Size Matters: Ryanodine Receptor Cluster Size Heterogeneity Potentiates Calcium Waves

Please cite this article in press as: Xie et al., Size Matters: Ryanodine Receptor Cluster Size Heterogeneity Potentiates Calcium Waves, Biophysical Journal (2019), https://doi.org/10.1016/j.bpj.2018.12.017

Article

Size Matters: Ryanodine Receptor Cluster Size Heterogeneity Potentiates Calcium Waves Yuanfang Xie,1 Yi Yang,1 Samuel Galice,1 Donald M. Bers,1 and Daisuke Sato1,* 1

Department of Pharmacology, University of California Davis, Davis, California

ABSTRACT Ryanodine receptors (RyRs) mediate calcium (Ca)-induced Ca release and intracellular Ca homeostasis. In a cardiac myocyte, RyRs group into clusters of variable size from a few to several hundred RyRs, creating a spatially nonuniform intracellular distribution. It is unclear how heterogeneity of RyR cluster size alters spontaneous sarcoplasmic reticulum (SR) Ca releases (Ca sparks) and arrhythmogenic Ca waves. Here, we tested the impact of heterogeneous RyR cluster size on the initiation of Ca waves. Experimentally, we measured RyR cluster sizes at Ca spark sites in rat ventricular myocytes and further tested functional impacts using a physiologically detailed computational model with spatial and stochastic intracellular Ca dynamics. We found that the spark frequency and amplitude increase nonlinearly with the size of RyR clusters. Larger RyR clusters have lower SR Ca release threshold for local Ca spark initiation and exhibit steeper SR Ca release versus SR Ca load relationship. However, larger RyR clusters tend to lower SR Ca load because of the higher Ca leak rate. Conversely, smaller clusters have a higher threshold and a lower leak, which tends to increase SR Ca load. At the myocyte level, homogeneously large or small RyR clusters limit Ca waves (because of low load for large clusters but low excitability for small clusters). Mixtures of large and small RyR clusters potentiates Ca waves because the enhanced SR Ca load driven by smaller clusters enables Ca wave initiation and propagation from larger RyR clusters. Our study suggests that a spatially heterogeneous distribution of RyR cluster size under pathological conditions may potentiate Ca waves and thus afterdepolarizations and triggered arrhythmias.

INTRODUCTION Delayed afterdepolarizations, activated by spontaneous calcium (Ca) waves, are one of the main causes of triggered arrhythmias and sudden cardiac death (1–3). Ryanodine receptors (RyRs) mediate Ca release from the sarcoplasmic reticulum (SR) during both excitationcontraction coupling and spontaneous Ca waves. RyRs are typically clustered in cardiac myocytes, particularly at the specialized junctional SR sites where SR Ca release primarily occurs. Spontaneous Ca sparks are the result of diastolic coherent stochastic recruitment of RyR channels via Ca-induced Ca release (CICR) that results in nearsimultaneous openings of RyRs from individual clusters (4,5). When sparks from local clusters are sufficiently large, neighboring clusters can be activated and propagate as arrhythmogenic Ca waves throughout the cell via ‘‘firediffuse-fire’’ CICR mechanism (6,7). In such a way, the ultrastructure of RyR clusters can be critical for intracellular

Submitted December 1, 2017, and accepted for publication December 12, 2018. *Correspondence: [email protected] Editor: Mark Cannell. https://doi.org/10.1016/j.bpj.2018.12.017

Ca dynamics in both normal and pathological conditions (8–10). In ventricular myocytes, the number of RyRs within the individual cluster (RyR cluster size) varies from a few to several hundred (8,11–13), presenting a nonuniform spatial distribution in RyR cluster size. The association between the size of RyR clusters and Ca spark morphology and frequency has been suggested by experimental studies (14–16) and in some simulation studies (17–20). Different RyR models or conditions used have left the relation between Ca spark frequency and RyR cluster size controversial (17–19). Moreover, how this heterogeneity in the RyR cluster size contributes to Ca wave initiation and propagation remains unclear. Our experimental results directly measuring RyR cluster size and Ca sparks simultaneously within ventricular myocytes showed that Ca spark frequency and the associated rate of Ca release steeply increase with RyR cluster size (21). In this study, using mathematical modeling, we further tested our hypothesis that the heterogeneity of RyR cluster size promotes Ca waves. First, we systematically quantified the effect of the RyR cluster size on Ca sparks and leak and then investigated how RyR cluster size heterogeneity promotes Ca waves.

Ó 2018 Biophysical Society.

Biophysical Journal 116, 1–10, February 5, 2019 1

Please cite this article in press as: Xie et al., Size Matters: Ryanodine Receptor Cluster Size Heterogeneity Potentiates Calcium Waves, Biophysical Journal (2019), https://doi.org/10.1016/j.bpj.2018.12.017

Xie et al.

