Predicting Local SR Ca²⁺ Dynamics during Ca²⁺ Wave Propagation in Ventricular Myocytes

Abstract

Of the many ongoing controversies regarding the workings of the sarcoplasmic reticulum (SR) in cardiac myocytes, two unresolved and interconnected topics are 1), mechanisms of calcium (Ca²⁺) wave propagation, and 2), speed of Ca²⁺ diffusion within the SR. Ca²⁺ waves are initiated when a spontaneous local SR Ca²⁺ release event triggers additional release from neighboring clusters of SR release channels (ryanodine receptors (RyRs)). A lack of consensus regarding the effective Ca²⁺ diffusion constant in the SR (D_Ca,SR) severely complicates our understanding of whether dynamic local changes in SR [Ca²⁺] can influence wave propagation. To address this problem, we have implemented a computational model of cytosolic and SR [Ca²⁺] during Ca²⁺ waves. Simulations have investigated how dynamic local changes in SR [Ca²⁺] are influenced by 1), D_Ca,SR; 2), the distance between RyR clusters; 3), partial inhibition or stimulation of SR Ca²⁺ pumps; 4), SR Ca²⁺ pump dependence on cytosolic [Ca²⁺]; and 5), the rate of transfer between network and junctional SR. Of these factors, D_Ca,SR is the primary determinant of how release from one RyR cluster alters SR [Ca²⁺] in nearby regions. Specifically, our results show that local increases in SR [Ca²⁺] ahead of the wave can potentially facilitate Ca²⁺ wave propagation, but only if SR diffusion is relatively slow. These simulations help to delineate what changes in [Ca²⁺] are possible during SR Ca²⁺release, and they broaden our understanding of the regulatory role played by dynamic changes in [Ca²⁺]_SR.

Introduction

In cardiac myocytes, the sarcoplasmic reticulum (SR) functions as the main Ca²⁺ storage organelle, and the concentration of Ca²⁺ within the SR ([Ca²⁺]_SR) is determined by the balance between release through ryanodine receptors (RyRs) and uptake via pumps in the SR membrane (sarcoendoplasmic reticulum Ca²⁺ ATPase (SERCA)). With each heartbeat, local increases in intracellular [Ca²⁺] ([Ca²⁺]_i) trigger release of SR Ca²⁺ in the form of thousands of individual units known as Ca²⁺ sparks, and the resulting cellular Ca²⁺ transient leads to myocyte contraction (1,2). Under pathological conditions such as Ca²⁺ overload, however, a spontaneous Ca²⁺ spark from a cluster of RyRs can trigger release from neighboring RyR clusters, and these sparks in turn trigger additional clusters, resulting in a regenerative Ca²⁺ wave (3). The global increase in [Ca²⁺]_i during a wave causes Ca²⁺ extrusion via the Na⁺-Ca²⁺ exchanger, and the resulting inward current depolarizes the cell membrane (4). Since these inappropriate depolarizations can trigger dangerous arrhythmias, understanding the mechanisms of Ca²⁺ wave propagation is essential for the treatment of pathologies such as heart failure.

Diffusion of Ca²⁺ within the cytoplasm and triggering of neighboring RyR clusters via Ca²⁺-induced Ca²⁺ release clearly plays an important role in the propagation of cardiac Ca²⁺ waves (3). However, increases in cytosolic [Ca²⁺] due to release are accompanied by dynamic local changes in [Ca²⁺]_SR (5–7). It is not clear how these changes might facilitate or hinder wave propagation. Recently, Keller et al. proposed that during Ca²⁺ waves, increases in [Ca²⁺]_SR at unactivated sites may increase the sensitivity of RyR clusters and promote propagation (8). This hypothesis was based on the observation, in Ca²⁺-overloaded guinea pig myocytes, that partial inhibition of SERCA caused a decrease in wave velocity, although there was no change in wave amplitude or resting [Ca²⁺]. Since SERCA inhibition would be expected to increase cytosolic [Ca²⁺], thereby accelerating Ca²⁺ waves (9), these authors reasoned that dynamic changes in [Ca²⁺]_SR influence wave propagation (8).

This hypothesis has been difficult to test directly, however, due to technical limitations. Small regions of junctional SR (JSR) associated with release sites have volumes (0.008 fL (6)) much smaller than the spatial resolution of confocal microscopes (imaging volume ∼0.06 fL), so fluorescence from neighboring SR contributes to the measured signal. In addition, because the dye most commonly used to measure [Ca²⁺]_SR, fluo-5N, has an affinity (400 μM (6)) two to three times less than diastolic [Ca²⁺]_SR, this indicator is operating near saturation during the Ca²⁺ overload conditions associated with waves. Given these considerations, it is clear that local and transient increases in [Ca²⁺]_SR would be very difficult to detect. Complicating matters further is the fact that the speed of Ca²⁺ diffusion within the SR is not known precisely. Two recent experimental estimates of the SR Ca²⁺ diffusion constant differed by nearly an order of magnitude (10,11).

