Statistical Approach to Incorporating Experimental Variability into a Mathematical Model of the Voltage-Gated Na+ Channel and Human Atrial Action Potential

Figures

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  Figure 1. Examples of three different biophysical models of the volgate-gated Na+ (Nav) channel. Open/Active states are indicated in green, while closed/inactive states are white. (A) Our novel hybrid Nav model with 3 independent components, an activation component (subscript m), and two inactivation components for fast and slow inactivation (subscripts f and s, respectively). The activation component consists of an open (Om) and closed (Cm) state, while the inactivation components each consist of an active state (A), a closed state (C) and an inactivated state (I). (B) Example of a Hodgkin-Huxley type model consisting of 4 independent gating variables: activation (M), fast inactivation (Hf), slow inactivation (Hs) and recovery (J) [20,24]. (C) A Markov model where channel gating is represented as transitions between coupled states, including inactivation (IX), closed (CX), and open states (O). Additionally, there are further states to allow for late current (LX), and for drug block (XB) [22].

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  Figure 2. Statistical approach for fitting model parameters. Schematic illustrating the workflow used to fit biophysical model parameters using multiple sources of data. Experimental Data on voltage-gated Na+ channel measurements in human atrial cells are curated and provided as input to the statistical model fitting procedure [Monte Carlo Markov Chain (MCMC) optimizer)]. A subset of the data are held back for validation. The MCMC fitting procedure generates an initial guess for the set of parameters corresponding to each dataset, which are fed to the biophysical model to generate a set of simulated results. Independent normal variables with means of simulated results are compared to experimental data in the model to update the model fit while experimental variances encompass the variability between the predictions and experimental results and is used to track goodness of fit. Finally, the Parameter Variance and Parameter Correlation capture the variability between different experiments and the relationships between different parameters.

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  Figure 3. Simulated results of individual model fits to specific experiments. (A–D). Current Voltage relations, steady-state activation, recovery from inactivation, and steady-state inactivation curves using a different set of fit model parameters predicted by the statistical model (solid lines) for each experiment (circles). Data from Feng et al. had 4 experimental groups which were the lengths of time the cells were kept before an experiment was performed: 0, 1, 3, 5 days; Cai et al. had two experimental groups: cells taken from adult or pediatric patients; 3 control IV curve experiments were taken from Lalevée et al.; Wettwer et al. had a single IV curve control experiment; Sakakibara et al. had one control activation curve and three recovery curves with different inter-pulse holding potentials: −100, −120, −140 mV; and two inactivation curve experiments were taken from Schneider et al. with different durations for the inactivation potential: 512 and 256 ms [33,34,35,36,37,38]. Data from different experiments taken from the same paper are indicated with a single color indicated in the legend (same for corresponding simulation results).

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  Figure 4. Simulated results of overall model fits to specific xperiments. (A–D). Current Voltage relations, steady-state activation, recovery from inactivation, and steady-state inactivation curves using a single set of fit model parameters with experiment-specific temperature and sodium concentrations reported for each study. Experimental data is the same as in Figure 3 from Feng et al., Cai et al., Lalevée et al., Wettwer et al., Sakakibara et al. and Schneider et al. [33,34,35,36,37,38]. Data from different experiments taken from the same paper are indicated with a single color indicated in the legend (same for corresponding simulation results). In the event that a different experimental protocol was used to measure the same property (e.g., recovery from inactivation) multiple simulation results are indicated with the same color corresponding to results using the same model parameters but different protocols.

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  Figure 5. Fitting of model parameters using maximum a posteriori (MAP) estimates. (A–E) The overall standard deviations (dark blue) and the UVEP effect standard deviations (light blue) for each model parameter. Estimates for standard deviations were made using maximum a posteriori values that maximize the likelihood of the statistical model (Appendix A). Model parameter factors are grouped according to functionality. (A) Parameter factors responsible for locations of steady state and rate curves. (B) Parameter factors responsible for the steepness parameters of the steady state curves. (C) The minimum and maximum values of the rate constants. Baseline is scaled for each rate constant by constants 1/15, 1/10, 1, 1, for gates m, hf, hs, j respectively. (D) The steepness parameters for the rate constants. (E) The conductance, GNa, and the proportion of fast to slow inactivation. (F) The estimated standard deviations for the error components in the model. Each error term corresponds to a different type of experiment. IV curve simulation error has units pA/pF, while all other errors are for normalized measurements and are unitless.

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  Figure 6. Simulation of model population at physiological temperature. (A–D) Current-voltage, steady-state activation, recovery from inactivation, and steady state inactivation curves from a population of models generated by randomly selecting 100 parameter sets from the fitted 25 dimensional multivariate normal distribution. The individual results from the 100 simulations (light blue lines) were used to generate a population mean (dark blue line) for each property and compared to the theoretical mean (black line) generated using the mean of the entire normal distribution. Simulations used the sodium concentrations and temperature used by the Grandi full sell AP model [9.1 and 140 mM, internal and external sodium, respectively, and temperature of 40 °C (unlike previous simulations (Figure 3 and Figure 4) that used room temperature to facilitate comparison to experiment (difficult to measure Nav1.5 at body temp due to its rapid kinetics and large amplitude)].

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  Figure 7. Model validation. The new voltage-gated Na+ channel model was incorporated into a detailed model of the human atrial action potential [8]. Validation for single cell dynamics was performed by comparing the maximum upstroke velocity and conduction velocity at different cycle lengths to published data, in single cells and in fiber. (A) Solid dark blue line is the mean, and the dark blue shaded region is the standard deviation for the maximum action potential upstroke velocity from the generated population of 100 cells. Similarly, the light blue line and shaded region are the mean and standard deviation, respectively, from a population of 40 fibers, each of which is 100 cells long. For the fiber, the upstroke velocity of the fiber is calculated as an average of cells, Xi, from the middle of the fiber: X20, X30, …, X70, X80. Solid dots are values reported in literature and error bars are standard deviations. Model was tuned by hand to the mean of Molina et al. (B) Conduction velocity, light blue line and shaded region use the same fibers as in A. Conduction velocity was calculated using the 30th and the 80th cells. Data from Hansson et al. is on individuals, so each point represents…