Ventricular myocyte model

We have developed a three-dimensional (3D) spatial model of a single myocyte based on the (non-spatial) Greenstein-Winslow canine ventricular myocyte model [13]. To enable reproduction of realistic Ca²⁺ waves and DADs, the original model was adapted to include spatial Ca²⁺ diffusion on a rectangular lattice of 25,000 Ca²⁺ release sites (Fig 1) distributed within the cell. The cell was divided into 25 and 20 lattice points in the two transverse directions and 50 lattice points in the longitudinal direction. Release sites were spaced 1 and 2 μm in the transverse and longitudinal directions, respectively [14]. The time constant for longitudinal Ca²⁺ diffusion was twice (i.e. 2x slower) that for the transverse direction such that the model exhibited symmetric Ca²⁺ wave propagation [15]. This difference in diffusion rates arises from differences in diffusion along versus across TTs.

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Fig 1. Multiscale cell and tissue model schematics.

(a) Diagram of intracellular Ca²⁺ compartments and transport. At each release site at coordinates (i,j,k), Ca²⁺ is released via RyRs from the JSR into the dyadic subspace (SS) and diffuses into a submembrane (SM) compartment. Ca²⁺ diffuses between SM compartments of adjacent release sites in the 3D lattice, as depicted in panel (b). Ca²⁺ can also diffuse from the SM into a single cell-averaged cytosolic compartment and can be transported by the SR Ca²⁺ ATPase (SERCA pump) from the cytosolic compartment into a single network SR (NSR), which refills the JSR. (c) Illustration of the different spatial scales incorporated in the tissue model including (from left to right) the 48 stochastic RyRs in each of the 25,000 release sites of a single cell. Hundreds of cells are coupled via gap junction currents to form the fiber model.

https://doi.org/10.1371/journal.pcbi.1005783.g001

A sub-membrane (SM) release site compartment (Fig 1A) was added to describe the volume under the TT membrane where the Ca²⁺ concentration ([Ca²⁺]) is elevated during Ca²⁺ sparks and cell-wide Ca²⁺ release [16]. Detailed imaging studies of Ca²⁺ release sites have revealed that RyR clusters exhibit edge-to-edge spacing of less than 100 nm [17, 18]. This suggests that neighboring sites may be functionally coupled through local Ca²⁺ diffusion. Therefore Ca²⁺ diffusion between SM compartments was implemented to reflect Ca²⁺ transport across steep [Ca²⁺] gradients on the periphery of the release site during release [19]. The SM compartment was modeled as a cylinder encircling the TT membrane with inner radius 100 nm, outer radius 180 nm, and 1 μm axis. It was assumed that 50% of NCX are located in the TT membrane of the SM compartment, and the remaining 50% are in the sarcolemmal membrane of the cytosolic compartment [20, 21]. Ca²⁺ in the SM is buffered by calmodulin and sarcolemmal binding sites. Ca²⁺ transport rates from the SM to the cytosol and between SM compartments were constrained to yield a realistic Ca²⁺ wave threshold (~100–150 μmol/[L cytosol]) [22] and velocity (50–100 μm/s) [4] in the baseline model.

Each spatial site, with location defined by coordinate (i,j,k), is represented by a set of ordinary differential equations describing local Ca²⁺ transport (Fig 1A). [Ca²⁺] is assumed to be uniform within the local JSR ([Ca²⁺]_JSR,i,j,k), dyadic subspace ([Ca²⁺]_d,i,j,k), and SM ([Ca²⁺]_SM,i,j,k) compartments. Spatial Ca²⁺ diffusion is modeled as transport between SM compartments of adjacent release sites in the 3D lattice (Fig 1B). Model equations and parameters are given in S1 Equations and S1 Table.

Global compartments are used to represent average cytosolic [Ca²⁺]_i and network SR [Ca²⁺] ([Ca²⁺]_NSR). The use of global compartments to represent these quantities eliminates two Ca²⁺ diffusion parameters and provides a three-fold reduction in the number of Ca²⁺ diffusion terms, thus substantially reducing model complexity and computational burden. While these simplifications result in over-estimation of the rise in [Ca²⁺]_i and fall of [Ca²⁺]_NSR at sites far away from an initiating Ca²⁺ wave, the model produces realistic spatiotemporal dynamics of spontaneous Ca²⁺ release and DADs.

Each release site contains a set of 48 RyRs and 8 LCCs that gate stochastically according to Markov chain models. The LCC model is as described in the Greenstein-Winslow model [13], with adjustments made to the rate of Ca²⁺-dependent inactivation (see S1 Text). Briefly, the LCC inactivation rate is a saturating function of [Ca²⁺]_SS, which was necessary to reproduce inactivation kinetics consistent with the original model. Note, however, that as a result AP duration does not substantially decrease at reduced SR Ca²⁺ loads as exhibited previously [13]. RyR gating is described by a minimal two-state Markov model based on the work of Williams et al. [23], described in detail in Walker et al. [19]. Briefly, mean open time of each channel is 2 ms, and the opening rate is given by (1) where k⁺ = 1.107 × 10⁻⁴ ms^-1 μM^-η is the opening rate constant, η = 2.1 is the Ca²⁺ Hill coefficient, and ϕ is a [Ca]_JSR,i,j,k-dependent regulation term given by (2)

Fiber model

The 3D cell model was incorporated into a tissue-scale model of a 1D fiber of myocytes. Fig 1C depicts the multiple biological scales represented in the model, from single ion channels to the multicellular fiber. The 3D cell model was augmented with a current carried by the gap junctions at either end of each cell (I_gap). The current from cell i into an adjacent cell i+1 is given by: (3) where g_gap is the gap junction conductance, which was adjusted to yield a conduction velocity of 55 cm/s. The membrane potential in the fiber was solved using the Crank-Nicolson method [24] with 50 μs time steps. Operator splitting was used to explicitly solve each cell model using an embedded adaptive Runge-Kutta method described previously [13].

