INTRODUCTION

Top Abstract Introduction Results Discussion References

We have previously modeled (Peskoff et al., 1992; Langer and Peskoff, 1996) the diffusion and binding of Ca in the region of the cell between the lateral cistern of the sarcoplasmic reticulum (SR) and the inner sarcolemmal (SL) membrane in cardiac ventricular cells. We have designated this region the "diadic cleft." In the earlier model, Ca is assumed to enter the diadic cleft from the SR "feet," from the Ca channel and/or via "reverse" Na/Ca exchange. Subsequently, most of this Ca diffuses from the cleft to the sarcoplasm, with a small fraction transported from the cell by Na/Ca exchange. Both components are markedly affected by Ca binding on the inner sarcolemmal surface (Post and Langer, 1992; Peskoff et al., 1992; Langer and Peskoff, 1996). In the computation, the boundary condition at the outer border of the cleft was assumed fixed at 100 nM. This did not introduce any significant error because the much higher, millimolar values of [Ca] that were predicted in the interior of the cleft were insensitive to the [Ca] assumed at the boundary, as long as it was small compared to the computed [Ca] inside the cleft.

In the present paper, we extend the model beyond the cleft into the remainder of the sarcomere, where [Ca] is in the micromolar range. The [Ca] at the cleft border, along with the [Ca] beyond the border in the sarcomere outside the cleft, now must be computed. The fixed 100 nM boundary condition of the earlier model is replaced by the condition that the [Ca] is continuous across the border, and the Ca flux leaving the cleft equals the Ca flux entering the region of the sarcomere beyond the cleft. Except for this change in the boundary condition, the region inside the cleft, assumed to have the idealized shape of a circular disk of 200 nm radius, is modeled as earlier. The one-half sarcomere is assumed to be a circular cylinder of 500-nm radius, 1000 nm length (Z-line to M-line), with the cleft disk located coaxially at the Z-line. In the region outside the cleft it is assumed that the magnitude of the Ca diffusion coefficient is between that of free solution and that in the cleft. Outside the cleft, the diffusion equation is coupled to the kinetic equations for binding to calmodulin and troponin, and when present, to the fluorescent dye fluo-3, using experimental values for the forward and backward rate constants. Inside the cleft, as earlier, we treat the binding reactions as instantaneous. We assume zero flux leaving the sarcomere through its ends or its cylindrical surface. Reuptake by the longitudinal SR is assumed to occur just inside the cylindrical surface of the sarcomere.

The computed spatial and temporal distribution of Ca allows us to relate an observed Ca distribution in the sarcomere to a particular source within the cleft. We also relate [Ca] measurements using fluorescent dyes to the condition without dye. We find that the concentrations predicted by the computation, particularly close to the border of the cleft, are considerably higher than have been observed. This can be explained by the lowering of the [Ca] by the presence of dye, the finite response time of the fluorescence, and the limited spatial resolution of the measurements. We also gain insight into the origin of measured Ca "sparks" by setting the magnitude of release by a ryanodine receptor within the cleft to yield a response in the sarcomere resembling the spark magnitude and configuration.

Initially we assume an SR Ca release sufficient to produce maximum force development (Fabiato, 1985). This is 2 × 10¹⁹ moles/cleft, which represents >150 µmol Ca/liter accessible cell water. This Ca is released over 10 ms from a source equivalent to a single ryanodine receptor (Cheng et al., 1996) at the center of the diadic cleft space. The subsequent Ca concentration, [Ca], is followed inside the cleft and in both radial and axial directions over the half-sarcomere (0.5 µm radius; 1 µm to the M-line) for 200 ms. In addition, we take simultaneous "snapshots" of free [Ca] and total Ca as it exists in the sarcomere at selected times (5, 10, 20, 50 ms) after release. These "snapshots" are three-dimensional representations of intrasarcomere [Ca], as might be revealed by high-speed digital imaging microscopy (Isenberg et al., 1996). We then calculate the percentage of SR Ca release that diffuses to the cytoplasm from the cleft; is exchanged via Na/Ca exchange, and is free and bound in the cleft over the 200 ms after release. Similarly, we also calculate the percentage of Ca entering the sarcomere from the cleft which is free, bound to troponin and calmodulin, and resequestered by the SR over the same time period. All of these distributions assume a 10-ms SR Ca release of 2 × 10¹⁹ moles Ca at the cleft center. We then reduce the 10-ms release by 50% to 1 × 10¹⁹ moles which, according to Fabiato (1985), would produce about 30% maximum force. We illustrate the effect of this reduction on [Ca] in the radial and axial directions and on the instantaneous intrasarcomeric distribution, as previously.

Finally, we examine a number of special cases: 1) the effect of adding 100 µM fluo-3 (Minta et al., 1989; Kao et al., 1989) on the sarcomeric [Ca] profile; 2) the [Ca] profile in the cleft and sarcomere after a 1-ms 0.3 pA Ca entry into the cleft center from a Ca channel (this is a revision of Langer and Peskoff, 1996); 3) SR Ca release of an amount (4 × 10²⁰ moles) and duration (2 ms) predicted to produce a sarcomeric "Ca spark" (Santana et al., 1996; Cheng et al., 1996) in both the absence and presence of fluo-3 (the [Ca] profiles in the cleft and sarcomere are followed for 50 ms after the time of release); 4) the effect on the cleft and sarcomeric [Ca] profiles of dividing 2 × 10¹⁹ moles Ca release equally between cleft center and corbular SR (CSR) placed at the sarcomere outer boundary at an axial distance 400 nm from the cleft (Jorgensen et al., 1985; Dolber and Sommer, 1984).

