Simulation of Arrhythmogenic Effect of Rogue RyRs in Failing Heart by Using a Coupled Model

Abstract

Cardiac cells with heart failure are usually characterized by impairment of Ca²⁺ handling with smaller SR Ca²⁺ store and high risk of triggered activities. In this study, we developed a coupled model by integrating the spatiotemporal Ca²⁺ reaction-diffusion system into the cellular electrophysiological model. With the coupled model, the subcellular Ca²⁺ dynamics and global cellular electrophysiology could be simultaneously traced. The proposed coupled model was then applied to study the effects of rogue RyRs on Ca²⁺ cycling and membrane potential in failing heart. The simulation results suggested that, in the presence of rogue RyRs, Ca²⁺ dynamics is unstable and Ca²⁺ waves are prone to be initiated spontaneously. These release events would elevate the membrane potential substantially which might induce delayed afterdepolarizations or triggered action potentials. Moreover, the variation of membrane potential depolarization is indicated to be dependent on the distribution density of rogue RyR channels. This study provides a new possible arrhythmogenic mechanism for heart failure from subcellular to cellular level.

1. Introduction

Calcium is considered to be the key ion in mediating the process of cardiac excitation-contraction coupling (E-C coupling). Since the discovery of Ca²⁺ sparks in 1993 [1], Ca²⁺ sparks have been widely accepted to be the stereotyped elementary Ca²⁺ release events in the intact myocyte. Sparks arise via clusters of ryanodine receptors (RyRs) localized in the junctional SR (jSR) which is in close apposition to transverse tubules (TTs) [2]. In a diastolic myocyte, spontaneous Ca²⁺ sparks occur randomly at very low frequency, even in the absence of Ca²⁺ influx. During a single muscle twitch, Ca²⁺ influx via sarcolemmal L-type Ca²⁺ channels will trigger synchronously occurrence of thousands of sparks, summation of which in space and time causes a global steep rise of Ca²⁺ concentration named Ca²⁺ transient. However under some pathological conditions, successive recruitment of Ca²⁺ sparks tends to evolve into Ca²⁺ waves propagating across the myocytes which might trigger ventricular arrhythmias [3].

With the improvement of optical methods and innovative techniques, microscopic Ca²⁺ signals at the subcellular level have been extensively investigated and characterized. In addition to Ca²⁺ sparks via clustered RyRs, nonspark Ca²⁺ release events, named Ca quarks, activated by low-intensity photolysis of Ca²⁺-caged compounds [4] or by inward Na⁺ current, 𝐼_Na [5], could elicit spatially homogeneous but small Ca²⁺ transient. These quarks are likely to be mediated via one or a few RyR channels called rogue RyRs [6]. Differing from RyR clusters that underly sparks, rogue RyRs are thought to be uncoupled with each other and behave in ways more like the characteristic of single RyR channels [7]. Although detection of these small rogue RyR channels is difficult by conventional instruments, some researchers have suggested that, besides sparks, the nonspark pathway via rogue RyRs explains a part of SR Ca²⁺ leak [8, 9]. Quantitatively, with the optical superresolution technique, Baddeley et al. have indicated that there are greater numbers of rogue RyR groups than large RyR clusters [10]. An experimental study that Ca²⁺ waves are inhibited without affecting Ca²⁺ sparks by ruthenium red suggests a nonspark producing RyR channels which are important to propagation of Ca²⁺ wave [11]. Direct visualization of small local release events has been made possible by recent technical innovations. Brochet et al. claimed that they have directly visualized quark-like or “quarky” Ca²⁺ release events which might depend on the opening of rogue RyRs (or small cluster of RyRs) in rabbit ventricular myocytes [12].

SR Ca²⁺ leak consists of two components: RyR-dependent leak and RyR-independent leak [8]. The former is thought to be comprised of spark-mediated leak (visible leak) and non-spark-mediated leak (invisible leak). Elevated SR Ca²⁺ leak would contribute to delayed afterdepolarizations (DADs) and consequently arrhythmia in heart failure (HF) [13]. Besides spark-mediated leak, additional Ca²⁺ leak via rogue RyRs may be an important factor in disturbing Ca²⁺ dynamics and triggering Ca²⁺ waves [11, 14]. However, how do these abnormal Ca²⁺ release events affect cellular electrophysiological properties? The precise relationships between property of rogue RyRs and Ca²⁺ handling as well as cellular electrophysiology in failing heart are not completely clear.

In this paper, we developed a coupled mathematical model including Ca²⁺ cycling processes from subcellular to cellular level and electrophysiology of the ventricular myocyte. The proposed coupled model was then applied to study the effects of Ca²⁺ release via rogue RyRs on subcellular spatiotemporal Ca²⁺ cycling and on the possible membrane potential changes in failing heart.

