INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION CONCLUSIONS REFERENCES

A critical step in cardiac excitation-contraction coupling is the activation of Ca²⁺ sparks (Cheng et al., 1993; Cannell et al., 1994) by voltage-gated Ca²⁺ influx through sarcolemmal L-type Ca²⁺ channels (Cannell et al., 1994, 1995) located in the sarcolemmal (SL) and transverse tubule (TT) membranes (Cheng et al., 1994, 1996a; Shacklock et al., 1995) (see Fig. 1). When an L-type channel opens, intracellular calcium ([Ca²⁺]_i) is elevated close to the channel, in the region between the sarcolemmal and sarcoplasmic reticulum (SR) membranes (also called the "fuzzy space" or the "subspace" (Lederer et al., 1990; Soeller and Cannell, 1997; Jafri et al., 1998)). This local elevation of [Ca²⁺]_i triggers SR Ca²⁺-release channels (i.e., ryanodine receptors or RyRs), located in the same subspace, to open and release a greater amount of Ca²⁺ from the SR. This Ca²⁺ released from the SR via RyRs underlies the large, local increase in [Ca²⁺]_i that is visualized as a Ca²⁺ spark. The process that activates Ca²⁺ sparks therefore displays both high gain (a small amount of "trigger Ca²⁺" activates a large amount of released Ca²⁺) and positive feedback (release of Ca²⁺ raises [Ca²⁺]_i and tends to trigger more release). These characteristics can, in principle, defeat controlled Ca²⁺ release from the SR (Stern, 1992). The fact that Ca²⁺ release is controlled locally (Niggli and Lederer, 1990) helps to overcome potential instabilities (Stern, 1992). However, the picture is currently incomplete because of the uncertainty surrounding the mechanisms that account for the termination of Ca²⁺ sparks.

View larger version (41K): [in this window] [in a new window]   FIGURE 1   Structural considerations and model schematic. (A) View of SR-TT junction showing cut section of TT membrane (transparent yellow) containing L-type Ca2+ channels (DHPR) (green). The TT containing DHPRs is closely apposed to an SR release unit composed of 100 RyRs (red). (B) Close-up view of 8 RyRs taken from (A). The green spheres represent FKBP12.6 proteins that mediate interactions between neighboring RyR homotetramers and are found close to points of contact. (C) Layout of the model elements. Ca2+ passing through a single L-type channel enters the subspace between the TT and SR membranes and can trigger release from a cluster of RyRs located in the junctional SR. In addition to channel fluxes of Ca2+, diffusion of Ca2+ from the subspace to the cytoplasm (Jefflux) and refilling of the junction from neighboring SR (Jrefill) determine Ca2+ concentrations in the subspace ([Ca2+]SS) and the local JSR lumen ([Ca2+]lumen). (D) RyR gating. Each RyR homotetramer is modeled with only a single open and a single closed state, without any adaptation or inactivation processes. Opening and closing rates depend on [Ca2+]SS, [Ca2+]lumen, and coupled gating of RyRs, as described in Methods.

To date, three distinctive explanations for Ca²⁺ spark termination have been put forward, but each hypothesis has specific weaknesses.

1.	The SR could simply run out of releasable Ca2+. Such SR exhaustion models, however, are contradicted at the global level by evidence of substantial SR Ca2+ reserves (nonzero SR Ca2+ content) following Ca2+ release (Varro et al., 1993; Bassani et al., 1995; Negretti et al., 1995), and, at the local level, by the existence of prolonged (lasting seconds) Ca2+ sparks (Cheng et al., 1993).
2.	The SR Ca2+ release process could undergo Ca2+-dependent or use-dependent inactivation or adaptation (Fabiato, 1985; Lukyanenko et al., 1998; Sham et al., 1998). However, examination of gating of RyRs shows that these processes occur too slowly (Györke and Fill, 1993; Valdivia et al., 1995; Fill et al., 2000b) or inadequately (Näbauer and Morad, 1990) to account for termination of the Ca2+ transient or the Ca2+ spark.
3.	All the RyRs in a cluster could by chance close at the same time (termed "stochastic attrition") (Stern, 1992). This mechanism is always present to some extent (Stern et al., 1999) and can work in concert with other mechanisms, but stochastic attrition fails to provide a robust control mechanism when more than a few RyRs are present in a cluster (Stern, 1992) (also see below). The absence of a viable model of Ca2+ spark termination is one of the biggest defects in our current concept of excitation-contraction coupling, as recent reviews have stressed (Niggli, 1999; Wier and Balke, 1999; Cannell and Soeller, 1999).

Three new results regarding the anatomy of the junctions between the SR and the TT in heart have considerable bearing on our understanding of Ca²⁺ sparks. First, the SR-TT junction contains nearly crystalline arrays of RyRs organized in clusters (Franzini-Armstrong et al., 1998, 1999). The number of RyRs in a cluster depends on species, but averages ~100. Second, the RyRs are organized as homotetrameric units, each touching four neighbors with a small protein, FK-binding protein (FKBP) at or near the point of contact (See Fig. 1, A and B) (Wagenknecht et al., 1997; Samso and Wagenknecht, 1998; Sharma et al., 1998). Third, RyRs incorporated into planar lipid bilayers can exhibit coupled gating, whereby two or more channels display synchronized openings and closings (Marx et al., 1998, 2001). These findings suggested to us that the clustering of the RyRs and the physical contact between homotetrameric RyRs may be functionally important.

An additional recent result is also important to our understanding of the regulation of cardiac calcium-induced calcium release (CICR). As suggested by experiments in heart cells (Cheng et al., 1996b; Lukyanenko et al., 1996), studies in planar lipid bilayers (Thedford et al., 1994; Gyorke and Gyorke, 1998; Ching et al., 2000) and cardiac vesicles (Ikemoto et al., 1989) have confirmed that Ca²⁺ in the SR lumen can influence RyR gating such that RyRs are more likely to be triggered by cytosolic Ca²⁺ when SR lumenal Ca²⁺ is elevated.