Beta-adrenergic stimulation

Sympathetic stimulation of the heart occurs through β-adrenergic receptor activation, which activates intracellular signaling pathways, most notably Protein Kinase A (PKA) and Ca²⁺/calmodulin-activated protein kinase II (CaMKII), that increase contractility [25]. β-adrenergic stimulation is also known to be pro-arrhythmic, and can contribute to spontaneous Ca²⁺ release [26] in pathological conditions. Cell model parameters were modified to reflect the effects of acute β-adrenergic stimulation. LCC open probability was increased [27] by changing the fraction of gating LCCs from 25% to 60% and setting 3–5% of the channels to gate in a high-activity mode in which the mean open time was increased from 0.5 to 5.8 ms [28, 29]. Enhanced activation of inward currents was implemented for I_Kr using modifications described previously [29] and for I_Ks by shifting the voltage-dependence of activation by -35 mV and increasing conductance by 40% [30]. SR Ca²⁺ loading was facilitated by reducing SERCA pump K_d for [Ca²⁺]_i by 50% [31]. RyR opening rate was increased by either 50% to reflect increased SR Ca²⁺ leak observed in experimental studies [32] or 400% to reproduce pathological behavior after ouabain overdose [33]. Unless otherwise noted, these conditions were applied to all simulations in this study.

Model properties

In order to reproduce protocols designed to measure ECC properties, membrane potential was stepped to varying test potentials for 200 ms from a holding potential of -80 mV. The dependence of normalized peak RyR and LCC Ca²⁺ fluxes on the test potential and corresponding ECC gain values are similar to those observed experimentally [34] (Fig 2A and 2B). The peak normalized RyR flux curve is right-shifted with respect to that of the LCC flux curve. The ECC gain is 13.8 at 0 mV and decreases monotonically with increasingly depolarized test potentials. The lack of high gain at potentials below -10 mV is a result of consolidating the four distinct dyadic subspace compartments utilized in the previous Greenstein-Winslow canine ventricular myocyte model [13] into one, a simplification of the Ca²⁺ release site model resulting in improved computational efficiency, but which attenuates the local [Ca²⁺]_d signal caused by the brief high-amplitude unitary LCC currents characteristic in this potential range.

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Fig 2. Ventricular myocyte model properties.

(a) Normalized peak LCC (triangles) and RyR (circles) flux when the membrane was stepped to test potentials between -20 and 40 mV. (b) Excitation-contraction coupling gain, defined as the ratio of the normalized peak RyR to LCC Ca²⁺ flux at each test potential. (c) Representative action potentials and (d) [Ca²⁺]_i transients under control conditions (solid black), with 50% I_K1 inhibition (dashed black), with β-adrenergic stimulation (solid red) conditions, and with β-adrenergic stimulation and 50% I_K1 inhibition (dashed red). (e) Cell-wide SR Ca²⁺ leak rate via RyRs at varying SR Ca²⁺ loads in the baseline model. The dotted line indicates the lowest SR Ca²⁺ load tested that exhibited spontaneous Ca²⁺ waves. (f) Example linescan plot showing a Ca²⁺ wave under β-adrenergic stimulation.

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Under control conditions, model APs and [Ca²⁺]_i transients are similar to those of normal canine ventricular myocytes [35], with an AP duration (APD), defined as the time to reach 90% repolarization, of approximately 320 ms (Fig 2C and 2D). The model reaches a stable steady state after ~ 10 seconds when paced at 1 Hz (S2 Fig). The NCX current during an action potential (S1 Fig) is in agreement with experimental and theoretical studies [36–38]. Reducing I_K1 density by 50% prolongs the action potential to 469 ms due to the reduction in repolarizing current. Note that this is shown for the first paced beat with identical initial conditions as the baseline model, as the model fails to repolarize under these conditions after continued pacing (see Discussion). Simulating the effect of β-adrenergic stimulation increases the amplitude of the AP plateau as well as [Ca²⁺]_i transient amplitude and decay rate in addition to decreasing the APD to ~255 ms [29]. Additionally reducing I_K1 density by 50% resulted in a marked increase in APD to ~300 ms at steady state. This prolonged AP increased Ca²⁺ loading, resulting in marginally higher systolic [Ca²⁺]_i.

[Ca²⁺]_NSR was clamped at increasing values to test the relationship between SR Ca²⁺ load and leak. The model exhibits an exponential leak-load relationship that is similar to experimental estimates [39, 40] (Fig 2E). Spontaneous Ca²⁺ waves form at a threshold SR Ca²⁺ load, at which wave fronts of propagating Ca²⁺ sparks emanate from random regions of high spark activity. Fig 2F shows a plot of SR Ca²⁺ leak along a cross-section through the center of the cell during a representative Ca²⁺ wave under β-adrenergic stimulation, this wave occurred at a lower SR Ca²⁺ load (82 μmol/L cytosol) compared to under baseline conditions due to greater RyR Ca²⁺ sensitivity. The wave shape and velocity of 68 μm/s are similar to those observed in experimental studies [4].

Delayed afterdepolarizations during pacing

Liu et al. demonstrated that ouabain overdose causes accumulation of [Na⁺]_i, leading to Ca²⁺ overload and DADs [33]. In addition, the authors showed that the production of reactive oxygen species, which are known to oxidize RyRs [41] and CaMKII [42], both of which enhance RyR activity, contributed to DAD generation. To induce Ca²⁺ overload, model parameters were modified to simulate β-adrenergic stimulation and to reflect the conditions in Liu et al. [33] by inhibiting the Na⁺/K⁺ ATPase by 90%, which leads to accumulation of intracellular Na⁺ (>20 mM).

To demonstrate the emergence of DADs in the model, we simulated a protocol in which the RyR opening rate and [Na⁺]_i were controlled over time. β-adrenergic stimulation was applied and the cell was paced at 1 Hz. First, [Na⁺]_i was fixed to 15 mM and the RyR opening rate was ramped from 1.5x to 5x that of baseline over t = 0 to 5 s. Intermittent Ca²⁺ waves, triggered by overload of JSR Ca²⁺, cause spontaneous [Ca²⁺]_i transients, which began to occur (Fig 3A). RyR sensitization initially causes an increase in [Ca²⁺]_i transient peak from 1.73 to 1.93 μM, followed by a decrease (Fig 3B) as the Ca²⁺ wave threshold decreases (Fig 3C) and cellular Ca²⁺ is extruded by NCX. Many of these events begin just prior to (<100 ms before) the next stimulus (Fig 3A inset). Prominent DADs occur at t = 4 and 6 s. This causes lower Ca²⁺ transient amplitude in the following beats due to the lost Ca²⁺ stores. The threshold SR load for Ca²⁺ waves was reduced to 80 μmol/[L cytosol] compared to the baseline model threshold of 140 μmol/[L cytosol] (see Fig 2E). This is due to high [Na⁺]_i and the greater RyR opening rate, which increases Ca²⁺ spark frequency and results in Ca²⁺ wave nucleation and propagation.