MATERIALS AND METHODS We carried out both probability analysis of SR Ca leak and computer simulations using a physiologically detailed ventricular myocyte model with spatial subcellular Ca dynamics developed in our previous study (see (4) and Supporting Materials and Methods for details). Briefly, in this model, each RyR is formulated in a four-state Markov model and regulated simultaneously by both [Ca] in the dyadic cleft space ([Ca]Cleft) and the SR lumen ([Ca]SR). Each RyR opens stochastically. There are 19,305 CRUs (65
27
11) in the cell. Each Ca release unit (CRU) contains only one RyR cluster. Here, we define the cluster as a functional release unit in which each RyR can see the same [Ca]Cleft. This functionally approximates CRU RyR clusters or superclusters of multiple clusters at a CRU that may work as a single functional release unit. CRUs are coupled by Ca diffusion in the cytosol and the network SR. Sarcoplasmic reticulum Ca2þ-ATPase (SERCA) pumps are distributed equally over the cell. Membrane voltage was fixed at 80 mV. At 80 mV, L-type Ca channels very rarely open and can be negligible (and for our analysis here, we set L-type channel conductance to zero). All simulations were evaluated after steady state was reached. To investigate the effect of RyR cluster size, we varied the number of RyRs in a cluster from 1 to 250 based on experimental observations (12,13,21). SR Ca releases can occur as Ca sparks or smaller amplitude nonspark leak events partly mediated via RyR (22). A local [Ca]i increase of 0.2 mM (relative to the diastolic level) was used as a threshold for Ca spark detection (versus non-spark-mediated SR Ca leak via RyR). [Ca]SR of the rabbit myocyte is relatively low when the cell is quiescent. To promote Ca waves, we increased extracellular Ca ([Ca]o) to 7 mM and raised SERCA2 uptake rate maximal (Vmax) to threefold. This combination enhances [Ca]SR in the resting myocyte in a way that mimics steady-state stimulation and/or stimulation of uptake and influx via b-adrenergic agonists.

RESULTS Effects of RyR cluster size on Ca sparks and nonspark leak Probability analysis

First, we analyze Ca spark and nonspark leak related to probability of RyR channel opening, which are extended in myocyte computer simulations in the next section. Ca released via RyRs is composed of spark-mediated and non-spark-mediated leak (22,23). Theoretical and experimental studies (4,22) have shown that we can differentiate Ca spark from nonspark RyR-mediated leak by assessing simultaneous RyR openings in an individual cluster. Briefly, we assumed that all RyRs of the same cluster sense the same local [Ca]Cleft and interact with each other only via [Ca]Cleft (and [Ca]SR) via CICR, without the allosteric coupling utilized by some groups (19). In this case, the process of Ca spark formation can be simplified into two steps: 1) initial single RyR opening and 2) recruitment of multiple secondary openings of additional RyRs (required for a spark). The initial single RyR opening is determined by diastolic [Ca]Cleft and [Ca]SR. With the typical diastolic [Ca]Cleft (0.1 mM), RyR open probability (PO) is very small (on the order of 105  106, Figs. S1 A and S2 A). The probability of the initial single RyR opening can be approximated as NPO, linearly dependent on RyR cluster size N (Fig. S1 B). The

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initial single RyR opening quickly increases [Ca]cleft, which rapidly approaches a steady-state value, whereas the RyR remains open (Fig. S2 B). Within the mean open time of RyR (1.5 ms), [Ca]Cleft increases to 27 mM for [Ca]SR ¼ 600 mM (Fig. S2 B). This elevated [Ca]Cleft can then recruit another (secondary) RyR opening in the same cluster through CICR (Fig. S2 C, for detailed calculation see (4)), which greatly enhances the probability of the close to open transition (PC / O) for the neighboring RyRs within the same cluster (to the range of 102, Fig. S2 C). Thus, the simultaneous opening of multiple RyRs in the secondary step becomes feasible, and the probability of n RyRs opening simultaneously (indicated by Pn ¼ m) obeys a binomial distribution with the assumption that RyRs open independently within an individual cluster (Eq. 1). Pn ¼ m ¼ N1 Cm PC/O ðtÞm ð1  PC/O ðtÞÞN1m ;

(1)

