Abstract
Understanding the roles of ion currents is crucial to predict the action of pharmaceuticals and also to guide clinical interventions in the heart, brain and other electrophysiological systems. Our ability to predict how ion currents contribute to cellular electrophysiology is in turn critically dependent on the characterization of ion channel kinetics. We present a method for rapidly exploring and characterizing ion channel kinetics, using the hERG channel, responsible for cardiac IKr current, as an example. We fit a mathematical model to currents evoked by a novel 8 second sinusoidal voltage clamp. The model is then used to predict over 5 minutes of recordings in the same cell in response to further voltage clamp protocols, including a new collection of physiological action potentials. Our technique allows rapid collection of data from single cells, produces more predictive ion current models than traditional approaches, and will be widely applicable to many ion currents.
1 Introduction
Mathematical models of ion channels are a quantitative expression of our understanding of the probability of the channel existing in different conformational states (typically, closed, open and inactivated) and the rates of transition between these states1,2. There have been some notable advances in deriving mathematical models for ion channel behavior3–7, with some stressing the need for validation/testing of the model using data from the same cell8. In this paper we present a new approach, based on novel short protocols and parameter inference techniques, which we demonstrate by constructing an improved ion channel model.
The KCNH2 gene (also known as hERG) has been shown to encode the primary subunit of the voltage-gated ion channel Kv11.1 that carries the rapid delayed rectifier potassium current (IKr)9,10. In this article we focus on mathematical modeling of hERG channel kinetics, demonstrating our approach by constructing an improved model of this ion channel. hERG plays important roles in the brain11; gastrointestinal tract12; uterine contractions13; cell-proliferation and apoptosis14 and cancer progression15, but IKT is best known as a repolarizing cardiac ion current. The channel is susceptible to binding and blockade by pharmaceutical compounds, which is strongly linked to many cases of drug-induced pro-arrhythmic risk16,17. Mathematical modeling of cardiac electrophysiology, including IKr, forms a core part of a new proposal for routine in vitro and in silico safety assessment to replace a human clinical drug safety study18,19. A wide range of different mathematical models have been proposed to describe IKr (details of literature models are given in Supplementary Material B, Table B1). We note that these models were developed to describe different species, cell types, temperatures and isoforms, so variation is to be expected. Figure 1 shows predicted current under three different voltage clamps for 29 literature models. Unfortunately, even models for the same conditions do not provide consistent predictions.
The first models of ion channel kinetics were proposed by Hodgkin & Huxley20, and relatively little has changed in the methods used for construction of mathematical models of ion channel gating since the original seminal work in 1952. The traditional approach is to fit peak currents and time constants of current activation/decay after clamping to fixed voltages; to assemble current/voltage (I–V) and time-constant/voltage (τ–V) curves; and to describe these curves with interpolating functions. Condensed voltage-step protocols have been suggested as the basis of optimized experiments that provide information about ion channel kinetics faster than experiments to construct I–V curves21,22; and optimized current and voltage step clamps have been used to optimize the fitting of maximal conductances in action potential models23. Single sinusoid voltage clamps have been previously been explored for choosing between possible Shaker channel models that were parameterized using traditional voltage step clamps24.
In this study, we extend these ideas and propose an 8second sinusoid-based voltage protocol, designed to explore the kinetics of the hERG channel, and use this protocol to record hERG currents from CHO-cells that are over-expressing hERG1a. These recordings are then used to parameterize a mathematical model of the ion current. We then validate the model by predicting the response to both standard voltage-step protocols and physiologically-relevant action potential clamps: an unprecedentedly thorough independent validation for a model of ion channel kinetics. Our approach uses a substantially shorter experimental recording to construct the ion channel model than the usual approach based on time constants and peak currents from a series of voltage-step protocols. As a consequence of the high information content of the short protocol, we are able to generate cell-specific models that advance our understanding of variability of ion currents between cells. Our methodology will be applicable to many ion channels, both in the heart and other electrophysiological systems.
2 Results
2.1 Experimental Protocol
In Figure 1B we provide an overview of the experimental approach, denoting the sequence of voltage clamp protocols we performed as Pr0–Pr7. In each cell we recorded: a series of more conventional voltage-step protocols designed to explore activation (Pr1–3), inactivation (Pr4) and deactivation (Pr5); a new protocol composed of a series of action potential clamps (Pr6 — composed from simulated action potentials from models representing diverse species, pacing frequencies, and in both healthy and repolarization-failure conditions); and our new 8s sinusoidal voltage protocol (Pr7, shown in Figure 2, full details of all protocols are given in the Online Methods 4.2).