	MATHEMATICAL MODEL

Geometry of the sarcomere

The "diadic cleft" is the narrow region between the lateral cistern of the sarcoplasmic reticulum (SR) and the inner sarcolemmal membrane (SL) of a cardiac cell. It contains "L" Ca channels, through which Ca enters from the interstitium, and the "feet" of the SR, from which Ca is released. The feet occupy about two-thirds of the volume of the cleft (Wibo et al., 1991; Langer and Peskoff, 1996). The SL contains Ca-binding sites of both low and high affinity (Post and Langer, 1992) and Na/Ca exchangers (Frank et al., 1992). In the present paper we extend our earlier model for Ca movement inside the diadic cleft to compute the [Ca] as it varies with position and time not only inside the cleft, but also on the boundary of the cleft, and in the remainder of the sarcomere outside the cleft. We use cylindrical coordinates (r, z) with the plane of the Z-line at z = 0 (Fig. 1). There is a half-sarcomere extending from the Z-line to an adjacent M-line, and its mirror image (not shown in Fig. 1) is extending to the M-line in the opposite direction. Ca diffusing out of the cleft splits equally between the two half-sarcomeres. The half-sarcomere is modeled as a circular cylinder of radius b = 0.5 µm and length l = 1.0 µm. The cleft, which is modeled as earlier (Langer and Peskoff, 1996), as a circular disk of radius a = 0.2 µm and thickness h = 12 nm, is located at a Z-line with its center at (r, z) = (0, 0), coaxial with the sarcomere.

View larger version (29K): [in this window] [in a new window]   FIGURE 1   Geometry of the model, showing one-half of a sarcomere extending from a Z-line to an adjacent M-line. The position in the sarcomere is expressed in cylindrical coordinates (r, z), with the plane of the Z-line defined as z = 0. The half-sarcomere is modeled as a circular cylinder of radius b = 0.5 µm and length l = 1.0 µm; the cleft is modeled as a coaxial circular disk of radius a = 0.2 µm and thickness h = 12 nm, with its center at (r, z) = (0, 0). The eight points shown in the z = 0 plane at differing radial locations (four points inside the cleft, one point on the cleft boundary, and three points outside the cleft) and the five points shown at r = a = 200 nm at differing axial locations are points at which graphs of [Ca] versus time will be shown. The shaded plane is the surface over which 3-D snapshots of the spatial distribution of [Ca] in the sarcomere outside the cleft will be shown.

The cleft is actually wrapped around a T-tubule and is irregular in shape; in the model it is assumed to be flattened into a circular disc of uniform thickness straddling the plane of the Z-line. We limit our consideration to rotationally symmetrical cases (independent of the azimuthal angle). Because diffusion across the transverse dimension a of the cleft is orders of magnitude slower than diffusion across the longitudinal dimension h (partly because h  a and partly because of the slowing of Ca movement in the r-direction by the presence of the SL binding sites), it is permissible to ignore the z-dependence and consider [Ca] within the cleft to be dependent only on the radial coordinate, r, and time, t. Outside the cleft it depends on r, z, and t.

The region of the sarcomere outside the cleft is modeled as a homogeneous medium with an effective diffusion coefficient lower than in free aqueous solution, but higher than in the cleft. Beyond lowering of the effective diffusion coefficient, the specific effects of geometric obstructions to Ca movement, for example, the mitochondria, are ignored. In particular, the volume occupied by the cleft itself, which extends 6 nm into each half of the sarcomere, is ignored, so that, for |z| < 6 nm and 0  r  200 nm, there is one value of [Ca] inside the cleft, denoted U(r, t), and one value outside the cleft, denoted C₀(r, z, t).

For the finite-difference computation, the region inside the cleft is divided into 80 annular elements. The continuous radial variable r is replaced by the discrete variable r_i = ir_cleft, with 0  i  80 and r_cleft = 2.5 nm. The region outside the cleft is divided into 20 annular elements in the radial direction and 40 slices in the axial direction. Outside the cleft the continuous variables r and z are replaced by r_j = j r and z_k = k z, respectively, with 0  j  20, 0  k  40, and r = z = 25 nm.

Diffusion and binding in the diadic cleft

The [Ca] inside the cleft, U(r, t), satisfies the nonlinear partial differential equation (Peskoff et al., 1992; Langer and Peskoff, 1996)

(1)

In Eq. 1, the second and third terms in square brackets times U/t are the number of moles of Ca per unit time per unit volume released from low- and high-affinity binding sites on the sarcolemmal surface. This is obtained by assuming that the bound Ca is in instantaneous equilibrium with the free Ca in the cleft, with the number of moles of bound Ca per unit volume given by

(2)

where N₁ and N₂ are the number of moles of low- and high-affinity sites per unit area and K₁ and K₂ are the dissociation constants of the low- and high-affinity sites (Table 1).

