Afterdepolarizations and abnormal calcium handling in atrial myocytes with modulated SERCA uptake: a sensitivity analysis of calcium handling channels

Abstract

Delayed afterdepolarizations (DADs) and spontaneous depolarizations (SDs) are typically triggered by spontaneous diastolic Ca²⁺ release from the sarcoplasmic reticulum (SR) which is caused by an elevated SR Ca²⁺-ATPase (SERCA) uptake and dysfunctional ryanodine receptors. However, recent studies on the T-box transcription factor gene (TBX5) demonstrated that abnormal depolarizations could occur despite a reduced SERCA uptake. Similar findings have also been reported in experimental or clinical studies of diabetes and heart failure. To investigate the sensitivity of SERCA in the genesis of DADs/SDs as well as its dependence on other Ca²⁺ handling channels, we performed systematic analyses using the Maleckar et al. model. Results showed that the modulation of SERCA alone cannot trigger abnormal depolarizations, but can instead affect the interdependency of other Ca²⁺ handling channels in triggering DADs/SDs. Furthermore, we discovered the existence of a threshold value for the intracellular concentration of Ca²⁺ ([Ca²⁺]_i) for abnormal depolarizations, which is modulated by the maximum SERCA uptake and the concentration of Ca²⁺ in the uptake and release compartments in the SR ([Ca²⁺]_up and [Ca²⁺]_rel). For the first time, our modelling study reconciles different mechanisms of abnormal depolarizations in the setting of ‘lone’ AF, reduced TBX5, diabetes and heart failure, and may lead to more targeted treatment for these patients.

This article is part of the theme issue ‘Uncertainty quantification in cardiac and cardiovascular modelling and simulation’.

1. Introduction

Atrial fibrillation (AF) leads to the rapid and irregular contraction of the upper chambers of the heart, characterized by the disorganized electrical activity within the atria [1]. It is the most encountered arrhythmia in clinical practice, and if left untreated, can lead to an increased risk of heart failure, stroke, and mortality [2]. AF is associated with an increased risk from conditions such as diabetes, obesity, and hypertension [3,4] or from old age [5]. Altered Ca²⁺ homeostasis is strongly implicated in the initiation of AF [6,7]. In paroxysmal AF, self-terminating episodes of arrhythmia occur in structurally normal atria. This has been linked with increased diastolic Ca²⁺ leak from the sarcoplasmic reticulum (SR) and triggered activity from upregulated SR Ca²⁺ ATPase2a (SERCA) and type 2 ryanodine receptor (RyR) dysregulation [7–10]. Similarly, an elevated SERCA uptake was observed in patients with persistent AF [10].

Recent studies from referral populations and in the community suggest that the familial clustering of AF may be genetically caused [11–15]. Furthermore, epidemiological studies have found that individuals who have a first-degree relative with lone AF carry an eightfold increased risk of AF [16]. Linkage and candidate gene sequencing methods have identified several rare genetic variants in genes encoding proteins which alter the kinetics of ion channels, leading to AF [17,18]. With the advent of the genome-wide association studies (GWAS) approach, studies were instead predominantly focused on identifying the common genetic variants associated with AF in the general population [19].

Among recent large-scale GWAS studies, multiple transcription factor genes including the T-box transcription factor gene TBX5 were found to be strongly associated with the arrhythmia [20–24]. TBX5 is known primarily for its role in cardiac development and rhythm control during embryogenesis [25]. Mutations in TBX5 have been reported to underlie Holt-Oram syndrome, features of which include conduction abnormalities and AF [26]. Through a crucial knockout mice study by Nadadur et al. [27], they demonstrated that impairment in TBX5 expression resulted in delayed afterdepolarizations (DADs) and spontaneous depolarizations (SDs) which can lead to AF (figure 1a) [28]. Dai et al. hypothesized that these abnormalities were caused by the remodelling of Ca²⁺ handling channels and they conducted a subsequent study to investigate their activity in TBX5 knockout mice [29]. Using an experimental protocol with caffeine-containing, Na⁺-free, Tyrode solution to block Ca²⁺ extrusion through I_NaCa, and then a protocol with Na⁺ containing caffeine solution to prevent uptake through SERCA in the subsequent experiment, they found that the activity of SERCA and I_NaCa had decreased and increased respectively. The conductivity of the L-type Ca²⁺ current (I_CaL) was also shown to have increased through a voltage-clamp experiment. In addition, they found that if the SERCA inhibitor phospholamban (PLN) was reduced in these TBX5 knockout atrial myocytes, then this instead restored SERCA activity and abolished DADs/SDs. This is intriguing, as these DADs and SDs were caused by a reduction in SERCA instead.

Figure 1.

(a) Example of experimental traces of the membrane potential from the experiment by Nadadur et al. [27]. Delayed afterdepolarizations (DADs) and spontaneous depolarizations (SDs) occur for a majority of (homogeneous TBX5 knockout) mice atrial myocytes, but none for control mice atrial myocytes. (b) Simplified schematic highlighting that reduced expression of TBX5 reduces sarcoplasmic reticulum Ca²⁺ ATPase 2a (SERCA) and increases I_NaCa and I_CaL, respectively. (Online version in colour.)

In relation to this, experimental studies on heart failure and diabetes have also reported a reduction in SERCA and an increase in I_NaCa [6,30–33], as well as the elevated occurrence of DADs/SDs [34,35]. Furthermore, restoration of SERCA function through viral transfection of SERCA into cardiac tissue in vivo or knockout of PLN has also been shown to improve cardiomyocyte contractility and suppress triggered activities in both animal and clinical studies of heart failure and diabetes [36–39]. Together, these studies highlight our incomplete understanding of the sensitivity of SERCA in DAD/SD genesis as well as its dependence on other key Ca²⁺ handling channels such as the RyR, I_NaCa, and I_CaL.

There has also been much debate into the actual mechanisms by which spontaneous diastolic Ca²⁺ release from the SR can occur to trigger DADs/SDs [40]. Several reports have hypothesized that this is due to the overload of Ca²⁺ in the SR, termed store-overload induced Ca²⁺ release [41–43]. Some experimental studies have instead indicated that a more loaded SR is not the only prerequisite for spontaneous diastolic Ca²⁺ release [44]. Using heart failure as an example, there have been reports that DADs/SDs can appear during this condition despite low Ca²⁺ levels in the SR [33,45–47].