In Figure 2 we present the novel sinusoidal protocol Pr7, the simulated predicted currents from existing models, and the responses we recorded using the whole-cell patch clamp technique with hERG 1a-transfected CHO cells (details of the experimental methodology can be found in the Online Methods 4.1.1–4.1.3). The protocol is comprised of simple voltage steps and a main sinusoidal section that is in the form of a sum of three sine waves of different amplitudes and frequencies, here optimized to rapidly explore hERG channel kinetics. The steps are included to measure leak current and to provoke a large current to help identify maximal conductance (found to be helpful in preliminary work, see Supplementary Material A2.1). The frequencies of the sine waves are selected to cover the range of characteristic time constants known to occur in hERG channel gating (from millisecond to second timescales)25,26, whilst avoiding harmonics that would lead to repetitive in-phase signals. The amplitudes of the sine waves are selected to sweep over physiological voltage ranges. Details of the protocol parameters and an equation for the protocol are given in Online Methods 4.2.
The sine wave protocol is of only eight seconds duration, which enables: efficient data collection, with training and validation data collected from the same cell; and the chance to make multiple interventions (such as the addition of drug compounds) since we can re-measure the full set of ion channel kinetics multiple times. The new protocol provokes from the existing literature IKr models an even wider array of different behaviors (middle panels in Figure 2A & B) than the existing voltage step or action potential clamps (Figure 1A); even among models constructed in/for similar conditions/species.
We recorded the full set of voltage protocols (Pr1–7) twice. After the first set of recordings, in vehicle conditions, we add a moderate dose of dofetilide (0.3 μM), allow the dofetilide-induced current block to reach equilibrium, and then repeat the full set of recordings. We leak-subtract each set of Pr1–7 recordings using the leak step at the beginning of Pr7 to estimate leak current resistance. Finally we subtract (the already leak-subtracted) second set of recordings from the first set, to obtain ‘dofetilide subtracted’ current traces, predominantly composed of dofetilide-sensitive hERG current. This procedure minimizes contributions from any endogenous currents (see Online Methods 4.4).
2.2 Model Calibration
We calibrate a mathematical model using only the sine wave protocol Pr7. The Hodgkin-Huxley20 structure of the model we use, and its corresponding model parameters, can be seen in Figure 3B. We independently fitted this model to each of the experimental current traces shown in Figure 2. For each cell, we obtain a probability distribution of estimates for each parameter that captures any observational uncertainty in the parameter values27,28. We used a global minimization algorithm29 followed by a custom-written Monte Carlo based Bayesian inference method (assuming uniform prior distributions, see Online Methods 4.7).
The result of the fitting procedure for one cell is shown in Figure 3. The parameter set with maximum posterior density is shown in Figure 3A, demonstrating an excellent fit between experimental and simulated data. The resulting posterior probability density for the parameters obtained from this Bayesian inference approach is projected across each parameter in Figure 3C. We also tested that our approach is theoretically appropriate for inferring all parameters by using synthetic data studies, as described in Supplementary Material C. The plausible parameter space is very narrow: if multiple parameter set samples are taken from the distribution shown in Figure 3C, the resulting simulated current traces are indistinguishable to the eye. To quantify this, taking 1000 samples we find that the 95% credible intervals for the simulated currents were always within at most either 3.45% or, in absolute terms, 0.0044 nA of the simulated current given by the maximum posterior density parameter set.
The results we present in Figure 3 are from a single cell with a good quality recording and a high signal:noise ratio (this choice of cell, and other cells’ predictions, are discussed later). We fit models on a cell-specific basis, and then also use averaged experimental data to create a single ‘averaged’ model as described in Supplementary Material F. We will compare these approaches below. We provide all parameter values with the maximum posterior density for all models in Supplementary Table F11.
2.3 Validation predictions
Having trained our model to eight seconds of experimental data from the sine wave protocol Pr7, we now test its ability to predict more than 5 minutes of independent experimental behavior. We predict the current in response to traditional voltage-step protocols Pr1–5 (adapted from those previously used in the literature1), and also to a novel physiologically-inspired voltage clamp protocol comprised of multiple action potentials (Pr6). All recordings shown in Figures 3–5 are from the same cell, using the experimental procedure shown in Figure 1B.
