MODELING IN PHYSIOLOGY
Predicted changes in concentrations of free and bound ATP and ADP during intracellular Ca2+ signaling

M. E. Kargacin and G. J. Kargacin

Department of Physiology and Biophysics, University of Calgary, Calgary, Alberta, Canada T2N 4N1

    ABSTRACT
Top
Abstract
Introduction
Methods
Results
Discussion
References

High Ca2+ concentrations can develop near Ca2+ sources during intracellular signaling and might lead to localized regulation of Ca2+-dependent processes. By shifting the amount of Ca2+ and other cations associated with ATP, local high Ca2+ concentrations might also alter the substrate available for membrane-associated and cytoplasmic enzymes. To study this, simultaneous equations were solved over a range of Ca2+ and Mg2+ concentrations to determine the general effects of Ca2+ on the concentrations of free and Ca2+- and Mg2+-bound forms of ATP. To obtain a more specific picture of the changes that might occur in smooth muscle cells, mathematical models of Ca2+ diffusion and regulation were used to predict the magnitude and time course of near-membrane Ca2+ transients and their effects on the free and bound forms of ATP near the membrane. The results of this work indicate that changes in free Ca2+ concentration over the range of 50 nM-100 µM would result in significant changes in free ATP concentration, MgATP concentration, and the CaATP-to-MgATP concentration ratio.

smooth muscle; striated muscle; computer modeling; calcium channels; restricted diffusion; adenosinetriphosphate-sensitive channels

    INTRODUCTION
Top
Abstract
Introduction
Methods
Results
Discussion
References

IN CELLS, AN INCREASE in intracellular Ca2+ concentration is often one of the first events in signal transduction. For Ca2+-dependent signaling to be effective, intracellular Ca2+ concentration must be precisely controlled. In resting cells, average free Ca2+ concentration is generally maintained below 100-200 nM and, in muscle, may rise to ~1 µM during contractile signaling. However, theoretical calculations (15, 23, 27-29, 33) and experimental measurements (8, 20, 25) indicate that localized Ca2+ levels much higher than this may develop near the plasma membrane of cells after cell stimulation as a result of Ca2+ influx through plasma membrane Ca2+ channels. Predictions of maximum near-membrane free Ca2+ concentrations during a Ca2+ transient range from several micromoles a few nanometers away from the membrane (15, 16, 23) to >100 µM in the immediate vicinity of Ca2+ channels or clusters of channels (7, 28, 29, 31, 33). Measurements of maximum near-membrane free Ca2+ concentration in cells range from 2-10 µM in muscle and cultured cells (8, 20) to 200-300 µM in presynaptic terminals (25).

It is becoming increasingly clear that Mg2+ is also regulated in cells (reviewed in Refs. 22 and 24). Determinations of free Mg2+ concentration in cardiac and smooth muscle cells range from ~0.1 to ~3 mM (13, 22), with more recent estimates falling in the lower part of this range (18, 22, 24). Corkey et al. (5) measured a total cytosolic Mg2+ concentration of 6.4 mM in isolated rat liver hepatocytes and free Mg2+ concentration of 0.38 mM; the total Mg2+ content of rabbit urinary bladder smooth muscle was determined by Kushmerick et al. (18) to be 6.6 µmol/g wet wt. Although Mg2+ itself may be involved in the regulation of cellular function, one of its primary roles in cells is in enzymatic processes, where it is complexed to ATP as a substrate for various ATPases. In muscle cells, estimates of the cellular content of ATP range from ~0.3-2 µmol/g wet wt in smooth muscle cells to ~5 mmol/kg wet wt in skeletal muscle cells (12). The dissociation constants (Kd) for Mg2+ binding to ATP4- reported in the literature are in the micromolar range, and those for binding to HATP3- are in the micromolar to millimolar range (9, 18, 21, 30). Mg2+ is also likely to bind to intracellular Ca2+ buffers with a lower affinity than Ca2+ (16, 22).

The fact that Ca2+ and Mg2+ can bind to a number of intracellular molecules suggests the possibility that in regions where high Ca2+ concentration develops in cells during signal transduction, the relative amounts of Mg2+ and Ca2+ associated with these molecules might be altered. Although Ca2+ binds to ATP4- and HATP3- with Kd values higher than those reported for Mg2+, changes in Ca2+ concentration might be expected to lead to changes in the amount of Mg2+ bound to ATP or in the relative amounts of Mg2+ and Ca2+ bound to ATP. This could have important physiological consequences, since MgATP is the preferred substrate for most intracellular enzymes. Any effect of Ca2+ on intracellular MgATP might be especially significant in smooth muscle cells, where levels of intracellular ATP (~1 mM) and free Mg2+ (<= 1 mM) are normally within an order of magnitude of the free Ca2+ concentration that might be reached near the plasma membrane during a Ca2+ transient. It is also thought that the MgATP-to-MgADP ratio may be an important factor in determining the ability of smooth muscle cells to develop and maintain force (see discussion in Ref. 13). Because ADP can also bind Ca2+, the development of high Ca2+ concentration in cells might be predicted to alter the MgADP-to-MgATP ratio and, as a consequence, influence contractility.

To examine the possible effects of high Ca2+ concentration on the amount of Ca2+ and Mg2+ associated with ATP, ADP, and intracellular Ca2+ buffers, we solved a set of simultaneous equilibrium equations at various Ca2+, Mg2+, ATP, ADP, and intracellular Ca2+ buffer concentrations. Our results predict that MgATP concentration is likely to remain fairly constant in the face of Ca2+ concentration changes of <1 µM but may then decrease significantly when higher concentrations are reached. The concentration of CaATP is likely to rise and may approach that of MgATP. The simulations also predict that the concentrations of unbound forms of ATP (ATP4-, HATP3-, and H2ATP2-) and the MgADP-to-MgATP ratio are likely to decrease. To further explore these predictions, a mathematical model of Ca2+ diffusion and regulation was used to simulate the changes in Ca2+ and accompanying changes in ATP that might occur in smooth muscle cells during contractile signaling. Our results indicate that a local high Ca2+ concentration may significantly influence local ATP-dependent processes after Ca2+ influx through the plasma membrane.

