Effects of rogue ryanodine receptors on Ca2+ sparks in cardiac myocytes

Ca2+ sparks and Ca2+ quarks, arising from clustered and rogue ryanodine receptors (RyRs), are significant Ca2+ release events from the junctional sarcoplasmic reticulum (JSR). Based on the anomalous subdiffusion of Ca2+ in the cytoplasm, a mathematical model was developed to investigate the effects of rogue RyRs on Ca2+ sparks in cardiac myocytes. Ca2+ quarks and sparks from the stochastic opening of rogue and clustered RyRs are numerically reproduced and agree with experimental measurements. It is found that the stochastic opening Ca2+ release units (CRUs) of clustered RyRs are regulated by free Ca2+ concentration in the JSR lumen (i.e. [Ca2+]lumen). The frequency of spontaneous Ca2+ sparks is remarkably increased by the rogue RyRs opening at high [Ca2+]lumen, but not at low [Ca2+]lumen. Hence, the opening of rogue RyRs contributes to the formation of Ca2+ sparks at high [Ca2+]lumen. The interplay of Ca2+ sparks and Ca2+ quarks has been discussed in detail. This work is of significance to provide insight into understanding Ca2+ release mechanisms in cardiac myocytes.

Zima et al. [11] demonstrated experimentally that single RyR opening could mediate Ca 2+ leak but fails to trigger Ca 2+ sparks when free Ca 2+ concentration in the JSR lumen ([Ca 2+ ] lumen ) is below a threshold level. On the other hand, mathematical models have been developed to investigate the dynamics of Ca 2+ sparks and quarks. Sobie et al. [12] proposed a model to solve the paradox of [Ca 2+ ] lumen to support the existence of rogue RyRs. Models of Ca 2+ leak from JSRs have been used to characterize differences between Ca 2+ sparks and quarks [13]. Walker et al. [14] included the computational nonspark-based Ca 2+ leak from JSRs to explain the exponential rise of Ca 2+ spark frequency. They further presented that Ca 2+ sparks could be triggered by spontaneous opening of a single RyR in a cluster [15]. A stochastic contact network model of the Ca 2+ initiation process was applied to realistic RyR cluster structures, which revealed that the Ca 2+ sparks probability depends on the position of the initial RyR in the cluster. Their works provided insight into the Ca 2+ release process in the heart and a framework for evaluating functional heterogeneity in populations of receptor clusters under normal and pathological conditions. Lu et al. [16,17] investigated 'nonjunctional rogue RyRs' located away from the release sites and showed their effects on Ca 2+ waves with heart failure. It suggested that Ca 2+ dynamics was unstable and Ca 2+ waves were likely to be triggered when 'non-junctional rogue RyRs' were taken into consideration. The variation of membrane potential depolarization was indicated to be dependent on the distribution density of rogue RyR channels, which is important for understanding the arrhythmogenic mechanism for heart failure from the subcellular to cellular level. Sato and Bers [18] used a mathematical model of JSR Ca 2+ releases to show that single RyR opening at low [Ca 2+ ] lumen could not recruit Ca 2+ sparks from Ca 2+ release units (CRUs). However, the effects of rogue RyRs on Ca 2+ sparks at high [Ca 2+ ] lumen remain unknown. The anomalous subdiffusion of cytoplasmic Ca 2+ and the random distribution of JSR RyRs, unnoticed in these models, could unveil the temporal and spatial properties of Ca 2+ sparks and quarks.
The objective of the study is to develop a mathematical model of JSR Ca 2+ release to quantify the interplay of rogue and clustered RyRs for a detailed explanation of spontaneous Ca 2+ sparks and quarks in cardiac myocytes under physiological conditions. A mathematical model was proposed to simulate the temporal and two-dimensional (2D) spatial distributions of Ca 2+ sparks and quarks in a cardiac myocyte with consideration of the distribution of clustered and rogue RyRs on the JSR membrane and the anomalous subdiffusion of Ca 2+ in the cytoplasm [19]. The model could explain various Ca 2+ release events from a JSR and predict the firing probability of clustered RyRs activated by Ca 2+ release through rogue RyRs. The stochastic opening of rogue and clustered RyRs was regulated by free Ca 2+ concentrations in both cytoplasm and JSR lumen of a cardiac myocyte. The line-scan experimental measurements were carried out in cardiac myocytes of rats to validate the present model. The significance was discussed to improve the understanding of Ca 2+ -induced-Ca 2+ -release events in cardiac myocytes.