2. Methods

Subcellular Ca²⁺ release events and cellular Ca²⁺ cycling as well as corresponding membrane potential were simulated synchronously by a coupled model. The model consists of two parts: a two-dimensional (2D) spatial Ca²⁺ reaction-diffusion model and an electrophysiological model of the ventricular myocyte.

2.1. A Subcellular Ca²⁺ Reaction-Diffusion Model

The shape of the cardiac myocyte in the model is represented as a circular cylinder 100 𝜇m in length and 10 𝜇m in radius. However, because of quasi-isotropic diffusion of Ca²⁺ on the transverse section [15], a 2D model was used in our simulation work (Figure 1), where x axis denotes the cell’s longitudinal direction and y axis is along the Z-line. It could still describe most of the key properties of Ca²⁺ waves, but needs much less computation work than a 3D model. The 2D spatiotemporal Ca²⁺ reaction-diffusion model is described based on a reaction-diffusion system proposed by Izu et al. [16]. Figure 1 shows the subcellular structural representation of RyRs network. The x-axis denotes the cell’s longitudinal direction and the y-axis is along the Z-line. The blue dots represent RyR clusters which account for Ca²⁺ sparks. The small red dots are the rogue RyR channels which raise Ca²⁺ quarks. Rogue RyRs are distributed in a stochastic manner. 𝑁_rogue is referred to the distributing density of rogue RyRs with the unit of rogue RyR/𝜇m².

Figure 1 

Geometry of RyRs distribution. The blue dots represent Ca2+ release units consisting of clusters of RyRs. 𝑙𝑥=2.0 𝜇m and 𝑙𝑦 = 1.0 𝜇m. The small red dots are rogue RyRs which scatter over the plane randomly. In this figure, 𝑁rogue is equivalent to 1.0 rogue RyR/μm2.

The free Ca²⁺ concentration [Ca²⁺]_𝑖 in the reaction-diffusion is described by a differential equation as follows: 𝜕Ca²⁺_𝑖𝜕𝑡=𝐷_𝑥𝜕²Ca²⁺_𝑖𝜕𝑥²+𝐷_𝑦𝜕²Ca²⁺_𝑖𝜕𝑦²+𝐽_dye+𝐽_buffers+𝐽_pump+𝐽_leak+𝐽_sub-rel, (1) where 𝐷_𝑥 and 𝐷_𝑦 are the diffusion coefficients; 𝐽_dye and 𝐽_buffers are due to fluorescent indicator dye and endogenous Ca²⁺ buffer, respectively; 𝐽_pump is pumping rate of SR Ca²⁺ ATPase; 𝐽_leak is defined as a RyR-independent leak flux which is small and invisible and persists in the presence of RyR inhibition [8]; 𝐽_sub-rel is summation of Ca²⁺ release fluxes in the 2D subcellular model which consists of two types of Ca flows as follows: 𝐽_sub-rel=_𝑖,𝑗𝐽_cluster𝑥_𝑖,𝑦_𝑗+_𝑚,𝑛𝐽_rogue𝑥_𝑚,𝑦_𝑛,𝐽_cluster𝑥_𝑖,𝑦_𝑗=𝑉_clusterCa²⁺_SR−Ca²⁺_{𝑖,(𝑥𝑖,𝑦𝑗)}, (2) where 𝐽_cluster(𝑥_𝑖,𝑦_𝑗) is Ca²⁺ release flux via a cluster of RyRs located on (𝑥_𝑖,𝑦_𝑗), 𝑉_cluster is maximal 𝐽_cluster conductance equivalent to 1.97×10⁻⁸ ms⁻¹, and 𝐽_rogue(𝑥_𝑚,𝑦_𝑛) is Ca²⁺ release flux via a rogue RyR channel located on (𝑥_𝑚,𝑦_𝑛), equivalent to 3.3166×10⁻⁹ pmol/ms.

Firings of the two types of RyR channels are considered to be stochastic processes and treated by the Monte Carlo simulation in our work. To evaluate the effects of SR luminal Ca²⁺ concentration ([Ca²⁺]_SR) on SR Ca²⁺ release, we integrate a new parameter 𝑘_CaSR into the probability of firing of Ca²⁺ sparks or quarks (𝑃_𝑗,𝑗 = cluster for RyR clusters, and j = rogue for rogue RyRs) as follows: 𝑘_CaSR=𝑘_max1+𝐷_SR/Ca²⁺_SR^𝑛SR,𝑃_𝑗=𝑃_max1+𝐾_𝑚𝑗/Ca²⁺_𝑖^𝑛𝑗𝑘_CaSR, (3) where 𝑘_max = 2.0, the Hill coefficient 𝑛_SR=4.5, 𝑃_max=0.3/event/ms, 𝑛_cluster=1.6, and 𝑛_rogue=1.0 for the less coupled gating of rogue RyRs than RyR clusters. 𝐷_SR is luminal Ca²⁺ sensitivity parameter of Ca²⁺ release events, and 𝐾_𝑚𝑗 is cytoplasmic Ca²⁺ sensitivity parameter of RyR clusters or rogue RyR channels. In our simulation work, 𝐾_𝑚rogue was always set to be of the same value as 𝐾_𝑚cluster; thus 𝐾_𝑚 was used to represent the value of 𝐾_𝑚cluster and 𝐾_𝑚rogue.