    METHODS
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Abstract
Introduction
Methods
Results
Discussion
References

Equilibrium equations. A computer program using a matrix inversion algorithm was written to solve a set of simultaneous equilibrium equations for the binding of various ionic species to ATP, ADP, and intracellular Ca2+ and Mg2+ buffers. The binding constants (Table 1) were from Martell and Smith (21, 30) and Fabiato (9). The equilibrium equations are linear if total Mg2+ concentration is calculated from a known free Mg2+ concentration. Thus it was necessary to use an iterative method of solution (17) for equations in which total Mg2+ concentration remained constant while free Mg2+ concentration varied as free Ca2+ concentration was changed in a given simulation. Before a simulation was started, a total Mg2+ concentration was determined for a desired starting free Mg2+ concentration at 50 nM free Ca2+. The total Mg2+ concentration corresponding to this free Mg2+ concentration was then used throughout the simulation as an end point for the iterative calculations. Briefly, as Ca2+ concentration was changed, an estimate of free Mg2+ concentration based on its previous value was entered into the program and used to calculate a new total Mg2+ concentration; if the calculated total Mg2+ concentration was not within a preset margin of error (0.01%) of the starting total Mg2+ concentration, the difference between the calculated and starting total Mg2+ concentration was used to make a corrected estimate of free Mg2+ concentration. For each free Ca2+ concentration, this iterative process was continued until the calculated total Mg2+ concentration was within the allowed margin of error of the starting total Mg2+ concentration. The concentrations of the other ionic species included in the simulations are given in Table 2 and in RESULTS. The equilibrium equations were solved using a program written in C language; simulations were run on a 486 personal computer, and results were plotted using commercially available graphics software (Sigma Plot, Jandel Scientific, San Rafael, CA).

                              
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Table 1.   Dissociation constants used in equilibrium equations

                              
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Table 2.   Parameters for equilibrium equations

Unless otherwise noted, [MgATP] is used to refer to the total concentration of all species of ATP bound to Mg2+ (i.e., [MgATP] = [MgATP2-] + [MgHATP-]). Similarly, [MgADP] = [MgADP2-] + [MgHADP-], [CaATP] = [CaATP2-] + [CaHATP-], [CaADP] = [CaADP2-] + [CaHADP-], and free ATP concentration ([ATP]free) = [ATP4-] + [HATP3-] + [H2ATP2-].

Smooth muscle cell model. To study the changes in [Ca2+] that are likely to develop in smooth muscle cells during contractile signaling and the effects of these changes on ATP concentration, a mathematical model of Ca2+ diffusion and regulation was used. The model was a higher-resolution version of the models used by Kargacin and Fay (15, 16). Details of the model and its method of solution are given in RESULTS and the APPENDIX. Simulations were run on a 486 personal computer or a personal computer with a Pentium processor.

    RESULTS
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Abstract
Introduction
Methods
Results
Discussion
References

Predicted effects of Ca2+ concentration on free and bound ATP at different starting Mg2+ and ATP concentrations. As discussed above (also see Ref. 12), measurements of the ATP content of smooth muscle cells range from 0.3 to 2 µmol/g wet tissue wt, and free Mg2+ concentration is generally thought to be <1 mM. To examine the effect of changes in Ca2+ concentration on the concentration of bound and free ATP in such cells, free Ca2+ concentration was varied in the equilibrium equations from 50 nM to 100 µM at various starting free Mg2+ concentrations ranging from 0.1 to 1 mM. For these simulations, total ATP concentration was 1 mM, the Kd Ca for the binding of Ca2+ to an intracellular Ca2+ buffer was assumed to be 1 µM (15, 16), and the Kd Mg for binding to the same buffer was 1 mM (the effects of using different values for Kd Mg are discussed below). To achieve the starting free Mg2+ concentrations listed above, starting total Mg2+ concentrations ranged from 0.77 to 2.07 mM. The equilibrium equations (Fig. 1A) predict that, under the conditions of the simulations, intracellular [MgATP] would change very little over free Ca2+ concentrations ranging from 50 nM to ~10 µM. For a starting free Mg2+ concentration of 0.5 mM, [MgATP] was predicted to increase slightly from 904 to 907 µM as free Ca2+ concentration increased from 50 nM to 1.5 µM; at 10 µM free Ca2+, [MgATP] was 898 µM. Between 10 and 100 µM free Ca2+, [MgATP] decreased significantly, and at 100 µM free Ca2+, [MgATP] had dropped to 797 µM. The ability of the model cell to maintain a fairly constant [MgATP] between 50 nM and 10 µM free Ca2+ was aided by the displacement of Mg2+ from the intracellular Ca2+ buffer as Ca2+ concentration increased (Fig. 1B). [ATP]free was also predicted to decrease significantly as free Ca2+ concentration increased (Fig. 1C). As expected, the most dramatic change occurred for the lowest starting free Mg2+ concentration (0.1 mM). In this case, [ATP]free declined by ~60% from ~260 to ~100 µM. The [CaATP]-to-[MgATP] ratio increased to 0.1 for the simulation with a starting free Mg2+ concentration of 1 mM and to 0.6 for the simulation with a starting free Mg2+ concentration of 0.1 mM.


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Fig. 1.   Effects of changes in Ca2+ concentration on MgATP ([MgATP]) and free ATP ([ATP]free) concentrations and CaATP-to-MgATP concentration ratio ([CaATP]/[MgATP]) on cells with 1 mM total ATP. A: [MgATP] vs. free Ca2+ concentration ([Ca2+]free) for starting free Mg2+ concentration ([Mg2+]free) = 0.1, 0.25, 0.5, and 1 mM. B: displacement of Mg2+ from intracellular Ca buffer as Ca2+ concentration increased (starting [Mg2+]free = 0.5 mM). C: [ATP]free vs. [Ca2+]free for starting [Mg2+]free = 0.1, 0.25, 0.5, and 1 mM. D: [CaATP]/[MgATP] vs. [Ca2+]free for starting [Mg2+]free = 0.1, 0.25, 0.5, and 1 mM. Numbers to right of traces in A, C, and D are starting [Mg2+]free.