Here, we address these controversies through computational modeling. We simulate spatiotemporal changes in cytosolic and SR [Ca²⁺] during Ca²⁺ waves and examine systematically how these dynamic changes are influenced by model parameters controlling SR Ca²⁺ diffusion and SERCA activity. Our results show that sensitization of RyR clusters by local increases in [Ca²⁺]_SR can indeed occur during Ca²⁺ waves, but only if Ca²⁺ movement within the SR is slow.

Methods

Model geometry

The model employed in this study (Fig. 1) simulates Ca²⁺ movements in two cellular compartments: release from SR to cytoplasm, uptake from cytoplasm to SR, and buffering and diffusion in each. The partial differential equations governing changes in free [Ca²⁺], described below, are therefore of reaction-diffusion form. The model contains a single RyR cluster/μm³. Each cluster is assumed to contain 56 channels and wrap around a transverse (T) tubule. RyR clusters are separated by distances of 2 μm in the longitudinal direction, and 0.5 μm and 1 μm in the two transverse directions. As our interest is in studying longitudinal propagation of Ca²⁺ waves, the model consists of an array of 20 sarcomeres placed end-to-end longitudinally. The overall dimensions are therefore 40 × 0.5 × 1 μm. The cytoplasm and network SR (NSR) are homogeneously distributed, with volumes (Table 1) calculated using standard cell volume percentages in rat ventricular myocytes (2). We assume that the T-tubule, which has a diameter of 200 μm and runs in the z direction, perpendicular to the plane of the figure, acts as a diffusion barrier.

Figure 1.

Ca²⁺ wave model schematic. Each release site is placed beside a transverse tubule and is located at a distance d from its neighbors. The default distance d between RyR clusters in the longitudinal direction, left to right in the figure, is 2 μm; results shown in Figs. 4, 6, and 7 also consider the case where d = 1 μm. A single cluster containing 56 RyRs is assumed to wrap completely around each T-tubule. For clarity of illustration, this is drawn as two clusters on either side of the T-tubule. NSR connects the regions of JSR associated with each release site. SERCA pumps are homogeneously distributed throughout the NSR.

Table 1.

Geometric parameters

Parameter	Definition	Value
VSS	Sub-space volume	1.0 × 10−12μL
VJSR	Junctional SR volume	3.2 × 10−12μL
VNSR	Network SR volume	3.2 × 10−10μL
VCYT	Cytoplasmic volume	4.6 × 10−9μL

In the two transverse directions, we apply reflective or no-flux boundary conditions. This is equivalent to assuming that in these two directions, RyR clusters spaced every 0.5 and 1 μm behave identically. Changes in [Ca²⁺] during longitudinal propagation of Ca²⁺ waves can therefore be calculated by considering the thin geometry described. Because the application of longitudinal no-flux boundary conditions at the 1st and 20th sarcomeres induces small boundary effects, only changes in [Ca²⁺] in the middle sarcomeres are displayed in the article (see Figs. 2–7).

Figure 2.

(A) Space-time image of cytosolic [Ca²⁺] (on a logarithmic scale) during a Ca²⁺ wave (left), with the corresponding image of SR [Ca²⁺] (right). Transverse tubules are indicated by horizontal white lines. (B) Spatiotemporal changes in [Ca²⁺]_SR within two sarcomeres are displayed for three values of D_Ca,SR: 12 μm²/s (slow; left), 60 μm²/s (medium; middle), and 300 μm²/s (fast; right) Slow SR diffusion causes the most dramatic increases in [Ca²⁺]_SR above the baseline value (1500 μM) in unactivated regions. (C) Recovery of total JSR [Ca²⁺] after release at an individual site depends on D_Ca,SR. Time constants range from 54 ms (fast) to 84 ms (slow).

Figure 3.

Logarithmic plots of [Ca²⁺]_i versus location (A) at different times during a Ca²⁺ wave for D_Ca,SR = 12 μm²/s (left), 60 μm²/s (middle), and 300 μm²/s (right), with the corresponding logarithmic plots of [Ca²⁺]_SR (B). The speed of SR Ca²⁺ diffusion can greatly influence [Ca²⁺]_SR but has little effect on [Ca²⁺]_i.

Figure 4.

Effects of SERCA inhibition on [Ca²⁺]_i (A) and [Ca²⁺]_SR (B) at the target site. Bars show percentage changes in [Ca²⁺] with normal and partially blocked SERCA for different values of D_Ca,SR and different spacing between release sites.

Figure 5.

Effects of SERCA stimulation on [Ca²⁺]_i and [Ca²⁺]_SR at the target site. Increasing SERCA activity leads to a decrease in [Ca²⁺]_i, and an increase in [Ca²⁺]_SR for all values of D_Ca,SR.

Figure 6.

Effects of SERCA pump [Ca²⁺]_i dependence with cluster spacing of 1 μm (left) and 2 μm (right). In either case, the exponent in the equation describing SERCA activity (η) was set to either 3, the control value of 1.78, or 0.75. An increased exponent leads to greater increases in [Ca²⁺]_SR, and slow diffusion favors accumulation of [Ca²⁺]_SR, in all cases.

Figure 7.

(A) Time course of JSR refilling for different values of NSR-to-JSR transfer rate v. (B) Percentage changes in target site [Ca²⁺]_SR for different values of v and D_Ca,SR.