Experimental methods
Similar to previous studies [20][21][22], eight Sprague-Dawley rats (about 2.5 months old, 225-300 g) were anaesthetized with pentobarbital sodium (40 mg kg −1 ) by peritoneal injection. Hearts were rapidly excised from animals. An isolated heart was immediately put into ice cold buffer, mounted in a Langendorff system, and perfused with a Ca 2+ -free buffer containing (in mM): 137 NaCl, 5.4 KCl, 1.2 MgCl 2 , 1.2 Na 2 HPO 4 , 20 HEPES, 10 taurine, and 10 glucose (at a pH value of 7.35, aerated with 95% O 2 and 5% CO 2 ) for 5 min. The heart was then digested in a buffer containing 0.5 mg ml −1 collagenase, 1 mg ml −1 bovine serum albumin, 0.06 mg ml −1 protease (type X-IV) and 50 µM CaCl 2 until becoming pale. The left ventricle was sectioned into small pieces and incubated in the digesting solution. Myocytes were harvested and stored in Tyrode's solution containing (in mM): 135 NaCl, 1 CaCl 2 , 4 KCl, 1.2 MgCl 2 , 1.2 Na 2 HPO 4 , 10 glucose, and 10 HEPES with a pH value of 7.35. Before imaging, myocytes were loaded with the dye Fluo-4-AM for 5 min and washed twice using the Tyrode's solution. A confocal microscope (Nikon A1+, Japan) equipped with a 40 × 1.3 NA oil immersion objective was used for line-scan images at a sample rate of 512 frames s −1 . The samples were excited at 488 nm. All line-scan measurements were performed at room temperature (23-25°C). Images were processed by the SPARKMASTER software [23].

Geometrical model
In a 2D model similar to Izu et al. [24], the regular intervals between Ca 2+ release sites are l x (= 2 µm) in the longitudinal direction (x-axis) and l y (= 0.8 µm) in the transverse direction (y-axis). Figure 1a shows the geometrical model of a cardiac myocyte. Each Ca 2+ release site represents a JSR. The schematic representative of a JSR is shown in figure 1b, which includes randomly distributed clustered RyRs and rogue RyRs. A CRU of clustered RyRs has 22 RyR channels and a rogue RyR has 3 RyR channels [5]. Figure 1c shows the distribution of clustered and rogue RyRs on a JSR for simulation. CRUs of clustered RyRs (blue dots, approx. 2 in a JSR) are surrounded by randomly distributed rogue RyRs (red dots, approx. 8 in a JSR). The number and location of clustered and rogue RyRs in each JSR are random in simulations.

Computational model
The balanced equation for free Ca 2+ concentration in the cytoplasm, [Ca 2+ ] cyto , with consideration of the anomalous subdiffusion of Ca 2+ in the cytoplasm and the distribution of clustered and rogue RyRs, can be written as: where t is time, x and y denote the spatial coordinates, D x (= 300 µm 2 s −1 ) and D y (= 150 µm 2 s −1 ) refer to the Ca 2+ diffusion coefficients. The anomalous subdiffusion order β is 2.25. Anomalous space subdiffusion corresponds to the short jump of the random walker and is defined though the relations [  where Γ denotes the Gamma function. J dye-cyto and J buffer-cyto are the fluxes due to the Ca 2+ fluorescent indicator dye (i.e. Rhod-2 or Fluo-4-AM, in the cytoplasm) and the endogenous stationary buffers respectively. J pump is the pumping rate of SR Ca 2+ -ATPase. SR pumps are started when [Ca 2+ ] cyto exceeds the resting Ca 2+ concentration level (0.1 µM) in the cytoplasm. These variables can all be defined as: and where the subscript 'n' refers to each buffer in the cytoplasm, the superscript 'h' refers to the Hill constant; [F] T and [B n ] T represent the total concentrations of indicator and buffers, respectively.
[CaF] and [CaB n ] are the concentrations of Ca 2+ -bound complexes; k + F , k − F , k + n and k − n are the reaction kinetic parameters. K pump is the affinity constant, and V max pump is the maximum rate for SR pumps. Values of the parameters are based on a previous study [26]. Moreover, J rogue and J clustered are the Ca 2+ fluxes released by rogue and clustered RyRs respectively, which can be written as: and where δ is the Dirac delta function and S is a stochastic function for the opening of clustered and rogue )are the positions of rogue RyRs and clustered RyRs in the 2D plane respectively. The release times for rogue and clustered RyRs are defined as T rogue = 20 ms [10] and T clustered = 10 ms [27]. The equivalent source strength of rogue RyRs and clustered RyRs [28]   Various parameters of dyes (Rhod-2 [30], Fluo-4-AM [26] and Fluo-5N [31]) and buffers [26,32] in the cytoplasm and JSR lumen are listed in table 1.