Predicted effects of Ca2+ concentration on various species of ATP at different Kd Ca-to-Kd Mg ratios for an intracellular Ca2+ buffer. The total Ca2+ buffer capacity of cells may be as high as 250 µM and the buffer Kd Ca approx  1 µM (1, 26, 28, 29, 34, 35; also see discussions in Refs. 15 and 16). Although these intracellular buffers are thought of primarily as Ca2+ buffers, they are likely to bind Mg2+. Kd Ca for striated muscle troponin range from 10-8 to 10-6 M, and Kd Ca for parvalbumin is ~10-8 M. Kd Mg for these proteins is ~10-4 M (26). In general, however, Kd Mg for various Ca2+-binding proteins have not been extensively studied; therefore, the equilibrium equations were solved for various Kd Mg values (1 µM-10 mM) for an intracellular Ca2+ buffer. To simplify the calculations, a single Ca2+/Mg2+ buffer was included, and the total buffer concentration and Kd Ca were kept constant at 250 and 1 µM, respectively; the starting free Mg2+ concentration for the simulations was 0.5 mM; [ATP]total was 1 mM (Fig. 2A). [MgATP] remained relatively constant between 50 nM and 1 µM free Ca2+ for all values of Kd Mg; however, it remained constant over the greatest range of Ca2+ concentrations for Kd Mg values of 100 µM or 1 mM. When Kd Mg was 10 mM, little Mg2+ was bound to the buffer, and at Kd Mg of 1 or 10 µM, Mg2+ was displaced from the buffer only at high free Ca2+ concentration (Fig. 2B). Free ATP was least affected by the change in free Ca2+ concentration for the Kd Mg values in which little Mg2+ was displaced from the buffer (Fig. 2C).


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Fig. 2.   Effects of ratio of dissociation constant for Ca2+ (Kd Ca) to dissociation constant for Mg2+ (Kd Mg) for an intracellular Ca2+ buffer on [MgATP] and [ATP]free. A: [MgATP] vs. [Ca2+]free for Kd Mg ranging from 1 µM to 10 mM. B: release of Mg2+ from buffer by Ca2+. C: [ATP]free vs. [Ca2+]free for Kd Mg ranging from 1 µM to 10 mM. For simulations in A-C, starting [Mg2+]free was 0.5 mM and [ATP]total was 1 mM. Kd Ca for buffer was 1 µM; total buffer concentration was 250 µM.

Effects of free Ca2+ concentration on ADP-to-ATP ratio. ADP reduces the rate of relaxation of tonic smooth muscle. This is presumably the result of the high affinity of the actomyosin cross bridge for MgADP, which competes with MgATP for dephosphorylated cross bridges (reviewed in Ref. 13). The most dramatic effects of ADP on muscle contraction are likely to occur under conditions of hypoxia or ischemia, when ATP declines and ADP increases. During hypoxia and inhibition or glycolysis, Allen et al. (2) noted a 44% decrease in ATP concentration and an increase in ADP concentration from 5 to 180 µM. Ischemia has also been shown experimentally (reviewed in Ref. 22) to be accompanied, at least initially, by an increase in free Mg2+ concentration, presumably as the result of the release of Mg2+ from ATP.

To examine possible effects of high Ca2+ concentration on the MgADP-to-MgATP ratio, equations were included in the equilibrium model for the binding of Mg2+, Ca2+, H+, and K+ to ADP. For the nonischemic condition (total ATP concentration = 1 mM, total ADP concentration = 5 µM, starting free Mg2+ concentration = 0.5 mM, total Mg2+ concentration = 1.49 mM), the MgADP-to-MgATP concentration ratio was 0.005 and decreased by 6% over the range of free Ca2+ concentrations of 50 nM-100 µM. With total Mg2+ concentration constant at 1.49 mM, the starting free Mg2+ concentration in the model increased to 0.82 mM when total ATP concentration was decreased to 0.5 mM and total ADP concentration increased to 0.1 mM. Under these conditions, the MgADP-to-MgATP concentration ratio was ~0.2 and decreased by only 5% when free Ca2+ concentration increased from 50 nM to 100 µM. Thus it is unlikely that changes in Ca2+ would have a significant physiological effect on the high-energy phosphate ratio.