In this study, the simulation of subcellular Ca²⁺ handling was performed on the longitudinal section of a cardiac myocyte with the size of 100𝜇m×20 μm along the cellular longitudinal direction (x-axis) and Z-line (y-axis), respectively. The number of RyR clusters was 49×19 along x and y axes, respectively, and the total number of rogue RyRs was 𝑁_rogue×2000 𝜇m². The diffusion partial differential equation was approximated by the finite difference method (FDM) with a time-step size of 0.01 ms and a mesh size of 0.1 𝜇m.

Because of the stochasticity of opening of RyR clusters and rogue RyRs, the properties of Ca²⁺ signalling were described by statistical results by carrying out repeated Monte Carlo simulations. All averaged data were expressed as mean±SEM. One-way analysis of variance (ANOVA) was used for comparison and 𝑃<0.05 was taken to indicate statistical significance.

2.2. A Cellular Electrophysiological Model

The electrophysiological behavior of a myocardial cell is modelled based on a cardiac action potential model proposed by Ten Tusscher and Panfilov [17]. The voltage across the cell membrane can be described with the following differential equation: 𝑑𝑉_𝑚1𝑑𝑡=−𝐶_𝑚𝐼_Na+𝐼_CaL+𝐼_to+𝐼_Kr+𝐼_Ks+𝐼_K1+𝐼_NaK+𝐼_NaCa+𝐼_pCa+𝐼_pK+𝐼_bCa+𝐼_bNa+𝐼_stim, (4) where 𝐶_𝑚 is the membrane capacitance, 𝐼_stim is a stimulus current, and 𝐼_𝑥 denotes all kinds of ionic currents across the sarcolemma.

However, different from the Ca²⁺ dynamical system by Ten Tusscher et al., global SR Ca²⁺ release current 𝐽_rel at the cellular level is calculated by the summation of local Ca²⁺ release fluxes in the 2D subcellular model: 𝐽_rel=𝑘_rel𝐽_sub-rel=𝑘_rel𝑖,𝑗𝐽_cluster𝑥_𝑖,𝑦_𝑗+_𝑚,𝑛𝐽_rogue𝑥_𝑚,𝑦_𝑛, (5) where 𝑘_rel is a constant multiplier equivalent to 22.25 in our coupled model.

2.3. Heart Failure Model

Changes of Ca²⁺ cycling as well as other ionic currents have been observed in failing heart; thus we modified the parameters of our coupled model to mimic abnormal Ca²⁺ dynamics and electrophysiological properties in heart failure from subcellular to cellular levels.

2.3.1. Ca²⁺ Handling

(a) SR 𝐶𝑎²⁺ Release Channels
In HF, RyR channels would become unstable due to phosphorylation of protein kinase A (PKA) [18] or Ca²⁺/calmodulin-dependent-protein-kinase-I- (CaMKI-) induced hyperphosphorylation [19] and be oversensitive to cytoplasmic Ca²⁺ and SR luminal Ca²⁺ [20]. In our simulation study, 𝐾_𝑚 was set to be 7.5 𝜇M and 𝐷_SR was 2.5 mM under the condition of heart failure, while 15 𝜇M and 3.25 mM, respectively, under control condition.

(b) SR 𝐶𝑎²⁺ Pump
Pumping activity of SR Ca²⁺ ATPase in failing heart is reduced as shown in experimental studies [21]. A 45% reduction in 𝐽_pump of a failing myocyte is incorporated into our HF model.

(c) SR 𝐶𝑎²⁺ Leak
Spontaneous openings of RyR clusters and rogue RyRs at diastole are the main contributors to SR Ca²⁺ leak as the form of Ca²⁺ sparks and quarks. Because of instability of RyR channels, RyR-mediated Ca²⁺ leak from SR increased in the resting HF myocyte. However, RyR-independent leak was unaltered in our HF model.

2.3.2. Ionic Current across the Sarcolemma

(a) Inward Rectifier Potassium Current: 𝐼_𝐾1
In heart failure, 𝐼_K1 was shown to be reduced in many studies [22, 23]. In our HF model the current density of 𝐼_K1 was assumed to decrease by 20%.