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Fig 3. DADs induced by Ca²⁺ overload during 1 Hz pacing under β-adrenergic stimulation in the myocyte model.

(a) Sub-threshold DADs (red arrows) and triggered APs (blue arrows) in between paced APs. Inset illustrates how the first DAD occurred just prior to the next stimulus. (b) Spontaneous [Ca²⁺]_i transients caused by underlying Ca²⁺ waves. Dotted lines indicate diastolic [Ca²⁺]_i. (c) Average [Ca²⁺]_JSR. Dotted lines indicate threshold for Ca²⁺ overload that resulted in spontaneous Ca²⁺ waves. This threshold decreased over the course of the protocol due to increased RyR sensitivity, increased [Na⁺]_i, and elevated diastolic [Ca²⁺]_i.

https://doi.org/10.1371/journal.pcbi.1005783.g003

To induce triggered APs, [Na⁺]_i was ramped from 15 to 23 mM over t = 10 to 15 s. This causes Ca²⁺ waves to form more readily by reducing extrusion of Ca²⁺ via NCX in the SM compartment, thus elevating [Ca²⁺]_d at sites adjacent to Ca²⁺ sparks and increasing the probability of Ca²⁺ spark propagation. Note that elevated diastolic [Ca²⁺]_i has been implicated in DAD formation in experimental studies [8, 43]. Diastolic [Ca²⁺]_i was at first ~140 nM when the DADs are sub-threshold. Higher [Na⁺]_i decreases the delay until the DADs, resulting in higher diastolic [Ca²⁺]_i. A DAD of sufficient amplitude to activate I_Na occurs at t = 18.7 s and triggers a spontaneous AP. Triggered APs result in greater spontaneous [Ca²⁺]_i transients due to activation of LCCs, and the cells exhibit elevated diastolic [Ca²⁺]_i in the range of ~470–520 nM. This causes a reduction in the SR Ca²⁺ load threshold for spontaneous release to 63 μmol/[L cytosol] due to the resulting increase in RyR opening rate and Ca²⁺ spark frequency. Note that under these conditions, triggered APs exhibit pacemaker-like automaticity, occurring every ~570 ms after cessation of pacing (S3 Fig) but stop when [Na⁺]_i is lowered below ~20 mM. This behavior is consistent with spontaneous contractions observed in mouse heart in the presence of ouabain [44].

The protocol also produced changes in AP duration (APD) (S3 Fig). The first beat has an APD of 235 ms, 27% shorter than under normal conditions due reduced inward NCX current in the presence of elevated [Na⁺]_i, [45]. Upon RyR sensitization, APD decreases to ~215 ms due to decreased inward NCX current accompanying the smaller [Ca²⁺]_i transients. Further increasing [Na⁺]_i to 23 mM reduced APD to 150 ms. A previous study showed that APD increases following spontaneous Ca²⁺ release due to slowed Ca²⁺-dependent inactivation of the LCCs [46]. The model does not reproduce this behavior because of the LCC Markov chain has a saturating dependence on [Ca²⁺]_d and is therefore not sensitive to Ca²⁺ load (see Discussion). Rather, the lower [Ca²⁺]_i transient results in less inward NCX current, thus reducing time to repolarization.

These results demonstrate how the model reproduces SR Ca²⁺ overload as a driver of DADs under pathophysiological conditions. Elevated RyR sensitivity and [Na⁺]_i accumulation led to DADs of sufficient amplitude to trigger action potentials during the diastolic intervals. In addition, these results illustrate the interplay between [Ca²⁺]_i and SR load dynamics during spontaneous Ca²⁺ release.

Effect of SR Ca²⁺ load on DADs

The relationship between SR Ca²⁺ load and spontaneous Ca²⁺ release was investigated. Simulations were run using initial conditions reflecting the cell state just prior to the moment when SR Ca²⁺ load reaches the Ca²⁺ wave threshold following an AP. Initial [Ca²⁺]_i was set to 150 nM, similar to the level during the late decay phase of a cytosolic Ca²⁺ transient. Fig 4A shows DADs occurring at the five different values of initial SR Ca²⁺ load shown in Fig 4B. At the highest SR Ca²⁺ load, DAD amplitude is large enough to trigger an AP, as shown in simulation (v) in Fig 4A. Elevating SR Ca²⁺ load reduces the delay until the spontaneous release event, consistent with the observations of Wasserstrom et al. [10]. The increase in DAD amplitude is consistent with a study by Schlotthauer and Bers, who demonstrated increased amplitude of caffeine-induced DADs at higher SR Ca²⁺ loads [47].

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Fig 4. Elevating SR Ca²⁺ load accelerated Ca²⁺ wave formation and increased DAD amplitude.

(a) DADs resulting from initializing SR Ca²⁺ load to five different values as shown in (b). (c) Volume renderings of Ca²⁺ in the simulations at three time points illustrating the greater spontaneous Ca²⁺ wave activity at higher SR Ca²⁺ loads. The number of Ca²⁺ wave nucleation sites (N_nuc) is tabulated for each simulation. All simulations initialized with [Na⁺]_i = 10 mM.

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Fig 4C shows volume renderings of [Ca²⁺]_SM at three time points in each simulation. The number of Ca²⁺ wave nucleation sites (N_nuc) was defined as the number of independently formed wave front epicenters and was estimated by inspection of the volume renderings (S1 Movie). N_nuc generally increases with SR Ca²⁺ load, in agreement with experimental studies in intact heart [9, 10]. Therefore, the increase in SR Ca²⁺ load also increases RyR Ca²⁺ release flux (J_RyR) by enhancing the synchrony of RyR opening and number of nucleation sites.