To make the predictions for Protocols Pr1–6 we performed simulations using the parameter set with the maximum posterior density in the fit to the sine wave (Pr7). As with the calibration protocol, all the predictions we will discuss below are indistinguishable by eye from the result of taking multiple samples from the distributions in Figure 3C and plotting a prediction for each of these parameter sets. We also compare the predictions from our new model with those from a sample of widely-used literature models25,30–33.
In Figure 4, we show traditional voltage step protocols, experimental recordings and the simulated predictions from the model. We also show some of the usual summary curves of the data, together with predicted summary curves from our model and a range of existing literature models (methods used to derive summary plots are given in the Online Methods 4.5, results for Pr1&2 in Supplementary Material E). We can predict a wide range of current behavior in response to the standard voltage-step protocols, without having used any of this information to fit the model. Many of the current-voltage relationships and time constant-voltage relationships we predict in response to the traditional voltage-step protocols are closer to the experimental data than similar model-experiment comparisons in the literature (even when existing literature models, with more parameters, were fitted to similar data).
Figure 5 shows the model prediction of the currents invoked in response to the physiologically-inspired action potential protocol Pr6, compared with the experimental recording (as shown in Figure 1B we used the first repeat of Pr6 for validation purposes, and the second as a quality control measure). Replicating behavior under action potentials is perhaps the most important requirement for a hERG channel model for use in physiological or pharmacological studies. The model is able to predict the response to all of the complex action potential protocol extremely well, and much better than existing models (even though we have scaled literature models’ maximal conductances (GKr) to fit this trace as well as possible in Figure 5).
We provide a quantitative comparison of predicted current traces for our model and each of the literature models for Pr3–7 in Supplementary Table D6. In each case, the worst-performing literature model is a Hodgkin-Huxley style model. Yet our simple model, with the same structure, is able to provide significantly better predictions than even the Markov-type models, which are usually considered to be better representations of hERG kinetics1. Our methodology has resulted in a simple and highly predictive mathematical model, able to describe a wide range of physiologically-relevant behavior.
2.3.1 Cell-specific validation
In Figure 6A we present the maximum posterior density parameter values when repeating the above approach using data from nine different cells. The clustered parameter values demonstrate that parameters derived from different cells take similar values, giving us confidence that the procedure is reproducible and biophysically meaningful. There is more cell-to-cell variability in some parameters than others, which may be related to variability in the underlying physiological processes that they represent; supporting the value, and perhaps necessity, of a cell-specific approach. We also acknowledge that some parameters may be more or less sensitive to variability in experimental conditions such as temperature, residual background/endogenous currents, and imperfect dofetilide and/or leak subtraction.
We order the cells in Figure 6 based on the lowest difference in leak resistance between the vehicle and dofetilide recordings of Pr7. This ordering gives a measure of recording stability, and is intended to be a surrogate for data quality. The cell presented above, in Figures 3–5, corresponds to Cell #5 of 9 under this ranking, so we obtain very good predictions even with our ‘median’ quality data. We show cell-specific predictions of the current-voltage relationship for the peak steady-state activation current for each cell-specific model in Figure 6B. While we focus on Cell #5 in the main text, Cells #1–4 also produce excellent cell-specific predictions (similar comparisons for other summary plots are in Supplementary Figures F8–F10).
We also investigated the benefit in a cell-specific approach by building a model using averaged experimental data from all nine cells. We describe this approach in Supplementary Material F, and summarize the results in Supplementary Table F12. Generally, for the cells with the highest data quality (Cells #1–5) the cell-specific models provide better predictions than the average model, as we see for Pr4 when comparing colored cell-specific predictions and experiment with the black line for the average model in Figure 6B. The same trend holds for the action potential protocol Pr6, in 6/9 cells the cell-specific model provides less error than the average cell model — the largest improvement was 37% less error; for the remaining 3/9 cells where the average cell model provided better predictions, this was by at most 6%.
3 Discussion
In this paper we have presented a novel method for constructing mathematical models of ion channel kinetics. We used a sinusoidal voltage protocol to construct a simple model of hERG channel kinetics using just 8 seconds of recording, as opposed to a traditional approach that requires several minutes of voltage-step data. All of our experimental data can be collected from a single cell, whereas a typical approach necessitates the collection of data from a number of different cells.
