In silico analysis of the dynamic regulation of cardiac electrophysiology by Kv11.1 ion‐channel trafficking

Cardiac electrophysiology is regulated by continuous trafficking and internalization of ion channels occurring over minutes to hours. Kv11.1 (also known as hERG) underlies the rapidly activating delayed‐rectifier K+ current (IKr), which plays a major role in cardiac ventricular repolarization. Experimental characterization of the distinct temporal effects of genetic and acquired modulators on channel trafficking and gating is challenging. Computer models are instrumental in elucidating these effects, but no currently available model incorporates ion‐channel trafficking. Here, we present a novel computational model that reproduces the experimentally observed production, forward trafficking, internalization, recycling and degradation of Kv11.1 channels, as well as their modulation by temperature, pentamidine, dofetilide and extracellular K+. The acute effects of these modulators on channel gating were also incorporated and integrated with the trafficking model in the O'Hara–Rudy human ventricular cardiomyocyte model. Supraphysiological dofetilide concentrations substantially increased Kv11.1 membrane levels while also producing a significant channel block. However, clinically relevant concentrations did not affect trafficking. Similarly, severe hypokalaemia reduced Kv11.1 membrane levels based on long‐term culture data, but had limited effect based on short‐term data. By contrast, clinically relevant elevations in temperature acutely increased IKr due to faster kinetics, while after 24 h, IKr was decreased due to reduced Kv11.1 membrane levels. The opposite was true for lower temperatures. Taken together, our model reveals a complex temporal regulation of cardiac electrophysiology by temperature, hypokalaemia, and dofetilide through competing effects on channel gating and trafficking, and provides a framework for future studies assessing the role of impaired trafficking in cardiac arrhythmias.


Introduction
Regulation of the number and activity of cardiac ion channels occurs over a wide range of timescales and enables dynamic changes in cardiac electrophysiology to adapt to varying circumstances. While post-translational changes in channel activity can occur within seconds, continuous turnover of ion channels happens over minutes to hours, through trafficking, internalization, degradation and recycling processes (Apaja et al., 2013;Dennis et al., 2011;Foo et al., 2019;Ghosh et al., 2018;Guo et al., 2009;Kanner et al., 2018;Ke et al., 2013;Li et al., 2022;Osterbur Badhey et al., 2017;Shi et al., 2015). Proteins are folded and glycosylated in the endoplasmic reticulum (ER) and Golgi complex (GC), after which the channels are transported to, and integrated into, the membrane. Once integrated, channels can gate and in their open state allow ions to pass into or out of the cell along the electrochemical gradient, producing an ion current. In the opposite direction, channels are internalized, with some subsequently undergoing lysosomal or proteasomal degradation, while the others are recycled back to the membrane. This complex dynamic flux of ion channels is a major regulator of cardiac electrophysiology (Balse & Boycott, 2017;Blandin et al., 2021;van der Heyden et al., 2018).
The K v 11.1 or hERG channel underlies the rapidly activating delayed-rectifier K + current (I Kr ). K v 11.1 plays a major role in the electrical recovery (repolarization) of human ventricular cardiomyocytes and its trafficking is highly dynamic (Apaja et al., 2013;Dennis et al., 2011;Foo et al., 2019;Guo et al., 2009;Kanner et al., 2018;Ke et al., 2013;Osterbur Badhey et al., 2017;Shi et al., 2015). K v 11.1 loss-of-function mutations result in prolonged repolarization and are associated with long-QT syndrome type-2 (LQTS2). LQTS2 is the second most prevalent type of LQTS and is associated with life-threatening polymorphic ventricular tachyarrhythmias, notably torsades de pointes (Wallace et al., 2019). Impaired trafficking is the dominant modus operandi of LQTS2-associated K v 11.1 missense mutations (Anderson et al., 2014;Smith et al., 2016). However, the mechanisms and dynamics underlying these trafficking deficiencies can differ markedly between mutations, highlighting the complex, highly dynamic nature of K v 11.1-channel trafficking 0 Stefan Meier obtained his Master's degree in 'Systems Biology' at Maastricht University in 2021 and is currently pursuing a PhD in computational electrocardiology at CARIM (Maastricht University) under the supervision of Dr Jordi Heijman and Prof. Dr Paul G.A. Volders. Previously, he primarily focused on heart failure with preserved ejection fraction, but he transitioned to cardiac electrophysiology and arrhythmias for his PhD. He is especially interested in the temporal dynamics of cardiac ion-channel trafficking and how to incorporate this in computer/mathematical models. (Apaja et al., 2013;Foo et al., 2019;Kanner et al., 2018;Ke et al., 2013;Osterbur Badhey et al., 2017). Adding to this complexity are several modulators of K v 11.1 transport like temperature, drugs and the concentration of extracellular potassium ([K + ]). Febrile temperatures can acutely speed up K v 11.1 gating, resulting in larger currents (Mauerhofer & Bauer, 2016;Vandenberg et al., 2006;Zhou et al., 1998), while long-term exposure to high temperatures reduces the number of K v 11.1 channels in the membrane (Foo et al., 2019;Zhao et al., 2016). The opposite is true for hypothermia, which acutely decreased I Kr due to slower K v 11.1 channel gating, but over the long term increased I Kr due to increased K v 11.1 trafficking. Moreover, hypokalaemia disrupts K v 11.1 channel gating and trafficking processes by promoting a non-conductive state which is prone to channel internalization and degradation (Guo et al., 2009;Massaeli et al., 2010). K v 11.1 drug block is an important factor in drug-induced proarrhythmia and therefore plays an important role in cardiac safety pharmacology (Heijman et al., 2014). Interestingly, while several drugs (e.g. E-4031, dofetilide, astemizole) acutely block the K v 11.1 channel and prolong repolarization, these drugs may rescue aberrant channel trafficking when applied for hours Dennis et al., 2012;Qile et al., 2020;Smith et al., 2013;Varkevisser et al., 2013). The characterization of these diverse modulating effects is experimentally challenging due to the different timescales involved (milliseconds to hours) and methodologies required (e.g. cell fixation and membrane-specific ion-channel staining vs. live-cell patch-clamp recording).
Computer models offer perfect observability and control (Heijman et al., 2021), which may help to characterize the above-mentioned trafficking processes and their modulation by various pathophysiological factors. From the initial computer models of cardiomyocyte electrophysiology developed in the second-half of the 20th century to the more advanced models of the 21st century, much progress has been made with respect to model complexity and scale (Heijman et al., 2016(Heijman et al., , 2021. Detailed Markov models (MMs) of ion-channel gating have been developed. For example, the five-state I Kr MM by Clancy and Rudy (Clancy & Rudy, 2001) can reproduce a wide range of experimental J Physiol 601.13 voltage-clamp data and, when embedded in an action potential (AP) model like the O'Hara-Rudy (ORd) human ventricular cardiomyocyte model (O'Hara et al., 2011), can simulate the genesis of early afterdepolarizations (EADs) and other arrhythmogenic responses (O'Hara et al., 2011). The ORd model currently serves as a basis for the 'comprehensive in vitro proarrhythmia assay' (CiPA) initiative proposed by the US Food and Drug Administration and pharmaceutical industry for improved screening of the risk of drug-induced ventricular arrhythmias, highlighting the real-world relevance of cellular electrophysiology models (Dutta et al., 2017;Li et al., 2020). However, none of the currently available AP models incorporate the dynamic trafficking of ion channels; instead they assume that the number of ion channels is fixed, precluding simulation of long-term regulation of cardiac electrophysiology by drugs, fever, hypokalaemia and other modulators.
Here, we developed a novel K v 11.1-trafficking component that enabled simulation of the regulation of K v 11.1 membrane levels over minutes to hours by temperature, two selected drugs (i.e. the trafficking blocker pentamidine and trafficking rescuer dofetilide) and extracellular [K + ], as well as their effects on cardiac cellular electrophysiology.

