenon has not been captured in any previous Kv1.5 Markov model.

Methods

The RNC model

Canine atrial myocytes displays regional variations in ionic current density (12). In this study, values for cells from the pectinate muscle (PM) were used, and as described in Ramirez et al. (12), the maximum conductance of L-type Ca ²⁺ channels, transient outward K⁺ current, and Na⁺ channels were set to 40%, 40%, and 50%, respectively, of their nominal values to account for the major changes in ion channel density that occurs during progression of AF in the dog (19). An often-encountered issue with models of electrophysiological systems is small drifts in ion concentrations. In our implementation, the ion concentrations of Na⁺, K⁺, and Cl^– were, for simplicity, fixed to their initial values as listed in Ramirez et al. (12). This did not alter the behavior of the model in any noticeable way except for the loss of dynamics for these ion concentrations.

Our Kv1.5 models

Transition rates between states in the models are assumed to have an exponential voltage dependence and are characterized by two parameters—the zero voltage rate and the equivalent charge movement up to the transition state. The parameters in our five-state Kv1.5 model were optimized to reproduce the behavior of the original RNC model. This optimization was performed using a multidimensional downhill simplex method (20) implemented in the Systems Biology Toolbox for MATLAB (The MathWorks, Natick, MA) (21,22). The model reduction used in connection with our 10-state model is based on an analytical equilibrium solution (23).

Simulations

Simulations were performed with the Systems Biology Toolbox for MATLAB (21,22), which uses the CVODE integrator (24). For all stimulation frequencies, a stimulation current of −2900 pA was applied during 2 ms. Simulation data used for analysis was always collected from the last of a series of 40 APs. This gave a maximum difference in membrane potential between the last two APs equal to 0.01 and 0.05 mV for AF remodeled cells stimulated at frequencies of 1 and 4 Hz, respectively. Visual inspections of APs and of the difference between consecutive APs was also performed to ensure that the AP waveforms really were converging toward stable limit cycles. As a measure of APD, we used APD₉₀, which was defined as the time it takes, after a stimulation, to reach a voltage level of −72 mV. This corresponds to a 90% repolarization of a normal cell stimulated at a frequency of 1 Hz.

Results

Modeling the Kv1.5 ion channel

Our strategy for implementing the continuous-time Markov model was in many respects similar to several modeling efforts of potassium channels found in the literature (13,15,16,23,25).

Kv1.5 lacks the so-called fast N-type inactivation but is affected by the slower C-type inactivation. From Fig. 3 in Ramirez et al. (12), it is apparent that the inactivation property becomes increasingly important for positive values of the membrane potential. However, for the range of voltages where the model will operate, inactivation is not very significant. At a membrane potential of 20 mV, the steady-state value of the inactivation variable, u_∞, is ∼0.95, which means a reduction in I_Kurd by only 5% at steady state. For membrane potentials at or below zero, steady-state inactivations are clearly negligible. Also considering that inactivation is a rather slow process, the effective reduction in I_Kurd is even less for the typically short periods of more strongly depolarized membrane potentials during an action potential. As a consequence, the inactivation property was, for simplicity, omitted from our model. This approximation was validated by simulating the AP both with the inactivation property in place and, as a comparison, with the inactivation turned off. For the normal cell, the difference in membrane potential did not exceed 1.5 mV during the course of the AP and the maximum difference in I_Kurd was 0.1 pA/pF at a stimulation frequency of 4 Hz. In the AF setting, which was most frequently used in this study, differences were even less. Less difference were also observed when all simulations were repeated using a stimulation frequency of 1 Hz.

Figure 3.

APD₉₀ as function of effective on-rate and off-rate of an uncharged drug at a stimulation frequency of 1 Hz (A). APD₉₀ as function of effective on-rate and off-rate of an uncharged drug at a stimulation frequency of 4 Hz (B). Absolute difference between APD₉₀ at stimulation frequencies of 1 and 4 Hz as function of effective on-rate and off-rate of an uncharged drug (C). Absolute difference between APD₉₀ calculated from the time- and voltage-independent model and from the six-state model, as function of effective on-rate and off-rate of an uncharged drug at a stimulation frequency of 4 Hz (D).