Ensemble properties of DADs

We hypothesized that stochastic gating of the RyRs drives variability in Ca²⁺ wave dynamics and thus DAD amplitude and timing. To test this, five independent realizations were generated, each of which had identical initial conditions similar to simulation (i) from Fig 4. The pseudorandom number generator seed was varied among the realizations in order to produce independent patterns of RyR gating. Fig 5A shows the resulting DADs, which exhibit marked variability in timing and amplitude. Time of occurrence of the DAD peak varies from 520 to 1209 ms and peak amplitudes range from 2.3 to 6.2 mV. These DADs appear qualitatively similar to experimental observations in rat myocytes, with delays ranging by ~1 s and amplitude ~2-fold, although the SR Ca²⁺ load was not reported [6]. The area under the curve of each DAD, measured relative to the resting potential, varied from 0.79 in (iii) to 1.18 in (ii) (50% greater). This roughly correlated with DAD amplitude, with the exception of the prolonged DAD (iv), which had the second greatest area under the curve despite having the lowest amplitude. Thus substantial DAD variability can be attributed to the stochastic nature of RyR gating.

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Fig 5. Variability of DAD timing and amplitude in five independent model realizations with identical initial conditions.

(a) Variability in DAD timing and amplitude. Inset shows the area under the curve above resting potential of each DAD. Strong correlations were observed between V_max and the maxima of both the spontaneous [Ca²⁺]_i transient (b) and the RyR Ca²⁺ release flux (c). (d) Volume renderings of Ca²⁺ wave dynamics at three time points in each simulation. Approximate locations of the nucleation sites are circled in white. N_nuc was higher in simulations i and ii which had greater DAD amplitude. All simulations initialized with [Na⁺]_i = 10 mM.

https://doi.org/10.1371/journal.pcbi.1005783.g005

Spontaneous Ca²⁺ release generates DADs by driving an inward current through NCX [6]. Imaging, electrophysiological, and modeling [38] studies have all suggested that NCX senses a [Ca²⁺]_SM that is greater than [Ca²⁺]_i because it can be localized near the release sites [16, 20, 21]. Therefore, the driving force for inward NCX current is likely to be determined, in part, by the aggregate number of concurrent Ca²⁺ sparks occurring across the cell. This is consistent with a study showing that peak Ca²⁺ release flux is strongly correlated with the likelihood of ectopic activity [48]. An ensemble of 98 independent simulations were performed to determine how peak [Ca²⁺]_i and J_RyR are related to DAD amplitude. Fig 5B and 5C show there is a strong linear correlation between the peak membrane potential during the DAD (V_max) and the maximum of [Ca²⁺]_i during the spontaneous [Ca²⁺]_i transient (r² = 0.950). There is an even stronger relationship between V_max and the peak J_RyR value (r² = 0.998). This confirms that the inward NCX current is primarily driven by J_RyR via the resulting rise in [Ca²⁺]_SM.

Note that J_RyR reflects the combined Ca²⁺ release flux associated with all Ca²⁺ sparks occurring at any time. It was therefore expected that the variability in DADs was the result of spatio-temporal variations in Ca²⁺ wave dynamics. Fig 5D and S2 Movie show volume renderings of the Ca²⁺ waves in each simulation. Waves emanate from nucleation sites of high Ca²⁺ spark activity. For example, in the simulation (iv) of Fig 5D at 700 ms, a large cluster of Ca²⁺ sparks is visible at the left end of the cell, and by 900 ms a wave front of Ca²⁺ is seen propagating radially away from this cluster site. The random nature of nucleation site locations results in DAD variability across simulations. Simulations i and ii are both associated with the highest-amplitude DADs as well as the greatest number of nucleation sites, as shown in Fig 5D. These nucleation sites are spaced widely across the cell, resulting in independent propagating wave fronts of high [Ca²⁺]_SM that drive inward I_NCX. The remaining three simulations have lower amplitudes due to fewer nucleation sites. Note that in simulation iv, two separate Ca²⁺ waves form over 100 ms apart, resulting in a prolonged low-amplitude DAD with two peaks. These results are consistent with the strong correlation between V_max and maximum J_RyR, which reflects the timing and pattern of Ca²⁺ wave formation.

Dependence of DAD distribution on Ca²⁺ load and I_K1 density

In this section, the statistical relationships between Ca²⁺ loading and DAD amplitude and timing in ensemble simulations are investigated. Fig 6A shows variability in sub-threshold DAD properties when initial SR Ca²⁺ load is varied. The dark lines indicate the median value of membrane voltage over time, and the shaded regions illustrate the range of the second and third quartiles (25^th to 75^th percentile). Consistent with findings from the individual cell simulations of Fig 4, DAD delay decreases and amplitude increases with increasing SR Ca²⁺ load. The peaks of the upper and lower bounds are -92.4 and -90.6, -82.7 and -79.5, and -73.2 and -68.1 mV for initial SR [Ca²⁺] values of 0.32, 0.36, and 0.40 mM, respectively. This suggests that there is an increase in DAD amplitude variability at higher SR Ca²⁺ loads. Note also that the width of the shaded regions decreases as SR load increased, reflecting increased DAD synchrony.

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Fig 6. Roles of cytosolic and SR Ca²⁺ and I_K1 density in DAD distribution.

DAD distributions when initial (a) [Ca²⁺]_SR and (b) [Ca²⁺]_i were varied in the presence of β-adrenergic stimulation. (c) DAD distributions when varying initial [Ca²⁺]_i after reducing I_K1 density by 50%. Shaded regions indicate the middle quartiles of the membrane potential for sub-threshold events out of 96 simulations. Dark lines indicate median values. Statistics of V_max (d) of sub-threshold events, phase plane plot of V and NCX current (e), and statistics of delay until DAD peak (f) are shown when SR [Ca²⁺] was varied (blue), [Ca²⁺]_i was varied (red), and [Ca²⁺]_i was varied with 50% reduction in I_K1 density (magenta) as a function of initial total cell Ca²⁺. Error bars indicate standard deviation (SD) of the estimates. Panel (e) shows representative sub- and supra-threshold DADs for the baseline model and I_K1 reduced by 50% with [Ca²⁺]_i initially at 3.5 and 2 μM, respectively. Arrows indicate direction of time. (g) Dependence of triggered AP probability on initial total cell Ca²⁺. All simulations initialized with [Na⁺]_i = 10 mM.