(b) Slowly Activated Delayed Rectifier Potassium Current: 𝐼_𝐾𝑠
𝐼_Ks is the slowly activated component of delayed rectifier potassium current. In the failing canine hearts 𝐼_Ks has been shown to be downregulated by nearly a half [24]. Therefore, maximal 𝐼_Ks conduction was changed to 50% of the value used in nonfailing myocytes.

(c) Transient Outward Potassium Current: 𝐼_𝑡𝑜
According to an experimental result, the current density of 𝐼_to in HF declined to 64% of the value in control cardiac cells [25], so that in our simulations 𝐼_to was reduced to 64% in failing myocytes.

(d) Fast Na Current: 𝐼_𝑁𝑎
It has been reported that the peak density of 𝐼_Na decreased significantly in heart failure [26]. Therefore, the maximal 𝐼_Na conductance 𝐺_Na was set to be 8.902 nS/pF in the failing myocytes, while equivalent to 14.838 nS/pF in the nonfailing myocytes.

(e) Na-Ca Changer Current: 𝐼_{𝑁𝑎𝐶𝑎}
The activity and/or gene expression of Na/Ca changer was found to be increased obviously in many experiments [27, 28]. Thus we upregulated 𝐼_NaCa by 65% in the failing myocytes.

(f) Na-K Pump Current: 𝐼_𝑁𝑎𝐾
As shown in the experimental study, the concentration of Na/K ATPase in the failing heart was reduced by 42% [29], so that reduction of 𝐼_NaK by the same proportion was incorporated in our HF model.

(g) Ca Background Current: 𝐼_𝑏𝐶𝑎
Inward 𝐼_bCa was considered to balance Ca²⁺ extrusion via Na/Ca exchanger and sarcolemmal Ca²⁺ pump at resting potential. In our HF model the conductance of 𝐼_bCa was increased due to the increase of 𝐼_NaCa.

The different values of parameters in nonfailing and failing myocyte models are shown in Table 1.

Table 1 

Parameters in nonfailing versus failing myocyte models.

3. Results

3.1. Ca²⁺ Cycling and 𝑉_𝑚 in HF

With the proposed coupled model, firstly we simulated the action potential and calcium cycling by applying a stimulus with a frequency of 1 Hz, duration of 1 ms, and an amplitude of 7 pA. Figure 2 shows the simulation results of membrane potential, cytoplasmic Ca²⁺ concentration, Ca²⁺ concentration in SR lumina, and the Na⁺/Ca²⁺ exchanger current after 10th stimulus. While blue curves in Figure 2 are obtained under the physiological conditions, red curves are under the pathological conditions, that is, heart failure. Compared with that in nonfailing myocytes, the plateau of action potential (AP) shows a larger amplitude and longer duration, causing a significant increase in AP duration (~45% longer than in normal condition) in heart failure. Meanwhile, decrease of the maximal conductivity of 𝐼_K1 in heart failure makes the resting potential elevate 2~3 mV. However, the amplitude of AP overshot is smaller in heart failure, which is due to the reduction of fast inward current 𝐼_Na. Then at the early stage of rapid repolarization, a weakened notch is observed in heart failure AP, which is caused by a decrease of 𝐼_to. Moreover, the prolonged plateau is mainly due to decease of maximal conductivity of 𝐼_Ks.

(a)

(b)

(c)

(d)

Figure 2 

Simulation results of membrane potential (𝑉𝑚) (a), cytoplasmic Ca2+ concentration ([Ca2+]𝑖) (b), SR luminal Ca2+ concentration ([Ca2+]SR) (c), and 𝐼NaCa (d) during a single twitch. The red curve represents the time course of those parameters in failing myocytes, while the blue curve is the time course of those parameters in control myocytes.

For the calcium handing in heart failure, it is mainly characterized by a significant impair of global Ca²⁺ transient and a much slower decay of calcium concentration. Moreover, the SR Ca²⁺ store is smaller in heart failure, that is, [Ca²⁺]_SR is ~15% lower in resting cells, and the restoring rate of SR calcium is slower than that on control condition. Due to the changes of AP morphology and calcium transient curves together with increase of the activity of Na/Ca exchanger, the curve of 𝐼_NaCa in heart failure differs significantly from that under normal condition. This can be seen in Figure 2(d); that is, in heart failure, both the inward and outward currents of 𝐼_NaCa are increased. However, it takes a longer time to reach the peak of inward current, and the amplitude of inward 𝐼_NaCa in resting stage is also larger compared with that under control conditions.