K v 11.1 trafficking model
A two-state model with sub-membrane (S) and membrane (M) states and four rates was created to model K v 11.1-channel trafficking (Fig. 1A). The state transitions were modelled by two ordinary differential equations: where ψ represents the channel production rate, α the forward trafficking rate, β the internalization rate and δ the degradation rate. The rates were optimized through stochastic single-channel simulations (Heijman et al., 2013), which generally aimed to minimize the sum of squared errors between the model trafficking behaviour and those observed in literature (Apaja et al., 2013;Dennis et al., 2011). The total number of K v 11.1 channels in the membrane was calibrated based on estimates from Heijman et al. (2013) which were based on whole-cell and single-channel conductance. The final rate configuration can be found in Table 1.  (4), (5), (9) and (10), respectively. Moreover, the κ bref and κ dref are the results from eqns (9) and (10) (Clancy & Rudy, 2001).
In particular, the α n , β n , α 2 , and μ rates in the MM were scaled with Q 10Activation and Q 10Deactivation , while α i and β i were scaled with Q 10Inactivation and Q 10Recovery , respectively ( Fig. 1B), whereby the scaling factor (sf) for each Q 10 was calculated as: The Q 10 values and temperature-dependent shift in V 1/2 of I Kr activation were calibrated based on experimental data (Mauerhofer & Bauer, 2016;Zhou et al., 1998) by replicating the experimental voltage-clamp protocols at different temperatures, calculating the corresponding Q 10 values, and optimizing the sum of squared errors between the model values and those found in literature.
Ion-channel trafficking is also affected by temperature, which was modelled using an asymmetrical sigmoid function: where a is the amplitude, b is the midpoint, c the steepness, s represents the asymmetry around the midpoint (with a value of 1 resulting in a symmetrical sigmoidal function) and d is a constant that prevents the number of channels from going to zero. Again, the temperature effects on trafficking were calibrated based on experimental data (Foo et al., 2019;Zhao et al., 2016). Finally, θ was used to scale the ψ rate in the trafficking model (Fig. 1A). The MM equations, parameters and their temperature-dependent changes can be found in Tables 2 and 3.
Modelling drug effects on K v 11.1 gating and trafficking The effects of pentamidine and dofetilide on acute gating and long-term trafficking were also incorporated in the model. Pentamidine is a K v 11.1-trafficking inhibitor that does not affect channel gating, whereas dofetilide is a potent K v 11.1 channel blocker but long-term exposure (e.g. hours) to dofetilide rescues impaired K v 11.1 trafficking (Asahi et al., 2019;de Git et al., 2013;Varkevisser et al., 2013). The opposing effects of pentamidine and dofetilide were modelled as: where [P] is the pentamidine concentration in μmol/l, [D] is the dofetilide concentration in μmol/l, h is the Hill factor related to pentamidine, h D is the Hill factor related to dofetilide, a is the magnitude of the dofetilide-induced promotion of trafficking, km D is the affinity of dofetilide, and km is the affinity of pentamidine. We defined km as: Figure 1. Model components required to simulate regulation of K v 11.1 trafficking and gating, and its effects on ventricular cardiomyocyte electrophysiology A, the K v 11.1 trafficking model consists of two states (M: Membrane, S: Sub-membrane) with four rates (ψ: production rate, α: forward trafficking rate, β: internalization rate and δ: degradation rate). The temperature (θ ), drugs (λ) and extracellular [K + ] (κ b and κ d ) parameters are used to scale the ψ, β and δ rates. B, the Clancy and Rudy (2001) I Kr Markov model was used to create a temperature-sensitive model of I Kr gating by shifting the voltage dependence of certain rates (see Table 2) and scaling each rate with their respective Q 10 values (Clancy and Rudy, 2001). In particular, α n , β n , α 2 , and μ rates were scaled with Q 10Activation and Q 10Deactivation , while α i and β i were scaled with Q 10Inactivation and Q 10Recovery , respectively.
The V 1/2Activation/Deactivation and V 1/2Inactivation/Recovery are abbreviated to V 1/2AD and V 1/2IR , respectively. Note, the temperature (T), voltage (V) and extracellular [K + ] are not pre-defined but inputted during simulation. The MM rates are defined in milliseconds −1 (ms).
where km is the baseline affinity of pentamidine in the absence of dofetilide, [D] is the dofetilide concentration in μmol/l, and R determines the impact of dofetilide on the affinity of pentamidine. Furthermore, we defined a as: where a is the magnitude of the dofetilide-induced promotion of trafficking in the presence of pentamidine, [P] is the pentamidine concentration in μmol/l, and b is the midpoint of the pentamidine dependence. The parameters from eqns (5) to (7) were calibrated based on the concentration dependence of both pentamidine and dofetilide, as well as the time-dependent effects of dofetilide reported in literature (Asahi et al., 2019;Varkevisser et al., 2013) and λ was used to scale ψ in the trafficking model. The mean plasma concentration of dofetilide over 24 h was extracted from literature and converted to μmol/l based on the molecular weight of dofetilide (Allen et al., 2000). This concentration was regarded as a 'clinical dose' and used to run simulations to model the effects of dofetilide-induced rescue of channel trafficking. Finally, the acute I Kr -blocking effect of dofetilide was modelled based on previously reported IC 50 values (Sutanto et al., 2019) as: 0.008 (8) where [D] is the dofetilide concentration in μmol/l. The final drug-modelling parameters can be found in Table 4.