Altered Ca²⁺ handling and its implications on the mechanisms of triggered activity can only be explored through a computer model as it is a powerful tool for quantitative dissection of the fundamental mechanisms and in investigating their individual role in DADs/SDs [48]. In this study, we use the Maleckar et al. model [49] to explore the complex relationship between SERCA and other Ca²⁺ handling channels, and how changes in this relationship affect the Ca²⁺ dynamics and the susceptibility of the atrial myocyte to DADs/SDs. We define an SD as a self-depolarization that has reached -50 mV or higher, and a DAD as a self-depolarization that is below −50 mV. Throughout this study, we consider only the occurrence of SDs in our results as it is easier to identify in simulation than DADs. However, all our analysis and results for SDs are applicable to DADs as well.

2. Methods

(a). The Maleckar et al. model

The Maleckar et al. model was based on the human atrial myocyte model of Nygren et al. [50,51], and was designed to accurately capture the repolarization phase of an action potential (AP) at different cycle lengths [49,52]. In the Maleckar et al. model (figure 2), the time and voltage-dependent currents I_Na (Na⁺ current), I_CaL, I_t, I_Kur, I_K1, I_Kr, and I_Ks (Ca²⁺ independent transient outward, ultra-rapid, inward rectifier and delayed rectifier K⁺ currents, respectively) are the main attributors to the generation of the human atrial AP.

Figure 2.

Schematic of the full Maleckar et al. model [49]. The sarcoplasmic reticulum Ca²⁺ ATPase2a (SERCA) pump is bidirectional, and the activation of the ryanodine receptor is modulated by [Ca²⁺]_d and [Ca²⁺]_i. (Online version in colour.)

Ca²⁺ cycling occurs through SERCA and RyR, located on the SR. The uptake of Ca²⁺ from the intracellular domain follows a bidirectional formulation [53], where if the concentration of Ca²⁺ in the uptake compartment ([Ca²⁺]_up) becomes high relative to the concentration of Ca²⁺ in the intracellular ([Ca²⁺]_i), then the pump will operate more prominently in the reverse direction to reduce the net flux from SERCA. In addition, Ca²⁺ is transferred from the uptake compartment to the release compartment through I_tr (transfer current). Activation of the RyR is modulated by the concentration of Ca²⁺ in the dyadic space and intracellular ([Ca²⁺]_d and [Ca²⁺]_i, respectively). Release of Ca²⁺ from the SR is triggered when [Ca²⁺]_d and/or [Ca²⁺]_i is sufficiently elevated, known as Ca²⁺-induced Ca²⁺ release. Once activated, the concentration difference between [Ca²⁺]_rel and [Ca²⁺]_i determines the magnitude of I_rel.

Currently, the only reported functional changes due to a reduction in TBX5 are from experimental studies by Dai et al. [29] and Laforest et al. [54]. Voltage-clamp experiments show an increase in I_CaL in atrial myocytes with reduced expression of TBX5, and separate caffeine-based protocols demonstrated a reduction in SERCA and an elevation in I_NaCa activity through measurements in the decay rate of [Ca²⁺]_i. Similar findings have also been reported in the setting of diabetes and heart failure [32,35,38,55]. Hence, we model functional changes in SERCA as simply a multiplicative change in maximum SERCA uptake, and changes in I_CaL and I_NaCa as a change in their conductivity. In addition, because the underlying commonality behind these conditions is disrupted Ca²⁺ handling, we have also included RyR in this study, and changes in its activity will also be modelled as a change in its conductivity. The study will be carried out across a wide parameter space to assess how changes in these parameters impact Ca²⁺ handling.

Throughout our experiments, we define the control as the original full Maleckar et al. model with its original parameters and equations, i.e. normal human atrial myocytes under physiological conditions. In each of our simulations with the full cellular model, a train of stimuli with a duration of 6 ms and magnitude of −15 pA/pF was applied at a frequency of 1 Hz up to 100 s into the simulation.

(b). Bifurcation analysis and sensitivity of SERCA

We first investigated how the individual sensitivity of SERCA compared with that of I_CaL, I_NaCa, and RyR in the genesis of SDs by running simulations where we individually varied the conductivities of these channels from a range of 0 to 2 times its control value. We then probed for the minimum increase or decrease required for each channel for an SD to occur in increments of 0.01 until it was identified.

Afterwards, we conducted a type of analysis known as a bifurcation analysis to investigate how changes in the parameter of a Ca²⁺ handling channel affected that of another channel in triggering SDs. The first bifurcation analysis we performed was between I_CaL and I_NaCa with SERCA at its control value. We varied the conductivity of I_CaL from a range of 0.2 to 2 times its control value in increments of 0.2, and for each of these values of I_CaL, we identified the minimum multiplier that must be applied to I_NaCa to trigger at least one SD. Following this, we then conducted three other bifurcation analyses and identified minimum multipliers for RyR, I_CaL, and I_NaCa for varying levels of SERCA using the same approach.

Minimum multipliers were determined through a trial and error approach. For example, if we were interested in finding the minimum multiplier that must be applied to RyR to trigger at least one SD, then we ran simulations of the full Maleckar et al. model with SERCA set at a certain condition, e.g. 120% of its original value, for varying levels of RyR until we ascertain a rough estimate of the parameter region where the SDs occur. After this, we honed in onto the minimum multiplier value for RyR that could cause at least one SD to occur for 120% SERCA. Through this approach, we derived the SD boundary lines for each of the corresponding bifurcation diagrams which illustrated the relationship between Ca²⁺ handling parameter pairs in the genesis of at least one SD.

(c). Calcium subsystem analysis

In the full Maleckar et al. model, it can be quite complex to identify the exact mechanisms to trigger an SD since many factors can influence this. However, by reducing the model down to only the essentials in Ca²⁺ handling, i.e. Ca²⁺ subsystem (electronic supplementary material, figure S1), we can instead probe for the conditions for Ca²⁺ oscillations instead which is far simpler to identify as they only take place when the myocyte is overloaded with Ca²⁺. These conditions can then be applied to the full Maleckar et al. model to investigate the mechanisms of SDs as such abnormal depolarizations are triggered by the occurrence of Ca²⁺ oscillations in the Ca²⁺ subsystem [56–60]. To carry out this Ca²⁺ subsystem analysis, we used the approach developed by Fink et al. [40].

To reduce the Maleckar et al. model down to its Ca²⁺ subsystem, we set all transmembrane currents apart from I_NaCa to zero, and set the membrane potential and intracellular and cleft space Na⁺ concentrations to a constant value as determined by their end-diastolic values after 3000 s of 1 Hz pacing in the full Maleckar et al. model. Note that I_CaL was also removed despite playing an important role in Ca²⁺ handling in the full model because since the membrane potential was clamped, I_CaL would not activate, and hence would provide a negligible amount of Ca²⁺ into the reduced model. Since the bulk space and K⁺ concentrations were not required in the Ca²⁺ subsystem, we removed them from the reduced model (electronic supplementary material, figure S1).

