An Investigative Mathematical Model of Calcium-Induced Calcium Release

Biophysical Journal Volume 83 July 2002 59–78 59

Termination of Cardiac Ca2؉ Sparks: An Investigative Mathematical Model of Calcium-Induced Calcium Release

Eric A. Sobie,* Keith W. Dilly,* Jader dos Santos Cruz,* W. Jonathan Lederer,* and M. Saleet Jafri† *Medical Biotechnology Center, University of Maryland Biotechnology Center, Baltimore, Maryland 21201, and †Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, Texas 75083 USA

ABSTRACT A Ca2ϩ 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 Ca2ϩ influx across the sarcolemmal or transverse tubule membrane neighboring the cluster, the Ca2ϩ spark has been shown to be the elementary Ca2ϩ signaling event of excitation–contraction coupling in heart muscle. However, the question of how the Ca2ϩ spark terminates remains a central, unresolved issue. Here we present a new model, “sticky cluster,” of SR Ca2ϩ release that simulates Ca2ϩ spark behavior and enables robust Ca2ϩ 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 [Ca2ϩ]. Using a Monte Carlo method, local Ca2ϩ-induced Ca2ϩ release from clusters containing between 10 and 100 RyRs is modeled. After release is triggered, Ca2ϩ flux from RyRs diffuses into the cytosol and binds to intracellular buffers and the fluorescent Ca2ϩ indicator fluo-3 to produce the model Ca2ϩ spark. Ca2ϩ sparks generated by the sticky cluster model resemble those observed experimentally, and Ca2ϩ spark duration and amplitude are largely insensitive to the number of RyRs in a cluster. As expected from heart cell investigation, the spontaneous Ca2ϩ spark rate in the model increases with elevated cytosolic or SR lumenal [Ca2ϩ]. Furthermore, reduction of RyR coupling leads to prolonged model Ca2ϩ sparks just as treatment with FK506 lengthens Ca2ϩ sparks in heart cells. This new model of Ca2ϩ spark behavior provides a “proof of principle” test of a new hypothesis for Ca2ϩ spark termination and reproduces critical features of Ca2ϩ sparks observed experimentally.

INTRODUCTION A critical step in cardiac excitation–contraction coupling is is controlled locally (Niggli and Lederer, 1990) helps to the activation of Ca2ϩ sparks (Cheng et al., 1993; Cannell et overcome potential instabilities (Stern, 1992). However, the al., 1994) by voltage-gated Ca2ϩ influx through sarcolem- picture is currently incomplete because of the uncertainty mal L-type Ca2ϩ channels (Cannell et al., 1994, 1995) surrounding the mechanisms that account for the termina- located in the sarcolemmal (SL) and transverse tubule (TT) tion of Ca2ϩ sparks. membranes (Cheng et al., 1994, 1996a; Shacklock et al., To date, three distinctive explanations for Ca2ϩ spark 1995) (see Fig. 1). When an L-type channel opens, intra- termination have been put forward, but each hypothesis has 2ϩ cellular calcium ([Ca ]i) is elevated close to the channel, in specific weaknesses. the region between the sarcolemmal and sarcoplasmic re- 2ϩ ticulum (SR) membranes (also called the “fuzzy space” or 1. The SR could simply run out of releasable Ca . Such the “subspace” (Lederer et al., 1990; Soeller and Cannell, SR exhaustion models, however, are contradicted at the 2ϩ 1997; Jafri et al., 1998)). This local elevation of [Ca2ϩ] global level by evidence of substantial SR Ca reserves i 2ϩ 2ϩ triggers SR Ca2ϩ-release channels (i.e., ryanodine receptors (nonzero SR Ca content) following Ca release or RyRs), located in the same subspace, to open and release (Varro et al., 1993; Bassani et al., 1995; Negretti et al., a greater amount of Ca2ϩ from the SR. This Ca2ϩ released 1995), and, at the local level, by the existence of pro- 2ϩ from the SR via RyRs underlies the large, local increase in longed (lasting seconds) Ca sparks (Cheng et al., [Ca2ϩ] that is visualized as a Ca2ϩ spark. The process that 1993). i 2ϩ 2ϩ activates Ca2ϩ sparks therefore displays both high gain (a 2. The SR Ca release process could undergo Ca -de- small amount of “trigger Ca2ϩ” activates a large amount of pendent or use-dependent inactivation or adaptation (Fa- released Ca2ϩ) and positive feedback (release of Ca2ϩ biato, 1985; Lukyanenko et al., 1998; Sham et al., 1998). 2ϩ However, examination of gating of RyRs shows that raises [Ca ]i and tends to trigger more release). These characteristics can, in principle, defeat controlled Ca2ϩ re- these processes occur too slowly (Gyo¨rke and Fill, 1993; lease from the SR (Stern, 1992). The fact that Ca2ϩ release Valdivia et al., 1995; Fill et al., 2000b) or inadequately (Na¨bauer and Morad, 1990) to account for termination of the Ca2ϩ transient or the Ca2ϩ spark. Submitted August 6, 2001 and accepted for publication March 11, 2002. 3. All the RyRs in a cluster could by chance close at the Drs. Sobie and Jafri are equal contributors to this paper. same time (termed “stochastic attrition”) (Stern, 1992). Address reprint requests to W. J. Lederer, Medical Biotechnology Center, This mechanism is always present to some extent (Stern UMBI, 725 W. Lombard Street, Baltimore, MD 21201. Tel: 410-706-4895; et al., 1999) and can work in concert with other mech- Fax: 253-660-4449; E-mail: [email protected]. anisms, but stochastic attrition fails to provide a robust © 2002 by the Biophysical Society control mechanism when more than a few RyRs are 0006-3495/02/07/59/20 $2.00 present in a cluster (Stern, 1992) (also see below). The 60 Sobie et al.

FIGURE 1 Structural considerations and model schematic. (A) View of SR–TT junction showing cut section of TT membrane (transparent yellow) containing L-type Ca2ϩ 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. Ca2ϩ 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 Ca2ϩ, 2ϩ 2ϩ diffusion of Ca from the subspace to the cytoplasm (Jefflux) and refilling of the junction from neighboring SR (Jrefill) determine Ca concentrations in 2ϩ 2ϩ the subspace ([Ca ]SS) and the local JSR lumen ([Ca ]lumen). (D) RyR gating. Each RyR homotetramer is modeled with only a single open and a single 2ϩ 2ϩ closed state, without any adaptation or inactivation processes. Opening and closing rates depend on [Ca ]SS, [Ca ]lumen, and coupled gating of RyRs, as described in Methods.

absence of a viable model of Ca2ϩ spark termination is 1999). The number of RyRs in a cluster depends on species, one of the biggest defects in our current concept of but averages ϳ100. Second, the RyRs are organized as excitation–contraction coupling, as recent reviews have homotetrameric units, each touching four neighbors with a stressed (Niggli, 1999; Wier and Balke, 1999; Cannell small protein, FK-binding protein (FKBP) at or near the and Soeller, 1999). point of contact (See Fig. 1, A and B) (Wagenknecht et al., Three new results regarding the anatomy of the junctions 1997; Samso and Wagenknecht, 1998; Sharma et al., 1998). between the SR and the TT in heart have considerable Third, RyRs incorporated into planar lipid bilayers can bearing on our understanding of Ca2ϩ sparks. First, the exhibit coupled gating, whereby two or more channels dis- SR–TT junction contains nearly crystalline arrays of RyRs play synchronized openings and closings (Marx et al., 1998, organized in clusters (Franzini-Armstrong et al., 1998, 2001). These findings suggested to us that the clustering of

Biophysical Journal 83(1) 59–78 Mechanism of Cardiac Ca2ϩ Spark Termination 61 the RyRs and the physical contact between homotetrameric TABLE 1 Geometry parameters RyRs may be functionally important. Parameter Definition Value

