Ryanodine receptor sensitivity governs the stability and synchrony of local calcium release during cardiac excitation-contraction coupling

Abstract

Calcium-induced calcium release is the principal mechanism that triggers the cell-wide [Ca²⁺]_i transient that activates muscle contraction during cardiac excitation-contraction coupling (ECC). Here, we characterize this process in mouse cardiac myocytes with a novel mathematical action potential (AP) model that incorporates realistic stochastic gating of voltage-dependent L-type calcium (Ca²⁺) channels (LCCs) and sarcoplasmic reticulum (SR) Ca²⁺ release channels (the ryanodine receptors, RyR2s). Depolarization of the sarcolemma during an AP stochastically activates the LCCs elevating subspace [Ca²⁺] within each of the cell’s 20,000 independent calcium release units (CRUs) to trigger local RyR2 opening and initiate Ca²⁺ sparks, the fundamental unit of triggered Ca²⁺ release. Synchronization of Ca²⁺ sparks during systole depends on the nearly uniform cellular activation of LCCs and the likelihood of local LCC openings triggering local Ca²⁺ sparks (ECC fidelity). The detailed design and true SR Ca²⁺ pump/leak balance displayed by our model permits investigation of ECC fidelity and Ca²⁺ spark fi-delity, the balance between visible (Ca²⁺ spark) and invisible (Ca²⁺ quark/sub-spark) SR Ca²⁺ release events. Excess SR Ca²⁺ leak is examined as a disease mechanism in the context of “catecholaminergic polymorphic ventricular tachycardia (CPVT)”, a Ca²⁺-dependent arrhythmia. We find that that RyR2s (and therefore Ca²⁺ sparks) are relatively insensitive to LCC openings across a wide range of membrane potentials; and that key differences exist between Ca²⁺ sparks evoked during quiescence, diastole, and systole. The enhanced RyR2 [Ca²⁺]_i sensitivity during CPVT leads to increased Ca²⁺ spark fidelity resulting in asynchronous systolic Ca²⁺ spark activity. It also produces increased diastolic SR Ca²⁺ leak with some prolonged Ca²⁺ sparks that at times become “metastable” and fail to efficiently terminate. There is a huge margin of safety for stable Ca²⁺ handling within the cell and this novel mechanistic model provides insight into the molecular signaling characteristics that help maintain overall Ca²⁺ stability even under the conditions of high SR Ca²⁺ leak during CPVT. Finally, this model should provide tools for investigators to examine normal and pathological Ca²⁺ signaling characteristics in the heart.

1. Introduction

The coupling of electrical stimuli to cardiac contraction requires the synchronous release of calcium (Ca²⁺) from thousands of Ca²⁺ release units (CRUs) across each cardiac myocyte. Local control of Ca²⁺ release at individual CRUs is an important feature of excitation-contraction coupling (ECC) that is required for the “graded” release of sarcoplasmic reticulum (SR) Ca²⁺ during systole [1,2]. Each CRU is composed of a number of (~6) L-type Ca²⁺ channels (LCCs) within the sarcolemmal (SL) transverse tubule (TT) membrane or exterior SL membrane that are positioned across a dyadic “subspace” (~15 nm) from a para-crystalline array of (~10–300) type 2 ryanodine receptors (RyR2) within the junctional SR (JSR) membrane [3,4]. During systole, the action potential (AP) depolarization of the sarcolemma (including the TTs) activates the LCC current which elevates local (i.e., subspace) Ca²⁺ to activate and open the clustered RyR2s in the JSR to produce a “triggered” Ca²⁺ spark. The Ca²⁺ spark is the fundamental unit of Ca²⁺-induced Ca²⁺-release (CICR) at a given CRU. The signal process is a high-gain positive feedback mechanism, which can operate with great stability due to the local Ca²⁺ signaling. It is the stochastic recruitment of triggered Ca²⁺ sparks that translates the electrical AP to the [Ca²⁺]_i transient and the cardiac contraction during ECC. Despite over 20 years of experimental and computational study, questions still abound regarding quantitative details of Ca²⁺ sparks and CICR dynamics at individual CRUs during systole and diastole.

Many mathematical models have been developed to investigate the mechanisms governing CICR and ECC. However, to our knowledge, a fully stochastic, local control computational model of mouse AP dynamics with a realistic CRU formulation and physiological SR Ca²⁺ pump/ leak balance has not been produced, until now. We have constrained the biophysical parameters of our model to match those of the mouse ventricular myocyte, as the mouse is by far the most commonly studied animal in biological research. To date, one of the most influential mouse cardiomyocyte models was developed by Bondarenko and colleagues in 2004 [5]. Their model has been adapted by numerous groups to study the molecular regulation of ECC, myosin cross-bridge cycling and contraction, as well as whole-heart function [6–11]. While these modeling efforts have provided exciting new data, they contain two critical features, common in many cardiac Ca²⁺ signaling models, that limit their mechanistic relevance: 1) They use a single “local [Ca²⁺]_i” level that is identical for all RyR2s and LCCs (i.e., common pool) [5,12–22]. 2) They use non-physiological RyR2 gating schemes [23–25] or terminate Ca²⁺ release from the SR using a non-physiological [Ca²⁺]_i-dependent inactivation scheme for RyR2 [2,21,26–32]. These two features are utilized in many cardiomyocyte computational models across species to either alleviate computational demand, enable Ca²⁺ spark termination, or both. We find that under normal conditions, using a physiological description of RyR2 gating, depletion of the [Ca²⁺]_jsr promotes the stochastic termination of Ca²⁺ sparks in a realistic timespan [33,34]. The model presented here does not contain either of these two non-physiological features; and thus, our mathematical description of cardiac ECC is the most mechanistic to date.