Under the assumption that transition rates between the open and closed conformations of a subunit are independent of the state of the other subunits and that all four subunits operate in an identical manner, the Kv1.5 channel can be represented by the five-state model in Fig. 1 A. It has four closed, nonconducting states, C₁-C₁, and an open, conducting state, O. Here, α and β are the forward and reverse rate constants of the transition to the open conformation, respectively. A mathematical description of the scheme in Fig. 1 A is given by a system of ordinary differential equations

$$\frac{d{C}_1}{dt}=-4\alpha C_1+\beta C_2,$$ (1a)

$$\frac{d{C}_2}{dt}=4\alpha C_1-\beta C_2-3\alpha C_2+2\beta C_3,$$ (1b)

$$\frac{d{C}_3}{dt}=3\alpha C_2-2\beta C_3-2\alpha C_3+3\beta C_4,$$ (1c)

$$\frac{d{C}_4}{dt}=2\alpha C_3-3\beta C_4-\alpha C_4+4\beta O,$$ (1d)

$$\frac{dO}{dt}=\alpha C_4-4\beta O,$$ (1e)

where

$$\alpha ={\alpha }_0\exp \left(\frac{Z_{\alpha }FV}{RT}\right),$$ (2a)

$$\beta ={\beta }_0\exp \left(\frac{-Z_{\beta }FV}{RT}\right),$$ (2b)

and C₁–C₄ and O represents the fraction of channels in the different states. The parameters α₀ and β₀ are the forward and reverse rate constants of the transition to the open conformation at zero membrane potential while Z_α and Z_β are the equivalent charge movements up to the transition state, defining the degree of voltage dependence of α and β. Together, the expressions in Eqs. 1 and 2 determine the time evolution of the fraction of open Kv1.5 ion channels. The conductance of the Kv1.5 channel was described by the voltage-dependent relation for the total conductance of Kv1.5 channel

$$g_{\text{Kurd}}=0.00855+\frac{0.0779}{1+\exp \left(\frac{V+11}{-16}\right)}$$ (3)

used in Ramirez et al. (12). This expression correctly relates experimental data of I_Kurd to the fraction of open channels (26). Multiplying the fraction of open channels with the conductance g_Kurd and with the deviation of the membrane potential from the Nernst potential of K⁺ ions, gave the expression for I_Kurd,

$$I_{\text{Kurd}}=g_{\text{Kurd}}O\left(V;{\alpha }_0,{\beta }_0,Z_{\alpha },Z_{\beta }\right)\left(V-E_K\right),$$ (4)

to be compared with the Hodgkin-Huxley expression

$$I_{\text{Kurd}}=g_{\text{Kurd}}{u}_a^3{u}_i\left(V-E_K\right)$$ (5)

used in Ramirez et al. (12).

Figure 1.

Five-state Kv1.5 model (A). Transitions between the states are determined by the rates α and β. Six-state Kv1.5 model with an open-channel block (B). Transitions between the open and the blocked state are determined by the rates γ and δ and by the drug concentration. Ten-state Kv1.5 model with several blocked states (C). Transitions between the blocked states are determined by the rates ζ and ɛ.

Determining parameters

To maintain the properties of the RNC model in absence of a blocking agent, values of the parameters in the expressions in Eq. 2 were chosen accordingly. In practice, this was done by generating artificial data of the membrane potential of a remodeled cell during an 2 Hz AP using the RNC model with the original equations for I_Kurd. The I_Kurd part of the model was then replaced by the expressions in Eqs. 1 and 2, and the parameters were optimized to fit the artificially generated data. Optimizing the parameters during a complete AP, as opposed to using an in silico voltage-clamp protocol for I_Kurd alone, is appealing because the data set is naturally weighted over the desired operating range of the model. In this way, the parameter values were found to be

$${\alpha }_0=0.6161{\text{s}}^{-1},{\beta }_0=0.1001{\text{s}}^{-1},Z_{\alpha }=0.9470,\text{and}{Z}_{\beta }=0.8129.$$

The quality of the obtained parameter set is demonstrated in Fig. S1 in the Supporting Material. There, the membrane potential and I_Kurd of the Markov model is compared to the original RNC model, using both the remodeled and normal cell.

Including blocked state

A simple open-channel block of Kv1.5 was introduced by adding a new state connected to the open state in Fig. 1 A. The new state represents a nonconducting conformation where the drug only interacts with the open state of the ion-channel protein. This is the so-called foot-in-the-door mechanism, which has been described for several Kv1.5 blockers including AVE0118 (27) and the diphenyl phosphine oxides DPO-1 and DPO-2 (2,28). Returning to the open or closed states requires that the bound drug dissociates from its binding site. The extended model is shown in Fig. 1 B. In the mathematical description, Eq. 1 e was modified and a differential equation for the blocked state was added,

$$\frac{dO}{dt}=\alpha C_4-4\beta O-\gamma \left[Drug\right]O+\delta B,$$ (6a)

$$\frac{dB}{dt}=\gamma \left[Drug\right]O-\delta B,$$ (6b)

where [Drug] denotes the concentration of the drug. The rates γ and δ are of the same form as α and β in the expressions in Eq. 2,

$$\gamma ={\gamma }_0\exp \left(\frac{Z_{\gamma }FV}{RT}\right),$$ (7a)

$$\delta ={\delta }_0\exp \left(\frac{-Z_{\delta }FV}{RT}\right).$$ (7b)