In the present study, we incorporate these new results into a mathematical model of cardiac Ca²⁺ sparks and put forward a new hypothesis, sticky cluster, to describe the behavior of Ca²⁺ sparks in heart. This hypothesis addresses two fundamental holes in our understanding of Ca²⁺ sparks, namely, how termination of Ca²⁺ sparks occurs, and what influence the large but variable number of RyRs in a cluster may have upon Ca²⁺ spark behavior. In the sticky cluster model, Ca²⁺ spark termination results from two factors: the influence of lumenal [Ca²⁺] on RyR gating, and coupled gating between the RyRs in a cluster. Furthermore, this model produces Ca²⁺ sparks that terminate robustly in a manner that is largely independent of the number of RyRs in a cluster. Additionally, the model simulates the occurrence of spontaneous sparks at a realistic rate, and is consistent broadly with experimental results, including those that reveal that Ca²⁺ sparks can be prolonged under certain conditions.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION CONCLUSIONS REFERENCES

Cell isolation and confocal imaging

Adult mice were killed by intraperitoneal injection of pentobarbital (100 mg/kg), and single ventricular myocytes were isolated (Cannell et al., 1995) and stored at room temperature (22-25°C) in Dulbecco's modified Eagle's medium (JRH Biosciences, Lanexa, KS) until used. Cells were loaded with fluo-3 by exposure to 5.5 µM fluo-3AM for 30 min at room temperature. Cells were superfused with NaCl 140 mM, KCl 5 mM, MgCl₂ 0.5 mM, CaCl₂ 1.8 mM, NaH₂PO₄ 0.33 mM, Glucose 5.5 mM, HEPES 5 mM. The pH of all extracellular solutions was kept at 7.4 with the temperature at 34-37°C. Confocal microscopy was used to measure Ca²⁺ sparks as described previously (Cheng et al., 1993; Cannell et al., 1994, 1995).

Model Layout

The model is separated into two independent components: 1) Mathematical description of SR Ca²⁺ release through an RyR cluster at the SR-TT junction. This component of the model contains the major novel features of our presentation. 2) Mathematical description of the Ca²⁺ spark produced by the Ca²⁺ efflux. This component of the model refines earlier descriptions of the diffusion, buffering, and imaging processes that produce the Ca²⁺ spark (Smith et al., 1998). Figure 1 C, which illustrates the model schematically, shows that it consists of a single L-type Ca²⁺ channel (also called a dihydropyridine receptor, DHPR) closely apposed to a cluster of RyRs in a region of junctional sarcoplasmic reticulum (JSR). The channels communicate through changes in [Ca²⁺] in a restricted subsarcolemmal space (subspace, SS). The following Ca²⁺ fluxes determine Ca²⁺ concentrations in the subspace and the local, JSR lumen ([Ca²⁺]_lumen): fluxes through the L-type channel (J_DHPR) and the RyRs (J_release), binding of Ca²⁺ to buffers in the subspace (J_buf), efflux from the subspace to the bulk myoplasm via diffusion (J_efflux), and refilling of the SR lumen by diffusion from neighboring network SR (J_refill). Thus, [Ca²⁺]_SS is described by the balance equation,

(1)

A description of how the individual fluxes are calculated follows.

Fluxes through the Ca²⁺ channels (DHPR and RyRs) depend on channel permeability, the concentration gradient for Ca²⁺, and whether the channels are open. Thus, J_DHPR = 0 when the channel is closed, and

(2)

when the channel is open, where _DHPR is the single-channel L-type current, F is Faraday's constant, and V_SS is the subspace volume (values given in Table 1). Similarly, flux through a single, open RyR is calculated as

(3)

where D_RyR reflects the ease with which Ca²⁺ passes through an open RyR. The total Ca²⁺ release flux, J_release, simply reflects the flux through all the open RyRs in the cluster, i.e.,

(4)

where RyR = 0 if channel i is closed, 1 if it is open, and N represents the number of channels in the cluster. We set N = 50 in control conditions. However, given the variability in cluster size observed experimentally, it is important to verify that Ca²⁺ sparks produced by clusters of various sizes can terminate robustly.

Ca²⁺ in the subspace can be bound to SL and SR membrane buffers and calmodulin (CaM), with buffer characteristics based on those reported previously (Smith et al., 1998). Thus,

(5)

where k is the Ca²⁺ on rate, k the Ca²⁺ off rate, [B_i] is the unbound buffer concentration, and [B_i]_tot is the total concentration (bound + unbound) of buffer i.

Efflux of Ca²⁺ from the subspace via diffusion to the myoplasm is calculated as

(6)

with [Ca²⁺]_myo held fixed at 0.1 µM, and _efflux is the time constant for Ca²⁺ transfer between the subspace and the bulk myoplasm. The different time constants for transfer to the bulk cytoplasm (_efflux) and within the SR (_refill) reflect the assumption that these two diffusion processes will typically occur over different characteristic distances. These time constants are roughly equivalent to diffusion over a spatial scale of, respectively, 8-13 nm and 0.5-1.58 µm (depending on whether the true or an effective diffusion constant for Ca²⁺ is assumed). This flux of Ca²⁺ from the subspace to the myoplasm is used as the input for the second model (described below) that simulates the production of Ca²⁺ sparks.

The balance equation for JSR lumenal Ca²⁺ concentration ([Ca²⁺]_lumen) is

(7)

where the JSR is depleted by RyR Ca²⁺ release scaled by the ratio of subspace volume (V_SS) to JSR volume (V_JSR) and refilled from the network sarcoplasmic reticulum (NSR) by J_refill,

(8)

with [Ca²⁺]_NSR held fixed at 1.0 mM and _refill is the time constant for Ca²⁺ transfer between the JSR and NSR. Buffering in the JSR by calsequestrin uses the rapid buffering approximation (Keizer and Levine, 1996) given by

(9)

where [CSQ]_tot, [CSQ], and K_CSQ represent the total concentration, unbound concentration, and Ca²⁺-dissociation constant, respectively, of calsequestrin.

Ryanodine receptor gating

The primary goal of the present study is to gain insight into Ca²⁺ spark behavior and the mechanisms of spark termination by investigating how the features of RyR gating in the sticky cluster influence Ca²⁺ spark characteristics. Because, in cardiac muscle, Ca²⁺ spark termination occurs independently of spark triggering (Cheng et al., 1993; Cannell et al., 1995), we did not simulate the gating of the DHPR. In most simulations, we initiated a Ca²⁺ spark with a stereotypical DHPR opening (0.5 pA, 0.5 ms) that could trigger a spark with >98% fidelity. The details of the gating of an RyR in a cluster follow.

The 50 RyRs are assumed to gate independently except for a "coupling" or "cooperativity" factor (CF_close and CF_open as defined below). Each RyR is modeled with only a single closed and a single open state and no adaptation or other inactivation processes over the time scale of the Ca²⁺ spark (Fig. 1 D). We exclude these features for the sake of simplicity and to demonstrate that they are not necessary for spark termination. We do not mean to argue against a role for these phenomena in regulating cardiac Ca²⁺ signaling. Thus, in this model, any intervention that alters the gating behavior of the RyR cluster must do so by modifying either the opening rate, k_open, or the closing rate, k_close.

