Our rabbit ventricular myocyte model consists of a 3-dimensional coupled network of CRUs and mitochondria. This network contains 21504 (64 2812) CRUs and 5376 (64 14 6) mitochondria. The membrane potential (V) of the cell is described by

where C_m=1 mF/cm² is the cell membrane capacitance, and I_sti is the stimulus pulse with the current density being −80 mA/cm² and the duration being 0.5 ms.

The oxidized CaMKII activation and the redox effect of ROS both increase the open probability of RyRs (Wehrens et al., 2004; Ai et al., 2005; Guo et al., 2006; Zima and Blatter, 2006; Wagner et al., 2013). To incorporate these effects, the close-to-open rate (k₁₂) of RyRs was modeled as follows:

where △k_CaMKII and △k_ROS are the CaMKII-dependent (Eq. 3) and ROS-dependent components (Eq. 4), respectively. k_base and k_u are the rate constants. [Ca^{2 +} ]_p is the Ca²⁺ concentration in the dyadic space of a CRU. [CaMKII]_act and [ROS]_cyt are the CaMKII activation level and the cytosolic ROS concentration in each CRU, respectively. The increase in CaMKII activation and ROS level increase k₁₂, which in turn increases the open probability of RyRs.

The formulation of SERCA is

where f_up,ATP, and f_up,ROS are ATP and ROS-dependent functions (Song et al., 2019):

v_up is the maximum SERCA strength and K_ithe half-maximum value. PLB([CaMKII]_act) is a CaMKII dependent function. [CaMKII]_act is CaMKII activation level in the cytosolic space of a CRU.

We used a 3-state (two close states C₀ and C₁, and an open state O) Markov model to simulate the stochastic opening and closing of the mPTP. The transition rate from the C₀ state to the C₁ state, k_c0c1, is set as:

where h_mPTP is the Hill coefficient, [Ca^{2 +}]_m is the mitochondrial free Ca²⁺ in the corresponding mitochondrion, and [Ca^{2 +} ]₀ is the half-maximum value. We assume that other transition rates are constants. To simulate different levels of mPTP open probability, we multiplied a factor, α_mPTP, to the transition rate from C₁ to O, kc⁢100, i.e.,

We investigated the impact of mitochondrial depolarizations on the occurrence of Ca²⁺ waves and DADs via mPTP opening. We performed simulations over a wide range of α_mPTP values at the PCL of 300 ms (Figure 1A). α_mPTP is a factor multiplied to transition rate of mPTP opening, and increasing its value results in higher mPTP opening. For each simulation, the cell was paced for 140 beats (42 s), following 3 s without pacing in order to observe Ca²⁺ waves and DADs. As shown in Figure 1A, the amplitude of DAD increases with α_mPTP, suggesting that mitochondrial depolarization due to openings of mPTP promotes spontaneous Ca²⁺ release and DADs. Also, when α_mPTP is greater than ∼50, the proarrhythmic effect appears to saturate. Under the control condition (α_mPTP=1), there is no occurrence of DADs (Figures 1A,B). The cytosolic ROS is ∼2 μM, CaMKII activation is ∼0.2%, the cytosolic ATP is ∼5 mM, and most of the mitochondria remain repolarized (-Δ⁢ψ¯=∼180 mV, and the mPTP open probability ∼0.8%, mitochondrial Ca²⁺ amplitude is ∼0.8 μM) (Figure 1B). However, with the higher open probability of mPTP (∼42%, for α_mPTP=60), we observed DADs. The corresponding line-scan image clearly shows enhanced spontaneous Ca²⁺ release as compared to a few scattered Ca²⁺ sparks under the control condition (Figure 1C). In this case, we should note that the mitochondrial Ca²⁺ amplitude increased to ∼1.2 M, and the cytosolic ROS drastically increased to ∼86 μM. Still, the CaMKII activation and the cytosolic ATP level insignificantly changed (∼0.6% and ∼4.8 mM, respectively).