Here, we advance our previous computational model formulation presented in Williams et al., 2011 from a local control, quiescent cardio-myocyte model with physiological RyR2 gating and realistic local Ca²⁺ signals to a full ECC model with systolic Ca²⁺ signaling and robust AP generation. We have previously shown how three SR Ca²⁺ leak pathways (Ca²⁺ sparks, Ca²⁺ quarks, and rogue RyR2s) combine to balance the activity of the SR Ca²⁺-ATPase (SERCA2a) during prolonged periods of diastole, or quiescent conditions [34]. During diastole, spontaneous Ca²⁺ sparks form the primary pathway for SR Ca²⁺ leak. Additional Ca²⁺ release occurs through “non-spark” events that represent the opening of single RyR2s (i.e., Ca²⁺ quarks) or the opening of only a few RyR2s (i.e., Ca²⁺ sub-sparks) which fail to induce a full Ca²⁺ spark and are invisible by conventional confocal imaging modalities [35]. There is also a small population of RyR2s distributed sporadically throughout the network SR, known as “rogue” RyR2s, that form a third route for SR Ca²⁺ leak [4,36,37]. SR Ca²⁺ pump/leak balance is a critical feature of cardiomyocyte Ca²⁺ homeostasis required for normal muscle function and overall survival. Realistic SR Ca²⁺ leak/pump dynamics gives the model significant relevance when studying pathological conditions related to changes in SR Ca²⁺ release (i.e., high “leak” conditions) or elevated SR Ca²⁺ content (i.e. load).

The pathophysiological Ca²⁺ signaling that arises from “leaky” RyR2s has been linked to diverse cardiac pathologies including catechol-aminergic polymorphic ventricular tachycardia (CPVT), an inherited ar-rhythmogenic disorder characterized by syncope and sudden cardiac death. While specific mutations may confer different pathologies in CPVT (e.g., autosomal dominant RyR2 mutations and autosomal recessive calmodulin, triadin, and calsequestrin (CASQ2) mutations), the development of overly sensitive RyR2s (assuming identical SR Ca²⁺ load) that display increased P_O appears to be a common theme in CPVT [38]. The model presented here is uniquely capable of investigating abnormal Ca²⁺ leak dynamics such as this because of the mechanistic formulation of Ca²⁺ sparks and SR Ca²⁺ leak it contains. Here we conduct a quantitative assessment of quiescent, diastolic, and systolic Ca²⁺ sparks and “invisible” Ca²⁺ leak pathways in the environment of enhanced RyR2 P_O and reduced JSR buffering, designed to simulate a genetic “knock-in” murine model of CASQ2-R33Q driven CPVT [39]. Our findings support those previously identified experimentally for the basal differences between wild-type (WT) and CASQ2-R33Q mutant cardiomyocytes (i.e., decreased amplitude of global [Ca²⁺]_i transients at slow, 0.2 Hz, pacing and decreased total SR Ca²⁺ content in CPVT myocytes) [40–43]. Our model identifies new critical differences in local calcium signals that may ultimately serve as the substrate for cellular arrhythmia (Ca²⁺ waves) and the rare, but deadly arrhythmia, CPVT.

The physiological, mathematical model of ECC presented here provides novel, important findings that change our understanding of Ca²⁺ signaling at both the local and global (cell-wide) levels. One key achievement in this model is a novel formulation for the LCC that is sensitive to changes in local Ca²⁺, a critical feature necessary for local control of CICR. Important new findings of this model include the following: 1) Quiescent, diastolic, and systolic Ca²⁺ sparks differ in a number of ways including frequency and amplitude; 2) Quantitative details describing the relative insensitivity of RyR2s (and, therefore, Ca²⁺ sparks) to LCC openings; 3) Simulated CPVT mutations lead to prolonged and at times, metastable Ca²⁺ sparks that increase the probability of activating neighboring CRUs; and 4) Ca²⁺ spark fidelity is significantly altered during CPVT and results in asynchronous systolic Ca²⁺ spark activity and increased diastolic SR Ca²⁺ leak. This model accounts for how local Ca²⁺ and realistic SR Ca²⁺ pump/leak balance influences global Ca²⁺ dynamics during ECC. These features offer a critical step toward development of a realistic whole cardiomyocyte spatial model of ECC.

1.1. Model formulation

Here, we present a whole-cell local control, Monte Carlo model of ECC in the mouse cardiac ventricular myocyte. The description of CICR from Williams et al. has been modified to include a novel stochastic model for LCCs that completes the physiologic CRU [34]. The stochastic RyR2 formulation has been updated to integrate the most recent biophysical characterization of the channel. Additionally, key SL membrane channels and transporters (see Fig. 1 A) have been incorporated into the model to facilitate robust AP generation. The resulting compartmental model of a mouse ventricular myocyte has numerous independent CRUs (i.e., N = 20,000), each containing multiple stochastically gated LCCs and RyR2s (i.e., N_L = 6 and N_R = 50, respectively) that are instantaneously coupled (see Eq. (S56)) to a local, dyadic subspace [Ca²⁺] ([Ca²⁺]_ds) [34]. Each CRU is associated with an individual junctional SR (JSR) compartment that is depleted during its own CICR and is refilled through transport from the network SR (NSR) compartment. Changes in bulk myoplasmic [Ca²⁺] ([Ca²⁺]_i) are generated via efflux of Ca²⁺ from each individual CRU subspace compartment (see ${\mathrm{J}}_{\text{efflux}}^{\mathrm{n}}$ in Fig. 1 A). The model contains physiological representations of SERCA2a (Eq. (S35)), Na⁺/Ca²⁺ exchanger (NCX) (Eq. (S37)), and the plasma membrane Ca²⁺ ATPase (PMCA) (Eq. (S38)) to facilitate SR Ca²⁺ re-uptake and myoplasmic Ca²⁺ extrusion. For this study, we assume no [Ca²⁺]_i buffering by the mitochondria and do not include a compartment for this organelle nor formulations for the associated Ca²⁺ handling proteins. Recent studies have shown that under physiological conditions there is minimal mitochondrial contribution to direct changes in [Ca²⁺]_i signals [44–47]. The Monte Carlo model presented here consists of 2N + 20 ordinary differential equations (ODEs) representing the time-evolution of membrane voltage (V), various in-tracellular ion concentrations (i.e., [Ca²⁺], [K⁺], [Na⁺]), and membrane current gating variables (see Eqs. (S9)–(S17)). The concentration balance equations, consistent with Fig. 1, can be found in the SI (Eqs. (S2)–(S7)). Each ODE was solved in Matlab using the first-order Euler method with variable time-step designed to ensure stability. A mathematical description of critical model components is presented below and includes: LCC, RyR2, SL membrane potential, and spatial sarcomere model.