Model equations

The following sequence of events dictates the differential equations that define the model. Ca²⁺ is released from clusters of RyRs into a small volume, or subspace, between the SR and T-tubule membranes. From the subspace, Ca²⁺ is transferred to the cytoplasmic region, where it diffuses, binds to buffers, and is taken into the NSR by SERCA pumps. At the same time, release causes [Ca²⁺] to decrease in the JSR, and this depletion leads to Ca²⁺ transfer from NSR to JSR. Important features of each process are mentioned briefly before the full equations are presented.

Ca²⁺ release flux, J_release, from each RyR cluster is calculated as

$$J_{release}=\frac{N_{RyR}{P}_{RyR}\left({\left[{\text{Ca}}^{2+}\right]}_{\text{JSR}}-{\left[{\text{Ca}}^{2+}\right]}_{\text{SS}}\right)}{V_{\text{SS}}},$$ (1)

where $P_{RyR}$ is the permeability of a single RyR channel, $N_{RyR}$ is the number of open channels in the cluster, and V_ss is the volume of the subspace. As described in more detail below, we assume for simplicity that at any time, either all RyRs are open or all RyRs are closed, in which case J_release = 0.

In each subspace, Ca²⁺ can be buffered by calmodulin, sarcolemmal membrane, and SR membrane. Parameters for these buffers, assumed to be immobile in the small subspace, are given in Table 2. Binding to each buffer is described by standard buffering equations, i.e., for a generic buffer B,

$$J_{\text{B}}=k_{on}\left[\text{B}\right]{\left[{\text{Ca}}^{2+}\right]}_{\text{SS}}-k_{off}\left({\left[\text{B}\right]}_{\text{Total}}-\left[\text{B}\right]\right),$$ (2)

where [B] is the concentration of free buffer, [B]_Total is the total concentration of buffer (constant at each grid volume), and k_on and k_off are the on and off rates, respectively. The rate of change of free buffer is therefore –J_B for each buffer. Fluxes corresponding to the individual buffers are added to compute, at each grid volume, the overall flux of Ca²⁺ onto or off of buffers.

$$J_{\text{buffer\_SS}}=\sum J_{\text{B}}\text{.}$$ (3)

Table 2.

Buffering parameters

Parameter	Definition	Value
[CaM]Total	Total calmodulin concentration	24 μM
$k_{on}^{\text{CaM}}$	Calmodulin Ca2+ on rate constant	100 μM−1 s−1
$k_{off}^{\text{CaM}}$	Calmodulin Ca2+ off rate constant	38 s−1
[TRPN]Total	Total troponin concentration	70 μM
$k_{on}^{\text{TRPN}}$	Troponin Ca2+ on rate constant	39 μM−1 s−1
$k_{off}^{\text{TRPN}}$	Troponin Ca2+ off rate constant	20 s−1
[SL]Total	Total SL membrane buffer concentration	900 μM
$k_{on}^{\text{SL}}$	SL membrane buffer Ca2+ on rate constant	115 μM−1 s−1
$k_{off}^{\text{SL}}$	SL membrane buffer Ca2+ off rate constant	1000 s−1
[SR]Total	Total SR membrane buffer concentration	47 μM
$k_{on}^{\text{SR}}$	SR membrane buffer Ca2+ on rate constant	115 μM−1 s−1
$k_{off}^{\text{SR}}$	SR membrane buffer Ca2+ off rate constant	100 s−1
[CSQ]Total	Total calsequestrin concentration	10.0 × 103μM
KCSQ	Calsequestrin Ca2+ dissociation constant	0.63 × 103μM
[JUN]Total	Total junctate concentration	10.0 × 103μM
KJUN	Junctate Ca2+ dissociation constant	0.217 × 103μM

The efflux (J_efflux) of Ca²⁺ from subspace to cytoplasm is calculated as

$$J_{efflux}=\frac{\left({\left[{\text{Ca}}^{2+}\right]}_{\text{SS}}-{\left[{\text{Ca}}^{2+}\right]}_{\text{CYT}}\right)}{{\tau }_{\text{SS\_CYT}}},$$ (4)

where τ_{SS_CYT} is the time constant of Ca²⁺ transfer between subspace and cytoplasm.

Changes in [Ca²⁺]_SS are dictated by the balance of the above-mentioned fluxes, i.e.,

$$\frac{d{\left[{\text{Ca}}^{2+}\right]}_{\text{SS}}}{dt}=J_{release}-J_{efflux}-J_{\text{buffer\_SS}}\text{.}$$ (5)

As in the subspace, Ca²⁺ in the cytoplasm can bind to buffers: a mobile buffer, calmodulin, and a stationary buffer, troponin. The equations governing binding of Ca²⁺ to these buffers are identical to those listed above for subspace buffers (Eq. 2). The quantities and affinities of these buffers (Table 2) are similar to those used in previous studies (12,13), and, as described below, the resulting effective diffusion constant is consistent with previous modeling studies and experimental measurements. This provides confidence that the results are not unique to our parameter choices. The rate of change of the free mobile buffer calmodulin depends on diffusion of buffer in addition to J_B, i.e.,

$$\frac{d\left[\text{B}\right]}{dt}=-J_{\text{buffer\_CYT}}+D_{\text{B}}{\nabla }^2\left[\text{B}\right],$$ (6)

where D_B is the diffusion coefficient of calmodulin.

