Depolarization of cardiac myocytes during the action potential elicits release of Ca²⁺ from the sarcoplasmic reticulum (SR), which, in turn triggers contraction. Coupling of depolarization to SR Ca²⁺ release occurs in structures called dyads. In the dyads, T-tubules and terminal cisternae of the SR meet in junctional patches where the two membranes are separated by a cleft of ~15 nm. The small subsarcolemmal fluid compartment in the dyadic cleft has also been coined “fuzzy space” (1,2). Here Ca²⁺ must diffuse from the inner mouth of the L-type Ca²⁺ channels (LTCC) that open during depolarization of the T-tubular membrane, to reach the ryanodine receptors (RyR) of the SR on the other side of the cleft. Binding of this “trigger Ca²⁺” to RyR initiates Ca²⁺ release from the SR. This is the classical concept of Ca²⁺-induced Ca²⁺ release (CICR). The process is self-amplifying, but due to restricted diffusion, Ca²⁺ release is limited to a local spark comprising a small number of ryanodine receptors (3). It is the sum of sparks that constitutes the Ca²⁺ transient, and the size of the Ca²⁺ current (number of channels that open) that determines the size of the transient.

This picture of the dyadic cleft and CICR is complicated by the presence of several other membrane proteins, including the Na⁺/Ca²⁺ exchanger (NCX). It has recently become clear that although the NCX proteins can be found throughout the sarcolemma, some NCX proteins are localized as close to the RyR as are the LTCCs (4). This raises the question of how the presence of NCX modifies the Ca²⁺ signal in the fuzzy space. The NCX can operate in both a normal forward mode and a reverse mode. Three Na⁺ ions are transported in exchange for one Ca²⁺ ion, so that in forward mode the current (I_NCX) will be inward, and Ca²⁺ will be pumped out of the cell. A rapid rise of Ca²⁺ in the fuzzy space can therefore be effectively “buffered” by the NCX (5). On the other hand, during an action potential, the combination of depolarization and accumulation of Na⁺ near the membrane will favor reversal of the NCX. It has been a longstanding question whether reverse mode NCX can contribute to CICR (6–9). Interestingly, cardiac-specific knockout of NCX1 only leads to a small decline of cardiac contractility (10), and overexpression has little effect unless it is substantial when it leads to hypertrophy and myocardial failure (11). Hence it seems clear that in species with a high reliance on SR for the Ca²⁺ transient, NCX predominantly plays a modulatory role for the [Ca²⁺]_i in the dyadic cleft.

This article focuses on the potential role of NCX in triggering Ca²⁺ release. There are reports that in the absence of an L-type Ca²⁺ current (I_Ca), close to normal contractions can be triggered by reverse mode exchange (12,8,13,7). However, recent experiments indicate that conditions for reverse mode exchange may have disappeared as soon as 5 ms after rapid depolarization and that accumulation of Ca²⁺ in the cleft rapidly switches the exchanger into forward transport mode (14,15). After 5 ms, some repolarization has occurred during a normal action potential and local accumulation of Na⁺ might already be dissipated. Several investigators have modeled the events that take place in the dyadic cleft treating the fuzzy space as one compartment (16–20). However, it is clear that there is a diffusion distance, and it has been argued that diffusion is slow in this fuzzy space. This makes it necessary to take both number and localization of individual channels and transporters into account.

The aim of this study was therefore to examine the role of NCX in the very early events of CICR both experimentally and by mathematical modeling. A diffusion model was developed for the dyadic cleft. Specifically, we examine the conditions required for NCX to modify the trigger Ca²⁺ in the dyadic cleft. The results indicate that reverse mode Na⁺/Ca²⁺ exchange can precede I_Ca, contributing to an early rise of the trigger Ca²⁺, an effect that subsides after a few milliseconds. The contribution of NCX to the trigger Ca²⁺ depends on two conditions that will require experimental verification. First, diffusion in the fuzzy space must be several orders of magnitude slower than in water, and second, a Na⁺ channel must be present within the dyad.