The RyR closing rate was independent of [Ca²⁺]_SS,

(10)

whereas the opening rate was a fourth-order function of [Ca²⁺]_SS,

(11)

Lumenal Ca²⁺ influences RyR gating through changes in K_m in the above equation. Sensitivity to [Ca²⁺]_SS is a linear, decreasing function of [Ca²⁺]_lumen, so that RyR opening is favored when [Ca²⁺]_lumen is high, as suggested by the literature (Thedford et al., 1994; Cheng et al., 1996b; Gyorke and Gyorke, 1998; Lukyanenko et al., 1998; Ching et al., 2000):

(12)

Coupled gating of RyRs is introduced by multiplying the opening and closing rate constants by cooperativity factors (CF_open for opening and CF_closed for closing) that depend, respectively, on the relative numbers of open and closed channels in the cluster,

(13)

(14)

where N_open is the number of open RyRs, and N_closed is the number of closed RyRs. CF_open and CF_close are both cooperative in their behavior because the value of either depends on the fraction of open channels. The scaling factor k_coop (equal to unity in control conditions) was introduced so that modification of a single parameter could simulate changes in the strength of coupling between RyRs.

A Monte Carlo method (Rice et al., 1999) was used to simulate the openings and closings of the RyRs with the Ca²⁺ balance equation solved at each time step using Euler's method. A time step of 10⁸ s was used in the simulations. The code was implemented in FORTRAN on an HP Visualize unix workstation.

Spark model

The model described above produces as its output a flux of Ca²⁺ from the SS into the myoplasm. To generate Ca²⁺ signals with similar spatiotemporal characteristics to those that would be measured experimentally, we simulated spark generation and detection, based on a method previously published (Smith et al., 1998). This model calculates Ca²⁺ diffusion in the myoplasm (spherical symmetry is assumed), Ca²⁺ binding to fluo-3 and stationary buffers, and blurring by the confocal microscope. The model implemented here is identical to that presented by Smith et al.(1998) with two exceptions: we assume that SL buffers are confined to within 300 nm of the cluster and the concentration of these buffers decreases linearly with distance from the RyR cluster, and we assume a fluo-3 Ca²⁺-dissociation constant of 0.5 µM, rather than 1.13 µM. These changes were made so that the time course of the model Ca²⁺ spark more closely resembled that typically recorded experimentally: specifically, the original Smith model produced sparks with an unrealistically large amplitude and an unrealistically slow decay after release termination. A semi-implicit algorithm was used to solve for Ca²⁺ dynamics in space and time, such that diffusion of Ca²⁺ and fluo-3 was treated implicitly with a Crank-Nicholson algorithm, and Ca²⁺ buffering was treated explicitly. The model was implemented in Matlab (The Mathworks, Natick, MA), and a time step of 2 µs was used. Results were visualized with IDL and Origin software, and CorelDraw was used to produce figures.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION CONCLUSIONS REFERENCES

To explain the behavior of Ca²⁺ sparks in intact cells, given the current knowledge of the anatomy, biochemistry, and biophysics of Ca²⁺ signaling in heart, two questions must be addressed: how do Ca²⁺ sparks terminate, and why are Ca²⁺ sparks so similar in duration?

Basic simulation

Figure 2 displays an example of simulated Ca²⁺ release under control conditions, with 50 RyRs in the sticky cluster and a cooperativity factor (k_coop) of 1. In this simulation, as in most of the simulations carried out in this study, a stereotypical DHPR opening (0.5 ms, 0.5 pA) triggers CICR in the RyR cluster. Our Monte-Carlo method simulates stochastic openings and closings of RyRs in the cluster (as seen by the noise or chatter in the records shown), and the temporal evolution of each Ca²⁺ release event is slightly different. The composite open probability (P_o) of the 50 RyRs, the resulting Ca²⁺ efflux from the subspace to the myoplasm, and the local JSR lumenal Ca²⁺ concentration ([Ca²⁺]_lumen) are displayed in Fig. 2, A, B, and C, respectively. Ca²⁺ entering the subspace (SS) through the DHPR activates release from the cluster by binding to the RyRs and increasing their P_o to a value close to 1. The opening of RyRs increases [Ca²⁺]_SS further, thereby contributing additional Ca²⁺ to the positive feedback of CICR. The composite P_o of the RyRs remains close to 1 for ~10 ms but declines as [Ca²⁺]_lumen (Fig. 2 C) and [Ca²⁺]_SS fall and exhibits larger fractional fluctuations until all RyRs close nearly simultaneously (P_o quickly declines to zero) after ~25 ms. The firm closure of the RyRs arises from the gating cooperativity of the RyRs in the cluster. Re-opening is prevented by hysteresis in the relationship between P_o and [Ca²⁺]_SS that is due to the influence of [Ca²⁺]_lumen on this relationship. The Ca²⁺ efflux (Fig. 2 B) during the release process reaches an early peak, declines as the local SR lumen becomes depleted of Ca²⁺, then decreases to zero when the RyRs in the cluster close.

View larger version (13K): [in this window] [in a new window]   FIGURE 2   Simulated RyR gating and Ca2+ fluxes during a typical Ca2+ spark. (A) Composite open probability (Po) of the 50 RyRs in the cluster. (B) Efflux from the subspace versus time. (C) Lumenal Ca2+ concentration ([Ca2+]lumen) versus time. The decrease in [Ca2+]lumen leads to increased flickering of RyRs and a slight decrease in Po until the influence of coupled gating causes all RyRs in the cluster to close within 3 ms and not reopen.

The efflux of Ca²⁺ through the cluster of RyRs, when used as an input to the buffering and diffusion model described above, produces a Ca²⁺ spark as shown in Fig. 3. Figure 3 A displays a simulated line-scan image of the resulting Ca²⁺ spark, similar to that which would be recorded experimentally (assuming the point of release is located directly on the scan line). The relative level of [Ca²⁺]_i (measured as F/F₀) is shown in time-profiles of the line-scan image in Fig. 3 B, with the color-coded tick marks on the right-hand edge of Fig. 3 A indicating that [Ca²⁺]_i time courses are displayed at the point of release and at distances 0.25 and 0.5 µm from the point of release. The spatial profiles that would be recorded during the Ca²⁺ spark are shown in Fig. 3 C. The tick marks on the bottom of Fig. 3 A show the times of the profiles, corresponding to 2, 5, 10, 20, and 50 ms after Ca²⁺ spark initiation.