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Recall that diastolic [Ca²⁺]_i plays a critical role in determining the SR Ca²⁺ wave threshold during pacing (Fig 3). The effect of increasing [Ca²⁺]_i on the DAD distribution with SR Ca²⁺ load held constant was next tested (Fig 6B). Elevating [Ca²⁺]_i yields identical effects as seen when increasing SR Ca²⁺ load, with both reducing the delay and increasing the amplitude of DADs.

The inward rectifier K⁺ current, I_K1, is the primary membrane current that stabilizes V at the cell’s resting potential and plays a critical role in protecting the cell from triggered APs. Down-regulation of I_K1 is associated with ventricular arrhythmias in diseases such as heart failure [49], Andersen’s syndrome [50], and long QT syndrome [51]. Fig 6C shows DAD distributions when I_K1 density is reduced by 50% and [Ca²⁺]_i is varied. These changes increase DAD amplitude and variability compared to cells with normal I_K1. Note that in the 350 nM [Ca²⁺]_i case, only one of the 98 realizations produces a sub-threshold DAD, while all others exhibit triggered APs.

Fig 6D shows the average and SD of V_max as a function of total cell Ca²⁺, defined as the total of buffered and free Ca²⁺ in the cytosol and SR. Changes in either initial [Ca²⁺]_i or [Ca²⁺]_SR (which implicitly include changes in Ca²⁺-bound buffer concentrations to their equilibrium values) are reflected in initial total cell Ca²⁺. The effect of 50% I_K1 density reduction on the [Ca²⁺]_i-dependence of V_max was also analyzed. These results demonstrate two important conclusions.

First, in the baseline model with normal I_K1, the distributions of V_max as a function of initial total cell Ca²⁺ are identical in both cases, as shown by overlap of the blue (initial [Ca²⁺]_i varied) and red (initial [Ca²⁺]_SR varied) traces. This strongly suggests that DADs are driven by both [Ca²⁺]_i and [Ca²⁺]_SR and is consistent with the notion that total cell Ca²⁺ plays a major role in setting the DAD distribution, as observed in other models [52].

Second, reducing I_K1 by 50% causes a marked increase in the average and SD of V_max. In this case, V_max was more sensitive to inward I_NCX during the DAD because I_K1 rectification reduces its outward current upon membrane depolarization which increases the contribution of I_NCX to the net membrane current, and this imbalance between the currents becomes greater with reduction in I_K1 density. This phenomenon is illustrated in a phase plane plot of example simulations at baseline and 50% I_K1 densities exhibiting both sub- and supra-threshold DADs (Fig 6E). For a given NCX current, V rises substantially higher at 50% I_K1 density compared to the baseline model. The rightward shift of the trajectories also illustrates the ~50% reduction in peak NCX current required to trigger an AP (~3.4 to ~1.7 pA/pF). Therefore, DAD amplitudes are greater for a given I_NCX amplitude, causing an increase in the mean. Similarly, amplitude SD is also increased. At the lowest Ca²⁺ load tested, 50% IK1 reduction caused ~3x greater SD, which is mainly accounted for by the fact that the amplitude is on average ~2.6x greater.

DAD delay and synchrony is also strongly correlated with total cell Ca²⁺, as measured by the distribution of the time until the DAD peak occurred (Fig 6F). Increasing initial total cell Ca²⁺ reduces DAD delay and increases synchrony, similar to the observations of Wasserstrom et al. [10]. In the baseline model, the standard deviation of the delay was 178 ms at the lowest SR load tested. This variability is similar to that measured by Wasserstrom et al. (162 ms) in whole rat heart paced at 1 Hz and elevated [Ca]_o = 7.0 mM. DAD delay does not considerably change with 50% I_K1 reduction because it is determined primarily by the timing of Ca²⁺ wave formation, which is not affected by I_K1 density.

With increasing initial total cell Ca²⁺, in the form of increased initial [Ca²⁺]_i or [Ca²⁺]_SR, the higher DAD amplitudes makes it more likely that the threshold membrane potential for triggering an AP (~ -55 mV) will be reached. The probability that a DAD-triggered beat occurred is shown in Fig 6G. Increasing cell Ca²⁺ results in a steep increase in trigger probability. This switch-like behavior is due to the DAD amplitude reaching threshold AP-triggering potential with high certainty. Reducing I_K1 by 50% reduces the critical Ca²⁺ load at which APs are triggered, consistent with the observed increase in DAD amplitude. The effect is significant: at normal I_K1 density zero of 96 cells exhibit triggered APs when initial [Ca²⁺] is 5.8 fmol, but reducing I_K1 density by 50% increases the probability to ~ 0.5. Thus I_K1 density plays a critical role in modulating the Ca²⁺ load at which arrhythmia-triggering events occur.

Probabilistic triggered activity in a paced fiber of myocytes

In the previous section, it was shown that triggered APs occur with a probability that depends on Ca²⁺ load and that varying I_K1 density shifts the Ca²⁺ load dependence. Simulations were performed to test whether the model could produce probabilistic triggered activity in a 1D fiber of myocytes during pacing, where electrotonic loading of any cell by adjacent cells becomes important. Fig 7A shows the membrane potential of a fiber paced at 0.5 Hz under conditions similar to those in Fig 3 with elevated [Na⁺]_i set to 19 mM, as may be observed in heart failure [53]. Note also that [Na⁺] sensed by membrane channels during contraction may be higher than [Na⁺]_i in the bulk cytosol due to local buildup of Na⁺ imported by NCX [54]. To reflect a state of pathological remodeling, I_K1 density and gap junction conductance were each reduced by 50% as observed in HF [49, 55]. To limit boundary effects, [Na⁺]_i was set to 10 mM in the outer twenty-four cells near each end of the cable in order to prevent DAD generation. DADs, appearing as faint bands of depolarization between the paced beats, and reach V_max values between approximately -70 and -60 mV. This is consistent with experimental observations of synchronized spontaneous Ca²⁺ release causing DADs in intact heart following rapid pacing [9].