The conceptual shift is that channel kinetics should be summarized by mathematical model parameters, not a series of current-voltage (I–V) and time constant-voltage curves. In essence, the model is the current characterization, rather than something designed to fit I–V and time constant curves, which only represent a certain subset of possible behaviors of the current. By fitting directly to the experimental current traces, instead of using summary curves, we can also reduce the possible influence of subjective choices during the time-constant fitting process.
We saw that our model is able to replicate the experimental training data very well (Figure 3). This is often the point at which literature approaches stop and conclude that a mathematical model is a good representation of ion channel kinetics (something that is also true more generally for mathematical models of biological processes). Instead, we performed an unprecedentedly thorough evaluation of the model by testing its ability to predict the behavior in response to a series of voltage clamp protocols it has not ‘seen before’ (both those traditionally used to characterize hERG channel kinetics, and also a new complicated series of action potential waveforms), all recorded from the same cell as the training data. The extremely good prediction from all our cell-specific models of the response to the complex action potential protocol is particularly remarkable (Figure 5). We are not aware of such a thorough, physiologically-relevant validation of an ion channel model having been performed before. Testing that we are able to predict the current response to a voltage pattern which may be observed in physiological or patho-physiological conditions is a particularly robust and useful way to validate a model, and critical if the IKr model is to be used to accurately predict cardiac electrical activity in healthy and potentially arrhythmic situations.
There are still some aspects of the experimental behavior that are not replicated by our model. In particular, there is only one time constant of deactivation, and low voltage-dependence in the inactivation time constant (Figure 4). But then neither is the full range of behavior captured by any of the existing, more complex, models available in the literature; and we have shown that our model can provide better predictions of all the raw currents than the literature models in the majority of cases, even where summary curves are not predicted as accurately. The inability of our model to replicate all of the experimental data may be a consequence of using a simple Hodgkin-Huxley model formulation, although it is a commonly used structure for currents within action potential models.
However, the simplicity of our model may also be the key to its success — with only eight kinetic parameters we have confidence that they are all being fitted well, and we have shown that there is low uncertainty in their values. As Pathmanathan & Gray27 and Mirams et al.28 have recently discussed, considering probabilistic uncertainty in our model parameters and predictions is evermore important as models begin to be used for safety-critical predictions such as the CiPA initiative18,19 and clinical applications34.
A key limitation of our approach is that experiments have been performed in expression line cells, creating a hERGla model; compared to native IKr current in cardiac cells which will have additional isoforms, subunits and regulation. To characterize IKr kinetics we plan to apply the methodology presented here in native myocytes, to make a model that is more applicable for use in cardiac safety testing and whole-organ simulations.
The success of our approach in different ion channels will be heavily dependent on the precise form of the sinusoidal protocol that is used, and in parallel work we are developing different strategies for optimizing the voltage protocol design for given currents. Previously, Bett et al.1 explored the behaviors of a subset of existing hERG channel models and concluded that the model proposed by Wang et al.25 was best able to replicate the activation kinetics of the hERG channel. In parallel work, we are extending the approach presented here for selecting and calibrating the most appropriate model structure for hERG channel kinetics (see Supplementary Material Figure B4 for the range of possibilities).
We have demonstrated the advantages of a cell-specific mathematical modeling approach, observing an overall improvement in model predictions using cell-specific models relative to a model made using averaged data. The approach therefore allows, for perhaps the first time, an exploration of both within-cell and between-cell variability in ion channel kinetics. The cell-specific predictions are particularly strong when using the highest quality data, highlighting the necessity of maintaining very high data quality for constructing accurate and robust mathematical models of ion channel kinetics.
The significant time saving of our short protocol opens up the possibility of taking more recordings in different experimental conditions within a single cell (e.g. drug concentrations35,36 or temperatures37), leading to datasets that are more consistent, and therefore of higher quality. These datasets will result in more accurate mathematical descriptions of ionic currents in these different conditions. The approach we have presented allows more predictive mathematical models of ion channel kinetics to be formulated, which will lead to more accurate predictions of ion currents in different organ systems.
Author Contributions
KAB, RB, YC, DJG, TdeB, GRM designed the study and modeling approach; KAB, RB & GRM designed and implemented the statistical methods; KAB, GRM, JIV, APH and TdeB designed and refined the experimental methods; KAB performed all the experiments, simulations and statistical analysis; KAB, TdeB, GRM wrote the manuscript; all authors approved the final version of the manuscript.