Modelling the effects of extracellular [K + ] on K v 11.1 gating and trafficking
The ORd model is sensitive to changes in extracellular [K + ], which, besides changing the driving force for all K + currents, also modulates the gating of I Kr and the inward-rectifier K + current (I K1 ). However, extracellular [K + ] also modulates K v 11.1 trafficking (Guo et al., 2009;Massaeli et al., 2010), which is not part of the original ORd model. Here, we modelled the trafficking effects of hypokalaemia through changes in β and δ, because experimental studies have shown that hypokalaemia primarily affects K v 11.1 channel internalization and degradation (Guo et al., 2009;Massaeli et al., 2010).
In particular, the β rate was scaled by a factor κ b (Fig. 1A), as follows: where s is a scalar determining the relative impact of extracellular [K + ] on δ versus β. The final parameters related to extracellular [K + ] can be found in Table 5 based on two different experimental data sets.

Embedding in the O'Hara-Rudy human ventricular action potential model
The trafficking effects of temperature, drugs and extracellular [K + ] were introduced as scaling factors to the appropriate rates in the trafficking model: where ψ base is the baseline production rate as shown in Table 1, λ represents the opposing effects of pentamidine and dofetilide, and θ represents the temperature-dependent regulation of K v 11.1-channel trafficking. In addition, where β base is the baseline internalization rate (Table 1), where δ base is the baseline degradation rate (Table 1) where γ is the acute I Kr -blocking effect of dofetilide, M represents the number of K v 11.1 channels in the membrane, M ref represents the number of K v 11.1 membrane channels in steady-state at baseline (i.e. in the absence of drugs, at 37°C, and with 5.4 mmol/l extracellular [K + ]), G Kr is the maximum conductance of K v 11.1 channels, O is the open probability of the I Kr MM model, V M is the membrane potential, and E rev is the reversal potential of I Kr .

Statistics, software and data availability
The experimental data are presented as means and standard deviations. All the simulations were performed through Myokit and Python (version 3.7.6.) (Clerx et al., 2016). The model code, optimization scripts and data can be found online at: https://github.com/HeijmanLab.

Calibration of the trafficking model
Stochastic simulation of the trafficking model allowed us to track single-channel transitions over time. Fig. 2A shows the state transitions over time for two channels, where the top panel shows a channel coming into existence after approximately 20 min and then continuously switching between the membrane and sub-membrane states. The lower panel shows another channel that has a much shorter lifespan of approximately 6.5 h. This single-channel resolution made it possible to replicate experimental data involving tagging of individual channels to quantify internalization and recycling rates (Apaja et al., 2013;Dennis et al., 2011). After parameter optimization, the model was in good agreement with the experimental behaviour (Fig. 2B), with the exception of the first recycling timepoint (Apaja et al., 2013;Dennis et al., 2011). We subsequently evaluated the simulated decay of K v 11.1 channels in the presence of forward trafficking block with these parameters and found substantial agreement with a wide range of experimental data (Fig. 2C), providing independent validation of the model fit (Apaja et al., 2013;Foo et al., 2019;Guo et al., 2009;Ke et al., 2013;Osterbur Badhey et al., 2017;Shi et al., 2015).