Male Wistar rats (Mollegaard Breeding and Research Centre, Skensved, Denmark) weighing ~400 g were intubated and ventilated on a Zoovent ventilator (Triumph Technical Services, Milton Keynes, UK) with 68% N₂O, 29% O₂, and 2–3% Isofluran (Abbott Laboratories, Abbott Park, IL). After injection of heparin i.v., hearts were quickly excised and placed in cold buffer-1 containing (in mM) 130 NaCl, 25 HEPES, 22 D-glucose, 5.4 KCl, 0.5 MgCl₂, 0.4 NaH₂PO₄, and 0.01 μg/ml insulin (pH 7.4). Hearts were then cannulated, mounted on a Langendorff setup, and perfused retrogradely (15–25 min) through the aorta with buffer-1 containing 200 U/ml collagenase Type-II and 0.1 mM Ca²⁺. Left ventricular free wall and septum were minced and gently shaken at 37°C for 3–4 min in the same solution plus 1% bovine serum albumin, 0.02 U/ml deoxyribonuclease I (Worthington Biochemical, Lakewood, NJ). After filtration (200 μm nylon mesh) and sedimentation, the cell pellet was washed three times in buffer-1 (1% bovine serum albumin) with progressively increasing [Ca²⁺] (0.1, 0.2, 0.5 mM). Isolated cardiomyocytes were stored at room temperature until use.

Whole-cell patch clamp experiments were conducted using 1–2 MΩ pipettes. The pipette solution contained (in mM) 120 K-aspartate, 0.5 MgCl₂, 6 NaCl, 0.06 EGTA, 10 HEPES, 10 Glucose, 25 KCl, and 4 K₂-ATP. Membrane potential and current recordings were made with an Axoclamp 2B amplifier (Axon Instruments, Foster City, CA) and pCLAMP software (Axon Instruments). [Ca²⁺]_i transients were elicited at 1 Hz by action potentials or voltage-clamp steps. Action potentials were elicited in bridge-mode using a 3 ms depolarizing current, ~20% above threshold. Discontinuous voltage clamp (sample rate 8 KHz) was used to elicit 100 ms voltage steps to 0 mV from the holding potential of −80 mV. In some experiments, myocytes were field-stimulated through a pair of platinum electrodes with a voltage 50% above threshold.

[Ca²⁺]_i transients were detected by confocal fluorescence imaging using an LSM 510 scanning system (Zeiss GmbH, Jena, Germany) with a 40× water immersion objective. Fluo-4 was excited at 488 nm, and emission intensity was measured at 510 nm. Myocytes were scanned along the longitudinal axis with a 512-pixel line, every 1.5 ms. Sequential scans were stacked to create two-dimensional images with time in the x axis.

Recordings in control conditions were collected for 1 min. Cells were treated with 5 μM KB-R7942 (KBR) (Tocris/Bio Nuclear AB, Bromma, Sweden), 200 nM tetrodoxin (TTX) (Alomone Labs, Jerusalem, Israel), 5 μM nifedipine (Sigma-Aldrich, St. Louis, MO), or a combination of KBR and TTX.

(1)

Since the focus of this article was the role of the Na⁺/Ca²⁺ exchanger, it was important that both Na⁺ and Ca²⁺ concentrations were modeled accurately. Consequently, the model in Eq. 1 is used for both of these ions. The equation for the Na⁺ concentration is the simplest since there is no internal sources, i.e., f is zero in Eq. 1. The situation is more complicated for Ca²⁺ since the Ca²⁺ buffers act as both drains and sources. Two types of buffers were included, troponin and calmodulin.

The domain Ω will in our case be an area consisting of the fuzzy space between a T-tubule and an SR unit together with some of the surrounding cytosol. The boundary Γ will then consist of the membrane of the T-tubule, the membrane of SR, and artificial boundaries in the cytosol, arising from the mathematical model.

The flux of ions across the membrane was included as boundary conditions. Only the most relevant transmembrane ionic transporters were included. For Na⁺, we included the fast inward current I_Na, the Na⁺/K⁺ pump current I_NaK, and the Na⁺/Ca²⁺ current I_NCX. For Ca²⁺, the included currents were the L-type current I_Ca and the Na⁺/Ca²⁺ current I_NCX. The conversion from current to flux is explained in Eq. 4. The release current from SR, I_rel, was not included since we only studied the process leading up to CICR, but not CICR itself.

The membrane currents do not only depend upon the ion concentrations but also on the transmembrane potential and the state of the gating proteins. To compute these currents, we used the model of Winslow et al. (16), which is a set of ordinary differential equations (ODEs). In addition, the Winslow model was changed according to Weber et al. (15) by including an allosteric NCX for activation by Ca²⁺ with k₀₅ = 152 nM.

For notational convenience, we define c = [Ca²⁺]_i and s = [Na⁺]_i. The state variables of the ODE model were lumped together in a vector denoted w. This includes the transmembrane potential, the gating variables, and different ionic concentrations, excluding intracellular Ca²⁺ and Na⁺, which were modeled separately. The complete model can be written as

(2)

(3)

(4)

There are three distinct buffers types: calmodulin and high and low affinity sites of troponin so

(5)

(6)

Here [T]_tot is the total concentration of buffer type T, and [Tb] is the corresponding concentration in bound form. The rate constants are k_T⁺ and k_T⁻. The values of the constants are given in Table 1.