Firing probability of rogue and clustered RyRs
The firing probability per unit time of RyRs is determined by Ca 2+ concentrations in both cytoplasm and JSR lumen [33][34][35], which can be expressed as P firing = P cyto · Φ lumen , (2.14) where P cyto refers to the firing probability per unit time of calcium release events controlled by [Ca 2+ ] cyto . Φ lumen represents a [Ca 2+ ] lumen -dependent regulation term of Ca 2+ release events. According to the coupled RyR gating model [5], P cyto can be expressed as where P RyR is the firing probability per unit time of a single RyR channel [36]. Here, Φ lumen is written as where φ is an empirical power function given in Walker et al.'s model [14], m is the regulation coefficient for rogue (m = 1) or clustered (m = 10) RyRs.

Results and discussion
3.1. Ca 2+ quarks and Ca 2+ sparks Figure 2a shows a computational Ca 2+ quark through a rogue RyR to mimic the line-scan measurements when the release time is set to 20 ms. The computational domain is a square of 5 × 5 µm 2 with the distribution of clustered and rogue RyRs on the JSR membrane in figure 1c. The dyes, Rhod-2 and Fluo-5N, were used to indicate Ca 2+ in the cytoplasm and JSR lumen, respectively. The shape of the Ca 2+ quark is consistent with that in a previous study [10]. Figure 2b plots the time courses of a QCR-QCD pair (i.e. a quarky Ca 2+ release-quarky Ca 2+ depletion pair) corresponding to figure 2a. The values of t 67 and F/F 0 were computed to be 22.0 and 22.5 ms and 0.065 and 0.025 for QCR and QCD, respectively. They are within 1 s.d. of experimental measurements, i.e. t 67 = 20.1 ± 1.1 ms for a QCR and 20.8 ± 1.9 ms for a QCD and F/F 0 = 0.069 ± 0.006 for a QCR and 0.025 ± 0.002 for a QCD [10]. Figure 2c shows the line-scan measurements of Ca 2+ release events in an isolated myocyte. The arrows refer to Ca 2+ sparks due to the firing of clustered RyRs after QCR events owing to the opening of rogue RyRs, which were further analysed using the SPARKMASTER software [23] in figure 2d. The peak F/F 0 of the two Ca 2+ spark is 1.02 and 1.44 in figure 2d (1) and (2), respectively. To avoid the background noise, we did not measure QCR events with F/F 0 < 0.2. The number of recorded Ca 2+ sparks is 1125 in all measurements. The proportion of sparks that are triggered by QCRs is approximately 11.6%. Accordingly, figure 2e shows the computational results of Ca 2+ sparks in a JSR with random distribution of clustered and rogue RyRs in figure 1c. The regular Ca 2+ spark (left) due to an opening CRU of clustered RyRs is initiated by three opening rogue RyRs. The large spark (right) results from two CRUs of clustered RyRs activated by four opening rogue RyRs. An agreement between experimental and computational results ( figure 2d versus figure 2e) validates the 2D mathematical model.

Interplay of rogue and clustered RyRs in a junctional sarcoplasmic reticulum
We simulated three modes of elemental Ca 2+ release events that could coexist at a Ca 2+ release site, as shown in figure 3a-c. Snapshots of elemental Ca 2+ release events in a computational domain of 5 × 5 µm 2 are taken at 10, 20 and 40 ms when three rogue RyRs are fired at the same time. Given the polymorphism of Ca 2+ sparks at a release site, there are three distinct modes: no activated Ca 2+ spark, a Ca 2+ spark with one quantal unit (i.e. a fired Ca 2+ spark from a CRU of clustered RyRs), and a Ca 2+ spark with two quantal units (i.e. a fired Ca 2+ spark from two CRUs of clustered RyRs). It demonstrates that QCRs from the opening rogue RyRs could activate the neighbour CRUs of clustered RyRs in a JSR to form a Ca 2+ spark with different quantal units. Moreover, rogue RyRs could be activated by the Ca 2+ release of clustered RyRs in the same JSR, which was recorded in the snapshots and shown in figure 3b,c. On the other hand, the amplitude of a Ca 2+ spark is mainly determined by the firing number of clustered RyRs regardless of rogue RyRs because QCR events have a low Ca 2+ flux.
This model explains the coexistence of the three modes of elemental Ca 2+ release in a Ca 2+ release site as well as predicting the properties of a Ca 2+ spark. QCRs do not always activate clustered RyRs because luminal Ca 2+ depletion reduces the probability of activating clustered RyRs. The probability of Ca 2+ sparks of the three modes activated by different initial opening numbers of rogue RyRs is shown in figure 4. The results reveal that clustered RyRs have the highest frequency of being triggered by four opening rogue RyRs simultaneously.