View larger version (21K): [in this window] [in a new window]   FIGURE 3   Simulated Ca2+ spark under control conditions. (A) Line scan image of simulated Ca2+ spark under control conditions (50 RyRs in cluster, kcoop = 1). (B) Time courses of Ca2+ sparks (plotted as F/F0) at locations 0, 0.25, and 0.5 µm from the source are displayed in black, red, and green, respectively. Locations on line scan are noted to right of the image. (C) Ca2+ spark spatial profiles 2, 5, 10, 20, and 50 ms after the beginning of the spark are displayed in black, red, green, blue, and, cyan respectively. Times are marked below the image.

Background noise

Experimentally measured line-scan images of Ca²⁺ sparks exhibit fluctuations in fluorescence intensity due in part to the noise of the system. This noise comes from the excitation laser, the photodetector, amplifiers, and the fluorescent light itself. We characterized the noise recorded in line-scan images of Ca²⁺ sparks and then simulated this noise as normally distributed fluctuations about a mean level. Figure 4 A (top) shows the line-scan image of a simulated Ca²⁺ spark with this realistic noise added. The increase in background reflects different scaling of the image after fluctuations around the background are added. The middle and bottom traces, respectively, display the time course of the noisy Ca²⁺ spark at the point of release and averaged over the width of the Ca²⁺ spark. (±0.5 µm centered on the point of Ca²⁺ release) Figure 4 B shows identical records produced by the model with noise added when the cluster comprises only a single RyR. One of the clear effects of the addition of a realistic amount of noise is the loss of the apparent shoulder produced as Ca²⁺ release is terminated. A second observation is that one cannot visually detect the [Ca²⁺]_i signal resulting from the opening of a single RyR. This is due to the small level of Ca²⁺ release and its short duration. This result, however, provides an explanation for the absence or near invisibility of Ca²⁺ sparks in cell systems with few or extremely small RyR clusters (Bhat et al., 1997; Haak et al., 2001).

View larger version (74K): [in this window] [in a new window]   FIGURE 4   Appearance of Ca2+ sparks with realistic noise. (A) Line-scan image of simulated Ca2+ spark that would result from the opening of 50 RyRs with realistic noise added (top), time course of [Ca2+]i as F/F0 (0.25 µm from source, middle) and time course of [Ca2+]i (averaged over central 1 µm of Ca2+ spark, bottom). The addition of realistic noise leads to the loss of the inflection in the time course of [Ca2+]i when release is terminated (see Fig. 3 B). See text for details. (B) Line-scan image of simulated Ca2+ spark with realistic noise that would result from the opening of a single RyR (top), time course of [Ca2+]i as F/F0 (0.25 µm from source, middle) and time course of [Ca2+]i (averaged over central 1 µm of point of Ca2+ release, bottom). The fluorescent Ca2+ signal is not readily distinguishable from the noise.

Cluster size and RyR flux

In Figs. 2-4, 50 RyRs were arranged in a cluster to simulate a control Ca²⁺ spark; however, because it is likely that the number of RyRs in a cluster may vary widely (Franzini-Armstrong et al., 1998, 1999), we simulated Ca²⁺ sparks produced by clusters of different sizes. Simulations from clusters containing 100, 50, 20, and 10 RyRs are shown in Fig. 5, A, B, C, and D, respectively. Each panel displays the SR release flux (top), [Ca²⁺]_lumen (middle), and the Ca²⁺ spark time course, measured at 0.25 µm from the source (bottom). Three features of these simulations are of particular interest. First, SR release terminates robustly at about the same time after it is initiated, leading to Ca²⁺ sparks with similar durations regardless of the number of RyRs in the cluster. Second, termination of Ca²⁺ release occurs at different levels of [Ca²⁺]_lumen as cluster size is varied. This helps to explain the small variation in the duration of release. With fewer RyRs in the cluster, it is more likely that the stochastic closing of a few RyRs will induce the remaining channels to close. However, the decreased peak Ca²⁺ efflux leads to slower depletion of [Ca²⁺]_lumen. Because P_o depends on [Ca²⁺]_lumen, the slower depletion tends to prolong the duration of release. These two effects essentially cancel each other, leading to the relatively constant Ca²⁺ spark duration observed. Third, although the peak of the Ca²⁺ release flux approximately scales with the number of RyRs in a cluster (top traces), the peak F/F₀ of a Ca²⁺ spark does not (bottom traces). Two explanations appear to account for this difference. With more RyRs in a cluster, the total amount of Ca²⁺ released during a spark does not scale with the peak level of Ca²⁺ efflux due to the more rapid lumenal depletion, and the Ca²⁺ spark amplitude reflects the amount of Ca²⁺ bound to fluo-3, which is nonlinearly related to the total amount of Ca²⁺ released (Izu et al., 2001).

View larger version (11K): [in this window] [in a new window]   FIGURE 5   Simulated Ca2+ sparks resulting from RyR clusters of various sizes. (A) 100 RyRs in cluster. Efflux from the subspace (top), [Ca2+]lumen (middle), and [Ca2+]i as F/F0 (measured 0.25 µm from source, bottom). (B) 50 RyRs in cluster (control conditions). (C) 20 RyRs in cluster. (D) 10 RyRs in cluster. Varying the number of RyRs in the cluster affects initial efflux significantly, spark amplitude modestly, and spark duration minimally (see Fig. 6).

Figure 6 displays the composite results from 500 simulated Ca²⁺ sparks at each cluster size. The histograms displayed in Fig. 6 A demonstrate that, although the mean SR release duration remains relatively constant, as noted above, variability increases as the cluster size is decreased from 100 to 10 RyRs. Figure 6 B shows the less than proportional increase in Ca²⁺ spark amplitude, and Fig. 6 C displays the biphasic changes in SR release duration produced by increases in cluster size. Because of stochastic attrition, SR release is quite short for very small clusters, but SR release duration increases rapidly as stochastic attrition ceases to be a significant factor (e.g., with ~10 RyRs). With an increase in cluster size beyond ~20 RyRs, spark duration decreases slightly due to the faster SR depletion noted above.

View larger version (25K): [in this window] [in a new window]   FIGURE 6   Effects of RyR number on Ca2+ spark amplitude and duration: population data. (A) Histograms of the duration of Ca2+ release for 100, 50, 20, and 10 RyRs in cluster. 500 Ca2+ sparks were simulated to generate each histogram. (B) Peak amplitude of Ca2+ sparks (averaged) versus RyR cluster size. An increase in cluster sizes causes a less-than-proportional increase in Ca2+ spark amplitude. (C) Mean Ca2+ spark duration and standard deviation plotted as a function of the number of RyRs in a cluster. Ca2+ spark duration reveals a biphasic dependence on number of RyRs in a cluster. For small clusters, Ca2+ sparks are short due to the influence of stochastic attrition. As RyRs increase in number for large clusters, increasingly rapid SR depletion underlies the shortening of the Ca2+ spark duration.