D_cleft, the diffusion coefficient inside the cleft, is assumed to be 0.1 µm²/ms, about one-sixth of the free aqueous diffusion coefficient (Langer and Peskoff, 1996). This reduction includes the effect of obstruction (Crank, 1975) to Ca movement by the feet of the SR, which occupy about two-thirds of the volume of the cleft.

J_SR is the Ca flux density into the cleft from the SR. We compute [Ca] as a function of position and time for two Ca inputs to the cleft from the SR: for a 10-ms pulse of magnitude corresponding to the SR release for near-maximum force and for an SR release of half that amount. We limit our consideration to cases where there is no azimuthal angular dependence, and assume that the spatial distribution of the Ca source is constant over the area of a 20-nm-radius circle coaxial with the sarcomere and cleft. A single Ryanodine receptor is approximately a 29 × 29 nm square (Radermacher et al., 1994).

The amount of Ca released to achieve near-maximum force is 2 × 10¹⁹ per cleft for each contraction cycle (Fabiato, 1985). This amount is assumed to be released uniformly over an idealized radial cross section of 20 nm at a constant rate, starting at t = 0 and ending at t = 10 ms. In terms of current, this is 3.86 pA per cleft for 10 ms or, in terms of number of Ca ions, a total release of 125,000 ions per cleft.

The flux density for the release from the SR in Eq. 1, therefore, is

(3)

where I_SR = 2 × 10²⁰ or 1 × 10²⁰ moles/ms in the two examples we will consider, and q = 20 nm and _SR = 10 ms.

J_chnl is the Ca flux density into the cleft from a Ca channel. We assume a 1-ms pulse corresponding to the Ca current from a single Ca channel. The radial extent of the Ca channel source is taken to be the minimum amount possible in the computation, covering the central, i = 0 element, which extends from the cleft center to a radius one-half of the 2.5-nm radial step size within the cleft, or 1.25 nm. This is much greater than the actual channel radius, so that the computation will yield a [Ca] much less than the actual [Ca] at distances less than 1.25 nm from the channel. The flux density for influx from the Ca channel is then

(4)

where I_chnl = 0.3 pA, p = 1.25 nm, and _chnl = 1 ms.

J_xch is the inward Ca flux density for the Na/Ca exchanger (negative for the exchanger operating in the "forward" direction with an outward Ca flux). We estimate 100 Na/Ca exchangers per cleft (Langer and Peskoff, 1996) or N_xch = 100/(a² × Avogadro's number) moles per unit area. We take a saturation value of V_xch = 1.5 cycles/ms. This is the value measured by Matsuoka and Hilgemann (1992), at 60 mV membrane, which is the mean transmembrane potential for a rat ventricular myocyte over a 200-ms time interval starting at the initiation of the action potential (Schanten and ter Keurs, 1985). A reversal concentration, U_rev = 100 nM, is introduced to prevent the continual extrusion of Ca from the cleft at Ca concentrations less than 100 nM. The Ca flux density through the Na/Ca exchanger is then given by

(5)

with K_xch = 5 µ M (Matsuoka and Hilgemann, 1992).

Diffusion, buffering, and reuptake in the sarcomere outside the cleft

Diffusion outside the cleft occurs with a diffusion coefficient assumed to be 0.3 µm²/ms, about half of the free aqueous diffusion coefficient (three times the coefficient assumed in the cleft). In the region of the sarcomere outside the cleft, we consider the effect of binding to fixed buffers calmodulin and troponin, using experimental values for their forward and backward rate constants and concentrations (Table 1). We also consider the effect of additional buffer, specifically the fluorescent dye fluo-3 (Minta et al., 1989; Kao et al., 1989). For simplicity, we model fluo-3 as being fixed. The effect of ignoring its mobility is to increase the computed peak [Ca] to some degree (Smith et al., 1996; Nowycky and Pinter, 1993). Completing the passage of Ca through the sarcomere, the model includes reuptake of the Ca by the longitudinal SR, assumed to be located just inside the outer cylindrical surface of the sarcomere. The diffusion equation and the three coupled kinetic equations for Ca buffering are

(6)

(7a)

(7b)

(7c)

C₀(r, z, t) is the concentration of free Ca. D_sarc = 0.3 µm²/ms is the diffusion coefficient in the sarcomere outside the cleft. C₁(r, z, t), C₂(r, z, t), and C₃(r, z, t) are the concentrations of Ca bound to calmodulin, troponin, and fluo-3, respectively. B₁, k₁₊, k₁, B₂, k₂₊, k₂, B₃, k₃₊, and k₃ are the total concentration and forward and backward rate constants for calmodulin, troponin, and fluo-3, respectively (Table 1).

The numerical computations of [Ca] within the cleft, U(r, t), and [Ca] in the sarcoplasm beyond the cleft, C₀(r, z, t), are coupled in the following way: the Ca flux across the cleft boundary at r = a, driven by the concentration gradient just inside the cleft boundary, is the source of Ca for the sarcomere outside the cleft, and the [Ca] is continuous across r = a.