Changes in Ca2+ and ATP in restricted diffusion spaces. A restricted diffusion space had been proposed to exist in smooth muscle cells, where the sarcoplasmic reticulum comes into close apposition to the plasma membrane (see discussion in Ref. 15). Free diffusion of Ca2+ into the central cytoplasm of the cell would be inhibited by the physical presence of the sarcoplasmic reticulum membrane, and one might expect high Ca2+ concentrations to develop in such spaces after Ca2+ influx through the plasma membrane. To study the magnitude and time course of Ca2+ signals that are likely to develop in restricted diffusion spaces and the effects of these signals on local ATP concentrations, the one-dimensional diffusion model described previously (15) was used. A diagram illustrating the model is shown in Fig. 3A. The equations describing radial diffusion into a cylindrical cell and the Ca2+ regulatory processes included in the model are given in the APPENDIX. The parameters used in the model are summarized in Table 3. The model was a higher-resolution version of the model described in detail by Kargacin and Fay (15, 16). Briefly, the model describes radial diffusion through a restricted diffusion space and into the central cytoplasm of a smooth muscle cell (3-µm radius). At the beginning of a simulation, Ca2+ moved into the cell through the plasma membrane. Plasma membrane Ca2+ influx was described by two exponentials (see APPENDIX and Table 3) and matched the time course of the Ca2+ current measured experimentally by Becker et al. (3) (see also Refs. 15 and 16). The maximum influx rate was adjusted so that the maximum average cytoplasmic free Ca2+ concentration reached in a simulation (<1 µM) was typical of the free Ca2+ level measured in smooth muscle cells during contractile signaling. A single immobile intracellular Ca2+ buffer (total concentration = 250 µM, rate constant for Ca2+ binding = 108 · M-1 · s-1, off rate for bound Ca2+ = 102/s, Kd = 1 µM; see also APPENDIX) was included in the model, as were kinetic equations (see APPENDIX) describing Ca2+ uptake by the sarcoplasmic reticulum Ca2+ pump and extrusion of Ca2+ through the plasma membrane. As discussed above, the total Ca2+ buffer concentration used in the simulations is consistent with estimates reported in the literature. The rate constants for Ca2+ binding to the buffer are typical of those reported for the binding of Ca2+ to various Ca2+-binding proteins (26, 29). The terms describing the velocities of the sarcoplasmic reticulum and plasma membrane Ca2+ pumps were adjusted to provide a rate of Ca2+ removal from the cytoplasm (~70 nM/s), in agreement with experimental estimates of the rate at which Ca2+ is removed from the smooth muscle cell cytoplasm (60-100 nM/s) (3) and with the percent contribution of the sarcoplasmic reticulum Ca2+ pump to this removal (75%) (17). The Na+/Ca2+ exchanger was not modeled explicitly, but its contribution to Ca2+ removal was assumed to be equal to that of the plasma membrane Ca2+ pump (see discussion in Ref. 15) and was included in the equation describing the plasma membrane pump. Release of Ca2+ from the sarcoplasmic reticulum was not included in the simulations described below. The diffusion coefficient for Ca2+ in the cell cytoplasm was 2.2 × 10-6 cm2/s on the basis of the measurements of Allbritton et al. (1). A barrier to the free diffusion of Ca2+ located 12.5 nm from the plasma membrane (Fig. 3A) was represented in the model as a region (125 nm thick) with a lower diffusion coefficient (0.22 × 10-6 or 0.022 × 10-6 cm2/s; see below). The presence of this barrier resulted in the formation of a restricted diffusion space between the plasma membrane and the barrier. The width of this space in the model (12.5 nm) is consistent with the spacing between the sarcoplasmic reticulum and the plasma membrane seen in electron micrographs of smooth muscle cells (10-20 nm) (11, 13). The magnitudes and time courses of Ca2+ transients predicted by the model [with a diffusion coefficient in the barrier region (Dbarrier) of 0.22 × 10-6 cm2/s] are shown in Fig. 3B. The free Ca2+ concentration reached a maximum of ~10 µM in the restricted diffusion space 15 ms after influx started (Fig. 3B, trace a). The free Ca2+ concentration transients at three locations in the central cytoplasm of the model cell are also shown in Fig. 3B. The initial rapid Ca2+ rise seen in the restricted space was not present in the central cytoplasm (Fig. 3B, traces b-d), but rather a more gradual increase in free Ca2+ concentration occurred over the time course of the simulation. The average central cytoplasmic free Ca2+ concentration in the cell reached a maximum of ~590 nM for this simulation (result not shown).


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Fig. 3.   A: smooth muscle cell model. Section of a portion of a smooth muscle cell is shown by light shading. Long axis of cell is in vertical direction. Areas of restricted diffusion (darker shading, not drawn to scale) are shown near plasma membrane on either side of cell. Cell was assumed to be cylindrical, and Ca2+ influx through plasma membrane from extracellular space and radial diffusion through cell cytoplasm are modeled. Regulatory processes incorporated into model are described in RESULTS. Ca2+ release by sarcoplasmic reticulum is not included in model; however, Ca2+ uptake into sarcoplasmic reticulum is included. Ca2+ pumps were located at surfaces of restricted diffusion space facing central cytoplasm and plasma membrane. To calculate central cytoplasmic [Ca2+]free in model, [Ca2+]free was averaged over volume included in 4.8-µm-diameter cylinder at center of cell (diameter of central cytoplasm = 80% of cell diameter). This approximates experimental measurements of Ca2+ concentration made with a Ca2+-sensitive dye through a mask centered over a cell. a-d, Approximate locations of Ca2+ transients in B. B: Ca2+ transients obtained with model in restricted diffusion space (a) and at 3 cytoplasmic locations (b-d). For simulation, diffusion coefficient in barrier region (Dbarrier) was 0.22 × 10-6 cm2/s. Trace b was obtained 2.4 µm from center of cell, trace c 1.5 µm from center of cell, and trace d at center of cell.

                              
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Table 3.   Parameters for diffusion model

Additional results from the simulation in Fig. 3B are shown in Fig. 4, along with those obtained for a second simulation with a lower Dbarrier. The permeability of the plasma membrane to Ca2+ was increased to the same extent at the start of each simulation. As noted above, Dbarrier for the first simulation was 0.22 × 10-6 cm2/s; for the second simulation Dbarrier was 0.022 × 10-6 cm2/s. Figure 4A compares the free Ca2+ concentration in the restricted diffusion space for the two simulations. With Dbarrier of 0.022 × 10-6 cm2/s, local free Ca2+ concentration reached ~80 µM in the space. The average cytoplasmic free Ca2+ concentration for the simulation with Dbarrier of 0.022 × 10-6 cm2/s (shown over a 3-s time interval in Fig. 4D) reached a maximum of ~600 nM. On the basis of previous work (15, 16), it appears unlikely that Ca2+ extrusion mechanisms would have a major influence on the time course or magnitude of the Ca2+ transient over an initial 100-ms time interval. The time course of the Ca2+ transients in the restricted space for simulations without Ca2+ uptake by the sarcoplasmic reticulum included in the model and the transients with Ca2+ removal processes included are shown in Fig. 4A. Figure 4B shows the amount of Ca2+ bound to an immobile intracellular Ca2+ buffer (see APPENDIX and Table 3) over the 100-ms time interval. With the rate constants used in the simulation, Ca2+ bound very rapidly to the buffer and was released relatively slowly. Thus neither Ca2+ removal processes nor Ca2+ binding to the buffer can account for the decline in free Ca2+ concentration with time seen in Fig. 4A. Instead, this decline is due primarily to the decrease in plasma membrane permeability and diffusion of Ca2+ away from the site of influx. The change in near-membrane [ATP]free predicted by the equilibrium equations for the Ca2+ transients shown in Fig. 4A are shown in Fig. 4C. In these simulations, starting free Mg2+ concentration was 0.5 mM and total ATP concentration was 1 mM. During the Ca2+ transient, [ATP]free dropped by ~10% when Dbarrier was 0.22 × 10-6 cm2/s and by ~30% when Dbarrier was 0.022 × 10-6 cm2/s. [MgATP] decreased by 7.6 and 9.6%, respectively, and the CaATP-to-MgATP ratio rose to 0.02 and 0.14, respectively, for the two diffusion coefficients (results not shown). Figure 4D shows the average free Ca2+ concentration in the central cytoplasm of the model cell for Dbarrier of 0.022 × 10-6 cm2/s and the corresponding change in [ATP]free in the same area. There was only a minimal decrease in [ATP]free from an initial level of ~70 µM in the central cytoplasm.