To induce an overload of Ca²⁺, a multiplier was applied to clamped [Na⁺]_i. This multiplier caused a rise in Ca²⁺ in the myocyte, which triggered I_NaCa to extrude Na⁺ out of the cell and Ca²⁺ into the cell. A minimum multiplier was required to sufficiently elevate the total Ca²⁺ in the myocyte (which we defined as [Ca²⁺]_tot= [Ca²⁺]_up + [Ca²⁺]_rel + [Ca²⁺]_d + [Ca²⁺]_i) so that Ca²⁺ oscillations occurred. During this phenomenon, [Ca²⁺]_up, [Ca²⁺]_rel, [Ca²⁺]_d, and [Ca²⁺]_i oscillated between a maximum and minimum value as time progressed. These oscillations would persist since [Na⁺]_i remained fixed at that elevated value.

We considered the system as having successfully generated Ca²⁺ oscillations if the average range during the simulation time-course of 5000 s was at least 0.2 mM for [Ca²⁺]_up, [Ca²⁺]_rel, and 0.00002 mM for [Ca²⁺]_d, and [Ca²⁺]_i. Note that if the multiplier for [Na⁺]_i could not sufficiently elevate [Ca²⁺]_tot to trigger SDs, then [Ca²⁺]_up, [Ca²⁺]_rel, [Ca²⁺]_d, and [Ca²⁺]_i would not oscillate and would immediately reach steady state instead.

Using the reduced model, we implemented the changes in SERCA, RyR, and I_NaCa from a range of 0.2 to 2 times their respective control values in increments of 0.2 individually. For each increase for a particular channel, we then found the minimum multiplier required for clamped [Na⁺]_i to subsequently elevate the total Ca²⁺ concentration in the atrial myocyte sufficiently to trigger Ca²⁺ oscillations. The corresponding minimum [Ca²⁺]_tot that triggered Ca²⁺ oscillations was derived by finding the steady state for [Ca²⁺]_up, [Ca²⁺]_rel, [Ca²⁺]_d, and [Ca²⁺]_i during these oscillations. Steady state for each compartment was obtained by computing their mean value over the simulation time-course of 5000 s. This approach was then used to investigate how changes in the Ca²⁺ handling channels along the points on the SD boundary line for SERCA-RyR and SERCA-I_NaCa bifurcations impacted the minimum [Ca²⁺]_tot for Ca²⁺ oscillations and hence SDs.

(d). The dependency test of ryanodine receptors and SERCA on the genesis of spontaneous depolarizations

As previously mentioned, there is much debate in experimental studies regarding the precise mechanisms by which diastolic spontaneous Ca²⁺ release occurs to trigger SDs. However, in the study by Fink et al., the importance of the modulation of [Ca²⁺]_i and [Ca²⁺]_up on SERCA, and [Ca²⁺]_i and [Ca²⁺]_d on RyR in the genesis of SDs was also investigated.

We conducted this dependency test on the full Maleckar et al. model to determine the necessary properties for the initiation of SDs in this model. To carry this out, we first applied a train of stimuli with a duration of 6 ms and magnitude of −15 pA/pF at a frequency of 1 Hz up to 3000 s into the simulation. At 3000 s, we recorded the values of [Ca²⁺]_up, [Ca²⁺]_i, and [Ca²⁺]_d, and labelled them as [Ca²⁺]_{up, 3000s}, [Ca²⁺]_{i, 3000s}, and [Ca²⁺]_{d, 3000 s,} respectively.

To illustrate the subsequent steps, we then used the RyR as an example, the modulation terms of the RyR in the full Maleckar et al. model were given by

$$\begin{array}{rl}{\mathrm{r}}_{{\mathrm{Ca}}_{\mathrm{d}\hspace{0.17em}\mathrm{term}}}& \hspace{0.17em}=\frac{{[\mathrm{C}{\mathrm{a}}^{2+}]}_{\mathrm{d}}}{{[{\mathrm{Ca}}^{2+}]}_{\mathrm{d}}+{\mathrm{k}}_{{\mathrm{rel}}_{\mathrm{d}}}},\end{array}$$ 2.1

$$\begin{array}{rl}{\mathrm{r}}_{{\mathrm{Ca}}_{\mathrm{i}\hspace{0.17em}\mathrm{term}}}& \hspace{0.17em}=\frac{{[{\mathrm{Ca}}^{2+}]}_{\mathrm{i}}}{{[{\mathrm{Ca}}^{2+}]}_{\mathrm{i}}+{\mathrm{k}}_{{\mathrm{rel}}_{\mathrm{i}}}},\end{array}$$ 2.2

$$\begin{array}{rl}{\mathrm{r}}_{{\mathrm{Ca}}_{\mathrm{d}\hspace{0.17em}\mathrm{factor}}}& \hspace{0.17em}={({\mathrm{r}}_{{\mathrm{Ca}}_{\mathrm{d}\hspace{0.17em}\mathrm{term}}})}^3,\end{array}$$ 2.3

$$\begin{array}{rl}\mathrm{and}{\mathrm{r}}_{{\mathrm{Ca}}_{\mathrm{i}\hspace{0.17em}\mathrm{factor}}}& \hspace{0.17em}={({\mathrm{r}}_{{\mathrm{Ca}}_{\mathrm{i}\hspace{0.17em}\mathrm{term}}})}^3.\end{array}$$ 2.4

To remove the modulation of [Ca²⁺]_i on RyR to test its dependence on the genesis of SDs, we substituted [Ca²⁺]_i in equation (2.2) with [Ca²⁺]_{i, 3000s}. This gave

$${\mathrm{r}}_{{\mathrm{Ca}}_{\mathrm{i}\hspace{0.17em}\mathrm{term}}}=\frac{{[{\mathrm{Ca}}^{2+}]}_{\mathrm{i},\hspace{0.17em}3000\mathrm{s}}}{{[\mathrm{C}{\mathrm{a}}^{2+}]}_{\mathrm{i},\hspace{0.17em}3000\mathrm{s}}+{\mathrm{k}}_{{\mathrm{rel}}_{\mathrm{i}}}}.$$ 2.5

To test the dependence of [Ca²⁺]_d on RyR, we substituted [Ca²⁺]_d in equation (2.1) with [Ca²⁺]_{d, 3000 s}. This gave

$${\mathrm{r}}_{{\mathrm{Ca}}_{\mathrm{d}}\mathrm{term}}=\frac{{[\mathrm{C}{\mathrm{a}}^{2+}]}_{\mathrm{d},\hspace{0.17em}3000\mathrm{s}}}{{[\mathrm{C}{\mathrm{a}}^{2+}]}_{\mathrm{d},\hspace{0.17em}3000\mathrm{s}}+{{\mathrm{k}}_{\mathrm{rel}}}_{\mathrm{d}}}.$$ 2.6

By a similar approach, the dependence of [Ca²⁺]_i and [Ca²⁺]_up on SERCA in the full Maleckar et al. model was also tested.

