Receptors, sparks and waves in a fire-diffuse-fire framework for calcium release

Calcium ions are an important second messenger in living cells. Indeed calcium signals in the form of waves have been the subject of much recent experimental interest. It is now well established that these waves are composed of elementary stochastic release events (calcium puffs or sparks) from spatially localised calcium stores. The aim of this paper is to analyse how the stochastic nature of individual receptors within these stores combines to create stochastic behaviour on long time-scales that may ultimately lead to waves of activity in a spatially extended cell model. Techniques from asymptotic analysis and stochastic phase–plane analysis are used to show that a large cluster of receptor channels leads to a release probability with a sigmoidal dependence on calcium density. This release probability is incorporated into a computationally inexpensive model of calcium release based upon a stochastic generalisation of the fire-diffuse-fire (FDF) threshold model. Numerical simulations of the model in one and two dimensions (with stores arranged on both regular and disordered lattices) illustrate that stochastic calcium release leads to the spontaneous production of calcium sparks that may merge to form saltatory waves. Illustrations of spreading circular waves, spirals and more irregular waves are presented. Furthermore, receptor noise is shown to generate a form of array enhanced coherence resonance whereby all calcium stores release periodically and simultaneously.     

Dynamics of calcium sparks and calcium leak in the heart

We present what we believe to be a new mathematical model of Ca²⁺ leak from the sarcoplasmic reticulum (SR) in the heart. To our knowledge, it is the first to incorporate a realistic number of Ca²⁺-release units, each containing a cluster of stochastically gating Ca²⁺ channels (RyRs), whose biophysical properties (e.g., Ca²⁺ sensitivity and allosteric interactions) are informed by the latest molecular investigations. This realistic model allows for the detailed characterization of RyR Ca²⁺-release properties, and shows how this balances reuptake by the SR Ca²⁺ pump. Simulations reveal that SR Ca²⁺ leak consists of brief but frequent single RyR openings (∼3000 cell⁻¹ s⁻¹) that are likely to be experimentally undetectable, and are, therefore, “invisible”. We also observe that these single RyR openings can recruit additional RyRs to open, due to elevated local (Ca²⁺), and occasionally lead to the generation of Ca²⁺ sparks (∼130 cell⁻¹ s⁻¹). Furthermore, this physiological formulation of “invisible” leak allows for the removal of the ad hoc, non-RyR mediated Ca²⁺ leak terms present in prior models. Finally, our model shows how Ca²⁺ sparks can be robustly triggered and terminated under both normal and pathological conditions. Together, these discoveries profoundly influence how we interpret and understand diverse experimental and clinical results from both normal and diseased hearts.     

Nonlinear propagation of spherical calcium waves in rat cardiac myocytes.

Spontaneous calcium waves in enzymatically isolated rat cardiac myocytes were investigated by confocal laser scanning microscopy (CLSM) using the fluorescent Ca2+-indicator fluo-3 AM. As recently shown, a spreading wave of enhanced cytosolic calcium appears, most probably during Ca2+ overload, and is initiated by an elementary event called a "calcium spark." When measured by conventional fluorescence microscopy the propagation velocity of spontaneous calcium waves determined at several points along the cardiac myocyte was previously found to be constant. More precise measurements with a CLSM showed a nonlinear propagation. The wave velocity was low, close to the focus, and increased with increasing time and propagation length, approaching a maximum of 113 microns/s. This result was surprising, inasmuch as for geometrical reasons a decrease of the propagation velocity might be expected if the confocal plane is not identical with that plane where the focus of the wave was localized. It is suggested that the propagation velocity is essentially dependent on the curvature of the spreading wave. From the linear relationship of velocity versus curvature, a critical radius of 2.7 +/- 1.4 microns (mean +/- SD) was worked out, below which an outward propagation of the wave will not take place. Once released from a sufficiently extended cluster of sarcoplasmic reticulum release channels, calcium diffuses and will activate its neighbors. While traveling away, the volume into which calcium diffuses becomes effectively smaller than at low radii. This effect is the consequence of the summation of elementary events (Ca2+ sparks) and leads to a steeper increase of the cytosolic calcium concentration after a certain diffusion path length. Thus the time taken to reach a critical threshold of [Ca2+]i at the neighboring calcium release sites decreases with decreasing curvature and the wave will propagate faster.     

