Exploring Impaired SERCA Pump-Caused Alternation Occurrence in Ischemia

Abstract

Impaired sarcoplasmic reticulum (SR) calcium transport ATPase (SERCA) gives rise to Ca²⁺ alternans and changes of the Ca2+release amount. These changes in Ca²⁺ release amount can reveal the mechanism underlying how the interaction between Ca²⁺ release and Ca²⁺ uptake induces Ca²⁺ alternans. This study of alternans by calculating the values of Ca²⁺ release properties with impaired SERCA has not been explored before. Here, we induced Ca²⁺ alternans by using an impaired SERCA pump under ischemic conditions. The results showed that the recruitment and refractoriness of the Ca²⁺ release increased as Ca²⁺ alternans occurred. This indicates triggering Ca waves. As the propagation of Ca waves is linked to the occurrence of Ca²⁺ alternans, the “threshold” for Ca waves reflects the key factor in Ca²⁺ alternans development, and it is still controversial nowadays. We proposed the ratio between the diastolic network SR (NSR) Ca content (Ca_nsr) and the cytoplasmic Ca content (Ca_i) (Ca_nsr/Ca_i) as the “threshold” of Ca waves and Ca²⁺ alternans. Diastolic Ca_nsr, Ca_i, and their ratio were recorded at the onset of Ca²⁺ alternans. Compared with certain Ca_nsr and Ca_i, the “threshold” of the ratio can better explain the comprehensive effects of the Ca²⁺ release and the Ca²⁺ uptake on Ca²⁺ alternans onset. In addition, these ratios are related with the function of SERCA pumps, which vary with different ischemic conditions. Thus, values of these ratios could be used to differentiate Ca²⁺ alternans from different ischemic cases. This agrees with some experimental results. Therefore, the certain value of diastolic Ca_nsr/Ca_i can be the better “threshold” for Ca waves and Ca²⁺ alternans.

1. Introduction

Cardiac arrhythmia has long been associated with abnormal intracellular Ca²⁺ handling dynamics [1–4]. One useful diagnostic marker of arrhythmias is electrical alternans [5–8], which is expressed as alternated action potential durations (APDs) at the cellular level [9] and T waves on the electrocardiogram (ECG) [5], where the T wave stands for the repolarization of the ventricles and T wave alternans (TWA) indicates that the amplitude or the morphology of the T wave alternates beat-to-beat. The link between ischemia and alternans has been extensively explored [10]. Our previous simulations identified that hyperkalaemia, one component of ischemia, results in depolarization alternans [11]. Other two ischemic components, hypoxia and acidosis, lead to repolarization alternans by causing instabilities in calcium cycling [4, 11, 12]. Ca²⁺ alternans in ischemia can be taken as the arrhythmic triggers leading to afterdepolarization and also as the substrate facilitating reentry by inducing electrical alternans [4].

Alternans depends on instabilities of membrane voltage (V_m) [13] or/and intracellular Ca²⁺ handling [1, 7, 9, 14–20], due to their bidirectional couplings [3, 9, 21]. For the latter, it is known that the Ca²⁺ handling includes Ca²⁺ influx and efflux [16], and its abnormality can arise from the dysfunction of sarcoplasmic reticulum (SR) calcium transport ATPase (SERCA) [11, 13, 14], ryanodine receptor (RyR2) [1, 4, 7, 14, 16], and Ca²⁺ leak [13]. Under ischemic conditions, both the Ca²⁺ release current (I_rel) [4] and the Ca²⁺ uptake current (I_up) decrease [4, 22] to facilitate the formation of Ca²⁺ alternans in an interactive manner. In this work, we will focus on Ca²⁺ alternans caused by an impaired SERCA pump in ischemia.