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Fig 7. Probabilistic triggered activity caused by DADs in the fiber model.

(a) Fiber of 192 cells paced at 0.5 Hz by applying a stimulus current for 2 ms to the first two cells in the fiber (black arrows). DADs occurred between paced beats (red arrows). (b) Ensemble simulations (i)-(iv) initialized to the state at time t_restart in panel (a). Realization (i) corresponds to the simulation in panel (a). Enhanced view of (iv) is shown at right.

https://doi.org/10.1371/journal.pcbi.1005783.g007

The model exhibited considerable variability in V_max along the length of the fiber. To investigate the source of this variability, the state of the model at time t_restart immediately after the third paced beat was recorded. A set of three independent realizations were run, each starting from this same model initial condition but with simulations performed using different initial pseudorandom number generator seeds. These realizations therefore differ in the particular pattern of LCC and RyR channel gating.

Fig 7B shows: (i) the DAD from the original simulation in Fig 7A at 4.9 s; and (ii)-(iv) the three additional simulations initialized at t_restart. Each simulation exhibits substantial differences in DAD amplitude along the fiber. In panel (iv), a spontaneous traveling action potential wave is observed. The wave originates from a region of locally high DAD amplitude near cell #50 and emanates in either direction along the fiber.

These results illustrate that arrhythmic events can occur probabilistically in tissue. By restarting the simulations immediately prior to the DADs, it became evident that the variability in DAD amplitude was due to variations in the stochastic events that occurred in this brief time window. Taken together with the results from Figs 5 and 6, the variability in DAD amplitude in the fiber can therefore be attributed to the underlying randomness of Ca²⁺ wave dynamics and thus stochastic RyR gating.

Roles of Ca²⁺ loading, I_K1 density, and g_gap in fiber DADs

Conditions reflecting pathological remodeling influenced the distribution of V_max in the fiber. Fig 8A shows fiber simulations when all cells are initialized to identical initial conditions with Ca²⁺ overload and β-adrenergic stimulation. The resulting DADs resemble those from the paced fiber shown in Fig 7. As in individual cells, V_max increases with initial [Ca²⁺]_i. At the highest [Ca²⁺]_i, V_max lies between -74.1 mV and -70.7 mV, a range of 3.4 mV. There is notably less variability than in individual cells at a similar Ca²⁺ load (SD of 0.7 compared to 4.6 mV, see Fig 6D). This reduction in variability is due to electrotonic coupling through inter-cellular gap junctional conductance, g_gap, which attenuates spatial gradients in V.

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Fig 8. Roles of initial [Ca²⁺]_i, I_K1 density, and g_gap in DAD variability in the fiber model.

(a) Spatiotemporal profile of V (left) and V_max profile (right) in a 480-cell fiber under baseline conditions. Similar simulations are shown with 50% I_K1 reduction (b) and both 50% I_K1 and 50% g_gap (c). The range of V_max values are indicated for the red traces, which correspond to the images on the left. To avoid boundary effects, the outer 24 cells on either end were initialized to normal SR loads. The inner cells were initialized to identical initial conditions to those of Fig 6. All simulations initialized with [Na⁺]_i = 10 mM.

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Fig 8B shows similar simulations where I_K1 density was reduced by 50%. This results in greater fluctuations of V_max over a range of 8.0 mV (SD 1.5 mV) compared to baseline. This is consistent with the increase in V_max SD from 1.7 to 6.1 mV when I_K1 density was decreased to 50% in isolated cells (see Fig 6D) and also reflects a substantial reduction in V_max variability due to electrotonic coupling in the fiber. Reducing g_gap by 50% in addition to the 50% I_K1 reduction further amplifies V_max spatial fluctuations, resulting in a range of 10.2 mV (SD 2.2 mV, Fig 8C). The increase in variability arises from the reduced electrotonic load experienced by each cell. These results demonstrate how perturbations of I_K1 and g_gap increase the likelihood of observing large DADs.

Overview of method for estimating rare event probabilities

The DAD results presented thus far indicate that spatial fluctuations in V_max can result in a triggered beat emanating from a cluster of cells in which V_max exceeds the threshold voltage (~ -55 mV). For a fiber of a given length, the probability of this event depends upon both the mean V_max and the likelihood of a deviation from the mean with sufficient amplitude to reach threshold. Under conditions where the mean V_max is far below threshold, however, it is unclear whether it would be plausible to observe a sufficiently large fluctuation that causes a triggered beat. We sought to characterize the likelihood of such events by estimating the upper tail of the V_max distribution.

In Fig 5, we showed that V_max was strongly correlated with J_max in DAD simulations of isolated cells. In tissue, however, electrotonic coupling attenuates DAD amplitude by drawing current to neighboring cells via gap junctions. However, if spontaneous Ca²⁺ release occurs synchronously in a cluster of cells, the potential gradient between cells is smaller, thus reducing the gap junction current and increasing DAD amplitude. Therefore the V_max of a given myocyte in tissue depends on the relative timing and amplitude of Ca²⁺ release in nearby cells.

We hypothesize that extreme deviations in V_max are caused by rare spatially clustered synchronized Ca²⁺ release events. However, estimating the probability of extremely rare events (e.g. 1 in 10⁶) by direct simulation is computationally prohibitive using the full biophysical model since it would require potentially millions of simulations. Therefore, a method was developed for estimating the probability of such events using the output from a limited set of simulations.

As shown in the previous sections, stochastic RyR activity gives rise to random Ca²⁺ wave patterns and thus variable profiles of J_RyR. Therefore J_RyR is a stochastic process that is dependent on complex microscopic events, which are computationally intensive to sample. The method leverages the fact that J_RyR has a relatively simple distribution by resampling J_RyR profiles from a set of simulations that is sufficiently large to approximate the distribution of J_RyR. Multiple realizations of release events (i.e., the J_RyR values from each separate simulation of the full cable model) are generated using a resampling method in which cell positions along the cable are shuffled to produce independent, distinct realizations. The key to this approach is the fact that spontaneous Ca²⁺ release events are decoupled across the cells in the fiber (see Discussion).