3.2. Dependence of DAD on Rogue RyR

In heart failure myocytes, the RyR channels become very unstable and are more likely to open with the same values of [Ca²⁺]_𝑖 and [Ca²⁺]_SR as those under normal conditions. However, the calcium release current through a RyR cluster decreases as the SR calcium store is partly unloaded, which is observed as reduction of amplitude and area of calcium sparks. Indeed, the simulation results by using our coupled model show that, although more spontaneous calcium sparks occur in failing myocytes, propagating Ca²⁺ waves are seldom found when there is no rogue RyRs on the 2D subcellular space without any stimulus. These spontaneous calcium sparks would slightly elevate global [Ca²⁺]_𝑖 on the cellular level (Δ[Ca²⁺]_𝑖 = (1.16 ± 0.06) × 10⁻⁴ mM, 𝑛=10) and depolarize transmembrane potential with a tiny amplitude (Δ𝑉_𝑚 = 4.21 ± 0.23 mV, 𝑛=10) as shown in Figure 3(a).

(a)

(b)

(c)

Figure 3 

(a) Effects of distribution density of rogue RyRs on global change in [Ca2+]𝑖 (Δ[Ca2+]𝑖, left image) and depolarization of membrane potential (Δ𝑉𝑚, right image ) without any external stimulus in heart failure. (b) Dependence of 𝑇peakCa and 𝑇peak𝑉𝑚 on 𝑁rogue. 𝑇peakCa represents the time to reach the peak of global [Ca2+]𝑖 from the end of resting stage, and 𝑇peak𝑉𝑚 is the time to reach the peak of 𝑉𝑚. For each 𝑁rogue, we simulated 10 times under the same conditions to get statistical results due to the randomness of opening of RyR channels. The values denoted by arrows in (a) and (b) are recorded when 𝐼Na = 0 pA. (c) Three groups of simulation results of 𝑉𝑚 when 𝑁rogue = 0.25, 1.0, and 1.75 rogue RyR/𝜇m2 (blue, red, and black curves, resp.). The inset is the enlarged view of depolarization of membrane potential during the period of 1100ms∼1500ms. 𝐼Na is also set to be 0 pA when 𝑁rogue = 1.75 rogue RyR/𝜇m2.

However, how do Ca²⁺ dynamics and electrophysiological properties of a failing myocyte change in the presence of rogue RyR channels? We integrate rogue RyR channels into the 2D RyR grid and vary their distribution density 𝑁_rogue to investigate the precise effect of rogue RyR channels on Ca²⁺ handling and membrane potential. When the density of rogue RyRs is relatively low, for example, 𝑁_rogue = 0.25 rogue RyR/𝜇m², spontaneously occurring calcium sparks are frequently observed in the subcellular region of HF myocytes during the resting state. Occasionally Ca²⁺ waves are formed, albeit at a small area, by recruiting several adjacent Ca²⁺ sparks. Those small Ca²⁺ waves could not propagate across the whole myocyte, but self-abort during a short time. Similar to the condition without rogue RyRs, global [Ca²⁺]_𝑖 and membrane potential are not affected severely by those Ca²⁺ release events under the condition with low density of rogue RyRs as shown in Figure 3(a) (Δ[Ca²⁺]_𝑖 = (1.61±0.08)× 10⁻⁴ mM, Δ𝑉_𝑚=5.95±0.29 mV, 𝑛=10).

Amplification and increase rate of [Ca²⁺]_𝑖 and 𝑉_𝑚 should be two groups of principal parameters to evaluate the effects of rogue RyRs on Ca²⁺ dynamics and electrophysiological properties. Besides Δ[Ca²⁺]_𝑖 and Δ𝑉_𝑚, two new parameters 𝑇_peakCa and 𝑇_peak𝑉𝑚 are used in our simulation. 𝑇_peakCa represents the mean time to reach the peak of [Ca²⁺]_𝑖 from the end of resting stage, and 𝑇_peak𝑉𝑚 is the time to reach the peak of 𝑉_𝑚. Increase rate of Ca²⁺ concentration and depolarization velocity could be estimated indirectly by the two parameters, which is shown in Figure 3(b). The simulated 𝑇_peakCa decreases significantly when 𝑁_rogue is upregulated from 0 to 0.25 rogue RyR/𝜇m² (𝑇_peakCa = 418 ± 19.7 ms and 325 ± 15.2 ms (𝑛=10), resp., 𝑛=10,𝑃<0.05). However, decrease of 𝑇_peak𝑉𝑚 is slight when 𝑁_rogue is from 0 to 0.25 rogue RyR/μm² (𝑇_peak𝑉𝑚 = 406 ± 38.5 ms and 340 ± 25.1 ms, resp., 𝑛=10,𝑃>0.05). In Figure 3(c) three blue curves when Time >1000 ms represent repeated simulation results of 𝑉_𝑚 without any stimulus when 𝑁_rogue = 0.25 rogue RyR/𝜇m². The results indicate a smooth variation without a significant peak in the membrane potential morphology.

