Gradient-based parameter optimization method to determine membrane ionic current composition in human induced pluripotent stem cell-derived cardiomyocytes

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Gradient‑based parameter
optimization method to determine
membrane ionic current
composition in human induced
pluripotent stem cell‑derived
cardiomyocytes
Hirohiko Kohjitani1, Shigeya Koda2, Yukiko Himeno2, Takeru Makiyama1, Yuta Yamamoto1,
Daisuke Yoshinaga3, Yimin Wuriyanghai1, Asami Kashiwa1, Futoshi Toyoda4, Yixin Zhang2,
Akira Amano2*, Akinori Noma2 & Takeshi Kimura1
Premature cardiac myocytes derived from human induced pluripotent stem cells (hiPSC-CMs) show
heterogeneous action potentials (APs), probably due to different expression patterns of membrane
ionic currents. We developed a method for determining expression patterns of functional channels
in terms of whole-cell ionic conductance (Gx) using individual spontaneous AP configurations. It has
been suggested that apparently identical AP configurations can be obtained using different sets of
ionic currents in mathematical models of cardiac membrane excitation. If so, the inverse problem of
Gx estimation might not be solved. We computationally tested the feasibility of the gradient-based
optimization method. For a realistic examination, conventional ’cell-specific models’ were prepared
by superimposing the model output of AP on each experimental AP recorded by conventional manual
adjustment of Gxs of the baseline model. Gxs of 4–6 major ionic currents of the ’cell-specific models’
were randomized within a range of ± 5–15% and used as an initial parameter set for the gradient-based
automatic Gxs recovery by decreasing the mean square error (MSE) between the target and model
output. Plotting all data points of the MSE–Gx relationship during optimization revealed progressive
convergence of the randomized population of Gxs to the original value of the cell-specific model with
decreasing MSE. The absence of any other local minimum in the global search space was confirmed
by mapping the MSE by randomizing Gxs over a range of 0.1–10 times the control. No additional local
minimum MSE was obvious in the whole parameter space, in addition to the global minimum of MSE
at the default model parameter.
Abbreviations
hiPSC-CMs Human induced pluripotent stem cell-derived cardiomyocytes
hVC model The human ventricular cell model
AP Action potential
MDP The maximum diastolic potential
SDD Slow diastolic depolarization
Im Membrane current
Vm Membrane voltage
orp Optimization of randomized model parameters
OS Overshoot potential
1
Department of Cardiovascular Medicine, Kyoto University Graduate School of Medicine, Kyoto, Japan. 2Graduate
School of Life Sciences, Ritsumeikan University, Kusatsu, Japan. 3Department Pediatrics, Kyoto University
Graduate School of Medicine, Kyoto, Japan. 4Department of Physiology, Shiga University of Medical Science, Otsu,
Japan. *email: a-amano@fc.ritsumei.ac.jp