where N is RyR cluster size, PC / O(t) is the probability of closed RyR to open within the time period t after the first initial opening, and m is the number of simultaneously opened RyRs. Here, we set t at the mean open time (1.5 ms; for 600 mM [Ca]SR; Fig. S2 B), which leads to a [Ca]cleft of 27 mM and a probability for a secondary opening PC / O(t ¼ 1.5 ms) equal to 0.01. As above, we can also determine the spark probability by calculating the simultaneous secondary RyR openings Pn ¼ m (Fig. 1 A, shows m ¼ 1–6). Pn ¼ m depends on RyR cluster size (Eq. 1). The probability of simultaneous openings (m R 2) increases with RyR cluster size. The probability of single RyR opening (m ¼ 1) increases at small cluster size but decreases at large RyR cluster size (Fig. 1 A, black curve) because the probability of simultaneous openings (m R 2) increases. We then set a simple threshold to separate Ca sparks from nonspark Ca leak as demonstrated in our previous study (4). That is, if at least two RyRs open simultaneously during the secondary opening (m R 2), the release is considered a spark; otherwise, it is considered a nonspark RyR leak. We then integrated the probability of each category and calculated their dependence on RyR cluster size (Fig.1, B and C, respectively). Thus, by multiplying the initial RyR opening probability (NPO) and the secondary accumulated openings (shown in Fig. 1, B and C, respectively), we predict the Ca spark and nonspark leak probability, respectively (black lines in Fig. 1, D and E, respectively). Because the secondary opening depends on PC / O, which is indirectly affected by [Ca]SR, we also calculated the spark and nonspark leak probabilities at a higher secondary open probability (e.g., PC / O ¼ 0.01 shown in Fig. S2, which can occur at long RyR open time and/or large [Ca]SR; Fig. 1, D and E, red). The dependence of secondary opening Pn ¼ m shows a similar pattern. With the same threshold, we found a similar dependence of probability of Ca spark and nonspark leak on RyR cluster size (Fig. 1, D and E, red). Despite the secondary opening, the

Please cite this article in press as: Xie et al., Size Matters: Ryanodine Receptor Cluster Size Heterogeneity Potentiates Calcium Waves, Biophysical Journal (2019), https://doi.org/10.1016/j.bpj.2018.12.017

RyR Cluster Size Heterogeneity

FIGURE 1 Theoretical prediction of RyR cluster size effects on the Ca spark and nonspark event probability. (A) The dependence of the probability for m channels (m ¼ 1  6 is shown) simultaneous opening on RyR cluster size in the secondary opening (PC / O ¼ 0.003, according to the black circle in Fig. S2 C). (B) The accumulated probability of zero and one channel opening from the data in (A). (C) The accumulated probability of more than one channel simultaneous opening from the data in (A). The dependence of nonspark event probability (D) and Ca spark probability (E) on RyR cluster size is shown. Black and red is calculated when the secondary RyR opening probability PC / O is equal to 0.003 and 0.01, respectively, corresponding to the black and red circle in Fig. S2 C. To see this figure in color, go online.

to small clusters (30 RyRs) at a given [Ca]SR (Fig. 2, red, solid versus dashed). This is consistent with our analysis that large RyR clusters increases the probability for the simultaneous openings of RyRs (Fig. 1 A, black). The nonspark Ca leak rate is larger in large RyR clusters than small ones (Fig. 2, green, solid versus dashed). This is due to the higher probability of openings of only one or two RyRs in large RyR clusters. We note that these qualitative effects of RyR cluster size still exist even without luminal Ca regulation of RyR, although the curves are less steep and slightly shifted along the abscissa (Fig. S3) compared to the steeper relationship when intrinsic [Ca]SR dependence is included. Thus, this phenomenon is an intrinsic property of the RyR activation process (even the most parsimonious versions). We then tested how Ca spark and nonspark Ca leak properties depend on RyR cluster size. In the detailed model, we cannot directly determine the probability from closed to open during the secondary opening (PC / O). Because PC / O is determined by [Ca]Cleft after the initial opening, which is governed also by [Ca]SR (via mean open time and driving force, Fig. S2), we can also assess PC / O as a function of fixed [Ca]SR. We simulated two [Ca]SR levels (580 and 660 mM) to illustrate Ca spark and nonspark leak at the small and large PC / O shown in our analysis (Fig. 1). The mean nonspark leak increases linearly with RyR cluster size but falls from linearity at large RyR cluster size with high [Ca]SR (Fig. 3 A). The mean Ca spark frequency increases steeply with RyR cluster size, with a steeper dependence of RyR cluster size at higher [Ca]SR (Fig. 3 B, red versus black). These are consistent with our theoretical predictions (Fig. 1, D and E). We also calculated the mean spark amplitude and duration for different RyR cluster sizes.

probability of Ca sparks steeply increases with RyR cluster size (Fig. 1 E); the probability of nonspark leak, however, increases almost linearly at small RyR cluster size (<50) but then decreases when cluster size becomes large (>150) (Fig. 1 D) as more Ca sparks occur. That is more obvious at large secondary RyR open probability (Fig. 1 D, red). In fact, this is expected from the probability distribution of secondary RyR opening, in which large RyR cluster size favors simultaneous RyR openings (Fig. 1 A, black). It suggests that nonspark Ca leak tends to turn into sparks at either high [Ca]SR or large RyR clusters. Quantification by computer simulations

Based on the above predictions, we extended our simulations to a physiologically detailed ventricular myocyte model of 19,305 CRUs of 100 RyR each. We first tested the Ca spark and nonspark leak rates on [Ca]SR in large versus small RyR clusters. Large clusters (100 RyRs) lead to larger spark rates and steeper release function compared

FIGURE 2 Ca flux rates and RyR cluster size. In large RyR clusters (100 RyRs, solid lines), the probability for the simultaneous openings of multiple RyRs increases, leading to larger spark rate and steeper release function than small clusters (30 RyRs, dashed lines) at a given [Ca]SR. Similarly, the nonspark leak rate is larger in large RyR clusters than small ones because of the higher probability of openings of only one or two RyRs. Black, red, and green indicate the total Ca release, Ca sparks release, and nonspark leak, respectively. To see this figure in color, go online.