In principle, in using this approach one could develop a modified tissue model in which J_RyR in each cell is fixed to one of the resampled J_RyR profiles rather than being simulated, thus greatly reducing the computational burden. However, this would still require immense computational power to simulate the remaining differential equations when estimating the probability of extreme events. The second part of the method further increases computational efficiency by fitting a linear spatial-averaging model to predict V_max.from J_RyR alone. This permits the rapid estimation of the greatest V_max in a 1D tissue model in millions of simulations and thus can estimate the probability that the cable will reach a proarrhythmic threshold potential.

The following sections describe this method and validate its accuracy for estimating DAD probabilities. We then apply this new technique to show how myocyte properties within a fiber, using I_K1 density and g_gap as examples, affect the likelihood of rare large amplitude DADs.

Filtering method for estimating V_max from J_RyR

A filtering method was developed for estimating V_max from the spatiotemporal profile of J_RyR in a single 1D fiber simulation. The left column of Fig 9A shows the simulation from Fig 8A where [Ca²⁺]_i was initialized to 300 nM, and the right column illustrates the steps used in the filtering method. The first step in the method is to apply a uniform averaging filter to J_RyR at each point in time to obtain a spatially smoothed profile (4) where x refers to cell index, t refers to time, and W is the width of the filter (an odd integer). The rational for doing this is that membrane potential fluctuations of the cell at position x is influenced not only by the way its own complement of NCX responding to the local J_RyR, but is also influenced by membrane potential fluctuations produced in response to J_RyR and NCX activity in neighboring cells. The filter width W will therefore depend on the strength of gap junction coupling. For each cell, the maximum J^f_RyR value over all time was computed as (5)

The value of W maximizing the correlation coefficient between V_max and J^f_max was calculated at each different value of gap junction coupling conductance used. J^f_max was then normalized to obtain estimates of V_max using the formula 6 where μ_V and σ_V are estimates of the mean and SD respectively of V_max, and μ_J and σ_J are estimates of the mean and SD respectively of J^f_max. These estimates are calculated from a single full 1D tissue simulation containing ~ 450 cells. Note how in the example of Fig 9A the estimated voltage profile V^f closely resembles that of V. This approach therefore leverages the strong linear correlation between J_RyR and V_max (Fig 5B and 5C) and the statistical independence of Ca²⁺ release events to dramatically reduce the number of simulations needed to estimate the probability distribution of membrane potential fluctuations.

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Fig 9. Filtering method for estimating V_max from the spatiotemporal J_RyR profile in a fiber.

(a) Illustration of the filtering method for an example fiber simulation. A uniformly-weighted spatial smoothing filter of width W was applied to the spatiotemporal J_RyR profile. The maximum values of the filtered profile were then normalized to obtain an estimate of the voltage profile (see Eq 6). (b) The filtering method was applied to three fiber simulations using the conditions indicated above each plot. The filtering method estimate (red) accurately reproduced V_max from the original simulation (black). Parameters are listed in Table 1.

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The filtering method was applied to simulations of DADs with baseline conditions, with 50% I_K1, and with both 50% I_K1 and 50% g_gap (Fig 9B). In the non-baseline conditions, initial [Ca²⁺]_i was adjusted from 300 to 220 nM so that μ_V would be approximately equal to that of the baseline. The width W of the smoothing filter is dependent on the fiber model parameters and therefore was optimized separately for each case. Table 1 shows the parameter fits for each condition. The resulting fits of V^f_max to V_max have high correlation coefficient values (r² ≥ 0.90). All values of μ_V fall within a narrow range from -77.0 mV to -79.1 mV, while the values of μ_J in the two non-baseline conditions are less than half of the baseline value due to their lower Ca²⁺ loads.

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Table 1. Parameters used for filtering method in Fig 9B.

https://doi.org/10.1371/journal.pcbi.1005783.t001

The increase in V_max variability in the two pathological conditions is reflected in the parameters of the filtering model. It can be shown that the quantity S_V = (σ_V/σ_J)/W scales with the SD of V^f_max (see S1 Text). Recall that 50% I_K1 reduction increased variability of V in the cell model (Fig 6). For this condition in the fiber, S_V is 28% larger compared to baseline, primarily reflecting the higher value of σ_V. Imposing 50% g_gap results in a filter width of only 27 cells compared to 43 and 49 in the other cases due to local decoupling of cells when gap junction conductance is decreased. This resulted in an S_V that is 87% larger than baseline and 46% larger than with 50% I_K1 alone.

These results show that the filtering method accurately estimates fluctuations in V_max based on the J_RyR profile. Moreover, the new empirical relationships derived here between J_RyR and V_max provide an approach to yield new insight and intuition into how changes in cellular properties and environment, such as reductions in outward currents and electrotonic coupling analyzed here, alter and in this case enhance spatial fluctuations in V_max.

Resampling method for estimating rare event probabilities

The filtering method enables studies of properties of V_max without the need to simulate the full biophysical model. Studying the statistical properties of V_max requires the generation of large numbers of realizations. To do this, independent samples of V_max were generated by shuffling cell positions in the J_RyR profile, as illustrated in Fig 10A. Note that boundary effects the outer 24 cells on either end of the fiber are initialized to normal SR Ca²⁺ loads and are not included in the shuffle. The filter method was then applied to the shuffled J_RyR profile to estimate V_max. An example V_max profile obtained using this method is shown on the right. Note that the fluctuations about the mean μ_V are qualitatively similar to those of the original simulation. In Fig 7, it was shown how a triggered propagating ectopic beat originated from a region of the fiber that experienced extreme DADs. Unfortunately it is impossible to perform the millions of cable simulations using the full biophysical model that are needed to study the probability of generating ectopic beats as model parameters are varied. However, our shuffling and filtering method does enable us to study the probability of a surrogate event of interest for each fiber simulation, with the surrogate event defined as the greatest value of V_max realized along the fiber, referred to here as V_peak.

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Fig 10. Method for estimation of rare extreme DAD probabilities.