Fig. 1.

Diagram of ECC model and Ca²⁺ channel models. (A) Model compartments, Ca²⁺ fluxes, and various membrane currents (black arrows). Only one of the cell’s N = 20,000 CRUs is shown for visual clarity and I_K+ represents all K⁺ currents described fully in Supporting Materials. (B) Transition-state diagram for the 7-state continuous-time, discrete-state Markov chain describing a single LCC. Transitions labeled as; v_a represent voltage-dependent activation transitions, v_i represent voltage-dependent inactivation transitions, v_d represent voltage-dependent deactivation transitions, and c_i represent Ca²⁺-dependent inactivation transitions (see SI for formulation of these rates). (C) Transition-state diagram for the 2-state continuous-time, discrete-state Markov chain representing a single RyR2 channel. Note, each CRU consists of 6 LCCs and 50 RyR2s.

1.1.1. LCC model

While numerous LCC models exist [14,25,29,48–52], few are capable of generating realistic LCC current in a local control model. Specifically, the dynamics of Ca²⁺-dependent inactivation must be modified when an LCC is experiencing realistic local Ca²⁺ concentrations (i.e., [Ca²⁺]_ds). We developed a novel LCC formulation where each voltage activated LCC is represented by a 7-state Markov chain containing six inactivated or closed states and one open state (state 7 represented in white in Fig. 1B). Voltage-dependent inactivation (VDI) is achieved by transitions from either the open state (state 7) or the closed state (state 2) to the VDI states (states 1 and 6) highlighted in blue (see Fig. 1B). Similarly, Ca²⁺-dependent inactivation (CDI) is modeled by transitions from either the open state or the closed state to the CDI states (states 3 and 4) highlighted in green (see Fig. 1B). State 5 highlighted in red (see Fig. 1B) represents a transient “activated” (but not open) state designed to achieve physiological LCC current rise times. Fig. S2 A shows that the model generates a realistic IV curve peaking between 0 and 10 mV, consistent with experimental results [53]. The LCC dynamics in response to both voltage- and Ca²⁺-dependent inactivation is displayed in Fig. S2B. While similar in number of states to Mahajan et al. [25], our novel description of transition rates between LCC states facilitates our ability to model local control of ECC by: 1) making the channels sensitive to realistic [Ca²⁺]_ds, and 2) permitting the use of a our vectorized channel gating approach to solving stochastic channels [54]. The second point is critical in order to reduce the computational demand of six 7-state LCCs present in each of our N = 20,000 CRUs. Junctional and non-junctional LCC formulations and the kinetics for LCC transitions can be found in the SI (see Eqs. (S25)–(S34) and Table S3).

1.1.2. RyR2 model

Recent work in lipid-bilayers by Michael Fill and co-workers [39] as well as others [55–57] has shown that under “near-physiological” conditions (e.g., Mg²⁺ and SR Ca²⁺) RyR2s are much less sensitive to Ca²⁺ than previously thought. We have modified our RyR2 model (depicted in Fig. 1C) to account for these updated experimental characterizations of RyR2 Ca²⁺ sensitivity. Our current RyR2 gating leads to a EC₅₀ of ~33 μM compared to our earlier work ([34]) where RyR2 EC₅₀ was closer to 10 μM. This change in RyR2 [Ca²⁺]_i sensitivity has allowed us to remove the allosteric coupling factors previously required to “quiet” the RyR2 channel gating under quiescent conditions. This assumption does not preclude a physical connection between neighboring RyR2s, but simply does not assert that this physical connection is the basis for propagating conformational changes between RyR2s. Additionally, we have updated our luminal Ca²⁺ sensitivity function, ϕ (see Eq. (S45) and Walker et al., 2014 [58]), which reflects the most recent biophysical characterizations of RyR2 gating with varied luminal Ca²⁺ levels. With normal or reduced SR Ca²⁺ content ϕ has minimal effect on RyR2 P_O, however, during SR Ca2+ overload ϕ results in a drastic increase in RyR2 P_O, see Fig. S1A. A complete description of the junctional and non-junctional RyR2 formulations can be found in the SI (Eqs. (S43)–(S47)). The Ca²⁺ spark dynamics (e.g., duration, amplitude, frequency, and fidelity) shown in Fig. 2, when using the modified (ϕ) and without inter-channel coupling remain remarkably similar to those shown in previous work [34]. Taken together, our LCC and RyR2 formulations are able to reproduce the graded nature of RyR2 flux, a hallmark of “local control” of CICR, in relation to membrane potential and high ECC gain (~20 fold at V = 0 mV) as measured by peak RyR2 flux divided by peak LCC flux (see Fig. S2C and D) [59].

Fig. 2.

Quiescent Ca²⁺ spark and non-spark Ca²⁺ leak dynamics. Shown are spontaneous Ca²⁺ sparks (large events), Ca²⁺ quarks (single RyR2 events), and Ca²⁺ sub-sparks (multi-RyR2 events). Each color represents the behavior from a unique CRU within the whole-cell model. Only 10% of the cell’s 20,000 CRUs are shown (for clarity). See Fig. S2 for $N_{R,O}^n$ and [Ca²⁺]_jsr dynamics. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

The RyR2 gating parameters for both WT and CPVT conditions are given in Table 1, where k⁺ is the RyR2 opening rate, k⁻ is the RyR2 closing rate, K_M is the half maximal point for RyR2 P_O as a function of [Ca²⁺]_i, and β_jsr is the total concentration of buffer within the JSR. The RyR2 steady-state open probability in response to changes in [Ca²⁺]_i for WT and CPVT simulations is shown in Fig. S1B. The effect of decreased k⁻ and slight rise in k⁺ on RyR2 P_O is consistent with the ~2 fold reduction in K_m caused by CASQ2-R33Q mutation examined in [39]. The 57% reduction in β_jsr is designed to simulate the altered CASQ2 expression seen in CASQ-R33Q knock-in mice [41].