The potential voltage dependence of these rates (for nonzero Z_γ and/or Z_δ) then reflects the fact that the drug binding site may be located somewhere within the membrane electrical field. If the drug has a net charge, it will sense part of the electrical field and the binding and dissociation rates will be altered. The extended model of Kv1.5 current is then given by

$$I_{\text{Kurd}}=g_{\text{Kurd}}O\left(V;{\alpha }_0,{\beta }_0,{\gamma }_0,{\delta }_0,Z_{\alpha },Z_{\beta },Z_{\gamma },Z_{\delta },\left[Drug\right]\right)\left(V-E_K\right),$$ (8)

reflecting the introduced dependence of the drug.

Sensitivity analysis

The model was first used to illustrate how the AP of a remodeled PM cell changes in response to different concentrations of a typical, uncharged drug with $\gamma =10$ μM^–1 s^–1 and $\delta =2$ s^–1, at a stimulation frequency of 1 Hz. The simulated membrane potential and I_Kurd current are shown in Fig. 2. In the absence of I_Kurd block, the AP of the remodeled cell has the characteristic triangular shape lacking a clearly defined plateau. At low drug concentrations, a plateau emerges and the triangular shape is gradually lost. As the drug concentration increases further, the plateau phase becomes wider and more elevated before the AP finally culminates in a spike and dome morphology for the highest concentrations. In this way, the addition of the drug delays the repolarization and therefore increases the duration time of the AP. These changes in the AP morphology are accompanied by an attenuation of I_Kurd during the peak and the first part of the plateau phase. The AP peak, shown in the inset of Fig. 2 A, is not affected by the drug. However, it can be noted that the peak is lower in our simulation of the AF-remodeled AP, reaching only 17 mV, compared to the normal cell simulated in Ramirez et al. (12). The above effects on the AP qualitatively resembles the effect of changing the maximum I_Kurd conductance on the normal canine AP (12). A prolonged duration of the AP, and an emerging plateau, have also been reported from simulations of the AF remodeled human atrial myocyte but no dome-shaped plateau phase was observed even though conductance of I_Kur was reduced to 20% (3) and 10% (9) of their nominal values, respectively.

Figure 2.

Action potential waveforms (A) and corresponding I_Kurd (B) generated by the remodeled PM-cell model stimulated at a frequency of 1 Hz. Bold traces corresponds to no drug, solid traces to drug concentrations of 1, 2, 3, 5, 8, and 13 μM, respectively, and dashed traces to 21 μM. The drug-blocking action was defined by the parameter values γ = 10 μM^–1 s^–1, δ = 2 s^–1, and Z_γ = Z_δ = 0. The APD₉₀ is marked by a dashed line in the plot of the membrane potential. (Insets) Magnification of the traces during the spike.

A drug-induced increase of the APD and, as a consequence, in the AERP, has been considered as an effective antiarrhythmic mechanism. To quantify the effects of Kv1.5 blocking drugs on the AP we therefore used the APD₉₀-measure, as defined in the Methods. For all simulation results presented in this section, the similarly defined measure APD₇₀ was also calculated. As it gave qualitatively very similar results, these results are not shown. However, this suggests that the choice of APD measure does not have a critical influence on the conclusions drawn as long as it is chosen somewhere in mid- or end-repolarization phase. This can also be seen in Fig. 2 where the traces run in parallel during the later part of repolarization. Because the proposed antiarrhythmic mechanism is an increase in the refractory period, the different concentrations of the drug in Fig. 2 was used to investigate how the refractory period varies with APD₉₀. Both for the remodeled and normal cell, this relationship was very well described by a linear function (not shown). This suggests that APD₉₀ can be used as a relevant measure of the refractory period.

Having tested the model for one specific parameter setup, it was used, subsequently, to examine systematically how the APD₉₀ of a remodeled PM cell changes with respect to the drug concentration, the free parameters of the expressions in Eq. 7, and AP stimulation frequency. The results were also compared to a simple model featuring a time- and voltage-independent block obtained by just reducing the conductance by a constant factor.

