Optimizing experimental designs for model selection of ion channel drug-binding mechanisms

Figures

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  Figure 1.. Markov diagrams of physiological hERG models (a,b) and a pharmacological binding model (c). The model in (a) is a four-state symmetric hERG model as used by Lei et al. [18]. The four states are IC (inactive closed), C (closed), I (inactive) and O (open). The model in (b) is a five-state model equivalent to that derived by Lu et al. [25]. This model has three closed states (C1, C2 and C3) as well as O (open) and I (inactive) states. The pharmacological drug-binding in (c) is Model 7 from Lei et al. [18]. The I and O states represent the inactive and open states in the underlying physiological hERG models (this model could be attached to the right-hand side of either physiological model shown in (a) or (b)). The binding model additionally has two drug-bound states ID (inactive drug-bound) and OD (open drug-bound), in which no current flows.

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  Figure 2.. Synthetic verapamil data under drug-binding Model 7 generated under a Milnes protocol with the Lei 37°C hERG model. On the left, we plot a single sweep of the Milnes protocol. In the top middle, we plot the control (in black) and drug currents (blue, orange, green and red corresponding to four different drug concentrations) for 10 sweeps of the Milnes protocol. We only plot the currents that occur during the 10 s pulse at 0 mV for each sweep. In the bottom middle, we plot the corresponding proportion open for each drug concentration, which is calculated by dividing each drug sweep by the control current. On the right-hand side, we include a zoomed-in look at the first 15 (2 s) of the first pulse for both currents and proportion open. This illustrates the currents starting at approximately 0 pA and the open proportion starting at approximately 1. The noise in the initial low currents contributes to the increased noise at the beginning of the open proportion sweeps. This motivates using a noise model for the ratio of two normally distributed variables when fitting drug-binding models to this data.

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  Figure 3.. Fits of the 15 binding models to the Milnes protocol proportion-open synthetic data shown in figure 2. Note that it is difficult to visually distinguish between the quality of the model fits for this protocol. The fitted models are shown with solid lines, and we plot the data shaded behind these fits.

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  Figure 4.. Maximized log-likelihoods for model fits to synthetic data generated under the modified Milnes protocol (left) and an optimized protocol (right). In each plot, we include a green dashed line at 104 below the largest maximized log-likelihood and a red dashed line at 105 below the largest maximized log-likelihood. We note that these lines act simply as visual aids to emphasize the increased spread in the quality of model fits under the optimized protocol. We also shade, in grey, around the true data-generating binding model.

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  Figure 5.. Synthetic data for drug-binding Model 7 generated under an optimized protocol with the Lei 37°C hERG model. On the left-hand side, we plot our optimized protocol with two 10 s pulses, each with three optimized voltage steps. Top right shows the control and drug currents for five sweeps of the optimized protocol. We only plot the currents that occur during the two 10 s pulses per protocol sweep. The bottom right-hand plot shows the corresponding proportion open for each drug concentration which is calculated by dividing each drug sweep by the control current.

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  Figure 6.. Fits of the binding models to the optimized protocol proportion open synthetic data shown in figure 5, in the same style as figure 3. Note how many fits are now visually distinguishable, we can immediately see that many models provide a worse fit to these data than others.

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  Figure 7.. A comparison of control currents between the Lei model (figure 1a) parameterized for 37°C and the Lu model (figure 1b). To the left, we plot the currents in response to one sweep of the Milnes protocol, and to the right in response to one sweep of the optimized protocol from figure 5. Conductances have been set to 33.3 nS for both models.

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  Figure 8.. Maximized log-likelihoods for model fits to synthetic data generated under the modified Milnes protocol (left) and an optimized protocol (right). This is for the discrepant hERG model case; the data-generating hERG model is the Lu model, while the Lei 37°C is used for model fitting. In each plot, we include a green dashed line at 104 below the largest maximized log-likelihood and a red dashed line at 105 below the largest maximized log-likelihood. We also shade, in grey, around the data-generating binding model.

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  Figure 9.. Top: heatmaps of maximized log-likelihoods for fitted models to Milnes (left) and optimized protocol (right) synthetic verapamil data. The fitted models within 104 of the largest log-likelihood (very good fits) in each row are highlighted in green squares. The fitted models that are between 104 and 105 below the largest log-likelihood (reasonable fits) in each row are highlighted in red squares. Black dots are plotted down the diagonal of each heatmap where the fitted model corresponds to the data-generating model. Note that our optimal protocol results in a far bigger spread of maximized log-likelihoods. Bottom: equivalent heatmaps of maximized log-likelihoods with discrepancy between the data-generating hERG model and the hERG model used to fit the data. The introduced hERG model discrepancy makes identifying the true data-generating drug-binding model difficult.

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  Figure 10.. Graph illustrating nesting between drug-binding models. An arrow pointing from Model A to Model B indicates that Model B is nested within Model A.

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  Figure 11.. Top: heatmaps of maximized log-likelihoods for fitted models to Milnes and optimized protocol synthetic bepridil data. Bottom: equivalent heatmaps of maximized log-likelihoods with discrepancy between the data-generating hERG model and the hERG model used to fit the data. For discussion of highlighting see the caption of figure 9.

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  Figure 12.. Top: heatmaps of maximized log-likelihoods for fitted models to Milnes and optimized protocol synthetic dofetilide data. Bottom: equivalent heatmaps of maximized log-likelihoods with discrepancy between the data-generating hERG model and the hERG model used to fit the data. For discussion of highlighting see the caption of figure 9.