The Ca flux leaving the cleft at r = a is  D_cleft (U/r)|_r=a × 2ah. This is assumed to reappear in the sarcomere outside the cleft distributed uniformly over the volume element at (r, z) = (a, 0) (extending into both halves of the sarcomere) of volume 2a r z, so that the source term in Eq. 6 from the flux exiting the cleft is

(8)

The condition for continuity of [Ca] at r = a is

(9)

It is assumed that just inside the outer r = b cylindrical surface of the sarcomere, reuptake of Ca by the longitudinal SR occurs such that

(10)

The value we use for the maximum reuptake rate of the SR is V_reup = 7.5 × 10²¹ moles/ms/sarcomere. The justification for this value is that it is a value which approximately balances the release and reuptake of Ca by the SR in a single contraction cycle. Also, it is essentially the same value used in the model of Luo and Rudy (1994). This is ~30 times the levels measured in vitro (Levitsky, 1981), but is probably a reasonable extrapolation to in vivo operation (MacLennon, personal communication). We use the values C_0rev = 100 nM and K_reup = 1 µM in Eq. 10.

S_corb is the source term in Eq. 6 for release from the corbular SR (CSR). The CSR is placed on the cylindrical surface of the sarcomere an axial distance 40% of the distance from the Z-line to the M-line (Dolber and Sommer, 1984). The axial extent of the release channel is taken to be 2z = 50 nm, and the radial extent to be r/2 = 12.5 nm. In the case where we include CSR release, we assume the junctional SR (JSR) and the CSR each release half the maximum force release amount, with the CSR release occurring during the 10-ms period immediately after the JSR release period. Thus

(11)

where I_SR = 1 × 10²⁰ moles/ms and _CSR = 10 ms.

Boundary and initial conditions

The boundary condition at the cylindrical surface of the sarcomere is such that no Ca passes through the surface. This boundary condition assumes that the same sequence of events is occurring in adjacent sarcomeres (Isenberg et al., 1996), so that the flux to adjacent sarcomeres is balanced by the flux from them. The half-sarcomere with its cleft is considered the functional unit of the cell, which is composed of many identical units. Similarly, there is no net flux in the axial direction at the two ends (M-lines) of the sarcomere, and by symmetry between the two halves of the sarcomere on either side of the Z-line, the same is true in the axial direction at the Z-line:

(12)

At t = 0 the free [Ca] everywhere is assumed to be 100 nM and the binding sites on the inner surface of the sarcolemma and the calmodulin and troponin in the sarcomere are assumed to be in equilibrium with the free Ca:

(13)

Numerical methods

Equation 1 for [Ca] inside the cleft is solved numerically using the explicit four-point finite-difference scheme with variable forward time steps, t_cleft, described in Peskoff et al. (1992). In this numerical procedure, U(r_i, t + t_cleft) is computed from U(r_i1, t), U(r_i, t) and U(r_i+1, t). The boundary condition at r = a, which previously was U(a, t) = 100 nM, now is replaced by Eqs. 8 and 9. This allows U(a, t) to vary to satisfy the conditions across the r = a border between the cleft and the sarcomere outside the cleft. For stability of the numerical computation, the condition that we use for the size of the variable time step inside the cleft, t_cleft, is

(14)

where we use the value of U at r = 0 because it is the location where U has its maximum value for all of the centrally positioned sources that we consider. When U(0, t) = 0.1 µM, Eq. 14 requires t_cleft  28 µs. When U(0, t) = 7 mM, the maximum concentration we will encounter, it requires a much smaller value, t_cleft  0.09 µs.

The coupled Eqs. 6 and 7a, b, and c for [Ca] outside the cleft are solved numerically using a six-point explicit finite-difference scheme, with fixed forward time steps, t. In this numerical procedure, C₀(r_j, z_k, t + t) is computed from C₀(r_j1, z_k, t), C₀(r_j, z_k1, t), C₀(r_j, z_k, t), C₀(r_j, z_k+1, t), and C₀(r_j+1, z_k, t). For stability of this numerical procedure outside the cleft, we take

(15)

When Eqs. 14 and 15 yield t_cleft > t, we take t_cleft = t. This is the case for U(0, t) less than ~2 mM.

The computation begins in the cleft at t = 0, with U(r, 0) given by the initial condition of Eq. 13 and, therefore, with t_cleft = t = 0.4167 µs. The numerical solution of Eq. 1 over the first time step yields U(r, t). The Ca that has diffused across the r = a border during the interval 0 < t < t is computed from Eq. 8, using a second-order approximation of (U/r)|_r=a. Next, the computation of diffusion, buffering, and reuptake by the SR in the region outside the cleft (Eqs. 6, 7, and 10) is done for the interval 0 < t < t. Finally, the Ca that diffused across the r = a border during the 0 < t < t interval is added to the Ca in the first volume element in the region outside the cleft: a  r/2 < r < a + r/2, z/2 < z < z/2. The [Ca] in this element, C₀(a, 0, t), is adjusted to account for the added Ca, and the [Ca] on the cleft side of the r = a border, U(a, t), is changed so that it equals the adjusted C₀(a, 0, t) (Eq. 8). The same procedure is then repeated for successive t intervals.