Ca²⁺ in the cytoplasm is also taken up into the NSR by SERCA pumps. The pump flux (J_uptake) is described by the formulation of Shannon and co-workers (14):

$$J_{uptake}=\frac{v_{\text{SERCA}}{\left(\frac{{\left[{\text{Ca}}^{2+}\right]}_{\text{CYT}}}{K_{mf}}\right)}^{\eta }-v_{\text{SERCA}}{\left(\frac{{\left[{\text{Ca}}^{2+}\right]}_{\text{NSR}}}{K_{mr}}\right)}^{\eta }}{1+{\left(\frac{{\left[{\text{Ca}}^{2+}\right]}_{\text{CYT}}}{K_{mf}}\right)}^{\eta }+{\left(\frac{{\left[{\text{Ca}}^{2+}\right]}_{\text{NSR}}}{K_{mr}}\right)}^{\eta }},$$ (7)

where ν_SERCA is the maximum rate of uptake, K_mf and K_mr are forward and reverse dissociation constants, and η is the Hill exponent. In most simulations we assume that SERCA pumps are homogeneously distributed throughout the network SR, implying spatially constant ν_SERCA. In other simulations (see Fig. S3 of the Supporting Material), we consider heterogeneous SERCA distributions with activity maxima either at the T-tubules or at the M-lines between the T-tubules.

Ca²⁺ leak from the SR through RyRs is calculated as

$$J_{\text{leak}}=v_{\text{leak}}\left({\left[{\text{Ca}}^{2+}\right]}_{\text{NSR}}-{\left[{\text{Ca}}^{2+}\right]}_{\text{CYT}}\right),$$ (8)

where ν_leak defines the rate of leak. Table 3 contains values for SERCA and leak parameters.

Table 3.

Sarcoplasmic reticulum Ca²⁺ flux parameters

Parameter	Definition	Value
νSERCA	Maximal SERCA pump rate	865 μM/s
Kmf	Forward half-saturation constant for Ca2+-ATPase	0.24 μM
Kmr	Reverse half-saturation constant for Ca2+-ATPase	1.7 × 103μM
η	Cooperativity constant for Ca2+-ATPase	1.78
vleak	Jleak rate constant	5.1 × 10−4 s−1
PRyR	Permeability of RyR channels	0.22 × 10−8μL/s

Hence, the rate of change for [Ca²⁺]_CYT is given by

$$\frac{d{\left[{\text{Ca}}^{2+}\right]}_{\text{CYT}}}{dt}=J_{efflux}\frac{V_{\text{SS}}}{V_{\text{CYT}}}+J_{\text{leak}}-J_{uptake}-J_{\text{buffer\_CYT}}+D_{\text{Ca},\text{CYT}}{\nabla }^2{\left[{\text{Ca}}^{2+}\right]}_{\text{CYT}}\text{.}$$ (9)

When Ca²⁺ release occurs, the depleting JSR is replenished via diffusion of Ca²⁺ from NSR to JSR (J_refill), calculated in the model as

$$J_{refill}=\frac{\left({\left[{\text{Ca}}^{2+}\right]}_{\text{NSR}}-{\left[{\text{Ca}}^{2+}\right]}_{\text{JSR}}\right)}{{\tau }_{\text{NSR\_JSR}}},$$ (10)

where τ_{NSR_JSR} is the time constant of transfer between NSR and JSR. In this case, [Ca²⁺]_NSR refers to the concentration in the grid cell adjacent to the JSR. In the manuscript, we refer to the rate of transfer, v, which is the reciprocal of τ_{NSR_JSR}. The balance equation for [Ca²⁺]_JSR is

$$\frac{d{\left[{\text{Ca}}^{2+}\right]}_{\text{JSR}}}{dt}={\beta }_{\text{JSR}}\left[-J_{release}\frac{V_{\text{SS}}}{V_{\text{JSR}}}+J_{refill}\right],$$ (11)

where V_SS and V_JSR are the subspace and JSR compartment volumes, respectively. The rapid buffering approximation (15) defines the buffering factor, β_JSR, and the volume ratio rescales the flux to account for the difference in compartment volumes. We have included calsequestrin and junctate (16) as the calcium binding proteins in the JSR (see Table 2 for buffer parameters). ${\beta }_{\text{JSR}}$ is defined as

$${\beta }_{\text{JSR}}={\left(1+\frac{{\left[\text{CSQ}\right]}_{\text{Total}}{K}_{\text{CSQ}}}{{\left(K_{\text{CSQ}}+{\left[{\text{Ca}}^{2+}\right]}_{\text{JSR}}\right)}^2}+\frac{{\left[\text{JUN}\right]}_{\text{Total}}{K}_{\text{JUN}}}{{\left(K_{\text{JUN}}+{\left[{\text{Ca}}^{2+}\right]}_{\text{JSR}}\right)}^2}\right)}^{-1},$$ (12)

where ${\left[\text{CSQ}\right]}_{\text{Total}}$ and ${\left[\text{JUN}\right]}_{\text{Total}}$ are the total concentrations, and $K_{\text{CSQ}}$ and $K_{\text{JUN}}$ are the dissociation constants, of calsequestrin and junctate, respectively.