The effects of [Ca 2+ ] lumen on Ca 2+ release events
The effects of [Ca 2+ ] lumen on the firing frequency of Ca 2+ sparks and quarks are quantified in a computational domain of 20 × 20 µm 2 , as shown in figure 5. The beginning levels of [Ca 2+ ] lumen is set from 0.2 to 1.0 mM. The computational results show a higher firing frequency of spontaneous QCR events than that of spontaneous Ca 2+ sparks consistent with previous experimental observations [10]. A computational study also showed a steep increase in the firing frequency of spontaneous Ca 2+ sparks despite a slight change in the firing frequency of spontaneous QCR events with the increase of [Ca 2+ ] lumen [18]. Moreover, we show a threshold value of [Ca 2+ ] lumen (i.e. less than 0.3 mM) where spontaneous Ca 2+ quarks become a major pathway of SR Ca 2+ leak. The present study shows the incidence of 12. of sparks that are triggered by QCRs is 34.7% because one or two openings of rogue RyRs were neglected in experimental measurements in figure 2c.

Effects of rogue RyRs on Ca 2+ sparks with consideration of [Ca 2+ ] lumen
We examined how rogue RyRs affect Ca 2+ sparks at different levels of [Ca 2+ ] lumen . A comparison of computational line-scan Ca 2+ release events with consideration of rogue RyRs or not is displayed in a square of 20 × 20 µm 2 (the intervals between JSRs are l x = 2 µm and l y = 0.8 µm) for 2 s (figure 6a versus figure 6b)

A comparison with previous models
Sato et al. [18] showed that a RyR channel in a cluster of RyRs could trigger adjacent RyR channels in the same cluster to form a Ca 2+ spark when [Ca 2+ ] lumen is above a threshold. Walker et al. [14,15] further investigated the structural effects of clustered RyRs on Ca 2+ sparks. Moreover, Lu et al. [16,17] indicated that non-junctional RyRs increased the probability of occurrence of spontaneous Ca 2+ waves. The present study showed the temporal and spatial properties of Ca 2+ quarks and sparks relevant to both rogue and clustered RyRs, where rogue RyRs randomly surround clustered RyRs within a single junctional space. We showed that Ca 2+ quarks and sparks coexist at a Ca 2+ release site, which agrees with experimental measurements from line-scan imaging. The opening of rogue RyRs leads to the formation of Ca 2+ sparks at high [Ca 2+ ] lumen , but not at low [Ca 2+ ] lumen . This supports the conclusion of Sato et al. [18].
On the other hand, the present model simulated Ca 2+ quarks and sparks based on the anomalous subdiffusion in comparison with previous models from Fickian diffusion [14][15][16][17][18]  paradox of full width at half-maximum (FWHM) due to Fickian diffusion. This study addresses the importance of rogue RyRs for understanding Ca 2+ release mechanisms from JSRs.

Potential implications
Ca 2+ sparks could trigger clustered RyRs in neighbour JSRs with the help of rogue RyRs. This mode is marked by the arrow in figure 6a and shown in figure 7 schematically. Ca 2+ quarks may trigger the opening of clustered RyRs in self-propagating succession along the length of a cell.  the changes in the number of rogue RyRs in a JSR may induce potential heart diseases. For example, a reduction of the number of rogue RyRs could lead to an inhibition of Ca 2+ waves and dyssynchronous Ca 2+ transients in myocytes of congestive heart failure [39]. Atrial fibrillation associated with overactive Ca 2+ release could be related to the increased number of rogue RyRs [40].

Critique of the study
In the study, the duration and current of Ca 2+ release events from JSRs were fixed similar to previous studies [10,27]. However, Ca 2+ release flux should be regulated by the SR structure, functional properties and the size of RyR cluster [41]. The impact of time-dependent Ca 2+ release flux from RyRs can give us new inspiration for the relation between Ca 2+ release events and the interplay of rogue and clustered RyRs. The spatial arrangement of RyRs within clusters influences the frequency of Ca 2+ sparks [14]. The detailed structure of clustered RyRs should be taken into consideration when a high-performance supercomputer is used to satisfy the requirement of large computation. Furthermore, the present study comes from the assumption that 3D geometry is simplified to a 2D model in healthy myocytes. Modelling 3D distribution of the JSRs in cardiac cells is more realistic and the 3D simulations of Izu et al. [42] indicated that it could reveal more complex RyR interactions between neighbour JSRs. Hence, a 3D model should be developed to investigate spontaneous Ca 2+ release events under both physiological and pathological conditions in future studies. Ethics. All animal experiments were performed in accordance with Chinese National and Peking University ethical guidelines regarding the use of animals in research, consistent with the NIH guidelines (Guide for the care and use of laboratory animals) on the protection of animals used for scientific purposes. The experimental protocols were approved by the Animal Care and Use Committee of Peking University, China.