When U(0, t) increases to the level for which Eqs. 14 and 15 require t_cleft < t, the computation in the cleft is done a number of times in succession until the sum of the t_cleft increments exceeds t. The last t_cleft increment is then adjusted downward, so that the sum of the t_cleft increments equals t, and the last computation inside the cleft within that t interval is done. During all of the t_cleft subintervals of this single t interval, the boundary condition at r = a is kept fixed, and the cumulative Ca that has diffused across the r = a border during the t interval is computed. Next, the computation of diffusion, buffering, and reuptake by the SR in the region outside the cleft is done for the same t interval; the Ca that diffused across the r = a border during the t interval is added to the Ca in the first volume element in the region outside the cleft; C₀(a, 0, t) is adjusted accordingly, and U(a, t) is changed to equal the adjusted C₀(a, 0, t). The same procedure is then repeated for successive t intervals. When the [Ca] drops below ~2 mM, the computation reverts to the procedure with t_cleft = t.

A Fortran 90 program has been written to implement this numerical procedure. The run time on a Dell Pentium Pro 200 MHz PC is ~15 min for a 200-ms Ca transient.

	RESULTS

Top Abstract Introduction Results Discussion References

Ca concentration after JSR release

For the computations illustrated in this section, Ca is assumed to be released from a single ryanodine receptor located at the center of the circular cleft. According to Eq. 3, the idealized receptor is assumed to release Ca distributed uniformly over a small circular area. This simplifying assumption is made so that the resulting Ca distribution is independent of the azimuthal angle, depending only on the radial and axial variables, r and z. Electron micrographs reveal the ryanodine receptor to be roughly square in cross section, with a side of ~29 nm, separated from adjacent receptors by ~7 nm (Langer and Peskoff, 1996; Wibo et al., 1991). In Eq. 3, we idealize the cross section of the receptor by a circle of radius 20 nm.

The maximum [Ca] attained inside the cleft in the immediate vicinity of the receptor increases as the radius of the receptor cross section decreases. However, some distance away at the outer boundary of the cleft, and everywhere outside the cleft, the computed concentration is found to be independent of the assumed receptor radius. For example, we have run the computation for a centrally located cluster of four receptors (40-nm radius) and nine receptors (60-nm radius). We found essentially no difference in the [Ca] outside the cleft compared to the results illustrated below for the case of a 20-nm receptor release radius, but significantly higher concentrations near the center of the cleft when the source is more concentrated. From these results, we can also expect that replacing the actual release from discrete points within an approximately square cross section, with uniform release over a circular cross section is inconsequential, except perhaps in the immediate vicinity of the receptor. In our previous paper modeling the cleft (Langer and Peskoff, 1996), we assumed that the radius of the SR release area was equal to the radius of the cleft, i.e., all ryanodine receptors in the cleft were releasing Ca simultaneously. The present assumption of a single active receptor (or perhaps several) leads to a more reasonable value of current from an individual receptor.

JSR release: 2 × 10¹⁹ moles

Fig. 2 A is a graph of the free [Ca] at eight radial locations in the Z-line plane: four points inside the cleft, one point on the border between the cleft and the rest of the sarcomere outside the cleft, and three points in the sarcomere outside the cleft. These are the points shown in Fig. 1 along the radial line in the Z-line plane, from the axis to the outer radius of the sarcomere. Fig. 2 B shows the first 20 ms of Fig. 2 A, on a 10-fold expanded time scale. Fig. 2 C is a three-dimensional picture of the same information as in Fig. 2 A, shown at 41 uniformly spaced (t = 5 ms) times from t = 0 to t = 200 ms and at 21 uniformly spaced (r = 25 nm) positions from r = 0 to r = 500 nm along the radial line in the Z-line plane. The shaded portion of the graphs represents the interior of the cleft; the unshaded portion represents the general sarcomere, outside the cleft. Note that, except for the r = 25 nm increments outside the cleft, the r and t increments in the graph in Fig. 2 C are larger than those used in the numerical computation. In the computation, r = 2.5 nm inside the cleft is one-tenth of the value in the graph. In the computation outside the cleft t = 0.4167 µs, and inside the cleft t  0.4167 µs, which are less than 10⁴ times the t = 5 ms time increments in the graph.

View larger version (40K): [in this window] [in a new window]   FIGURE 2   [Ca] for an SR release of 2 × 1019 moles. (A) [Ca] versus t at the eight points in the z = 0 plane shown in Fig. 1. (B) The first 20 ms of A on a 10-fold expanded time scale. (C) Three-dimensional picture of the same information as in A, shown at 41 uniformly spaced times (t = 5 ms) and at 21 uniformly spaced positions (r = 25 nm) along a radial line (0  r  500 nm) in the z = 0 plane. The shaded portion of the graphs represents the interior of the cleft; the unshaded portion represents the sarcomere outside the cleft.

The graphs show that the [Ca] rises rapidly from its initial value of 0.1 µM at t = 0, and reaches a peak value of 7.4 mM at the center of the cleft (r = 0) at the end of the release period (t = 10 ms). After t = 10 ms, at which time the SR release ceases, the concentration begins to fall rapidly at the center of the cleft. Meanwhile, at the periphery of the cleft (r = a = 200 nm) there is a delay, and the rise begins at about t = 3.5 ms and reaches a peak value of 42.6 µM at t = 10.7 ms, followed by a more gradual fall than at the central release site. Further out radially in the general sarcomere, the delay increases somewhat and the peak concentration decreases. At the outer limit of the sarcomere (r = b = 500 nm), the rise begins at ~4.0 ms and peaks at a concentration of 5.6 µM at t = 11.8 ms. The concentration everywhere returns to a value close to the initial value of 100 nM after ~200 ms.