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Fig. 4.   Predicted near-membrane Ca2+ transients and accompanying changes in [ATP]free in smooth muscle cells. A: time course of [Ca2+]free (solid lines and open circle ) in a restricted diffusion space near plasma membrane. Dbarrier values in model were 0.022 × 10-6 cm2/s (top trace) and 0.22 × 10-6 cm2/s (bottom trace). Near-membrane Ca2+ transient is shown with (solid lines) and without (open circle ) sarcoplasmic reticulum Ca2+ uptake included in model. B: Ca2+ bound to intracellular Ca2+ buffer. Solid line, Dbarrier = 0.022 × 10-6 cm2/s; dashed line, Dbarrier = 0.22 × 10-6 cm2/s. C: changes in near-membrane [ATP]free calculated from equilibrium equations for Dbarrier = 0.022 × 10-6 cm2/s (solid line) and Dbarrier = 0.22 × 10-6 cm2/s (dashed line). D: average central cytoplasmic [Ca2+]free (left, solid line) and [ATP]free (right, dashed line) changes for a simulation with Dbarrier = 0.022 × 10-6 cm2/s (average [Ca2+]free was calculated as described in METHODS; [ATP]free was calculated from average [Ca2+]free curve using equilibrium equations). Time scale in D is different from time scale in A-C.

The results shown in Fig. 4C were derived from the Ca2+ transients in Fig. 4A using equilibrium equations. They indicate that the most significant changes in the restricted diffusion space were in [ATP]free and CaATP-to-MgATP ratio. Because of the dynamic nature of the transient, however, it might be argued that equilibrium conditions are unlikely to be met in a cell; therefore, in the results presented below, rate equations for the binding of Ca2+ to ATP were incorporated into the diffusion model itself (using Eqs. 5 and 6 in the APPENDIX) to better approximate the situation in a living cell. To simplify the calculations, only the binding and release of Ca2+ from free ATP were modeled, and diffusion of CaATP out of the restricted space was not included. On the basis of the results obtained with the equilibrium equations, starting [ATP]free was set at 70 µM. Two different sets of on and off rates for the binding of Ca2+ to ATP were used (on rate = 108 · M-1 · s-1, off rate = 1.7 × 104/s; on rate = 106 · M-1 · s-1, off rate = 170/s), both giving a Kd equal to that for Ca2+ binding to ATP4- (log Kd = -3.77; Table 1). Figure 5 shows that the magnitude of the change in near-membrane [ATP]free predicted using either of the two sets of rate constants was quite similar to that obtained with the equilibrium calculation. The time courses of the equilibrium trace and that determined with the faster on and off rate constants were also similar, although the rate equations predicted a slightly faster return of [ATP]free toward its starting level. The decrease in [ATP]free seen with the slower on and off rates was delayed in onset and was slightly lower in magnitude. One possible reason for the differences between the results obtained with the equilibrium equations and those obtained with the faster rate constants is that the equilibrium equations included binding of Ca2+ to other ionic species (e.g., HATP3-) and the release of Mg2+ from the Ca2+ buffer as Ca2+ concentration increased, which were not included in the simplified calculation with the rate constants. This indeed appeared to be the case. When the equilibrium equations were modified to include only the binding of Ca2+ to the intracellular Ca2+ buffer and to ATP4- (log Kd = -3.77, starting [ATP]free = 70 µM), a trace virtually identical to that calculated with the rate equations was obtained (Fig. 5).


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Fig. 5.   Comparison of predicted changes in near-membrane [ATP]free calculated with equilibrium equations or determined from rate constants incorporated into diffusion model. Changes in [ATP]free were calculated from near-membrane [Ca2+]free transient shown by top trace in Fig. 4A using equilibrium equations (solid line: starting [Mg2+]free = 0.5 mM, [ATP]total = 1 mM) or rate constants for Ca2+ binding to ATP (dotted line: on rate = 106 · M-1 · s-1, off rate = 170/s; dashed line: on rate = 108 · M-1 · s-1, off rate = 1.7 × 104/s). Trace shown by open circle  was obtained from near-membrane [Ca2+]free transient shown by top trace in Fig. 4A and equilibrium equations that included only Ca2+ binding to cellular Ca2+ buffer and to a single species of ATP (starting [ATP]free = 70 µM, log Kd = -3.77). For simulations with diffusion model, Dbarrier = 0.022 × 10-6 cm2/s.

    DISCUSSION
Top
Abstract
Introduction
Methods
Results
Discussion
References

Our results predict that the changes in Ca2+ concentration that occur during signal transduction may significantly alter the free and bound concentrations of ATP and the CaATP-to-MgATP concentration ratio in cells. Although during Ca2+ signaling inhomogeneities in Ca2+ concentration are likely to be present throughout cells (15, 16), the steepest gradients and highest Ca2+ concentrations occur near plasma membrane Ca2+ channels or clusters of channels and near Ca2+-release sites on the sarcoplasmic reticulum. In smooth muscle cells, high Ca2+ concentrations would be especially prominent in the restricted diffusion spaces that have been postulated to exist where the sarcoplasmic reticulum is in close apposition to the plasma membrane. In the simulations shown in Fig. 4, two diffusion coefficients for Ca2+ diffusion through a barrier region were used to predict the Ca2+ concentrations that might develop in the restricted diffusion space between this barrier and the plasma membrane. The higher Dbarrier (0.22 × 10-6 cm2/s, 1/10th of that in the absence of the barrier) predicted a maximum free Ca2+ concentration of ~10 µM in the restricted diffusion space and models diffusion into a restricted space that has fairly good communication with the central cytoplasm of the cell. A lower Dbarrier (0.022 × 10-6 cm2/s, 1/100th of that in the absence of a barrier) predicted a maximum free Ca2+ concentration of ~80 µM. A region such as this might be found where influx occurs into a space where access to the central cytoplasm is limited. Electron micrographs of smooth muscle cells obtained from longitudinal or transverse sections can show regions of close contact between the plasma membrane and the sarcoplasmic reticulum that extend for >= 1 µm (11). Ca2+ moving into such a region would be expected to have very limited access to the rest of the cell and could reach very high levels. The magnitude of the Ca2+ transient predicted in our simulations may indeed be conservative on the basis of the measurements and predictions made by others for the free Ca2+ concentrations near single channels or clusters of Ca2+ channels. Models developed by a number of investigators (23, 28, 29, 31, 33, 34) predict free Ca2+ concentrations of this magnitude or greater near the pore of single Ca2+ channels, near clusters of channels, in presynaptic terminals, and in the diadic region of cardiac muscle cells. Llinás et al. (20) measured 200-300 µM Ca2+ in the presynaptic terminal of the squid giant axon.