(e). Intracellular calcium injection protocols

The first protocol was designed to investigate the threshold value for [Ca²⁺]_i in triggering SDs, and whether this threshold followed a biphasic trend as SERCA decreased from 200% to 20% of its control value. The second protocol was designed to assess whether [Ca²⁺]_up and [Ca²⁺]_rel were also determining factors in [Ca²⁺]_i SD threshold for various levels of SERCA, and to ascertain how changes in SERCA impacted Ca²⁺ handling and hence gave rise to the biphasic trend in the [Ca²⁺]_i SD threshold.

(i). Intracellular calcium injection protocol 1

We applied a train of stimuli with a duration of 6 ms, magnitude of −15 pA/pF and frequency of 1 Hz up to 100 s into the simulation. At exactly 100 s into the simulation, we recorded the value of [Ca²⁺]_i, then applied a multiplier to this value. This new value was then used to replace the original value of [Ca²⁺]_i, and the simulation was then resumed except no external stimulus was applied. This protocol was repeated until we found the minimum multiplier required to increase [Ca²⁺]_i to the threshold value needed to trigger an SD at 100 s. The protocol was performed for values of SERCA from 20% to 200% control SERCA in increments of 20%.

(ii). Intracellular calcium injection protocol 2

For this protocol, we followed the same steps as outlined above, except exactly 100 s into the simulation, we adjusted [Ca²⁺]_up and [Ca²⁺]_rel to their value at 20% SERCA at 100 s in protocol 1 before we applied the multiplier to [Ca²⁺]_i. The same protocol in protocol 1 was repeated but with [Ca²⁺]_up and [Ca²⁺]_rel adjusted to their value at 200% at 100 s instead.

3. Results

(a). The impact of individual calcium handling channels on spontaneous depolarizations

From our series of simulations where we individually implemented the remodelling of each Ca²⁺ handling channel from a range of 0% to 200% of their control value, we identified a minimum value that I_NaCa, I_CaL, RyR, and SERCA must be increased or decreased to trigger SDs. For I_NaCa, SDs occurred when it was decreased by 11%, while for I_CaL and RyR, they instead occurred when they were increased by 13% and 38%, respectively (figure 3a). We were unable to identify a minimum value by which we needed to increase or decrease SERCA alone to generate SDs. In figure 3b–f, we show the time-course of the membrane potential, I_rel, [Ca²⁺]_i, [Ca²⁺]_up, and [Ca²⁺]_rel after the last stimulated AP (black triangle) and the SDs occurred at their respective minimum percentage for I_NaCa, I_CaL, and RyR.

Figure 3.

(a) The minimum percentage change to control individual I_NaCa, I_CaL and RyR to trigger SDs. (b–f) Time-course of the membrane potential, I_rel, [Ca²⁺]_i, [Ca²⁺]_up, and [Ca²⁺]_rel, for control and for 89% I_NaCa, 113% I_CaL, and 138% RyR after the last stimulated AP (black triangle). SERCA, sarcoplasmic reticulum Ca²⁺-ATPase; SD, spontaneous depolarization; RyR, ryanodine receptors. (Online version in colour.)

(b). The sensitivity of spontaneous depolarizations to SERCA

We examined the relationship between I_CaL and I_NaCa in SDs through a bifurcation analysis (figure 4a). As shown by the blue SD boundary quadratic line in figure 4a, as we increased the multiplier to I_CaL, the minimum multiplier for I_NaCa required to generate SDs also increased. The region below the blue SD boundary line illustrates the parameter range between I_CaL and I_NaCa where SDs can occur.

Figure 4.

(a) Bifurcation diagram of I_CaL and I_NaCa at 100% SERCA. Shaded region illustrates the parameter range where SDs occur. (b–f) Time-course of the membrane potential, I_rel, [Ca²⁺]_i, [Ca²⁺]_up, and [Ca²⁺]_rel for control and at 140% I_CaL and 125% I_NaCa for 100%, 54%, and 107% SERCA after the last stimulated AP (black triangle). SERCA, sarcoplasmic reticulum Ca²⁺-ATPase; SD, spontaneous depolarization; AP, action potential. (Online version in colour.)

We then chose a data point on the blue line of the I_CaL-I_NaCa bifurcation (figure 4a), e.g. 1.4 × I_CaL, 1.25 × I_NaCa, and ran a series of simulations where we changed SERCA from 0% to 200% of its control value to ascertain how changes in SERCA affected the capacity of I_CaL, and I_NaCa in generating SDs. As expected, SDs were observed when we kept SERCA at its control value with 140% I_CaL and 125% I_NaCa since we used a pair of values on the SD threshold of I_CaL and I_NaCa. However, SDs were abolished when SERCA was reduced below 55% or increased above 106% of its control value (figure 4b). This illustrates an interdependency between SERCA with I_CaL and I_NaCa, as SDs only occurred within a certain parameter range for SERCA. In figure 4c–f, we show the time-course of I_rel, [Ca²⁺]_i, [Ca²⁺]_up, and [Ca²⁺]_rel after the last stimulated AP with SERCA at 100%, 54%, 107% of its control value.

(c). Bifurcation analysis between SERCA and other calcium handling channels

As observed in the previous subsection, changes in SERCA can have an impact on I_CaL, RyR, and I_NaCa in generating SDs. To investigate this issue systematically, we conducted a SERCA-I_CaL, SERCA-RyR and SERCA-I_NaCa bifurcation. For both SERCA-RyR and SERCA-I_CaL bifurcations (figure 5a,b), at less than approximately 80% SERCA, the minimum multiplier for I_CaL and RyR required to generate at least an SD decreased linearly with increasing SERCA. However, at greater than approximately 80%, the minimum multiplier for I_CaL and RyR instead increased linearly with increasing SERCA. For the SERCA-I_NaCa bifurcation, at less than approximately 80% SERCA, the minimum multiplier for I_NaCa instead increased linearly with increasing SERCA, but decreased linearly at greater than approximately 80% (figure 5c). Together, this implied that at approximately 80% SERCA, there was a shift in the Ca²⁺ handling of the atrial myocyte since that was roughly the point where the trend changed direction for all three bifurcation diagrams.

Figure 5.