First, the effect of an uncharged drug was investigated for two stimulation frequencies. Combining 21 values of the effective on-rate, the concentration of the drug times the binding rate, between 0 and 200 s^–1,

$$\left[Drug\right]\gamma =10n,n=0,1,\mathrm{..},20$$

with 17 values of the dissociation rate between 0.25 and 64 s^–1,

$$\delta =2^n,n=-2,-1.5,\mathrm{..},6$$

produced 375 different test situations. Results are shown as combined surface and contour plots in Fig. 3. At a stimulation frequency of 1 Hz, APD₉₀ increased for both decreasing values of the off-rate and for increasing values of the effective on-rate (see Fig. 3 A). At 4 Hz, the trend is comparable to 1 Hz, but saturation at high effective on-rates is not as evident and values of APD₉₀ are substantially lower overall (see Fig. 3 B). A very low frequency, 0.1 Hz, was also tested, and was found to be very similar to 1 Hz (results not shown). The difference in APD₉₀ between 1 Hz and 4 Hz, in absolute numbers, is shown in Fig. 3 C. It shows that APD₉₀ is 17 ms longer at 1 Hz compared to 4 Hz in absence of drug (points along the off-rate axis where the effective on-rate is zero). For almost all other points in the plane, the difference is larger, with a maximum difference of 41 ms.

Evidently, the presence of Kv1.5 targeting drugs in a remodeled PM cell tend to extend the duration of the AP more at lower frequencies than at higher. This reverse use-dependence also holds in terms of the relative increase in APD₉₀ (not shown). The frequency dependence was investigated further by looking at APD₉₀ and seeing how the fraction of open and blocked Kv1.5 developed over time in both the remodeled and the normal cell. Four different frequencies, 0.5, 1, 2, and 4 Hz, was used with and without 8 μM of the drug used in Fig. 2. The highest frequency, 4 Hz, could not be tested for the normal cell in presence of the drug because no stable limit cycle with a period of 250 ms was present. In Fig. 4 A the frequency dependence of APD₉₀ is shown for the remodeled cell (left), and for the normal setting (right). Interestingly, in the normal setting, the reverse use-dependence seen for the remodeled cell is not present. Instead, the increase in APD₉₀ becomes larger at higher frequencies. Fig. 4 B shows the fraction of Kv1.5 in the open state during the first 100 ms of the AP in absence of the drug. The morphological changes in the AP due to rate adaption have the effect that lower fractions of Kv1.5 are in the conducting, drug-susceptible state at higher frequencies compared to lower. This observation holds both for the remodeled and normal cell. The same plot, but in the presence of the drug, is shown in Fig. 4 C. Because of the block, smaller fractions of conducting Kv1.5 are now seen. Finally, the fraction of blocked channels at different frequencies is shown in Fig. 4 D for the first 100 ms of the remodeled (left) and normal (right) AP. At the lowest frequency, there were virtually no channels blocked at the point of stimulation. At higher frequencies, significant fractions of channels were still blocked at the onset of stimulation.

Figure 4.

AP frequency dependence of 8 μM test drug defined by γ = 10 μM^–1 s^–1, δ = 2 s^–1, and Z_γ = Z_δ = 0. Left column uses the AF setting, right column uses the normal cell setting. Row A shows APD₉₀ as function of frequency in absence of the test drug (dashed) and with the test drug (solid). Row B shows the fraction open Kv1.5, O, during the first 100 ms of the AP in absence of the test drug while rows C and D show the fraction of open, O, and blocked, B, Kv1.5, respectively, in the presence of the test drug during the first 100 ms of the AP. Bold traces correspond to a stimulation frequency of 0.5 Hz, solid traces to 1 and 2 Hz, respectively, and dashed traces to 4 Hz.

The parameters Z_γ and Z_δ in Eq. 7 are the equivalent charge movements up to the transition state of a drug in the electrical field. By setting Z_γ and Z_δ to values other than zero, we examined the effects of various types of charged drugs on the APD. Assuming a drug with a net charge of +1, its binding site located at a point 20% into the electrical field, and a symmetrical barrier, we set Z_γ = Z_δ = 0. These values are similar to experimentally determined values for two blockers of the human Kv1.5 (13,16). However, introducing a voltage-dependence of the interaction between the drug and its binding site had limited effect on APD₉₀. At 4 Hz, there was virtually no change in APD₉₀, except for very small off-rates, where the duration was decreased by ∼3–4 ms. The effect was even more marginal at 1 Hz, producing only a small decrease in duration of 4 ms for combinations of low effective on-rate and low off-rate. These results suggest that Z_γ and Z_δ may only have a significant impact if the receptor site is located deep within the electrical field and/or the drug has several charged groups at physiological pH.