ATP-dependent reactions near the plasma membrane and/or the sarcoplasmic reticulum would be the most affected by alterations in the relative amount of ATP bound to different cations. In addition to effects on ATP-dependent enzymes in this region, changes in Ca2+ concentration could also alter the permeability of ATP-sensitive channels. In ventricular muscle, ATP-sensitive potassium (KATP) channel permeability is decreased by an increase in [MgATP] or ATP4- concentration (10, 19). KATP channels are also activated by nucleotide diphosphates (19). Thus local changes in [ATP]free, [MgATP], and/or the ADP-to-ATP concentration ratio induced by changes in Ca2+ concentration could locally alter KATP channel activity. Bezprozvanny and Erlich (4) found that MgATP and Na2ATP, in the presence of inositol trisphosphate (IP3), increased the open probability of IP3-sensitive Ca2+ channels in planar lipid bilayers. The effect was maximum at ~1 mM ATP but was apparent at concentrations as low as 10 µM. At higher concentrations, ATP inhibited the channels. ATP has also been shown to increase the open probability of ryanodine receptor channels (32). Hofer et al. (14) recently reported an effect of ATP concentration on the leak of Ca2+ through the membrane of the endoplasmic reticulum of fibroblasts. Thus the presence of high Ca2+ near the plasma membrane and the sarcoplasmic or endoplasmic reticulum could alter the permeability of plasma membrane K+ channels and/or IP3 receptor, ryanodine receptor, and Ca2+ leak channels in the sarcoplasmic reticulum. This could result in changes in plasma membrane permeability, membrane potential, or the efficiency of Ca2+ uptake into intracellular storage sites.

The effects of a decrease in [MgATP] in the presence of a local high Ca2+ concentration would depend on the concentration requirements of the enzymes near the high-Ca2+ site. The most important effect of Ca2+, however, might be on the CaATP-to-MgATP ratio. Our results indicate that a significant increase in this ratio is likely to occur in cells as Ca2+ concentration increases. Because MgATP appears to be the preferred substrate for most ATP-dependent processes in cells, an increased availability of CaATP could have an important inhibitory influence on enzymes that can bind but cannot use CaATP efficiently as a substrate.

The results of this study also suggest that the ability of intracellular Ca2+ buffers to also bind Mg2+, but with a lower affinity than they bind Ca2+, could stabilize [MgATP] in the presence of changes of free Ca2+ concentration from 50 nM to ~10 µM. Given a Kd Ca of 1 µM for the buffer, Kd Mg-to-Kd Ca ratios of 100-1,000 would provide the cell with relatively stable [MgATP] over the greatest range of free Ca2+ concentrations. These Kd ratios are generally consistent with those reported by Robertson et al. (26) for the binding of Ca2+ and Mg2+ to intracellular proteins. Because the precise nature of the Ca2+ buffers in cells and their capacities remains the subject of ongoing research, the role of these buffers as regulators of Mg2+ and MgATP levels remains to be determined. In the present work, all the cellular buffers were considered as a single immobile buffer. It is more realistic, however, to assume that different proteins would have different relative affinities for Ca2+ and Mg2+ and that mobile and immobile buffers are present (see discussions in Refs. 28, 29, 34, and 35) Different buffers are also likely to be localized to different cellular regions. Relatively high concentrations of Mg2+ associated with buffers on or near the plasma membrane of cells might tend to better stabilize [MgATP] in these regions, provided Ca2+ is able to displace Mg2+ from these buffers.

In considering the implications of the present study, it is important to keep in mind that changes in Ca2+ concentration may be highly localized in cells. Ca2+ signaling is also a dynamic process, and the high Ca2+ concentrations and steep gradients achieved near sites of Ca2+ influx and/or release may be present only for short periods of time before they are dissipated by diffusion, extrusion, and sequestration. Comparison of Figs. 3-5, however, indicates that high Ca2+ concentrations of short duration, such as those predicted by our models, could significantly alter local ATP concentrations. The results obtained with the equilibrium equations and those obtained with the two sets of rate constants were qualitatively and quantitatively quite similar. The equilibrium equations, which can provide more detailed information about specific ionic species, appear able to provide a good starting point for evaluating questions such as those posed in this study. The simplifying assumption made in the present simulations that the cytoplasmic Ca2+ buffers and ATP were immobile did not allow us to explore the effects of the diffusion of Ca2+-bound species on local Ca2+ or ATP concentrations. On the basis of the work of others (29, 35), it is, nevertheless, possible to make some qualitative predictions. One would expect the diffusion of CaATP away from the restricted space to increase the concentration of this species and Ca2+ at more central sites. Because of this, free Ca2+ concentration in the restricted space would also be expected to decline somewhat faster than predicted by our model. The diffusion of unbound ATP back into the restricted space would tend to bring [ATP]free toward its resting level. This effect would be partially offset, however, by additional Ca2+ binding to ATP. Using Ca2+ and buffer concentrations and other parameters quite similar to those used in our model, Smith et al. (29) predicted that a mobile Ca2+ buffer [diffusion coefficient for the buffer (Dbuffer) = 0.75 × 10-6 cm2/s] near a Ca2+ pore would be ~15% less saturated than immobile buffers near the pore. We would, therefore, expect that the magnitude of the decrease in [ATP]free would be slightly less than that shown in Fig. 5. Zhou and Neher (35) and Smith et al. (29) estimated that the presence of a mobile buffer (mobile-to-immobile buffer concentration ratio approx  1:4; Dbuffer approx  0.6 and 0.7 × 10-6 cm2/s) can increase the effective diffusion coefficient for Ca2+ by a factor of 2-3. On the basis of these results, the rate of return of [ATP]free toward baseline levels obtained with the rate equations (dashed line in Fig. 5) might be a good approximation of the rate when diffusion of CaATP away from the restricted space is considered.