TABLE 1 Buffer constants used in the model were taken from Winslow et al. (16), except the rate constants for calmodulin, which were taken from Smith et al. (73)

The computational domain Ω consists of two distinct types, the bulk space, denoted Ω_bulk, and the fuzzy space, denoted Ω_fuzzy. A bar over the variable means an average and the subscript describes over which domain, e.g., is the average concentration of Ca²⁺ in the fuzzy space:

The appropriate values for the diffusion coefficients are not known and we have used a wide range of values to investigate the consequences for the mathematical model. It is clear that diffusion is slower in the cell than in water. E.g., Despa et al. (21) obtained a value for Na⁺ diffusion in the cytoplasma to be two orders of magnitude lower than in water. Crowding in the cleft space might reduce this number further. For our purpose, it is the diffusion in cleft space that is critical, and in the simulation we have used values 10²–10⁴ lower than that in water. The results are quite insensitive to the diffusion level in the bulk space and we have simply used the values from Despa et al. (21) in this domain.

According to Langer and Peskoff (22), the Ca²⁺ diffusion is half that of Na⁺. Thus, the default diffusion parameters in the bulk myoplasma was set to k_c = 0.5·10⁻⁷ cm²/s and k_s = 1·10⁻⁷ cm²/s, which is equivalent to k_c = 5000 nm²/ms and k_s = 10000 nm²/ms. This is ~1% of the diffusion in water (which for Na⁺ is 1.2·10⁶ nm²/ms). The intracellular Na⁺ and Ca²⁺ concentrations were assumed to be in equilibrium before the onset of the stimulus, i.e., the concentrations in the bulk space and the fuzzy space were equal. For Na⁺, the initial condition was set to 12.0 mM and for Ca²⁺, the initial condition was 0.1 μM. The rest of the initial conditions followed the model by Winslow et al. (16).

(7)

(8)

TABLE 2 Channel densities used in the model

Here, Γ_i are all the channel openings for current i. The set of Γ_i forms a partitioning of Γ, i.e., ⁿ_{i = 0}Γ_i = Γ and Γ_i ∩ Γ_j = Ø if i ≠ j.

Let us now verify that the flux of ions over the boundary causes a change in the concentration, consistent with the original compartmental formulation. From mass conservation and Fick's law we have that

(9)

(10)

Here we have defined A_i to be the area of Γ_i. Observe that A_i = m_i·a_i. Using Eq. 7, we have that

(11)

(12)

The names in capital letters refer to procedures that solve the subproblems. For the ODE problem, we use an explicit fourth order Runge-Kutta method with internal adaptive time stepping. The two procedures CALCIUM and SODIUM are implicit partial differential equation (PDE) solvers. Both of these problems are nonlinear due to the boundary condition, i.e., the influx over the boundary depends nonlinearly on the concentration. For the CALCIUM problem, there is additional nonlinearity due to the buffering. We only describe the CALCIUM algorithm, as SODIUM might be seen as a special case of CALCIUM.

Equation 2 has the structure

(13)

To ensure numerical stability we use a implicit scheme to discretize this equation in time:

(14)

This is a nonlinear problem in cⁿ⁺¹. There are several approaches to solving these kinds of equations. We have chosen to use successive substitution due to its simplicity and robustness. Another possible choice is Newton's method. It is typically faster but requires expressions for the derivatives, which in our case will be quite complex. Also the convergence depends upon a good initial guess for the solution.

Using subscript “l” to represent the iteration counter, the iteration is

(15)

Thus for each iteration, we solved Eq. 15, which is a linear problem in c_l+1. The finite element method is used. First a weak form of Eq. 15 is derived and the subsequent problem has been implemented using Diffpack (23).

The T-tubules are found nearly every 1.8 μm along the Z-line and enter the heart cell perpendicular to the surface membrane. The T-tubule invaginations contain extracellular fluid (24). The data on T-tubule dimensions are highly dependent on the species and also vary considerably within a single species. Most studies have been conducted with conventional transmission or scanning electron microscopy. In the guinea pig, diameters in the range of 200–360 nm (25) and 170–850 nm (26) have been found. Rat ventricular myocytes have been the subject of several studies, with T-tubule diameters ranging from 70 to 114 nm (27–29) . Soeller and Cannell (30) point out that methodological problems with conventional electron microscopy may cause erroneous estimates. In our study, we have used their T-tubule diameter estimate of 255 nm, which was obtained using two-photon laser scan microscopy on living rat myocytes.