In several studies (3,11,29,30), the effect of an ion channel blocking drug has been modeled as a time- and voltage-independent decrease in channel conductance. If the degree of decrease in conductance in such a model is assumed to be the result of a simple state-independent bimolecular reaction in equilibrium (between the drug and its binding site), a comparison of this approach to our dynamic modeling is possible. The constant degree of block in the simpler model, B_const, would then be a function of the effective on-rate k_on (the concentration of the drug times the binding rate) and the off-rate k_off (dissociation rate),

$$B_{\text{const}}=\frac{k_{\text{on}}}{k_{\text{off}}+k_{\text{on}}}.$$ (9)

To see how the two models differ, the difference in their APD₉₀ at a stimulation frequency of 4 Hz was calculated from simulations. The result is shown in Fig. 3 D. For large effective on-rates and small off-rates, as well as for the opposite situation with small effective on-rates and large off-rates, the two models gave similar results. In the other parts of the on-off plane the difference was bigger, peaking at 32 ms. We also looked at the difference between the models at a stimulation frequency of 1 Hz. There was a similarly located peak of the same height, but the differences in other parts were roughly halved (not shown).

Extending the model

It is important to emphasize that the drug dissociation rate in our open-channel block model only accounts for voltage dependence arising from the charge of the drug. For the human Kv1.5, voltage-dependent dissociation rates not related to the charge of the drug has been reported (2,15). The dissociation rates in Lagrutta et al. (2) were determined at two different membrane potentials and denoted k_on for the open state and k_off for the closed state. For one particular compound, the DPO-1, these rates differed by a factor of 60. Apparently, at least for the hKv1.5 channel, the state of the blocked channel can change with the membrane potential producing a strong voltage dependence of the dissociation rate in addition to any voltage dependence due to the charge of the drug.

To describe this, additional blocked states could be added in sequence to the blocked state of Fig. 1 B. This would introduce a state-dependent recovery from block, similar to the already existing state-dependent formation of block. The model outlined in Fig. 1 C is one out of many possible extensions to our model that could account for the state-dependent type of voltage dependence discussed above. A similar model structure has been used for the hERG channel (31). Four new blocked states were added, creating symmetry to the four closed states in the upper part. The reason for introducing precisely four new blocked states was consequently to reflect an assumption that the four subunits may fully or partially close independently and identically also when a drug is bound. Because these conformational changes would be expected to be different than in the absence of a drug, two new rates of the same type as earlier, ζ and ε, was introduced. Compared to the six-state model, the 10-state model can be seen as a more elaborate implementation of the foot-in-the-door mechanism.

For the rates ζ and ε to be determined, four parameter values are required. To reduce complexity of the extended model, it was assumed that the transition rates between the closed states were faster than drug binding and dissociation. Under this assumption, the blocked branch of the system can be reduced to a single lumped state. Let the blocked lumped state be defined as

$$L_B=B_1+B_2+B_3+B_4+B.$$ (10)

The occupancy of the original states are now determined from the equilibrium relations in each of the lumped states. In particular, the fraction of L_B in the blocked state B, f_B, is

$$f_B\left(V\right)=\frac{1}{1+4Q+6Q^2+4Q^3+Q^4},$$ (11)

where Q is the ratio of ζ and ε. In the same way as O determines the fraction of unblocked channels accessible for drug binding, f_B determines the fraction of blocked channels that are subject to drug dissociation. Using this variable, the changes in O and in the lumped state L_B with respect to time can then be described by

$$\frac{dO}{dt}=\alpha C_4-4\beta O+\delta f_B{L}_B-\gamma \left[Drug\right],$$ (12a)

$$\frac{d{L}_B}{dt}=\gamma \left[Drug\right]-\delta f_B{L}_B,$$ (12b)

to be compared with Eq. 6.

Results from extended model

To explore further the impact of state-dependent recovery from block, literature values for voltage-dependent apparent association and dissociation rates (2) were implemented in the 10-state model. In particular, data from two compounds were chosen, DPO-1 and DPO-2, because of their similarity in all parameters but the apparent dissociation rate, where they displayed a large difference. These inhibitors have been characterized by measuring the apparent association rate at +40 mV, here denoted k_on, and the apparent dissociation rate at +40 mV and at −80 mV, here denoted $k_{\mathrm{off}}^{\mathrm{open}}$ and $k_{\mathrm{off}}^{\mathrm{closed}}$, respectively. Although the kinetic parameters were determined for hKv1.5 channels, they will be assumed to be the same for the canine Kv1.5. In the 10-state model described above, the drug association rate is γ. Because basically all channels will open at +40 mV, the measured association rate, k_on, translates directly to our parameter γ. The apparent dissociation rate for the collection of blocked states in the 10-state model is δf_B, with f_B being a voltage-dependent function containing the variable Q. Written out, Q depends on the membrane potential, V, on Q₀, the ratio of ζ₀ and ε₀, and on Z_Q, the sum of Z_ɛ and Z_ζ,

$$Q=\frac{\zeta }{\mathit{\varepsilon }}=\frac{{\zeta }_0}{{\mathit{\varepsilon }}_0}\exp \left(\frac{-\left(Z_{\mathit{\varepsilon }}+Z_{\zeta }\right)FV}{RT}\right)=Q_0\exp \left(\frac{-Z_{Q}FV}{RT}\right).$$ (13)

