Sneyd, James, Brian T. R. Wetton,. Andrew C. Charles, and Michael. J. Sanderson. Intercellular calcium waves mediated by diffusion of inositol trisphosphate:.
Intercellular calcium waves mediated by diffusion of inositol trisphosphate: a two-dimensional model
Sneyd, James, Brian T. R. Wetton, Andrew C. Charles, and Michael J. Sanderson. Intercellular calcium waves mediated by diffusion of inositol trisphosphate: a twodimensional model. Am. J. PhysioZ. 268 (CeZZ PhysioZ. 37): C1537-C1545, 1995.-In response to mechanical stimulation of a single cell, airway epithelial cells in culture exhibit a wave of increased intracellular free Ca2+ concentration that spreads from cell to cell over a limited distance through the culture. We present a detailed analysis of the intercellular wave in a two-dimensional sheet of cells. The model is based on the hypothesis that the wave is the result of diffusion of inositol trisphosphate (IP3) from the stimulated cell. The twodimensional model agrees well with experimental data and makes the following quantitative predictions: as the distance from the stimulated cells increases, 1) the intercellular delay increases exponentially, 2) the intracellular wave speed decreases exponentially, and 3) the arrival time increases exponentially. Furthermore, 4) a proportion of the cells at the periphery of the response will exhibit waves of decreased amplitude, 5) the intercellular membrane permeability to IP3 must be - 2 pm/s or greater, and 6) the ratio of the maximum concentration of IP3 in the stimulated cell to the K, of the IP3 receptor (with respect to IPS) must be - 300 or greater. These predictions constitute a rigorous test of the hypothesis that the intercellular Ca2+ waves are mediated by IP3 diffusion. intercellular gap junctions;
signaling; calcium oscillations; mechanical stimulation
epithelial
cells;
CELL in a culture of airway epithelial cells is mechanically stimulated, a wave of increased free cytosolic calcium concentration ( [Caz+]) is initiated in the stimulated cell and spreads from cell to cell via gap junctions to form an intercellular Ca2+ wave (7, 27). It has been proposed that this intercellular Ca2+ wave serves to coordinate a multicellular response to a local stimulus. For example, a Ca2+ wave may be an important initial step in coordinating mucociliary transport or in the wound-healing processes that follow local trauma (26). Simila r intercellular communication of Ca2+ signals is observed in a wide variety of cells (for a review, see Ref. 26). The intercellular wave propagates from cell to cell in a sequential manner. Each cell exhibits an intracellular Ca2+ wave that propagates across the cell at -20 pm/s and stops at the cell boundary. After a WHEN A SINGLE
0363-6143/95
$3.00 Copyright
short delay, a similar intracellular Ca2+ wave is initiated in neighboring cells. Repetition of this process results in an intercellular wave moving across the culture. The distance the wave propagates appears to depend on the magnitude of the initial stimulus. Inositol trisphosphate (IP,) is a second messenger that releases Ca 2+ from the endoplasmic reticulum via IP3 receptor Ca2+ channels that are sensitive to both Ca2+ and IP3 (5, 6, 14, 16, 23), and evidence indicates that the intercellular wave is mediated by the movement of IP3 through gap junctions. Iontophoretic injection of IP3 initiates an intercellular Ca2+ wave (27), and blockage of gap junctions with halothane in epithelial cells prevents the propagation of the intercellular wave (27). Similar Ca2+ waves in C6 glioma cells require the expression of the gap junction protein connexin 43 (11). Furthermore, addition of heparin, an antagonist of the IP3 receptor, prevents the appearance of a Ca2+ wave (7). An imposed fluid flow across the culture does not bias the direction of the intercellular wave in epithelial cells (15), suggesting that, unless the stimulated cell is ruptured, there appears to be no extracellular diffusing messenger helping to propagate the wave. Mechanical stimulation in the absence of extracellular Ca2+ initiates an intercellular wave of slightly decreased amplitude and speed (27), suggesting that the bulk of the Ca2+ wave consists of Ca2+ released from internal stores. However, in the absence of extracellular Ca2+, the stimulated cell shows no increase in [Ca2+], although an intercellular wave is still initiated in the neighboring cells (27). This indicates that a