Competing Financial Interests
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Disclaimer
The opinions presented here are those of the authors. No official support or endorsement by the Food & Drug Administration is intended nor should be inferred.
Materials & Correspondence
All computational codes, and the experimental current recordings that were used for calibration and validation (leak and dofetilide subtracted), are openly available in a Supplementary Data repository at https://github.com/mirams/sine-wave. A permanently archived version is available on Figshare at https://doi.org/10.6084/m9.figshare.4704550.v2 alongside the full raw data (in both plain text and PClamp formats) at https://doi.org/10.6084/m9.figshare.4702546.v1. K.A.B. is the corresponding author.
4 Online Methods
4.1 Experimental methods
4.1.1 Cell Culture
Chinese Hamster Ovary (CHO) cells stably expressing Kv11.1 were used in the patch clamp experiments performed in this study. Cells were cultured in Ham’s F12 nutrient mix containing 5% fetal bovine serum and maintained at 37°C with 5% CO2.
4.1.2 Electrophysiology Solutions
The bath solution was composed of: NaCl (137mM), KCl (4mM), MgCl2 (1 mM), HEPES (10mM), glucose (10mM), and CaCl2 (1.8mM). The pH of the solution was adjusted to 7.4 with NaOH. Borosilicate glass micropipettes were pulled and fire polished to final tip resistances of approximately 2–5 MΩ when filled with pipette solution containing: KCl (130mM), MgCl2 (1 mM), HEPES (10mM), EGTA (5mM), and MgATP (5mM). pH of the solution was adjusted to 7.2 with KOH. All experiments were performed at room temperature (21–22°C). Using this temperature and the composition of the bath and pipette solutions, a K+ reversal potential of approximately -88.4 mV was calculated using the Nernst potential (equation (8)), the exact value depending on the particular temperature of each experimental recording.
4.1.3 Recording Techniques
Current recordings were made using an Axopatch 200B amplifier in whole-cell patch clamp mode. Data acquisition was performed using pClamp 10 software (Molecular Devices, Sunnyvale, USA). The protocols were first created as text files and then converted to .abf stimulus files to make corresponding .pro protocol files in the pClamp 10 software. A CV 203BU amplifier headstage and a Digidata 1440A were used. A Sutter MP225 micromanipulator was used for positioning of the microelectrode. The current signal was sampled at a rate of 10 kHz. 75–80% series resistance compensation was applied and data were 5 kHz low pass Bessel filtered by the hardware. No software filtering was applied. Whole-cell capacitance compensation was applied electronically. Leak subtraction was applied offline by using a 50 ms leak step to allow correction. To make a series of successive recordings using different protocols on the same cell, the pClamp “Sequencing Keys” tool was utilized, with a .sks file detailing the sequence the protocols should be performed in.
4.2 Experimental Protocols
For each cell we recorded a series of standard voltage-step protocols, a protocol comprised of a series of action potentials and the sine wave protocol, as shown schematically in Figure 1B. In all protocols the holding potential was initially held at –80 mV before applying a 50 ms leak step to –120 mV before returning back to –80 mV, with this step being used to estimate the leak current (as described in Section 4.3). Note that the ‘standard’ voltage-step protocols we have used to test our approach are shorter than some of those which have previously been used to calibrate mathematical models (with fewer test voltages/timings used) in order to perform them all in a single cell, with and without dofetilide subtraction, so a ‘traditional approach’ would take longer than the experiment performed here, perhaps requiring multiple cells. The voltage step to –120 mV at the end of all the protocols ensures that channels close quickly, reducing the time needed between protocols to regain a steady closed state.
Protocol 0 Repeated activation step
Before the start of each set of recordings on each cell an activation step protocol with a start-to-start interval of 12 seconds was repeated several times until consistent currents were observed on each repeat. From an initial holding potential of –80mV, this protocol comprised a 5s step to 10mV followed by a 5s step to –50mV before returning again to a holding potential of –80mV. This protocol is depicted in Supplementary Figure A1. We repeated this protocol while dofetilide was added (see Figure 1B) and the current traces recorded from this protocol were used to assess when a steady level of dofetilide block had been reached.
Protocol 1,2 Activation Kinetics
After the initial period at holding potential incorporating the –120 mV leak step, a step to Vstep1 followed and was held at that voltage for Tstep ms, before a step to –120 mV for 2.5 s, before returning to holding potential of –80 mV for 1 second. The protocol was repeated 6 times with a different Tstep on each repeat. Tstep took the values of 3, 10, 30, 100, 300 and 1000 ms.