With the parameters we have chosen for this model, the peak current through a single RyR (i.e., before local SR depletion occurs) is 0.07 pA. Although this is considerably smaller than any i_RyR that has been measured in bilayers, we feel that it may represent a realistic value for the single RyR current that occurs in cells (see Discussion). However, to ensure that these specific parameter choices did not significantly affect the overall model behavior, we ran additional simulations in which i_RyR was increased by a factor of 5, to 0.35 pA. Figure 7 compares the Ca²⁺ sparks observed with the control value of current (A) and with this increased value (B). With greater single-channel RyR Ca²⁺ flux, the peak Ca²⁺ release flux is larger, but SR depletion occurs more quickly, so the Ca²⁺ spark is not five times larger (similar to the changes that occur with increased RyR number). Interestingly, the SR release duration is almost identical even with this increased i_RyR. Thus, our model is robust enough that similar behavior occurs even with substantial changes in single-channel RyR current.

View larger version (30K): [in this window] [in a new window]   FIGURE 7   Effect of increasing single-channel RyR current. (A) Efflux from the subspace (top), F/F0 0.25 µm from the source (middle), and simulated Ca2+ spark line-scan image (bottom) under control conditions, assuming a single-channel RyR current of 0.07 pA. (B) Simulated Ca2+ spark assuming an increased single-channel RyR current of 0.35 pA. Although the initial efflux is considerably larger, the Ca2+ spark amplitude is only larger by a factor of ~1.5. Ca2+ spark duration is almost identical.

Effects of changes in [Ca²⁺]_SS and [Ca²⁺]_lumen

As a test of the model, we examined how changes in [Ca²⁺]_SS and [Ca²⁺]_lumen affect simulated Ca²⁺ sparks. Because Ca²⁺ sparks are triggered by the large increase in [Ca²⁺]_SS resulting from a DHPR opening, smaller steady-state increases in global [Ca²⁺]_i will increase [Ca²⁺]_SS. In the absence of Ca²⁺ spark triggering by the opening of DHPRs, this increase in [Ca²⁺]_SS will slightly increase the RyR opening rate and therefore should result in more frequent background or spontaneous sparks. Experimental studies have suggested (Cheng et al., 1996b) that such an increase in Ca²⁺ sparks occurs with elevated [Ca²⁺]_i (Satoh et al., 1997). We tested this idea by repeatedly running simulations with no DHPR opening and recording how often spontaneous Ca²⁺ sparks occurred. Figure 8, A and B, which displays simulated line-scan images that would be recorded from quiescent cells, shows that spontaneous sparks occur more frequently when [Ca²⁺]_SS is increased from 100 to 150 nM. On average (600 simulations, 800 ms each), this elevation in [Ca²⁺]_SS increased Ca²⁺ spark frequency by six times. Figure 8 C plots the spontaneous spark rate (defined as number of sparks per cluster per second) over a much wider range of intracellular Ca²⁺ concentrations. The Ca²⁺ spark rate increases by approximately four orders of magnitude as [Ca²⁺]_SS increases from 100 nM to ~1 µM.

View larger version (12K): [in this window] [in a new window]   FIGURE 8   Effect of cytosolic [Ca2+] on spontaneous sparks. (A) Simulated Ca2+ line-scan image showing spontaneous sparks at cytosolic [Ca2+] of 100 nM. 600 simulations of 800 ms each were run without L-type channel openings to determine the rate at which spontaneous sparks occurred. The results of these simulations were translated into line-scan images by assuming that a longitudinal line scan that covered 50 µm of a ventricular myocyte would be able to detect Ca2+ sparks from 100 separate clusters of RyRs. (B) Simulated line-scan images showing spontaneous sparks with cytosolic [Ca2+] increased to 150 nM. More spontaneous sparks occur in this example. On average, the spontaneous spark rate was increased by six times upon increasing [Ca2+]SS to 150 nM (5 spontaneous sparks in 4.8 s at [Ca2+]i = 100 nM; 30 sparks at 150 nM). (C) Plot of normalized Ca2+ spark rate versus [Ca2+]i = [Ca2+]SS. The spontaneous spark rate (e.g., sparks per cluster per second) increases by four orders of magnitude as [Ca2+]i increases from a resting level of 100 nM to 1 µM.

Figure 9 A displays simulated Ca²⁺ sparks obtained at three different initial levels of [Ca²⁺]_lumen. For these simulations, we fixed the network SR Ca²⁺ concentration at 1 mM to keep the [Ca²⁺]_lumen refilling rate relatively constant. Increasing the SR load increases the spark amplitude as one would expect but has little effect on Ca²⁺ spark duration. The SR Ca²⁺ fluxes that produce the sparks shown in Fig. 9 A are plotted on a normalized scale in Fig. 9 B. Consistent with experiments that back-calculated release flux from Ca²⁺ spark records (Lukyanenko et al., 1998), we observe an increase in the release flux decay rate with increasing SR load. Fig. 9, C and D, displays how changes in [Ca²⁺]_lumen affect the spontaneous spark rate and the spark duration, respectively. Spark rate is elevated nonlinearly with increasing [Ca²⁺]_lumen, as experiments have demonstrated (Cheng et al., 1996b; Satoh et al., 1997). This increased occurrence of spontaneous sparks is thought to be responsible for the Ca²⁺ waves and arrhythmogenic currents that are seen in Ca²⁺ overload (E. A. Sobie, W. J. Lederer, and M. S. Jafri, work in progress). Ca²⁺ spark duration is extremely insensitive to [Ca²⁺]_lumen with refilling as we have modeled it. However, at extremely low SR Ca²⁺ loads, decreasing [Ca²⁺]_lumen further causes Ca²⁺ sparks to terminate prematurely.

View larger version (16K): [in this window] [in a new window]   FIGURE 9   Effects of [Ca2+]lumen on Ca2+ sparks. (A) Simulated Ca2+ sparks at different beginning levels of [Ca2+]lumen. With constant refilling rate, increasing [Ca2+]lumen increases Ca2+ spark amplitude but does not change spark duration. (B) SR Ca2+ release fluxes that produce the sparks shown in (A), plotted on a normalized scale. Release flux decays more quickly with increased SR load. (C) Spontaneous spark rate increases nonlinearly with increases in [Ca2+]lumen. (D) Ca2+ release flux duration is generally insensitive to [Ca2+]lumen except at extremely low SR load.