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Xie et al.

FIGURE 3 Dependence of spark and nonspark leak on RyR cluster size in simulations. (A) Nonspark leak rate linearly increases with RyR cluster size but tends to decrease at large RyR cluster and high [Ca]SR (red). (B) Spark frequency steeply increases with RyR cluster size, showing a steeper release function in higher [Ca]SR (red versus black). Both spark amplitude (C) and duration (D) increases with RyR cluster size as more RyRs are recruited during a spark. (E) Total Ca leak rate increases steeply with RyR cluster size. Red and black indicate the average cell [Ca]SR fixed at 660 and 580 mM, respectively. The error bars show the SD over # simulations/clusters. To see this figure in color, go online.

Spark amplitude increases with RyR cluster size but flattens or even decreases in large RyR cluster size (>200) when [Ca]SR is high (Fig. 3 C). The dependence of spark amplitude on RyR cluster size is affected by [Ca]SR via driving force and RyR sensitization. Therefore, spark amplitude increases gradually with RyR cluster size at small [Ca]SR, and this dependence becomes steeper at a large [Ca]SR (Fig.3 C, red versus black). The mean spark duration also gradually increases with RyR cluster size but decreases at high [Ca]SR when RyR cluster size is large (Fig. 3 D). These results altogether lead to a steep dependence of total Ca leak rate on RyR cluster size (Fig. 3 E). We also simulated line-scan measurements and compared it with experimental results. In the simulated line scan of a model myocyte in which large (Fig. 4 A, red circles) and small RyR clusters coexist, Ca release shows different spark amplitude and frequency at large clusters versus small ones. More frequent and larger sparks occur at large RyR cluster sites (Fig. 4 A, red circles; a typical spark is shown in Fig. 4 Ac). Small Ca sparks and nonspark leak tend to occur at small cluster sites less frequently (Fig. 4 A, a and b). We also examined experimental line scans of Ca sparks (experimental methods are as in our companion article (21)). We found similar phenomena (i.e., large Ca sparks) (Fig. 4 B) occurred more frequently in large RyR clusters (marked by higher red fluorescent FKBP12.6 that labels RyRs stoichiometrically). Spark amplitude saturated at a smaller RyR cluster size in experiments than in simulations (21) but was in general agreement with the theoretical curves in Fig. 3 C. Small Ca sparks tend to occur at small RyR cluster sites (indicated by low intensity of FKBP12.6 fluorescence) with fewer incidences (Fig. 4 Ba; see further experimental data analysis in (21)). Effect of RyR cluster size on the diastolic cleft [Ca] SR Ca leak is critical for Ca wave generation. An increased Ca leak can be caused by elevated [Ca]SR or local diastolic

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[Ca]Cleft, and this can precipitate Ca waves and arrhythmias in many cardiac pathologies, such as heart failure and catecholaminergic polymorphic ventricular tachycardia (CPVT). Elevated diastolic [Ca]Cleft is also expected to influence the

FIGURE 4 Line scans showing sparks occur more frequently at large RyR cluster sites. (A) The longitudinal direction of the myocyte model is taken for line scan. Red circles indicate the large RyR cluster sites, and the others are the small ones. More frequent and larger sparks occur at large RyR cluster sites. (a) Small Ca spark and (b) quark nonspark leak at small cluster sites and (c) large Ca spark at large cluster site. More detailed spatial and temporal profiles are shown in Fig. S4. (B) Line scan shows similar phenomena in experiment. Left is an FKBP12.6 image of a myocyte, which shows the longitudinal direction for line scan (white line); in the middle, the line scan shows different Ca spark release (green) at different RyR cluster size (indicated by the intensity of red lines). The area inside the white frame was zoomed in to show the difference of RyR cluster size and Ca spark release (right panel). Small and large Ca sparks were shown on the top; the associated number indicates their incidences. To see this figure in color, go online.

Please cite this article in press as: Xie et al., Size Matters: Ryanodine Receptor Cluster Size Heterogeneity Potentiates Calcium Waves, Biophysical Journal (2019), https://doi.org/10.1016/j.bpj.2018.12.017

