Exploring Impaired SERCA Pump-Caused Alternation Occurrence in Ischemia

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 (Cansr) and the cytoplasmic Ca content (Cai) (Cansr/Cai) as the “threshold” of Ca waves and Ca2+ alternans. Diastolic Cansr, Cai, and their ratio were recorded at the onset of Ca2+ alternans. Compared with certain Cansr and Cai, 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 Cansr/Cai can be the better “threshold” for Ca waves and Ca2+ alternans.


Introduction
Cardiac arrhythmia has long been associated with abnormal intracellular Ca 2+ handling dynamics [1][2][3][4]. One useful diagnostic marker of arrhythmias is electrical alternans [5][6][7][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. e 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]. Ca 2+ 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 Ca 2+ handling [1,7,9,[14][15][16][17][18][19][20], due to their bidirectional couplings [3,9,21]. For the latter, it is known that the Ca 2+ handling includes Ca 2+ 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 Ca 2+ leak [13]. Under ischemic conditions, both the Ca 2+ release current (I rel ) [4] and the Ca 2+ uptake current (I up ) decrease [4,22] to facilitate the formation of Ca 2+ alternans in an interactive manner. In this work, we will focus on Ca 2+ alternans caused by an impaired SERCA pump in ischemia. e sarcoplasmic reticulum (SR) Ca 2+ release curve describes the nonlinear relationship between SR Ca 2+ release content and diastolic SR Ca 2+ content (Ca sr ). e steep slope of this curve indicates that more Ca 2+ is released at high diastolic SR Ca 2+ content. In heart failure (HF), we have identified the primary role of the steep SR Ca 2+ 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 Ca 2+ alternans [19]. e onset of Ca 2+ 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 Ca 2+ release, where released Ca 2+ 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 Ca 2+ release. According to the above description, the steep slope of SR Ca 2+ 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][24][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 . en, 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. e "3R theory" defines three critical properties (α for "randomness", β for "refractoriness," and c for "recruitment") of a Ca spark, and we use the properties to analyze Ca 2+ alternans. ese properties are further introduced in the Materials and Methods section. e propagation of Ca waves is linked to the onset of Ca 2+ 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 Ca 2+ concentration (Ca i ) [30][31][32]. We propose the ratio of diastolic network SR (NSR) Ca 2+ content (Ca nsr ) to diastolic Ca i (Ca nsr /Ca i ) as the "threshold" for Ca waves and Ca 2+ alternans, which highlights both of their roles, and finally verify it by simulations. e "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.

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 Ca 2+ alternans [33].
e ORd model can reproduce the rate dependence of Ca 2+ in experiments. e SERCA pump model is built based on experimental data of the rabbit and other animals. We used it to obtain the Ca 2+ uptake rate per pump and multiplied a scale factor to calculate I up . e appropriate coefficient was determined by comparing the I up amplitude produced by the original ORd epi cell model and the Ca 2+ uptake rate per pump at steady state under normal conditions (this method was described in detail in our previous study [11]). e SERCA pump model incorporates the regulation effect of phospholamban (PLB) on Ca 2+ uptake. Similar with the effect of increased pH (Figure 4 in [22]), PLB phosphorylation decreases the half-maximum Ca 2+ uptake rate K 0.5 [34] and increases SR Ca 2+ 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 Ca 2+ 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.
As shown below, two Ca 2+ are translated from the cytoplasm to the SR during Ca 2+ uptake [22]: 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 . is investigation explained how I up cooperated with I rel to cause Ca 2+ alternans.

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 Ca 2+ alternans. After 1000 beats, action potentials (APs) and Ca transients were taken to be stable. en, we analyzed them in the subsequent 1000 beats. e CL also affects whether Ca 2+ alternans can occur or not in different ischemic conditions. We attempted to find the ranges of CL in which Ca 2+ 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+ Release Curve. In our simulations, the total amount of diastolic SR Ca 2+ (Ca sr_total (mmol)) comprised the amount of diastolic NSR Ca 2+ and the junctional SR (JSR) Ca 2+ . e amount of Ca 2+ release (Ca release(k) (mmol)) was expressed as the integral of the Ca 2+ release flux (J rel(k) (mmol/L/ms)) on the k th beat (equation (4)). en, we used the ratio of Ca release(k) to Ca sr_total(k− 1) to represent the fraction of SR Ca 2+ release:

e SR Ca
where JSR Ca 2+ included free and buffered Ca 2+ (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 Ca 2+ content on the k th beat.