The concentration of the cytosolic ROS depends on the level of mitochondrial depolarization, and thus, increasing α_mPTP increases the open probability of mPTP, which, in turn, elevates the level of cytosolic ROS. To further identify the role of cytosolic ROS in the genesis of DADs, we performed simulations for a clamped ROS level at the PCL 300 ms. In the free-running ROS case as shown in Figure 1, the ROS dynamics in the model remains intact, while in the clamped ROS condition, the cytosolic ROS is clamped to a constant regardless the level of mitochondrial depolarization is. Here we clamped ROS to be 1.0 μM, which is close to the level under the normal control condition. We then measured the amplitude of DAD with different α_mPTP values for the clamped ROS (Figure 2, red) condition. We observed that when the ROS was clamped at the control level (1.0 μM), no DADs occurred, suggesting that the cytosolic ROS plays a critical role in inducing DADs during mitochondrial depolarization.

Furthermore, the cytosolic ROS is known to alter the characteristics of both SERCA and RyRs (Zima and Blatter, 2006; Wagner et al., 2013). Therefore, we investigated the redox effect of ROS on the RyRs and SERCA pump separately to dissect out its individual role. We observed that DADs disappeared when we removed the redox effect of ROS on the RyRs (Figure 2, green). However, removing the redox effect on SERCA did not significantly impact the amplitude of DADs (Figure 2, blue), suggesting that the redox effect on SERCA may not play a critical role in the genesis of DADs. In fact, the direct redox effect of ROS inhibits the SERCA pump activity. Thus, removing the redox effect on SERCA increased the SR Ca²⁺ load instead, causing higher amplitudes of DADs. For instance, at α_mPTP=20, the amplitude of DAD is 7.17 mV when the redox effect of ROS is only exerted on RyRs (Figure 2, blue), but 6.84 mV under control as shown in Figure 1A.

Several studies have shown that under certain pathological conditions, MCU activity is enhanced (Santulli et al., 2015; Xie et al., 2018), which may elevate the mitochondrial free Ca²⁺. Our previous study (Song et al., 2019) showed that the increase of MCU up to 20-fold does not alter cytosolic Ca²⁺ markedly at the steady-state. However, we hypothesize that the higher mitochondrial Ca²⁺ due to MCU overexpression could increase the mPTP open probability and cause higher ROS production in the cytosol (Korge et al., 2011). To test this hypothesis, we performed simulations to examine the effect of MCU overexpression on the genesis of DADs.

We multiplied a factor, denoted as α_MCU, to the maximal MCU conductance. α_MCU=1 represents the control case and higher α_MCU values are used to represent the different levels of MCU activity. We plotted in Figure 3A the amplitude of DADs for different α_MCU and α_mPTP. We observed that at α_mPTP = 1, increasing the MCU activity did not result in DADs even for α_MCU=50. When α_mPTP becomes greater, the effect of MCU on promoting DADs appears to be more significant. Time traces of membrane voltage, the whole-cell averaged cytosolic Ca²⁺ and the mitochondrial free Ca²⁺ for the three marked locations in the phase map (Figure 3A) are shown in Figure 3B. These results indicate that increasing MCU activity could promote spontaneous Ca²⁺ release and DADs. The mechanism revealed in the model is that increasing MCU activity elevates the mitochondrial free Ca²⁺, which increases the open probability of mPTP, resulting in the elevation of the cytosolic ROS, which in turn promotes the spontaneous Ca²⁺ release and DADs.

As seen in Figure 1B, the CaMKII activation and ATP appeared to change slowly during the simulations due to the slow kinetics in the governing equations of their dynamics. It is computationally cumbersome to run long simulations (up to several thousand beats) for these variables to reach their steady states. Therefore, to evaluate the individual role of CaMKII activation and ATP in the genesis of DADs, we clamped them to different constant values, respectively.

Figure 4A shows the relationship between the amplitude of DADs and the CaMKII activation level. Our results clearly show that increasing CaMKII activation promotes DADs. However, due to its slow kinetics, CaMKII activation insignificantly changes during the acute phase of mitochondrial depolarization (Figure 1B), suggesting that CaMKII activation may not play a primary role in the genesis of DADs during the short period immediately after mitochondrial depolarization. Still, it may promote DADs chronically due to its regulation on SERCA, LCC, and RyRs (Wang et al., 2020).