RyR Cluster Size Heterogeneity

activation of calmodulin, calcineurin, and Ca-Calmodulindependent protein kinase that are also concentrated in this domain. Indeed, time-averaged [Ca]Cleft was directly measured to be at least twofold higher than [Ca]Bulk even during rest (24). We thus further examined how RyR cluster size affects diastolic [Ca]Cleft and bulk [Ca] ([Ca]Bulk). In resting myocytes, [Ca]Cleft is determined by the rates of local SR Ca leak, extrusion via Na/Ca exchange (NCX), and cytosolic Ca diffusion out of the cleft, as well as SR Ca uptake (in which SERCA is more globally distributed). This creates a standing [Ca]i gradient that has been directly measured (24). In the absence of L-type Ca channel influx at the resting membrane potential, Fig. 3 E shows that the SR Ca leak rate depends on the RyR cluster size. Therefore, the time average of [Ca]Cleft rises as the cluster size increases (Fig. 5 A, red lines). Because of Ca diffusion between the cleft space and the bulk cytosol, time-averaged [Ca]Bulk also increases with RyR cluster size but much more gradually (Fig. 5 A, black lines). [Ca]Bulk in this scenario is limited in part by Ca extrusion from the cell via NCX. The SR Ca leak becomes larger as the SR Ca load increases (22,23,25–27), and our simulations also showed that as the SR Ca load increases (here by increasing SERCA activity), the divergence between [Ca]Cleft and [Ca]Bulk is larger (Fig. 5 A, solid versus dashed curves, and Fig. S5 A). This is due to an increased Ca leak rate at both higher [Ca]SR and larger RyR clusters (Fig. 3, A and E). The higher SERCA activity explains why [Ca]Bulk and [Ca]Cleft start lower at the left in the 700 vs. 600 mM simulation (i.e., net Ca was shifted from cytosol to SR).

Spatial distribution of NCX molecules can affect SR Ca leak via effects on [Ca]Cleft (28). Fig. 5 B illustrates how NCX distribution can modify the ratio of [Ca]Cleft to [Ca]Bulk as a function of RyR cluster size. In our rabbit ventricular myocyte model, 11% of NCX is at the cleft (assuming uniform NCX sarcolemmal distribution; i.e., 11% of the sarcolemma is in clefts) with 89% of NCX in sarcolemma outside the cleft. Because RyR cluster size increases [Ca]Cleft much faster than [Ca]Bulk, the [Ca]Cleft/[Ca]Bulk ratio increases (Fig. 5 B, red). Shifting all NCX out of the cleft has only a slight effect on this result, and in this case, efflux from cleft is exclusively via Ca diffusion. Therefore, the [Ca]Cleft/[Ca]Bulk ratio becomes slightly larger (Fig. 5 B, black and Fig. S5 B). In the unlikely extreme that all of the cell’s NCX is packed into the cleft, efflux via NCX would increase and the [Ca]Cleft/[Ca]Bulk ratio would be smaller, such that [Ca]Cleft could even fall below [Ca]Bulk, particularly at small RyR clusters (Fig. 5 B, green and Fig. S5 D). [Ca]Cleft/[Ca]Bulk is greater than 1.0 for typical RyR cluster (#RyRs100) (Fig. 5 B). For clusters or [Ca]SR larger than the average, the [Ca]Cleft/[Ca]Bulk can easily reach 2, as was measured experimentally (24). The distribution of RyR cluster size is not uniform. For example, the distribution may be Gaussian (8,21). Because the increase of [Ca]Cleft along with the increase of RyR cluster size is relatively linear, the distribution of the timeaverage [Ca]Cleft reflects the distribution of the RyR cluster size, showing a similar Gaussian distribution (Fig. 5 C). Here, we note that the average [Ca]Cleft at any moment is mainly determined by many (>1000) junctions having

FIGURE 5 Cluster size affects [Ca]Cleft. (A) The time average of both [Ca]Cleft and [Ca]Bulk increases as the cluster size becomes larger. [Ca]Cleft increases much more steeply with RyR cluster than [Ca]Bulk, and this difference between the time average of [Ca]Cleft and [Ca]Bulk becomes larger at higher [Ca]SR ([Ca]SR ¼ 600 vs. 700 mM). Solid lines represent [Ca]SR ¼ 700 mM. Dashed lines represent [Ca]SR ¼ 600 mM. Black represents [Ca]Bulk. Red represents [Ca]Cleft. (B). NCX location affects the dependence of [Ca]Cleft on RyR cluster size. As the percentage of NCX in the cleft space increases, the increase of [Ca]Cleft (by comparing to [Ca]Bulk) with RyR cluster becomes smaller. Black represents 0% NCX in the cleft space and 100% NCX in the submembrane space. Red represents 11% NCX in the cleft space and 89% NCX in the submembrane space. This is a physiological distribution. Green represents 100% NCX in the cleft space and 0% NCX in the submembrane space. (C) The time-averaged [Ca]Cleft distribution. When a spontaneous Ca spark occurs, large [Ca]Cleft can be observed. ((D) shows a snapshot [Ca]Cleft distribution when a spontaneous Ca spark occurs). To see this figure in color, go online.

