Modeling Calcium Wave Based on Anomalous Subdiffusion of Calcium Sparks in Cardiac Myocytes

sparks and waves play important roles in calcium release and calcium propagation during the excitation-contraction (EC) coupling process in cardiac myocytes. Although the classical Fick’s law is widely used to model sparks and waves in cardiac myocytes, it fails to reasonably explain the full-width at half maximum(FWHM) paradox. However, the anomalous subdiffusion model successfully reproduces sparks of experimental results. In this paper, in the light of anomalous subdiffusion of sparks, we develop a mathematical model of calcium wave in cardiac myocytes by using stochastic release of release units (CRUs). Our model successfully reproduces calcium waves with physiological parameters. The results reveal how concentration waves propagate from an initial firing of one CRU at a corner or in the middle of considered region, answer how large in magnitude of an anomalous spark can induce a wave. With physiological currents (2pA) through CRUs, it is shown that an initial firing of four adjacent CRUs can form a wave. Furthermore, the phenomenon of calcium waves collision is also investigated.     

A mathematical model of spontaneous calcium release in cardiac myocytes

In cardiac myocytes, calcium (Ca) can be released from the sarcoplasmic reticulum independently of Ca influx from voltage-dependent membrane channels. This efflux of Ca, referred to as spontaneous Ca release (SCR), is due to Ryanodine receptor fluctuations, which can induce spontaneous Ca sparks, which propagate to form Ca waves. This release of Ca can then induce delayed after-depolarizations (DADs), which can lead to arrhythmogenic-triggered activity in the heart. However, despite its importance, to date there is no mathematical model of SCR that accounts for experimentally observed features of subcellular Ca. In this article, we present an experimentally based model of SCR that reproduces the timing distribution of spontaneous Ca sparks and key features of the propagation of Ca waves emanating from these spontaneous sparks. We have coupled this model to an ionic model for the rabbit ventricular action potential to simulate SCR within several thousand cells in cardiac tissue. We implement this model to study the formation of an ectopic beat on a cable of cells that exhibit SCR-induced DADs.     

Calcium-dependent inactivation and the dynamics of calcium puffs and sparks

Localized intracellular Ca²⁺ elevations known as puffs and sparks arise from the cooperative activity of inositol 1,4,5-trisphosphate receptor Ca²⁺ channels (IP₃Rs) and ryanodine receptor Ca²⁺ channels (RyRs) clustered at Ca²⁺ release sites on the surface of the endoplasmic reticulum or sarcoplasmic reticulum. When Markov chain models of these intracellular Ca²⁺-regulated Ca²⁺ channels are coupled via a mathematical representation of a Ca²⁺ microdomain, simulated Ca²⁺ release sites may exhibit the phenomenon of “stochastic Ca²⁺ excitability” reminiscent of Ca²⁺ puffs and sparks where channels open and close in a concerted fashion. To clarify the role of Ca²⁺ inactivation of IP₃Rs and RyRs in the dynamics of puffs and sparks, we formulate and analyze Markov chain models of Ca²⁺ release sites composed of 10–40 three-state intracellular Ca²⁺ channels that are inactivated as well as activated by Ca²⁺. We study how the statistics of simulated puffs and sparks depend on the kinetics and dissociation constant of Ca²⁺ inactivation and find that puffs and sparks are often less sensitive to variations in the number of channels at release sites and strength of coupling via local [Ca²⁺] when the average fraction of inactivated channels is significant. Interestingly, we observe that the single channel kinetics of Ca²⁺ inactivation influences the thermodynamic entropy production rate of Markov chain models of puffs and sparks. While excessively fast Ca²⁺ inactivation can preclude puffs and sparks, moderately fast Ca²⁺ inactivation often leads to time-irreversible puffs and sparks whose termination is facilitated by the recruitment of inactivated channels throughout the duration of the puff/spark event. On the other hand, Ca²⁺ inactivation may be an important negative feedback mechanism even when its time constant is much greater than the duration of puffs and sparks. In fact, slow Ca²⁺ inactivation can lead to release sites with a substantial fraction of inactivated channels that exhibit puffs and sparks that are nearly time-reversible and terminate without additional recruitment of inactivated channels.     

