An in Silico Investigation into the Role of SK Channels in Failing VentricularMyocytes
Marta Gomez1 , Jesus Carro1 , Esther Pueyo2,3 , Violeta Monasterio1
12Universidad San Jorge, Villanueva de Gallego, Spain
Aragon Institute for Engineering Research, University of Zaragoza, IIS Aragon, Spain
3CIBER in Bioengineering, Biomaterials & Nanomedicine (CIBER-BBN), Spain
Abstract

SK channels are small conductance (∼10 pS) calciumactivated potassium channels. Although they have beensuggested not to play a relevant role in healthy ventricles,
SK channel upregulation has been reported in failing ventricles, with potential implications for arrhythmogenesis.
In this work, we aimed at uncovering the contribution of
SK channels to ventricular repolarization in failing myocytes. To that end, we extended an in silico electrophysiological model of human ventricular failing myocytes byintroducing the equations representing SK channel activity. To determine the value of the maximal SK current conductance, GSK , we simulated action potentials (APs) at different pacing frequencies and we fitted the changes in AP
duration induced by SK channel inhibitions to available
experimental data recorded under pharmacological interventions. The SK block-induced effects in our simulations
were consistent with experimental evidence. Early afterdepolarizations were observed at pacing frequencies below 0.7 Hz only when SK channels were blocked. In conclusion, our presented model of human ventricular failing
myocytes integrating an SK channel formulation can allow
dissecting the contribution of SK channels to ventricularrepolarization and may help in understanding their role inarrhythmogenesis.

1.

Introduction

The small conductance (∼10 pS) calcium-activatedpotassium (SK) channels are an important group ofpotassium-selective ion channels. Some studies haveshown that SK channels significantly contribute to the repolarization phase of the atrial action potential (AP), although their beneficial or adverse effects remain debated.
SK channels were initially considered not to play a relevantrole in ventricular repolarization. However, different studies in ventricular myocytes, tissues and whole hearts haveshown that SK channel blockade has remarkable ventric-Computing in Cardiology 2022; Vol 49

ular effects in certain pathological states [1], highlighting
these channels as a possible backup mechanism in diseasedventricles.
In heart failure (HF), a chronic, progressive condition inwhich the heart muscle cannot pump enough blood to meet
the body’s needs for blood and oxygen [1], blockade of SK
channels has been experimentally reported to prolong AP
duration (APD). It has been suggested that upregulationof SK channels under pathological conditions could be anadaptive physiological response to shorten the APD underconditions of reduced repolarization reserve [2]. Nevertheless, there is conflicting evidence on the pro- or antiarrhythmic role of these channels [3]. The development ofmathematical models that incorporate descriptions of SK
channel activity in failing ventricles could help to shedlight on their role in cardiac pathophysiology.
The aim of this work was to uncover the contribution of
SK channels to ventricular repolarization in human failing
myocytes. We extended a mathematical model of cellularelectrophysiology in failing myocytes by incorporating aformulation for the current generated by the SK channels,
ISK , based on experimental data available in the literature.
We simulated APs with and without SK channel blockade
and investigated the occurrence of proarrhythmic early afterdepolarizations (EADs).

2.

Methods

2.1.

Model of failing myocytes with SK
channel formulation

The O’Hara et al. model (ORd) [4], which is the mostwidely used, validated human ventricular myocyte modelin the literature, was used as a basis. We replaced the original formulation of the fast sodium current, IN a , in the ORdmodel with the formulation in the ten Tusscher et al. model[5] to better represent the AP upstroke characteristics. HFassociated electrophysiological remodeling was simulatedby scaling the maximal conductances and time constants

Page 1 ISSN: 2325-887X DOI: 10.22489/CinC.2022.228

of various ionic currents and fluxes as in Gomez et al. [6].
A formulation for the ISK current was integrated in thefailing ventricular model by using the equations proposedin Landaw et al. [7]:= GSK xSK (V − EK )ISKxSK,∞ − xSKdx
SK= dtτSK(1)
1
xSK,∞ =n
1 + (Kd /CaSS )τ1= τ0 +τSK
1 + (CaSS /0.1)
where CaSS is the calcium concentration in the subspacecompartment (submembrane space near t-tubules), which
is sensed by SK channels. In the equations, the following
parameter values were set in concordance with experimental observations [7, 8]: τ0 = 4 ms, τ1 = 20 ms, n = 3.14,
and Kd = 0.000345 mM .

2.2.

Fitting of SK current conductance

The maximal conductance GSK of the SK current wasadjusted so that the resulting model reproduced the experimental AP prolongation induced by ISK block in human ventricular failing myocytes [1]. In the experiments,
APD at 50% and 90% repolarization, APD50 and APD90 ,were measured in midmyocardial ventricular cells fromseven HF patients, at baseline and after apamine-induced
ISK block while pacing at 0.5, 1 and 2 Hz (Figure 1).
From the experimental APD values, we calculated therelative change (R) in APD50 and APD90 induced by SK
channel block, for 1 Hz and 2 Hz pacing:
R=APDapamin − APDbaseline
100APDbaseline(2)where APD represents APD50 or APD90 . Experimental
data at 0.5 Hz pacing were not used for GSK fitting sincevarious AP traces presenting EADs had been excluded
from the data in [1]. The four experimental R values arepresented in Table 1.
The optimal value of GSK was found by minimizing thedifference between the experimental and simulated relativechanges in APD50 and APD90 using the method proposedin [9]:4X
f (GSK ) =(Ri − Si (GSK ))2 ,(3)i=1where Ri represents each of the four experimental relativechanges and Si (GSK ) represents the corresponding simulated relative change for a given conductance value GSK .
Simulations were ran for twenty different values of GSK
and the resulting S values were interpolated by quadratic
polynomials, from which the search for the minimum wasperformed using Brent’s method. A search range for optimal GSK was established from 0 to 0.01 based on experimental reports [7]. The value of GSK associated with thelowest error f was selected.
Simulations were carried out at baseline (with the optimal GSK value) and under full SK channel block (GSK = 0)at pacing frequencies varying from 0.5 to 2 Hz in 0.1-Hzsteps. For comparison purposes, additional simulations
were performed using the non-failing ORd model, bothwith and without SK channels.
All simulations were ran using the cardiac electrophysiology simulator DENIS [10]. Forward Euler wasused for numerical integration with a time step of0.002 ms. Monophasic current pulses with an amplitudeof −80 µA/µF were used for stimulation. The model waspaced for 20 cycles after steady-state was reached. The
effect of apamin was simulated as full SK channel block(GSK = 0), as the concentration used in the experiments(i.e. 100 nM ) was an order of magnitude higher than theIC50 value reported in the literature [11].