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Please cite this article in press as: Xie et al., Size Matters: Ryanodine Receptor Cluster Size Heterogeneity Potentiates Calcium Waves, Biophysical Journal (2019), https://doi.org/10.1016/j.bpj.2018.12.017

Xie et al.

single RyR openings (or nonspark leak) and small [Ca]Cleft elevation (<400 nM) versus the rare capture of a Ca spark occurrence (in which [Ca]Cleft reaches >50 mM for a few milliseconds; right side of Fig. 5 D). Heterogeneous distribution of RyR clusters influences Ca wave formation Our results above showed that larger RyR clusters promote Ca spark fluxes by increasing spark frequency, amplitude, and duration as well as the diastolic [Ca]Cleft and [Ca]Bulk. This large Ca flux from the large RyR cluster in intact myocytes can in turn decrease [Ca]SR, which would tend to limit Ca waves (26,29). We calculated the steady-state [Ca]SR in myocytes in which all RyR clusters were the same size. As expected, steady-state [Ca]SR decreases as the RyR cluster size increases (Fig. 6 A). This can be understood by Ca flux balance (25,30), such that at steady state, the Ca uptake rate (influx) and leak from the SR (efflux) must be balanced. The increased total Ca leak rate with large RyR clusters (Fig. 3 E) initially exceeds the uptake rate, causing a decline in [Ca]SR, which reduces SR Ca leak (and increases SR Ca uptake because of higher [Ca]Bulk) until uptake and leak reach steady-state balance. At this point, SERCA flux rises to balance the leak, but the leak is a bit lower because [Ca]SR is lower. The extent to which [Ca]Bulk and [Ca]Cleft are elevated depends upon competition between SERCA and NCX (and Ca influx across the sarcolemma). Therefore, similar Ca spark rates occur at a lower [Ca]SR in myocytes with large RyR clusters than those with small RyR clusters (Fig. 6 B, red versus blue line).

As a result, the low [Ca]SR in myocytes with homogeneous distribution of large RyR clusters may limit Ca wave initiation and propagation. On the other hand, in myocytes with homogeneous distribution of small RyR clusters, neither the spark amplitude nor frequency is sufficient to initiate Ca waves despite high [Ca]SR. To test our hypothesis that a mixture of both large and small RyR clusters can potentiate Ca waves, we simulated myocytes with a mixture of larger (#RyR ¼ 100) and smaller (#RyR ¼ 60) RyR clusters (Fig.6 C, red and blue, respectively). As expected, Ca waves initiate (Fig. 6 D, top) from locations in which large RyR clusters are concentrated, such that CICR can successfully propagate to small RyR clusters, forming a cell-wide Ca wave (Fig. 6 D). In contrast, Ca waves fail to initiate in myocytes with either all large (#RyR ¼ 100) or all small (#RyR ¼ 60) RyR clusters (Fig. 6 E). To demonstrate that Ca wave initiation and propagation in the mixture are due to the heterogeneity of RyR cluster size rather than the difference in the total number of RyRs in the cell, we also simulated a homogeneous myocyte with an intermediate RyR cluster size (#RyR ¼ 75) so that the total number of RyRs is the same as in the mixture (Fig. 6 B, green). At steady state, the intermediate Ca spark rate sets the SR load at an intermediate level, which is not sufficiently high to initiate Ca waves in this homogeneous myocyte because of the inadequate Ca spark rate and relatively shallow release curve (Fig. 6 B, green circle). However, in myocytes with mixed RyR cluster size, this level of Ca load (because of the small Ca efflux from small RyR clusters) sets a sufficiently large spark rate for the large clusters (on the steep part of the

FIGURE 6 Ca waves occur in a myocyte model with heterogeneous RyR cluster size. (A) Because more sparks and nonspark leaks occur in large RyR clusters, [Ca]SR decreases as the cluster size increases. Blue, red, and green circles, respectively, indicate homogeneous cells with small, large, or medium (which is the average between large and small, as shown in D) RyR clusters. (B) In a myocyte with homogeneous distribution of large RyR clusters, [Ca]SR is too low to initiate waves (red circle), whereas in a myocyte with homogeneous distribution of small ones, sparks are too small to initiate waves (blue circle). A mixture of large and small clusters increases [Ca]SR enough for large RyR clusters to initiate waves (light green circle). This is not because of the difference in total number of RyRs in the cell but the heterogeneity in RyR clusters. A homogeneous distribution of RyR clusters with the same total RyRs also fails to initiate Ca waves (green circle). Red, green, and blue circles are corresponding to those in (A). (C) The corresponding twodimensional distribution of underlying RyR clusters that potentiates Ca waves; red indicates large clusters, and blue indicates small ones. (D) Ca waves initiate from large RyR cluster sites and propagate into the whole cell. The time of the Ca snapshots is indicated to the right. (E) Probability of full waves. We simulated 20 times for each case with different random numbers. No full waves were observed when the size distribution is homogeneous. On the other hand, full waves were always observed when small and large clusters were mixed. To see this figure in color, go online.