The sarcoplasmic reticulum (SR) Ca²⁺ release curve describes the nonlinear relationship between SR Ca²⁺ release content and diastolic SR Ca²⁺ content (Ca_sr). The steep slope of this curve indicates that more Ca²⁺ is released at high diastolic SR Ca²⁺ content. In heart failure (HF), we have identified the primary role of the steep SR Ca²⁺ release curve in the genesis of alternans through simulation study [23]. Furthermore, the steep slope of the curve is also able to explain the impaired SERCA pump-caused Ca²⁺ alternans [19]. The onset of Ca²⁺ alternans in HF and in this study can be described as follows [9, 16]: when the slope of the curve is steep at certain Ca_sr, a small increment of diastolic Ca_sr will result in a larger Ca²⁺ release, where released Ca²⁺ cannot be completely refilled back to the SR by impaired SERCA pumps. In the following heartbeat, the decreased Ca_sr gives rise to a smaller subsequent Ca²⁺ release. According to the above description, the steep slope of SR Ca²⁺ release curve provides the substrate for alternans onset and impaired SERCA pump enhances the susceptibility. Previous studies attribute the steep slope to Ca wave propagation [23–25] or the saturation of buffered Ca_sr [26]. In fact, the steep slope of the curve is directly linked to the change of I_rel. Then, what are the detailed changes of I_rel to increase slopes? What is the factor that brings change to the I_rel? To investigate these questions, we took use of “3R theory” [27] to find the answers. The “3R theory” defines three critical properties (α for “randomness”, β for “refractoriness,” and γ for “recruitment”) of a Ca spark, and we use the properties to analyze Ca²⁺ alternans. These properties are further introduced in the Materials and Methods section.

The propagation of Ca waves is linked to the onset of Ca²⁺ alternans [25, 27]. Although experimental and theoretical studies have investigated the development of Ca waves, there is a dispute regarding the definition of the “threshold” for Ca waves. Some experimental studies indicated Ca_sr as the “threshold” [24, 28, 29], while others highlighted the role of intracellular Ca²⁺ concentration (Ca_i) [30–32]. We propose the ratio of diastolic network SR (NSR) Ca²⁺ content (Ca_nsr) to diastolic Ca_i (Ca_nsr/Ca_i) as the “threshold” for Ca waves and Ca²⁺ alternans, which highlights both of their roles, and finally verify it by simulations. The “threshold” of diastolic Ca_nsr/Ca_i is determined by thermodynamic constraints, which provides the theoretical basis for our new “threshold.” Moreover, this new “threshold” theory may help us better understand alternans and potentially provide a novel therapeutic strategy for alternans.

2. Materials and Methods

A thermodynamic model of SR Ca pump (SERCA pump model) [22] was integrated into the human epicardial (epi) ventricular cell model (O'Hara-Rudy dynamic (ORd) model) to simulate Ca²⁺ alternans [33]. The ORd model can reproduce the rate dependence of Ca²⁺ in experiments. The SERCA pump model is built based on experimental data of the rabbit and other animals. We used it to obtain the Ca²⁺ uptake rate per pump and multiplied a scale factor to calculate I_up. The appropriate coefficient was determined by comparing the I_up amplitude produced by the original ORd epi cell model and the Ca²⁺ uptake rate per pump at steady state under normal conditions (this method was described in detail in our previous study [11]). The SERCA pump model incorporates the regulation effect of phospholamban (PLB) on Ca²⁺ uptake. Similar with the effect of increased pH (Figure 4 in [22]), PLB phosphorylation decreases the half-maximum Ca²⁺ uptake rate K_0.5 [34] and increases SR Ca²⁺ uptake rate. During early phase of ischemia, the increased PLB phosphorylation helps to maintain the function of the SERCA pump [4]. After 20–30 minutes of ischemia, PLB dephosphorylation reduces SR Ca²⁺ uptake rate [4, 35]. In this work, we simulated the membrane voltage and ion concentrations after 10–20 minutes of ischemia, where phosphorylation level of PLB was kept the same as in control and the SERCA pump was impaired by ischemic components.

Figure 4.

Diastolic Ca_nsr, Ca_i, and Ca_nsr/Ca_i at the onset of Ca²⁺ alternans. Ca²⁺ alternans arises at different ranges of CLs in three ischemic cases. The ranges of CLs in ischemic cases 1, 2, and 3 are from 250 ms to 280 ms, from 250 ms to 380 ms, and from 250 ms to 300 ms.