The T-tubule may be flattened or rounded in the junctional regions, e.g., see Nakamura et al. (27) and Forbes and Sperelakis (31). Junctional sarcoplasmic reticulum usually forms extensive flat cisternae. Zhang et al. reported an average height of the cisternae of 18.6 nm in mice myocardium (32).

Quantitative data on the size of the junctional patches are scarce. Assuming a circular structure, Franzini-Armstrong et al. (29) estimated the size of a junction to be 75,690, 224,547, and 107,648 nm² in dog, rat, and mouse myocardium, respectively. In chick myocardium, peripheral couplings have been found to be 16,600–17,700 nm² (33).

Zhang et al. (32) have reported an average length of 334 nm of junctions in the mouse ventricle. Assuming a circular morphology, this would correspond to a patch size of 87,571 nm², which is in reasonable accordance with the older data of Franzini-Armstrong et al.

In our model, SR was modeled as a cap with 18.6 nm thickness and a mean diameter of 334 nm. The cap is not flat but curves around the T-tubule. The thickness of the fuzzy space was set to 15.5 nm (34), i.e., the distance from the T-tubule to SR. The RyR, which consists of four identical monomers, is 29 × 29 × 12 nm and protrudes into the gap between the junctional sarcoplasmic reticulum and the T-tubule (35). The receptors were organized with a center to center distance of 31.5 nm (36).

Despa et al. (37) found that 63% of the NCX and 59% of the Na⁺/K⁺-pump (NKA) resides in the T-tubule with a corresponding T-tubule/sarcolemmal density of 3.6 and 3.1, respectively. By using a sarcolemmal NCX density of 400/μm² (38), NKA density of 750/μm² (39), Na⁺ channel density of 7.5/μm² (40), and LTCC density of 6/μm² (41), we calculated the channel spacing to be 29.5 nm and 16.5 nm for NCX and NKA. The density of the fast Na⁺ channel is thought to be uniformly distributed throughout the whole cardiomyocyte (42), which gives the possibility that one or no Na⁺ channel resides in the dyad.

By assuming symmetry around the T-tubule, the three-dimensional problem could be reduced to a two-dimensional computation. Fig. 1 shows a schematic of the geometry. A channel opening in the symmetry plane represents a band of channels at that radial distance from the axis of symmetry. Channel placement is made such that the distance between these bands corresponds to the typical interchannel distances stated above. For example, when the RyRs are spaced with 31.5 nm on the SR surface, they are represented by six sites in the model. It is convenient to refer to these sites by the angle they make in radial coordinates; these are 33.2, 45.8, 58.4, 71.1, and 83.7° (Fig. 1). The NCXs and the NKAs are distributed in a similar fashion on the T-tubule surface, with the spacing 29.5 nm and 16.6 nm, respectively. This gives a total of six NCX sites and nine NKA sites. One LTCC is placed at 90°, i.e., at the axis of symmetry.

FIGURE 1 Schematic picture of channel placement in the fuzzy space. The blue channel represents the LTCC, the black and green channels represent NCX and RyR, respectively. The red channel represents the Na+ channel (see text for details).

The LTCC, RyRs, NCXs, and NKAs were fixed with the above geometry. The Na⁺ channel, however, was subjected to three scenarios: 1), no Na⁺ channel in the dyad (due to the small volume in the fuzzy space, we will not exclude this possibility), 2), one Na⁺ channel midway between two NCXs and, 3), one Na⁺ channel adjacent to an NCX.

The bulk space surrounding the T-tubule, SR, and fuzzy space was set to 4000 nm × 4000 nm. Na⁺ and Ca²⁺ ions were allowed to diffuse across the artificial cytosolic boundaries.

FIGURE 2 Line scan images taken before (A) and during (B) KBR application show that NCX inhibition increases SR Ca2+ release latency. Mean changes in latency during treatment with 5 μM KBR, 200 nM TTX, and KBR + TTX are shown in C.