Sensitivity analysis
A sensitivity analysis was performed to obtain a better understanding of the impact of individual rates on K v 11.1 membrane levels and their dynamic interplay. Each parameter was scaled separately (i.e. 4-, 2-, 1-, 0.5-and 0.25-fold) while the other parameters were kept at their optimized value. As expected, the number of channels in the membrane state increased after augmenting either the production rate or the forward trafficking rate, while augmenting the internalization or degradation rates decreased the number of membrane channels (Fig. 3, left panels). Increasing or decreasing the production and degradation rates (ψ and δ, respectively) had similar effects on the sub-membrane state as the membrane state, with increased production or decreased degradation increasing the total number of channels and vice versa. By contrast, scaling the α and β rates only had a transient effect, with a relatively quick convergence to the original number of channels in the sub-membrane state (Fig. 3, right panels). Thus, ψ and δ modulate total K v 11.1 levels, whereas α and β primarily affect the distribution between sub-membrane and membrane states.
Modelling trafficking-deficient K v 11.1 mutations K v 11.1 missense mutations associated with LQTS2 are generally associated with trafficking deficiencies in the forward trafficking direction, although reduced channel stability, increased retrograde trafficking and increased degradation have also been reported (Apaja et al., 2013;Foo et al., 2019;Kanner et al., 2018;Ke et al., 2013). We hypothesized that fitting the K v 11.1-trafficking model to experimental data on a K v 11.1 trafficking-deficient mutation would provide information on the primary underlying pathophysiological mechanism, reflected by the most-affected model parameter. The K v 11.1-p.(Ala57Pro) missense mutation (p.(A57P)) characterized by Kanner et al. (2018) was chosen as a representative example to test this hypothesis. K v 11.1-p.(A57P) is a forward trafficking-deficient mutation with normal internalization (Fig. 4A), resulting in an approximately 35% reduction of K v 11.1 membrane levels (Kanner et al., 2018). Because the temporal dynamics of both the wild-type (WT) and mutant channel measured by Kanner et al. (2018) were substantially faster (half-time: approximately 12 min) than the majority of the other experimental sources (half-time: approximately 8 h; Fig. 2B), we focused on the relative differences in dynamics between the WT and p.(A57P) mutant. Four starting points for parameter optimization were created by scaling each model rate individually to approximate the 35% reduction of K v 11.1 membrane channel levels, while keeping the other rates constant. For example, parameter set 1 was obtained by reducing α by approximately 35% while the other rates were kept at their WT values. Initial values for parameter sets 2−4 were obtained similarly by scaling β, δ and ψ. Thereafter, all four parameter sets were optimized by updating all the rates from each parameter set. An overview of the optimized parameter sets is presented in Table 6. Despite substantial differences in parameter values, all optimized parameter sets could accurately reproduce the observed . B, comparison of internalization and recycling rates derived from single-channel simulations (black bars) to experimental data from biotinylation assays and ELISA methods (Apaja et al., 2013;Dennis et al., 2011). C, decrease in the relative number of K v 11.1 channels in response to forward trafficking block in the model (black lines, with ψ = 0) and experimental data (symbols) (Apaja et al., 2013;Foo et al., 2019;Guo et al., 2009;Ke et al., 2013;Osterbur Badhey et al., 2017;Shi et al., 2015). Note that some experiments solely focused on membrane channels, whereas others quantified the amount of mature K v 11.1 channels (155 kDa) or total amount of K v 11.1 channels (e.g. mature and immature bands at 155 kDa and 135 kDa, respectively). However, model results based on membrane channels only or membrane and sub-membrane channels combined were superimposable (not shown  difference in membrane stability between WT and mutant K v 11.1 (Fig. 4A and B). Furthermore, each parameter set resulted in a similar reduction of membrane channels compared with WT (∼35%; Fig. 4C). As expected, the reduction in membrane channels also resulted in AP prolongation (Fig. 4D), consistent with the clinical LQTS phenotype of carriers of the K v 11.1-p.(A57P) mutation (Anderson et al., 2014;Kapplinger et al., 2009). A similar I Kr reduction in the original ORd model resulted in a comparable APD prolongation of approximately 100 ms (data not shown). Together, the data from Figs 3 and 4 indicate that data on forward trafficking and membrane stability are sufficient to reproduce the changes in K v 11.1 membrane levels and clinical phenotype, but cannot identify the underlying molecular defect.