Since the SERCA pump requires ATP for its normal function, a lower level of ATP directly impairs the SERCA pump activity (Eq. 5). However, ATP depletion is a slow process, which is evident from Figure 1B. Hence, we clamped ATP at different levels for α_mPTP = 30 from the beginning of the simulations. Figure 4B plots the relationship between the amplitude of DADs and the cytosolic ATP concentration, and it shows that ATP depletion suppresses DADs during mitochondrial depolarization (α_mPTP = 30). Our results demonstrate that the cytosolic ATP level has a significant impact on the genesis of DADs. However, the depletion of the cytosolic ATP concentration during mitochondrial depolarization is a relatively slow process. Therefore, ATP depletion should not play a central role in the genesis of DADs during the acute phase of mitochondrial depolarization. Thus, similar to CaMKII activation, ATP depletion may only suppress DADs in a much longer time scale.

Experimental findings reported that the mitochondria depolarization through the application of FCCP promotes Ca²⁺ waves (Zhao et al., 2013). Furthermore, the effects of FCCP were counteracted by the application of mPTP blocker cyclosporine A (Zhao et al., 2013). Also, our previous experimental observations using cyclophilin D knockout mouse model showed attenuation of Ca²⁺ waves (Gordan et al., 2016). Elevation of cytosolic ROS during mPTP opening could be a significant factor, and experimental evidence showed that oxidative stress during mitochondrial depolarization slightly alters SR Ca²⁺ leaks (Ca²⁺ spark) amplitude but drastically increases its frequency (Yan et al., 2008; Zhou et al., 2011; Williams et al., 2013). Furthermore, ROS can oxidize CaMKII and enhance its activation. It has been shown that ROS and CaMKII activation act on the major Ca²⁺ handling proteins, such as RyRs and SERCA (Hund and Rudy, 2004; Wehrens et al., 2004; Ai et al., 2005; Guo et al., 2006; Zima and Blatter, 2006; Wagner et al., 2013). The direct redox effect of ROS increases the leakiness of RyRs and decreases the strength of SERCA (Zima and Blatter, 2006; Wagner et al., 2013), whereas CaMKII activation increases both the leakiness of RyRs and the strength of SERCA (Hund and Rudy, 2004; Ai et al., 2005; Mattiazzi and Kranias, 2014). Thus, the observed Ca²⁺ waves in experiments during mitochondrial depolarization are the consequences of the combined effects of the above factors. However, what is the primary player remains unclear. Here, by using our previously established physiologically detailed computer model, we show that our simulation results agree with the experimental observations that mitochondrial depolarization could induce spontaneous Ca²⁺ release and DADs. Furthermore, we found that it is the redox effect on RyRs that causes the DADs under the acute effect of mitochondrial depolarization, and the redox effect of ROS on reducing SERCA strength may not be sufficient to suppress DADs (Figure 2).

In addition, we indeed observed an increase in CaMKII activation due to the mPTP opening (Figure 1B), but the dynamics of CaMKII activation appeared much slower than that of the ROS. The CaMKII activation was increased from ∼0.2% under control to ∼0.6% during the mitochondrial depolarization for a 45-sec long simulation (Figure 1B). A further simulation showed that CaMKII activation could reach up to ∼40% for a much more extended duration (1200 s). These results suggest that CaMKII activation may be too slow to play an essential role in inducing DADs under the acute effect of mitochondrial depolarization. However, simulations with different clamped CaMKII activation levels reveal that CaMKII activation may play a vital role in causing spontaneous Ca²⁺ release and DADs chronically, since a higher CaMKII activation level caused a greater DAD amplitude (Figure 4A).