Table 1.

RyR2 gating dynamics and JSR buffering under normal and pathological conditions.

	k+ (s−1 μM−η,ryr)	k− (s−1)	RyR2 Km (μM)	βjsr (mM)
WT	0.2	425	33	30
CPVT	0.275	120	16	13

1.1.3. Sarcolemmal membrane potential

In this study, two forms of electrical stimuli are utilized to trigger LCC opening and CICR, “voltage clamp” and “current clamp”. The “voltage clamp” protocol forces a step change in SL membrane potential while a small inward current, I_app, is applied in the “current clamp” protocol to elicit an action potential. The SL membrane potential during “current-clamp” mode is governed by a single ODE (Eq. (S1)). The formulation of the K⁺ currents used here can be found in the SI (see Eqs. (S12)–(S22)) and are primarily based on the currents from the mouse action potential model by Bondarenko and co-workers [5]. The fast Na⁺ current is adapted from [60] and is also outlined in the SI (see Eqs. (S9)–(S11), (S22), and (S23)).

1.1.4. Spatial sarcomere model

To further elucidate the physiologic consequences of “leaky” RyR2s we developed a 3 dimensional (3D) spatial implementation of our local control model that represents a transverse subsection of a cardio-myocyte (see Eq. (S57)). This spatial model represents a single sarco-mere (M-line to M-line) centered on a Z-line and contains equally distributed CRUs, separated from one another by 600 nm [61]. Linescan images were generated by assessing levels of Ca²⁺-bound Fluo4 concentration [CaF] (see Eq. (S58)) over time with optical blurring and Gaussian white noise added afterwards, consistent with previous work by Smith et al. [62]. While a whole cell 3D spatial model integrating 20,000 CRUs would be ideal, it is beyond the scope of the current work, however the model presented here could serve as the backbone of such a model in the future.

2. Results

2.1. Quiescent Ca²⁺ Sparks and SR Ca²⁺ Leak

Ca²⁺ leak dynamics, at physiological RyR2 sensitivity, are shown in Fig. 2 for quiescent conditions (prolonged rest/diastole) and include experimentally visible Ca²⁺ sparks and invisible Ca²⁺ sub-sparks and Ca²⁺ quarks. The Ca²⁺ sparks shown in Fig. 2 are consistent with quiescent Ca²⁺ sparks observed in small rodents in both frequency (~ 150 cell⁻¹ s⁻¹) and duration (~20 ms). Note that Ca²⁺ spark duration is measured by the duration of the number of open RyR2s within a CRU (N_R,O, see Fig. S3). Prior work has shown that RyR2 currents of this duration yield F/F₀ signals consistent with experimental recordings [33,62]. We observe here, and previously (see [34]), that Ca²⁺ quarks occasionally lead to Ca²⁺ sparks via the regenerative nature of CICR. Note that these Ca²⁺ quarks while frequent are very brief (~2–5 ms) and account for a small fraction of SR Ca²⁺ leak under normal SR Ca²⁺ load conditions (see [34]). While a Ca²⁺ quark represents the opening of a single RyR2 we observe that sometimes multiple RyR2s, open but fail to trigger a Ca²⁺ spark, we term these events, “Ca²⁺ sub-sparks”. This detailed, molecular mechanism for SR Ca²⁺ leak gives this model of ECC increased relevance for studying local regulation of Ca²⁺ signaling as well as pathophysiological modulation of Ca²⁺ dynamics.

2.1.1. Action potential and global CICR dynamics

The mouse action potential, shown in Fig. 3A, is more effective at recruiting LCC current, shown in Fig. 3B, than a voltage clamp to 0 mV. This surprising result is due to the combination of the rapid upstroke of the AP (activating the voltage-activated states of the LCC) followed by the rapid repolarization (due to the outward K⁺ currents) which increases the driving force for Ca²⁺ entry [63]. The time to peak for the [Ca²⁺]_i transients during current clamp is influenced by the rapid recruitment of LCC current during an AP (see Fig. 3B). Despite synchronous and uniform exposure of each LCC within the model to the same membrane potential, not all CRUs activate to trigger a Ca²⁺ spark during systole. The subsection on ECC fidelity, the probability that an LCC opening will activate CICR and trigger a Ca²⁺ spark, investigates this phenomenon in greater detail. The membrane dynamics and the underlying membrane currents for our mouse AP formulation are depicted in Fig. S7 and are consistent with Bondarenko et al. [5]. The resulting AP has an APD₉₀ (time to 90% recovery of depolarization) of ~40 ms, which is consistent with whole cell ruptured patch clamp and micro-electrode recordings of mouse APs [64,65]. Additional [Ca²⁺]_i and AP dynamics are shown in Figs. S7, S10, and S11. Global Ca²⁺ dynamics in response to both current- (Fig. 3C–D) and voltage-clamp (Fig. 3E–F) conditions highlight key differences of these excitation methods. The dashed lines represent the initial [Ca²⁺]_i transient after a prolonged period of quiescence. The SR content is equivalent between the current-and voltage-clamp protocols. While [Ca²⁺]_nsr only depletes moderately (15–25%) during an AP, the fraction of total SR Ca²⁺ released (~50% during an AP) is greater due to the strong depletion of [Ca²⁺]_jsr (see Fig. S3) and significant buffering (β_jsr) present in the JSR compartment (see Table 1). [Ca²⁺]_i and [Ca²⁺]_nsr dynamics shown in Fig. 3C–F are provided for both simulated WT and CPVT conditions, using gating parameters and total JSR buffering, described in Table 1. Consistent with high ECC gain observed experimentally, during a single AP, ECC gain is 12–25 depending on SR Ca²⁺ content.

Fig. 3.