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RyR Cluster Size Heterogeneity

leak-load curve, red) that promotes the initiation of reproducible Ca waves (Fig. 6 B, dashed light green circle). We note that Ca sparks and nonspark leak deplete SR Ca locally. However, these local depletions are quickly equalized by Ca flux from neighboring sites. As intra-SR Ca diffusion is fast (31,32), SR Ca nonuniformity is small before the onset of Ca waves (Fig. S6). The Ca waves typically initiate near the center of larger clusters, in which sufficiently large releases occur. If we artificially slow intra-SR Ca diffusion dramatically, waves can still occur but sometimes initiate at the border between large and small clusters. DISCUSSION In this study, we demonstrated that heterogeneous distribution of RyR cluster size can potentiate Ca sparks and waves using both theoretical analysis and computational modeling. Compared to small RyR clusters, large RyR clusters result in higher Ca leak rates because of much more frequent Ca sparks with somewhat large amplitudes and higher nonspark leak, leading to higher diastolic [Ca] at both cleft and bulk spaces. This sets a high vulnerability to Ca waves for large RyR clusters, but this tendency is counterbalanced by the reduced [Ca]SR that is a consequence of this high leak. That is, the lower [Ca]SR reduces the driving [Ca]i gradient for CICR propagation and also RyR sensitivity, which reduce the Ca that drives CICR and the excitability of neighboring junctions, respectively. Thus, having all large RyR clusters is not particularly arrhythmogenic. On the other hand, small RyR clusters lead to higher [Ca]SR because of the lower Ca leak rate and low diastolic Ca level, which also limits Ca wave occurrence (high [Ca]SR is counterbalanced by low excitability). A heterogeneous distribution of both large and small RyR clusters can, however, potentiate Ca waves in such a way that smaller RyRs help maintain cell average [Ca]SR, allowing larger clusters to initiate Ca waves. Ca waves propagate via a ‘‘fire-diffuse-fire’’ CICR mechanism (6,7). To initiate and maintain a Ca wave, the Ca spark at the initial firing site must be sufficiently large so that the Ca diffusing to the neighboring sites can recruit new sparks. Also, higher Ca spark frequency can enhance Ca wave propensity by increasing the probability of both waves and macrosparks that facilitate wave propagation (6,33). Diastolic [Ca] is critical for Ca wave initiation and propagation. An increased Ca leak is associated with Ca waves in many pathological conditions, such as heart failure (25,27,34,35) and CPVT (25,36), in which RyR sensitivity is increased. The higher [Ca] in both bulk and cleft spaces (Fig. 5) for large RyR clusters (because of large nonsparkmediated leak) sets a similar scenario for Ca waves, even without pathological increases of RyR sensitivity. If all RyR clusters are homogeneously large, steady-state [Ca]SR may be below the Ca wave threshold (preventing Ca waves; Fig. 6 B) because of the large leak rate. If many or

most of the RyR clusters are small, that can elevate the cell average [Ca]SR because of their small Ca leak and thus push [Ca]SR to exceed the Ca wave threshold for large clusters. This elevation of [Ca]SR is also required to cause Ca waves and arrhythmias in heart failure and CPVT, usually due to the sympathetic challenges (36,37). Our study shows how heterogeneous distribution of both large and small clusters of normal RyRs can also potentiate Ca waves. In normal myocytes, a large variation of the RyR cluster size has been shown by electron microscopy data, from 30 to 260, regardless of species (11). Recent experimental data also found a wide range in RyR cluster size (from 0 up to 500) in data of both humans and rats (12). Our study showed that this heterogeneity itself can potentiate Ca waves. Experimental measurements often exhibit Ca ‘‘hotspots,’’ in which Ca sparks occur more frequently than elsewhere, which are prone to become Ca wave initiation sites (15,38). Our study provides a structural mechanism that may underlie Ca waves initiation at hotspots. Moreover, our experiments show that larger RyR clusters are prone to be Ca spark hotspots (21). In some pathological conditions, detubulation may increase cluster heterogeneity (39,40) or the endogenous spatial heterogeneity in RyR cluster size can be further amplified by local [Ca]SR, local RyR phosphorylation, or local geometric preferences (38). In these conditions, the RyR cluster heterogeneity may significantly decrease their tolerance to sympathetic challenge and potentiate Ca waves and thus arrhythmias. In parallel to the wide range of RyR cluster size, Ca sparks often exhibit a wide range of variation in both amplitude and duration (14–16). Although variations of spark dynamics can be due to stochastic variance of RyR opening (16,41), different RyR cluster sizes could play a role (15), and repeated sparks from a single site tend to be consistent in size and shape (suggesting structure as a determinant (42)). Our study provides a quantitative framework to understand the variation of RyR cluster size and Ca spark dynamics, in which a large RyR cluster results in larger and more frequent Ca sparks (Figs. 1 D and 3 A). This is consistent with a previous modeling study in which spark amplitude increases with the number of opened RyRs in a cluster (17), and the decreased RyR availability by ruthenium red (functionally decreasing RyR cluster size) reduces spark frequency and amplitude (43). Recent modeling results suggest that Ca spark frequency steeply increases with RyR cluster size at a small size but then saturates at large RyR cluster size (19) or decreases monotonically with RyR cluster size (18). Discrepancies with our results might be partly due to their inclusion of allosteric RyR coupling in their cluster models. Our modeling results have also been verified by our experimental data (21), which demonstrated that the CICR among RyRs in individual clusters is sufficient to form the synergistic effect on Ca release and thus monotonic increase in Ca spark frequency with cluster size.