rise in [Ca2+] is not necessary for the transmission of the intercellular wave. In Ca2+-free conditions, the response in the stimulated cell can be restored by the addition of the Ca2+ channel blocker, Gd 3+ (S . Boitano E R. Dirksen, and M. J. Sanderson, unpublished observations). This indicates that the lack of response in Ca2+-free conditions is due to the rapid loss of Ca2+ through Ca2+ channels to the extracellular space. Also, a rise in [Ca2+] is not sufficient for intercellular wave transmission, since spontaneous oscillations in [Ca2+] occurring in airway epithelial cells do not initiate intercellular Ca2+ waves (7). Because the peak magnitude of the oscillations is similar to that of the intercellular wave, this cannot be explained by the
o 1995 the American
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JAMES SNEYD, BRIAN T. R. WETTON, ANDREW C. CHARLES, AND MICHAEL J. SANDERSON Departments of Biomathematics, Neurology, and Anatomy and Cell Biology, School of Medicine, University of California: Los Angeles, California 90024; and Department of Mathematics, University of British Columbia, Vancouver V6T lY4, Canada
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Mechanical Fig. 1. Schematic diagram of passive-diffusion hypothesis. Mechanical stimulation produces inositol trisphosphate (IPS), which binds to IP3 receptors (IPR) on endoplasmic reticulum (ER). This releases Ca2+, which then modulates Ca2+ flux through the IPS receptor in a biphasic manner. At low Ca2+ concentration ([Ca2+]) an increase in [Ca2+] increases Ca2+ flu; through the receptor, whereas at high [Ca2+], an increase in [Ca2+] decreases the flux. For the sake of clarity, inactivation of IPR by Ca2+ has been omitted. IP3 diffuses through gap junctions (GJ) to initiate a Ca2+ wave in neighboring cells. Although Ca2+ may move through gap junctions, it plays no role in transmission of the wave. Also, intracellular diffusion of Ca2+ does not play any role in transmission of intercellular wave. Slow diffusion of Ca2+ is swamped by more rapid diffusion of relatively large amounts of IP3. The wave in the model is essentially a diffusive wave, rather than an actively propagated one.
stimulation
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release via ryanodine receptors. However, it has been shown recently that airway epithelial cells do not have an elevated [Ca2+] in response to caffeine or ryanodine (M. Hansen, E. R. Dirksen, and M. J. Sanderson, unpublished observations), which suggests the absence of ryanodine receptors in these cells. The new model is thus based only on the kinetic properties of IP3 receptors and the existence of a single intracellular Ca2+ pool. With the two-dimensional model we are able to make a number of quantitative predictions about the consequent properties of the intercellular wave. These predictions provide a rigorous test of the initial hypothesis. Glossary k flux b
k1
k2
kY Y P kP VP
Ca2+ flux when all IP3 receptors are open and activated; essentially just a scaling factor (3 pM/s) Fraction of activated IP3 receptors at zero calcium; determined by fitting to experimental steady-state data (0.11) (4,24) K, for activation of IP3 receptors by Ca2+; determined by fitting to experimental steady-state data (0.7 PM) (4,24) K, for inactivation of IP3 receptors by Ca2+; determined by fitting to experimental steadystate data (0.7 FM) (4,24) Time constant for inactivation of IP3 receptor by Ca2+ (0.2 s) (14) K, of endoplasmic reticulum Ca2+ pumps (0.27 J-LW (20) Maximum rate of pumping of endoplasmic reticulum Ca2+ pumps (1 PM/S) Rate of Ca2+ leakage into cytoplasm (0.15 PM/S) K, for binding of IP3 to its receptor (0.01 FM) (21) Rate of IP3 degradation when [IP,] is small (0.08 s-l) (29, 30)
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existence of a threshold for intercellular wave initiation. Although Ca2+ may go through gap junctions, our experimental data indicate that this is not an important factor in the intercellular transmission of the Ca2+ wave. The intercellular Ca2+ waves seen in endothelial and glial cells (9-12) have very similar properties to those seen in airway epithelial cells, and, thus, although our principal focus here is on epithelial cells, we discuss results from these other cell types where appropriate. From these experiments, we postulated the following mechanism (which we call the passive-diffusion hypothesis) for intercellular wave propagation (Fig. 1). Mechanical stimulation of a single cell initiates the production of IP3 in that cell and consequent release of Ca2+. Some of this IP3 moves through gap junctions to neighboring cells, releasing Ca 2+ from internal stores there. A small amount of IP3 can stimulate a large release of Ca2+ via a positive-feedback process, whereby Ca2+ stimulates its own release through the IP3 receptor. The sequential movement of IP3 through more distal cells in the culture results in a corresponding intercellular Ca2+ wave. This hypothesis for the propagation of intercellular Ca2+ waves relies on the passive diffusion of IP3 between cells via gap junctions. However, it is unknown whether sufficient IP3 can be produced in the stimulated cell to drive the Ca2+ wave by diffusion of IP3 from the stimulated cell or whether the regeneration of IP3 is required as it moves through the culture. We have previously investigated this question using a onedimensional model (28). This model reproduced the intercellular wave and was able to account for the subsequent Ca2+ oscillations in glial cells. We confirmed that, if IP3 can move through the culture quickly enough, the passive-diffusion hypothesis provides an excellent qualitative explanation of the intercellular wave. Here, we extend this work to two spatial dimensions and incorporate new experimental evidence. The previous model was based on a model of intracellular Ca2+ wave propagation that relied on Ca2+-induced Ca2+
CALCIUM
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kP D D GC
When [IP,] = h,, rate of IP3 degradation is half maximal (1 PM) Diffusion coefficient of Ca2+ (20 pm2/s) (2) Diffusion coefficient of IP3 (300 km2/s) (2) Intercellular permeability to IP3 (2 pm/s>
CALCIUM
The model equations dC jj
a2c =
MODEL
EQUATIONS
Th
I& dynamics. We assume that IP3 moves through the culture by passive diffusion, moving from cell to cell via gap junctions, and is degraded with saturable kinetics. Thus, within each cell, we have D
2
dt
-=
k;
each cell) are
d2c +
2
a3/
k”2 +
(within
+
Jflux
-
Jpurnp
+
Jleak
La
1
h
(3)
~2
YC2 J pump = k; + 3
Yf?PkP
k, + P
where P is the intracellular concentration of IP3. Note that, when P -K k,, it decays with time constant l/VP. This is the case in most of the culture, as IP3 concentration ([IP,]) is high only in the stimulated cell. Furthermore, the intercellular flux of IP3 is assumed to be proportional to the concentration difference between cells, i.e., if cell n has P = P,andceZZn + 1 hasp = Pn+l, then the flux of IP3 from cell n to ceZZ n + 1 is given by or intercelluHP, - P, + J, where Fis the permeability lar flux coefficient (Fig. 2). Equation 1 is then coupled to equations describing the dynamics of intracellular Ca2+, as described below. Ca2+ dynamics. There is a great deal of evidence from other cell types that Ca2+ oscillations and waves may be the result of Ca2+ modulation of the IP3 receptor (6, 14, 16,23,24,31), and there exist a number of models based on these results (4, 13, 18, 19). Here we use the one due to Atri et al. (4). The basis of the model is the experimental evidence that the Ca2+ flux through the IP3 receptor depends on Ca2+ concentration (Ca2+) in a biphasic manner. When [Ca2+] is low, an increase in [Ca2+] increases Ca 2+ flux through the IP3 receptor, whereas, when [Ca2+] is high, an increase in [Ca2+] decreases the Ca2+ flux (6, 14). The inactivation of the IP3 receptor by Ca2+ occurs on a slower time scale than activation (14). Thus an IP3 receptor responds to an increase in [Ca2+] by an initial increase in Ca 2+ flux, followed by a slower recovery to a lower steady flux. Following Atri et al. (4), we model the IP3 receptor in a phenomenological manner by assuming that the IP3 receptor has three binding domains, one for IP3 and two for Ca2+. Each binding domain may contain more than one binding site and is merely a convenient way of grouping the binding sites according to functionality. The binding of IP3 to domain one occurs very quickly and increases the Ca 2+ flux through the receptor. The released Ca2+ may bind quickly to domain two, increasing the Ca2+ flux still further and generating an autocatalytic release of Ca 2+ . However, over a longer time period, Ca2+ will bind to domain three, inactivating the receptor. The steady-state properties of the binding domains are chosen to match the experimentally determined steady-state Ca 2+ flux as a function of [Ca2+] (4,24).