An additional recent result is also important to our un- Ϫ13 VSS Subspace volume 1.0 ϫ 10 ␮L derstanding of the regulation of cardiac calcium-induced Ϫ11 VJSR JSR volume 1.0 ϫ 10 ␮L Ϫ7 calcium release (CICR). As suggested by experiments in ␶efflux SS efflux time constant 7.0 ϫ 10 s heart cells (Cheng et al., 1996b; Lukyanenko et al., 1996), ␶refill JSR refilling time constant 0.01 s 2ϩ 2ϩ studies in planar lipid bilayers (Thedford et al., 1994; [Ca ]myo Bulk myoplasmic Ca 0.1 ␮M Gyorke and Gyorke, 1998; Ching et al., 2000) and cardiac concentration [Ca2ϩ] NSR Ca2ϩ concentration 1.0 mM vesicles (Ikemoto et al., 1989) have confirmed that Ca2ϩ in NSR the SR lumen can influence RyR gating such that RyRs are more likely to be triggered by cytosolic Ca2ϩ when SR 2ϩ lumenal Ca is elevated. RyRs in a region of junctional sarcoplasmic reticulum (JSR). The channels In the present study, we incorporate these new results into communicate through changes in [Ca2ϩ] in a restricted subsarcolemmal 2ϩ a mathematical model of cardiac Ca sparks and put for- space (subspace, SS). The following Ca2ϩ fluxes determine Ca2ϩ concen- 2ϩ ward a new hypothesis, sticky cluster, to describe the be- trations in the subspace and the local, JSR lumen ([Ca ]lumen): fluxes 2ϩ through the L-type channel (J 2ϩ havior of Ca sparks in heart. This hypothesis addresses DHPR) and the RyRs (Jrelease), binding of Ca to buffers in the subspace (J two fundamental holes in our understanding of Ca2ϩ sparks, buf), efflux from the subspace to the bulk 2ϩ myoplasm via diffusion (Jefflux), and refilling of the SR lumen by diffusion namely, how termination of Ca sparks occurs, and what 2ϩ from neighboring network SR (Jrefill). Thus, [Ca ]SS is described by the influence the large but variable number of RyRs in a cluster balance equation, may have upon Ca2ϩ spark behavior. In the sticky cluster 2ϩ 2ϩ model, Ca spark termination results from two factors: the d͓Ca ͔SS ϭ J ϩ J ϩ J ϩ J .(1) influence of lumenal [Ca2ϩ] on RyR gating, and coupled dt release DHPR efflux buf gating between the RyRs in a cluster. Furthermore, this model produces Ca2ϩ sparks that terminate robustly in a A description of how the individual fluxes are calculated follows. Fluxes through the Ca2ϩ channels (DHPR and RyRs) depend on chan- manner that is largely independent of the number of ϩ nel permeability, the concentration gradient for Ca2 , and whether the RyRs in a cluster. Additionally, the model simulates the channels are open. Thus, JDHPR ϭ 0 when the channel is closed, and occurrence of spontaneous sparks at a realistic rate, and is consistent broadly with experimental results, including I៮DHPR those that reveal that Ca2ϩ sparks can be prolonged under JDHPR ϭϪ (2) 2FVSS certain conditions.

when the channel is open, where I៮DHPR is the single-channel L-type current, F is Faraday’s constant, and VSS is the subspace volume (values METHODS given in Table 1). Similarly, flux through a single, open RyR is calculated as Cell isolation and confocal imaging 2ϩ 2ϩ JRyR ϭ DRyR͑[Ca ͔lumen Ϫ ͓Ca ͔SS)(3) Adult mice were killed by intraperitoneal injection of pentobarbital (100 mg/kg), and single ventricular myocytes were isolated (Cannell et al., 2ϩ where DRyR reflects the ease with which Ca passes through an open 1995) and stored at room temperature (22–25°C) in Dulbecco’s modified 2ϩ RyR. The total Ca release flux, Jrelease, simply reflects the flux through Eagle’s medium (JRH Biosciences, Lanexa, KS) until used. Cells were all the open RyRs in the cluster, i.e., loaded with fluo-3 by exposure to 5.5 ␮M fluo-3AM for 30 min at room temperature. Cells were superfused with NaCl 140 mM, KCl 5 mM, MgCl2 N 0.5 mM, CaCl 1.8 mM, NaH PO 0.33 mM, Glucose 5.5 mM, HEPES 5 i 2 2 4 Jrelease ϭ RyRopenJRyR (4) mM. The pH of all extracellular solutions was kept at 7.4 with the i͸ϭ1 temperature at 34–37°C. Confocal microscopy was used to measure Ca2ϩ i sparks as described previously (Cheng et al., 1993; Cannell et al., 1994, where RyRopen ϭ 0 if channel i is closed, 1 if it is open, and N represents 1995). the number of channels in the cluster. We set N ϭ 50 in control conditions. However, given the variability in cluster size observed experimentally, it is important to verify that Ca2ϩ sparks produced by clusters of various sizes can terminate robustly. Model Layout Ca2ϩ in the subspace can be bound to SL and SR membrane buffers and The model is separated into two independent components: 1) Mathematical calmodulin (CaM), with buffer characteristics based on those reported description of SR Ca2ϩ release through an RyR cluster at the SR–TT previously (Smith et al., 1998). Thus, junction. This component of the model contains the major novel features of i 2ϩ i 2ϩ ϭ Ϫ Ϫ our presentation. 2) Mathematical description of the Ca spark produced Jbuf ͸ kon͓Ca ͔SS͓Bi͔ koff͓͑Bi͔tot ͓Bi͔͒,(5) by the Ca2ϩ efflux. This component of the model refines earlier descrip- iϭCaM,SR,SL tions of the diffusion, buffering, and imaging processes that produce the 2ϩ i 2ϩ i 2ϩ Ca spark (Smith et al., 1998). Figure 1 C, which illustrates the model where kon is the Ca on rate, koff the Ca off rate, [Bi] is the unbound 2ϩ schematically, shows that it consists of a single L-type Ca channel (also buffer concentration, and [Bi]tot is the total concentration (bound ϩ un- called a dihydropyridine receptor, DHPR) closely apposed to a cluster of bound) of buffer i.

Biophysical Journal 83(1) 59–78 62 Sobie et al.

Efflux of Ca2ϩ from the subspace via diffusion to the myoplasm is TABLE 3 Buffering parameters calculated as Parameter Definition Value [Ca2ϩ] Ϫ ͓Ca2ϩ͔ myo SS [BCSQ]tot Total calsequestrin concentration 10.0 mM Jefflux ϭ ,(6)K Calsequestrin Ca2ϩ dissociation 0.8 mM ␶efflux CSQ constant 2ϩ with [Ca ]myo held fixed at 0.1 ␮M, and ␶efflux is the time constant for [BCaM]tot Total calmodulin concentration 24.0 ␮M 2ϩ CaM 2ϩ Ϫ1 Ϫ1 Ca transfer between the subspace and the bulk myoplasm. The different kon Calmodulin Ca on-rate constant 100.0 ␮M s CaM 2ϩ Ϫ1 time constants for transfer to the bulk cytoplasm (␶efflux) and within the SR koff Calmodulin Ca off-rate constant 38.0 s (␶refill) reflect the assumption that these two diffusion processes will [BSR]tot Total SR membrane buffer 47.0 ␮M typically occur over different characteristic distances. These time constants concentration SR 2ϩ Ϫ1 Ϫ1 are roughly equivalent to diffusion over a spatial scale of, respectively, kon SR membrane buffer Ca on-rate 115.0 ␮M s 8–13 nm and 0.5–1.58 ␮m (depending on whether the true or an effective constant 2ϩ 2ϩ CaM 2ϩ Ϫ1 diffusion constant for Ca is assumed). This flux of Ca from the koff SR membrane buffer Ca 100.0 s subspace to the myoplasm is used as the input for the second model off-rate constant 2ϩ (described below) that simulates the production of Ca sparks. [BSL]tot Total SL membrane buffer 1124.0 ␮M 2ϩ 2ϩ The balance equation for JSR lumenal Ca concentration ([Ca ]lumen) concentration SL 2ϩ Ϫ1 Ϫ1 is kon SL membrane buffer Ca on-rate 115.0 ␮M s constant 2ϩ d[Ca ]lumen VSS kSL SL membrane buffer Ca2ϩ off-rate 1000.0 sϪ1 ϭ ␤ Ϫ J ϩ J ,(7) off dt JSRͩ RyR V refillͪ constant JSR where the JSR is depleted by RyR Ca2ϩ release scaled by the ratio of subspace volume (VSS) to JSR volume (VJSR) and refilled from the network sarcoplasmic reticulum (NSR) by Jrefill, adaptation or other inactivation processes over the time scale of the Ca2ϩ 2ϩ 2ϩ ͓Ca ͔NSR Ϫ ͓Ca ͔lumen spark (Fig. 1 D). We exclude these features for the sake of simplicity and Jrefill ϭ ,(8)to demonstrate that they are not necessary for spark termination. We do not ␶refill mean to argue against a role for these phenomena in regulating cardiac 2ϩ 2ϩ Ca signaling. Thus, in this model, any intervention that alters the gating with [Ca ]NSR held fixed at 1.0 mM and ␶refill is the time constant for Ca2ϩ transfer between the JSR and NSR. Buffering in the JSR by calse- behavior of the RyR cluster must do so by modifying either the opening rate, kopen, or the closing rate, kclose. questrin uses the rapid buffering approximation (Keizer and Levine, 1996) 2ϩ given by The RyR closing rate was independent of [Ca ]SS, Ϫ1 Ϫ1 kclose ϭ CFclose ϫ 480.0 s ,(10) ͓CSQ]totKCSQ ␤JSR ϭ 1 ϩ 2 ,(9) 2ϩ ͫ ͑KCSQ ϩ ͓CSQ͔͒ ͬ whereas the opening rate was a fourth-order function of [Ca ]SS,