In the absence of any dependence of RyR gating on [Ca²⁺]_lumen, the model produces long Ca²⁺ sparks that do not terminate as illustrated in Fig. 10. Figure 10 shows how the composite P_o in the cluster changes (A) as the Ca²⁺ efflux (B) and [Ca²⁺]_lumen (C) decline. Panels D and E display the relative [Ca²⁺], 0.25 µm from the point of release and the line-scan image of the Ca²⁺ spark, respectively. The fluctuations in P_o during the Ca²⁺ spark arise because the decrease in [Ca²⁺]_lumen leads to a decrease in J_release and hence a decrease in [Ca²⁺]_SS. However, because RyR gating does not depend on [Ca²⁺]_lumen, the fluctuations in P_o remain relatively minor, the RyRs are continually activated by the small but sustained J_release, and the Ca²⁺ spark does not terminate.

View larger version (19K): [in this window] [in a new window]   FIGURE 10   Ca2+ sparks when [Ca2+]lumen has no effect on RyR gating. (A) Composite RyR Po versus time. For this simulation, Km in Eq. 3 was set to 3.6 µM, the value at the control diastolic [Ca2+]lumen of 1 mM. Release does not terminate before the end of the 800-ms simulation and was not observed under these conditions. (B) Jrelease. (C) [Ca2+]lumen. (D) [Ca2+]i as F/F0 (0.25 µm from source). (E) Line-scan image of Ca2+ spark. Note that, during the prolonged spark, F/F0 is maintained at ~60% of its peak level, although the release flux declines by more than 90%. This occurs because of the presence of cytosolic Ca2+ buffers.

Stickiness of the RyRs in the cluster at the SR-TT junction

Because any specific assumption about the nature of RyR interactions in a cluster, other than their tendency to open and close together, would require arbitrary assumptions that would be difficult to verify experimentally (see Discussion), we modeled coupled gating of RyRs with the minimal number of assumptions. This allowed us to vary the strength of coupling between RyRs by modifying a single parameter, k_coop in Eq. 14. Figure 11 exhibits the changes in Ca²⁺ sparks that occur when k_coop is decreased from its control value of 1.0, reducing the tendency of one RyR to influence the P_o of another RyR. Typical Ca²⁺ sparks simulated with k_coop = 0.5 and k_coop = 0.4 are presented in Fig. 11, A and B, respectively. With decreased coupling between RyRs, there are large fluctuations in P_o. However, the coupling between channels is not strong enough for the closed channels to induce closure of the remaining channels while [Ca²⁺]_SS remains high. This leads to an increase in the duration of J_release, prolonged Ca²⁺ sparks, and noticeable fluctuations in fluorescence. Figure 12 A compares the durations of Ca²⁺ sparks for 500 simulations under control conditions (k_coop = 1, left) to those observed with decreased coupling (k_coop = 0.4, right). The duration of Ca²⁺ sparks (mean and standard deviation) is plotted against k_coop in Fig. 12 B over the range 0.4-1.5. The spark duration begins to rise rapidly, and variability increases, as k_coop declines to values less than ~0.7. When coupling between RyRs is removed completely (k_coop = 0), Ca²⁺ sparks do not terminate (data not shown), revealing Ca²⁺ sparks that look roughly like the one shown in Fig. 10. Due to the maintained dependence on [Ca²⁺]_lumen, there are greater fluctuations in P_o. However, in the absence of coupled gating, the activation of RyRs in the cluster is sufficient to produce a maintained Ca²⁺ spark.

View larger version (31K): [in this window] [in a new window]   FIGURE 11   Ca2+ sparks under conditions of reduced coupling between RyRs. Time course of Ca2+ spark components when coupling between RyRs is reduced. (A) Reduced coupled gating with kcoop = 0.5. Po (top), Jrelease (second from top), [Ca2+]lumen (middle), [Ca2+]i as F/F0 (0.25 µm from source, second from bottom), and line-scan image of Ca2+ spark (bottom). (B) Reduced coupled gating with kcoop = 0.4. The release flux lasts for ~60 ms (kcoop = 0.5) and ~200 ms (kcoop = 0.4), leading to prolonged Ca2+ sparks. Reduced coupling also leads to increased flickering of RyRs during the prolonged release, which is apparent in the spark time course and image. These fluctuations vary greatly from simulation to simulation as does the Ca2+ spark duration (see Fig. 12).

View larger version (21K): [in this window] [in a new window]   FIGURE 12   Ca2+ spark duration when RyR coupling is reduced. (A) Histograms of Ca2+ release duration generated by 500 simulations with kcoop = 1 (control conditions, left) and kcoop = 0.4 (right). Bins are 2.5 ms wide for control and 25 ms wide for kcoop = 0.4. (B) Mean release duration and standard deviation (error bars) versus decreasing strength of coupling (kcoop). Reducing the coupling between RyRs greatly increases both the mean value and the variability of the Ca2+ spark duration.

FK506 and Rapamycin

Because FK506 and rapamycin have been shown to reduce coupled gating between RyRs in planar lipid bilayer experiments (Marx et al., 2001), but conflicting effects on Ca²⁺ sparks in intact heart cells have been reported, (McCall et al., 1996; Xiao et al., 1997), we carried out experiments in mouse heart cells. Consistent with some prior reports (Xiao et al., 1997; Lukyanenko et al., 1998) we observed prolonged Ca²⁺ sparks in the presence of FK506 or rapamycin as shown in Fig. 13. Figure 13 D shows that about half of the Ca²⁺ sparks obtained in FK506 lasted longer than 40 ms, whereas only ~10% of control Ca²⁺ sparks did. These findings are consistent with the hypothesis that RyR interactions, including those involving RyR-associated proteins like FKBP12.6, contribute to Ca²⁺ spark termination. Other actions of FK506 and related agents are discussed below.