In other experiments, we attempted to indirectly inhibit reverse NCX using low-dose TTX. Maier et al. (44) have reported that the highly TTX-sensitive brain-type Na⁺ channels are preferentially localized in the T-tubules in cardiomyocytes and can be blocked by a low-dose (200 nM) of TTX, which does not affect cardiac-type Na⁺ channels. Thus, locally inhibiting Na⁺ influx in the T-tubules should prolong latency of Ca²⁺ release. We observed that treatment with 200 nM TTX prolonged the latency of Ca²⁺ transients triggered by action potentials. Maximal effects were reached after 30 s of TTX application when mean latency values were increased by 1.1 ± 0.2 ms (P < 0.05, n = 4) from control values (Fig. 2 C). However, in cells pretreated with KBR, treatment with TTX did not alter latency values, suggesting these drugs increase latency by the same mechanism. As well, alterations in latency did not result from altered dV/dt max values for the upstroke of the action potential (control = 241 ± 47 V/s, n = 8; TTX = 234 ± 78 V/s, n = 3, P = NS; KBR = 204 ± 37 V/s, n = 6, P = NS; KBR + TTX = 211 ± 40 V/s, n = 6, P = NS). Other experiments were conducted with voltage-clamp steps in which the NCX would be expected to be directly activated by membrane depolarization. In these experiments, treatment with 200 nM TTX did not alter latency of Ca²⁺ release during 4 min of application (control = 2.4 ± 0.4 ms, 4 min = 2.7 ± 0.4 ms, P = NS, n = 4). This indicates that the NCX is likely involved in the TTX effects observed during stimulation with action potentials. Blockade of the L-type Ca²⁺ channel with 5 μM nifedipine did not alter latency in field-stimulated cells (control = 2.4 ± 0.2 ms, nifedipine = 3.2 ± 0.3, n = 5, P = NS).

FIGURE 3 Traces from the underlying set of ODEs generated from a modified model by Winslow et al. (16) (see text for explanation). Membrane potential (upper left panel), [Ca2+]ss (upper right panel), INa (middle left panel), ICa (middle right panel), (more ...)

In this section, we will present a set of simulations that will demonstrate under which circumstances the model predicts that NCX will contribute to CICR.

Fig. 4 shows the Ca²⁺ distribution in the fuzzy space 1.1 ms after the initial stimulus has been applied. By gradually increasing Ca²⁺ influx into fuzzy space using a formulation of the model in which the RyRs were activated, we obtained a threshold value for CICR to be 0.5 μM in the fuzzy space. This value was used throughout the whole set of simulations. In the figure, areas with deep red color have reached this level. We see that Ca²⁺ has entered through the L-type channel, creating a pool of high Ca²⁺ concentration on the intracellular side. Note that Ca²⁺ is also entering via the exchangers. Even though no Na⁺ channel is present, the basal concentration of intracellular Na⁺ concentration (12 mM) is sufficient to generate a small reverse current in the early phase of the action potential. In this simulation, the fuzzy space diffusion is assumed to be three orders of magnitude lower than in water. The concentration of Ca²⁺ at the SR side of the fuzzy space has not yet reached the threshold value necessary for CICR occur.

FIGURE 4 Snapshot of Ca2+ concentration in the fuzzy space with kc = 500 nm2/ms. Values are in millimolars, i.e., the color range is 0.1–0.5 μM. Note the elevated Ca2+ concentration in the fuzzy space around the NCXs.

To demonstrate the temporal evolution of the concentration, we have in Fig. 5 plotted the time development of [Ca] at the five RyR sites in our model in the absence of any Na⁺ channel and with [Na⁺]_i equal to 12 mM. In the figure, the RyR sites are referred to by the angle of their location with respect to the x axis in Fig. 1, e.g., 90° corresponds to the y axis where the L-type channel is located. The larger the angle the closer the RyR is to the L-type channel. The thin blue horizontal line represents threshold for CICR. We see that the RyR closest to LTCC, i.e., the one labeled 83.7° is triggered just after 1 ms, in accordance with Fig. 4. The other sites are triggered shortly thereafter. The concentrations at the other sites rise more slowly, indicating that Ca²⁺ also diffuses out into the bulk space instead of accumulating in the fuzzy space. Note that the actual Ca²⁺ release is not included in this simulation. If it were, then the release from the first receptor would cause a sufficient increase in the fuzzy space concentration to also trigger the rest of the sites. As mentioned in the Materials and Methods section, we are studying the role of the NCX in CICR, not CICR in itself. The basic result from Fig. 5 is that the conditions were sufficient for CICR to occur at 1 ms. In the lower panel of Fig. 5, the exchanger currents at the corresponding sites are plotted. There is only a very small transient phase of reverse mode NCX current before the exchanger sitting close to the LTCC switches into strong forward mode.

FIGURE 5 (Upper panel) Time course of Ca2+ at the different RyR locations with kc = 500nm2/ms and no Na+ channel present in the dyad. The thin blue horizontal line is the activation threshold for the RyRs. (Lower panel) Time course for (more ...)