4 where [CSQ]tot, [CSQ], and KCSQ represent the total concentration, un- [Ca]ss 2ϩ 4 Ϫ1 bound concentration, and Ca -dissociation constant, respectively, of kopen ϭ 3.0 ϫ 10 ⅐ CFopen 4 4 s .(11) calsequestrin. ͓Ca]ss ϩ Km

2ϩ Lumenal Ca influences RyR gating through changes in Km in the above 2ϩ equation. Sensitivity to [Ca ]SS is a linear, decreasing function of Ryanodine receptor gating 2ϩ 2ϩ [Ca ]lumen, so that RyR opening is favored when [Ca ]lumen is high, as The primary goal of the present study is to gain insight into Ca2ϩ spark suggested by the literature (Thedford et al., 1994; Cheng et al., 1996b; behavior and the mechanisms of spark termination by investigating how Gyorke and Gyorke, 1998; Lukyanenko et al., 1998; Ching et al., 2000): 2ϩ the features of RyR gating in the sticky cluster influence Ca spark 2ϩ characteristics. Because, in cardiac muscle, Ca2ϩ spark termination occurs Km ϭ 6.0 Ϫ 0.0024͓Ca ͔lumen ␮M. (12) independently of spark triggering (Cheng et al., 1993; Cannell et al., 1995), Coupled gating of RyRs is introduced by multiplying the opening and we did not simulate the gating of the DHPR. In most simulations, we closing rate constants by cooperativity factors (CF for opening and initiated a Ca2ϩ spark with a stereotypical DHPR opening (0.5 pA, 0.5 ms) open CF for closing) that depend, respectively, on the relative numbers of that could trigger a spark with Ͼ98% fidelity. The details of the gating of closed open and closed channels in the cluster, an RyR in a cluster follow. The 50 RyRs are assumed to gate independently except for a “coupling” Nopen ϩ 1 or “cooperativity” factor (CF and CF as defined below). Each RyR close open CFopen ϭ 1 ϩ ,(13) is modeled with only a single closed and a single open state and no Nopen ϩ Nclosed

Nclosed ϩ 1 CFclose ϭ kcoop 1 ϩ ,(14) TABLE 2 Channel parameters ͫ Nopen ϩ Nclosedͬ

Parameter Definition Value where Nopen is the number of open RyRs, and Nclosed is the number of closed RyRs. CF and CF are both cooperative in their behavior F Faraday’s constant 96480 C ⅐ molϪ1 open close because the value of either depends on the fraction of open channels. The I៮ DHPR single-channel current Ϫ0.5 pA DHPR scaling factor k (equal to unity in control conditions) was introduced so D Ca2ϩ diffusion through open 4000.0 sϪ1 coop RyR that modification of a single parameter could simulate changes in the RyR parameter strength of coupling between RyRs.

Biophysical Journal 83(1) 59–78 Mechanism of Cardiac Ca2ϩ Spark Termination 63

A Monte Carlo method (Rice et al., 1999) was used to simulate the openings and closings of the RyRs with the Ca2ϩ balance equation solved at each time step using Euler’s method. A time step of 10Ϫ8 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 Ca2ϩ from the SS into the myoplasm. To generate Ca2ϩ signals with similar spatiotem- poral 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 Ca2ϩ diffusion in the myoplasm (spherical symmetry is assumed), Ca2ϩ 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 Ca2ϩ-dissociation constant of 0.5 ␮M, rather than 1.13 ␮M. These changes were made so that the time course of the model Ca2ϩ 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 Ca2ϩ dynamics in space and time, such that diffusion of Ca2ϩ and fluo-3 was treated implicitly with a Crank–Nicholson algorithm, and Ca2ϩ buffering was treated explicitly. The model was implemented in Matlab (The Mathworks, Natick, MA), and a time step of 2 ␮s was used. Results were visualized with IDL and Origin software, and CorelDraw was used to produce figures. FIGURE 2 Simulated RyR gating and Ca2ϩ fluxes during a typical Ca2ϩ

spark. (A) Composite open probability (Po) of the 50 RyRs in the cluster. (B) Efflux from the subspace versus time. (C) Lumenal Ca2ϩ concentration 2ϩ 2ϩ RESULTS ([Ca ]lumen) versus time. The decrease in [Ca ]lumen leads to increased 2ϩ flickering of RyRs and a slight decrease in Po until the influence of coupled To explain the behavior of Ca sparks in intact cells, given gating causes all RyRs in the cluster to close within 3 ms and not reopen. the current knowledge of the anatomy, biochemistry, and biophysics of Ca2ϩ signaling in heart, two questions must be addressed: how do Ca2ϩ sparks terminate, and why are posite Po of the RyRs remains close to 1 for ϳ10 ms but Ca2ϩ sparks so similar in duration? 2ϩ 2ϩ declines as [Ca ]lumen (Fig. 2 C) and [Ca ]SS fall and exhibits larger fractional fluctuations until all RyRs close nearly simultaneously (P quickly declines to zero) after Basic simulation o ϳ25 ms. The firm closure of the RyRs arises from the Figure 2 displays an example of simulated Ca2ϩ release gating cooperativity of the RyRs in the cluster. Re-opening under control conditions, with 50 RyRs in the sticky cluster is prevented by hysteresis in the relationship between Po and 2ϩ 2ϩ and a cooperativity factor (kcoop) of 1. In this simulation, as [Ca ]SS that is due to the influence of [Ca ]lumen on this in most of the simulations carried out in this study, a relationship. The Ca2ϩ efflux (Fig. 2 B) during the release stereotypical DHPR opening (0.5 ms, Ϫ0.5 pA) triggers process reaches an early peak, declines as the local SR CICR in the RyR cluster. Our Monte-Carlo method simu- lumen becomes depleted of Ca2ϩ, then decreases to zero lates stochastic openings and closings of RyRs in the cluster when the RyRs in the cluster close. (as seen by the noise or chatter in the records shown), and The efflux of Ca2ϩ through the cluster of RyRs, when the temporal evolution of each Ca2ϩ release event is slightly used as an input to the buffering and diffusion model 2ϩ different. The composite open probability (Po) of the 50 described above, produces a Ca spark as shown in Fig. 3. RyRs, the resulting Ca2ϩ efflux from the subspace to the Figure 3 A displays a simulated line-scan image of the myoplasm, and the local JSR lumenal Ca2ϩ concentration resulting Ca2ϩ spark, similar to that which would be re- 2ϩ ([Ca ]lumen) are displayed in Fig. 2, A, B, and C, respec- corded experimentally (assuming the point of release is tively. Ca2ϩ entering the subspace (SS) through the DHPR located directly on the scan line). The relative level of 2ϩ activates release from the cluster by binding to the RyRs [Ca ]i (measured as F/F0) is shown in time-profiles of the and increasing their Po to a value close to 1. The opening of line-scan image in Fig. 3 B, with the color-coded tick marks 2ϩ 2ϩ RyRs increases [Ca ]SS further, thereby contributing ad- on the right-hand edge of Fig. 3 A indicating that [Ca ]i ditional Ca2ϩ to the positive feedback of CICR. The com- time courses are displayed at the point of release and at

Biophysical Journal 83(1) 59–78 64 Sobie et al.

FIGURE 3 Simulated Ca2ϩ spark under control conditions. (A) Line scan image of simulated Ca2ϩ spark under control conditions (50 RyRs in cluster, 2ϩ kcoop ϭ 1). (B) Time courses of Ca sparks (plotted as F/F0) at locations 0, 0.25, and 0.5 ␮m from the source are displayed in black, red, and green, respectively. Locations on line scan are noted to right of the image. (C) Ca2ϩ 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.