When 𝑁_rogue is increased to 0.5 rogue RyR/𝜇m², similar to above, small spontaneous calcium waves cannot propagate in a long distance and quickly decay as well. The depolarization of membrane potential caused by calcium release events has bigger amplitude compared with that when 𝑁_rogue = 0.25 rogue RyR/𝜇m² (𝑃<0.05), but is also relatively weak with an average of 7.48 ± 0.25 mV. Again, the membrane potential morphology is smooth.

However, as 𝑁_rogue is further increased, specifically when 𝑁_rogue≥0.75 rogue RyR/𝜇m², large calcium waves could be initiated spontaneously in the 2D subcellular region. Moreover, our Monte-Carlo simulation results show that these large calcium release events cause a significant larger calcium transient at the whole cell level and subsequently depolarize the membrane potential to a larger extent when 𝑁_rogue is increased by a step of 0.25 rogue RyR/𝜇m² (𝑃<0.05) as shown in Figure 3. Figure 4 shows typical simulation results when 𝑁_rogue = 1.0 rogue RyR/𝜇m². After 3 action potentials paced by the cycle length of 1000 ms, a relatively large Ca²⁺ transient is observed, as well as a big inward 𝐼_NaCa and consequently a DAD without external stimulus in the heart failure myocytes. The linescan image in Figure 4 indicates the underlying microcosmic Ca²⁺ cycling on the subcellular level.

Figure 4 

Occurrence of DAD following three action potentials in failing myocyte when 𝑁rogue = 1.0 rogue RyR/𝜇m2. Five figures from top to bottom represent membrane potential, cytoplasmic [Ca2+]𝑖, SR luminal [Ca2+]SR, 𝐼NaCa, and line-scan image along longitudinal direction of the cell, respectively, in failing ventricular myocyte.

Besides, as 𝑁_rogue is increased to a larger value, Ca²⁺ transient elicited by spontaneous Ca²⁺ release and the depolarization of membrane potential enlarge further (𝑃<0.05), while the values of 𝑇_peakCa and 𝑇_peak𝑉𝑚 decrease significantly (𝑃<0.05) (as shown in Figure 3). All these results together suggest close relationship between rogue RyRs and DADs.

3.3. Triggered Action Potential

As demonstrated above, the spontaneous Ca²⁺ release from SR causes a Ca²⁺ transient in cytoplasm and subsequently depolarizes the membrane potential even in the resting stage without any stimulus. Furthermore, the amplitude of calcium transient and the degree of depolarization are positively correlated to the distribution density of rogue RyR channels. Therefore, once the density of rogue RyRs is large enough, the membrane potential may be depolarized to reach the threshold that will trigger an action potential.

Figure 5 shows the time course of simulated membrane potentials: cytoplasmic Ca²⁺ concentration, SR Ca²⁺ store, the Na/Ca exchange current, as well as the Ca²⁺ dynamics at subcellular level. From the line-scan image we can see that, along the cellular longitudinal direction, many spontaneous Ca²⁺ waves are initiated nearly at the same time. These Ca²⁺ waves could propagate and diffuse to finally form a large wave. These Ca²⁺ releases rise the [Ca²⁺]_𝑖 quickly and drive a strong inward component of 𝐼_NaCa, which causes the depolarization of the membrane potential. When the membrane potential is depolarized to reach the threshold for activation of fast Na⁺ channel, the large 𝐼_Na is produced very fast and induces an upstroke of the membrane potential. Then, the L-type Ca²⁺ channels will be subsequently activated and a flux of extracellular calcium ions burst into cytoplasm via the L-type calcium channels. This inflow of Ca²⁺ together with the previously released Ca²⁺ from SR can activate the remaining available RyR channels and elicit an even larger Ca²⁺ transient in the cytoplasm. The other ionic channels at the membrane are successively opened and determine the morphology of AP together. Particularly, for the 𝐼_NaCa, it switches to a weak outward current during the plateau stage, but turns to a strong inward current in the repolarization stage of AP, by which the Ca²⁺ is ejected to the extracellular space.

Figure 5 

Triggered action potential in failing myocyte when 𝑁rogue = 1.75 rogue RyR/𝜇m2. Five figures from top to bottom represent membrane potential, cytoplasmic [Ca2+]𝑖, SR luminal [Ca2+]SR, 𝐼NaCa, and line-scan image along longitudinal direction of the myocyte, respectively, in failing ventricular cell.