Calculating Values of α, β, and c according to "3R" eory.
In the spatially distributed calcium cycling model developed by Rovetti et al. [27], SR Ca 2+ 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), N 0 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 Ca 2+ 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; c 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]. e number of CRUs activated on the (k + 1) th beat is given as follows [27]: where <ΔCa> is the average SR Ca 2+ depletion of each CRU and Ca sr is the average Ca 2+ content of each CRU before release [27]. <Ca b > refers to the average Ca 2+ 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). e left-hand side is SR Ca 2+ concentration depletion. us, Ca release(k) and Ca sr_total(k− 1) , calculated from our simulations (equations (2) and (4)), were linked with N K and N 0 .
N K and N 0 in equation (8) were replaced by Ca release(k) , Ca sr_total(k− 1) , and <Ca b >.
us, the relationship between properties of RyRs and our simulation results was built.
where α, β, c, 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 c 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 ). en, 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). us, the number of unknowns is decreased, and the remaining three unknowns are able to be obtained during the short period of alternans formation.

Definition of the Occurrence of Ca 2+
Alternans. Ca 2+ alternans was supposed to occur when the following criteria were met: where Ca amplitude(k) is defined as the amplitude of Ca 2+ transient on the k th beat.
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 Ca 2+ alternans-caused fluctuation of I CaL . A larger Ca 2+ 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 voltagedependent repolarization currents cause different repolarization phases.
As shown in Figure 2, Ca 2+ 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. e values of Ca sr at which bifurcations occur decrease with the ischemic degree at the same CLs.
In Figure 3, α, β, and c were obtained during the formation of Ca 2+ alternans under different ischemic conditions. Compared with the control group, β and c increase obviously in all ischemic conditions. In control condition, average β and c are 0 and 0.42, respectively. ey both increase to 1 in ischemic case 2 with CL of 300 ms. Nonetheless, α does not vary a lot.    e values of diastolic Ca nsr , Ca i , and Ca nsr /Ca i in Figure 4 were recorded once Ca 2+ alternans occurred in ischemia. On the one hand, Ca 2+ 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 Ca 2+ alternans ( Figure 5). Ca 2+ alternans lasts for some beats and gradually disappears in ischemic case 3 when CL � 250 ms. In the whole process of Ca 2+ alternans development, diastolic Ca nsr , Ca i , and their ratio fluctuate and increase ( Figure 5). After Ca 2+ 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. is will facilitate the comparison in Figure 5(a).