2ϩ distances 0.25 and 0.5 ␮m from the point of release. The [Ca ]i signal resulting from the opening of a single RyR. spatial profiles that would be recorded during the Ca2ϩ This is due to the small level of Ca2ϩ release and its short spark are shown in Fig. 3 C. The tick marks on the bottom duration. This result, however, provides an explanation for of Fig. 3 A show the times of the profiles, corresponding to the absence or near invisibility of Ca2ϩ sparks in cell 2, 5, 10, 20, and 50 ms after Ca2ϩ spark initiation. systems with few or extremely small RyR clusters (Bhat et al., 1997; Haak et al., 2001). Background noise Cluster size and RyR flux Experimentally measured line-scan images of Ca2ϩ sparks exhibit fluctuations in fluorescence intensity due in part to In Figs. 2–4, 50 RyRs were arranged in a cluster to simulate the noise of the system. This noise comes from the excita- a control Ca2ϩ spark; however, because it is likely that the tion laser, the photodetector, amplifiers, and the fluorescent number of RyRs in a cluster may vary widely (Franzini- light itself. We characterized the noise recorded in line-scan Armstrong et al., 1998, 1999), we simulated Ca2ϩ sparks images of Ca2ϩ sparks and then simulated this noise as produced by clusters of different sizes. Simulations from normally distributed fluctuations about a mean level. Figure clusters containing 100, 50, 20, and 10 RyRs are shown in 4 A (top) shows the line-scan image of a simulated Ca2ϩ Fig. 5, A, B, C, and D, respectively. Each panel displays the 2ϩ 2ϩ spark with this realistic noise added. The increase in back- SR release flux (top), [Ca ]lumen (middle), and the Ca ground reflects different scaling of the image after fluctua- spark time course, measured at 0.25 ␮m from the source tions around the background are added. The middle and (bottom). Three features of these simulations are of partic- bottom traces, respectively, display the time course of the ular interest. First, SR release terminates robustly at about noisy Ca2ϩ spark at the point of release and averaged over the same time after it is initiated, leading to Ca2ϩ sparks the width of the Ca2ϩ spark. (Ϯ0.5 ␮m centered on the with similar durations regardless of the number of RyRs in point of Ca2ϩ release) Figure 4 B shows identical records the cluster. Second, termination of Ca2ϩ release occurs at 2ϩ produced by the model with noise added when the cluster different levels of [Ca ]lumen as cluster size is varied. This comprises only a single RyR. One of the clear effects of the helps to explain the small variation in the duration of addition of a realistic amount of noise is the loss of the release. With fewer RyRs in the cluster, it is more likely that apparent shoulder produced as Ca2ϩ release is terminated. A the stochastic closing of a few RyRs will induce the remain- second observation is that one cannot visually detect the ing channels to close. However, the decreased peak Ca2ϩ

Biophysical Journal 83(1) 59–78 Mechanism of Cardiac Ca2ϩ Spark Termination 65

FIGURE 4 Appearance of Ca2ϩ sparks with realistic noise. (A) Line-scan image of simulated Ca2ϩ spark that would result from the opening of 50 RyRs 2ϩ 2ϩ with realistic noise added (top), time course of [Ca ]i as F/F0 (0.25 ␮m from source, middle) and time course of [Ca ]i (averaged over central 1 ␮m 2ϩ 2ϩ of Ca spark, bottom). The addition of realistic noise leads to the loss of the inflection in the time course of [Ca ]i when release is terminated (see Fig. 3 B). See text for details. (B) Line-scan image of simulated Ca2ϩ spark with realistic noise that would result from the opening of a single RyR (top), time 2ϩ 2ϩ 2ϩ course of [Ca ]i as F/F0 (0.25 ␮m from source, middle) and time course of [Ca ]i (averaged over central 1 ␮m of point of Ca release, bottom). The fluorescent Ca2ϩ signal is not readily distinguishable from the noise.

2ϩ efflux leads to slower depletion of [Ca ]lumen. Because Po spark duration observed. Third, although the peak of the 2ϩ 2ϩ depends on [Ca ]lumen, the slower depletion tends to pro- Ca release flux approximately scales with the number of 2ϩ long the duration of release. These two effects essentially RyRs in a cluster (top traces), the peak F/F0 of a Ca spark cancel each other, leading to the relatively constant Ca2ϩ does not (bottom traces). Two explanations appear to ac-

2ϩ 2ϩ FIGURE 5 Simulated Ca sparks resulting from RyR clusters of various sizes. (A) 100 RyRs in cluster. Efflux from the subspace (top), [Ca ]lumen 2ϩ (middle), and [Ca ]i as F/F0 (measured 0.25 ␮m 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).

Biophysical Journal 83(1) 59–78 66 Sobie et al.

FIGURE 6 Effects of RyR number on Ca2ϩ spark amplitude and duration: population data. (A) Histograms of the duration of Ca2ϩ release for 100, 50, 20, and 10 RyRs in cluster. 500 Ca2ϩ sparks were simulated to generate each histogram. (B) Peak amplitude of Ca2ϩ sparks (averaged) versus RyR cluster size. An increase in cluster sizes causes a less-than-proportional increase in Ca2ϩ spark amplitude. (C) Mean Ca2ϩ spark duration and standard deviation plotted as a function of the number of RyRs in a cluster. Ca2ϩ spark duration reveals a biphasic dependence on number of RyRs in a cluster. For small clusters, Ca2ϩ 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 Ca2ϩ spark duration. count for this difference. With more RyRs in a cluster, the feel that it may represent a realistic value for the single RyR total amount of Ca2ϩ released during a spark does not scale current that occurs in cells (see Discussion). However, to with the peak level of Ca2ϩ efflux due to the more rapid ensure that these specific parameter choices did not signif- lumenal depletion, and the Ca2ϩ spark amplitude reflects icantly affect the overall model behavior, we ran additional 2ϩ the amount of Ca bound to fluo-3, which is nonlinearly simulations in which iRyR was increased by a factor of 5, to related to the total amount of Ca2ϩ released (Izu et al., 2001). 0.35 pA. Figure 7 compares the Ca2ϩ sparks observed with Figure 6 displays the composite results from 500 simu- the control value of current (A) and with this increased value lated Ca2ϩ sparks at each cluster size. The histograms (B). With greater single-channel RyR Ca2ϩ flux, the peak displayed in Fig. 6 A demonstrate that, although the mean Ca2ϩ release flux is larger, but SR depletion occurs more SR release duration remains relatively constant, as noted quickly, so the Ca2ϩ spark is not five times larger (similar above, variability increases as the cluster size is decreased to the changes that occur with increased RyR number). from 100 to 10 RyRs. Figure 6 B shows the less than Interestingly, the SR release duration is almost identical 2ϩ proportional increase in Ca spark amplitude, and Fig. 6 C even with this increased iRyR. Thus, our model is robust displays the biphasic changes in SR release duration pro- enough that similar behavior occurs even with substantial duced by increases in cluster size. Because of stochastic changes in single-channel RyR current. attrition, SR release is quite short for very small clusters, but SR release duration increases rapidly as stochastic attrition [Effects of changes in [Ca2؉] and [Ca2؉ ceases to be a significant factor (e.g., with ϳ10 RyRs). With SS lumen an increase in cluster size beyond ϳ20 RyRs, spark duration As a test of the model, we examined how changes in 2ϩ 2ϩ 2ϩ decreases slightly due to the faster SR depletion noted above. [Ca ]SS and [Ca ]lumen affect simulated Ca sparks. With the parameters we have chosen for this model, the Because Ca2ϩ sparks are triggered by the large increase in 2ϩ peak current through a single RyR (i.e., before local SR [Ca ]SS resulting from a DHPR opening, smaller steady- 2ϩ 2ϩ depletion occurs) is 0.07 pA. Although this is considerably state increases in global [Ca ]i will increase [Ca ]SS. In 2ϩ smaller than any iRyR that has been measured in bilayers, we the absence of Ca spark triggering by the opening of

Biophysical Journal 83(1) 59–78 Mechanism of Cardiac Ca2ϩ Spark Termination 67

FIGURE 7 Effect of increasing single-channel RyR current. (A) Efflux from the subspace (top), F/F0 0.25 ␮m from the source (middle), and simulated Ca2ϩ spark line-scan image (bottom) under control conditions, assuming a single-channel RyR current of 0.07 pA. (B) Simulated Ca2ϩ spark assuming an increased single-channel RyR current of 0.35 pA. Although the initial efflux is considerably larger, the Ca2ϩ spark amplitude is only larger by a factor of ϳ1.5. Ca2ϩ spark duration is almost identical.