Modelling dofetilide-induced rescue of channel trafficking
After parameter optimization, the model could accurately reproduce the concentration-dependent effect of pentamidine on K v 11.1 membrane levels (Fig. 5A), as well as the concentration and time-dependent rescue of mature K v 11.1 levels (155 kDa) by dofetilide in the presence of 10 μmol/l pentamidine ( Fig. 5B and C).
We subsequently employed the model to investigate the combined effect of acute channel inhibition and long-term trafficking promotion by dofetilide in the presence of 5 μmol/l pentamidine (Fig. 6, dashed lines) and absence of pentamidine (Fig. 6, solid lines). A supraphysiological (1 μmol/l) concentration of dofetilide acutely completely MT) compared with wild-type K v 11.1 (black squares, Exp. WT) in experimental recordings (Kanner et al., 2018). B, similar to panel A for simulations with the default parameters representing wild-type K v 11.1 (black line, model WT) and four different parameter sets obtained through optimization on the relative differences between wild-type and K v 11.1-p.(A57P) after a perturbation in each of the four rates. C, the number of membrane channels for the wild-type model and each of the four parameter sets that was optimized to mimic the K v 11. inhibited I Kr , while promoting an approximately 50% increase in membrane channels over 24 h in the presence of pentamidine (Fig. 6A, dashed lines). By contrast, in the absence of pentamidine, dofetilide had a negligible effect on K v 11.1 membrane trafficking (Fig. 6A, solid lines). The acute inhibition resulted in repolarization failure during dofetilide treatment (25 th hour; Fig. 6A, bottom panel). However, the rescue of K v 11.1 membrane channels after 24 h of dofetilide caused a slight AP shortening shortly after dofetilide application was stopped, counteracting the effects of pentamidine (50 th hour; Fig. 6A, bottom panel). By contrast, a more clinically relevant dofetilide concentration (3.4 nmol/l) produced a modest acute I Kr inhibition that prolonged AP duration (APD), while having a minimal rescuing effect on membrane channel numbers after 24 h, independent of the presence of pentamidine (Fig. 6B). As such, there was only a minimal rebound in APD after cessation of simulated dofetilide application. Thus, while dofetilide can rescue K v 11.1 trafficking under pathological conditions (e.g. in the presence of the trafficking-blocker pentamidine), this effect appears negligible at clinically relevant concentrations. However, other drugs and trafficking modulators such as temperature may alter K v 11.1 membrane levels over a physiological range.
Modelling temperature-dependent modulation of K v 11.1 channel trafficking Similar to dofetilide, temperature acutely affects K v 11.1 gating, while modulating channel trafficking over hours. Fig. 7A shows the calibration of the acute effects of temperature on gating through Q 10 values and shifts in V 1/2 (Mauerhofer & Bauer, 2016;Zhou et al., 1998). Combined, these effects resulted in faster gating and larger I Kr at elevated temperatures (Fig. 7B), consistent with experimental data (Amin et al., 2008) (Fig. 7C). To model the long-term effects of temperature on trafficking, the parameters of eqn (4) were calibrated to experimental data with temperatures ranging from 30°C to 41°C (Fig. 8A) (Foo et al., 2019;Zhao et al., 2016). Overall, higher temperatures resulted in fewer channels in the membrane, while lower temperatures increased the number of membrane channels. The combined acute and long-term effects of altered temperatures are shown in Fig. 8B. Higher temperatures (fever) produced a slight acute increase in I Kr due to faster channel gating. However, after 24 h, I Kr decreased substantially due to reduced membrane expression of K v 11.1 channels. This was also reflected in a slight acute shortening of APD and extreme prolongation of APD after 24 h (Fig. 8C). The opposite was true for lower temperatures, albeit less pronounced.

Modulation of K v 11.1 gating and trafficking by hypokalaemia
Extracellular [K + ] is a prominent regulator of ventricular electrophysiology, with both hyper-and hypokalaemia being associated with an increased risk of cardiac arrhythmias. Previously, hypokalaemia has been shown to negatively regulate K v 11.1 channel gating and membrane stability in a concentration-dependent manner through increased internalization and degradation (Guo et al., 2009;Massaeli et al., 2010). The model's extracellular [K + ] dependence was calibrated to experimental data from Guo et al. (2009), which revealed a distinct half-maximal concentration after overnight (i.e. 12 h) incubation compared with incubation for a week (Fig. 9A, left vs. right panel). The rate of decrease in K v 11.1 membrane levels in the presence of low (0.1 mmol/l) extracellular [K + ] and the rate of recovery of K v 11.1 membrane expression after switching back to 5.0 mmol/l extracellular [K + ] following overnight incubation at 0.1 mmol/l were also calibrated based on experimental data (Fig. 9B). The corresponding 'overnight' and 'week' parameter sets can be found in Table 5. Subsequently, we performed simulations similar to those in Fig. 6 to evaluate the combined acute and long-term (trafficking) effects of hypokalaemia. After 24 h, the [K + ] was reduced from 5.4 mmol/l to 2.5 mmol/l, reflecting a clinically relevant hypokalaemia. The 'overnight' parameter set resulted in an approximately 20% reduction in I Kr ; however, the amount of K v 11.1 membrane channels remained stable, reflecting the acute effects of hypokalaemia on channel-gating over time (Fig. 9C). For the 'week' parameter set, the reduction in I Kr was much more pronounced (e.g. approximately 45%) due to an additional 25% reduction in K v 11.1 membrane channels (Fig. 9C). This is also reflected in differences between APD prolongation immediately after extracellular [K + ] was reduced to 2.5 mmol/l (25 th hour; Fig. 9D) and towards the end of the hypokalaemic period (47 th hour). With the 'overnight' parameters, APD remained mostly stable after the first hour of hypokalaemia (Fig. 9D, left panel). By contrast, the APD related to the 'week' parameters substantially increased during hypokalaemia (Fig. 9D, right panel).

Discussion
Here, we developed a novel computational model of ion-channel trafficking that can reproduce a wide range of experimental data, including the temperature, extracellular [K + ] and/or drug-dependent regulation of K v 11.1 gating and trafficking. Our sensitivity analyses revealed that multiple mechanisms, all producing a similar steady-state decrease in K v 11.1 membrane levels, can underlie the phenotypic effects of LQTS2-associated missense mutations. Moreover, the model indicates that simulated application of dofetilide (in the presence of pentamidine), changes in temperature or hypokalaemia produce complex dynamic changes in I Kr , and consequently APD, due to interactions between acute gating and long-term trafficking effects. These results underscore the importance of evaluating the time course of dynamic regulation of cardiac electrophysiology, rather than only studying acute or steady-state effects.