At the measured membrane potentials +40 mV and −80 mV, the apparent dissociation rate is known. The parameters Q₀, Z_Q, and δ should therefore be set so that the two conditions

$$\delta f_B\left(40\right)=k_{\mathrm{off}}^{\mathrm{open}}\text{and}\delta f_B\left(-80\right)=k_{\mathrm{off}}^{\mathrm{closed}}$$

are satisfied. With three parameters and only two relations, there is no unique solution. However, the solution can be parameterized in terms of, e.g., Z_Q. From the parameterized solution (not shown), it can be noted that the lower the value of Z_Q, the more linear the apparent dissociation rate will be as function of V between the known values at +40 mV and −80 mV. Conversely, the larger the value of Z_Q, the more nonlinear the apparent dissociation rate will be between the known values.

The different combinations of Q₀ and Z_Q that reproduce the kinetic parameters for the two compounds have different interpretations. The parameter Z_Q should be interpreted as the total equivalent charge moved during transition of one of the Kv1.5 subunits, when the drug is bound to the receptor site. In the absence of a bound drug molecule, the parameter optimization resulted in a total equivalent charge movement, Z_α + Z_β, equal to 1.76. This sum constitutes an upper bound for Z_Q provided that binding of the drug, per se, does not induce a conformational change or that the interaction of the drug with the receptor site does not increase the distance translocated by a subunit during opening/closing. However, the binding of the drug may act as an obstacle, decreasing the distance moved in the electrical field, resulting in a value of Z_Q lower than Z_α + Z_β. In any case, nothing is said about the location of the barrier, the exact values of Z_ε and Z_ζ, because of the model reduction. While Z_Q has the interpretation of equivalent charge movement, Q₀ should be interpreted as the relative affinity for the closed state of a subunit in the absence of an electrical force. One can hypothesize that the presence of a bound drug alters this relative affinity. Taken together, in the interpretation of the 10-state model a blocking drug leads to one or two things besides stopping the potassium current through the channel. The subunits may be hindered by the drug, making them move a shorter distance in the electrical field, and the bound drug may alter the subunits relative affinity for the open and closed states. Both events will affect the apparent dissociation rate through their impact on f_B.

Four versions of each compound were considered. For the values 1.75, 1.3, 0.85, and 0.4 of Z_Q, we calculated Q₀ and δ. We did not consider values of Z_Q lower than 0.4 because f_B at this point is already quite linear with respect to V, and little further change is anticipated. All parameter values, including the experimentally determined ones in Lagrutta et al. (2), are shown in Table S1 in the Supporting Material.

Having derived parameter sets for the four versions of each drug, their effect on the APD₉₀ of a remodeled PM-cell was tested using a broad range of concentrations. The results from simulations are shown in Fig. 5. Upper and lower rows are results for a stimulation frequency of 1 Hz and 4 Hz, respectively. Column A shows APD₉₀ for the four versions of compound 1, column B shows APD₉₀ for the four versions of compound 2, and column C shows the differences in ADP₉₀ between columns A and B. The colors black, blue, green, and red corresponds to the different versions with Z_Q set to the values 1.75, 1.3, 0.85, and 0.4. The concentrations tested are K_d2ⁿ, where K_d is the dissociation constant, and where n = −3, −2,…, 7 is shown on the x axis. At 1 Hz, APD₉₀ was 134 ms in the absence of a drug, increasing to values close to 200 ms at high concentrations for all parameter versions. For compound 1 (column A) the particular version was of considerable importance at intermediate concentrations. The fourth version (red curve), with Z_Q = 0.4, gave the strongest prolongation of the AP, and the first version (black curve) gave the weakest. This is reasonable, considering that the voltage dependence of f_B give rise to smaller apparent dissociation rates in the range between +40 mV and −80 mV for Z_Q = 0.4 than for Z_Q = 1.75. For compound 2 (column B), however, the dose-response curve was virtually independent of the parameter setup. The reason for this is that even though the behavior of f_B also differs for the different Z_Q values for compound 2, it has a ratio of $k_{\text{off}}^{\text{open}}$ to $k_{\text{off}}^{\text{closed}}$ equal to 2.27, while this number is 61.7 for compound 1. This means that the relative size of f_B at different Z_Q is less for compound 2 than for compound 1, and hence that the variation of the apparent dissociation rate is less in its different versions. At 4 Hz, APD₉₀ was 118 ms in absence of a drug, increasing to values close to 170 ms at high concentrations for all parameter setups. Also at this frequency, the versions of compound 2 showed practically no differences, in contrast to compound 1 where the parameter setup was again critical for the resulting APD₉₀. In column C, the differences in APD₉₀ of all versions of the two compounds are displayed. At 1 Hz, the maximum difference ranged from 31 ms to 19 ms, and at 4 Hz from 26 ms to 10 ms. Based on these simulations it can be concluded, at least when looking at concentrations on a K_d scale, that DPO-1 is a more efficacious blocker than DPO-2, irrespective of the version implemented. This conclusion is consistent with the fact that DPO-1 is a more potent blocker of human Kv1.5 than DPO-2 (2). DPO-1 has also been shown to increase dog atrial refractoriness and to increase human atrial AP duration (2,8). It should also be noted from the simulations that both compounds display a reverse use-dependence, showing a larger increase in APD₉₀ at 1 Hz than at 4 Hz.