Finally, the rate of change for ${\left[{\text{Ca}}^{2+}\right]}_{\text{NSR}}$ is calculated as

$$\frac{d{\left[{\text{Ca}}^{2+}\right]}_{\text{NSR}}}{dt}=\left[\left(J_{uptake}-J_{\text{leak}}\right)\frac{V_{\text{CYT}}}{V_{\text{NSR}}}-J_{refill}\frac{V_{\text{JSR}}}{V_{\text{NSR}}}+D_{\text{Ca},\text{SR}}{\nabla }^2{\left[{\text{Ca}}^{2+}\right]}_{\text{NSR}}\right],$$ (13)

where D_{Ca_SR} is the diffusion constant for Ca²⁺ in the NSR. Diffusion constants and parameters controlling intercompartment transfer are provided in Table 4. J_refill is calculated from Eq. 10 in NSR elements that are adjacent to JSR and is zero elsewhere.

Table 4.

Diffusion coefficients

Parameter	Definition	Value
$D_{\text{Ca},\text{CYT}}$	Diffusion coefficient of Ca2+ within cytoplasm	300 μm2 s−1
$D_{\text{Ca},\text{SR}}$	Diffusion coefficient of Ca2+ within NSR	12–300 μm2 s−1
$D_{\text{CaM}}$	Diffusion coefficient of calmodulin and Ca2+-calmodulin	70 μm2 s−1
$dx,dy,dz$	Spatial step sizes	0.1 μm
${\tau }_{\text{SS\_CYT}}$	Transfer rate of Ca2+ from subspace to cytoplasm	4 × 10−5 s
${\tau }_{\text{NSR\_JSR}}$	Transfer rate of Ca2+ from NSR to JSR	0.012 s

Simulation protocol

Free and bound [Ca²⁺] in all compartments are initially assumed to be at steady state. Free cytoplasmic [Ca²⁺] and SR [Ca²⁺] are set at 0.215 and 1500 μM, respectively, to simulate a state of Ca²⁺ overload associated with frequent spontaneous Ca²⁺ waves. Because we use the bidirectional SERCA pump formulation of Shannon et al. (14) and assume a very low rate of passive leak, changes in ν_SERCA have a negligible effect on the steady-state levels of cytosolic and SR [Ca²⁺]. Ca²⁺ waves are simulated using a simplified protocol that eliminates the complexities caused by stochastic gating of RyRs. All RyR channels in a cluster are opened for a fixed duration of 22 ms, and RyR clusters are opened sequentially, at intervals determined by the assumed cluster spacing and wave velocity. For instance, under control conditions, we assume that waves propagate at 91 μm/s (8), which implies that sites spaced 2 μm apart open at 22-ms intervals. Our simplified protocol provides for precise control of the wave velocity, which allows us to determine how the velocity itself affects the resulting changes in cytsosolic and SR [Ca²⁺]. For instance, we perform two sets of simulations with partially inhibited SERCA, one in which wave velocity remains 91 μm/s (22-ms intervals), and a second set for which wave velocity is reduced to 69 μm/s (8), implying an interval of 33 ms between cluster openings.

To verify that our parameters for compartment volumes, buffers, and intercompartment fluxes are reasonable, we simulated a triggered Ca²⁺ transient by opening 90% of RyR clusters over a period of 120 ms. The decay and recovery of SR [Ca²⁺] obtained in this simulation (Fig. S1) was very similar to experimental measurements of Ca²⁺ scraps (6), thereby providing confidence in our parameter choices.

Comparison of effective D_Ca,CYT with previous models

We assume that the cytosolic diffusion constant of free Ca²⁺, D_Ca,CYT, is 300 μm²/s. After taking into account the capacities and mobilities of Ca²⁺ buffers, this translates to an effective D_Ca,CYT of ∼38 μm²/s. This is consistent with the classic measurements of Allbritton et al. (17), as well as with other computational studies of Ca²⁺ movements in heart cells (see Table 5). We calculated effective diffusion coefficients using the equation (18)

$$D_{\text{CYT}}^{eff}=\frac{D_{\text{Ca},\text{CYT}}+\sum _1^N\frac{D_{\text{B}}{\left[\text{B}\right]}_{Mobile}}{K_{\text{B}}}}{1+\sum _1^M\frac{{\left[\text{B}\right]}_{\text{Total}}}{K_{\text{B}}}},$$ (14)

where ${\left[\text{B}\right]}_{Mobile}$ refers to the mobile buffers listed in Table 2, N is the number of mobile buffers, and M is the number of total buffers, mobile and stationary.

Table 5.