Bifurcation analysis for SERCA-RyR (a) and SERCA-I_Cal (b), respectively. (c) Bifurcation analysis for SERCA-I_NaCa. Shaded region illustrates the parameter range where SDs occur. The cross marks roughly where the shift in the Ca²⁺ handling took place. SERCA, sarcoplasmic reticulum Ca²⁺-ATPase; SD, spontaneous depolarization; RyR, ryanodine receptors. (Online version in colour.)

(d). The calcium subsystem sensitivity analysis of SERCA

Under this subsection, we conducted a Ca²⁺ subsystem sensitivity analysis to investigate how changes in SERCA, RyR, and I_NaCa alone affected the minimum multiplier to clamped [Na⁺]_i (and hence the total Ca²⁺ concentration threshold) to generate Ca²⁺ oscillations. For either SERCA or I_NaCa, as it decreased from 200% of its control value to 40% of its control value, the minimum multiplier to clamped [Na⁺]_i also decreased (electronic supplementary material, figure S2). By contrast, the minimum multiplier to clamped [Na⁺]_i instead increased as RyR decreased. Since the existence of Ca²⁺ oscillations in the Ca²⁺ subsystem corresponded to the occurrence of SDs in the full model, this showed that reducing SERCA or I_NaCa alone decreased the minimum total Ca²⁺ concentration required for Ca²⁺ oscillations, while reducing RyR instead has the opposite effect.

We then conducted a subsequent analysis to find how changes along the SD boundary in the direction of increasing SERCA for both SERCA-RyR and SERCA-I_NaCa bifurcations affected the minimum multiplier for clamped [Na⁺]_i for Ca²⁺ oscillations to occur. The obtained data points for the minimum multiplier for clamped [Na⁺]_i from the changes in SERCA and I_NaCa, and changes in SERCA and RyR along the SD boundary line in their respective bifurcation diagrams were then compared with the points obtained from changes in SERCA alone (electronic supplementary material, figure S2).

As shown in electronic supplementary material, figure S3A, as we went in the direction of decreasing SERCA, the minimum multiplier for clamped [Na⁺]_i for Ca²⁺ oscillations decreased for all three cases. Correspondingly, decreasing SERCA uptake function reduced the minimum [Ca²⁺]_tot needed for SDs to occur, and thus increased the myocyte's susceptibility to SDs in the full Maleckar et al. model (electronic supplementary material, figure S3B). Interestingly, electronic supplementary material, figure S3B also showed that only the minimum [Ca²⁺]_tot for SDs was significantly different for changes in both SERCA and RyR based upon the points in the SD boundary of the SERCA-RyR bifurcation. This, along with the linear decreasing tend among the three cases suggested that reducing the minimum [Ca²⁺]_tot was not the only requirement for SDs to occur, and that a deeper mechanism existed alongside this.

(e). The necessary components of the genesis of spontaneous depolarizations

One plausible explanation for the biphasic trend in the SERCA-RyR, SERCA-I_CaL, and SERCA-I_NaCa bifurcation diagrams was due to the Ca²⁺ threshold required to trigger SDs changing in a biphasic manner as SERCA changes, and thus different minimum increases in RyR and I_CaL, or decreases in I_NaCa were required to meet this Ca²⁺ threshold for various levels of SERCA. To be able to determine if this was the case, we first needed to identify the necessary components in the Maleckar et al. model to give rise to an SD. Hence we performed the dependency test proposed by Fink et al. where we tested the modulation of [Ca²⁺]_i and [Ca²⁺]_up on SERCA, and [Ca²⁺]_i and [Ca²⁺]_d on RyR to see which ones were essential for the genesis of SDs. Through this test, we found that only the modulation of RyR by [Ca²⁺]_i was crucial for the initiation of SDs (figure 6b).

Figure 6.

The dependency test of SERCA and RyR in the genesis of SDs. (a) Time-course of the membrane potential, [Ca²⁺]_up, [Ca²⁺]_rel, and I_rel for modulated SERCA and RyR, removed modulation of [Ca²⁺]_i on SERCA, removed modulation of [Ca²⁺]_up on SERCA, and removed modulation of [Ca²⁺]_d on RyR. (b) Time-course of the membrane potential, [Ca²⁺]_up, [Ca²⁺]_rel, and I_rel for removed modulation of [Ca²⁺]_i on RyR. The black triangle indicates the time when a stimulus was applied. SERCA, sarcoplasmic reticulum Ca²⁺-ATPase; SD, spontaneous depolarization; RyR, ryanodine receptors. (Online version in colour.)

(f). The impact of SERCA on the intracellular calcium threshold for spontaneous depolarizations

Having determined that it was the modulation of [Ca²⁺]_i on RyR that was essential in the genesis of SDs in the full Maleckar et al. model, we were then interested in ascertaining whether there existed a threshold value for [Ca²⁺]_i in triggering SDs, and whether this threshold followed a biphasic trend as SERCA decreased from 200% to 20% of its control value. Hence, we performed the intracellular Ca²⁺ injection protocol 1 for various levels of SERCA. As shown in figure 7, the [Ca²⁺]_i threshold to trigger SDs followed a biphasic trend as SERCA decreased from 200% to 20% of its control value, demonstrating that this threshold was different for varying levels of SERCA. This also confirmed that when SERCA was sufficiently inhibited or elevated, to trigger SDS, an increase in RyR, I_CaL, or decrease in I_NaCa was required to meet the elevated [Ca²⁺]_i threshold to trigger SDs.

Figure 7.

The [Ca²⁺]_i SD threshold for various levels of SERCA. The cross represents the point where the trend changed. SERCA, sarcoplasmic reticulum Ca²⁺-ATPase. (Online version in colour.)

(g). The relationship between calcium concentrations in the uptake and release compartment and the intracellular calcium threshold

Having confirmed that there existed a threshold value for [Ca²⁺]_i in triggering SDs and that the threshold followed a biphasic trend as SERCA decreased, we were then interested in knowing the Ca²⁺ handling mechanisms which gave rise to the biphasic relationship between the [Ca²⁺]_i SD threshold and SERCA. We hypothesized that this mechanism not only involved SERCA but also [Ca²⁺]_up and [Ca²⁺]_rel, because as we were conducting the experiment under protocol 1, we noted that they kept decreasing as SERCA decreased (electronic supplementary material, figure S4A and B). Hence, there was the possibility that different levels in [Ca²⁺]_up and [Ca²⁺]_rel could therefore also have an impact on the [Ca²⁺]_i SD threshold.