Modulation of local Ca2+ release sites by rapid fluid puffing in rat atrial myocytes

Atrial myocytes that lack t-tubules appear to have two functionally separate sarcoplasmic Ca2+ stores: a peripheral store associated with plasmalemmal L-type calcium channels and a central store with no apparent proximity to L-type calcium channels. Here we describe a set of calcium sparks and waves that are triggered by puffing of pressurized (200-400 mmH2O) bathing solutions onto resting isolated rat atrial myocytes. Puffing of pressurized (200 mmH2O) solutions, identical to those bathing the myocytes from distances of approximately 150 microm onto the surface of a single myocyte triggered or enhanced spontaneously occurring peripheral sparks by five- to six-fold and central Ca2+ sparks by two- to three-fold, without altering the unitary spark properties. Exposure to higher pressure flows (400 mmH2O) often triggered longitudinally spreading Ca2+ waves. These results suggest that pressurized flows may directly modulate Ca2+ signaling of atrial myocytes by activating the intracellular Ca2+ release sites.     

Detection, properties, and frequency of local calcium release from the sarcoplasmic reticulum in teleost cardiomyocytes

Calcium release from the sarcoplasmic reticulum (SR) plays a central role in the regulation of cardiac contraction and rhythm in mammals and humans but its role is controversial in teleosts. Since the zebrafish is an emerging model for studies of cardiovascular function and regeneration we here sought to determine if basic features of SR calcium release are phylogenetically conserved. Confocal calcium imaging was used to detect spontaneous calcium release (calcium sparks and waves) from the SR. Calcium sparks were detected in 16 of 38 trout atrial myocytes and 6 of 15 ventricular cells. The spark amplitude was 1.45±0.03 times the baseline fluorescence and the time to half maximal decay of sparks was 27±3 ms. Spark frequency was 0.88 sparks µm⁻¹ min⁻¹ while calcium waves were 8.5 times less frequent. Inhibition of SR calcium uptake reduced the calcium transient (F/F₀) from 1.77±0.17 to 1.12±0.18 (p = 0.002) and abolished calcium sparks and waves. Moreover, elevation of extracellular calcium from 2 to 10 mM promoted early and delayed afterdepolarizations (from 0.6±0.3 min⁻¹ to 8.1±2.0 min⁻¹, p = 0.001), demonstrating the ability of SR calcium release to induce afterdepolarizations in the trout heart. Calcium sparks of similar width and duration were also observed in zebrafish ventricular myocytes. In conclusion, this is the first study to consistently report calcium sparks in teleosts and demonstrate that the basic features of calcium release through the ryanodine receptor are conserved, suggesting that teleost cardiac myocytes is a relevant model to study the functional impact of abnormal SR function.     

Size Matters: Ryanodine Receptor Cluster Size Heterogeneity Potentiates Calcium Waves

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

Life and death of a cardiac calcium spark

Calcium sparks in cardiac myocytes are brief, localized calcium releases from the sarcoplasmic reticulum (SR) believed to be caused by locally regenerative calcium-induced calcium release (CICR) via couplons, clusters of ryanodine receptors (RyRs). How such regeneration is terminated is uncertain. We performed numerical simulations of an idealized stochastic model of spark production, assuming a RyR gating scheme with only two states (open and closed). Local depletion of calcium in the SR was inevitable during a spark, and this could terminate sparks by interrupting CICR, with or without assumed modulation of RyR gating by SR lumenal calcium. Spark termination by local SR depletion was not robust: under some conditions, sparks could be greatly and variably prolonged, terminating by stochastic attrition–a phenomenon we dub “spark metastability.” Spark fluorescence rise time was not a good surrogate for the duration of calcium release. Using a highly simplified, deterministic model of the dynamics of a couplon, we show that spark metastability depends on the kinetic relationship of RyR gating and junctional SR refilling rates. The conditions for spark metastability resemble those produced by known mutations of RyR2 and CASQ2 that cause life-threatening triggered arrhythmias, and spark metastability may be mitigated by altering the kinetics of the RyR in a manner similar to the effects of drugs known to prevent those arrhythmias. The model was unable to explain the distributions of spark amplitudes and rise times seen in chemically skinned cat atrial myocytes, suggesting that such sparks may be more complex events involving heterogeneity of couplons or local propagation among sub-clusters of RyRs.     