When 𝑁_rogue = 1.5 rogue RyR/μm², triggered APs are observed in 11 simulations (totally 22 Monte Carlo simulations), that is, the probability of triggered AP is 50%. Moreover, when 𝑁_rogue = 1.75 rogue RyR/𝜇m², triggered APs are found in 18 out of 21 Monte Carlo simulations, that is, the probability of triggered AP rises to 87.5%. On the contrary, when 𝑁_rogue < 1.5 rogue RyR/𝜇m², no triggered AP is seen in our simulations.

To quantitatively investigate the effect of high dense rogue RyRs on Ca²⁺ handling, we also recorded the variations of [Ca²⁺]_𝑖 and membrane potential (Δ[Ca²⁺]_𝑖 and Δ𝑉_𝑚) as well as 𝑇_peakCa and 𝑇_peak𝑉𝑚. However, Δ𝑉_𝑚 elicited by intensive Ca²⁺ release events is often large enough to activate 𝐼_Na. Consequently, the voltage upstroke induced by 𝐼_Na would overlap the original Δ𝑉_𝑚. Then, Δ[Ca²⁺]_𝑖 by Ca²⁺ release events is also overlapped by the following inward-flowing 𝐼_CaL during a triggered AP. Under those conditions, the measurement of Δ[Ca²⁺]_𝑖 and Δ𝑉_𝑚 as well as 𝑇_peakCa and 𝑇_peak𝑉𝑚 becomes very difficult. Thus, in our simulation, when 𝑁_rogue⩾ 1.5 rogue RyR/𝜇m², we set the 𝐼_Na to be zero after removing the external stimulus. By doing this, triggered AP will not be formed even when a DAD makes the membrane potential more positive than the threshold for 𝐼_Na. Therefore, we can calculate these parameters easily. Actually, our simulation results are shown in Figure 3 marked by arrows, from which we can conclude that, as 𝑁_rogue is increased from 1.5 to 1.75 rogue RyR/𝜇m², the amplitude of [Ca²⁺]_i enlarges (𝑃<0.05), whereas the time to reach the peak does not change obviously (𝑃>0.05). However, the amplitude of a DAD increases significantly (𝑃<0.05), while the time to observe the DAD decreases (𝑃<0.05).

4. Discussion

4.1. Mechanism of Ca Handling in HF

Heart failure, a syndrome caused by significant impairments in cardiac function, has become one of the biggest human killers with a poor prognosis [30]. Ca²⁺ handling of cardiac cells in heart failure is always characterized by reduction in the amplitude as well as by slowed decay of Ca²⁺ transient [31]. The primary reason for decrease in the amplitude of Ca²⁺ transient is the partly unloaded Ca²⁺ store in SR. Three factors mainly accounting for the smaller store are (1) increased Ca²⁺ leak in the resting myocyte, (2) decreased activity of SR Ca²⁺ pump, and (3) increase in expression and/or activity of Na⁺-Ca²⁺ exchanger.

Despite the increased activity of Na/Ca exchanger, at the early stage of [Ca²⁺]_𝑖 decay, the net current of Na/Ca exchanger might be outward current (i.e., Ca²⁺ influx) or weak inward current, due to longer AP plateau and higher plateau potential in failing myocyte. Therefore, slowed decay of Ca²⁺ transient is mainly due to decreased SR Ca²⁺ pump, which removes major amount of Ca²⁺ at the early stage of [Ca²⁺]_𝑖 decay. Only when the membrane potential is repolarized to a relatively negative voltage and [Ca²⁺]_𝑖 is still high that 𝐼_NaCa turns to a strong inward current and accelerates decay of Ca²⁺ transient at the late stage of [Ca²⁺]_𝑖 decay.

4.2. Arrhythmogenic Effect of Rogue RyRs

Besides pump failure, patients with severe heart failure are at high risk of sudden cardiac death generally triggered by a lethal arrhythmia [23, 32]. DADs are thought to be the primary mechanism underlying arrhythmia in failing heart [33]. In our simulation work, although the probability of firing of RyR clusters increases in resting failing cardiocytes, the spontaneous Ca²⁺ sparks could not elicit enough amplitude of Ca²⁺ transient to induce an obvious DAD in the absent of rogue RyRs. The existence of rogue RyR channels is of importance in initiation and propagation of spontaneous Ca²⁺ waves in ventricular myocytes with heart failure [14].