Figure 5.

APD₉₀ values and difference between APD₉₀ values generated by the remodeled PM-cell model using the 10-state blocking model and four versions of compound 1 and compound 2, respectively. Versions 1–4 are encoded by black, blue, green, and red. Upper and lower row are for frequencies 1 Hz and 4 Hz, respectively. Columns A and B correspond to compound 1 and compound 2, respectively. Column C shows the difference between A and B. Drug concentration is given by the expression K_d2ⁿ , where K_d is the dissociation constant, and where n = −3, −2,…, 7.

Discussion

To enable more-realistic studies on the effects of Kv1.5 blockers on the canine atrial AP in silico, two continuous-time Markov models of the guarded receptor type were set up for the Kv1.5 channel. Ion-channel model parameters were determined by using artificially generated data from the RNC model and available experimental data on the kinetics of drug-ion channel interaction. The Kv1.5 Markov models were subsequently inserted to the simple AF-remodeled version of the RNC model.

The main findings from this study are:

• 1.Time- and state-dependent Markov models of open-channel Kv1.5 block gave significantly different results compared to a time- and state-independent model with a downscaled conductance.
• 2.The outcome of Kv1.5 block on the macroscopic system variable APD₉₀ was dependent on the precise mechanism of block.
• 3.Open-channel block produced a reverse use-dependent prolongation of the APD₉₀.

Our data clearly suggest that the choice of ion channel model has a large impact on the simulation results. When the dynamic six-state model was compared to a time- and state-independent model, differences in the simulated APD₉₀ as large as 32 ms were observed. This suggests that the simple down-scaling of I_Kurd conductance commonly used in many studies may be too unsophisticated for quantitative modeling of drug effects.

To be able to simulate drugs showing an apparent voltage dependence in their dissociation rates, a 10-state Markov model was formulated. To our knowledge, this is the first Markov model of Kv1.5 able to capture this phenomena. It was shown how the experimentally measured kinetic parameters of two clinically relevant drugs (2) could be implemented and the model was subsequently used to simulate the effect of these compounds. The simulations showed that the slower recovery of block at polarized potential of DPO-1 made it a more efficacious drug than DPO-2.

There are two frameworks commonly used for modeling state-dependent block—the modulated receptor theory and the guarded receptor theory. Sections reviewing these concepts and the means by which they have been applied in ion-channel modeling can be found in Brennan et al. (32). Briefly, in the modulated receptor theory, drugs interact with their receptor for all conformational states of a channel, each state differing in its kinetics of drug binding and dissociation. In the guarded receptor theory, drug interaction is limited to certain channel states only. The 10-state guarded receptor model of Kv1.5 developed in this study can be considered a special case of the more-general modulated receptor model. Hypothesizing nonzero affinities for all of the closed states in Fig. 1 C would transform the model into a modulated receptor. While adding vertical transitions between the closed and blocked states might allow for more complex and sophisticated behaviors, it would also make system identification from experimental data more demanding. As was seen for the two compounds, all parameters in our model could not even be uniquely determined. Because of the identifiability issue, and the fact that the 10-state model developed contained enough complexity to describe state-dependent recovery from block, a general modulated receptor model was not considered in this study.

Tsujimae et al. (9) took another approach to incorporate the effects of voltage- and time-dependent block, including the voltage-dependent recovery from block. They extended a Hodgkin-Huxley formulation of human I_Kur by introducing a new multiplicative variable, describing the fraction of unblocked channels, in the expression for this current. Their modified formulation is given by

$$I_{\text{Kur}}=y_{\text{Kur}}{g}_{\text{Kur}}{u}_a^3{u}_i\left(V-E_{\text{K}}\right),$$ (14)

where y_Kur is the fraction of I_Kur that is not blocked by a drug. The remaining variables have the same meaning as those of Eq. 5. Because the variables of Hodgkin-Huxley formulations are independent, open-state block was mimicked by designing the voltage-dependent steady-state profile for the fraction of unblocked channels to resemble, qualitatively, the opposite of the activation profile. Time constant profiles describing the kinetics of formation of and recovery from block were set up based on the same compounds investigated by us in this study. The authors connected known values of the time constant of block at two membrane potentials with a Boltzmann function with a half-value and slope of −40 mV and 5 mV, respectively. However, compared with our implementation of the two DPOs, it was not discussed in detail how to determine the voltage-dependent profiles for the relaxation time-constant and for the steady-state fraction of unblocked channels for some particular drug. Neither did their implementation address the effects from charged drugs. Furthermore, our model allows for any value of the drug concentration, whereas the Tsujimae model only considers one particular steady-state profile for the fraction of unblocked channels.