As shown below, two Ca²⁺ are translated from the cytoplasm to the SR during Ca²⁺ uptake [22]:

$$\begin{array}{c}2{\text{Ca}}_{\text{i}}^{2+}+\text{MgATP}+{\text{H}}_2\text{O}\rightleftharpoons 2{\text{Ca}}_{\text{sr}}^{2+}+\text{MgADP}+\text{Pi}+{\text{H}}^{+}\end{array}$$ (1)

To investigate the effect of the ischemia-impaired SERCA pump on I_rel at stable state, we first compared calcium transients between control and ischemia and then applied the “3R theory” to analyze the changes of I_rel achieved by decreased I_up. This investigation explained how I_up cooperated with I_rel to cause Ca²⁺ alternans.

2.1. Conditions Setting in the Simulations

Ischemic conditions contribute to compromised metabolism and thus lead to decreased function of SERCA pumps. Specifically, hypoxia decreases intracellular ATP concentration ([ATP]_i) and increases intracellular ADP concentration ([ADP]_i) [36]. Meanwhile, inorganic phosphate (Pi) in the cytoplasm increases [37] and pH is decreased by acidosis [12]. According to these experimental data, we simulated three cases of ischemia (Table 1) with the cycle length (CL) of 250 ms and 350 ms, respectively to obtain Ca²⁺ alternans. After 1000 beats, action potentials (APs) and Ca transients were taken to be stable. Then, we analyzed them in the subsequent 1000 beats.

Table 1.

Parameters setting under different conditions.

Conditions	[ATP]i (mmol/L)	pH
Ischemic case 1	3	6.5
Ischemic case 2	3.5	6.6
Ischemic case 3	4	6.7
Control condition	9.8	7

The values of [ATP]_i and pH were in the ranges during early phase of ischemia [12, 38]. In ischemia, [ADP]_i and [Pi]_i were set to 0.2 mmol/L [36] and 30 mmol/L [37], respectively. In control, the two were set to 0.015 mmol/L and 0 mmol/L, respectively.

The CL also affects whether Ca²⁺ alternans can occur or not in different ischemic conditions. We attempted to find the ranges of CL in which Ca²⁺ alternans could arise in these three cases. In our simulations, the starting CL is 250 ms and the increasing step is 10 ms. Finally, we determined the ranges of CL in ischemic cases 1, 2, and 3, which are from 250 ms to 280 ms, from 250 ms to 380 ms, and from 250 ms to 300 ms, respectively.

2.2. The SR Ca²⁺ Release Curve

In our simulations, the total amount of diastolic SR Ca²⁺ (Ca_{sr_total} (mmol)) comprised the amount of diastolic NSR Ca²⁺ and the junctional SR (JSR) Ca²⁺. The amount of Ca²⁺ release (Ca_release(k) (mmol)) was expressed as the integral of the Ca²⁺ release flux (J_rel(k) (mmol/L/ms)) on the k^th beat (equation (4)). Then, we used the ratio of Ca_release(k) to Ca_{sr_total(k−1)} to represent the fraction of SR Ca²⁺ release:

$$\begin{array}{c}{\text{Ca}}_{\text{sr}\_\text{total}\left(k\right)}=v_{\text{nsr}}\ast {\text{Ca}}_{\text{nsr}\left(k\right)}+v_{\text{jsr}}\ast \left({\text{Ca}}_{\text{jsr}\_\text{free}\left(k\right)}+{\text{Ca}}_{\text{jsr}\_\text{buff}\left(k\right)}\right),\end{array}$$ (2)

$$\begin{array}{c}{\text{Ca}}_{\text{sr}\left(k\right)}=\frac{{\text{Ca}}_{\text{sr}\_\text{total}\left(k\right)}}{v_{\text{nsr}}+v_{\text{jsr}}},\end{array}$$ (3)

$$\begin{array}{c}{\text{Ca}}_{\text{release}\left(k\right)}=v_{\text{jsr}}{\int }_0^T{J}_{\text{rel}\left(k\right)}\text{d}t\text{,}\end{array}$$ (4)

where JSR Ca²⁺ included free and buffered Ca²⁺ (Ca_{jsr_free} (mmol/L) and Ca_{jsr_buff} (mmol/L)); v_nsr and v_jsr represented the volume of NSR and JSR; and Ca_sr(k) (mmol/L) and Ca_nsr(k) (mmol/L), respectively, referred to the diastolic SR and NSR Ca²⁺ content on the k^th beat.