J leak
=
P
where c denotes [Ca2+] and h, the inactiv ,ation variable, denotes the fraction of IP3 receptors that have not been inactivated by Ca2+. The term Joux denotes the flux of Ca2+ into the cytoplasm through the IP3 receptor. It is the product of a number of terms. The function p(P) models the activation of the IP3 receptor by IP3 and is given by pm
P3 = ~ k; + P3
(4)
Thus we are assuming that IP3 binds to the receptor cooperatively, with a Hill coefficient of 3 and that the half-maximal binding occurs at k, = 10 nM (21). The term b + [(1 - b)c/(kl + c)] models how the Ca2+ flux through the IP3 receptor is an increasing function of [Ca2+] when [Ca2+] is low. The term h serves to inactivate the receptor. The steady state of h is given by hi/k; + c2, a decreasing function of c. Hence, as c increases, h gets smaller, decreasing Jaux. However, h is not an instantaneous function of c but decreases to steady state with time constant ?h. Thus, as c increases, JRux will activate instantaneously but will inactivate on a slower time scale. The term Jpump models Ca2+ pumping out of the cytoplasm and has been chosen to agree with the experimental data of Lytton et al. (20), whereas the represents a constant leak of Ca2+ into the term Jleak cytoplasm, either from the extracellular space or from the endoplasmic reticulum. The model parameters are summarized in the GZossary. Many of the parameters can be estimated directly from experimental data, and the relevant references are given in the Glossary. The time constant for receptor inactivation (Th) is chosen to be 200 ms, a factor of 10 smaller than that used by Atri et al. (4). The smaller value is based on the data of Finch et al. (14). There are no direct experimental data on the maximum pumping rate (y) or the leak (p). However, since the balance between these two terms sets the steady state, and the balance between y, p, and kHux sets the wave peak, these three parameters were chosen so as to obtain agreement with experimentally observed steady states and wave peaks. There is some discrepancy in the literature
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aP
-at-
DC (
dh THE
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RESULTS
In Fig. 4 we show the results of a typical simulation. An intercellular wave propagates through the culture in a manner similar to that observed experimentally. An intracellular wave propagates across each cell, and there is a significant delay in the transmission of the wave between cells. The intercellular wave covers an area with a radius of approximately four cells. Over this area, [Ca2+] remains elevated for some time after the wave front has passed, and recovery from the wave begins in the periphery of the response. As we mentioned in the introduction, in the absence of extracellular Ca2+ the stimulated cell shows no response. The model is not able to reproduce this feature of the experimental response,
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No-flux
conditions on perimeter
for Ca2’ and IP, of grid
,/ /
/
\ \
6
1 \ \
/ /
7
/
/ /
1
’
\ \ A \
/
m31=p,
II&l =P, IP, flux = F (P1-P2)
Fig. 2. Diagram of numerical setup. Ca2+ and IPZ dynamics within each cell are given by Eqs. l-4 in text. Cells are coupled by intercellular flux conditions to form a 2-dimensional sheet, and, on the perimeter of the sheet, no-flux conditions for Ca2+ and IP3 were imposed. IP3 was added to shaded cell over a period of 15 s as described in text. Details of numerical method are given in APPENDIX.