Computational modelling of ion-channel trafficking
The trafficking model was designed for K v 11.1 channels because the data on the temporal dynamics of ion-channel trafficking are most abundant for this channel. However, the molecular basis of ion-channel trafficking is highly complex, with numerous regulators that are crucial for normal trafficking (Blandin et al., 2021). These include, but are not limited to, the motifs involved in the ER-associated degradation system, the Ras-associated binding proteins, and the anchoring, tethering and scaffolding proteins (Basheer & Shaw, 2016;Blandin et al., 2021). Moreover, native K v 11.1 channels are composed of two α-subunits (K v 11.1a and K v 11.1b) and channel trafficking depends on the exact subunit composition (Phartiyal et al., 2008). In particular, K v 11.1b subunits are retained in the ER, unless they are co-assembled with K v 11.1a (Phartiyal et al., 2008). Our model presents a simplified representation of this complex trafficking paradigm, where all processes taking place at the nucleus, ER and GC are lumped together in a single rate (ψ) and no other regulating proteins/processes of forward trafficking besides temperature, pentamidine and dofetilide were considered. In this model structure, ψ reflects both channel production and the first part of forward trafficking (e.g. microtubule-mediated trafficking between ER and GC), so we cannot distinguish between the effects of modulators on these two components. Finally, we also ignored the dynamics of subunit composition and channel assembly. Nevertheless, this Figure 7. Temperature-dependent regulation of K v 11.1 gating A, calibration of the shift in midpoint of voltage dependence and Q 10 values for activation, deactivation, inactivation and recovery of I Kr in experimental recordings (Mauerhofer & Bauer, 2016;Zhou et al., 1998) and model. B, combined effects of temperature-dependent changes in midpoint and Q 10 on I Kr at 30, 37 and 40°C. Inset shows voltage-clamp protocol for steady-state and tail I Kr . C, relative tail current amplitudes at 23, 35 and 40°C normalized to 35°C obtained with the voltage-clamp protocol from panel B in experimental recordings (Amin et al., 2008) and model. [Colour figure can be viewed at wileyonlinelibrary.com] simplified and partially phenomenological representation of the trafficking paradigm is able to reproduce the vast majority of the experimentally observed characteristics of K v 11.1 trafficking in a computationally efficient manner. In the future, our model could be extended by compartmentalizing the ψ rate into an ER and GC state, but this would require experimental data on the trafficking dynamics between these compartments, which, to the best of our knowledge, are currently not available for K v 11.1.
Of note, the underlying structure of this trafficking model can be applied more broadly, given that most channels rely on a similar trafficking paradigm. For example, Ghosh et al. (2018) and Li et al. (2022) showed that L-type Ca 2+ channels and K ir 2.1 channels also dynamically circulate between the cytoplasm and plasma membrane. Both studies also highlighted the importance of the cytoskeleton in trafficking, where disruption of actin and tubulin was associated with impaired trafficking of both K ir 2.1 and L-type Ca 2+ channels, while inhibition of dynamin motor-proteins resulted in reduced K ir 2.1 channel internalization. Similarly, small-conductance Ca 2+ -activated K + channels display highly dynamic trafficking behaviour (Heijman & Dobrev, 2017), which is Figure 8. Temperature-dependent regulation of K v 11.1 gating and trafficking and its effect on ventricular cardiomyocyte repolarization A, time course of changes in K v 11.1 membrane or membrane and sub-membrane levels in response to temperature changes in experiments (symbols) (Foo et al., 2019;Zhao et al., 2016) and model (lines). B, combined effect of temperature-dependent changes on K v 11.1 gating and trafficking. Higher temperatures acutely increase I Kr due to faster gating, but after 24 h I Kr is decreased due to reduced membrane expression of K v 11.1 channels, producing significant prolongation of action potential (AP) duration. C, AP morphology for the first AP after a change in temperature (dashed lines) or after 24 h (solid lines) at 30, 37, 40 and 41°C, showing the opposing acute and long-term effects of temperature-dependent regulation of K v 11.1. [Colour figure can be viewed at wileyonlinelibrary.com] J Physiol 601.13 partially regulated by atrial rate (Ozgen et al., 2007) and dependent on cytoskeletal proteins (Rafizadeh et al., 2014;Zhang et al., 2017), adding another layer of complexity to the trafficking paradigm that could be incorporated in future versions of the model when quantitative data on the impact on channel trafficking are available. The integration of trafficking models for multiple ion channels would also make it possible to model reciprocities due to α-α subunit interactions between channels. For example, K v 11.1-channel trafficking and functioning can be modulated by K v 7.1 (KCNQ1), the α-subunit of the slow-activating delayed-rectifier K + current (I Ks ) (Ehrlich et al., 2004;Guo et al., 2011), whereas Na v 1.5 expression reduces K ir 2.1 internalization in rodents (Milstein et al., 2012).