Comparison of effective diffusion coefficients

DCa,CYT (μm2/s)	Reference
3.7	(13)
13.6	(38)
42	(39)
594	(40)

Results

The model consists of an array of 20 RyR clusters arranged linearly, with constant spacing between them (Fig. 1). Under the assumption of uniform wave propagation in the longitudinal direction, this is equivalent to simulating many identical clusters in the transverse plane. [Ca²⁺] in the cytosol and SR depends on release through RyR clusters, diffusion, buffering, and uptake via SERCA pumps. To avoid variability due to stochastic gating of RyRs, we simulate Ca²⁺ waves by opening the channels for fixed durations and at fixed intervals, with intervals determined by the cluster spacing and wave velocity (see Methods). We further assume that 1), each RyR cluster opens for 22 ms; and 2), initial concentrations of [Ca²⁺]_SR and [Ca²⁺]_i are 1500 and 0.215 μM, respectively (to simulate Ca²⁺ overload). With this simple protocol, we investigate how model parameters affect [Ca²⁺]_SR at the unactivated target site immediately ahead of the Ca²⁺ wave.

Fig. 2 A, which shows cytosolic [Ca²⁺] in an ∼10-μm region during a simulated Ca²⁺ wave, illustrates that the wave results from sequential openings of RyR clusters along the linear array. Cytosolic [Ca²⁺] is shown on a logarithmic scale, because the relative changes are much more extreme than those in [Ca²⁺]_SR. Along with [Ca²⁺]_i, the model also simulates the accompanying spatiotemporal changes in [Ca²⁺]_SR (Fig. 2 A; right). Fig. 2 B shows [Ca²⁺]_SR as a function of space and time within two sarcomeres for three values of D_Ca,SR: 12 μm²/s (slow; left), 60 μm²/s (medium; middle), and 300 μm²/s (fast; right). The first two values are similar to the recent experimental estimates (10,11), whereas the last represents an extreme case. For comparison, the free and effective Ca²⁺ diffusion coefficients in the cytosol are 300 μm²/s and 38.8 μm²/s, respectively. In the color scheme used to display the images, the resting level of [Ca²⁺]_SR is orange, red represents an increase above the diastolic level, and colder colors indicate local depletion of [Ca²⁺]_SR. T-tubules are indicated in white because [Ca²⁺]_SR and [Ca²⁺]_i are undefined at these locations along this particular line. Ca²⁺ in the cytosol and SR can, however, diffuse around the T-tubules (see Fig. 1).

The results show that extremely rapid diffusion (D_Ca,SR = 300 μm²/s) leads to a shallow Ca²⁺ gradient and lower [Ca²⁺]_SR ahead of the wave. For intermediate or slow diffusion, however, [Ca²⁺]_SR at the target site can increase above the initial value, as predicted by Thul et al. (19), with a D_Ca,SR of 5 μm²/s. Since [Ca²⁺]_SR depletes at the activated site, this clearly does not result from diffusion within the SR. Instead, released Ca²⁺ travels toward the target site in the cytosol and then is pumped into the SR by SERCA, as previously proposed by Jafri and Keizer (9). Slow SR diffusion allows this local increase in [Ca²⁺]_SR to persist. To estimate how SR diffusion influences measurable quantities such as restitution of Ca²⁺ spark amplitude, we performed simulations in which release occurred from a single RyR cluster. The recovery after release of total [Ca²⁺] (free plus bound to calsequestrin) in the JSR (Fig. 2 C) shows that slow diffusion is consistent with measurements of Ca²⁺ spark amplitude restitution (5,7), whereas medium and fast diffusion lead to somewhat faster recovery.

Fig. 3 displays semilogarithmic plots of cytosolic and SR [Ca²⁺] versus location at different times, with the breaks in each plot indicating T-tubules. These confirm the impression from the images in Fig. 2: D_Ca,SR has little effect on cytosolic [Ca²⁺] during the wave but can influence local, time-dependent changes in [Ca²⁺]_SR. Fast diffusion attenuates gradients in SR [Ca²⁺], whereas slow diffusion enhances these gradients. Moreover, when diffusion is slow, Ca²⁺ in front of the wave can transiently increase above diastolic levels, potentially allowing for sensitization of RyRs before they are activated by cytosolic [Ca²⁺].

We next investigated the effects of partial (50%) inhibition of SERCA. Since experiments have demonstrated that this reduces Ca²⁺ wave velocity (8), we increased the interval between consecutive activations from 22 to 33 ms. Fig. 4 compares the percentage changes in [Ca²⁺]_i (Fig. 4 A) and [Ca²⁺]_SR (Fig. 4 B) at the target site for three cases: 1), complete SERCA activity just before activation of the target site at 22 ms; 2), partial SERCA inhibition, but assuming 22-ms intervals between activations; and 3), partial SERCA inhibition just before activation of the target site at 33 ms. The same three values of D_Ca,SR were considered, and the results confirm that SR Ca²⁺ diffusion has little effect on cytosolic [Ca²⁺]. Regardless of D_Ca,SR, however, inhibition of SERCA leads to an increase in cytosolic [Ca²⁺] and a decrease in [Ca²⁺]_SR. If dynamic local changes in [Ca²⁺]_SR played no role in Ca²⁺ wave propagation, these changes in cytosolic Ca²⁺ would be expected to accelerate rather than retard Ca²⁺ waves. The most striking observation in Fig. 4 concerns how the interval between release site activations influences [Ca²⁺]_SR. Extending the interval from 22 to 33 ms leads to a decline in target site [Ca²⁺]_SR for medium or fast diffusion, but an increase when diffusion is slow. The results therefore suggest that if slowed propagation due to SERCA inhibition results from a longer time required for an increase in [Ca²⁺]_SR at the target site, then slow diffusion in the SR is the most plausible scenario.