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Xie et al.

In our simulations, the spark amplitude tends to saturate as the cluster size increases (Fig. 3 C, red). This is due to partial local depletion of [Ca]SR near the junction as measured directly (32) that contributes to the termination of local SR Ca release by two known mechanisms (4,25): 1) RyR deactivation as local [Ca]SR declines (luminal Ca effect) and 2) decreased [Ca]cleft-regulated CICR activation (because of reduced [Ca]SR driving force to maintain [Ca]cleft). The first is well supported by much compelling bilayer data of luminal [Ca]-dependence of Po at fixed cytosolic [Ca] (44–49). The second is an implicit consequence of local [Ca]SR decline as the release proceeds, as included in multiple models (4,20,50), in which the impact depends upon the severity of local [Ca]SR depletion. This combination provides a robust shutoff of SR Ca release without fully depleting the SR. As we show here, increasing RyR cluster size has the same qualitative effect on SR Ca leak and sparks even when the luminal [Ca] dependence of gating is turned off, but having both mechanisms in play steepens the impact of RyR cluster size on arrhythmogenic SR Ca leak. In this study, all RyRs in the same cluster sense the same [Ca]SR. If more details of RyR location within clusters are included in the model, the saturation of the spark amplitude and the shortening of duration may be altered. However, our experimental data show remarkable Ca sparks amplitude stability versus RyR cluster size (21). Another structural factor that may limit local SR Ca release is how well a particular junctional SR region is diffusionally coupled to the network of SR because that also governs the extent of local [Ca]SR decline for a given release flux (32). Ca spark properties can be affected by many factors. For example, the different termination mechanisms could affect spark duration and thus leak rates and SR Ca load. However, our conclusions are driven mainly by the positive feedback between RyRs within the cluster during the initiation process, so termination mechanisms do not change our qualitative conclusions. This remains true for the luminal regulation of RyR gating in our model because of similar reasons. The RyR luminal regulation is strong at high cytosolic [Ca] compared to some experimental data, but the RyR open probability at diastolic [Ca]i of 100 nM changes only two to threefold ((from 3 to 6
107) with increasing [Ca]SR from 200 to 800 mM) (4). As we are focused on initial conditions in which [Ca]Cleft is resting (100 nM), and our conclusions are drawn from the positive feedback in the activation process, the luminal Ca regulation of RyR does not impact our mechanistic conclusions. We verified this in simulations without [Ca]SR dependence (Fig. S3), in which only quantitative not qualitative differences were observed. RyR gating is also modulated by several factors, including binding of calmodulin or FKBP12.6, oxidation, and phosphorylation (51–53). In this sense, the higher RyR sensitivity seen in heart failure (22,27) or CPVT (36) would promote arrhythmogenic Ca waves by mimicking high RyR cluster sizes. Moreover, if

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RyR distribution, phosphorylation, or oxidation are spatially heterogeneous in pathological states, that could also increase the propensity for initiation sites for arrhythmogenic Ca waves in the same way as the wave initiation in Fig. 6 D. The new framework that we have developed here, focusing on the functional influence of RyR cluster size, is thus translatable to additional modes of RyR functional modulation. We did not consider the spatial arrangement of RyRs within a cluster but made the simplifying assumption that each RyR within a CRU senses the same [Ca]cleft. Recently, Walker et al. (19) modeled such local RyR arrangement derived from super-resolution microscopy data to test effects on Ca sparks. They showed that this effect dominates only among similarly sized RyR clusters but that the number of RyRs plays a major role on spark rates even when spatial arrangements of RyRs and Ca gradients within dyads are simulated. Because different spatial distributions (e.g., Gaussian and exponential) of RyR cluster size may exist in different cells, species, or conditions (8,13), and Ca diffusion and the distance between RyR clusters may change as T-Tubules reorganize in pathological states (39,40), future studies could investigate how these factors affect Ca wave initiation and propagation. SUPPORTING MATERIAL Supporting Materials and Methods, six figures, and three tables are available at http://www.biophysj.org/biophysj/supplemental/S0006-3495(18)34531-4.

AUTHOR CONTRIBUTIONS All authors contributed ideas and discussion. Y.X. performed computer simulations and mathematical analysis. Y.Y. and S.G. performed wet experiments. All authors wrote the manuscript, approved the final version of the manuscript, and agree to be accountable for all aspects of the work. All persons designated as authors qualify for authorship, and all those who qualify for authorship are listed.

ACKNOWLEDGMENTS This work was supported by National Institutes of Health grants R01HL30077 (D.M.B.), R01-HL105242 (D.M.B.), and R00-HL111334 (D.S.), American Heart Association grant 16GRNT31300018 (D.S.), and Amazon AWS Cloud Credits for Research (D.S.).

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