Parameter Estimation of Ion Current Formulations Requires Hybrid Optimization Approach to Be Both Accurate and Reliable

Computational models of cardiac electrophysiology provided insights into arrhythmogenesis and paved the way toward tailored therapies in the last years. To fully leverage in silico models in future research, these models need to be adapted to reflect pathologies, genetic alterations, or pharmacological effects, however. A common approach is to leave the structure of established models unaltered and estimate the values of a set of parameters. Today’s high-throughput patch clamp data acquisition methods require robust, unsupervised algorithms that estimate parameters both accurately and reliably. In this work, two classes of optimization approaches are evaluated: gradient-based trust-region-reflective and derivative-free particle swarm algorithms. Using synthetic input data and different ion current formulations from the Courtemanche et al. electrophysiological model of human atrial myocytes, we show that neither of the two schemes alone succeeds to meet all requirements. Sequential combination of the two algorithms did improve the performance to some extent but not satisfactorily. Thus, we propose a novel hybrid approach coupling the two algorithms in each iteration. This hybrid approach yielded very accurate estimates with minimal dependency on the initial guess using synthetic input data for which a ground truth parameter set exists. When applied to measured data, the hybrid approach yielded the best fit, again with minimal variation. Using the proposed algorithm, a single run is sufficient to estimate the parameters. The degree of superiority over the other investigated algorithms in terms of accuracy and robustness depended on the type of current. In contrast to the non-hybrid approaches, the proposed method proved to be optimal for data of arbitrary signal to noise ratio. The hybrid algorithm proposed in this work provides an important tool to integrate experimental data into computational models both accurately and robustly allowing to assess the often non-intuitive consequences of ion channel-level changes on higher levels of integration.


Measurement data
In this study, measured current traces were used besides synthetic traces to parametrize the Courtemanche et al. (1998) I Kr , I Kur , and I Ks formulations. Here, the details of the experimental protocols are given. All currents were recorded using a Warner OC-725A (Warner Instruments, Hamden, CT, USA) amplifier, low-pass filtered at 1 to 2 kHz (-3dB, four-pole Bessel filter) and digitized at 5 to 10 kHz (Digidata 1322A, Axon Instruments, Union City, CA, USA). The currents recorded in different cells were normalized to their maximum value, averaged, and scaled to the average maximum value.

hERG measurements
Human ether-à-go-go-related gene (hERG; alternative nomenclature KCNH2) encodes the α-subunit of the Kv11.1 protein carrying the rapid delayed rectifier potassium current (I Kr ). Wildtype hERG channels were expressed in Xenopus oocytes after injection of 46 nl cRNA solution per oocyte. After 3 to 4 days incubated at a temperature of 16 • C, double micro-electrode voltage clamp experiments were performed in n = 8 cells. The tip resistances of the microelectrodes were in the range of 1 to 5 MΩ. The voltage clamp recordings were performed at room temperature (23 to 25 • C). The bathing solution consisted of 5 mM KCl, 100 mM NaCl, 1.5 mM CaCl 2 , 2 mM MgCl 2 , and 10 mM HEPES (pH adjusted to 7.4 with NaOH) and the pipette solution contained 3 M KCl. The applied voltage clamp protocol and the corresponding current traces are depicted in Figure 7B in the paper.

KCNA5 measurements
KCNA5 encodes the Kv1.5 protein carrying the ultra-rapid delayed rectifier potassium current (I Kur ). Wildtype KCNA5 was transfected into chinese hamster ovary (CHO) cell using Fugene reagent (Promega, Madison, WI, USA) (3 µg DNA per bowl). The CHO cells were incubated at 37 • C in minimum essential medium α and an atmosphere of 95% humidified air and 5% CO 2 . The medium was supplemented with 100 µg/ml streptomycin sulphate, 10% fetal bovine serum, and 100 U/ml penicillin G sodium. Resistances ranged between 38 and 98 MΩ. The bathing solution consisted of 140 mM NaCl, 5 mM KCl, 1 mM MgCl 2 *6H 2 O, 10 mM HEPES, 1.8 mM CaCl 2 *2H 2 O, and 10 mM glucose monohydrate. pH was adjusted to 7.4 using NaOH. The pipette solution contained 100 mM K aspartate, 20 mM KCl, 2 mM MgCl 2 *6H 2 O, 1 mM CaCl 2 *2H 2 O, 10 mM HEPES, 10 mM EGTA, and 2 mM Na 2 ATP. pH was adjusted to 7.2 using KOH; patch clamp recordings were performed at a temperature of 37 • C in n = 3 cells. The applied voltage protocol and the corresponding current traces are depicted in Figure 7D in the paper.

KCNQ1+KCNE1 measurements
KCNQ1 encodes the α-subunit of the Kv7.1 protein carrying the slow delayed rectifier potassium channel (I Ks ), KCNE1 encodes the β-subunit of Kv7.1. KCNQ1 was coexpressed with KCNE1 in Xenopus oocytes after injection of 46 nl cRNA solution per oocyte. Double micro-electrode voltage clamp experiments were performed at room temperature (20 to 25 • C) 2 days after injection in n = 5 cells. The tip resistances of the microelectrodes were in the range of 1 to 5 MΩ. The bathing solution consisted of 5 mM KCl, 100 mM NaCl, 1.5 mM CaCl 2 , 2 mM MgCl 2 , and 10 mM HEPES (pH adjusted to 7.4 with NaOH). The current and voltage electrodes were filled with 3 M KCl solution. The applied voltage protocol and the corresponding current traces are depicted in supplementary Figure S3.

Ion current formulations
This section gives the equations for the ion currents as formulated by Courtemanche et al. (1998). The adjustable parameters are marked by red color in the equations. They are named and classified as additive or multiplicative in Tables S1-S3. Moreover, their values in the original formulation together with the corresponding wide and narrow range are given. The intracellular potassium concentration [K] i was estimated for all currents: with x r being the gating variable. Its steady state value x r∞ , the two rate constants α xr and β xr and the time constant τ xr are defined as follows: (3)

SUPPLEMENTARY TABLES
Supplementary Table 1. I Kr parameters: besides the parameter names and units, their classification as additive (±) or multiplicative (*), the standard value from Courtemanche et al. (1998) and the parameter ranges are given.
Parameter Unit Type Standard value Narrow range Wide range        Supplementary Figure 5. Resulting I Ks current using the estimated parameters together with the corresponding voltage protocol (A). Solid lines indicate measured currents used for parameter estimation. Crosses represent the best fit obtained using the "high" setup of the hybrid (PSO+TRR)+TRR approach (every 15 th sample is shown). (B) shows the difference between the simulated and the measured currents. Samples directly adjacent to voltage steps were ignored in the cost function for the optimization and not plotted. Note the different scales in (A) and (B).