Although further experimental and theoretical work is required to better determine the precise magnitudes and time courses of the changes in Ca2+ concentration, [MgATP], [ATP]free, CaATP-to-MgATP concentration ratio, and ADP-to-ATP ratio, the present study raises a number of interesting possibilities concerning ways in which these changes could fine tune regulatory processes in cells. This regulation would be especially significant near the plasma membrane and near the sarcoplasmic reticulum in muscle cells. The results obtained in this study, however, are quite general and should be applicable to a number of other cell types that rely on Ca2+-dependent signal transduction events.

    APPENDIX

In Eqs. 1-10, Ca denotes free Ca2+ concentration.

The time rate of change of Ca2+ due to radial diffusion and the uptake and release of Ca2+ by the Ca2+ regulatory processes in a cylindrical cell at radial position r can be described by the following equation
<FR><NU>∂Ca</NU><DE>∂<IT>t</IT></DE></FR> = <FR><NU>1</NU><DE><IT>r</IT></DE></FR> <FR><NU>∂</NU><DE>∂<IT>r</IT></DE></FR> <FENCE><IT>rD</IT> <FR><NU>∂Ca</NU><DE>∂<IT>r</IT></DE></FR></FENCE> + <IT>S</IT>(Ca, <IT>t</IT>, <IT>r</IT>) (1)
The first term on the right in Eq. 1 is derived from Fick's laws of diffusion for radial diffusion in a medium with cylindrical symmetry. The second term on the right, S(Ca, r, t), includes the Ca2+ concentration-, position-, and time-dependent processes (described below) that act as sources or sinks of Ca2+ in the cell. In the work described here, only radial diffusion was modeled. This assumes that there was no dependence of Ca2+ concentration on the axial (z) or angular ( theta ) dimensions in the region modeled. A numerical solution for the diffusion part of Eq. 1 was obtained using the explicit finite differences method described by Crank (6). For this solution, the cell was divided into concentric annuli of thickness Delta r. The rate of change of Ca2+ due to diffusion (diffusion coefficient, D) into and out of the ith annulus is
<FR><NU>&Dgr;Ca<SUB><IT>i</IT></SUB></NU><DE>&Dgr;<IT>t</IT></DE></FR> = <FR><NU><IT>D</IT></NU><DE>(&Dgr;<IT>r</IT>)<SUP>2</SUP></DE></FR> [(Ca<SUB><IT>i</IT>+1</SUB> − 2Ca<SUB><IT>i</IT></SUB> + Ca<SUB><IT>i</IT>−1</SUB>)
+ (Ca<SUB><IT>i</IT>+1</SUB> − Ca<SUB><IT>i</IT>−1</SUB>)/2(<IT>i</IT> − 1)] for <IT>i</IT> > 1 (2)
<FR><NU>&Dgr;Ca<SUB><IT>i</IT></SUB></NU><DE>&Dgr;<IT>t</IT></DE></FR> = <FR><NU>4<IT>D</IT></NU><DE>(&Dgr;<IT>r</IT>)<SUP>2</SUP></DE></FR> (Ca<SUB>2</SUB> − Ca<SUB>1</SUB>) for <IT>i</IT> = 1 (center of cell) (3)
Equations 2 and 3 were applied for diffusion within the restricted diffusion space near the plasma membrane, the barrier region (where the diffusion coefficient was lower), and the central cytoplasm of the model cell diagramed in Fig. 3. As discussed by Crank, finite differences solutions agree with analytic solutions of the diffusion equation, provided DDelta t/(Delta r)2 < 0.5. This condition was met by adjusting the time interval (Delta t) for the calculations once D and Delta r were chosen for the cytoplasmic and barrier spaces. As noted, diffusion within each of the spaces shown in Fig. 3 could be calculated using Eqs. 2 and 3; however, because the diffusion coefficient was lower in the barrier region, it was necessary to include expressions describing the movement of Ca2+ across the boundaries between the restricted space and the barrier region and between the barrier region and the central cytoplasm, where the diffusion coefficient changed. To do this, the method described by Crank for determining the diffusion of a substance through the boundary separating regions with different diffusion coefficients was used. For radial diffusion in a cylinder, the change in Ca2+ at the boundary between two such regions is
<FR><NU>&Dgr;Ca</NU><DE>&Dgr;<IT>t</IT></DE></FR> = <FENCE><FENCE>1 + <FR><NU>1</NU><DE>2<IT>m</IT></DE></FR></FENCE> (Ca<SUB><IT>n</IT>+1</SUB> − Ca) <FR><NU><IT>D<SUB>n</SUB></IT></NU><DE>&Dgr;<IT>r<SUB>n</SUB></IT></DE></FR> − <FENCE>1 − <FR><NU>1</NU><DE>2<IT>n</IT></DE></FR></FENCE></FENCE>
× (Ca − Ca<SUB><IT>m</IT>−1</SUB>) <FENCE><FR><NU><IT>D</IT><SUB><IT>m</IT></SUB></NU><DE>&Dgr;<IT>r<SUB>m</SUB></IT></DE></FR></FENCE><FENCE><FR><NU>1</NU><DE>2</DE></FR> <FENCE><FENCE>1 − <FR><NU>1</NU><DE>2<IT>n</IT></DE></FR></FENCE> &Dgr;<IT>r</IT><SUB><IT>m</IT></SUB> + <FENCE>1 + <FR><NU>1</NU><DE>2<IT>m</IT></DE></FR></FENCE>&Dgr;<IT>r</IT><SUB><IT>n</IT></SUB></FENCE></FENCE> (4)
where Dm and Delta rm are the diffusion coefficient and thickness of the annuli on the central side of the boundary, respectively, and Dn and Delta rn are the equivalent parameters on the other side of the boundary. The number m is the number of annuli of thickness Delta rm between the center of the cell and the boundary, and n is the number of annuli of thickness Delta rn between the center of the cell and the boundary (i.e., if b is the radial distance between the center of the cell and the boundary, b = mDelta rm = nDelta rn). [The method for deriving this equation for radial diffusion in a cylinder of composite media is given by Crank (6); details of the derivation are provided in the first edition of the same volume.]