For Protocol 1, Vstep1 is 0 mV. This protocol is depicted in Supplementary Figure A2.
For Protocol 2, Vstep1 is +40 mV. This protocol is depicted in Supplementary Figure A3.
Protocol 3 Steady-State Activation
From the initial period at holding potential incorporating the –120 mV leak step, a step to Vstep was applied for 5 seconds, followed by a 1 s step to –40 mV, before a 500 ms step to –120 mV, and then returning back to holding potential for one second. This process was repeated 7 times with a different Vstep on each repeat. Vstep ranged from –60 mV to +60 mV in 20 mV increments. This protocol is depicted in Figure 4A (left column).
Protocol 4 Inactivation
From the initial period at holding potential incorporating the –120 mV leak step, a step to 50 mV for 600 ms, and a step to –90 mV for 60 ms, followed by a step to Vstep for 150 ms, before a 500 ms step to –120 mV, and a 1s step back to holding potential of –80 mV; This was repeated 16 times with a different Vstep on each repeat. Vstep ranged from –100 mV to 50 mV in 10 mV increments. This protocol is depicted in Figure 4A (middle column).
Protocol 5 Deactivation
From the initial period at holding potential incorporating the –120 mV leak step, a step to 50 mV for 2 s was applied, followed by a step to Vstep for 6 s, before a 500 ms step to –120 mV, and then returning back to holding potential for one second. This process was repeated 9 times with a different Vstep on each repeat. Vstep ranged from –120 mV to –40 mV in 10 mV increments. This protocol is depicted in Figure 4A (right column).
Protocol 6 Action Potentials
This protocol was formed by combining a series of different simulated action potentials from the Cardiac Electrophysiology Web Lab38 (and we added some simulated action potentials where early after depolarizations and delayed after depolarizations had been induced). The range of models we used for the simulations encompassed different cell types and species, the action potentials were shifted slightly so that their resting potentials were exactly –80 mV (see the Supplementary Code for full details and code to reproduce this protocol).
Protocol 7 Sine Wave Protocol
The training protocol took the form of 250 ms at holding potential of –80 mV, followed by a 50 ms leak step to –120 mV, and then 200 ms back at –80 mV. This was followed by a 1s step to 40 mV, and a 500 ms step to –120 mV, before returning to –80 mV for 1 second. The 3.5 s sine wave portion of the protocol then followed (the form of which is described below), before a 500 ms step to –120 mV, and a return to –80 mV for 1s.
The sine wave portion of the protocol takes the form of a sum of three sine waves as shown in Equation 1. where A1 = 54 mV, A2 = 26 mV, A3 = 10 mV, ω1 = 0.007/(2π), ω2 = 0.037/(2π) and ω3 = 0.19/(2π), and t is time measured in milliseconds.
The protocol was initially designed with just the –120 mV leak step and not the additional steps to 40 mV and –120 mV (which were included after preliminary experiments as described in Supplementary Material A2.1) and so the sine wave was shifted by –2500 ms (as shown in equation (1)) to begin at the same phase after we incorporated the additional steps.
All of the protocols described in this section were adjusted on the amplifier to account for the liquid junction potential which was calculated to be 4.1 mV from the ionic composition of our physiological solutions which are described in Section 4.1.2. The liquid junction potential was calculated using the junction potential calculator in the pClamp software.
4.3 Leak Correction
We used the leak-step from –80 mV to –120 mV in order to leak-correct the experimental data, according to:
Leak subtraction was performed using a MatLab script written for this purpose. We identified the most appropriate Rleak value to minimize the difference between the mean current value during the leak step (to –120 mV) compared to the mean value at a holding potential of –80 mV, whilst ensuring that the trace was not over-corrected (which would result in negative currents during the initial stages of activation). We manually selected leak resistances to correct the current evoked by the sine wave protocol in both vehicle and dofetilide conditions. We then applied this leak resistance to the remaining protocols performed in the same condition on each cell. The mean current during the –80 mV step was calculated from 200 ms of the –80 mV holding period before the –120 mV leak step (not including the capacitive spike at the point at which the step occurs). The baseline current at a holding potential of –80 mV was then adjusted back to 0 nA with a constant additive current if required.