2.3. Calculating Values of α, β, and γ according to “3R” Theory

In the spatially distributed calcium cycling model developed by Rovetti et al. [27], SR Ca²⁺ is released through CRUs. One CRU is set to have six neighbors in the 3D-distribution cell simulation [27]. As shown in equations (5) and (6), N₀ represents the total number of CRUs and N_K is the number of that activated on the k^th beat [27], where α represents the probability of a Ca spark being activated spontaneously or by the L-type Ca²⁺ current (I_CaL); β is the probability of a Ca spark triggered on the k^th beat being unavailable during the (k + 1)^th beat; γ indicates the probability of a Ca spark recruiting one of its neighboring; and f represents the percentage of secondary Ca sparks in the remaining available CRUs [27]. The number of CRUs activated on the (k + 1)^th beat is given as follows [27]:

$$\begin{array}{c}{N}_{K+1}=\left(N_0–\beta \ast N_K\right)\ast \left(\alpha +1-\alpha \ast f\right),\end{array}$$ (5)

$$\begin{array}{c}f=1-{\left(1-\alpha \ast \gamma \ast \left(1-\beta \ast \frac{N_K}{N_0}\right)\right)}^6,\end{array}$$ (6)

$$\begin{array}{c}〈\mathrm{\Delta }\text{Ca}〉=\left({\text{Ca}}_{\text{SR}}-〈{\text{Ca}}_{\text{b}}〉\right)\ast \frac{N_K}{N_0},\end{array}$$ (7)

where <ΔCa> is the average SR Ca²⁺ depletion of each CRU and Ca_sr is the average Ca²⁺ content of each CRU before release [27]. <Ca_b> refers to the average Ca²⁺ content of these N_k CRUs after they sparked [27]. We put Ca_release(k) and Ca_{sr_total} to replace <ΔCa> and Ca_SR to obtain equation (8). The left-hand side is SR Ca²⁺ concentration depletion. Thus, Ca_release(k) and Ca_{sr_total(k−1)}, calculated from our simulations (equations (2) and (4)), were linked with N_K and N₀.

$$\begin{array}{c}\frac{{\text{Ca}}_{\text{release}\left(k\right)}}{v_{\text{nsr}}+v_{\text{jsr}}}=\left(\frac{{\text{Ca}}_{\text{sr\_total}\left(k-1\right)}}{v_{\text{nsr}}+v_{\text{jsr}}}-\frac{〈{\text{Ca}}_{\text{b}}〉}{v_{\text{nsr}}+v_{\text{jsr}}}\right)\ast \frac{N_K}{N_0}.\end{array}$$ (8)

N _K and N₀ in equation (8) were replaced by Ca_release(k), Ca_{sr_total(k−1)}, and <Ca_b>. Thus, the relationship between properties of RyRs and our simulation results was built.

$$\begin{array}{c}\frac{{\text{Ca}}_{\text{release}\left(k+1\right)}}{{\text{Ca}}_{{\text{sr}}_{\text{total}\left(k\right)}}-〈{\text{Ca}}_{\text{b}}〉}=\left(1-\beta \ast \frac{{\text{Ca}}_{\text{release}\left(k\right)}}{{\text{Ca}}_{{\text{sr}}_{\text{total}\left(k-1\right)}}-〈{\text{Ca}}_{\text{b}}〉}\right)\ast \left(\alpha +\left(1-\alpha \right)\ast \left(1-{\left(1-\alpha \ast \gamma \ast \left(1-\beta \ast \frac{{\text{Ca}}_{\text{release}\left(k\right)}}{{\text{Ca}}_{\text{sr\_total}\left(k-1\right)}-〈{\text{Ca}}_{\text{b}}〉}\right)\right)}^6\right)\right),\end{array}$$ (9)