2.3.

Sensitivity analysis

A sensitivity analysis was conducted to determine theinfluence of variations in ionic conductances on the generation of EADs at a low pacing rate of 0.5 Hz. Each conductance in the model was varied by ±5% and ±10% andthe absence or presence of EADs was assessed both withand without ISK block.

Figure 1. Experimental APD50 and APD90 with and without ISK block. Figure replicated from [1].

3.

Results

3.1.

Model development and validation

The optimal value for the SK current conductance was
GSK = 0.0038 mS/µS. Relative changes in APD50 andAPD90 for simulations calculated with that GSK value differed from the experimental relative changes, R, by lessPage 2

Table 1. Relative changes in APD50 and APD90 induced
by ISK block in simulations and experiments.
Simulated Reference (R) Error (%)
APD50 (1Hz) 27.27
26.43
0.84
APD50 (2Hz) 19.85
21.22
1.37
APD90 (1Hz) 21.77
25.26
3.49
APD90 (2Hz) 20.51
19.211.3

Figure 3. APs computed with and without ISK block atpacing frequencies of (A) 2 Hz, (B) 1 Hz and (C) 0.5 Hz.

Figure 2. Simulated APD50 and APD90 with and without
ISK block. EADS appear below 0.7 Hz (*).
than 3.5% (Table 1). Both APD50 and APD90 were prolonged under ISK block, in line with experimental observations for pacing frequencies above 0.7 Hz (Figure 2). For
frequencies below 0.7 Hz, simulated ISK block led to EAD
generation. Simulated AP traces at 0.5, 1 and 2 Hz are
depicted in Figure 3, both with and without ISK block.
Simulated APD50 and APD90 values for the non-failingventricular myocyte model are shown in Figure 4. In thiscase, ISK block prolonged both AP durations, but did notlead to EADs at any pacing frequency.

3.2.

Ionic variability

When variability in ionic currents other than ISK wassimulated in our extended failing myocyte model, EADs
could not be observed for any pacing frequency. Following simulation of ISK block, EADs developed for most of
the ionic current variations (Figure 5).

4.

Discussion and conclusions

In this work, we extended an existing cardiac electrophysiology model of failing human ventricular myocytesby adding the equations for the ISK current. To do so, aformulation without rectification was chosen for ISK . Alternative formulations with rectification, as in [12], would
require the estimation of a larger number of parameters,
which, given the lack of sufficient experimental data, could

Figure 4. Simulated APD50 and APD90 with and without
ISK block.
introduce a high degree of uncertainty in the model. Inour selected formulation, the values for the time constants
and the dissociation constant were taken from the literature. The value for the conductance GSK was determining by solving an optimization problem so that the model
best replicated the experimental prolongation of APD50
and APD90 induced by SK channel block at pacing frequencies of 1 and 2 Hz (Table 1). Although lower pacing
frequencies, like 0.5 Hz, were simulated, the results werenot used to fit GSK because AP traces exhibited EADs when
SK channels were blocked (see Figures 2 and 3), in agreement with experiments in [1].
By defining a set of failing myocyte models built by
varying ionic current conductances up to ±10%, the resultsof our sensitivity analysis suggested that ISK plays a major
protective role from EADs formation at low pacing frequencies. No EADs were found when ISK was active, while
EADs developed for most ionic variations when ISK wasPage 3

Figure 5. EAD development (dark red) under ionic current
variations and ISK block.
blocked (Figure 5). These findings are consistent with theexperimental observations in [1], where EADs were frequent after apamin infusion but did not occur at baseline,
and suggest that SK channels may exert an antiarrhythmic
effect by compensating the low repolarization reserve infailing ventricles, as hypothesized in the literature [8].
As limitations of this work, it should be noted that theadjusted value for the conductance GSK was estimated for
midmyocardial cells and may not apply for epi- or endocardial cells, as transmural differences in ISK have been reported experimentally [8]. Also, while the proposed parameter values in the ISK formulation reproduce the APD
prolongation in failing myocytes, it is not applicable tonon-failing myocytes. Indeed, results in Figure 4 showthat including exactly the same ISK formulation in the nonfailing model would lead to APD prolongation under SK
channel block, which would disagree with the understanding that SK channels do not contribute to AP repolarizationin ventricular cells under physiological conditions [13].

Acknowledgments
This work was supported by Ministerio de Ciencia eInnovación (Spain) through project PID2019-105674RBI00 and by Aragón Government through BSICoS group(T39 20R) and projects LMP94 21 and LMP141 21.
Computations were carried out in DENIS (Volunteer
Computer platform supported by Universidad San Jorge
through Proyectos Internos 2021-22).

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Address for correspondence:
Name: Marta Gomez
Address: A.Mudejar, km.299, 50830 Villanueva de Gallego
E-mail address: magomez@usj.esPage 4