Potential limitations
Experimental data on ion-channel trafficking are relatively scarce and the data that are available were obtained with different methodologies, under varying conditions, and often in distinct heterologous expression systems (e.g. HeLa,and H9C2 cells). Despite these sources of heterogeneity, the model was consistent with the vast majority of data, suggesting that the temporal dynamics are in the same order of magnitude regardless of cell type and detection method. The only notable exception for K v 11.1 was the high-throughput flow cytometry data from Kanner et al. (2018), where the half-time for both forward trafficking and internalization were in the order of minutes rather than hours. The field of ion-channel trafficking  (Guo et al., 2009) (black line/symbols) and corresponding model versions (red lines). Experimental data were based on I Kr recordings in 5 mmol/l [K + ] after incubation at the indicated concentration for 12 h or one week, which were used as a proxy for K v 11.1 membrane levels. B, time course of reduction in K v 11.1 membrane levels in response to incubation in low (0 mmol/l in experiments, 0.1 mmol/l in model) extracellular [K + ] (left) or recovery after 12 h at low extracellular [K + ] following re-exposure to 5 mmol/l extracellular [K + ] (right) in experimental data (Guo et al., 2009) as well as 'overnight' and 'week' model configurations. C, simulated time course of I Kr (left) and K v 11.1 membrane levels (right) during 24 h at baseline (i.e. 5.4 mmol/l [K + ]), followed by 24 h with hypokalaemia (2.5 mmol/l), and 24 h at baseline, revealing acute inhibition for both the 'overnight' and 'week' model configurations and long-term decrease of K v 11.1 membrane levels for the 'week' parameters. The dashed vertical lines (grey) indicate the start and end of hypokalaemia. D, action-potential morphology at various time points from the simulations in panel C for the 'overnight' parameters (dashed lines) and 'week' parameters (solid lines), showing the acute prolongation of repolarization duration for both parameter sets and subsequent additional APD prolongation for the 'week' parameters, which remains present after cessation of hypokalaemia due to the decrease in K v 11.1 membrane levels (compare blue vs. black curves in right panel). [Colour figure can be viewed at wileyonlinelibrary.com] is evolving rapidly and novel methodologies with higher spatial and temporal resolution may cause reconsideration of the current conceptual framework (Ghosh et al., 2018;Kanner et al., 2018;Li et al., 2022), warranting updates to the model. Moreover, the distinct acute and long-term effects identified in the present study would likely still apply even if future experiments would show faster time courses, just for different time points (e.g. 1 vs. 12 h could be 10 vs. 60 min). The 'long-term' effects of trafficking modulators would then be observable after a couple of hours instead of 24−48 h. However, at present our model provides a parsimonious representation of the current understanding of K v 11.1 trafficking.
Despite the model's simplicity, our sensitivity analyses (Figs 3 and 4) revealed that similar phenotypic behaviour of LQTS2 mutations can be obtained through markedly different parameter combinations, even for a mutation (p.(A57P)) for which experimental data indicate that only forward trafficking is impaired. Mutations with more complex phenotypes (Kanner et al., 2018) are even less likely to provide a unique parameter set. It cannot be excluded that these different parameter sets show distinct behaviour under other (patho)physiological conditions, or a distinct response to interventions. This aspect could be assessed in more detail through a population-of-models approach, which would make it possible to assess the likelihood of outcomes such as the formation of afterdepolarizations, rather than considering only a binary (yes/no) result (Heijman et al., 2021;Ni et al., 2018;Sobie, 2009).
In general, the model was able to closely mimic experimental data, with only minor deviations from the experimentally observed range in Figs 2B, 5B and 9B. In particular, the experimental data from Dennis et al. (2011) in Fig. 2B showed approximately 60% K v 11.1 recycling within 3 min after 30 min of experimental channel internalization. Thereafter, the amount of channel recycling remains stable. Our model shows a more sigmoidal increase in channel recycling, with the model recycling rate falling within the experimental standard deviation after 10 min. This difference might lead to an underestimate of the short-term (occurring within 5 min) effects of modulators of channel recycling and should be taken into consideration when interpreting our findings. The model also slightly underestimated the effect of dofetilide on K v 11.1 trafficking at low doses (Fig. 5B), since we emphasized the 1 μmol/l concentration, which is what is primarily used experimentally. As such, the effect of clinically relevant concentrations may have been slightly underestimated. In Fig. 9B, the reduction in membrane K v 11.1 channels during hypokalaemia was slightly faster and less pronounced in the model than in the experiments, whereas the recovery after hypokalaemia was a bit too slow, although it was still within the standard deviation of the experimental data. More complex models might be able to approximate these data even more closely, but the general agreement with numerous data sets is noteworthy given the relative simplicity of the model.
The effects of temperature, pentamidine, dofetilide and hypokalaemia were implemented in a phenomenological way by scaling the production, internalization and degradation rates. There are many other known pharmacological modulators of K v 11.1 trafficking, including cardiac glycosides, tricyclic anti-depressants and E-4031 (Apaja et al., 2013;de Git et al., 2013;Dennis et al., 2011;van der Heyden et al., 2008). Similarly, other (patho)physiological factors such as hyperglycaemia are known to modulate ion-channel trafficking, which may contribute to their proarrhythmic risk (Shi et al., 2015).