Spatial non-uniformities in [Ca2+]i during excitation-contraction coupling in cardiac myocytes.

The intracellular calcium ([Ca2+]i) transient in adult rat heart cells was examined using the fluorescent calcium indicator fluo-3 and a laser scanning confocal microscope. We find that the electrically evoked [Ca2+]i transient does not rise at a uniform rate at all points within the cell during the [Ca2+]i transient. These spatial non-uniformities in [Ca2+]i are observed immediately upon depolarization and largely disappear by the time the peak of the [Ca2+]i transient occurs. Importantly, some of the spatial non-uniformity in [Ca2+]i varies randomly in location from beat to beat. Analysis of the spatial character of the non-uniformities suggests that they arise from the stochastic nature of the activation of SR calcium-release channels. The non-uniformities in [Ca2+]i are markedly enhanced by low concentrations of Cd2+, suggesting that activation of L-type calcium channels is the primary source of activator calcium for the calcium transient. In addition, the pattern of calcium release in these conditions was very similar to the spontaneous calcium sparks that are observed under resting conditions and which are due to spontaneous calcium release from the SR. The spatial non-uniformity in the evoked [Ca2+]i transient under normal conditions can be explained by the temporal and spatial summation of a large number of calcium sparks whose activation is a stochastic process. The results are discussed with respect to a stochastic local control model for excitation-contraction (E-C) coupling, and it is proposed that the fundamental unit of E-C coupling consists of one dihydropyridine receptor activating a small group of ryanodine receptors (possibly four) in a square packing model.     

Eavesdropping on the social lives of Ca(2+) sparks

Ca²⁺ sparks arise from the stochastic opening of spatially discrete clusters of ryanodine receptors called a Ca²⁺ release unit (CRU). If the RyR clusters were not spatially separated, then Ca²⁺ released from one RyR would immediately diffuse to its neighbor and lead to uncontrolled, runaway Ca²⁺ release throughout the cell. While physical separation provides some isolation from neighbors, CRUs are not incommunicado. When inter-neighbor interactions become large enough, Ca²⁺ waves spontaneously emerge. A more circumscribed interaction shows up in high-speed two-dimensional confocal images as jumping Ca²⁺ sparks that seem to be sequentially activated along the Z-line and across Z-lines. However, since Ca²⁺ sparks are stochastic events how can we tell whether two sparks occurring close together in space and time are causally related or appeared simply by coincidence? Here we develop a mathematical method to disentangle cause and coincidence in a statistical sense. From our analysis we derive three fundamental properties of Ca²⁺ spark generation: 1), the “intrinsic” spark frequency, the spark frequency one would observe if the CRUs were incommunicado; 2), the coupling strength, which measures how strongly one CRU affects another; and 3), the range over which the communication occurs. These parameters allow us to measure the effect RyR regulators have on the intrinsic activity of CRUs and on the coupling between them.     