To investigate this, we conducted two more series of simulations under intracellular calcium injection protocol 2. These series of simulations respectively assessed how the [Ca²⁺]_i SD threshold was impacted when [Ca²⁺]_up and [Ca²⁺]_rel was reduced or increased beyond their intrinsic value for a particular level of SERCA. As shown in figure 8, when [Ca²⁺]_rel and [Ca²⁺]_up were elevated to their values for 200% SERCA, the [Ca²⁺]_i SD threshold decreased relative to the threshold obtained in the default case for all values of SERCA. By contrast, when [Ca²⁺]_rel and [Ca²⁺]_up were reduced to their values for 20% SERCA, the [Ca²⁺]_i SD threshold instead increased relative to the threshold obtained in the default case for all values of SERCA.

Figure 8.

The [Ca²⁺]_i SD threshold for various levels of SERCA for three different cases. The default case is when no change was made to [Ca²⁺]_up and [Ca²⁺]_rel at 100 s into the simulation (blue). The other two cases are when at 100 s into the simulation, prior to intracellular Ca²⁺ injection, [Ca²⁺]_up and [Ca²⁺]_rel were elevated and reduced to their values at 200% SERCA (bottom) and 20% SERCA (top) at 100 s respectively in the default case (see electronic supplementary material, figure S4A and B). The coloured crosses indicate where a change in the trend occurs in the corresponding case. SERCA, sarcoplasmic reticulum Ca²⁺-ATPase; SD, spontaneous depolarization. (Online version in colour.)

4. Discussion

It is generally accepted that SDs/DADs are caused by spontaneous release of Ca²⁺ from the SR due to an upregulation in SERCA uptake and dysfunctional RyR in atrial cells [8,9]. However, recent mice knockout experiments of TBX5 have demonstrated that SDs could also occur through a reduced SERCA uptake, and increased I_NaCa and RyR [27,29,54]. Similar findings have also been reported in experimental and clinical studies of diabetes [6,31] and heart failure [30,32,33]. Surprisingly, in all these cases, the rescue of impaired SERCA uptake through the knockout of PLN or SERCA2a gene therapy was able to suppress these SDs [29,36–39]. To our knowledge, this is the first computational study that systematically investigates the sensitivity of SERCA in relation to other Ca²⁺ handling channels in the genesis of SDs/DADs and the mechanisms which are involved.

Though the results shown in this study are specific for the occurrence of SDs as they are easier to identify in simulations than DADs, the conclusions on the dynamic relationship between the key Ca²⁺ proteins, i.e. SERCA, I_CaL, I_NaCa, and RyR can be drawn similarly for DADs. Generally, in our simulations, DADs occur only in the form of a slight depolarization if changes to channels are close to the changes required to trigger an SD. For example, if we increased I_CaL to 112% of its control value instead (figure 3), then after the last stimulated AP, a self-depolarization up to −72 mV was observed. By contrast, at 113% of the control value of I_CaL and greater, SDs were observed.

Our four bifurcation analyses together illustrate that variations in SERCA can have a pronounced impact on Ca²⁺ handling. When SERCA is significantly inhibited (i.e. less than approx. 80% SERCA) or elevated (i.e. greater than approx. 100%), the minimum increase in RyR or I_CaL, or decrease in I_NaCa that is required to trigger SDs is greater than when SERCA is only slightly inhibited (i.e. greater than approx. 80% SERCA and less than approx. 100% SERCA). This highlights that the myocyte's susceptibility to abnormal depolarizations is greatest when SERCA is slightly inhibited, because changes in RyR, I_CaL, and I_NaCa needed to trigger SDs/DADs in this parameter range for SERCA are at a minimum. These bifurcation diagrams also illustrate that increased RyR or I_CaL triggers SDs/DADs, but increased I_NaCa suppresses them instead. This is interesting, because in the setting of reduced TBX5, diabetes, and heart failure, SDs/DADs was observed alongside an elevation in I_NaCa. Therefore, for SDs/DADs to occur in conditions with an increased I_NaCa, there must also be an increase in I_CaL to overcome the suppression effects of I_NaCa.

In the dependency test by Fink et al., we found that in the full Maleckar et al. model, only the modulation of the RyR by [Ca²⁺]_i was crucial. This differs from the 37 previously published mathematical models of atrial and ventricular myocytes that they tested, where they found that in most models, it was the modulation of RyR through Ca²⁺ in the dyadic space ([Ca²⁺]_d) that triggered SDs [40]. This could be because in most models they tested, the RyR is connected to the dyadic space rather than the intracellular space as is the case in the Maleckar et al. model.

During our experiment with intracellular Ca²⁺ injection protocol 1, we observed that diastolic [Ca²⁺]_up and [Ca²⁺]_rel at the time point where intracellular Ca²⁺ is injected to trigger spontaneous Ca²⁺ release decreased as SERCA decreased. This is consistent with experimental findings by Fernandez-Tenorio and Niggli, where they applied a range of pharmacological SERCA stimulators and a PLB antibody to mouse cardiomyocytes, and found that mice with non-stimulated SERCA had a lower diastolic [Ca²⁺]_SR prior to spontaneous Ca²⁺ release. Interestingly, they considered [Ca²⁺]_SR as being the threshold for spontaneous Ca²⁺ release, and this threshold increased as SERCA was elevated [61].

With the second intracellular Ca²⁺ injection protocol, we probed whether [Ca²⁺]_up and [Ca²⁺]_rel were indeed also factors in determining the [Ca²⁺]_i SD threshold. Results from this experiment revealed that in general, if [Ca²⁺]_up and [Ca²⁺]_rel are increased above their intrinsic value for a particular value of SERCA, then the [Ca²⁺]_i SD threshold will be lowered. However, if [Ca²⁺]_up and [Ca²⁺]_rel are decreased below their intrinsic value instead, then the [Ca²⁺]_i SD threshold will be elevated instead. From the point of view of this study, it is hence better to view [Ca²⁺]_up and [Ca²⁺]_rel alongside SERCA to be regulators of the [Ca²⁺]_i SD threshold rather than act as a threshold itself for spontaneous Ca²⁺ release and hence SDs/DADs as proposed by Fernandez-Tenorio and Niggli.