We will now turn to the two scenarios where a Na⁺ channel is adjacent to an NCX unit and when it is located midway between two NCX units. Fig. 6 shows the case with a Na⁺ channel located adjacent to the NCX unit closest to the LTCC, as in Fig. 1. Comparing Fig. 6 with Fig. 5, we see that the Ca²⁺ concentration rises more rapidly and to a higher level when the Na⁺ channel is present, the reason being that the elevated Na⁺ concentration greatly enhances the reverse mode of the exchanger located closest to the Na⁺ channel. This can be seen by comparing the lower panels of Fig. 5 and Fig. 6. However, the difference in timing is quite small; the CICR level is reached 0.3 ms earlier with the Na⁺ channel in place. With the channel placed further away from the NCX channel the difference was even less. Plots from this latter simulation are not shown. In the simulation above, the maximum [Na⁺] seen by the NCX was 74 mM.

FIGURE 6 (Upper panel) As in Fig. 5, but with a Na+ channel present in the dyad. In the lower panel, we can clearly observe the effect of including a Na+ channel in the dyad. The NCX current peaks very rapidly in reverse mode before it return to (more ...)

The diffusion constant in the bulk space of the cytosol is fixed in all the simulations in this article. The assumption that the diffusion constant in fuzzy space was three orders of magnitude lower than in water was used in the simulations presented above. If we instead assume equal diffusion in the two compartments, the results with respect to CICR triggering are virtually unchanged, except that the more distal RyR sites are triggered earlier.

The appropriate value for the diffusion constant in fuzzy space is uncertain, but if it is larger than 1000 nm²/ms, our simulations suggest that the NCX does not play an important role in CICR. Below we shall see that provided the diffusion value in fuzzy space is lower, the NCX mechanism becomes important. With slow diffusion, the Ca²⁺ entering through the LTCC will use a longer time to reach the RyRs, thus causing a delay in CICR through this mechanism. On the other hand, with reduced diffusion, more Na⁺ will build up around the Na⁺ channel and enhance the reverse mode of neighboring NCX units. Thus, we lowered the diffusion constants in the fuzzy space to four orders of magnitude lower than in water (corresponding to k_c = 50 nm²/ms and k_s = 100 nm²/ms). The quantitative effect of this is seen clearly in Fig. 7. In this case, the maximum [Na⁺] seen by the NCX was 177 mM. Triggering through LTCC alone (Case 1 in Fig. 7) occurs at 6.5 ms, whereas it occurs at 1.4 ms in the case where the Na⁺ channel lies adjacent to the NCX unit (Case 3). When the Na⁺ channel lies farther from the NCX unit (Case 2), the exchanger current is weaker and the triggering occurs at 5.2 ms. Depending on the channel location, we thus get a time lag of up to 5 ms in the absence of a Na⁺ channel. In Fig. 8, NCX currents for the three cases are drawn. In Case 3, a strong, but short lasting reverse mode is present. This should be compared to the blue trace in the lower panel of Fig. 6, which shows the current under the same conditions but with 10 times the rate of diffusion. In the low diffusion case (Fig. 8), the reverse mode is strong due to larger accumulation of Na⁺, and the forward mode is reduced, due to a longer lasting Na⁺ transient. In Fig. 9 we have repeated the experiment, but with the LTCC blocked. The only Ca²⁺ source is the NCX. With no fast Na⁺ channel present (Case 1), there is not sufficient driving force for the exchanger. With the Na⁺ channel lying adjacent to the NCX unit (Case 3), the latency is quite similar to the situation with an open LTCC. When the Na⁺ channel lies farther from the NCX unit (Case 2), the exchanger current is too weak on its own to trigger release. Simulations with the reverse mode of the NCX blocked have also been performed (not shown). The effect is very similar to not having a Na⁺ channel present. In all cases, the rise in [Na⁺]_i was only transient, with no buildup from beat to beat.

FIGURE 7 Calcium concentration at the RyR site closest to the LTCC using kc = 50nm2/ms The right-hand panel shows the channel placement for the three different cases. The blue channel at the y axis represents the LTCC, the black channel at 30 nm is the (more ...)

FIGURE 8 Exchanger current at the RyR site closest to the LTCC. The right-hand panel is repeated from Fig. 7.

FIGURE 9 LTCC blocked. The exchanger current is very similar to the tracings in Fig. 8 and thus not repeated. The symbols are as described in Fig. 7.