Calcium Oscillations and Waves Generated by Multiple Release Mechanisms in Pancreatic Acinar Cells

We explore the dynamic behavior of a model of calcium oscillations and wave propagation in the basal region of pancreatic acinar cells [Sneyd, J., et al., Biophys. J. 85: 1392–1405, 2003]. Since it is known that two principal calcium release pathways are involved, inositol trisphosphate receptors (IPR) and ryanodine receptors (RyR), we study how the model behavior depends on the density of each receptor type. Calcium oscillations can be mediated either by IPR or RyR. Continuous increases in either RyR or IPR density can lead to the appearance and disappearance of oscillations multiple times, and the two receptor types interact via their common effect on cytoplasmic calcium concentration and the subsequent effect on the total amount of calcium inside the cell. Increases in agonist concentration can stimulate oscillations via the RyR by increasing calcium influx. Using a two time-scale approach, we explain these complex behaviors by treating the total amount of cellular calcium as a slow parameter. Oscillations are controlled by the shape of the slow manifold and where it intersects the nullcline of the slow variable. When calcium diffusion is included, the existence of traveling waves in the model equation is strongly dependent on the interplay between the total amount of calcium in the cell and membrane transport, a feature that can be experimentally tested. Our results help us understand the behavior of a model that includes both receptors in comparison to the properties of each receptor type in isolation.     

Putting out the fire: what terminates calcium-induced calcium release in cardiac muscle?

The majority of contractile calcium in cardiac muscle is released from stores in the sarcoplasmic reticulum (SR), by a process of calcium-induced calcium release (CICR) through ryanodine receptors. Because CICR is intrinsically self-reinforcing, the stability of and graded regulation of cardiac EC coupling appear paradoxical. It is now well established that this gradation results from the stochastic recruitment of varying numbers of elementary local release events, which may themselves be regenerative, and which can be directly observed as calcium sparks. Ryanodine receptors (RyRs) are clustered in dense lattices, and most calcium sparks are now believed to involve activation of multiple RyRs. This implies that local CICR is regenerative, requiring a mechanism to terminate it. It was initially assumed that this mechanism was inactivation of the RyR, but during the decade since the discovery of sparks, no sufficiently strong inactivation mechanism has been demonstrated in vitro and all empirically determined gating schemes for the RyR give unstable EC coupling in Monte Carlo simulations. We consider here possible release termination mechanisms. Stochastic attrition is the spontaneous decay of active clusters due to random channel closure; calculations show that it is much too slow unless assisted by another process. Calcium-dependent RyR inactivation involving third-party proteins remains a viable but speculative mechanism; current candidates include calmodulin and sorcin. Local depletion of SR release terminal calcium could terminate release, however calculations and measurements leave it uncertain whether a sufficient diffusion resistance exists within the SR to sustain such depletion. Depletion could be assisted by dependence of RyR activity on SR lumenal [Ca(2+)]. There is substantial evidence for such lumenal activation, but it is not clear if it is a strong enough effect to account for the robust termination of sparks. The existence of direct interactions among clustered RyRs might account for the discrepancy between the inactivation properties of isolated RyRs and intact clusters. Such coupled gating remains controversial. Determining the mechanism of release termination is the outstanding unsolved problem of cardiac EC coupling, and will probably require extensive genetic manipulation of the EC coupling apparatus in its native environment to unravel the solution.     

A mathematical analysis of the generation and termination of calcium sparks

Calcium sparks are local regenerative releases of Ca(2+) from a cluster of ryanodine receptors on the sarcoplasmic reticulum. During excitation-contraction coupling in cardiac cells, Ca(2+) sparks are triggered by Ca(2+) entering the cell via the T-tubules (Ca(2+)-induced Ca(2+) release). However under conditions of calcium overload, Ca(2+) sparks can be triggered spontaneously. The exact process by which Ca(2+) sparks terminate is still an open question, although both deterministic and stochastic processes are likely to be important. In this article, asymptotic methods are used to analyze a single Ca(2+) spark model, which includes both deterministic and stochastic biophysical mechanisms. The analysis calculates both spark frequencies and spark duration distributions, and shows under what circumstances stochastic transitions are important. Additionally, a model of the coupling of the release channels via the FK-binding protein is analyzed.     