2ϩ 2ϩ DHPRs, this increase in [Ca ]SS will slightly increase the duration. The SR Ca fluxes that produce the sparks shown RyR opening rate and therefore should result in more fre- in Fig. 9 A are plotted on a normalized scale in Fig. 9 B. quent background or spontaneous sparks. Experimental Consistent with experiments that back-calculated release studies have suggested (Cheng et al., 1996b) that such an flux from Ca2ϩ spark records (Lukyanenko et al., 1998), we 2ϩ 2ϩ increase in Ca sparks occurs with elevated [Ca ]i (Satoh observe an increase in the release flux decay rate with et al., 1997). We tested this idea by repeatedly running increasing SR load. Fig. 9, C and D, displays how changes 2ϩ simulations with no DHPR opening and recording how in [Ca ]lumen affect the spontaneous spark rate and the often spontaneous Ca2ϩ sparks occurred. Figure 8, A and B, spark duration, respectively. Spark rate is elevated nonlin- 2ϩ which displays simulated line-scan images that would be early with increasing [Ca ]lumen, as experiments have dem- recorded from quiescent cells, shows that spontaneous onstrated (Cheng et al., 1996b; Satoh et al., 1997). This 2ϩ sparks occur more frequently when [Ca ]SS is increased increased occurrence of spontaneous sparks is thought to be from 100 to 150 nM. On average (600 simulations, 800 ms responsible for the Ca2ϩ waves and arrhythmogenic cur- 2ϩ 2ϩ 2ϩ each), this elevation in [Ca ]SS increased Ca spark fre- rents that are seen in Ca overload (E. A. Sobie, W. J. quency by six times. Figure 8 C plots the spontaneous spark Lederer, and M. S. Jafri, work in progress). Ca2ϩ spark 2ϩ rate (defined as number of sparks per cluster per second) over duration is extremely insensitive to [Ca ]lumen with refill- amuchwiderrangeofintracellularCa2ϩ concentrations. The ing as we have modeled it. However, at extremely low SR 2ϩ 2ϩ 2ϩ 2ϩ Ca spark rate increases by approximately four orders of Ca loads, decreasing [Ca ]lumen further causes Ca 2ϩ magnitude as [Ca ]SS increases from 100 nM to ϳ1 ␮M. sparks to terminate prematurely. Figure 9 A displays simulated Ca2ϩ sparks obtained at In the absence of any dependence of RyR gating on 2ϩ 2ϩ 2ϩ three different initial levels of [Ca ]lumen. For these simu- [Ca ]lumen, the model produces long Ca sparks that do lations, we fixed the network SR Ca2ϩ concentration at 1 not terminate as illustrated in Fig. 10. Figure 10 shows how 2ϩ 2ϩ mM to keep the [Ca ]lumen refilling rate relatively con- the composite Po in the cluster changes (A) as the Ca 2ϩ stant. Increasing the SR load increases the spark amplitude efflux (B) and [Ca ]lumen (C) decline. Panels D and E as one would expect but has little effect on Ca2ϩ spark display the relative [Ca2ϩ], 0.25 ␮m from the point of

Biophysical Journal 83(1) 59–78 68 Sobie et al.

FIGURE 8 Effect of cytosolic [Ca2ϩ] on spontaneous sparks. (A) Simulated Ca2ϩ line-scan image showing spontaneous sparks at cytosolic [Ca2ϩ] 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 ␮m of a ventricular myocyte would be able to detect Ca2ϩ sparks from 100 separate clusters of RyRs. (B) Simulated line-scan images showing spontaneous sparks with cytosolic [Ca2ϩ] increased to 150 nM. More spontaneous sparks occur in this example. On average, the spontaneous spark rate was increased by six times upon increasing 2ϩ 2ϩ 2ϩ 2ϩ [Ca ]SS to 150 nM (5 spontaneous sparks in 4.8 s at [Ca ]i ϭ 100 nM; 30 sparks at 150 nM). (C) Plot of normalized Ca spark rate versus [Ca ]i ϭ 2ϩ 2ϩ [Ca ]SS. The spontaneous spark rate (e.g., sparks per cluster per second) increases by four orders of magnitude as [Ca ]i increases from a resting level of 100 nM to 1 ␮M. release and the line-scan image of the Ca2ϩ spark, respec- and close together, would require arbitrary assumptions that 2ϩ tively. The fluctuations in Po during the Ca spark arise would be difficult to verify experimentally (see Discussion), 2ϩ because the decrease in [Ca ]lumen leads to a decrease in we modeled coupled gating of RyRs with the minimal 2ϩ Jrelease and hence a decrease in [Ca ]SS.However,be- number of assumptions. This allowed us to vary the strength 2ϩ cause RyR gating does not depend on [Ca ]lumen,the of coupling between RyRs by modifying a single parameter, 2ϩ fluctuations in Po remain relatively minor, the RyRs are kcoop in Eq. 14. Figure 11 exhibits the changes in Ca continually activated by the small but sustained Jrelease, sparks that occur when kcoop is decreased from its control and the Ca2ϩ spark does not terminate. value of 1.0, reducing the tendency of one RyR to influence 2ϩ the Po of another RyR. Typical Ca sparks simulated with k ϭ 0.5 and k ϭ 0.4 are presented in Fig. 11, A and Stickiness of the RyRs in the cluster at the coop coop B, respectively. With decreased coupling between RyRs, SR–TT junction there are large fluctuations in Po. However, the coupling Because any specific assumption about the nature of RyR between channels is not strong enough for the closed chan- interactions in a cluster, other than their tendency to open nels to induce closure of the remaining channels while

Biophysical Journal 83(1) 59–78 Mechanism of Cardiac Ca2ϩ Spark Termination 69

2ϩ 2ϩ 2ϩ 2ϩ FIGURE 9 Effects of [Ca ]lumen on Ca sparks. (A) Simulated Ca sparks at different beginning levels of [Ca ]lumen. With constant refilling rate, 2ϩ 2ϩ 2ϩ increasing [Ca ]lumen increases Ca spark amplitude but does not change spark duration. (B) SR Ca 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 2ϩ 2ϩ 2ϩ increases in [Ca ]lumen.(D) Ca release flux duration is generally insensitive to [Ca ]lumen except at extremely low SR load.

2ϩ [Ca ]SS remains high. This leads to an increase in the al., 1996; Xiao et al., 1997), we carried out experiments in 2ϩ duration of Jrelease, prolonged Ca sparks, and noticeable mouse heart cells. Consistent with some prior reports (Xiao fluctuations in fluorescence. Figure 12 A compares the du- et al., 1997; Lukyanenko et al., 1998) we observed pro- rations of Ca2ϩ sparks for 500 simulations under control longed Ca2ϩ sparks in the presence of FK506 or rapamycin conditions (kcoop ϭ 1, left) to those observed with decreased as shown in Fig. 13. Figure 13 D shows that about half of 2ϩ 2ϩ coupling (kcoop ϭ 0.4, right). The duration of Ca sparks the Ca sparks obtained in FK506 lasted longer than 40 2ϩ (mean and standard deviation) is plotted against kcoop in Fig. ms, whereas only ϳ10% of control Ca sparks did. These 12 B over the range 0.4–1.5. The spark duration begins to findings are consistent with the hypothesis that RyR inter- rise rapidly, and variability increases, as kcoop declines to actions, including those involving RyR-associated proteins values less than ϳ0.7. When coupling between RyRs is like FKBP12.6, contribute to Ca2ϩ spark termination. Other 2ϩ removed completely (kcoop ϭ 0), Ca sparks do not termi- actions of FK506 and related agents are discussed below. nate (data not shown), revealing Ca2ϩ sparks that look roughly like the one shown in Fig. 10. Due to the main- 2ϩ tained dependence on [Ca ]lumen, there are greater fluctu- DISCUSSION ations in P . However, in the absence of coupled gating, the o Overview activation of RyRs in the cluster is sufficient to produce a maintained Ca2ϩ spark. We have developed a relatively simple mathematical model of the cardiac Ca2ϩ spark to test a new hypothesis that addresses one of the remaining major unresolved questions in cardiac EC FK506 and Rapamycin coupling: how Ca2ϩ sparks are able to terminate. In the model, Because FK506 and rapamycin have been shown to reduce aclusterofuptoahundredRyRsinanSRjunctionis coupled gating between RyRs in planar lipid bilayer exper- responsible for a Ca2ϩ spark, and the RyR Ca2ϩ release iments (Marx et al., 2001), but conflicting effects on Ca2ϩ channels can assume only one of two conformations: open or sparks in intact heart cells have been reported, (McCall et closed. However, each RyR is influenced by the ensemble