where α, β, γ, and <Ca_b> are unknown parameters and others could be obtained from our simulation results. To obtain these unknown parameters, we solved equation (9) by using the MATLAB built-in lsqcurvefit function. First, the inputs of Ca_release(k) and Ca_{sr_total(k−1)} were calculated from simulations. Meanwhile, initial α, β, and γ were set as random values from zero to one and the initial <Ca_b>/(v_nsr + v_jsr) was from zero to the maximum Ca_{sr_total}/(v_nsr + v_jsr). Then, these values were input to solve equation (9). Specifically, when we calculated these parameters during the short period of alternans formation, the groups of inputs were too few to obtain accurate values of these unknown parameters. We solved <Ca_b> in equation (9) before and after alternans onset in advance and take the value of it as a constant to input equation (9). Thus, the number of unknowns is decreased, and the remaining three unknowns are able to be obtained during the short period of alternans formation.

2.4. Definition of the Occurrence of Ca²⁺ Alternans

Ca²⁺ alternans was supposed to occur when the following criteria were met:

$$\begin{array}{c}\frac{\left({\text{Ca}}_{\text{amplitude}\left(k+1\right)}-{\text{Ca}}_{\text{amplitude}\left(k\right)}\right)}{{\text{Ca}}_{\text{amplitude}\left(k\right)}}\ge 5\%,\end{array}$$ (10)

where Ca_amplitude(k) is defined as the amplitude of Ca²⁺ transient on the k^th beat.

3. Results

According to equation (1), increased [ADP]_i, [Pi]_i, and [H⁺]_i and decreased [ATP]_i result in less Ca²⁺ transported from the cytoplasm to the SR. The function of the SERCA pump is impaired by ischemia. As shown in Figure 1, the amplitudes of Ca_i and Ca_sr alternate obviously. In contrast, diastolic Ca_i alternates slightly (inset of Figure 1(e)). APs also show slight alternans (Figure 1(d)), due to the Ca²⁺ alternans-caused fluctuation of I_CaL. A larger Ca²⁺ release decreases I_CaL, makes the transient outward current (I_to) more prominent, and leads to a slightly deeper notch of the AP. Subsequently, the voltage-dependent repolarization currents cause different repolarization phases.

Figure 1.

Alternans of APs and Ca transients and the relevant currents in ischemic case 2. Alternated APs, Ca_i, and Ca_sr are shown in (a)–(c), respectively. Aligned I_to, APs, Ca_i, J_rel, and I_CaL between two continuous APs are compared in (d)–(h), respectively, where the solid black lines stand for ischemic conditions and the dashed red lines stand for control condition. CL is 300 ms.

As shown in Figure 2, Ca²⁺ alternans can be observed in all three ischemic cases when CL = 250 ms. However, when the CL increases to 300 ms, it can only be observed in cases 2 and 3. In case 1, the maximum Ca_sr with CL of 300 ms does not reach the value of Ca_sr at which bifurcations occur with CL of 250 ms. In addition, the slopes of curves change slightly before alternans onset (inset of Figure 2), but the values of Ca_sr change obviously when bifurcations occur. The values of Ca_sr at which bifurcations occur decrease with the ischemic degree at the same CLs.

Figure 2.

SR Ca²⁺ release curves under control and ischemic conditions (cases 1, 2, and 3). (a) CL = 250 ms. (b) CL = 300 ms.

In Figure 3, α, β, and γ were obtained during the formation of Ca²⁺ alternans under different ischemic conditions. Compared with the control group, β and γ increase obviously in all ischemic conditions. In control condition, average β and γ are 0 and 0.42, respectively. They both increase to 1 in ischemic case 2 with CL of 300 ms. Nonetheless, α does not vary a lot.

Figure 3.