On closer inspection, this experiment also provides some insights into why the [Ca²⁺]_i SD threshold increases when SERCA is significantly inhibited (i.e. less than approx. 80% SERCA) or elevated (i.e. greater than approx. 100% SERCA). To illustrate the former situation, we consider the default case starting at 80% of its control value of SERCA in figure 8. If we reduce SERCA from 80% to 60% of its control value, the [Ca²⁺]_i SD threshold will increase slightly (80% SERCA to 60% SERCA on the blue line). If at 60% SERCA, [Ca²⁺]_up and [Ca²⁺]_rel are artificially reduced below its intrinsic value, the increase in the [Ca²⁺]_i SD threshold from 80% SERCA to 60% will be much greater (80% SERCA on the blue line to 60% SERCA on the red line). However, if at 60% SERCA, [Ca²⁺]_up and [Ca²⁺]_rel are artificially elevated above its intrinsic value, the [Ca²⁺]_i SD threshold from 80% SERCA to 60% will instead decrease (80% SERCA on the blue line to 60% SERCA on the green line). This seems to illustrate that in the default case, the [Ca²⁺]_i SD threshold starts to increase at less than approximately 80% SERCA because the maximum SERCA uptake is no longer high enough to maintain [Ca²⁺]_up and [Ca²⁺]_rel at sufficient levels. As a result, greater [Ca²⁺]_i is required to compensate this for spontaneous Ca²⁺ release and subsequently SDs to occur. In a similar fashion, we can illustrate the latter situation. However, the key difference here is that raising [Ca²⁺]_up and [Ca²⁺]_rel has a negligible impact on the [Ca²⁺]_i SD threshold (160% SERCA on the blue line to 160% SERCA on the green line).

This latter situation likely occurs because of the properties of the bidirectional SERCA pump. If maximum SERCA uptake is elevated, then this would result in an increase of [Ca²⁺]_up and [Ca²⁺]_rel. Because [Ca²⁺]_up and [Ca²⁺]_rel are now elevated, the reverse mode of the bidirectional pump activates, reducing the net Ca²⁺ uptake into the SR by SERCA. Further increasing [Ca²⁺]_up and [Ca²⁺]_rel would therefore have a negligible effect on the net Ca²⁺ uptake into the SR by SERCA. In this situation, an elevation in [Ca²⁺]_i is required to reduce the reverse direction of SERCA and restore SERCA uptake function such that spontaneous Ca²⁺ release and hence SDs/DADs can occur. In electronic supplementary material, figure S7, we have demonstrated that by replacing the bidirectional SERCA pump in the Maleckar et al. model with the unidirectional SERCA pump in the CRN model and a generic leak in the uptake compartment (electronic supplemental materials, figure S6), the minimum multiplier for RyR and I_CaL to trigger at least one SD is significantly less than in the bidirectional case, and significantly greater for I_NaCa for elevated SERCA (greater than approx. 100% SERCA), illustrating that the [Ca²⁺]_i SD threshold is much lower with a unidirectional SERCA pump than a bidirectional one in this parameter range.

Based on these findings together, we therefore hypothesize that there are three main phases in the Ca²⁺ handling involving SERCA, [Ca²⁺]_up, and [Ca²⁺]_rel which attribute to the [Ca²⁺]_i SD threshold in the full Maleckar et al. model at varying levels of maximum SERCA uptake. If SERCA is significantly inhibited (i.e. less than approx. 80% SERCA), the bidirectional SERCA pump becomes too inhibited to uptake enough calcium into the SR necessary to aid in the initiation of spontaneous Ca²⁺ release and hence SDs, and so an elevation in [Ca²⁺]_i is required to compensate (figure 9a).

Figure 9.

Schematic illustrating the general differences in Ca²⁺ handling for maximum SERCA uptake ranges between (a) approximately 20–80% (b) approximately 80–100% and (c) greater than approximately 100% of its control value in the Maleckar et al. model. Pie graph inserts show the relative magnitude of the forward and reverse mode of the bidirectional SERCA pump in each range. Bar graph inserts qualitatively highlight the relative magnitude of the net SERCA uptake and [Ca²⁺]_SR in each range, and how this impacts the magnitude of the [Ca²⁺]_i threshold for SDs. SERCA, sarcoplasmic reticulum Ca²⁺-ATPase; SD, spontaneous depolarization. (Online version in colour.)

On the other hand, when SERCA is elevated (e.g. PLN knockout or SERCA2a gene therapy) (i.e. greater than approx. 100% SERCA), the properties of the bidirectional SERCA pump becomes pronounced. The resultant [Ca²⁺]_up and [Ca²⁺]_rel become relatively high, which trigger the reverse direction of the pump. This in turn reduces the net uptake of Ca²⁺ into the SR and disrupts the ability of SERCA to contribute to spontaneous Ca²⁺ release and hence SDs. To promote the forward direction of the bidirectional SERCA pump in this situation, there needs to be an increase in [Ca²⁺]_i such that [Ca²⁺]_up and [Ca²⁺]_rel do not appear relatively high (figure 9c).

Figure 9b illustrates when SERCA is in the ideal range to trigger SDs (i.e. greater than approx. 80% SERCA and less than approx. 100% SERCA). In this range, SERCA has sufficient uptake into the SR such that [Ca²⁺]_up and [Ca²⁺]_rel reach appropriate levels without triggering the reverse mode of the SERCA pump and without a compensatory increase in [Ca²⁺]_i. As a result, the [Ca²⁺]_i SD threshold is at a minimal in this range, and only small increases in RyR and I_CaL, or decreases in I_NaCa are required.

5. Limitations

There are several limitations to this modelling study. The first issue concerns the insufficient details of spatial Ca²⁺ handling in the Maleckar et al. model. The internal structure of Ca²⁺ handling in cardiac myocytes is vastly more complicated than is represented in common pool models. The SR is a complex structural network within the myocyte and consists of clusters of RyRs which can open and close collectively. To be able to expand the scope of this study to include the mechanisms by which a modulation of SERCA can increase the incidence and spatial dynamics of Ca²⁺ waves, and how normalization or elevation of SERCA can reduce this, the descriptions of Ca²⁺ handling in the Maleckar et al. model need to be further developed to include these structural and spatial aspects of Ca²⁺ handling, such as the Voigt et al. [7] and Thul et al. models [62,63]. Additionally, the peak systolic [Ca²⁺]_i level and decay of [Ca²⁺]_i in our simulated results of the Maleckar et al. model are different from those in the experimental studies by Neef et al. and Voigt et al. [7,64,65]. This is because the Maleckar et al. model was based on the Nygren et al. model, which was validated through a different experimental study on human atrial myocytes. Further investigation is required to ascertain the modifications which are required in the formulations of the ionic and Ca²⁺ handling channels such that the model can accurately capture the experimental results by Neef et al. and Voigt et al. as well as the phenomena in SERCA as detailed in this study.