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and hysteresis that allow for Ca2ϩ 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 Ca2ϩ sparks and is consistent with simple experimental interventions. Model Ca2ϩ sparks terminate readily, and many features of Ca2ϩ sparks are largely insensitive to the number of RyRs in the cluster (Figs. 5 and 6). In 2ϩ 2ϩ addition, increases in either [Ca ]SS or [Ca ]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 Ca2ϩ 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 Ca2ϩ 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 Ca2ϩ transient reflects the recruitment of individual units of release (Ca2ϩ sparks), which are triggered by local increases in Ca2ϩ and do not themselves trigger regenerative, cell-wide Ca2ϩ release under normal conditions. However, the stability provided by local control would be lost without a reliable mechanism for the termination of Ca2ϩ sparks. Three primary explanations for how Ca2ϩ sparks terminate have been proposed in the literature. First, it is possible that the SR exhausts its supply of Ca2ϩ, and the duration of 2ϩ 2ϩ 2ϩ a Ca spark reflects the time it takes to empty the local SR FIGURE 10 Ca sparks when [Ca ]lumen has no effect on RyR gating. 2ϩ (A) Composite RyR P versus time. For this simulation, K in Eq. 3 was Ca store. This explanation appears unlikely because the o m 2ϩ 2ϩ 2ϩ SR retains much Ca set to 3.6 ␮M, the value at the control diastolic [Ca ]lumen of 1 mM. after a [Ca ]i transient (Varro et al., Release does not terminate before the end of the 800-ms simulation and 1993; Negretti et al., 1995; Bassani et al., 1995) and because 2ϩ 2ϩ was not observed under these conditions. (B) Jrelease.(C) [Ca ]lumen.(D) extremely long (seconds) Ca sparks can be observed [Ca2ϩ ␮ 2ϩ ]i as F/F0 (0.25 m from source). (E) Line-scan image of Ca under certain experimental conditions (Cheng et al., 1993). spark. Note that, during the prolonged spark, F/F0 is maintained at ϳ60% 2ϩ of its peak level, although the release flux declines by more than 90%. This Second, it is possible that the RyRs undergo Ca -depen- occurs because of the presence of cytosolic Ca2ϩ buffers. dent or use-dependent inactivation after they open. How- ever, in planar lipid bilayer experiments, simple steady-state inactivation only occurs at Ca2ϩ concentrations above ϳ10 behavior of the other RyRs in the cluster through a cooperat- mM (Sitsapesan and Williams, 2000) not at the concentra- ivity factor (kcoop)therebyallowingthemodeltosimulate tions that are thought to be present near the RyRs during a coupled gating among RyRs. We term this the sticky cluster Ca2ϩ spark (10–100 ␮M) (Meissner et al., 1988; Rousseau model to emphasize the central importance of physical contact and Meissner, 1989; Ashley and Williams, 1990). RyRs in between neighboring RyRs. The other key feature of the model bilayer experiments display adaptation (a complicated form is that RyR gating is influenced not only by the local cytosolic of inactivation), but this process is incomplete and occurs 2ϩ 2ϩ 2ϩ [Ca ]inthesubspace([Ca ]SS), but also by the Ca relatively slowly (Gyo¨rke and Fill, 1993; Valdivia et al., 2ϩ concentration in the local JSR lumen ([Ca ]lumen). The com- 1995). These observations argue against a primary role for bination of these two factors provides the negative feedback adaptation or inactivation in the termination of Ca2ϩ sparks.

Biophysical Journal 83(1) 59–78 Mechanism of Cardiac Ca2ϩ Spark Termination 71

FIGURE 11 Ca2ϩ sparks under conditions of reduced coupling between RyRs. Time course of Ca2ϩ spark components when coupling between RyRs is 2ϩ 2ϩ reduced. (A)Reducedcoupledgatingwithkcoop ϭ 0.5. Po (top), Jrelease (second from top), [Ca ]lumen (middle), [Ca ]i as F/F0 (0.25 ␮mfromsource,second 2ϩ from bottom), and line-scan image of Ca spark (bottom). (B)Reducedcoupledgatingwithkcoop ϭ 0.4. The release flux lasts for ϳ60 ms (kcoop ϭ 0.5) and ϳ200 2ϩ ms (kcoop ϭ 0.4), leading to prolonged Ca 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 Ca2ϩ spark duration (see Fig. 12).

A third possibility is “stochastic attrition” (Stern, 1992), or adaptation processes to demonstrate that these are not whereby all of the RyRs in a cluster happen to close at the necessary for Ca2ϩ spark termination, but these results do same time. Although this would be a plausible hypothesis if not necessarily argue against an important role for either of very small RyR clusters produced Ca2ϩ sparks, stochastic these mechanisms in the regulation of cardiac CICR. Add- attrition becomes increasingly unlikely as the number of ing either of these phenomena to the model would presum- RyRs in a cluster increases (Stern, 1992; Stern et al., 1999) ably assist the termination of the Ca2ϩ spark by closing and its probability is vanishingly small when more than some of the RyRs in the cluster, thus causing the SR release ϳ20 RyRs are present. flux to decay more steeply and subspace [Ca2ϩ] to be The model presented here overcomes the limitations of reduced. Finally, coupled gating as we have modeled it is other hypotheses while incorporating some of their impor- similar in spirit to the allosteric interactions between RyRs tant features. Our proposed mechanism is similar to stochas- that have been modeled by Stern et al. (1999). However, tic attrition in the sense that some RyRs must close proba- although allosteric interactions could greatly stabilize the bilistically for coupled gating to induce the remaining activation of Ca2ϩ release in that model, their effects on channels to close. It resembles SR exhaustion in that sub- termination of release were less well-explored and only stantial local SR depletion is required for the spark to occurred through the transition to an inactivated state. Ad- terminate. We modeled RyR gating without any inactivation ditional work is clearly necessary to better explore the

Biophysical Journal 83(1) 59–78 72 Sobie et al.

2ϩ 2ϩ FIGURE 12 Ca spark duration when RyR coupling is reduced. (A) Histograms of Ca release duration generated by 500 simulations with kcoop ϭ 1 (control conditions, left) and kcoop ϭ 0.4 (right). Bins are 2.5 ms wide for control and 25 ms wide for kcoop ϭ 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 Ca2ϩ spark duration. multiple effects that interactions between RyRs can have on refractoriness of Ca2ϩ release due to Ca2ϩ-dependent or use- cluster behavior. dependent inactivation, it is also possible that this refractoriness results from the time it takes for partially depleted SR 2ϩ 2؉ release sites to be refilled with Ca .IfRyRsaremorelikely Restitution and SR Ca content 2ϩ to open when [Ca ]lumen is high, as we have modeled here Following a Ca2ϩ release event, time must elapse before CICR and as experiments have indicated, Ca2ϩ sparks will be more can be triggered again, at both the local (Ca2ϩ spark) and difficult to trigger during the refilling time that follows a spark. global (cell-wide Ca2ϩ transient) levels, a feature called “res- This factor makes it difficult to interpret experiments in titution.” For instance, a Ca2ϩ transient evoked by field stim- which the cell contains a high concentration of exoge- ulation after a spontaneous Ca2ϩ wave caused less Ca2ϩ re- nous intracellular Ca2ϩ buffer, because these buffers lease in locations that the wave had just passed than in compete with the SR Ca2ϩ pump and slow reuptake of locations at which more time had elapsed (Cheng et al., Ca2ϩ into the SR (Sham et al., 1998; DelPrincipe et al., 1996b). Consistent with this, Tanaka et al. (1998) found that, 1999). Indeed, studies that have used high Ca2ϩ buffer during a Ca2ϩ transient, sparks were not triggered at locations concentration have observed an unrealistically slow re- where a spontaneous Ca2ϩ spark had occurred within the covery of Ca2ϩ release (DelPrincipe et al., 1999). Clearly previous 25 ms. In addition, Sham et al. (1998) observed a more experiments are necessary to resolve the roles negative correlation between the amounts of Ca2ϩ release played by SR refilling, recovery from inactivation, and triggered by Ca2ϩ current upon depolarization and by Ca2ϩ tail possibly other mechanisms in the restitution process. current upon repolarization. In other words, sites that released Ca2ϩ early during a depolarizing pulse tended to not release ?What is the shape of the Ca2؉ release flux Ca2ϩ when a second Ca2ϩ influx occurred at the end of the pulse, even though the SR still contained Ca2ϩ (Sham et al., Most previous models of the cardiac Ca2ϩ spark (see be- 1998). Although these results can be interpreted to indicate a low), have, for simplicity, assumed that a spark results from