In principle, our model predicts that NCX can contribute to early Ca²⁺ release from RyR in rat cardiomyocytes. However, after a few milliseconds, the reverse mode is rapidly turned off and the NCX switches into equilibrium or forward mode. The forward mode will be amplified by the subsequent Ca²⁺ release. However, Ca²⁺ release was not included in our model. We focused on the very early trigger Ca²⁺ that enters the cell through LTCCs and NCX. The model corroborates the experimental finding that inhibition of reverse mode exchange by KBR delays the onset of the Ca²⁺ transient by 3–4 ms. The conditions for this contribution of NCX to early Ca²⁺ release are twofold. First, a Na⁺ channel must be present in the dyad very close to the NCX protein, and second, diffusion in the dyadic cleft (fuzzy space) must be several orders of magnitude slower than in water, and slower even than in the bulk cytoplasm. Thus, CICR seems to depend on a multi-molecular complex involving membrane proteins of both the SR and T-tubular membranes between which ionic gradients can transiently exist due to slow diffusion. The model also predicts that NCX alone in the absence of I_Ca can trigger Ca²⁺ transients as shown by several investigators (12,8,13,7).

The distribution of the various channels within the dyad is a matter of debate. Scriven et al. (48), using wide-field fluorescence microscopy and image deconvolution, found that even though the fast Na⁺ channels and the NCXs were distributed largely within the T-tubules, they were unable to detect any colocalization to the RyRs. On the other hand, in a more recent study by Thomas et al. (4), using immunogold labeling and electron microscopy, they found that some NCX proteins were as close to the RyRs as the LTCCs and thus, could contribute to CICR. The results from Thomas et al. served as a basis for our geometric data to perform the PDE modeling within the dyad.

This study is one of a few that examine the diffusion in the fuzzy space in a three-dimensional model. The model includes sets of ODEs and PDEs to solve both the temporal and spatial distribution of the diffusing ions in the fuzzy space. Langer and Peskoff (22) pioneered this approach in their excellent article. However, we have used a different geometry and the underlying currents (I_Ca, I_Na, and I_NCX) were taken from the Winslow model providing more accurate formulations. Of particular note, Langer and Peskoff have used I_Ca = 0.3 pA (1 ms ramp) and I_Na = 2 pA (trapezoidal pulse), which is not in accordance with data showing that peak I_Na is ~50 times higher than peak I_Ca (see, e.g., Winslow et al. (16)). Furthermore, they focused on SR-Ca²⁺ release through the RyR complex, whereas our focus is on Ca²⁺ fluxes that trigger release. Also, we have tested effects of various channel and transporter localizations as well as various diffusion constants for Na⁺ and Ca²⁺ in the fuzzy space. Indeed, as we can see in Fig. 7, the [Ca²⁺] elicited from LTCC alone reached the threshold value of 0.5 μM within 6.5 ms, whereas in combination with reverse NCX, the same concentration was reached at 1.4 ms. The conditions for this contribution of the exchanger were a close association between NCX and a Na⁺ channel in combination with slow diffusion.

Shannon et al. (55) developed a detailed mathematical model for Ca²⁺ handling and ionic currents in the rabbit ventricular myocyte, but only modeled diffusion between different compartments, not in the restricted fuzzy space. They used a diffusion constant for Na⁺ from the dyadic cleft to the submembrane compartment (defined as a space beneath the sarcolemma through the whole cell) very close to Na⁺ diffusion in water. Thus, their diffusion constant between these compartments were two orders of magnitude higher than the bulk diffusion from the study of Despa et al. (37). On the other hand, it might not be correct to compare these constants since they are from partially different compartments. Our focus, however, was diffusion in the fuzzy space alone, i.e., one three-dimensional compartment, which included solving nonlinear PDEs. The model predicts that diffusion constants for Na⁺ and Ca²⁺ must be considerably slower than previously thought, in accordance with the experimental data and in support of the “fuzzy space” hypothesis of Lederer et al. (1).

Our model showed that with the presence of the fast Na⁺ channel in the dyad, the diffusion constant in fuzzy space had to be four orders of magnitude slower than in water, i.e., 100 nm²/ms for Na⁺ and 50 nm²/ms for Ca²⁺ to reproduce the experimental data. With higher diffusion rates, not enough Na⁺ would accumulate locally to allow for reversal of the exchanger. A word of caution is required for the estimate of the diffusion constant for Ca²⁺. A large fraction of cytosolic Ca²⁺ and probably also in the dyadic cleft is bound to buffers. As has been reported for protons, buffers will slow the apparent diffusion constant (56). We have no direct data to substantiate that the diffusion of Ca²⁺ is 50% of that for Na⁺. It may be even slower. Our result strongly indicates the presence of a fuzzy space with restricted diffusion in cardiac ventricular cells.