Termination of cardiac Ca(2+) sparks: an investigative mathematical model of calcium-induced calcium release

A Ca(2+) spark arises when a cluster of sarcoplasmic reticulum (SR) channels (ryanodine receptors or RyRs) opens to release calcium in a locally regenerative manner. Normally triggered by Ca(2+) influx across the sarcolemmal or transverse tubule membrane neighboring the cluster, the Ca(2+) spark has been shown to be the elementary Ca(2+) signaling event of excitation-contraction coupling in heart muscle. However, the question of how the Ca(2+) spark terminates remains a central, unresolved issue. Here we present a new model, "sticky cluster," of SR Ca(2+) release that simulates Ca(2+) spark behavior and enables robust Ca(2+) spark termination. Two newly documented features of RyR behavior have been incorporated in this otherwise simple model: "coupled gating" and an opening rate that depends on SR lumenal [Ca(2+)]. Using a Monte Carlo method, local Ca(2+)-induced Ca(2+) release from clusters containing between 10 and 100 RyRs is modeled. After release is triggered, Ca(2+) flux from RyRs diffuses into the cytosol and binds to intracellular buffers and the fluorescent Ca(2+) indicator fluo-3 to produce the model Ca(2+) spark. Ca(2+) sparks generated by the sticky cluster model resemble those observed experimentally, and Ca(2+) spark duration and amplitude are largely insensitive to the number of RyRs in a cluster. As expected from heart cell investigation, the spontaneous Ca(2+) spark rate in the model increases with elevated cytosolic or SR lumenal [Ca(2+)]. Furthermore, reduction of RyR coupling leads to prolonged model Ca(2+) sparks just as treatment with FK506 lengthens Ca(2+) sparks in heart cells. This new model of Ca(2+) spark behavior provides a "proof of principle" test of a new hypothesis for Ca(2+) spark termination and reproduces critical features of Ca(2+) sparks observed experimentally.     

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.     

A preferred amplitude of calcium sparks in skeletal muscle

In skeletal and cardiac muscle, calcium release from the sarcoplasmic reticulum, leading to contraction, often results in calcium sparks. Because sparks are recorded by confocal microscopy in line-scanning mode, their measured amplitude depends on their true amplitude and the position of the spark relative to the scanned line. We present a method to derive from measured amplitude histograms the actual distribution of spark amplitudes. The method worked well when tested on simulated distributions of experimental sparks. Applied to massive numbers of sparks imaged in frog skeletal muscle under voltage clamp in reference conditions, the method yielded either a decaying amplitude distribution (6 cells) or one with a central mode (5 cells). Caffeine at 0.5 or 1 mM reversibly enhanced this mode (5 cells) or induced its appearance (4 cells). The occurrence of a mode in the amplitude distribution was highly correlated with the presence of a mode in the distribution of spark rise times or in the joint distribution of rise times and spatial widths. If sparks were produced by individual Markovian release channels evolving reversibly, they should not have a preferred rise time or amplitude. Channel groups, instead, could cooperate allosterically or through their calcium sensitivity, and give rise to a stereotyped amplitude in their collective spark.     

Velocity-curvature relationship of colliding spherical calcium waves in rat cardiac myocytes.

Colliding spherical calcium waves in enzymatically isolated rat cardiac myocytes develop new wavefronts propagating perpendicular to the original direction. When investigated by confocal laser scanning microscopy (CLSM), using the fluorescent Ca2+ indicator fluo-3 AM, "cusp"-like structures become visible that are favorably approximated by double parabolae. The time-dependent position of the vertices is used to determine propagation velocity and negative curvature of the wavefront in the region of collision. It is evident that negatively curved waves propagate faster than positively curved, single waves. Considering two perfectly equal expanding circular waves, we demonstrated that the collision of calcium waves is due to an autocatalytic process (calcium-induced calcium release), and not to a simple phenomenon of interference. Following the spatiotemporal organization in simpler chemical systems maintained under conditions far from the thermodynamic equilibrium (Belousov-Zhabotinskii reaction), the dependence of the normal velocity on the curvature of the spreading wavefront is given by a linear relation. The so-called velocity-curvature relationship makes clear that the velocity is enhanced by curvature toward the direction of forward propagation and decreased by curvature away from the direction of forward propagation (with an influence of the diffusion coefficient). Experimentally obtained velocity data of both negatively and positively curved calcium waves were approximated by orthogonal weighted regression. The negative slope of the straight line resulted in an effective diffusion coefficient of 1.2 x 10(-4) mm2/s. From the so-called critical radius, which must be exceeded to initiate a traveling calcium wave, a critical volume (with enhanced [Ca2+]i) of approximately 12 microm3 was calculated. This is almost identical to the volume that is occupied by a single calcium spark.     