In this work, we propose a coupled mathematic model by integrating the spatiotemporal Ca²⁺ reaction-diffusion system into the cellular electrophysiological model. Rogue RyR channels are then incorporated into the coupled model to simulate subcellular Ca²⁺ dynamics and global cellular electrophysiology simultaneously under heart failure conditions. Our simulation results show that, in the presence of rogue RyRs, Ca²⁺ dynamics is more unstable and Ca²⁺ waves are more likely to be initiated than the condition without rogue RyRs. Different from sporadic sparks in a resting myocyte without Ca²⁺ waves, a number of SR Ca²⁺ release events occur intensively during the process of spontaneous occurrence of Ca²⁺ waves. These release events could elevate the amplitude of Ca²⁺ transient effectively and thus induce Ca²⁺-dependent inward current (mainly via Na/Ca exchanger) which depolarizes the sarcolemma and lends to a DAD, or a triggered AP sometimes. For a given level of Ca²⁺ release in failing myocytes, inward depolarizing current becomes larger due to increased activity of Na/Ca exchanger. And increased membrane resistance owing to reduction of 𝐼_K1 enables the same inward current to produce greater depolarization. Once a DAD elevates membrane potential to the threshold for activation of 𝐼_Na, a triggered AP is then formed. DADs and triggered AP are the primary triggered activities accounting for arrhythmias in heart failure.

4.3. Dependence of 𝑉_𝑚 on Density of Rogue RyRs

Without rogue RyR channels or with low distribution density, occurrence of spontaneous Ca²⁺ sparks and/or quarks is independent in time and space, which is unlikely to evolve into propagating Ca²⁺ waves with partially unloaded SR Ca²⁺ store. [Ca²⁺]_𝑖 variation elicited by these Ca²⁺ release events is slight. In our simulation, when 𝑁_rogue = 0 or 0.25 rogue RyR/𝜇m², the amplitude of membrane potential depolarization is very small, only several mV, and the average 𝑇_peak𝑉𝑚 is big and extensive with large SEM. The morphology of membrane potential is very smooth without a distinct peak, so that this type of depolarization could not be referred to a genuine DAD.

However, as more rogue RyRs are distributed over the 2D plane, Δ𝑉_𝑚 increases gradually, while the value of 𝑇_peak𝑉𝑚 decreases but becomes more intensive. When the value of 𝑁_rogue is elevated to 1.5 rogue RyR/𝜇m² or bigger, the large membrane potential depolarization tends to evoke a triggered AP, and the probability of occurrence of triggered APs increases with the larger 𝑁_rogue. The reason is that larger number of rogue RyRs would increase the amplitude and rate of DADs by initiating more Ca²⁺ waves which occur more synchronously. Therefore, depolarization of 𝑉_𝑚 is indicated to be dependent on the distribution density of rogue RyR channels.

4.4. Limitations and Further Work

Because the rogue RyR remains to be a hypothetical channel rather than a determinate concept, experimental parameters of rogue RyRs are lacked. Some assumptions were made regarding the density, distribution, and kinetic of rogue RyR channels. In our work, a constant was used to represent Ca²⁺ release flux via a rogue RyR channel, while Ca²⁺ release flux via a cluster of RyRs was dependent on global luminal Ca²⁺ concentration ([Ca²⁺]_SR) and local Ca²⁺ concentration ([Ca²⁺]_{𝑖(𝑥𝑖,𝑦𝑖)}). Besides, different 𝑁_rogue values were used to evaluate the effect of rogue RyR on Ca²⁺ cycling and membrane potential in failing heart.

Numerous key regulatory proteins, such as protein kinase A(PKA), Calstalin, CaMKII, and phosphatase, are bound to RyR, thus forming the junctional complex. RyR channels would be regulated via different signaling pathways [34]. For example, PKA phosphorylation dissociates FKBP12.6 from the RyR and thus makes the RyR channel unstable in failing hearts [35]. These regulating processes could not be mimiced by our present model. Besides, defective Ca²⁺ handling also occurs in many cardiac diseases, such as myocardial infarction, atrial fibrillation, and various arrhythmogenic paradigms. The coupled model is planned to be improved, and more relevant parameters should be added to investigate the potential mechanisms of Ca²⁺ dynamics in various kinds of cardiac diseases.

5. Summary

By integrating the spatiotemporal Ca²⁺ reaction-diffusion model into the cellular electrophysiological model, appearance of subcellular Ca²⁺ release events and evolution of waves together with dynamics of ionic concentration and membrane potential on the cellular level could been monitored simultaneously. By using the coupled model we investigate the effects of rogue RyRs on Ca²⁺ handling from subcellular to cellular level as well as electrophysiological properties in failing heart. The simulation results suggest that rogue RyR with tiny Ca²⁺ release flux should be an important factor in triggering arrhythmia in failing cardiac cells. Our work suggests the importance of rogue RyRs in initiation of Ca²⁺ release events (especially Ca²⁺ waves) and consequently DADs or triggered APs. Our study indicates the arrhythmogenic effect of rogue RyRs and helps to elucidate a possible arrhythmia mechanism in failing heart.

Acknowledgments

This project is supported by the 973 National Basic Research & Development Program (2007CB512100) and the National Natural Science Foundation of China (81171421).