We considered the possibility that three simplifications in our model could have influenced the results. One is the fact that we treat the cell as a closed system and ignore the effects of Na⁺-Ca²⁺ exchange (NCX), which will extrude Ca²⁺ from the myocyte during waves (4). Simulations that include either low levels of NCX, as in rat myocytes (20), or high levels of NCX, as in rabbit or guinea pig myocytes (20), show that including NCX has little effect on target site [Ca²⁺]_SR (Fig. S2). Two other simplifications we considered were a uniform distribution of SERCA pumps in the SR membrane, and the use of the rapid buffering approximation in the JSR. Relaxing these simplifications had virtually no effect on the model predictions (Fig. S3 and Fig. S4).

Although RyR clusters are primarily located at Z-lines, implying a longitudinal spacing of ∼2 μm, the effective distance between spark sites may be considerably shorter due to nonjunctional RyR clusters (21,22) or irregularities in cellular structure (23,24). We therefore performed additional simulations in which the spacing between release sites was 1 μm rather than 2 μm. Fig. 4 (right) shows that in this case, dynamic changes in [Ca²⁺]_SR are more dramatic, although the overall trends are the same. This further confirms that RyR sensitization by increased [Ca²⁺]_SR requires relatively slow SR diffusion.

We also performed simulations in which SERCA activity was increased rather than decreased (Fig. 5). For all values of D_Ca,SR, this condition leads to decreased cytosolic [Ca²⁺] and increased SR [Ca²⁺] at the target site. The overall trend is the same, that sensitization of unactivated RyR clusters by increased [Ca²⁺]_SR is more likely with slow than with fast SR diffusion. In a similar way, this general trend holds regardless of the exponent (η) in the equation describing SERCA uptake (Eq. 7). For η = 3, 1.78, or 0.75, fast SR diffusion leads to either a decrease or a smaller increase in [Ca²⁺]_SR at the target site (Fig. 6), whether RyR clusters are spaced 1 μm (left) or 2 μm (right) apart. Based on biochemical data, an exponent of ∼2 is generally considered most realistic (25).

Further simulations altered the rate of Ca²⁺ transfer between network SR and junctional SR, referred to here as v. This parameter influences both the speed of JSR refilling (Fig. 7 A) and [Ca²⁺]_SR at the target site (Fig. 7 B). Consistent with the effects of altering D_Ca,SR (Figs. 2 and 3), slower NSR-to-JSR transfer (decreased v) causes an increase in [Ca²⁺]_SR at the target site for either 1-μm or 2-μm spacing. Of particular interest, though, is how target site [Ca²⁺]_SR is influenced by the interplay between D_Ca,SR and v. When SR Ca²⁺ diffusion is slow, changes in v have little effect on target site [Ca²⁺]. However, when D_Ca,SR is higher, changes in v can cause more pronounced effects. This result implies that if NSR to JSR refilling is highly heterogeneous, as recent evidence suggests (26), then slow SR diffusion would allow for greater robustness in the changes in [Ca²⁺]_SR experienced at remote sites.

Discussion

In this study, we used mathematical modeling to address two current interrelated controversies in cardiac cellular physiology: 1), can sensitization wave fronts in the SR contribute to the propagation of Ca²⁺ waves?; and 2), what is the speed of Ca²⁺ diffusion within the SR? These issues were investigated by simulating Ca²⁺ release from regularly spaced RyR clusters, as would occur during Ca²⁺ waves, and monitoring the resulting changes in cytosolic and SR [Ca²⁺]. Our results indicate that, consistent with the hypothesis proposed by Keller et al. (8), local increases in [Ca²⁺]_SR can occur ahead of a propagating wave, thereby potentially sensitizing unactivated RyRs. These local increases in [Ca²⁺]_SR do not result from diffusion within the SR; instead, Ca²⁺ released from an activated RyR cluster diffuses in the cytosol toward an unactivated target site, then is taken into the SR by SERCA pumps. Slow rather than fast SR Ca²⁺ diffusion therefore promotes the accumulation of [Ca²⁺]_SR at unactivated sites. Partial inhibition of SERCA attenuates these local increases in [Ca²⁺]_SR, possibly accounting for the slower Ca²⁺ wave propagation observed under these conditions (8), although alternative explanations are possible (see below).

Work performed in recent years has provided compelling evidence for the hypothesis that dynamic local changes in SR [Ca²⁺] play a major role in the regulation of SR Ca²⁺ release. Sobie et al. proposed that local depletion of [Ca²⁺]_SR was responsible for the termination of Ca²⁺ sparks (27), and experiments performed at roughly the same time (28–30) produced data that supported this hypothesis. More recent studies have further documented the importance of local SR depletion in the regulation of both Ca²⁺ sparks (26,31) and Ca²⁺ waves (32,33). This study extends this general idea by exploring the possibility that local increases in [Ca²⁺]_SR, in addition to local decreases, may influence SR Ca²⁺ release dynamics.