Large currents generate cardiac Ca2+ sparks

Previous models of cardiac Ca2+ sparks have assumed that Ca2+ currents through the Ca2+ release units (CRUs) were approximately 1-2 pA, producing sparks with peak fluorescence ratio (F/F(0)) of approximately 2.0 and a full-width at half maximum (FWHM) of approximately 1 microm. Here, we present actual Ca2+ sparks with peak F/F(0) of >6 and a FWHM of approximately 2 microm, and a mathematical model of such sparks, the main feature of which is a much larger underlying Ca2+ current. Assuming infinite reaction rates and no endogenous buffers, we obtain a lower bound of approximately 11 pA needed to generate a Ca2+ spark with FWHM of 2 microm. Under realistic conditions, the CRU current must be approximately 20 pA to generate a 2- microm Ca2+)spark. For currents > or =5 pA, the computed spark amplitudes (F/F(0)) are large (approximately 6-12 depending on buffer model). We considered several factors that might produce sparks with FWHM approximately 2 microm without using large currents. Possible protein-dye interactions increased the FWHM slightly. Hypothetical Ca2+ "quarks" had little effect, as did blurring of sparks by the confocal microscope. A clusters of CRUs, each producing 10 pA simultaneously, can produce sparks with FWHM approximately 2 microm. We conclude that cardiac Ca2+ sparks are significantly larger in peak amplitude than previously thought, that such large Ca2+ sparks are consistent with the measured FWHM of approximately 2 microm, and that the underlying Ca2+ current is in the range of 10-20 pA.     

Fast calcium waves

Calcium waves are propagated in five main speed ranges which cover a billion-fold range of speeds. We define the fast speed range as 3-30μm/s after correction to a standard temperature of 20°C. Only waves which are not fertilization waves are considered here. 181 such cases are listed here. These are through organisms in all major taxa from cyanobacteria through mammals including human beings except for those through other bacteria, higher plants and fungi. Nearly two-thirds of these speeds lie between 12 and 24μm/s. We argue that their common mechanism in eukaryotes is a reaction-diffusion one involving calcium-induced calcium release, in which calcium waves are propagated along the endoplasmic reticulum. We propose that the gliding movements of some cyanobacteria are driven by fast calcium waves which are propagated along their plasma membranes. Fast calcium waves may drive materials to one end of developing embryos by cellular peristalsis, help coordinate complex cell movements during development and underlie brain injury waves. Moreover, we continue to argue that such waves greatly increase the likelihood that chronic injuries will initiate tumors and cancers before genetic damage occurs. Finally we propose numerous further studies.     

Impact of testosterone on cardiac L-type calcium channels and Ca2+ sparks: acute actions antagonize chronic effects