Similarly, we have observed a slight change in cytosolic ATP under the acute effect of mitochondrial depolarizations in our simulations (Figure 1B). Therefore, within a relatively short duration after mitochondrial depolarization, ATP may not be involved in the genesis of spontaneous Ca²⁺ release and DADs. Clamped ATP simulations showed that a lower cytosolic ATP level is linked to a smaller amplitude of DADs (Figure 4B). This is because the reduction of ATP impaired SERCA activity and suppress the DADs, which agree well with experimental evidence that ATP synthase inhibitor, oligomycin, does not promote DADs (Zhao et al., 2013). Although ATP reduction seems to suppress Ca²⁺ waves and DADs, our simulations and other’s experimental work suggest that ATP reduction could promote Ca²⁺ alternans, which is still arrhythmogenic (Hüser et al., 2000; Kockskämper et al., 2005; Zima and Blatter, 2006; Pandey et al., 2021).

Mitochondrial Ca²⁺ uptake has been reported to rise in heart failure (Santulli et al., 2015; Xie et al., 2018) and can promote EADs (Xie et al., 2018) and Ca²⁺ alternans (Pandey et al., 2021). Our previous work demonstrated MCU upregulation could promote EADs in heart failure conditions without mPTP openings (Xie et al., 2018). And MCU upregulation promotes Ca²⁺ alternans through the Ca²⁺ dependent opening of mPTPs (Pandey et al., 2021). The previous experiment by Zhao et al. (2013) reported that the mitochondrial Ca²⁺ efflux in the proximity of the junctional SR played an essential role in the regulation of Ca²⁺ waves. Furthermore, our previous study has shown that MCU overexpression can lead to Ca²⁺ overload in mitochondria (Song et al., 2019). Also, there is evidence that mitochondrial Ca²⁺ overload can cause the Ca²⁺-dependent opening of mPTP (Kwong and Molkentin, 2015), resulting in mitochondrial depolarization (Santulli et al., 2015). Here, our simulation study shows that increasing MCU activity promotes spontaneous Ca²⁺ release and DADs (Figure 3). The underlying mechanism revealed in our simulations is that increasing MCU activity enhanced mitochondrial depolarization through the Ca²⁺-dependent openings of mPTP, which resulted in spontaneous Ca²⁺ release primarily due to the direct redox effect of ROS on RyRs.

Mitochondrial dysfunction has been associated with increased arrhythmic risk (Santulli et al., 2015; Shimizu et al., 2015; Xie et al., 2015, 2018; Gordan et al., 2016), which could account for mortality in many cardiac diseases such as cardiomyopathy, heart failure, and ischemia/reperfusion injury (IRI). We have demonstrated that the direct redox effect of ROS on RyRs plays a critical role in promoting Ca²⁺ waves and DADs under the acute effect of mitochondrial depolarization. Furthermore, the upregulation of MCU can promote DADs through Ca²⁺-dependent opening of mPTPs. These findings suggest that pharmacological interventions targeted at avoiding ROS buildup and MCU upregulation may provide novel therapeutics to prevent or treat cardiac arrhythmias.

This detailed model coupled AP, CRUs, and mitochondria to capture excitation-contraction-metabolism coupling in ventricular myocytes. However, it has some limitations. For instance, heterogeneities in T-tubule networks and distributions of ion channels and Ca²⁺ handling proteins are a few examples (Soeller and Cannell, 1999; Baddeley et al., 2009) that should be considered in the future analysis. These heterogeneities in T-tubule networks may alter the genesis of DADs (Song et al., 2018).

We note that in Figure 3A, for α_mPTP=1, there was virtually no DAD occurring even with α_MCU=50, suggesting that the proposed mechanism of increasing MCU activity inducing DADs in this study requires a certain basal level of mPTP opening. In this model, we consider the mPTP gating kinetics only mitochondrial Ca²⁺ dependent. However, the ROS-induced ROS release mechanism also impacts the mPTP open probability (Zorov et al., 2000, 2006), which is essential for modeling mitochondrial depolarization waves (Yang et al., 2010; Zhou et al., 2010; Nivala et al., 2011). Thus, this ROS-induced ROS release mechanism may provide another critical positive feedback loop between mitochondrial and cytosolic Ca²⁺ instability. In the future, the ROS-induced ROS release will be added to our model to study the role of mitochondrial depolarization waves in the genesis of arrhythmias in diseased conditions, such as heart failure.