Global [Ca²⁺]_i dynamics during ECC. Shown is A) membrane potential and B) respective whole-cell LCC currents, where solid black are representative results during current-clamp conditions following an applied current to generate an AP and dashed is representative of a voltage-clamp stimulus to 0 mV. C) [Ca²⁺]_i transients and D) NSR Ca2+ dynamics shown for current-clamp stimulus. E) [Ca²⁺]_i transients and F) NSR Ca2+ dynamics shown for voltage-clamp stimulus. For C–F, blue traces represent WT conditions while red traces depict CPVT. Additionally, solid lines represent [Ca²⁺]_i and [Ca²⁺]_nsr dynamics at steady-state while dashed traces represent the first transient after quiescence. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

2.1.2. Ca²⁺ sparks and SR Ca²⁺ leak during ECC

Systolic and diastolic Ca²⁺ sparks (and Ca²⁺ quarks) elicited by LCC openings during (and following) an AP are shown in Fig. 4. Each color indicates activity from a different CRU within the 20,000 CRU compartmental model. The bulk cytosol and NSR dynamics displayed in Fig. 3 result from the integration of local Ca²⁺ signals at the dyadic subspace and JSR level, see Fig. 4, synchronized during the AP interval (indicated by the blue bar between 10 and 50 ms). We show that at steady-state, 1 Hz pacing, ~7000 Ca²⁺ sparks are recruited during an action potential, or roughly 35% of the CRUs within a cardiomyocyte are activated for a given electrical stimulus. Systolic Ca²⁺ sparks displayed in Fig. 4 are roughly the same amplitude and duration as quiescent Ca²⁺ sparks (see Fig. 2), even though they are initiated at a much lower SR Ca²⁺ content, [Ca²⁺]_nsr ~ 700 μM at 1 Hz pacing vs. [Ca²⁺]_nsr ~ 950 μM during quiescence. Further comparison of systolic Ca²⁺ release and quiescent Ca²⁺ leak as functions of [Ca²⁺]_nsr can be found in the subsection, Dependence of spontaneous and triggered SR Ca²⁺ release on SR Ca²⁺ content. Interestingly, diastolic Ca²⁺ sparks, shown in Fig. 4, are less frequent and noticeably smaller (with a peak [Ca²⁺]_ds b 100 μM due to lower SR Ca²⁺ content, see Fig. 3D) than Ca²⁺ sparks displayed during quiescent conditions (see Fig. 2). See Restitution of Ca²⁺ spark fidelity subsection below for analysis of how Ca²⁺ sparks and Ca²⁺ quark dynamics “restore” following an AP.

Fig. 4.

Systolic and diastolic Ca²⁺ spark and non-spark Ca²⁺ release dynamics during an AP (similar to the solid blue line in Fig. 3A). Each colored line represents the behavior from a different CRU within the whole-cell model. Only 10% of the cell’s 20,000 CRUs are shown (for clarity in the display). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

2.1.3. Implications of leaky RyR2s on CICR

The influence of leaky RyR2s, a hallmark of CPVT [66], on Ca²⁺ sparks, Ca²⁺ quarks, SR Ca²⁺, and systolic [Ca²⁺]_i transients is shown in Fig. 5 and S6. Fig. 5A shows Ca²⁺ spark and Ca²⁺ quark dynamics of the model during quiescence under simulated CPVT conditions. These “leaky” RyR2s result in longer Ca²⁺ sparks (~50 ms). In some cases, Ca²⁺ sparks fail to fully terminate and even attempt to reactivate (see thick purple line in Fig. 5 A). These “unstable” Ca²⁺ release events are reminiscent of the “metastable” Ca²⁺ sparks described by Stern et al., 2013 [67]. While each quiescent Ca²⁺ spark is smaller in peak [Ca²⁺]_ds (due to lower [Ca²⁺]_nsr) they more than make up for it in both increased frequency (~500 Ca²⁺ sparks cell⁻¹ s⁻¹) and duration. If only RyR2 P_O is altered, see Fig. S4, we again see “unstable” Ca²⁺ sparks as well as increased Ca²⁺ spark frequency, while Ca²⁺ spark frequency and termination are normal with lower JSR buffering alone, see Fig. S5. To better visualize these “metastable” Ca²⁺ sparks we incorporated our CRU formulation into a 3D model of Ca²⁺ diffusion within a single sarcomere (centered on z-line). A representative simulated linescan shown in Fig. 6 (see SI for more details) dramatically shows the resulting unstable CICR during simulated CPVT conditions. A representative linescan for normal, WT conditions is shown in Fig. S9.

Fig. 5.

CPVT Ca²⁺ spark and Ca²⁺ leak dynamics. Influence of increased RyR2 open probability and reduced JSR buffering on Ca²⁺ dynamics during (A) quiescent conditions (see Fig. S5 for ${\mathrm{N}}_{\mathrm{R},\mathrm{O}}^{\mathrm{n}}$ and [Ca²⁺]_jsr dynamics) and (B) ECC. Each color represents activity from a different CRU within the whole-cell model. For clarity only 10% of the cell’s 20,000 CRUs are shown. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 6.

Pro-arrhythmic Ca²⁺ release behavior during CPVT. A) Stochastic RyR2 gating during quiescent, CPVT conditions and B) simulated transverse line-scan of quiescent Ca²⁺ release dynamics (as F/Fo) during CPVT conditions. Realistic noise and confocal blurring are added after simulation (see SI).

In addition to altered quiescent Ca²⁺ spark and Ca²⁺ leak dynamics observed under CPVT conditions (see Fig. 5A), we have examined the influence of CPVT channels on CICR dynamics during ECC (see Fig. 3 and Fig. 5B). After prolonged quiescence, WT [Ca²⁺]_i transient amplitude is greater than CPVT (see Fig. 3C and E), similar to what was shown experimentally at 0.2 Hz pacing frequency of CASQ-R33Q knock-in cardiomyocytes [40]. This is likely due to greater [Ca²⁺]_nsr under WT conditions ([Ca²⁺]_nsr = ~950 μM vs. ~800 μM, WT and CPVT respectively). As expected, the increased sensitivity of RyR2s to [Ca²⁺]_i under CPVT conditions corresponds to more Ca²⁺ sparks (though smaller in mass) recruited during a 1 s interval of steady-state 1 Hz AP pacing, ~11,000 (see Fig. 5B). There is a clear increase in late systolic and diastolic Ca²⁺ sparks under CPVT conditions. At steady-state 1 Hz pacing, we show little difference between WT and CPVT [Ca²⁺]_i transient amplitude (see Fig. 3C and E). If only RyR2 P_O is altered, the steady-state [Ca²⁺]_i transient is larger than WT with asynchronous late systolic Ca²⁺ spark activity, while reduced JSR buffering results in a smaller [Ca²⁺]_i transient than WT, see Fig. S8. The rate that Ca²⁺ sparks occur and the restitution of Ca²⁺ spark fidelity (the probability that opening of a single RyR2 will trigger a Ca²⁺ spark) following an AP is discussed below.