Frequency and release flux of calcium sparks in rat cardiac myocytes: a relation to RYR gating

Cytosolic calcium concentration in resting cardiac myocytes locally fluctuates as a result of spontaneous microscopic Ca²⁺ releases or abruptly rises as a result of an external trigger. These processes, observed as calcium sparks, are fundamental for proper function of cardiac muscle. In this study, we analyze how the characteristics of spontaneous and triggered calcium sparks are related to cardiac ryanodine receptor (RYR) gating. We show that the frequency of spontaneous sparks and the probability distribution of calcium release flux quanta of triggered sparks correspond quantitatively to predictions of an allosteric homotetrameric model of RYR gating. This model includes competitive binding of Ca²⁺ and Mg²⁺ ions to the RYR activation sites and allosteric interaction between divalent ion binding and channel opening. It turns out that at rest, RYRs are almost fully occupied by Mg²⁺. Therefore, spontaneous sparks are most frequently evoked by random openings of the highly populated but rarely opening Mg₄RYR and CaMg₃RYR forms, whereas triggered sparks are most frequently evoked by random openings of the less populated but much more readily opening Ca₂Mg₂RYR and Ca₃MgRYR forms. In both the spontaneous and the triggered sparks, only a small fraction of RYRs in the calcium release unit manages to open during the spark because of the limited rate of Mg²⁺ unbinding. This mechanism clarifies the unexpectedly low calcium release flux during elementary release events and unifies the theory of calcium signaling in resting and contracting cardiac myocytes.     

Calcium-induced calcium release in smooth muscle: loose coupling between the action potential and calcium release

Calcium-induced calcium release (CICR) has been observed in cardiac myocytes as elementary calcium release events (calcium sparks) associated with the opening of L-type Ca(2+) channels. In heart cells, a tight coupling between the gating of single L-type Ca(2+) channels and ryanodine receptors (RYRs) underlies calcium release. Here we demonstrate that L-type Ca(2+) channels activate RYRs to produce CICR in smooth muscle cells in the form of Ca(2+) sparks and propagated Ca(2+) waves. However, unlike CICR in cardiac muscle, RYR channel opening is not tightly linked to the gating of L-type Ca(2+) channels. L-type Ca(2+) channels can open without triggering Ca(2+) sparks and triggered Ca(2+) sparks are often observed after channel closure. CICR is a function of the net flux of Ca(2+) ions into the cytosol, rather than the single channel amplitude of L-type Ca(2+) channels. Moreover, unlike CICR in striated muscle, calcium release is completely eliminated by cytosolic calcium buffering. Thus, L-type Ca(2+) channels are loosely coupled to RYR through an increase in global [Ca(2+)] due to an increase in the effective distance between L-type Ca(2+) channels and RYR, resulting in an uncoupling of the obligate relationship that exists in striated muscle between the action potential and calcium release.     

Model of intracellular calcium cycling in ventricular myocytes

We present a mathematical model of calcium cycling that takes into account the spatially localized nature of release events that correspond to experimentally observed calcium sparks. This model naturally incorporates graded release by making the rate at which calcium sparks are recruited proportional to the whole cell L-type calcium current, with the total release of calcium from the sarcoplasmic reticulum (SR) being just the sum of local releases. The dynamics of calcium cycling is studied by pacing the model with a clamped action potential waveform. Experimentally observed calcium alternans are obtained at high pacing rates. The results show that the underlying mechanism for this phenomenon is a steep nonlinear dependence of the calcium released from the SR on the diastolic SR calcium concentration (SR load) and/or the diastolic calcium level in the cytosol, where the dependence on diastolic calcium is due to calcium-induced inactivation of the L-type calcium current. In addition, the results reveal that the calcium dynamics can become chaotic even though the voltage pacing is periodic. We reduce the equations of the model to a two-dimensional discrete map that relates the SR and cytosolic concentrations at one beat and the previous beat. From this map, we obtain a condition for the onset of calcium alternans in terms of the slopes of the release-versus-SR load and release-versus-diastolic-calcium curves. From an analysis of this map, we also obtain an understanding of the origin of chaotic dynamics.     