Biophysical Journal 83(1) 59–78 Mechanism of Cardiac Ca2ϩ Spark Termination 73

FIGURE 13 Experimental results: effects of FK506 and rapamycin. (A) A typical Ca2ϩ spark (line-scan image: position shown vertically, time 2ϩ horizontally) under control conditions (top), and plot of F/F0 (bottom). (B) Three examples of Ca sparks following application of 25 ␮M FK506 displayed as in (A). (C) Three examples of Ca2ϩ sparks following application of 20 ␮M rapamycin displayed as in (A). (D) Fraction of total Ca2ϩ sparks Ͻ40 ms and Ն40 ms. ** p Ͻ 0.05. Calibration: position ϭ 10 ␮m; fluorescence ϭ 0.5 F/F0; time ϭ 100 ms.

a constant SR Ca2ϩ release flux. In skeletal muscle, Schnei- local SR depletion could occur over the time scale of the der and colleagues have used curve-fitting of high temporal Ca2ϩ spark. resolution Ca2ϩ spark recordings to argue that the Ca2ϩ The idea of a decaying release flux accounting for the release flux underlying the Ca2ϩ spark is constant (Lacam- cardiac Ca2ϩ spark is consistent with the results of Luky- pagne et al., 1999), but a comparable study on cardiac Ca2ϩ anenko et al. (1998), who back-calculated the responsible sparks has not yet been done. For a variety of reasons, we fluxes from line-scan Ca2ϩ spark recordings. However, the believe that the cardiac Ca2ϩ spark likely results from a accuracy of their calculations could have been compromised decaying SR Ca2ϩ release flux. First, the extremely small by the simple buffering approximation that they used. They volume of the junctional SR suggests that, even with con- assumed, as we have here, that binding of fluo-3 to station- siderable Ca2ϩ buffering power, Ca2ϩ release through a ary cytoplasmic proteins can be modeled by simply reduc- cluster of RyRs could empty this volume rather quickly. For ing the fluo-3 diffusion constant. However, the results of example, a local JSR with a diameter of 300 nm and a depth Harkins et al. (1993) suggest that fluo-3 bound to Ca2ϩ has of 10 nm would have a volume of only 7 ϫ 10Ϫ4 ␮m3 a different affinity for proteins than does free fluo-3. For (assuming a cylindrical geometry). Even with 50 mM of this reason, we performed additional simulations with a buffer-bound Ca2ϩ and 1 mM free Ca2ϩ, this volume more complex buffering-diffusion scheme derived from the wound only contain ϳ21000 Ca2ϩ ions. Because a 1-pA Harkins et al. data (detailed in Hollingworth et al., 2000). current is equivalent to ϳ3000 Ca2ϩ ions per millisecond, The Ca2ϩ sparks produced by stereotyped Ca2ϩ release the Ca2ϩ in the local JSR would only be able to provide the fluxes with the complex buffering scheme are compared Ca2ϩ spark release flux a few milliseconds. Thus, even if with those generated with the simple buffering scheme (i.e., refilling of the JSR is fast, it seems likely that significant the model used in the other simulations) in Fig. 14. Three

Biophysical Journal 83(1) 59–78 74 Sobie et al.

FIGURE 14 Effects of diffusion-buffering model on Ca2ϩ spark characteristics. Ca2ϩ sparks produced by stereotyped Ca2ϩ release fluxes (top) were computed with both the simple buffering and the complex buffering approximations (see text for descriptions). The middle panels present the Ca2ϩ spark time course (0.25 ␮m from the source) and line-scan image for simple buffering, whereas the bottom panels display these for complex buffering. features of these simulations are of interest. One is that time. However, because flux depends on both the RyR open Ca2ϩ sparks produced by decaying fluxes are rounded at the probability and the Ca2ϩ gradient, a decay in the release peak, whereas those resulting from constant fluxes come to flux can theoretically result from severe local depletion, as a sharp peak when release shuts off. A second observation we have modeled here, from strong inactivation of RyRs, or is that the choice of buffer model causes only subtle changes from a combination of the two. in the Ca2ϩ spark shape. In other words, the Ca2ϩ spark time courses produced by constant and decaying sources look similar with either buffer model. Finally, the extended Other spark models Ca2ϩ sparks displayed show that a maintained flux of 2ϩ only10% of the peak level can produce a maintained ⌬F/F0 Earlier models of the Ca release process (Stern, 1992; Stern 2ϩ of ϳ50% of the peak level. These maintained F/F0 levels, et al., 1999; Rice et al., 1999) or Ca spark characteristics which are similar to those observed experimentally (e.g., (Pratusevich and Balke, 1996; Smith et al., 1998; Izu et al., Fig. 13), suggest that a very small Ca2ϩ flux can maintain a 2001) have provided useful interpretations of experimental significant plateau during long Ca2ϩ sparks. Because of the data and predictions that have motivated important new exper- presence of slow Ca2ϩ buffers in the Ca2ϩ spark model, the iments. Our results complement these studies by examining peak ⌬F/F0 that is achieved during the ϳ20 ms of release in issues that earlier models did not thoroughly address. Specif- the normal Ca2ϩ spark is considerably less than the steady- ically, our study extends previous work in the following ways: state level that would be achieved if this flux were main- 1) the model provides a hypothesis for the termination of Ca2ϩ tained indefinitely. Thus, a small maintained flux can pro- release that is experimentally justified; 2) simple experimental duce a relatively larger ⌬F/F0. Simulated sparks in which interventions have been modeled and are consistent with pub- the maintained flux was 50% of the peak produced Ca2ϩ lished data; 3) Ca2ϩ spark properties are maintained roughly sparks with plateau levels close to the peak levels, incon- the same when the RyR number is varied; and 4) the Ca2ϩ sistent with the experimental data (results not shown). sparks produced by our release fluxes are rounded at their peak Taken together, these observations strongly suggest cardiac rather than coming to a sharp peak (Smith et al., 1998) or Ca2ϩ sparks result from Ca2ϩ release fluxes that decay with having a flat-top shape (Izu et al., 2001). However, one weak-