Limited availability of data for SERCA and other Ca²⁺ handling channels in the setting of reduced TBX5, diabetes, and heart failure is also an obstacle of this study. Across these settings, studies only provide the voltage-clamp recordings of I_CaL and caffeine-based protocol results of SERCA and I_NaCa for a select few myocytes [6,29,54,55,66], and so it would be difficult to verify if any parameters that we assigned would fall under physiological ranges. In our study, we instead conducted a modelling analysis of how changes in the conductivity of I_CaL, I_NaCa, and RyR, and the uptake of SERCA impacts Ca²⁺ handling in the atrial myocyte across a wide parameter space to qualitatively examine the mechanisms by which a reduction in SERCA can trigger SDs, and how normalization or elevation of SERCA through knockout of PLN or SERCA2a gene therapy can suppress SDs.

While this approach provides insights as to how Ca²⁺ handling is altered as we vary maximum SERCA uptake and the relationships between SERCA and other Ca²⁺ handling channels in triggering SDs, it can only make predictions into the physiological ranges where these phenomena can occur. This may be where it would be more advantageous to use a population-based approach combined with a complementary animal study [67] to investigate if deviations from the limited data that we have on Ca²⁺ handling channels under reduced TBX5, diabetes, and heart failure can still give rise to SDs, and for a better estimate of the parameter ranges where SDs can be triggered or suppressed. Experimental studies in reduced TBX5, diabetes, and heart failure used varying experimental and pacing protocols [6,27,29,54,55,66]. In our simulations with the Maleckar et al. model, a pacing frequency of 1 Hz was used instead as we are interested in how reduced SERCA impacts Ca²⁺ handling to trigger SDs, and how normalization or elevation of SERCA can suppress SDs in human atrial myocytes. While we currently lack the experimental data to verify our results, it would, however, be of benefit to this study if simulations were conducted at multiple different pacing frequencies. This could be because changing the pacing frequency would likely also affect [Ca²⁺]_up and [Ca²⁺]_rel, causing an increase in the [Ca²⁺]_i SD threshold.

In terms of the suppression of SDs due to the knockout of PLN, this was regarded as an increase in maximum SERCA uptake back to normal or elevated levels. This is analogous to changing the maximum conductance through agents such as a pharmacological inhibitor or through SERCA2a gene therapy. However, disruption of PLN-mediated SERCA inhibition can also cause an increased affinity to [Ca²⁺]_i and therefore likely shift the balance between the forward and reverse mode of SERCA. Hence for our hypothesis to hold, it is assumed that changes in SERCA activity are exclusive to changes in its maximum SERCA uptake. Disruption of PLN-mediated SERCA inhibition could be more rigorously modelled as changes in the phosphorylation rates of SERCA and could potentially shed more insight into the Ca²⁺ handling mechanisms attributing to the suppression of SDs/DADs if these parameters were also investigated through a bifurcation analysis and intracellular Ca²⁺ injection protocol to gauge how changes in phosphorylation rates impacted the [Ca²⁺]_i threshold. Similarly, the impact of Ca²⁺ buffering on Ca²⁺ handling warrants investigation through these analytical methods.

The second caveat in our hypothesis is that for spontaneous Ca²⁺ release to occur, there must also be sufficient diffusion of Ca²⁺ in the SR. Diffusion of Ca²⁺ through the SR has been shown to be a critical factor of spontaneous Ca²⁺ release in atrial myocytes [68]. Rapid Ca²⁺ diffusion within the SR is necessary to stabilize and balance [Ca²⁺]_SR within the myocyte. In the context of this study, reducing Ca²⁺ diffusion could potentially cause an overall reduction in [Ca²⁺]_up and [Ca²⁺]_rel, which subsequently increases the [Ca²⁺]_i SD threshold.

The other intermediate pathway that was not considered in the Maleckar et al. model or in our mathematical analyses was the Ca²⁺/calmodulin-dependent protein kinase II (CaMKII). A reduction in SERCA uptake would increase diastolic [Ca²⁺]_i, and thus lead to greater phosphorylation of PLN on SERCA or RyR. This in turn could mitigate the reduced SERCA uptake and increase the open probability of RyR and could potentially also explain the onset of SDs from a reduced SERCA, and the suppression of SDs from PLN knockout or SERCA2a gene therapy [69]. More recently, Alsina et al. discovered an intermediate pathway involving a novel protein phosphatase 1 (PP1)- regulatory subunit PPP1R3A (PP1 regulatory subunit type 3A) within the RyR2 channel complex. Through the knockout of PPP1R3A, they demonstrated that reduced expression of this novel PP1 regulatory subunit reduces the binding of PP1 to both RyR2 and PLN, which subsequently increases SR Ca²⁺ release and hence increases the susceptibility of the myocyte to SDs [70]. Changes in Ca²⁺ handling channels can therefore have an impact on the dynamics of this intermediate pathway and thus the [Ca²⁺]_i SD threshold, and would need to be incorporated along with the intermediate pathway involving CaMKII in a future study.

6. Conclusion

In summary, a reduction in SERCA alone cannot cause SDs/DADs, but instead affects the [Ca²⁺]_i threshold required to trigger SDs/DADs. Compensatory changes in other Ca²⁺ handling channels, such as increasing RyR/ I_CaL or decreasing I_NaCa alongside SERCA are therefore required to raise [Ca²⁺]_i to its threshold level to trigger spontaneous Ca²⁺ release and hence SDs/DADs. Furthermore, this study also demonstrates that a bidirectional SERCA pump is necessary to capture the effects of SD/DAD initiation particularly due to a reduction in SERCA and [Ca²⁺]_SR load. For the first time, our modelling study reconciles different mechanisms of abnormal depolarizations in the setting of ‘lone’ AF, TBX5, diabetes, and heart failure, and it may lead to more targeted treatment for these patients, such as the suppression of SDs/DADs from the normalization or elevation of SERCA from PLN knockout or SERCA2a gene therapy.

Data accessibility

All supporting data and computer simulation source code can be accessed at the CELLML depository (https://models.physiomeproject.org/exposure/bbd802c6a6d6e69b746244f83b4fb89b/maleckar_greenstein_trayanova_giles_2009.cellml) and the GitHub repository (https://github.com/SirAndii/AndyHub/blob/master/MaleckarCaSubsystemControl.m).

Authors' contributions

A.L. and J.Z. conceived and designed the study. A.L. conducted the experiments and drafted the manuscript. A.L., J.B., P.A.G., V.V.F. and J.Z. interpreted the data, and reviewed, revised and approved the final version of this manuscript.

Competing interests

The authors have no competing interests.

Funding

This work was supported by the Health Research Council of New Zealand (16/385), the National Institutes of Health grant nos (HL115580 and HL135109), the National Natural Science Foundation of China (grant no. 61901192) and the National Key R&D Program of China (grant nos 2019YFC0120100 and 2019YFC0121900).