The return to 100 nM and the speed of return is governed to a large extent by four parameters in the model: the saturation rate and reversal concentration of the sodium-Ca exchanger, and the rate and reversal concentration of the SR reuptake. The saturation rate of the exchanger is an experimentally derived parameter (Matsuoka and Hilgemann, 1992). The concentration at which the exchanger reverses depends on the intra- and extracellular sodium concentrations and the membrane potential, all of which vary during the course of an action potential. We do assume, however, that reversal occurs at 100 nM and, at this point, ignore its voltage and concentration dependence. This seems a reasonable assumption at ~200 ms, when the membrane potential is about 90 mV and the computed [Ca] within the cleft is approaching 100 nM. This modification in the model, compared to our earlier model (Langer and Peskoff, 1996), prevents the exchanger from continuing to extrude Ca from the cleft when it should be acting in "reverse," and thereby prevents the concentration inside the cleft from falling to unphysiological concentrations less than 100 nM.

The two reuptake parameters were selected somewhat arbitrarily to force the return to 100 nM in a time interval of ~200 ms. In the absence of SR release, the reuptake must stop at some level, which we take to be 100 nM, to prevent the [Ca] from dropping to a nonphysiological level. Equation 10, governing the reuptake process, is obviously a simplification of the actual process, which is outside the scope of the present model, but we would not expect this simplification to have any serious impact on the computed intracellular concentrations.

Fig. 3 A shows graphs of the time dependence of free [Ca] at five points equally spaced between the Z-line and the M-line along a line parallel to the sarcomere axis at a distance from the axis equal to the cleft radius, shown in Fig. 1. The uppermost curve is the variation at the boundary of the cleft, and is a repeat of that curve in Fig. 2 B for the same point, which peaks at 43 µM. The middle solid curve at (r, z) = (200 nm, 500 nm) is almost identical to the bottom curve in Fig. 2 A for (r, z) = (500 nm, 0 nm). It reaches a peak value of 5.4 µM at t = 12 ms. The bottom curve in Fig. 3 A at (r, z) = (200 nm, 1000 nm) at the M-line has a peak of 1.8 µM at t = 13 ms. Also shown, in the dashed curve, is the average of the [Ca] over the volume of the sarcomere outside the cleft. It peaks at 6.5 µM at t = 11.5 ms. Thus the average concentration is approximately one-sixth of the maximum concentration that the model predicts at the boundary between the cleft and the rest of the sarcomere and indicates the magnitude of the intrasarcomeric Ca gradient. Fig. 3 B shows the first 20 ms in Fig. 3 A on a 10-fold expanded time scale.

View larger version (35K): [in this window] [in a new window]   FIGURE 3   [Ca] for an SR release of 2 × 1019 moles. (A) [Ca] versus t at the five axially spaced points at r = a = 200 nm shown in Fig. 1. The uppermost curve is of [Ca] at the boundary of the cleft, and is the same as the curve in Fig. 2 B for the same point. The dashed curve is the average of the [Ca] over the volume of the sarcomere outside the cleft. (B) The first 20 ms in A on a 10-fold expanded time scale.

Fig. 4 A shows four "snapshots" of the free [Ca] in the sarcomere outside the cleft at t = 5, 10, 20, and 50 ms, as a function of radial and axial position in the shaded plane pictured in Fig. 1. Note that the portion of the curves at z = 0 from r = 0 to r = 0.2 µm are in the Z-line plane, outside the cleft, unlike the shaded portions of Fig. 2, A-C, from r = 0 to r = 0.2 µm, which are inside the cleft.

View larger version (41K): [in this window] [in a new window]   FIGURE 4   [Ca] for an SR release of 2 × 1019 moles. (A) Four snapshots of the free [Ca] in the sarcomere outside the cleft at t = 5, 10, 20, and 50 ms, as a function of radial and axial position in the shaded plane pictured in Fig. 1. Note that the portion of the curves at z = 0 from r = 0 to r = a = 200 nm are of [Ca] in the Z-line plane, outside the cleft, unlike the shaded portions of Figs. 2 A-C, which are of [Ca] inside the cleft (see text: Geometry of the Model). (B) The same as in A, but for total [Ca] (i.e., the sum of free Ca and Ca bound to troponin and calmodulin).

Again, the peak concentration is seen to occur at the boundary point between the cleft and the general sarcomere, falling off fairly rapidly in any direction away from that point, and diminishing as time progresses. At 20 and 50 ms, an approximately parabolic radial distribution has developed at the outer axial limit of the sarcomere, resulting from the Ca "sink" at the location of the SR reuptake. Fig. 4 B, for the same conditions as Fig. 4 A, shows the total [Ca] (i.e., the sum of free Ca and Ca bound to troponin and calmodulin). Note that at t = 50 ms, the free [Ca] ranges between 0.2 and 0.9 µM over the sarcomere, whereas total [Ca] ranges between 40 and 70 µM.