2.1.4. Restitution of Ca²⁺ spark fidelity

Following the burst of Ca²⁺ sparks produced by an AP there is an immediate, rapid drop in the overall rate of Ca²⁺ sparks. Fig. 7A–B shows the average rate of Ca²⁺ sparks over time following an AP binned into 100 ms (WT and CPVT, respectively). Ca²⁺ spark fidelity is near 60% during the AP (see Fig. 9H), but falls below steady-state diastolic levels immediately after the AP with Ca²⁺ quarks and Ca²⁺ sub-sparks dominating SR Ca²⁺ leak for WT conditions during diastole (see Fig. 4). Under normal conditions, at steady-state, the probability that a Ca²⁺ quark will induce a Ca²⁺ spark is <2% during diastole. There is a slow restitution of Ca²⁺ spark fidelity that corresponds to the time course for [Ca²⁺]_nsr refilling after ECC (see Fig. 3D). This suggests that [Ca²⁺]_nsr levels are responsible for governing the balance between invisible (Ca²⁺ quarks) and visible (Ca²⁺ sparks) leak. This is consistent with prior work that showed that visible Ca²⁺ leak transitions to invisible leak as SR Ca²⁺ content declines [34,68]. The decline in the total number of Ca²⁺ leak events, Ca²⁺ sparks, Ca²⁺ sub-sparks, and Ca²⁺ quarks corresponds well with the decline of the [Ca²⁺]_i transient (see Figs. 3 C and 7) and suggests that [Ca²⁺]_i plays a critical role in governing the overall Ca²⁺ leak rate. The restitution of Ca²⁺ spark fidelity under simulated CPVT conditions (see Fig. 7B) is similar to WT but with two critical quantitative differences. 1) The number of Ca²⁺ sparks following the AP spark burst is much higher and 2) the Ca²⁺ spark fidelity is significantly greater (about 10-fold in this example) with ~30% of RyR2 openings resulting in a Ca²⁺ spark (both likely due to the higher RyR2 P_O). The increased Ca²⁺ spark rate and higher RyR2 P_O both increase the likelihood of aberrant activity (e.g., early after depolarizations (EADs), Ca²⁺ waves, etc.).

Fig. 7.

Restitution of Ca²⁺ spark fidelity following an action potential. For WT (n = 400, blue, A) and CPVT (n = 65, red, B) The total number Ca²⁺ release events (Ca²⁺ Sparks and Ca²⁺ Sub-Sparks) are plotted (left y-axis) with the black cross-hatch showing the number of Ca²⁺ sparks. Ca²⁺ spark fidelity is represented by black circles (far right y-axis). Ca²⁺ release events and Ca²⁺ spark fidelity during an AP initiated [Ca²⁺]_i transient are binned into 100 ms bins. Note that Ca²⁺ spark fidelity during the AP bin (0–100 ms) is not shown, but is ~0.6 for WT and ~0.8 for CPVT conditions. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 9.

Effect of SR Ca²⁺ load on spontaneous and triggered SR Ca²⁺ release. A–B) Number of Ca²⁺ sparks during quiescence and systole (the first 100 ms after an AP), respectively. C–D) Number of Ca²⁺ quarks and sub-sparks (invisible leak events) during quiescence systole, respectively. E–F) Average Ca²⁺ spark duration during quiescence and systole, respectively. G–H) Ca²⁺ spark fidelity during quiescence and systole, respectively For all tests (A–H) WT results are shown in blue and CPVT conditions are shown in red, with steady-state quiescent [Ca²⁺]_nsr marked by * and diastolic [Ca²⁺]_nsr marked by #. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

2.1.5. ECC fidelity

The sensitivity of RyR2s to LCC openings has been a focus of numerous studies since the first experimental measurements by Santana et al. [69]. Note that the likelihood of a LCC opening will trigger a Ca²⁺ spark, termed “ECC fidelity”, is distinct from “Ca²⁺ spark fidelity”. The inverse relationship between ECC fidelity (defined as the number of Ca²⁺ sparks divided by the number of LCC openings) and membrane potential is shown in Fig. 8A is primarily due to declining driving force for Ca²⁺ entry via the LCC and the brief duration of an LCC opening (~0.5 ms). When the membrane is significantly depolarized (V ≥ 0 mV) numerous LCC openings are required to trigger a single Ca²⁺ spark. ECC fidelity is lower in normal conditions than during pathological states such as CPVT (see red line), with a pronounced increase in RyR2 sensitivity to brief LCC openings at more negative membrane potentials (−40 to −10 mV), see Fig. 8A. As expected, there are no significant differences in the number of LCC openings between WT and CPVT conditions, see Fig. 8B. However, “leaky” RyR2s result in an increased number of Ca²⁺ sparks, shown in Fig. 8C, across all membrane potentials. The substantial increase in Ca²⁺ spark sensitivity to brief LCC openings and resulting increase in Ca²⁺ spark frequency under CPVT conditions likely creates an environment that could promote instability and arrhythmia.

Fig. 8.