View larger version (45K): [in this window] [in a new window]   FIGURE 13   Experimental results: effects of FK506 and rapamycin. (A) A typical Ca2+ spark (line-scan image: position shown vertically, time horizontally) under control conditions (top), and plot of F/F0 (bottom). (B) Three examples of Ca2+ sparks following application of 25 µM FK506 displayed as in (A). (C) Three examples of Ca2+ sparks following application of 20 µM rapamycin displayed as in (A). (D) Fraction of total Ca2+ sparks <40 ms and 40 ms. ** p < 0.05. Calibration: position = 10 µm; fluorescence = 0.5 F/F0; time = 100 ms.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION CONCLUSIONS REFERENCES

Overview

We have developed a relatively simple mathematical model of the cardiac Ca²⁺ spark to test a new hypothesis that addresses one of the remaining major unresolved questions in cardiac EC coupling: how Ca²⁺ sparks are able to terminate. In the model, a cluster of up to a hundred RyRs in an SR junction is responsible for a Ca²⁺ spark, and the RyR Ca²⁺ release channels can assume only one of two conformations: open or closed. However, each RyR is influenced by the ensemble behavior of the other RyRs in the cluster through a cooperativity factor (k_coop) thereby allowing the model to simulate coupled gating among RyRs. We term this the sticky cluster model to emphasize the central importance of physical contact between neighboring RyRs. The other key feature of the model is that RyR gating is influenced not only by the local cytosolic [Ca²⁺] in the subspace ([Ca²⁺]_SS), but also by the Ca²⁺ concentration in the local JSR lumen ([Ca²⁺]_lumen). The combination of these two factors provides the negative feedback and hysteresis that allow for Ca²⁺ sparks to terminate robustly and not be immediately re-triggered. The model thus takes into account new findings on both the anatomy and the gating behavior of cardiac RyRs.

The sticky cluster model successfully reproduces many features of cardiac Ca²⁺ sparks and is consistent with simple experimental interventions. Model Ca²⁺ sparks terminate readily, and many features of Ca²⁺ sparks are largely insensitive to the number of RyRs in the cluster (Figs. 5 and 6). In addition, increases in either [Ca²⁺]_SS or [Ca²⁺]_lumen increase the frequency with which spontaneous sparks occur (Figs. 8 and 9), as experiments have shown (Cheng et al., 1996b; Satoh et al., 1997; Lukyanenko et al., 2001). Finally, decreasing coupling between RyRs prolongs Ca²⁺ spark duration (Fig. 12), similar to the experimental effects of FK506 (Fig. 13). The model should therefore prove useful for generating new predictions and extending our understanding of Ca²⁺ signaling in heart. The work does, nevertheless, raise some specific points of interest that can only be addressed with additional experiments and computer modeling.

Termination of Ca²⁺ sparks

As a high-gain, positive-feedback system, CICR is potentially unstable. One way that the cardiac myocyte minimizes the risk of instability is by placing CICR under local control. By this it is meant that the Ca²⁺ transient reflects the recruitment of individual units of release (Ca²⁺ sparks), which are triggered by local increases in Ca²⁺ and do not themselves trigger regenerative, cell-wide Ca²⁺ release under normal conditions. However, the stability provided by local control would be lost without a reliable mechanism for the termination of Ca²⁺ sparks.

Three primary explanations for how Ca²⁺ sparks terminate have been proposed in the literature. First, it is possible that the SR exhausts its supply of Ca²⁺, and the duration of a Ca²⁺ spark reflects the time it takes to empty the local SR Ca²⁺ store. This explanation appears unlikely because the SR retains much Ca²⁺ after a [Ca²⁺]_i transient (Varro et al., 1993; Negretti et al., 1995; Bassani et al., 1995) and because extremely long (seconds) Ca²⁺ sparks can be observed under certain experimental conditions (Cheng et al., 1993). Second, it is possible that the RyRs undergo Ca²⁺-dependent or use-dependent inactivation after they open. However, in planar lipid bilayer experiments, simple steady-state inactivation only occurs at Ca²⁺ concentrations above ~10 mM (Sitsapesan and Williams, 2000) not at the concentrations that are thought to be present near the RyRs during a Ca²⁺ spark (10-100 µM) (Meissner et al., 1988; Rousseau and Meissner, 1989; Ashley and Williams, 1990). RyRs in bilayer experiments display adaptation (a complicated form of inactivation), but this process is incomplete and occurs relatively slowly (Györke and Fill, 1993; Valdivia et al., 1995). These observations argue against a primary role for adaptation or inactivation in the termination of Ca²⁺ sparks. A third possibility is "stochastic attrition" (Stern, 1992), whereby all of the RyRs in a cluster happen to close at the same time. Although this would be a plausible hypothesis if very small RyR clusters produced Ca²⁺ sparks, stochastic attrition becomes increasingly unlikely as the number of RyRs in a cluster increases (Stern, 1992; Stern et al., 1999) and its probability is vanishingly small when more than ~20 RyRs are present.

The model presented here overcomes the limitations of other hypotheses while incorporating some of their important features. Our proposed mechanism is similar to stochastic attrition in the sense that some RyRs must close probabilistically for coupled gating to induce the remaining channels to close. It resembles SR exhaustion in that substantial local SR depletion is required for the spark to terminate. We modeled RyR gating without any inactivation or adaptation processes to demonstrate that these are not necessary for Ca²⁺ spark termination, but these results do not necessarily argue against an important role for either of these mechanisms in the regulation of cardiac CICR. Adding either of these phenomena to the model would presumably assist the termination of the Ca²⁺ spark by closing some of the RyRs in the cluster, thus causing the SR release flux to decay more steeply and subspace [Ca²⁺] to be reduced. Finally, coupled gating as we have modeled it is similar in spirit to the allosteric interactions between RyRs that have been modeled by Stern et al. (1999). However, although allosteric interactions could greatly stabilize the activation of Ca²⁺ release in that model, their effects on termination of release were less well-explored and only occurred through the transition to an inactivated state. Additional work is clearly necessary to better explore the multiple effects that interactions between RyRs can have on cluster behavior.

Restitution and SR Ca²⁺ content

Following a Ca²⁺ release event, time must elapse before CICR can be triggered again, at both the local (Ca²⁺ spark) and global (cell-wide Ca²⁺ transient) levels, a feature called "restitution." For instance, a Ca²⁺ transient evoked by field stimulation after a spontaneous Ca²⁺ wave caused less Ca²⁺ release in locations that the wave had just passed than in locations at which more time had elapsed (Cheng et al., 1996b). Consistent with this, Tanaka et al. (1998) found that, during a Ca²⁺ transient, sparks were not triggered at locations where a spontaneous Ca²⁺ spark had occurred within the previous 25 ms. In addition, Sham et al. (1998) observed a negative correlation between the amounts of Ca²⁺ release triggered by Ca²⁺ current upon depolarization and by Ca²⁺ tail current upon repolarization. In other words, sites that released Ca²⁺ early during a depolarizing pulse tended to not release Ca²⁺ when a second Ca²⁺ influx occurred at the end of the pulse, even though the SR still contained Ca²⁺ (Sham et al., 1998). Although these results can be interpreted to indicate a refractoriness of Ca²⁺ release due to Ca²⁺-dependent or use-dependent inactivation, it is also possible that this refractoriness results from the time it takes for partially depleted SR release sites to be refilled with Ca²⁺. If RyRs are more likely to open when [Ca²⁺]_lumen is high, as we have modeled here and as experiments have indicated, Ca²⁺ sparks will be more difficult to trigger during the refilling time that follows a spark. This factor makes it difficult to interpret experiments in which the cell contains a high concentration of exogenous intracellular Ca²⁺ buffer, because these buffers compete with the SR Ca²⁺ pump and slow reuptake of Ca²⁺ into the SR (Sham et al., 1998; DelPrincipe et al., 1999). Indeed, studies that have used high Ca²⁺ buffer concentration have observed an unrealistically slow recovery of Ca²⁺ release (DelPrincipe et al., 1999). Clearly more experiments are necessary to resolve the roles played by SR refilling, recovery from inactivation, and possibly other mechanisms in the restitution process.