(a) Example illustrating how independent fiber realizations are generated by shuffling cell positions and applying the filtering method. The resulting V_max profile of the original simulation (black) and V^f_max obtained after shuffling (red) are plotted on the right. V_peak is defined as the maximum potential achieved along the length of the fiber. (b) Histograms of V_peak from model simulations (black), the filtering-only method (blue), and filtering and shuffling method (red). See text for details. (c) Upper tail of the distributions from (b). (d) Predicted tail of V_peak distributions plotted relative to μ_V for fibers with the baseline model (blue), with 50% I_K1 reduction (red), and with both 50% I_K1 reduction and 50% g_gap (magenta).

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The shuffling of cell positions makes two assumptions. The first is that the J_RyR profiles of the cells are decoupled stochastic processes. The second assumption is that the fiber contains a sufficient number of cells such that the true distribution of J_RyR is well represented by the collection obtained from a single simulation.

Previous work has shown that sub-threshold membrane depolarization induces Ca²⁺ sparks and waves [56]. It is unclear whether local depolarizations in the membrane potential caused by spontaneous release affects release in neighboring cells, which would invalidate the independence assumption. This was tested by computing the peak amplitude and peak time of [Ca²⁺]_i in the fiber simulation represented by the red trace in Fig 8C. The absolute difference in the peak amplitude and peak time were then computed for all adjacent cell pairs and all cell pairs separated by a 50-cell distance along the fiber. Assuming that Ca²⁺ release in 50^th neighbors are decoupled, if release in adjacent cells were coupled then one would expect the distributions of peak amplitude and timing to differ from those of the 50^th neighbor pairs. In particular, if local Ca²⁺ release events were more synchronized, one would expect the difference in timing to be smaller. However, no substantial differences in Ca²⁺ release peak or timing were observed (S4 Fig), as these distributions also were not significantly different according to non-parametric Kruskal-Wallis tests (p = 0.77, 0.66). Therefore the coupling effect of adjacent cells does not substantially affect spontaneous Ca²⁺ release, which can thus be treated as a decoupled process in each cell in the fiber.

Validation of these assumptions also requires that the distribution of V_peak generated using this method match that obtained when using the detailed biophysical model. This distribution was estimated from 1,000 independent realizations. Simulating the ~500-cell fiber with 25,000 release sites in each cell is computationally prohibitive. For the purposes of the validation, the number of release sites was reduced to 2,500 and the fiber length to 96 cells. The values of μ_V and σ_V were computed using the V_max values from all cells in all fibers.

Fig 10B compares the true distribution of V_peak computed using the full biophysical model to that obtained from applying only the filtering method to the simulated J_RyR profiles (without shuffling). While this distribution exhibits a very small bias of ~ -0.05 mV, it is not significantly different from the true distribution on the basis of a Kruskal-Wallis test after correcting for the bias by subtracting their means (p = 0.89). The shuffling method was then validated by resampling from the J_RyRs in all 1,000 fibers to produce 1,000 96-cell fiber realizations and computing V_peak with the filtering method. While the resulting distribution also exhibits a very small bias of ~ -0.08 mV, it is not significantly different from the true distribution after adjusting the means (p = 0.69) (Fig 10B). The upper tails of the distributions containing the extreme DADs of interest are also similar (Fig 10C). To validate the second assumption that the sampling population of J_RyR is sufficiently large, these tests were repeated using a subset of 5 of the 1,000 fibers to compute μ_V, σ_V, and the population of resampled J_RyR’s (bias +0.05 mV, p = 0.68). Therefore, the method is accurate using a total of ~ 500 simulated cells. In summary, the method outlined here is a computationally efficient approach that permits the rapid estimation of the V_peak distribution using output from a single 500-cell fiber simulation.

Prediction of rare events

Fig 10D plots the probability that V_peak—μ_V exceeds a given potential estimated from an ensemble of 10⁶ realizations. Reducing I_K1 by 50% shifts the distribution tail to greater amplitude DADs compared to the baseline model. The probability of observing a V_peak 3 mV greater than μ_V (-77.1 mV) is 9.9 × 10⁻⁵ in the baseline model compared to 0.096 with 50% reduction in I_K1, a nearly 1000-fold increase. The most extreme event observed with 50% I_K1 was 6.1 mV greater than μ_V compared to 3.4 mV in baseline, corresponding to DAD amplitudes 41% and 21% above average, respectively. Reducing g_gap by 50% further increases the likelihood of large-amplitude DADs, with the probability of a 3 mV DAD reaching 0.50, a 5000-fold increase over baseline, and the most extreme event observed at 7.5 mV, corresponding to a DAD amplitude 53% higher than average. These results demonstrate how changes ion channel expression or function can alter the probability of supra-threshold membrane potential fluctuations. In this case, reductions in I_K1 and g_gap considerably widen the distribution of V_peak and increase the probability of occurrence of larger V_peak values by multiple orders of magnitude.

To illustrate the nature of these rare events, the realizations exhibiting the greatest V_peak values were examined in each condition. Fig 11A plots V_max—μ_V. Because V_max at any position is determined by an average of the Ca²⁺ release profiles over a window of cells, each rare depolarization occurs at a cluster of cells whose width corresponds to the window size. Fig 11B shows the underlying J_RyR profiles and the filter window centered on the V_peak. In each case, the extreme event occurs at a cluster of cells within the window where J_RyR tends to be greater and more synchronized than in the rest of the fiber. This result demonstrate that rare (occurring ~1 in 10⁶ beats in a 500-cell fiber) proarrhythmic events result from spontaneous Ca²⁺ release exhibiting high (1) flux magnitude, (2) synchronization, and (3) spatial clustering. None of the 999,999 other simulations result in such an extreme event due to a lack of one or more of these properties. Note that because the window size is smaller in the case of 50% g_gap, fewer cells need to fulfill these three criteria and thus the probability of extreme events is much higher.

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Fig 11. Realizations of rare extreme DADs.

(a) Plots of V_max—μ_V in the realizations from Fig 10D containing the most extreme DADs out of 10⁶ trials. Curves correspond to the baseline model (blue), 50% reduction of I_K1 (red), and both 50% I_K1 and 50% g_gap (magenta). (b) Spatiotemporal J_RyR profiles in each realization from panel (a). Brackets and dotted lines mark the location of the filter window centered on the extreme DAD.

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