While androgens generally have been associated with an increased cardiovascular risk, recent studies indicate potential beneficial acute effects of testosterone. However, detailed evaluation of chronic and acute actions of testosterone on the function of cardiac I(Ca,L) and intracellular Ca2+ handling is limited. To clarify this situation we performed whole-cell and single-channel analysis of I(Ca,L), recordings of Ca2+ sparks, measurements of contractility and quantitative real-time RT-PCR in rat cardiomyocytes following testosterone pretreatment and acute testosterone application. Pretreatment with testosterone 100 nM for 24-30 h increased whole-cell I(Ca,L) from 3.8+/-0.8 pA/pF (n=10) to 10.1+/-0.31 pA/pF (n=9) at +10 mV (p<0.001). Increase of I(Ca,L) density was caused by both, increased expression levels of the alpha 1C subunit of L-type calcium channel and a pronounced increment of the single-channel activity (availability 81.8+/-3.15% versus 37.1+/-7.01%; open probability 12.8+/-3.09% versus 1.0+/-0.62%, p<0.01). Moreover, testosterone pretreatment significantly increased the frequency of Ca2+ sparks and improved myocytes contractility without altering SR Ca2+ load. All chronic effects could be inhibited by flutamide. In contrast acute testosterone administration significantly reduced I(Ca,L) density. Indeed, on the single-channel level acute testosterone application completely reversed the chronic testosterone-mediated effects, and antagonized the chronic testosterone effects on Ca2+ spark frequency, which was unaffected by flutamide. Thus, testosterone pretreatment activates I(Ca,L) via nuclear receptor-mediated pathways, while testosterone acutely blocks I(Ca,L) in a direct manner. Thus, testosterone chronically affects the basal level of intracellular Ca2+ handling, which in addition rapidly may be modulated by acute changes of hormone levels.     

Ryanodine receptor allosteric coupling and the dynamics of calcium sparks

Puffs and sparks are localized intracellular Ca²⁺ elevations that arise from the cooperative activity of Ca²⁺-regulated inositol 1,4,5-trisphosphate receptors and ryanodine receptors clustered at Ca²⁺ release sites on the surface of the endoplasmic reticulum or the sarcoplasmic reticulum. While the synchronous gating of Ca²⁺-regulated Ca²⁺ channels can be mediated entirely though the buffered diffusion of intracellular Ca²⁺, interprotein allosteric interactions also contribute to the dynamics of ryanodine receptor (RyR) gating and Ca²⁺ sparks. In this article, Markov chain models of Ca²⁺ release sites are used to investigate how the statistics of Ca²⁺ spark generation and termination are related to the coupling of RyRs via local [Ca²⁺] changes and allosteric interactions. Allosteric interactions are included in a manner that promotes the synchronous gating of channels by stabilizing neighboring closed-closed and/or open-open channel pairs. When the strength of Ca²⁺-mediated channel coupling is systematically varied (e.g., by changing the Ca²⁺ buffer concentration), simulations that include synchronizing allosteric interactions often exhibit more robust Ca²⁺ sparks; however, for some Ca²⁺ coupling strengths the sparks are less robust. We find no evidence that the distribution of spark durations can be used to distinguish between allosteric interactions that stabilize closed channel pairs, open channel pairs, or both in a balanced fashion. On the other hand, the changes in spark duration, interspark interval, and frequency observed when allosteric interactions that stabilize closed channel pairs are gradually removed from simulations are qualitatively different than the changes observed when open or both closed and open channel pairs are stabilized. Thus, our simulations clarify how changes in spark statistics due to pharmacological washout of the accessory proteins mediating allosteric coupling may indicate the type of synchronizing allosteric interactions exhibited by physically coupled RyRs. We also investigate the validity of a mean-field reduction applicable to the dynamics of a ryanodine receptor cluster coupled via local [Ca²⁺] and allosteric interactions. In addition to facilitating parameter studies of the effect of allosteric coupling on spark statistics, the derivation of the mean-field model establishes the correct functional form for cooperativity factors representing the coupled gating of RyRs. This mean-field formulation is well suited for use in computationally efficient whole cell simulations of excitation-contraction coupling.     

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.     

Transition from stochastic to deterministic behavior in calcium oscillations

Simulation and modeling is becoming more and more important when studying complex biochemical systems. Most often, ordinary differential equations are employed for this purpose. However, these are only applicable when the numbers of participating molecules in the biochemical systems are large enough to be treated as concentrations. For smaller systems, stochastic simulations on discrete particle basis are more accurate. Unfortunately, there are no general rules for determining which method should be employed for exactly which problem to get the most realistic result. Therefore, we study the transition from stochastic to deterministic behavior in a widely studied system, namely the signal transduction via calcium, especially calcium oscillations. We observe that the transition occurs within a range of particle numbers, which roughly corresponds to the number of receptors and channels in the cell, and depends heavily on the attractive properties of the phase space of the respective systems dynamics. We conclude that the attractive properties of a system, expressed, e.g., by the divergence of the system, are a good measure for determining which simulation algorithm is appropriate in terms of speed and realism.     