4.4 Dofetilide Subtraction
In preliminary work in HEK cells we observed that our sine wave protocols could elicit endogenous currents, potentially interfering with the predictive ability of the resulting mathematical models. To overcome this technical issue we first constrained the design of the sine wave protocol so that only voltages within a physiological range of –120 mV to +60 mV were explored. In addition, we repeated all the voltage protocols described above both in vehicle conditions and in the presence of 0.3 μM dofetilide. We then subtracted the currents remaining in the presence of the moderate dose of dofetilide from those recorded in the vehicle to remove any contribution of endogenous currents (and to produce what we refer to as ‘dofetilide subtracted’ data). Prior to performing this subtraction, we first leak subtracted both the vehicle and dofetilide recordings individually, as described above.
The required dose of dofetilide was obtained by serial dilution. Dofetilide was first dissolved in Dimethyl sulfoxide (DMSO) before being added to the bath solution to produce the required concentrations.
We observed that the levels of endogenous currents the protocols elicited varied from cell to cell (and were generally much lower in the CHO cells used in this study than in the HEK cells used in pilot studies). It may not always be necessary for dofetilide subtraction to be performed on the data, but we applied this method nonetheless to generate a gold-standard dataset.
4.5 Deriving I–V Curves and Time constant-V Curves
To derive time constant-voltage relationships from experimental data and simulated data traces, we used the Levenberg-Marquardt algorithm with a tolerance of 10−6 within Clampfit v10.5. To derive the instantaneous inactivation time constant curves shown in Figure 4 (inactivation column, row D) we fitted a single a single exponential to the current responses during the 150 ms Vstep, as defined in the inactivation protocol (Pr4) description above.
To produce the deactivation and recovery from inactivation rate time constant–voltage relationship for the experimental data traces, we fitted a triple exponential through the experimental data trace from the deactivation protocol (Pr 5). The section of the data used for fitting is the current in response to the 6 second Vstep. Both this region of experimental data used for fitting and that for the instantaneous inactivation time constant described above are highlighted in row B of Figure 4. The fastest time constant from the triple exponential fit to each test step corresponded to the recovery from inactivation time constant. We then used the weights of the remaining two time constants from each triple exponential fit to produce a single weighted time constant for deactivation39. To derive the deactivation and recovery from inactivation time constants from simulated data we fitted a double exponential through the current in response to the 6 second Vstep section of the deactivation protocol. Again, we used the faster time constant as the recovery from inactivation time constant and the slower time constant as that for deactivation.
To produce the peak current-voltage relationship for the steady state activation protocol for the simulated data traces we wrote MatLab code to identify the peak current in the region between 5.6292 and 5.7292 seconds on each sweep of the protocol, which corresponds to the current response just after the 5 second Vstep when the voltage is stepped to –40 mV. We then normalized the peak current data to the maximum overall peak identified in this region to produce the current-voltage relationship curve. For the simulated data we wrote a MatLab script (for additional details see Supplementary Code) to identify the peak-current voltage relationship for this protocol but for the experimental traces we verified these peak points manually to avoid incorrect peaks being identified due to noise or capacitive effects. We also identified the peak currents in the currents evoked by the activation kinetics protocols manually for the same reason. In the activation kinetics protocol we identified the peak currents during the Vstep for each interval of Tstep duration.
4.6 Mathematical Model Equations
The system of ordinary differential equations underlying the mathematical model structure shown in Figure 3B is as follows; where
The eight parameters P1 to P8 determine the rates k1 to k4 as shown in Figure 3B. The current, IKr, is modeled with a standard Ohmic expression: where GKr is the maximal conductance, EK is the Nernst potential for potassium ions, and O is the open probability, given by the solution to the system of equations above. EK is calculated from the ratio of ion concentrations on each side of the cell membrane: where R is the ideal gas constant, T is the temperature, F is the Faraday constant, z is the valency of the ions (in this case 1), and [K] represents the concentration of potassium ions. Note that this expression has a temperature dependence, and the temperature of the bath was recorded for each cell and used in relevant simulations.
All simulations were performed in MatLab. Mex functions were used to define and simulate each hERG channel model with CVODE40 used to solve the systems of differential equations, with both absolute and relative tolerances set to 10−8. Code is available to download as described at the end of the main manuscript.
4.7 Parameter Inference
Parameters were estimated using a Bayesian Inference scheme similar to that described in Johnstone et al.41. Here we outline the likelihood function formulation. In Supplementary Material A2 we give details of how this likelihood is used to generate a posterior distribution for the parameter set.