Values of α, β, and γ during the formation of Ca²⁺ alternans in control and ischemic cases. Twenty groups of parameter values are included in each box plot. In each panel, the left three box plots are in control conditions and the right three are under ischemia. The ranges of Ca_sr are the same in every two contrasting groups. (a) Ca_sr ranges from 3 to 3.7 mmol/L with the CL of 300 ms. (b) Ca_sr ranges from 3.2 to 3.9 mmol/L with the CL of 300 ms. (c) Ca_sr ranges from 2.8 to 3.15 mmol/L and the CL is 250 ms. (d) Ca_sr ranges from 3 to 4 mmol/L and the CL is 250 ms. (e) Ca_sr ranges from 3.5 to 4.4 mmol/L and the CL is 250 ms.

The values of diastolic Ca_nsr, Ca_i, and Ca_nsr/Ca_i in Figure 4 were recorded once Ca²⁺ alternans occurred in ischemia. On the one hand, Ca²⁺ alternans in case 1 or case 2 cannot be distinguished by the recorded values of Ca_nsr (Figure 4(a)). On the other hand, there is a small difference between the values of the recorded Ca_i in case 2 and those in case 3 (Figure 4(b)). In contrast, the ratios vary obviously with ischemic cases. Compared with the effect of CLs, the degree of ischemia (values of [ADP]_i, [Pi]_i, [H⁺]_i, and [ATP]_i) affects the ratios more effectively. Furthermore, how diastolic Ca_sr, Ca_i, and their ratio change with sequent heartbeats is analyzed under transient Ca²⁺ alternans (Figure 5). Ca²⁺ alternans lasts for some beats and gradually disappears in ischemic case 3 when CL = 250 ms. In the whole process of Ca²⁺ alternans development, diastolic Ca_nsr, Ca_i, and their ratio fluctuate and increase (Figure 5). After Ca²⁺ alternans disappears, the ratio remains a constant value (Figure 5(b)) while the other two continue to increase (Figure 5(a)), where Ca_sr and Ca_i are divided by their maximum values, respectively, to get normalized values, which are no bigger than one. This will facilitate the comparison in Figure 5(a).

Figure 5.

Normalized diastolic Ca_i (dashed red line in (a)), Ca_sr (dashed blue line in (a)), and diastolic Ca_nsr/Ca_i (b) under transient Ca²⁺ alternans. These values are recorded under ischemic case 3 during the 1000 consecutive beats.

4. Discussion

Consistent with previous study [25], fluctuations of Ca_sr are observed during the impaired SERCA pump-caused Ca²⁺ alternans (Figures 1(c) and 5(a)). However, the slight fluctuations of Ca_sr alone are insufficient to maintain Ca²⁺ alternans without the steep slope of Ca²⁺ release curve. The large fraction of Ca²⁺ release is demonstrated to generate Ca²⁺ alternans [16, 23, 25, 39]. Figure 2 shows that the curve slopes change slightly before the onset of alternans in different ischemic conditions. Subsequently, the obvious bifurcations occur in the curve. These obvious changes are the dominant factors to cause alternans. To elucidate how these bifurcations happen, we analyze how I_rel is affected by impaired SERCA pump in the period of bifurcations occurrence.

I _rel can be regarded as a collective effect of Ca sparks. During the formation of alternans, changes in properties of Ca sparks reflect how I_rel is affected by the impaired SERCA pump. The values of β and γ increase obviously in ischemic groups compared to control without Ca²⁺ alternans (Figure 3). Rovetti et al. [27] concluded that large β and γ together with properly chosen α promote alternans. Our results confirmed their prediction (Figure 3).

Large β indicates long refractory period of RyRs when the CL is unchanged [27], implying a long time required for complete recovery of RyRs. Our results show that large β can be induced by the impaired SERCA pump. This can be easily understood through introducing a Ca²⁺ cycling hypothesis [16]: cytosolic Ca²⁺, taken up by the SERCA pump in the NSR, is released by the RyRs channels in the JSR. Thus, the process of transporting Ca²⁺ from the NSR to the JSR results in a delayed Ca²⁺ release after the uptake. The impaired SERCA pump slows the Ca²⁺ recycling process and increases RyRs refractory period. During the slow Ca²⁺ recycling process, the amount of Ca²⁺ reaching the release sites fluctuates, leading to alternated large and small Ca²⁺ releases. This hypothesis presents a possible mechanism underlying how SERCA pump modulates the refractoriness of the RyRs.