Fig. 5 A illustrates the partition of Ca ions among the various possible states in or departed from the cleft. The figure shows the fraction of the Ca ions, released by the SR up to the time t, that have left the cell via the exchanger, that have diffused radially out of the cleft and entered the sarcoplasm, or that are still within the cleft volume either free or bound to low- or high-affinity sites. The free and bound fractions were obtained by integrating the free and bound concentrations at time t, over the volume of the cleft; the exchanged fraction was obtained by integrating the exchanger flux density over the sarcolemmal surface of the cleft and over the time interval from 0 to t; the diffused fraction was obtained by integrating the diffusional flux density, the product of the diffusion coefficient and the radial [Ca] gradient at r = a = 200 nm, over the circumferential boundary of the cleft and over the time interval from 0 to t. All amounts are normalized with respect to the total SR release during the time interval 0-10 ms and multiplied by 100%.

View larger version (42K): [in this window] [in a new window]   FIGURE 5   (A) The partition of Ca ions among the various possible states in or departed from the cleft: the fraction of the Ca ions released by the SR up to the time t that have left the cell via the Na-Ca exchanger, that have diffused radially out of the cleft and entered the sarcoplasm, or that are still within the cleft volume either free, or bound to low- or high-affinity sites. All amounts are normalized with respect to the total SR release and multiplied by 100%. Note that at t = 200 ms, 92% of the Ca has diffused to the general sarcoplasm and 8% has been transported out through the exchanger. (B) The fraction of Ca ions released from the SR up to the time t that are free in the sarcomere outside the cleft, that are bound to troponin or calmodulin, or that have diffused to the cylindrical boundary of the sarcoplasm, r = b = 500 nm, and have been taken up by the SR. At any instant the sum of the fractions equals the percentage of the total 10-ms-long SR release that has diffused out of the cleft up to that instant. Note that at t = 200 ms, ~3% of the Ca is bound to troponin, 1% is bound to calmodulin, 88% has been taken up by the SR, and 0.016% is free in the sarcoplasm. The remaining 8% has exited from the cell via the Na-Ca exchanger (see A).

At any instant the sum of all fractions equals the percentage of the total SR Ca that has been released up to that instant, which equals 100% after the release is over at t = 10 ms. At the end of the SR release (t = 10 ms), ~55% of the Ca is bound to low-affinity sites, 9% is bound to high-affinity sites, 34% has diffused out of the cleft into the general sarcoplasm, 1% has been removed by the exchanger, and 1% is free in the cleft. At t = 60 ms, ~91% of the Ca has diffused to the sarcoplasm, 6% has been exchanged, 2.7% is bound to the high-affinity sites, 0.6% is bound to the low-affinity sites, and 0.004% is free in the cleft. At t = 200 ms, 92% of the Ca has diffused to the general sarcoplasm, and 8% has been transported out through the exchanger.

In Fig. 5 B we have integrated over the volume of the general sarcomere outside the cleft to get the fraction of Ca ions released from the SR, up to the time t, that are free in the general sarcomere outside the cleft, that are bound to troponin or calmodulin, or that have diffused to the cylindrical boundary of the sarcoplasm, r = 500 nm, and have been taken up by the SR. The amounts are again normalized with respect to the total SR release. At any instant the sum of the fractions equals the percentage of the total 10-ms-long SR release that has diffused out of the cleft up to that instant. At the end of the SR release (t = 10 ms), ~13.9% of the released Ca is bound to troponin, 7.4% is bound to calmodulin, 8.3% has diffused to the cylindrical boundary of the sarcomere and been taken up by the SR, and 4.0% is free in the sarcoplasm. At t = 60 ms, ~19% of the Ca is bound to troponin, 6% is bound to calmodulin, 66% has been taken up by the SR, and 0.15% is free in the sarcoplasm. At t = 200 ms, ~3% of the Ca is bound to troponin, 1% is bound to calmodulin, 88% has been taken up by the SR, and 0.016% is free in the sarcoplasm. The remaining 8% has exited from the cell via the Na/Ca exchanger (see Fig. 5 A).

JSR release: 1 × 10¹⁹ moles

In Figs. 6 and 7, the results for an SR release of 1 × 10¹⁹ moles, half the amount of release in Figs. 2-5, are shown. This amount of Ca released from the SR will produce ~30% maximum force (Fabiato, 1985). Fig. 6 A shows that [Ca] rises to a peak value of 2.5 mM at the center of the cleft at the end of the release period. This value is less than half the value shown above for double the release because, at these concentrations, the buffering capacity of the low-affinity sites is near its saturation value. At the periphery of the cleft the rise in concentration begins after a delay of ~5 ms and reaches a peak value of 13.9 µM at 11.6 ms (compared to 3.5 ms, 42.6 µM, and 10.7 ms for an SR release of 2 × 10¹⁹ moles). At the outer radius of the sarcomere, the rise begins at ~6 ms and peaks at ~1.0 µM at t = 12.7 ms (compared to 4.0 ms, 5.6 µM, and 11.8 ms).

View larger version (36K): [in this window] [in a new window]   FIGURE 6   [Ca] for an SR release of 1 × 1019 moles. (A) [Ca] versus t at the eight points in the z = 0 plane shown in Fig. 1. Compare to Fig. 2 A. (B) [Ca] versus t at the five axially spaced points at r = a = 200 nm, shown in Fig. 1. The dashed curve is the average of the [Ca] over the volume of the sarcomere outside the cleft. Compare to Fig. 3 A. (C) Four snapshots of the free [Ca] in the sarcomere outside the cleft at t = 5, 10, 20, and 50 ms, as a function of radial and axial position in the shaded plane pictured in Fig. 1. Compare to Fig. 4 A for an SR release of 2 × 1019 moles.