ECC fidelity. A) ECC fidelity (measured as number of Ca²⁺ sparks divided by the number of LCC openings during a 200 ms voltage pulse to the various test potentials). B) Number of LCC openings per CRU (N⁻¹) during same conditions as (A). C) Number of Ca²⁺ sparks per N CRUs during same conditions as (A). For all tests (A–C) WT results are shown in blue and simulated CPVT conditions are shown in red. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

2.1.6. Dependence of spontaneous and triggered SR Ca²⁺ release on SR Ca²⁺ content

Consistent with our prior work, we show that as SR Ca²⁺ content declines, Ca²⁺ quarks and Ca²⁺ sub-sparks (i.e. “non-spark” based leak), begin to account for the majority of SR Ca²⁺ leak, see Figs. 9A and C. Under normal conditions, quiescent Ca²⁺ sparks are infrequent at [Ca²⁺]_nsr < 750 μM and reach ~150 Ca²⁺ sparks per cell per second at steady-state quiescent [Ca²⁺]_nsr of ~950 μM. Ca²⁺ spark frequency during CPVT was significantly greater across a broad range of SR Ca²⁺ content and compares well with experimental results [70,71]. Experimentally invisible Ca²⁺ leak events, Ca²⁺ quarks and Ca²⁺ sub-sparks, quantified in Fig. 9C as non-sparks, were stable across SR Ca²⁺ load for WT conditions, but decreased with increasing [Ca²⁺]_nsr under CPVT conditions. Average Ca²⁺ spark duration (ASD) for WT conditions was relatively constant, ~20 ms, regardless of SR Ca²⁺ load, see Fig. 9E. However in the CPVT state, Ca²⁺ sparks fail to efficiently terminate above a [Ca²⁺]_nsr of 1 mM resulting in ASD > 100 ms. Near quiescent steady-state SR Ca²⁺ content, WT Ca²⁺ spark fidelity is ~0.15 and increases above a [Ca²⁺]_nsr of 1 mM, see Fig. 9G. Leaky RyR2s have greater Ca²⁺ spark frequency and decreased invisible Ca²⁺ leak events which contribute to the overall increase in Ca²⁺ spark fidelity of, ~0.55, at quiescent steady-state SR Ca²⁺ content of ~800 μM.

Systolic Ca²⁺ release is also sensitive to SR Ca²⁺ load, with Fig. 9B showing the increase in number of Ca²⁺ sparks during the first 100 ms following the start of an AP with increasing [Ca²⁺]_nsr. RyR2s during simulated CPVT promote greater recruitment of Ca²⁺ sparks during an AP, while “invisible” Ca²⁺ release events decrease with increasing [Ca²⁺]_nsr under normal and CPVT conditions, see Fig. 9D. Ca²⁺ spark duration during systole, shown in Fig. 9F, is relatively stable at ~20 ms and ~50–60 ms, for WT and CPVT respectively. The depletion of the [Ca²⁺]_nsr during the [Ca²⁺]_i transient likely underlies the added stability of systolic ASD even under “leaky” RyR2 conditions. While Ca²⁺ spark fidelity under normal conditions is ~0.5 at [Ca²⁺]_nsr of ~700 μM (steady-state 1 Hz pacing) and can increase to >0.7 at high SR Ca²⁺ content (Fig. 9H), Ca²⁺ spark fidelity is much less dynamic during CPVT. At 1 Hz steady-state pacing [Ca²⁺]_nsr is ~650 μM and Ca²⁺ spark fidelity is already >0.7 and only slightly rises with increasing SR Ca²⁺ content (Fig. 9 H).

3. Discussion

A novel mathematical model of ECC in mouse heart has been presented. The model builds upon our previous stochastic description of quiescent RyR2 mediated SR Ca²⁺ leak where Ca²⁺ spark and “invisible”, Ca²⁺ sub-spark events combine to balance the activity of SERCA2a [34]. Importantly, it includes a novel, 7-state Markov chain formulation of the LCC where transitions depend on membrane potential and also on [Ca²⁺]_ds thereby allowing for the generation of a realistic LCC current. The model also includes a quantitative description of the electrical behavior of the myocyte with detailed descriptions for the SL current components found in the external sarcolemma and in the transverse and axial tubule system. With LCC and AP membrane current components in place, the model can simulate systolic and diastolic Ca²⁺ dynamics while capturing key details of individual CRU activity (e.g., LCC and RyR2 openings, [Ca²⁺]_ds, and [Ca²⁺]_jsr). This unique tool allows us to explore characteristic properties of elementary Ca²⁺ sparks, the summation of local signals during ECC, as well as pathological Ca²⁺ dynamics derived from conditions associated with “leaky” RyR2s.

3.1. Local Ca²⁺ release underlies [Ca²⁺]_i transients

The Ca²⁺ spark is the fundamental unit for both diastolic and systolic SR Ca²⁺ release in cardiomyocytes. The stochastic opening of SR RyR2 channels mediates “invisible” sub-spark Ca²⁺ release which has the potential to induce a full Ca²⁺ spark through CICR. As shown in [34] Ca²⁺ sparks are the major contributor to SR Ca²⁺ leak within the resting car-diomyocyte and play an important role in cellular Ca²⁺ homeostasis by balancing SERCA2a activity. Ca²⁺ sparks observed after long periods of rest have equivalent characteristics (i.e. frequency, amplitude, duration, and decay constant) whether they occur due to spontaneous RyR2 opening or are evoked experimentally through transient LCC opening during membrane depolarization. We show that Ca²⁺ sparks elicited through ECC are of similar size and duration to those observed during quiescence. In the diastolic period after a [Ca²⁺]_i transient the cardio-myocyte CRUs are less capable of participating in CICR. The induction of a Ca²⁺ spark quickly results in depletion of [Ca²⁺]_jsr for that CRU, and when taken across the ~7000 CRUs activated during ECC leads to a large decrease in SR Ca²⁺ content. This leads to a decrease in diastolic Ca²⁺ spark amplitude and Ca²⁺ spark fidelity, as diastolic Ca²⁺ spark frequency is diminished dramatically and the proportion of “invisible” leak, Ca²⁺ quarks, rises. Our findings highlight the critical importance of SR Ca²⁺ content and RyR2 Ca²⁺ sensitivity in controlling Ca²⁺ spark frequency and, while not the focus of this study, may help to explain the low quiescent Ca²⁺ spark rate in other animals including larger mammals.