Evolution of cardiac calcium waves from stochastic calcium sparks

We present a model that provides a unified framework for studying Ca2+ sparks and Ca2+ waves in cardiac cells. The model is novel in combining 1) use of large currents (approximately 20 pA) through the Ca2+ release units (CRUs) of the sarcoplasmic reticulum (SR); 2) stochastic Ca2+ release (or firing) of CRUs; 3) discrete, asymmetric distribution of CRUs along the longitudinal (separation distance of 2 microm) and transverse (separated by 0.4-0.8 microm) directions of the cell; and 4) anisotropic diffusion of Ca2+ and fluorescent indicator to study the evolution of Ca2+ waves from Ca2+ sparks. The model mimics the important features of Ca2+ sparks and Ca2+ waves in terms of the spontaneous spark rate, the Ca2+ wave velocity, and the pattern of wave propagation. Importantly, these features are reproduced when using experimentally measured values for the CRU Ca2+ sensitivity (approximately 15 microM). Stochastic control of CRU firing is important because it imposes constraints on the Ca2+ sensitivity of the CRU. Even with moderate (approximately 5 microM) Ca2+ sensitivity the very high spontaneous spark rate triggers numerous Ca2+ waves. In contrast, a single Ca2+ wave with arbitrarily large velocity can exist in a deterministic model when the CRU Ca2+ sensitivity is sufficiently high. The combination of low CRU Ca2+ sensitivity (approximately 15 microM), high cytosolic Ca2+ buffering capacity, and the spatial separation of CRUs help control the inherent instability of SR Ca2+ release. This allows Ca2+ waves to form and propagate given a sufficiently large initiation region, but prevents a single spark or a small group of sparks from triggering a wave.     

Calcium microdomains and oxidative stress

The phenomenon of calcium microdomains is firmly established in the field of subcellular physiology. These regions of localized, transient calcium increase are exemplified by the spontaneous 'sparks' released through the ryanodine receptor in myocytes, but include subplasmalemmal microdomains, focal calcium oscillations and microdomains enclosed within organelles, such as the endoplasmic reticulum, golgi and mitochondria. Increasing evidence suggests that oxidative stress regulates both the formation and disappearance of microdomains. Calcium release channels and transporters are all modulated by redox state, while several mechanisms that generate oxidative or nitrosative stress are regulated by calcium. Here, we discuss the evidence for the regulation of calcium microdomains by redox state, and, by way of example, demonstrate that the frequency of calcium sparks in cardiomyocytes is increased in response to oxidative stress. We consider the evidence for the existence of analogous microdomains of reactive oxygen and nitrogen species and suggest that the refinement of imaging techniques for these species might lead to similar concepts. The interaction between Ca(2+) microdomains and proteins that modulate their formation results in a complex and dynamic, spatial signaling mechanism, which is likely to be broadly applicable to different cell types, adding new dimensions to the calcium signaling 'toolkit'.     

心肌细胞兴奋-收缩偶联的微观机制

心肌细胞的兴奋 收缩偶联 (ECC)本质上是胞膜上的电压门控L 型钙通道 (LCCs)和胞内ryanodine受体 (RyRs)之间通过钙诱导钙释放 (CICR)机制进行沟通进而引发肌细胞收缩的过程。最近的研究进一步揭示了微观水平上LCCs和RyRs之间的信息联系。在钙偶联位点 (couplons)上 ,LCCs因膜去极化而随机开放 ,在局部产生高强度的钙脉冲 (即钙小星 ,Ca2 sparklet) ,作用于邻近肌质网终末池上的RyRs。钙偶联位点通过由钙小星随机激活的RyRs(即钙释放通道 )以钙火花 (Ca2 spark)的形式释放钙。这些钙在全细胞水平上总和即形成钙瞬变 (Ca2 transient)。因此 ,钙小星触发钙火花就构成了ECC中的基本事件。本文重点阐述LCCs和RyRs分子间的信号转导机制 ,也即从微观水平上探讨CICR及ECC的形成机制。