View larger version (59K): [in this window] [in a new window]   FIGURE 7   The partition of Ca among the various states outside the cleft. Same as Fig. 5 B, but for an SR release of 1 × 1019 moles. Note that at t = 200 ms, ~4% of the Ca is bound to troponin, 1% is bound to calmodulin, 80% has been taken up by the SR, and 0.02% is free in the sarcoplasm. The remaining 15% has exited from the cell via the Na-Ca exchanger.

Fig. 6 B illustrates the [Ca] for an SR release of 1 × 10¹⁹ moles at five points on a line parallel to the sarcomere axis pictured in Fig. 1, and averaged over the sarcomere volume. Compared to the results for a 2 × 10¹⁹ mole release, the peak concentrations are generally lower by somewhat more than a factor of 2, and the delay times for the start of the rise in concentration and for the occurrence of the peak are somewhat longer. Both of these effects are attributable to the greater effectiveness of the buffers at the lower concentrations.

Fig. 6 C shows snapshots of the free [Ca], at t = 5, 10, 20, and 50 ms, as a function of radial and axial position in the shaded plane pictured in Fig. 1.

Fig. 7, showing the partition of Ca among the various states outside the cleft, is a repeat of Fig. 5 B for the lower SR release of 1 × 10¹⁹ moles. It shows that at t = 10 ms, ~13.9% of the Ca is bound to troponin, 7.4% is bound to calmodulin, 3.7% has diffused to the cylindrical boundary of the sarcomere and been taken up by the SR, and 4.0% is free in the sarcoplasm. At t = 60 ms, ~19% of the Ca is bound to troponin, 7% is bound to calmodulin, 57% has been taken up by the SR, and 0.16% is free in the sarcoplasm. At t = 200 ms, ~4% of the Ca is bound to troponin, 1% is bound to calmodulin, 80% has been taken up by the SR, and 0.02% is free in the sarcoplasm. The remaining 15% has exited from the cell via the sodium-Ca exchanger.

SPECIAL CASES

JSR release with 100 µM fluo-3 present

19

JSR release: 2 × 10¹⁹ moles

19

View larger version (37K): [in this window] [in a new window]   FIGURE 8   [Ca] for an SR release of 2 × 1019 moles, with 100 µM fluo-3 added to the sarcoplasm. (A) [Ca] versus t at the five axially spaced points at r = a = 200 nm shown in Fig. 1. The dashed curve is the average of the [Ca] over the volume of the sarcomere outside the cleft. Compare to Fig. 3 A for the same conditions without the added fluo-3. (B) Same as in A, but for [Ca-fluo-3] and volume average of [Ca-fluo-3].

View larger version (62K): [in this window] [in a new window]   FIGURE 9   The partition of Ca among the various states outside the cleft. Same as Figs. 5 B and 7, but for an SR release of 2 × 1019 moles, with 100 µM fluo-3 added to the sarcoplasm. Note that at t = 200 ms, ~3.1% of the Ca is bound to fluo-3, 4.3% is bound to troponin, 1.3% is bound to calmodulin, 83% has been taken up by the SR, and 0.03% is free in the sarcoplasm.

JSR release: 1 × 10¹⁹ moles

19

View larger version (36K): [in this window] [in a new window]   FIGURE 10   Same as in Fig. 8, but for an SR release of 1 × 1019 moles, with 100 µM fluo-3 added to the sarcoplasm. (A) [Ca] versus t at the five axially spaced points at r = a = 200 nm, shown in Fig. 1. The dashed curve is the average of the [Ca] over the volume of the sarcomere outside the cleft. Compare to Fig. 6 B for the same conditions without the added fluo-3. (B) Same as in A, but for [Ca-fluo-3] and volume average of [Ca-fluo-3].

View larger version (61K): [in this window] [in a new window]   FIGURE 11   The partition of Ca among the various states outside the cleft. Same as in Figs. 5 B and 9, but for an SR release of 1 × 1019 moles, with 100 µM fluo-3 added to the sarcoplasm. Note that at t = 200 ms, ~3.6% of the Ca is bound to fluo-3, 5.0% is bound to troponin, 1.6% is bound to calmodulin, 75% has been taken up by the SR, and 0.03% is free in the sarcoplasm. The remaining 15% has exited from the cell via the Na-Ca exchanger.

[Ca] after entry from the Ca channel

View larger version (58K): [in this window] [in a new window]   FIGURE 12   Free [Ca] versus t at nine radial distances from the cleft center, r = 0, to the cleft outer radius, r = a = 200 nm after 1 ms of L-channel opening. The r = 0 curve, which represents the mean concentration over a circle of 1.25 nm radius, has a peak concentration of 0.94 mM. The r = 200 nm curve rises only a small amount above 100 nM, is still rising at t = 20 ms, and reaches a peak of 111 nM at about t = 30 ms (not shown).

View larger version (53K): [in this window] [in a new window]   FIGURE 13   Snapshot of [Ca] on the shaded plane in Fig. 1. at t = 30 ms after channel opening, the time at which [Ca] at r = a = 200 nm reaches its maximum value. It indicates that the release from a single Ca channel at the center of the cleft has a trivial effect on [Ca] outside the cleft.

Ca spark