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PO method Parameter optimization method
PS method Pattern search method
BP Base point for searching minimum MSE in the Pattern Search
NP Searching point in reference to BP in the Pattern Search
MSE Mean square error between two different Vm records
Stp Step size to move NP
x Subscript to represent membrane current, such as INa, ICaL, IK1, Iha, IKr, IKur, IKs, and IbNSC
Over the past half-century, the biophysical characteristics of ion-transporting molecules (channels and ion
exchangers) have been extensively analyzed. Biophysical models of each functional component have largely been
­detailed1–4 , including human induced pluripotent stem cells (hiPSC-CMs)5–7. In addition, various composite cell
models, including membrane excitation, cell contraction, and intracellular ionic composition homeostasis, have
been developed by integrating mathematical models at the molecular level into cardiac cell m
­ odels8–11. These
models have been useful for visualizing individual currents underlying the action potential (AP) configuration
under various experimental conditions in mature cardiac myocytes. However, the utility of these mathematical
cell models is limited because of the lack of extensive validation of the model output accuracy. This is a drawback
of the subjective manual fitting method used in almost all published mathematical cardiac cell models. A new
challenge of mechanistic models of cardiac membrane excitation might be an examination in a very different
paradigm to assess if the many, but continuous, variety of cardiac AP configurations, such as those recorded in
hiPSC-CMs, can be reconstructed by applying the automatic parameter optimization method to the hiPSC-CM
version of human cardiac cell models. We do not intend to propose a new hiPSC-CM model.
The automatic parameter optimization technique objectively determines parameters in a wide range of biological models, including cardiac e­ lectrophysiology12–15, systems p
­ harmacology16–20, and other models. Because of
this utility, many improvements in information technology have been r­ ealized21,22. However, in electrophysiology,
different combinations of model parameters may produce very similar ­APs13,23–25. The determination of current
density at high fidelity and accuracy likely requires additional improvements to the optimization method in the
cardiac cell model because of the complex interactions among ionic currents underlying membrane e­ xcitation23,26.
The final goal of our study is to develop an objective and accurate method for determining the current profile
(i.e., the expression level of functional ionic currents) underlying individual AP configurations. As a case study,
we chose a large variety of AP configurations in hiPSC-CMs, which are difficult to classify into the conventional
nodal, atrial, or ventricular types. The molecular bases of the ion channels expressed in hiPSC-CMs correspond
to those in adult cardiac myocytes in the GSE154580 Gene Expression Omnibus Accession viewer. Electrophysiological findings suggest that the gating of ionic currents is quite similar to that observed in mature ­myocytes27.
Thus, we modified the ion channel gating kinetics of the human ventricular cell (hVC) ­model11 according to the
prior experimental m
­ easurements27 for a hiPSC-CM type baseline model of the parameter optimization (PO)
method. For simplicity, we assumed that the opening/closing kinetics of ion channels expressed by the same
human genome remains the same among hiPSC-CMs. We also assumed that the heterogeneity of the electrical
activities of hiPSC-CMs might be determined by the variable expression levels of ion channels in the cell membrane. We computationally examined the feasibility of one of the basic gradient-based optimization methods, the
pattern search (PS) ­algorithm21,22,28, in a model of cardiac AP generation. We prepared a given AP configuration
using each ’cell-specific model’ prepared by the conventional manual fitting of the hVC model to the respective
experimental recordings. To assess the accuracy of the PS method for parameter optimization, the AP waveform
generated by the cell-specific model was used as a target of the optimization. The initial set of parameters for the
optimization was then prepared by uniform randomization centered around the default values of the model. The
PS algorithm should return the original parameter values by decreasing the mean squared error (MSE) function
between the modified model output and target AP waveforms. The accuracy of the optimization was determined
by recovering the original values of each ionic current amplitude as the MSE progressively decreased toward zero.

Materials and Methods

Baseline model of hiPSC‑CM membrane excitation.  The baseline model of hiPSC-CMs was essen-

tially the same as the hVC model, which has been ­detailed10,11 and which shares many comparable characteristics
with other published human m
­ odels8,9. The hVC model consists of a cell membrane with a number of ionic channel species and a few ion transporters, the sarcoplasmic reticulum equipped with the C
­ a2+ pump (SERCA), and
the refined C
­ a2+ releasing units coupled with the L-type C
­ a2+ channels on the cell membrane at the nanoscale
dyadic ­space4,29, contractile fibers, and three cytosolic ­Ca2+ diffusion spaces containing several ­Ca2+-binding
proteins (Fig. S1). All model equations and abbreviations are described in the Supplemental Materials.
The source code of the hiPSC-CM model was written in VB.Net and is available from the archive site (https://​
doi.​org/​10.​1101/​2022.​05.​16.​492203).
The kinetics of the ionic currents in the baseline model were readjusted according to new experimental
measurements if available in hiPSC-CMs27 (Fig. S2). In the present study, the net membrane current (Itot_cell) was
calculated as the sum of nine ion channel currents and two ion transporters (INaK and INCX) (Eq. 1).

Itot_cell = INa + ICaL + Iha + IK1 + IKr + IKs + IKur + IKto + IbNSC + INaK + INCX

(1)

The membrane excitation of the model is generated by charging and discharging the membrane capacitance
(Cm) using the net ionic current (Itot_cell) across the cell membrane (Eq. 1). The driving force for the ionic current
is given by the potential difference between Vm and the equilibrium potential (E x) (Eq. 3). The net conductance of the channel is changed by the dynamic changes in the open probability (pO) of the channel, which is

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mostly Vm-dependent through the Vm-dependent rate constants ( α , β ) of the opening and closing conformational
changes of the channel (Eq. 4 and 5). ...

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