Biophysical Journal 83(1) 59–78 Mechanism of Cardiac Ca2ϩ Spark Termination 75 ness of the previous studies, which is shared by models of RyRs, sticky clusters, and spatial organization skeletal muscle Ca2ϩ sparks (Jiang et al., 1999), has not been with the heart cell overcome in the present model: Ca2ϩ spark width (full-width at half-maximum (FWHM) is about half of that observed The large number of RyRs in a cluster in heart (Franzini- experimentally. This is due in part to the way that fluo-3 Armstrong et al., 1998, 1999) along with evidence of cou- diffusion was modeled with the simple buffering scheme, pled gating among RyRs in heart (Marx et al., 2001) sug- because sparks simulated with the complex buffering scheme gests that sticky clusters may be functionally important. The (Fig. 14) are somewhat wider (FWHM at the peak 1.39 versus two-dimensional organization of such clusters would depend on how they were arranged with respect to the TT and 1.32 ␮m). It is also interesting to note that Ca sparks with a the size of the TTs at the SR–TT junction. Current estimates decaying release flux have larger FWHM than those produced are that TTs in mammalian hearts are ϳ300 nm in diameter, by the equivalent constant release flux (e.g., 1.66 ␮minthe permitting a half-circumference of ϳ500 nm. If RyRs were Smith buffering model). However, simulated sparks still have spaced 50 nm apart within a cluster, such a TT could readily asmallerFWHMthanthatobservedexperimentally.Another 2ϩ accommodate a cluster of 100 RyRs arranged in a square hypothesis for the unreasonable narrowness of model Ca symmetrical array. If each cluster contained 100 RyRs and sparks is that the fluxes thought to produce sparks are too small the cell contained 106 by approximately an order of magnitude (Izu et al., 2001). total RyRs, the clusters would be separated by 2 ␮ ␮ More experiments and modeling will be necessary to test this m in the longitudinal direction and 1 m ϫ provocative idea. along the other directions, assuming a cell size of 200 20 ϫ 10 ␮m. A 1-␮m-thick image would therefore contain 10 clusters per Z line. If each cluster contained fewer RyRs, this would require closer spacing between clusters and a Relationship to RyR gating in planar lipid bilayers greater number of Ca2ϩ spark sites. These issues become extremely important when exploring how restitution occurs To explore easily the mechanisms by which Ca2ϩ release and how Ca instability arises, which is why morphometry terminated in the model, we implemented a simple RyR data that examines this organization and new experiments gating scheme that did not include features explored in other that examine functional characteristics of the SR will be studies (e.g., adaptation, inactivation, or modal gating). invaluable in advancing our understanding of calcium sig- Thus, detailed comparisons between these models of RyR naling in heart. gating and our integrated model of Ca2ϩ spark behavior are Here we should note that it is possible that some of the impossible. We can, however, compare our results to other RyRs present in a cluster are inactive and do not contribute published RyR schemes in terms of sensitivity to activation to the release flux that produces a Ca2ϩ spark. We think it by subspace [Ca2ϩ]. In the absence of coupled gating in our is one of the major strengths of the model that robust model and with 1 mM [Ca2ϩ] , a single RyR tetramer lumen termination, and relatively constant Ca2ϩ spark character- displays half-maximal activation at 1.28 ␮M, which is sim- istics, occur for RyR clusters of various sizes. Thus, if a ilar to the sensitivity (0.6 ␮M) of the relatively simple particular cluster contained 100 RyRs, the same general scheme of Cannell and Soeller (1997). Without running mechanism for spark termination could be operable whether extensive simulations, it is not straightforward to compute all 100 or only 20 of those channels were active. As new half-maximal activation in more complex models, but the information becomes available, we may need to adjust fact that other models (Zahradnikova and Zahradnik, 1996; model parameters, such as the strength of coupling between Keizer and Smith, 1998; Fill et al., 2000) show activation of RyRs, the single channel RyR conductance, or the lumenal RyRs by steps of Ca2ϩ to 1 ␮M indicates that these models dependence of RyR gating to simulate accurately the de- have similar sensitivity to [Ca2ϩ] as ours. However, we SS tailed characteristics of Ca2ϩ sparks. However, the general should note that, because RyRs are involved in the locally mechanism we have proposed here should still provide positive-feedback process of CICR, in both our integrated robust termination of Ca2ϩ release and realistic Ca2ϩ model and in cells, more than just the equilibrium values of sparks. Ca2ϩ activation are important. A small initial RyR opening 2ϩ can increase [Ca ]SS and increase the open probability of the other channels in the cluster (even without coupled Model choices gating), which means that the dynamics of RyR opening will be extremely important in determining how Ca2ϩ Because our aim was to incorporate the most recent exper- sparks are triggered and terminate. In addition, factors such imental observations into our new Ca2ϩ spark model, we as Mg2ϩ, ATP, and phosphorylation by cellular kinases can simulated phenomena that are incompletely characterized influence RyR gating, but our understanding of these effects and were therefore forced to make certain somewhat arbi- is still incomplete. Despite these limitations, we feel that our trary choices. The reasons for some of our choices, and how gating scheme is consistent with critical features of RyR the model may need to be adjusted as new data become behavior. available, are detailed here.

Biophysical Journal 83(1) 59–78 76 Sobie et al.

It is unclear exactly how many RyRs are present in each restitution or how Ca2ϩ sparks produce Ca2ϩ waves. One cluster, and this number may be species-dependent (Fran- possible approach is suggested by modeling of conforma- zini-Armstrong et al., 1998, 1999). Although we defined a tional spread in clusters of receptors involved in bacterial control cluster as containing 50 RyRs, we consider it a chemotaxis. This formulation, in which the presence of a strength of the model that Ca2ϩ spark duration is relatively receptor in one state makes that state more energetically insensitive to the number of RyRs in the cluster. However, favorable for the neighboring receptors, has been shown to as additional morphological data become available, param- influence the behavior of clusters in several potentially eters such as RyR number, single RyR Ca2ϩ flux, and the useful ways (e.g., greater sensitivity) (Duke and Bray, 1999; strength of coupling between RyRs may need to be modi- Duke et al., 2001). In general, then, the broad outlines of our fied for the model to produce realistic sparks. The value that model are clearly consistent with the majority of the exper- we chose for iRyR, 0.07 pA, is smaller than any single imental data available, but several minor adjustments will channel RyR current that has been measured experimen- likely need to be made as these phenomena are character- tally, but we feel it may be a realistic estimate of the ized in greater detail. physiological RyR flux. Our parameters were based on the single channel bilayer study of Mejia-Alvarez et al. (1999) FKBP/FK506/Rapamycin under quasi-physiological conditions but were adjusted to account for the slightly different ionic conditions that we FK506 has been shown to bind to FKBP12.6 and FKBP12 modeled. These authors recorded an iRyR of 0.35 pA with and remove them from the RyRs to which they are associ- 150 mM cytosolic Csϩ, 0 mM cytosolic Mg2ϩ, and 2 mM ated (Marks, 1996; Yano et al., 2000; Marx et al., 2000, lumenal Ca2ϩ (their Fig. 2). If we adjust for the smaller 2001; Bultynck et al., 2001; Prestle et al., 2001). The current that would be present with 1 mM lumenal Ca2ϩ, and application of FK506 reversibly disrupts coupled gating for the ϳ67% reduction in current caused by 1 mM Mg2ϩ over time in planar lipid bilayer experiments. In experi- (as in their Fig. 5), this yields approximately 0.07 pA, close ments on isolated cells, FK506 has been shown by others to the value we used. A similar result is obtained by linear (Xiao et al., 1997; Lukyanenko et al., 1998) to cause long interpolation of their Fig. 6 A to account for the different Ca2ϩ sparks similar to those presented here (Fig. 13), but conditions present. one group reported no change in Ca2ϩ spark characteristics The notion that lumenal Ca2ϩ affects RyR gating, al- (McCall et al., 1996). This conflicting observation may though initially controversial, is now supported by both result from the fact that Ca2ϩ spark half-amplitude dura- indirect evidence from cellular studies of CICR and direct tions were reported by McCall et al. (1996). With such a evidence from bilayer studies. However, it remains uncer- definition of Ca2ϩ spark duration, sparks that decline to a tain whether this effect is mediated through free Ca2ϩ in the plateau level below 50% of the peak and are then extended lumen or through Ca2ϩ binding to an associated protein would not be considered to be longer than normal. The such as calsequestrin. Thus, our mathematical representa- collection of data thus suggests that the proposed model is tion of this effect should not be considered unique. Indeed, reasonable. However it should be noted that FK506 and in preliminary simulations, we were able to produce Ca2ϩ rapamycin have additional effects on cellular function (du- release fluxes that decayed much less severely but still Bell et al., 1997, 1998, 2000), but the effect they have in terminated robustly by varying the sensitivity of RyR gating common is to bind FKBPs and change RyR behavior. 2ϩ to [Ca ]lumen and the rate of JSR refilling (results not shown). Also, although the Marks (1996) lab has observed CONCLUSIONS coupled gating between both skeletal muscle (Marx et al., 1998) and cardiac (Marx et al., 2001) RyRs, these results We have proposed a new model of the cardiac Ca2ϩ release have not yet been confirmed by others and remain contro- process in which a Ca2ϩ spark results from the action of a versial. This discrepancy probably results from slight dif- sticky cluster of RyRs. It succeeds in characterizing this ferences in the experimental protocols used to isolate RyRs high-gain, positive-feedback CICR signaling system under for bilayer studies and emphasizes that caution needs to be many conditions. Although experimental results obtained to exercised when interpreting behavior of ion channels ob- date are consistent with the model, many features of Ca2ϩ served under conditions far from their native state. Finally, signaling have not yet been tested or investigated. Similar- because there is not enough experimental data currently ities in other CICR systems may enable modified versions available to provide an accurate mathematical representa- of the model of Ca2ϩ spark behavior to be applied to smooth tion of coupled gating, we decided to simplify our modeling muscle and skeletal muscle. of coupled gating by not considering any spatially resolved interactions between RyRs. Although this representation We would like to thank Dennis Bray for discussion and comments, Greg has proven to be adequate for the present considerations, it Smith for code sharing and discussion on modeling of Ca2ϩ sparks, and is undoubtedly an oversimplification, and a more complex Andrew R. Marks for discussion on coupled gating. We would also like to treatment may be necessary to examine phenomena such as thank the reviewers of the manuscript for useful suggestions.

Biophysical Journal 83(1) 59–78 Mechanism of Cardiac Ca2ϩ Spark Termination 77

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