According to “3R theory,” large γ indicates frequent spark-induced sparks, which probably produce Ca waves. The propagation of Ca waves requires Ca diffusion to take effect. Ca diffusion is also shown to influence another behavior of synchronizing local Ca²⁺ release [40]. The degree of synchronization increases obviously when the “threshold” CL for alternans is approaching [40]. Because of the different dependence of pacing, Ca waves differ with the synchronization. However, when the synchronization occurs, Ca diffusion is also more likely to propagate Ca waves. The occurrence of Ca²⁺ alternans also promotes Ca²⁺ wave to propagate. On the other hand, the prolonged refractory period can produce alternated Ca waves by regulating the numbers of available CRUs. In all, large γ can either directly result from Ca²⁺ alternans or be induced by the prolonged refractory period.

Although we have investigated how I_rel changes during the formation of Ca²⁺ alternans, the timing for these changes taking place is still unknown. Large γ is associated with the propagation of Ca waves. Ca waves are also linked to the onset of Ca²⁺ alternans. Figure 2 shows that Ca²⁺ alternans begin at some value of Ca_sr. That is, that value of Ca_sr is able to be regarded as the “threshold” for Ca waves [24, 28, 29]. This idea is also supported by the fact that increased Ca_sr increases α and initiates Ca waves [24, 28]. However, if Ca_sr is believed to be the “threshold” for Ca waves and Ca²⁺ alternans, then Ca²⁺ alternans should not disappear as Ca_sr keeps rising (Figure 5(a)). In fact, whether the “threshold” of Ca_sr determines Ca waves onset or not is also debated in other studies that tried to link Ca_i to the occurrence of Ca waves [30–32]. Moreover, although previous experimental study [28] supports the idea that the “threshold” of Ca_sr determines Ca waves onset, diastolic Ca_i has also been associated with the frequency of release (Figures 2(a) and 2(b) of [28]). Undoubtedly, diastolic Ca_i also exerts influence in producing Ca waves and Ca²⁺ alternans. Therefore, we propose diastolic Ca_nsr/Ca_i as the “threshold” of Ca waves and Ca²⁺ alternans, which reflects the roles of both Ca_sr and Ca_i in the formation of Ca waves and Ca²⁺ alternans.

SERCA pumps contribute to maintain the Ca²⁺ concentration gradient between the SR and the cytoplasm. Since the Ca²⁺ uptake sites are in the NSR, diastolic Ca_nsr/Ca_i is related to the Ca²⁺ uptake. According to equation (1), the maximum uptake rate is modulated by ischemic conditions ([ADP]_i, [ATP]_i, [Pi]_i, and [H⁺]_i). This means the maximum diastolic Ca_nsr/Ca_i can be affected by ischemic cases. In addition, the onset of alternans is induced by ischemia, and thus its “threshold” is taken to be different with ischemic degrees. In Figure 4, the “threshold” of diastolic Ca_nsr/Ca_i differs with ischemic conditions. As a contrast, neither Ca_i nor Ca_nsr is able to distinguish different ischemic cases. In Figure 5, before the onset of Ca²⁺ alternans, Ca_sr and Ca_i increase. Correspondingly, I_up goes on retaking the increasing released Ca²⁺ and diastolic Ca_nsr/Ca_i keeps rising. However, to what extent I_up and diastolic Ca_nsr/Ca_i can increase is limited by the thermodynamic constrains. Ca²⁺ alternans and Ca waves form when the unbalance between the Ca²⁺ uptake and Ca²⁺ release occurs. Subsequently, as diastolic Ca_i and Ca_nsr keep on increasing, the Ca²⁺ uptake rate increases enough to uptake all released Ca²⁺ and Ca²⁺ alternans disappears. This final constant value of diastolic Ca_nsr/Ca_i indicates the new balance between the Ca²⁺ release and Ca²⁺ uptake. On the other hand, if diastolic Ca_i or Ca_nsr is the “threshold” for Ca waves and Ca²⁺ alternans, these two increasing values will initiate larger Ca waves and Ca²⁺ alternans will not disappear.