Implications for cardiac arrhythmogenesis
Here, we focused on dofetilide as a modulator of K v 11.1 trafficking because of its clinical relevance in the treatment of cardiac arrhythmias, in particular atrial fibrillation. Previous experimental work has shown that high concentrations of dofetilide can rescue K v 11.1-trafficking deficiencies induced by pentamidine (Varkevisser et al., 2013). Similarly, our model only rescues channels when pentamidine is present, but our results also suggest that the impact of dofetilide on trafficking is likely limited at clinically relevant concentrations, even in the presence of pentamidine. In general, drugs that rescue ion-channel trafficking primarily seem to have an effect during aberrant conditions (e.g. trafficking-deficient mutations, trafficking blockers), but may not be able to increase K v 11.1 trafficking under physiological conditions (Anderson et al., 2006;Varkevisser et al., 2013;Wible et al., 2005). However, a recent study has shown an approximately 60% increase in WT K v 11.1 levels after 24 h of incubation with 5 μmol/l E-4031 (Al-Moubarak et al., 2020) in the absence of pentamidine or other compounds impairing K v 11.1 trafficking. More research on the effects of trafficking rescuers under physiological conditions is therefore warranted. Moreover, distinct short-and long-term effects of dofetilide have been reported and were attributed to modulation of phosphoinositide 3-kinase signalling, promoting an increase in late Na + current over hours (Yang et al., 2014). For other drugs, the effects on K v 11.1 trafficking could manifest at physiological concentrations and may similarly lead to differences between acute and long-term behaviour. Alternatively, combined application of dofetilide and LUF7244, an allosteric K v 11.1 activator that counteracts the direct inhibitory effects of high concentrations of dofetilide, has been proposed as an approach to rescue impaired K v 11.1 trafficking without drug-induced proarrhythmia due to excessive I Kr inhibition (Qile et al., 2020). J Physiol 601.13 It is known that fever can be an important trigger of arrhythmias when a vulnerable substrate is present; for example, in Brugada Syndrome (Adler et al., 2013;Roterberg et al., 2020) or for certain LQT2 mutations (Amin et al., 2008). Our model predicted extreme APD prolongation after 24 h of simulated febrile temperatures (Fig. 8C), which would be expected to promote arrhythmogenesis. By contrast, a study comparing ECG parameters in patients presenting with fever to the emergency department with a comparison ECG obtained within 30 days without fever suggested a shortening of QTc interval duration in the presence of fever (Drew et al., 2017). Conversely, long-term hypothermia decreased APD in our model, whereas clinical studies in patients with therapeutic hypothermia after cardiac arrest instead suggest QTc prolongation (Khan et al., 2010;Nishiyama et al., 2012). This inconsistency is likely due to the fact we only implemented the temperature-dependent regulation of K v 11.1 in the ORd model, while other channels also undergo dynamic temperature-dependent modulation of gating and trafficking. When sufficient quantitative data become available, these effects on other channels should also be implemented in the ORd to gain more realistic insights into the temperature-dependent effects on cardiac repolarization and arrhythmogenesis. In addition, fever is often associated with inflammation which itself has complex direct and indirect electrophysiological effects. For example, inflammatory mediators such as interleukin-1, interleukin-6 and tumour necrosis factor α have been shown to induce oxidative stress and proarrhythmic calcium-handling abnormalities, as well as affecting K + -and Ca 2+ -channel functioning (Dobrev et al., 2023;Heijman et al., 2020;Lazzerini et al., 2015). Similarly, therapeutic hypothermia is performed on severely ill patients and has numerous systemic effects, including the induction of hypokalaemia (Khan et al., 2010), which, as shown in Fig. 9, may also dynamically modulate repolarization and arrhythmogenic risk. Thus, the presence of several other confounding factors during changes in body temperature preclude a direct comparison of our results with clinical data. As such, the goal of Fig. 8 was not to show the macroscopic effects of fever, but rather the temporal effects of temperature on gating and trafficking, and their combined effect on APD.
Hypokalaemia affects several key repolarizing K + channels and is a known risk factor for cardiac arrhythmogenesis (Pezhouman et al., 2015). Our simulations show an acute prolongation of APD and hyperpolarization of the resting membrane potential in response to hypokalaemia (Fig. 9), in line with experimental data (Pezhouman et al., 2015). In addition, severe hypokalaemia may induce additional APD prolongation over time due to a decrease in K v 11.1 membrane levels. Whether this effect occurs at clinically relevant concentrations depends on the affinity of K v 11.1 trafficking for extracellular [K + ]. Guo et al. (2009) identified a half-maximal effect on K v 11.1 internalization of 0.5 mmol/l for 12 h incubation and 2.1 mmol/l for 1 week incubation. Thus, while short periods of clinically relevant hypokalaemia are unlikely to affect K v 11.1 membrane levels, longer periods may reduce K v 11.1 levels, potentially contributing to excessive APD prolongation.
Our results highlight the complexity of the many factors affecting cardiac electrophysiology over timescales from milliseconds to hours, with several factors having opposing acute and long-term effects. As such, the acute evaluation of drug effects alone may not be sufficient for safety screening of compounds that will be administered over longer periods of time in clinical practice, as repolarization abnormalities may only show up after some hours.

Conclusion
In conclusion, we presented a simple and computationally efficient mathematical framework of K v 11.1-channel trafficking that makes it possible to study mediumto long-term regulation of cardiac electrophysiology. This framework highlights the distinct effects of acute modulation of channel gating and long-term regulation of channel trafficking induced by temperature changes, pharmacological interventions and hypokalaemia. It provides a foundation to integrate other modulators of ion-channel trafficking (e.g. hyperglycaemia and other drugs) and study trafficking-deficient mutations and/or interventions that rescue ion-channel trafficking, which may facilitate a better understanding of arrhythmogenic disorders. Finally, our model can be used to optimize experimental protocols which are normally challenging to perform over longer timescales (e.g. hours to days) and to generate new hypotheses about arrhythmia mechanisms.