Stability of cardiac waves

Cardiac waves can fail to propagate when the membrane potential of the cells in the wavefront rises too slowly. The sodium channel inactivation gates play an important role in this process of propagation block. Simple models including inactivation gates can have travelling waves of constant form with two possible velocities. A stability analysis demonstrates that the slower velocity is always unstable, and in limited parameter regimes the faster velocity can also be unstable. Waves with the lower velocity propagate a finite distance before they dissipate due to this instability and this distance is calculated. The distance can be large suggesting that they might be seen in certain pathological conditions. The analytical results are compared with numerical simulations of the simplified model and a detailed cardiac ionic model.     

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.     

Phosphatidate releases calcium from cardiac sarcoplasmic reticulum

Phosphatidate (PA) inhibits calcium accumulation by cardiac sarcoplasmic reticulum (SR) and enhances its Ca⁺⁺ ATPase activity. These effects seem to be related to a phosphatidate-induced increase in the calcium permeability of the SR membrane with resultant calcium release. The amount of calcium released by phosphatidate is dependent both on the calcium concentration outside the SR vesicles and the internal calcium concentration. The ionophoric effects of phosphatidate on the sarcoplasmic membrane provide a novel pathway for controlling Ca⁺⁺ transport in the cardiac cell.     

Surfing on calcium waves

A key question in brain development is how migration of neuronal precursors is guided to establish the ordered laminar layers. In the April 20, 2007 issue of Cell, Guan et al. show that the leading process of migrating cerebellar granule neurons senses repulsive Slit molecules by generating a Ca(2+) wave that propagates to the soma to cause reversal of cell polarity and migration.     

Saltatory propagation of Ca2+ waves by Ca2+ sparks.

Punctate releases of Ca2+, called Ca2+ sparks, originate at the regular array of t-tubules in cardiac myocytes and skeletal muscle. During Ca2+ overload sparks serve as sites for the initiation and propagation of Ca2+ waves in myocytes. Computer simulations of spark-mediated waves are performed with model release sites that reproduce the adaptive Ca2+ release observed for the ryanodine receptor. The speed of these waves is proportional to the diffusion constant of Ca2+, D, rather than D, as is true for reaction-diffusion equations in a continuous excitable medium. A simplified "fire-diffuse-fire" model that mimics the properties of Ca2+-induced Ca2+ release (CICR) from isolated sites is used to explain this saltatory mode of wave propagation. Saltatory and continuous wave propagation can be differentiated by the temperature and Ca2+ buffer dependence of wave speed.     

Cooperative and stochastic calcium releases from multiple calcium puff sites generate calcium microdomains in intact Hela cells

Ca(2+) microdomains or locally restricted Ca(2+) increases in the cell have recently been reported to regulate many essential physiological events. Ca(2+) increases through the inositol 1,4,5-trisphosphate (IP(3)) receptor (IP(3)R)/Ca(2+) release channels contribute to the formation of a class of such Ca(2+) microdomains, which were often observed and referred to as Ca(2+) puffs in their isolated states. In this report, we visualized IP(3)-evoked Ca(2+) microdomains in histamine-stimulated intact HeLa cells using a total internal reflection fluorescence microscope, and quantitatively characterized the spatial profile by fitting recorded images to a two-dimensional Gaussian distribution. Ca(2+) concentration profiles were marginally spatially anisotropic, with the size increasing linearly even after the amplitude began to decline. We found the event centroid drifted with an apparent diffusion coefficient of 4.20 ± 0.50 μm(2)/s, which is significantly larger than those estimated for IP(3)Rs. The sites of maximal Ca(2+) increase, rather than initiation or termination sites, were detected repeatedly at the same location. These results indicate that Ca(2+) microdomains in intact HeLa cell are generated from spatially distributed multiple IP(3)R clusters or Ca(2+) puff sites, rather than a single IP(3)R cluster reported in cells loaded with Ca(2+) buffers.