Xie et al. [15] demonstrated the SR Ca²⁺ efflux cooperates with the influx to affect the “threshold” for alternans. An intermediate SR Ca²⁺ uptake rate and a larger SR Ca²⁺ release work synergistically to produce alternans at longer CLs [16]. According to our new “threshold” theory, the limited increase of I_up contributes to unbalanced I_up and I_rel and promotes abnormal intracellular Ca²⁺ handling. In addition, other studies demonstrated that the properties of RyRs affect the “threshold” of Ca_sr for alternans [24, 41, 42]. When the open probability of RyRs channels increases at the same Ca_sr [41], the Ca²⁺ uptake rate becomes larger accordingly. As the physiological conditions ([ADP]_i, [ATP]_i, [Pi]_i, and [H⁺]_i) are identical, the maximum Ca²⁺ uptake rate is kept the same. Thus, maximum I_up occurs at a lower Ca_sr, resulting in a reduced “threshold” of Ca_sr for alternans (Figure 3 of [24] and Figure 4 of [41]).

This “threshold” theory provides a new idea for changes of I_rel during the formation of Ca²⁺ alternans. Our new “theory” shows that the unbalance between I_up and I_rel begins at the “threshold” of diastolic Ca_nsr/Ca_i. These two currents interact with each other and result in Ca²⁺ alternans. We should also note that Ca²⁺ alternans of case 3 is transient (Figure 5), which suggests that when I_up and I_rel get balanced, Ca²⁺ alternans is suppressed. Therefore, this “threshold” theory may disclose a novel therapeutic strategy for Ca²⁺ alternans. In theory, keeping the ratio below the “threshold” stops Ca²⁺ alternans occurrence. In clinical, the treatment aiming at the interaction between these two currents may have a promising effect. Although direct therapeutic tools modulating the SR release channels have not been fully developed [2], the new proposed “threshold” theory can be regarded as a strong guideline for searching for new therapeutic targets.

5. Conclusion

The integrated cell model can be used to simulate the SERCA pump function in specific ischemic conditions, whereas the original ORd model cannot be used. The simulated results indicate that Ca waves can be induced by impaired SERCA pump and thus give rise to Ca²⁺ alternans. That is, these components of Ca cycling interact with each other to affect Ca²⁺ alternans development. In addition, compared with isolated changes of diastolic Ca_sr and Ca_i, the value of diastolic Ca_nsr/Ca_i is more appropriate to function as the “threshold” for alternans. By defining this new “threshold,” we can better explain how the interplay between the I_up and I_rel causes alternans. Furthermore, this proposed “threshold” theory may help find therapeutic targets for suppressing Ca²⁺ alternans.

6. Limitations

This model is just used to simulate the impaired SERCA pump function during ischemia, whereas L-type calcium and other currents are also affected under ischemia, and these changes are not included in our model. We need to further improve this integrated model to simulate more accurate ischemic conditions. It is noted that the values of the three properties can be different by using different groups of inputs. The range of Ca_i chosen in our simulation can also determine the results by influencing the inputs. We also need to take use of other cell models to calculate values of these properties during Ca²⁺ alternans occurrence and identify our “threshold” theory. In addition, although we conclude that I_up and I_rel interact to produce Ca²⁺ alternans, we just identify how the Ca release is affected by impaired SERCA pump, and the effect of I_rel on I_up should also be identified in detail in the future.

Data Availability

The data used to build graphs in this study are available from the corresponding author upon request.

Disclosure

Jiaqi Liu and Xiaoye Zhao are the co-first authors.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Supplementary Materials

Supplementary Materials

Our supplementary files include the modified human ventricular cell model, codes, and data for plotting figures in our papers.