The outcome of the simulations with the 10-state model leads to important conclusions regarding the experimental protocols used to probe potential I_Kurd blockers. Because the precise choice of Q₀ and Z_Q had a major impact on the AP-prolonging effects of one of the compounds, additional measurements of the apparent dissociation rate at other membrane potentials may be crucial. Such additional measurements of the apparent dissociation rate would allow Q₀ and Z_Q to be determined. To increase the chances of good parameter identifiability besides just increasing the number of measurements, the preceding analysis of the 10-state model can be used for optimal experiment design with respect to the voltages at which measurements are to be performed, thereby increasing the predictive power of the model.

It was also found that open-channel block produced larger increases of the canine APD₉₀ at lower frequencies than at higher. This was observed for the six-state model as well as for two specific blockers encoded in the 10-state model, using the remodeled setting. On the other hand, when the six-state model was used with the normal cell setting, a positive frequency dependence was observed. Because an open-state block can, in itself, never give rise to reverse use-dependence, it must be understood that it is an emergent property of the ensemble of all currents and the role played by Kv1.5 in this context. The environment of other ionic currents can be thought to interact with Kv1.5 and Kv1.5 block in two ways.

First, the voltage-dependent activation of Kv1.5 means that the fraction of Kv1.5 susceptible to block is dependent on how V develops during the AP. This, in turn, depends on the density and activity of all other currents. The rate adaption to higher stimulation frequencies, described in Ramirez et al. (12), leads to a decreased APD and a lower fraction of open Kv1.5 channels susceptible to block during the AP. This was seen both in the AF setting and for the normal cell (see Fig. 4 B) However, a lower fraction of open channels did not lead to a lower fraction of blocked channels when the drug was added. This was seen in Fig. 4 D where the fraction of blocked channels is shown at different frequencies. Because the fraction of blocked channels during the initial phase of the AP in fact increases with frequency, it appears that the use-dependence of open-state block itself overcomes the decreasing fraction of susceptible channels at higher frequencies shown in Fig. 4 B.

Second, the impact of actually blocking Kv1.5 again depends on all other currents and the relative importance of I_Kurd compared to them. In Fig. 4 C, it was shown that in addition to the increase of the blocked fraction with higher frequencies in Fig. 4 D, the fraction of open channels is decreasing. Despite the decrease in the fraction of conducting Kv1.5 at higher frequencies observed for both the remodeled and normal cell, the increase in APD has a positive frequency dependence only in the normal cell. This illustrates the fact that there are several factors in addition to the drug-ion channel interaction itself that determine the outcome of ion channel block on a particular systems property, i.e., APD₉₀. It is most likely that such properties are different depending on species, tissue-type, the degree of electrical remodeling, etc. The finding that that DPO-1 produced a positive frequency-dependent increase of APD in human atrial cells (2) does thus not necessarily imply a contradiction to our simulation data. DPO-1 has been shown to increase atrial AERP and thus most likely APD₉₀ in a canine disease model, but no information is available regarding frequency dependency or effects in normal tissue (8). Interestingly, Wu et al. (33) showed that a Kv1.5 block produced a reverse use-dependent increase in pig atrial AERP in vivo.

The changes in conductances used in the AF setting were chosen as to mimic, in a simplistic way, the major ionic alterations observed during tachycardia-dependent remodeling in the dog according to references given. These ionic changes were experimentally observed and model AP morphology was then fitted to experimental data from cells where these ionic changes had been observed. It is clear from our data and Ramirez et al. (12) that a block of I_Kurd in the native canine myocyte model, apart from increasing the duration of the plateau phase of the AP, most likely also prolongs late repolarization phases of the AP, suggesting an increased effective refractory period and, hence, less likelihood of AF initiating. This prolongation has also been observed experimentally in isolated canine myocytes (34). Although the simplistic remodeled atrial cell model used in this study does not fully capture all changes that occur during the remodeling process and the predictive value, thus, is uncertain, our data clearly suggest that more-complex ion-channel models are a prerequisite for quantitative modeling of drug effects.