because it does not include terms to describe the opening and closing of stretch-activated channels and subsequent Ca2+ transport to and from the extracellular space. A convenient way of summarizing the intercellular wave movement is shown in Figs. 5A and 6, where we plot the position of the wave front against time. The wave front is here defined to be the place at which of the curves [Ca2+] = 0.3 PM. The rising portions correspond to the movement of the wave across a cell, whereas the flat portions correspond to the intercellular delay. In Figs. 5 and 6, we show the effect of varying9 and k,, two parameters with significant effects on model behavior. As 9- decreases, the intercellular wave moves more slowly, due principally to an increase in the intercellular delay, and the wave traverses fewer cells (Fig. 5A). An increase in k, has a similar effect (Fig. 6). As the wave moves across the culture, the intercellular
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between the affinity of the IP3 receptor measured by binding studies, which is -5-20 nM, and the [IP,] required for half-maximal Ca2+ release (the EC&, which is > 600 nM (see discussion in Ref. 25). It is therefore not clear which value should be used in the present model. There is also no general agreement on the cooperativity of the receptor. Although earlier data indicate that the Hill coefficient for the IP3 receptor is - 3 (21), it is not clear that these experiments eliminated Ca2+ feedback on the receptor. The question is complicated by recent data (Y. Yao, V. Ilyin, and I. Parker, unpublished observations; J. Sneyd, E. A. Finch, S. M. Goldin, and A. Atri, unpublished observations), which show that, when [Ca2+] is clamped, the EC& of the IP3 receptor flux is - 650 nM, with a Hill coefficient of - 1, but that, when [Ca2+] is allowed to vary in the whole cell, Ca 2+ feedback on the receptor results in a higher apparent cooperativity and a lower ECsO. Because of this uncertainty, we present model results for a range of values for the K, of the IP3 receptor (k,) and for different values of the Hill coefficient for the action of IP3 on the IP3 receptor. The value ofFis also unknown, and we present model results for a range of values forTalso. Numerical methods. We solved the model equations on a two-dimensional grid of square cells, each 30 x 30 pm, chosen to be consistent with experimental data (Fig. 2). Within each cell, the Ca2+ and IP3 dynamics were described by Eqs. l-4, and the cells were connected by boundary conditions. At the cell boundaries, the IP3 flux was assumed to be proportional to the difference in concentration across the boundary as described above. Explicit gap junctions were not built into the numerical simulations; it was assumed that IP3 could move from cell to cell at any point of contact. In view of experimental data, it was also assumed that Ca2+ does not cross the cell boundaries. The equations were solved on a grid of 8 x 8 cells with 16 x 16 grid points within each cell. A detailed presentation of the method is given in the APPENDIX. Initially, every cell was set to the steady-state value corresponding to P = 0. At t = 0, we started to add P to the stimulated cell (the grey cell in Fig. 2) at the rate of 0.72 FM/S and continued to do so for 15 s. P was then allowed to relax to steady state. Typical time courses of P from the stimulated cell and ceZZs 1 and 2 are shown in Fig. 3. Note that, due to the degradation of P, it attains a maximum value of - 3 PM.
CALCIUM
INTERCELLULAR
10
Fig. 3. Concentration of IPs as shaded cell in Fig. 2, and cells was added at 0.72 FM/S for maximum of - 3 PM in first 15 degradation and diffusion.