Discussion
Consistent with previous study [25], fluctuations of Ca sr are observed during the impaired SERCA pump-caused Ca 2+ alternans (Figures 1(c) and 5(a)). However, the slight fluctuations of Ca sr alone are insufficient to maintain Ca 2+ alternans without the steep slope of Ca 2+ release curve. e large fraction of Ca 2+ release is demonstrated to generate Ca 2+ 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. ese 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. e values of β and c increase obviously in ischemic groups compared to control without Ca 2+ alternans (Figure 3). Rovetti et al. [27] concluded that large β and c 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. is can be easily understood through introducing a Ca 2+ cycling hypothesis [16]: cytosolic Ca 2+ , taken up by the SERCA pump in the NSR, is released by the RyRs channels in the JSR. us, the process of transporting Ca 2+ from the NSR to the JSR results in a delayed Ca 2+ release after the uptake. e impaired SERCA pump slows the Ca 2+ recycling process and increases RyRs refractory period. During the slow Ca 2+ recycling process, the amount of Ca 2+ reaching the release sites fluctuates, leading to alternated large and small Ca 2+ releases. is hypothesis presents a possible mechanism underlying how SERCA pump modulates the refractoriness of the RyRs.
According to "3R theory," large c indicates frequent spark-induced sparks, which probably produce Ca waves. e propagation of Ca waves requires Ca diffusion to take effect. Ca diffusion is also shown to influence another behavior of synchronizing local Ca 2+ release [40]. e 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.
e occurrence of Ca 2+ alternans also promotes Ca 2+ 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 c can either directly result from Ca 2+ alternans or be induced by the prolonged refractory period.
Although we have investigated how I rel changes during the formation of Ca 2+ alternans, the timing for these changes taking place is still unknown. Large c is associated with the propagation of Ca waves. Ca waves are also linked to the onset of Ca 2+ alternans. Figure 2 shows that Ca 2+ alternans begin at some value of Ca sr . at is, that value of Ca sr is able to be regarded as the "threshold" for Ca waves [24,28,29].
is 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 Ca 2+ alternans, then Ca 2+ 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][31][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 Ca 2+ alternans. erefore, we propose diastolic Ca nsr /Ca i as the "threshold" of Ca waves and Ca 2+ alternans, which reflects the roles of both Ca sr and Ca i in the formation of Ca waves and Ca 2+ alternans.
SERCA pumps contribute to maintain the Ca 2+ concentration gradient between the SR and the cytoplasm. Since the Ca 2+ uptake sites are in the NSR, diastolic Ca nsr /Ca i is related to the Ca 2+ uptake. According to equation (1), the maximum uptake rate is modulated by ischemic conditions ([ADP] i , [ATP] i , [Pi] i , and [H + ] i ). is 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 Ca 2+ alternans, Ca sr and Ca i increase. Correspondingly, I up goes on retaking the increasing released Ca 2+ 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. Ca 2+ alternans and Ca waves form when the unbalance between the Ca 2+ uptake and Ca 2+ release occurs. Subsequently, as diastolic Ca i and Ca nsr keep on increasing, the Ca 2+ uptake rate increases enough to uptake all released Ca 2+ and Ca 2+ alternans disappears. is final constant value of diastolic Ca nsr /Ca i indicates the new balance between the Ca 2+ release and Ca 2+ uptake. On the other hand, if diastolic Ca i or Ca nsr is the "threshold" for Ca waves and Ca 2+ alternans, these two increasing values will initiate larger Ca waves and Ca 2+ alternans will not disappear.
Xie et al. [15] demonstrated the SR Ca 2+ efflux cooperates with the influx to affect the "threshold" for alternans. An intermediate SR Ca 2+ uptake rate and a larger SR Ca 2+ 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 Ca 2+ 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 Ca 2+ uptake rate becomes larger accordingly.  Figure 3 of [24] and Figure 4 of [41]). is "threshold" theory provides a new idea for changes of I rel during the formation of Ca 2+ alternans. Our new "theory" shows that the unbalance between I up and I rel begins at the "threshold" of diastolic Ca nsr /Ca i . ese two currents interact with each other and result in Ca 2+ alternans. We should also note that Ca 2+ alternans of case 3 is transient ( Figure 5), which suggests that when I up and I rel get balanced, Ca 2+ alternans is suppressed. erefore, this "threshold" theory may disclose a novel therapeutic strategy for Ca 2+ alternans. In theory, keeping the ratio below the "threshold" stops Ca 2+ 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.

Conclusion
e integrated cell model can be used to simulate the SERCA pump function in specific ischemic conditions, whereas the original ORd model cannot be used. e simulated results indicate that Ca waves can be induced by impaired SERCA pump and thus give rise to Ca 2+ alternans. at is, these components of Ca cycling interact with each other to affect Ca 2+ 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 Ca 2+ alternans.

Limitations
is 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. e 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 Ca 2+ alternans occurrence and identify our "threshold" theory. In addition, although we conclude that I up and I rel interact to produce Ca 2+ 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 e 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
e authors declare that there are no conflicts of interest.