3.1.1. Insights into Pathological Conditions Related to “Leaky” RyR2s

We observe significant abnormalities affecting Ca²⁺ pump/leak balance, local Ca²⁺ sparks, and whole-cell Ca²⁺ homeostasis in our simulation of CASQ2-R33Q mutant cardiomyocyte. We incorporated two critical features in the pathogenesis of CASQ2-R33Q mediated CPVT, reduced JSR Ca²⁺ buffering and enhanced RyR2 P_O and [Ca²⁺]_i sensitivity. Decreased JSR Ca²⁺ buffering seemed to limit global [Ca²⁺]_i transient abnormalities, while the reduction in k⁻ and slight increase in k⁺ that increased RyR2 [Ca²⁺]_i sensitivity and P_O under CPVT conditions were the critical alterations that lead to aberrant Ca²⁺ handling. Under these conditions Ca²⁺ sparks are not well synchronized by the AP and persist into “late” systole as the RyR2s are much more sensitive to openings of neighboring RyR2s and LCCs. While Ca²⁺ spark fidelity and frequency decline after ECC under both WT and CPVT conditions, the decline is far less pronounced for the “leaky” RyR2 state. In fact, the Ca²⁺ spark fidelity for CPVT conditions is always much greater than that of WT. Individual CPVT Ca²⁺ sparks display all the hallmarks of unstable Ca²⁺ release such as delayed termination and even reactivation of the CRU. The “metastable” sparks are capable of activating neighboring transverse CRUs which has the potential to induce an arrhythmogenic Ca²⁺ wave. The dynamic difference between ECC fidelity displayed by CPVT conditions compared to WT at more negative potentials (−40 to −10 mV) might prime the cell for EADs and possibly delayed afterdepolarizations (DADs). Under normal conditions there should be little SR Ca²⁺ release at these potentials. However, increased release of SR Ca²⁺ due to “leaky” RyR2s may lead to buildup of [Ca²⁺]i over time and activate [Ca²⁺]_i-dependent currents, notably I_ncx. The baseline differences in local and global Ca²⁺ signals between simulated WT and CPVT mutation cardiomyocytes in the current study may form the molecular basis for cellular Ca²⁺ waves and possibly the rare but lethal ventricular arrhythmia during CPVT.

3.1.2. Mouse model to human translation

We designed our model to recapitulate the electrophysiology and Ca²⁺ handling properties of the mouse as it is the most commonly used animal in biomedical research. It should, however, be noted the Ca²⁺ signaling components in our model respond appropriately to different AP waveforms. While our model was designed to reproduce mouse AP dynamics during ruptured patch conditions, Fig. S11A shows minimal difference in Ca²⁺ signals after steady-state stimulation with an AP based on those recorded from perforated patch measurements [72]. Additionally, our model can respond to longer duration APs (APD₉₀ of ~250–350 ms) typical of larger mammals such as rabbit, see Fig. S11B [73]. While not the focus of the present study, future work could adapt our model to fit large mammal (human, canine, or rabbit) cardiomyocyte biophysical characteristics (i.e. SL membrane channels, transporters, and detailed Ca²⁺ handling properties). Importantly, the principles of Ca²⁺ pump/leak balance, inherent in our model design, are critical for modeling Ca²⁺ signaling regardless of species. SL Ca²⁺ fluxes regulate [Ca²⁺]_i, while the balance of SERCA2a [Ca²⁺]_i reuptake and RyR2 based SR Ca²⁺ leak control [Ca²⁺]_nsr levels. Any change to the delicate balance of SL or SR Ca^{2+ fl}uxes will shift steady-state [Ca²⁺]_i and [Ca²⁺]_nsr. When comparing mice and humans, the increased NCX activity in cardiomyocytes of larger mammals promotes extrusion of [Ca²⁺]_i and results in depletion of [Ca²⁺]_nsr of during quiescence. As shown in Williams et al., 2011, the lower resting [Ca²⁺]_nsr of large mammals likely leads to a shift of SR Ca²⁺ leak from primarily visible (e.g., Ca²⁺ sparks) toward more invisible leak pathways (e.g., Ca²⁺ quarks and sub-sparks). Furthermore, the increased I_ncx may also pre-dispose human cardiomyocytes to EADs and possibly DADs, especially under conditions characterized by “leaky” RyR2s as presented here.

3.2. Future challenges

While the present work provides much insight into cardiac CICR dynamics during both quiescence and ECC under physiological and pathological conditions, it is still just a stepping stone in a sequence of model development. We think a whole-cell spatially resolved model of the entire cardiomyocyte will be necessary to more completely investigate CICR instabilities (e.g., Ca²⁺ waves, Ca²⁺ alternans, etc.). Such a model could better investigate the alterations in Ca²⁺ signaling in an environment that promotes CPVT arrhythmia. A comprehensive whole-cardiomyocyte spatial model with robust β-adrenergic signaling may provide important insight into the conditions that elicit deadly arrhythmia in patients with CPVT and account for the rarity of sudden cardiac death. The physiological Ca²⁺ spark and Ca²⁺ leak dynamics presented here which now provide novel insights into normal and pathological cardiac Ca²⁺ signals in mouse heart, should serve as a solid base for that new work.

Abbreviations

CICR

calcium induced calcium release

ECC

excitation contraction coupling

AP

action potential

APD₉₀

duration to 90% repolarization of action potential

LCC

L-type calcium channels

VDI

voltage dependent inactivation

CDI

calcium dependent inactivation

RyR2

type 2 ryanodine receptor

SERCA2a

sarcoplasmic reticulum calcium ATPase

CASQ2

calsequestrin

NCX

sodium/calcium exchanger

PMCA

plasma membrane calcium ATPase

SR

sarcoplasmic reticulum

JSR

junctional sarcoplasmic reticulum

CRU

calcium release unit

N

number of CRUs

ASD

average calcium spark duration

CPVT

catecholaminergic polymorphic ventricular tachycardia

Appendix A. Supplementary data

Supplementary data to this article can be found online at http://dx.doi.org/10.1016/j.yjmcc.2016.01.024.