As was done previously (15, 16), intracellular Ca2+ buffers were treated as a single immobile buffer distributed uniformly throughout the cell. Buffering of Ca2+ by the intracellular buffer was described by the following equations
<FR><NU>&Dgr;Ca</NU><DE>&Dgr;<IT>t</IT></DE></FR> = −<IT>K</IT><SUB>on</SUB>([buffer]<SUB>free</SUB>)(Ca) + <IT>K</IT><SUB>off</SUB>[Ca · buffer] (5)
[buffer]<SUB>total</SUB> = [buffer]<SUB>free</SUB> + [Ca · buffer] (6)
where Kon and Koff are the rate constants for the binding of Ca2+ to free buffer and the release of bound Ca2+ from the buffer, respectively. Similar equations were used to describe the binding of Ca2+ to free ATP for the simulations shown in Fig. 5. Values for the rate constants used in the simulations are given in Table 3.

Ca2+ influx into the cell through the plasma membrane was described by the following equations
<FR><NU>&Dgr;Ca</NU><DE>&Dgr;<IT>t</IT></DE></FR> = <IT>K</IT>(Ca<SUB>out</SUB> − Ca) (7)
<IT>K</IT> = <IT>K</IT><SUB>0</SUB>(1 − <IT>e</IT><SUP>−<IT>t</IT>/<IT>t</IT><SUB>on</SUB></SUP>)(<IT>e</IT><SUP>−<IT>t</IT>/<IT>t</IT><SUB>off</SUB></SUP>) (8)
The time constants ton and toff (Table 3) were adjusted so that the time course of Ca2+ influx in the model cell matched the time course of the Ca2+ current measured by Becker et al. (3) in voltage-clamped smooth muscle cells. The constant K0 was adjusted so that the average central cytoplasmic free Ca2+ concentration reached in the model cell was <1 µM (in agreement with the free Ca2+ levels measured experimentally in smooth muscle cells during Ca2+ transients).

Extrusion of Ca2+ out of the model cell was described by the Hill equation
<FR><NU>&Dgr;Ca</NU><DE>&Dgr;<IT>t</IT></DE></FR> = <FR><NU><IT>V</IT><SUB>max</SUB>(Ca)<SUP><IT>n</IT></SUP></NU><DE><IT>K</IT><SUP><IT>n</IT></SUP><SUB><IT>m</IT></SUB> + (Ca)<SUP><IT>n</IT></SUP></DE></FR> (9)
where Vmax is the maximum velocity of extrusion, Km is the Ca2+ concentration at half-maximal velocity, and n is the Hill coefficient. The values for Km are given in Table 3 and are consistent with the values reported in the literature (see discussion in Ref. 16). Vmax was adjusted as described in RESULTS. To balance the Ca2+ efflux across the plasma membrane when the cell was at rest, an inward Ca2+ leak was included in the simulations so that no net removal of Ca2+ from the cytoplasm occurred. The leak was described by the following equation
<FR><NU>&Dgr;Ca</NU><DE>&Dgr;<IT>t</IT></DE></FR> = <IT>K</IT><SUB>leak</SUB>(Ca<SUB>extracellular</SUB> − Ca<SUB>cytoplasm</SUB>) (10)
and the leak constant (Kleak) was adjusted so that the Ca2+ influx resulting from the leak was equal to the resting extrusion of Ca2+ through the plasma membrane (determined from Eq. 9 with 150 nM cytoplasmic Ca2+). A similar pair of equations was used to describe Ca2+ uptake by the sarcoplasmic reticulum and a leak out of the sarcoplasmic reticulum membrane when these elements were included in the model. For the sarcoplasmic reticulum uptake and leak, extracellular Ca2+ (Caextracellular) in Eq. 10 was replaced by the free Ca2+ concentration in the sarcoplasmic reticulum (1.5 mM) and Kleak was adjusted (as described above) to balance resting sarcoplasmic reticulum Ca2+ uptake. The values used for Km and n for sarcoplasmic reticulum Ca2+ uptake are given in Table 3. Vmax was adjusted as described in RESULTS. The equations describing Ca2+ influx through the plasma membrane, Ca2+ buffering, extrusion out of the cell, and the inward leak of Ca2+ through the plasma membrane were included in all simulations. As described in RESULTS, uptake of Ca2+ by the sarcoplasmic reticulum was included in the simulations except when the contribution of sarcoplasmic reticulum Ca2+ uptake to the time course of the near-membrane Ca2+ transient was examined (Fig. 4A, open circle ). Sarcoplasmic reticulum uptake sites were located on both surfaces of the barrier region.

    ACKNOWLEDGEMENTS

This work was supported by grants from the Heart and Stroke Foundation of Alberta and the Medical Research Council of Canada to G. J. Kargacin and a grant from the Heart and Stroke Foundation of Alberta to M. E. Kargacin. G. J. Kargacin is an Alberta Heritage Foundation for Medical Research Scholar.

    FOOTNOTES

Address for reprint requests: M. E. Kargacin, Dept. of Physiology and Biophysics, University of Calgary, 3330 Hospital Dr. NW, Calgary, AB, Canada T2N 4N1.

Received 9 January 1997; accepted in final form 2 June 1997.

    REFERENCES
Top
Abstract
Introduction
Methods
Results
Discussion
References

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AJP Cell Physiol 273(4):C1416-C1426
0363-6143/97 $5.00 Copyright © 1997 the American Physiological Society




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