What is the shape of the Ca²⁺ release flux?

Most previous models of the cardiac Ca²⁺ spark (see below), have, for simplicity, assumed that a spark results from a constant SR Ca²⁺ release flux. In skeletal muscle, Schneider and colleagues have used curve-fitting of high temporal resolution Ca²⁺ spark recordings to argue that the Ca²⁺ release flux underlying the Ca²⁺ spark is constant (Lacampagne et al., 1999), but a comparable study on cardiac Ca²⁺ sparks has not yet been done. For a variety of reasons, we believe that the cardiac Ca²⁺ spark likely results from a decaying SR Ca²⁺ release flux. First, the extremely small volume of the junctional SR suggests that, even with considerable Ca²⁺ buffering power, Ca²⁺ release through a cluster of RyRs could empty this volume rather quickly. For example, a local JSR with a diameter of 300 nm and a depth of 10 nm would have a volume of only 7 × 10⁴ µm³ (assuming a cylindrical geometry). Even with 50 mM of buffer-bound Ca²⁺ and 1 mM free Ca²⁺, this volume wound only contain ~21000 Ca²⁺ ions. Because a 1-pA current is equivalent to ~3000 Ca²⁺ ions per millisecond, the Ca²⁺ in the local JSR would only be able to provide the Ca²⁺ spark release flux a few milliseconds. Thus, even if refilling of the JSR is fast, it seems likely that significant local SR depletion could occur over the time scale of the Ca²⁺ spark.

The idea of a decaying release flux accounting for the cardiac Ca²⁺ spark is consistent with the results of Lukyanenko et al. (1998), who back-calculated the responsible fluxes from line-scan Ca²⁺ spark recordings. However, the accuracy of their calculations could have been compromised by the simple buffering approximation that they used. They assumed, as we have here, that binding of fluo-3 to stationary cytoplasmic proteins can be modeled by simply reducing the fluo-3 diffusion constant. However, the results of Harkins et al. (1993) suggest that fluo-3 bound to Ca²⁺ has a different affinity for proteins than does free fluo-3. For this reason, we performed additional simulations with a more complex buffering-diffusion scheme derived from the Harkins et al. data (detailed in Hollingworth et al., 2000). The Ca²⁺ sparks produced by stereotyped Ca²⁺ release fluxes with the complex buffering scheme are compared with those generated with the simple buffering scheme (i.e., the model used in the other simulations) in Fig. 14. Three features of these simulations are of interest. One is that Ca²⁺ sparks produced by decaying fluxes are rounded at the peak, whereas those resulting from constant fluxes come to a sharp peak when release shuts off. A second observation is that the choice of buffer model causes only subtle changes in the Ca²⁺ spark shape. In other words, the Ca²⁺ spark time courses produced by constant and decaying sources look similar with either buffer model. Finally, the extended Ca²⁺ sparks displayed show that a maintained flux of only10% of the peak level can produce a maintained F/F₀ of ~50% of the peak level. These maintained F/F₀ levels, which are similar to those observed experimentally (e.g., Fig. 13), suggest that a very small Ca²⁺ flux can maintain a significant plateau during long Ca²⁺ sparks. Because of the presence of slow Ca²⁺ buffers in the Ca²⁺ spark model, the peak F/F₀ that is achieved during the ~20 ms of release in the normal Ca²⁺ spark is considerably less than the steady-state level that would be achieved if this flux were maintained indefinitely. Thus, a small maintained flux can produce a relatively larger F/F₀. Simulated sparks in which the maintained flux was 50% of the peak produced Ca²⁺ sparks with plateau levels close to the peak levels, inconsistent with the experimental data (results not shown). Taken together, these observations strongly suggest cardiac Ca²⁺ sparks result from Ca²⁺ release fluxes that decay with time. However, because flux depends on both the RyR open probability and the Ca²⁺ gradient, a decay in the release flux can theoretically result from severe local depletion, as we have modeled here, from strong inactivation of RyRs, or from a combination of the two.

View larger version (25K): [in this window] [in a new window]   FIGURE 14   Effects of diffusion-buffering model on Ca2+ spark characteristics. Ca2+ sparks produced by stereotyped Ca2+ release fluxes (top) were computed with both the simple buffering and the complex buffering approximations (see text for descriptions). The middle panels present the Ca2+ spark time course (0.25 µm from the source) and line-scan image for simple buffering, whereas the bottom panels display these for complex buffering.

Other spark models

Earlier models of the Ca²⁺ release process (Stern, 1992; Stern et al., 1999; Rice et al., 1999) or Ca²⁺ spark characteristics (Pratusevich and Balke, 1996; Smith et al., 1998; Izu et al., 2001) have provided useful interpretations of experimental data and predictions that have motivated important new experiments. Our results complement these studies by examining issues that earlier models did not thoroughly address. Specifically, our study extends previous work in the following ways: 1) the model provides a hypothesis for the termination of Ca²⁺ release that is experimentally justified; 2) simple experimental interventions have been modeled and are consistent with published data; 3) Ca²⁺ spark properties are maintained roughly the same when the RyR number is varied; and 4) the Ca²⁺ sparks produced by our release fluxes are rounded at their peak rather than coming to a sharp peak (Smith et al., 1998) or having a flat-top shape (Izu et al., 2001). However, one weakness of the previous studies, which is shared by models of skeletal muscle Ca²⁺ sparks (Jiang et al., 1999), has not been overcome in the present model: Ca²⁺ spark width (full-width at half-maximum (FWHM) is about half of that observed experimentally. This is due in part to the way that fluo-3 diffusion was modeled with the simple buffering scheme, because sparks simulat