Two important features of our model are that 1), SR Ca²⁺ release occurs at discrete sites; and 2), each Ca²⁺ spark is treated as a discrete, all-or-none event. This general mechanism has been termed fire-diffuse-fire in mathematical models of waves (34). In a continuum model that did not take the discrete-site nature of Ca²⁺ release into account, we would expect different predictions about local changes in [Ca²⁺]_SR. Ca²⁺ released into the cytosol through a single RyR in a continuum model will diffuse to a neighboring channel very quickly, since the channels are infinitely close. This second channel will immediately begin to open, because continuum models treat RyR open probability as a fraction between zero and one. SERCA at the second site will take Ca²⁺ into the SR at the same time that the partially open RyR is releasing Ca²⁺, and the change in local [Ca²⁺]_SR will depend on whether release or uptake is greater. Because the maximum flux of Ca²⁺ through RyRs is orders of magnitude greater than the maximum rate of SERCA activity in cardiac myocytes, it is likely that realistic models will predict Ca²⁺ depletion rather than Ca²⁺ accumulation at the second RyR. This highlights the importance of considering the fire-diffuse-fire nature of Ca²⁺ waves in mathematical treatments of this phenomenon.

It has been difficult to unambiguously determine the effects of SERCA activity on Ca²⁺ wave velocity in experiments, and apparently contradictory results are present in the literature. For instance, O'Neill et al. observed that SERCA inhibition slowed Ca²⁺ waves (35), whereas Lukyanenko et al. reported an increase in Ca²⁺ wave velocity (36). The difficulty of interpretation arises from the fact that SERCA inhibition (stimulation) is likely to cause a decrease (increase) in the level of [Ca²⁺]_SR just before wave initiation. Since wave velocity depends on the quantity of Ca²⁺ released during each Ca²⁺ spark (37), and this in turn depends on the prewave SR load, it is difficult in most experiments to delineate the separate contributions to Ca²⁺ wave velocity of SERCA activity and prewave [Ca²⁺]_SR. Keller et al. (8) modulated SERCA activity rapidly using brief flashes of ultraviolet light combined with a photosensitive SERCA inhibitor. It therefore seems likely that these authors were able to measure decreased Ca²⁺ wave velocity before substantial changes in prewave [Ca²⁺]_SR. However, decreased wave velocity due to lower [Ca²⁺]_SR, rather than to SERCA inhibition per se, must still be considered as an alternative explanation for these results.

An important limitation of our simulation approach should be mentioned. Because we impose on our model the time and location-dependent SR Ca²⁺ release flux responsible for the Ca²⁺ wave, we cannot make predictions about how the time course of Ca²⁺ release itself may be altered by different interventions. RyR gating may be modified by pharmacological agents, phosphorylation, or mutations, and this model cannot capture how these may influence the dynamics of Ca²⁺ waves. The benefit of our strategy, on the other hand, is that the conclusions do not depend on specific assumptions regarding the factors that influence RyR gating. Our results do, of course, depend on SERCA activity as well as physical parameters such as diffusion constants, Ca²⁺ buffers, and the distance between RyR clusters. Because our simulation protocol is relatively simple, we could explore the influence of the important parameters rather systematically. Although the extent of the change in [Ca²⁺]_SR in front of the Ca²⁺ wave depended on the parameter values in a particular simulation, a general trend was observed in which slow SR Ca²⁺ movements favored RyR sensitization at the target site, whereas fast Ca²⁺ movements led to depletion ahead of the wave. In future work, we plan to integrate this spatial model of multiple RyR clusters with a stochastic simulation of RyR gating to more thoroughly explore the factors that influence Ca²⁺ wave velocity, local changes in [Ca²⁺]_SR, and the transition between stable Ca²⁺ sparks and unstable Ca²⁺ waves.

In conclusion, the simulations presented in this study investigated the dynamic local changes in [Ca²⁺]_SR that occur in cardiac myocytes during Ca²⁺ waves. We can safely conclude that for the experimentally measured D_Ca,SR (10,11), it is possible that [Ca²⁺]_SR increases near unactivated RyR clusters. Moreover, slow SR Ca²⁺ diffusion seems easiest to reconcile with the experimental observations of Keller et al. (8). Although the model predicts relatively small (<8%) increases in target site [Ca²⁺]_SR, these occur at the same time that [Ca²⁺] is elevated on the cytosolic side. The combined effect on RyR open probability of these two changes in [Ca²⁺] is likely to be considerably greater than the effect of either in isolation. Future studies that combine experimental measurements with mathematical analyses will provide additional insight into the mechanisms underlying Ca²⁺ waves, and hopefully suggest strategies for preventing these potentially arrhythmogenic events.

Supporting Material

Four figures are available at http://www.biophysj.org/biophysj/supplemental/S0006-3495(10)00316-4.