4.7.1 Likelihood formulation
For an observed experimental recording which we will denote y, we can infer the probability of different combinations of model parameters θ. Bayes’ rule underpins this approach which is expressed as
P(θ|y) is a probability density that encodes our belief that the parameters of the model are in a neighborhood of θ after observing the experimental data y, and is termed the posterior probability density. P(y|θ) is the probability density that corresponds to the probabilistic generation of the experimental data y given a model parameterized with parameters θ. P(θ) encapsulates our beliefs about θ before observing any experimental data and is termed the prior distribution (details of the prior that we used are in Supplementary Material A2.2). P(y) is a normalizing term which is the integral of all possible probabilities P(y|θ) and ensures that the posterior density P(θ|y) integrates to 1. In practice this normalizing term is calculated by
A Bayesian inference approach to parameter estimation combines beliefs about the parameters in the prior distribution P(θ) with the likelihood P(y|θ) to determine the posterior probability distribution P(θ |y)
We define the likelihood to insist on the fact that we consider it as a function of θ, with y kept fixed at the observation values. Bayes’ rule (in Equation (9)) can be rewritten in terms of likelihood as
When the prior distribution is assumed to be uniform (as it is in this study), we can make inferences based on just the likelihood, as the prior P(θ) is either constant or zero. If a proposed parameter is outside our chosen prior then likelihood is 0 and we simply record that this parameter set has a likelihood of 0 and propose another parameter set.
We assume that the errors at each time point are independent and so the conditional probability density of observing the whole experimental trace from time sample 0 to time sample T given the model parameter set θ is
We assume that the experimental noise is independently and normally distributed with a mean of zero and variance of σ2. The likelihood is then expressed as
In our case ft(θ) is the predicted current at each time point given the parameters, this is given by equation (7) after solving the model system (equations (3)–(6)). Calculating equation (14) requires the evaluation of the product of many numbers less than 1, so it is more numerically convenient to calculate the log-likelihood instead. As our aim is to identify parameter sets θ which maximize the likelihood in equation (14), maximizing the likelihood is equivalent to maximizing the log-likelihood:
In practice, the sums over time in equation (15) are formulated so that we exclude time points from regions where the data are affected by capacitive spikes. To be precise, we exclude 10 ms intervals following step-changes in the imposed voltage clamp. In the sine wave protocol (Pr7) these step-changes occur at 0.25 seconds, 0.3 seconds, 0.5 seconds, 1.5 seconds, 2 seconds, 3 seconds, 6.5 seconds and 7 seconds (spikes are seen in experimental recordings at these times in Figure 2).
4.8 Note on Normalization
Where existing literature model simulations were plotted alongside experimental traces, or one experimental trace was compared with another, we first had to normalize to account for differences in conductance values and allow clearer comparison. This was achieved by selecting a scaling factor for the conductance value for each model simulation (or experimental trace) that minimized the square difference between each trace and a reference experimental trace.
For literature models the reference trace was the experimental current from the action potential clamp Pr6. Note this provides a best-case fit for Pr6 all of the models, removing the possibility that some models open ‘half as much’ because they have ‘twice the conductance’. For the new model, no scaling was applied and conductance was directly fitted to the experimental current from the sine wave protocol (along with other parameters).
Acknowledgments
Our thanks to Prof. Gail Robertson of University of Wisconsin–Madison for assistance in acquiring the cell line used in pilot stages of this study. We would also like to thank the following people for technical assistance, access to facilities, support and encouragement: Jim Louttit, Nick McMahon, Carol Wilson, Sam Turner, Kate Harris and Sara Graham of GSK Safety Assessment; Jules Hancox of University of Bristol; Monique Windley and Mark Hunter of Victor Chang Cardiac Research Institute; Rianne Rijken and Birgit Goversen of UMC Utrecht. Thanks to Ross Johnstone (University of Oxford) for removing singularities from the Zeng model, and to Frank Ball (University of Nottingham) for comments on a paper draft.
KAB was supported by the EPSRC and GlaxoSmithKline Plc (grant numbers EP/G037280/1, EP/I017909/1 and EP/K503769/1). JIV and APH acknowledge funding from the NHMRC. RB acknowledges support from ANR grant BoB ANR-16-CE23-0003. GRM gratefully acknowledges support from a Sir Henry Dale Fellowship jointly funded by the Wellcome Trust and the Royal Society (grant number 101222/Z/13/Z).