There may be multiple reasons for the slow diffusion in the fuzzy space. The dyadic cleft seems to be filled with proteins that are part of the signaling cascades or serve to stabilize the adjacent channel proteins both in the T-tubule and in the SR membranes. Hence it is quite likely that macromolecular crowding as described by Ellis (57) in a recent review contributes significantly to slowing diffusion. Hence, tortuous “electrolyte pathways” will be present (58). Also, the protrusion of the RyR contributes to the macromolecular crowding, and unless the Ca²⁺ binding sites are exactly opposite to the LTCC, there will be a longer distance for the ions to move. Also, there is little knowledge of the structure of the water molecules between the proteins. Various degrees of crystallization can importantly influence the diffusion of solutes.

We chose to simulate a dyad with only one LTCC. It is possible that there are more LTCCs within a single dyad, and Bondarenko et al. (61) in their model have recently simulated graded Ca²⁺ release from the SR by including a variable number of LTCCs in the dyads. They point out that there may be a large heterogeneity between dyads, and that this may explain graded release. It is quite possible that there is dyad heterogeneity also with regard to both presence and location of NCX as well as of Na⁺-channels. Hence our model may have restricted applicability to a small subpopulation of dyads, although the experiments with inhibition of reverse mode NCX point to early trigger of Ca²⁺ release as a general phenomenon achieved by the NCX.

Another important factor is the formulation of the NCX kinetics. Luo and Rudy (59) presented a model for the NCX, which was also adapted by Winslow et al. (16). Other formulations of the NCX have also been proposed. Weber et al. (63) included allosteric regulation of the NCX. Again, we used the original current expressions from Winslow et al. (16) but modified it to include an allosteric component for activation by Ca²⁺ with k₀₅ = 152 nM.

Interestingly, the effect of including the Na⁺/K⁺ pump in the model was surprisingly minimal since a very high number of pump molecules close to the NCX would be required to dampen the local rise of Na⁺. In a study by Silverman et al. (53), no transient activation of Na⁺/K⁺ pump current was observed during the first 10 ms after the onset of I_Na in isolated ventricular myocytes from guinea pigs. However, the Na⁺/K⁺ pump will be very important in setting the average Na⁺ concentration, and elevation of diastolic [Na⁺] will amplify the contribution of the NCX to trigger Ca²⁺ release. It may be of interest that James et al. (69) suggested that the two isoforms of the Na⁺/K⁺ pump may be spatially and functionally separate. They further claim that the α1 isoform, which is the dominant isoform in cardiac cells, has no influence on excitation-contraction coupling and NCX. Hence, it may be important in future models to study more closely various placement of the two isoforms in cardiac cells.

The experimental data showed that with a blockade of the reverse NCX by KBR the latency for SR Ca²⁺ release was increased by 3–4 ms compared to the control situation. Also, a partial inhibition of the fast Na⁺ channel by TTX revealed an increased latency of >1 ms compared to control, which also supports that inhibition of reverse mode NCX leads to a more pronounced lag between depolarization and Ca²⁺ release. The mathematical model constitutes an SR unit wrapped around a T-tubule. The mathematical formulation was a nonlinear version of the diffusion equation derived from Fick's law. We modified the currents from the model of Winslow et al. (16) to obtain a rat AP, thus avoiding the prominent spike and dome AP typically present in larger mammals. The model is very robust and tolerates any changes in time step, channel placement, and diffusion constants. Due to the complicated equations, the time for one run (10 ms) ranged from 10–30 min depending on the time step. The equations were solved with the Diffpack program (23), running on a Linux operating system.

Taken together, the simulation results show that to reproduce the experimental data, the diffusion rate in the fuzzy space must be in the range of four orders of magnitude lower than in water. In addition, provided a Na⁺ channel is present in the dyad close to an NCX molecule, the trigger Ca²⁺ from the reversal of the NCX will precede the Ca²⁺ from the LTCCs in reaching the RyRs. Thus, NCX is important for CICR. With a normal operating NCX, the SR Ca²⁺ release is up to 5 ms faster compared to a situation where the reverse NCX is not contributing to trigger Ca²⁺. Hence, both the experimental and mathematical results clearly indicate the presence of a fuzzy space with restricted diffusion and an important role for local Na⁺ provided by I_Na in reversing the NCX.

The authors gratefully acknowledge Morten Eriksen and Carsten Lund for animal care and Fredrik Swift for isolation of cardiomyocytes.

This study was supported by grants from the Research Council of Norway and the Anders Jahres Fund for Promotion of Science and Research Forum, Ullevaal University Hospital.