20
30
Time (s)
40
50
function of time in stimulated cell, i.e., 1 and 2. In numerical simulation, IPs IPs concentration ([IPa]) to reach a s. Subsequent decline of [IPs] is due to
delay increases, the arrival time increases, and the intracellular wave speeds all decrease with distance from the stimulated cell. The intercellular delay and the arrival time are sensitive to changes in,Fand h,, but the intracellular wave speed is much less so (Fig. 5, B-D). Although the fitting curves are not shown here, all the curves in Fig. 5, B-D, can be fitted well by exponentials. For the purpose of comparison, experimental data curves are also included in Fig. 5, C and D. The dotted line in Fig. 5C reproduces the data of Demer et al. (Ref. 12; Fig. 51, which show that the arrival time of the intercellular wave in endothelial cells is an exponentially increasing
WAVES
Cl541
function of distance from the stimulated cell, with a space constant of 0.92/tell. The two dotted curves in Fig. 50 are obtained from Charles et al. (Table II of Ref. 11) and are measurements of the intercellular lag for the wave in glioma cells, at two different levels of gap junction expression. The behavior of the wave in the periphery of the response is significant. As can be seen from the time courses in Fig. 7A, the response of cell 4 is not as large as the maximal response. Furthermore, the difference in the responses of cells 6 and 7 shows that, in some regions, the intercellular wave will appear to stop abruptly. Thus the model predicts that, although the wave will have a sharp boundary in some regions, in other regions the boundary will be more blurred and some cells in the periphery will show a partial response. This agrees well with experimental results from epithelial cells (data not shown). DISCUSSION
These results can be understood intuitively. Enough IP3 is produced in the stimulated cell to saturate the IP3 receptors (note that k, = 0.01 FM). Due to the high cooperativity of the action of IP3 on its receptor (Hill coefficient of 3, see Eq. 4), each IP3 receptor has a thresholdlike response. As IP3 diffuses out, the point at which [IPJ is above threshold moves out also. This causes the advancing edge of the intercellular wave. The threshold response of the IP3 receptor, combined with Ca2+-induced Ca2+ release on the receptor, results in a distinct intracellular wave. However, as with all realistic excitable systems, the responses are not truly all-or-
Fig. 4. Density plots of a typical numerical simulation, using same parameter values as in Fig. 3 and Glossary.
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0
CALCIUM
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INTERCELLULAR
CALCIUM
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B n
E
n +
100
25
60
Q-4
0
20
40
s .d *
l r(
g &
15
20
10 5
IO
20
15
35
a E
g
25 20
3
-kP=5nM ........ k, = 10 nM -m-m. = 100 nM kP
60
3
D
C
nothing; there is a range of intermediate behavior. Thus, as IP3 diffuses out, there is necessarily a slowing down of the wave rise time, an increase in the intercellular delay, and a “smeared out” wave boundary. It is important to emphasize that these waves are essentially just diffusive waves, caused by the diffusion of IP3, and are not actively propagated. Thus, as was
3 *0
2
Cell Number
Cell Number
80
25
Time (s)
2
z DA c)
F =8 pm s-l
30
80
d)
s 3
+
35
J]
10 15 Time (s)
20
25
Fig. 6. Model simulations for 3 different values for k,. As in Fig. 5A, position of wave front is plotted against time. As k, increases, intercellular wave slows down, due principally to an increase in intercellular delay.
I
3
Cell Number
pointed out to us by J. Keizer, the diffusion of Ca2+ is playing only a minor role in the model simulations (17). This was tested by repeating the model simulations with D = 0, which gave waves of identical nature to those presented here. Ca 2+ feedback on the IP3 receptor is important in generating and sharpening the wave front but is not actually helping to propagate the wave, as it is essentially swamped by the rapid diffusion of large amounts of IP3. From the above results it can be seen that this model provides a number of ways in which the passivediffusion hypothesis may be tested in a quantitative fashion. First, the intracellular wave speed, time of arrival, and intercellular delay must be exponential functions of distance from the stimulated cell as shown in Figs. 5 and 6. Demer et al. (12) have shown that endothelial cell cultures exhibit an intercellular Ca2+ wave similar to that seen in epithelial cells. In endothelial cells the arrival time is an exponentially increasing function of distance from the stimulated cell, which is consistent with the passive-diffusion hypothesis (Fig. 5C, dotted line). However, the intracellular wave speed appeared to remain constant as the intercellular wave progressed, which is not consistent with the present model. Charles et al. (11) show that, in glioma cell cultures, the intercellular delay is an increasing function of distance from the stimulated cell, which is consistent with the present model (Fig. 50, dotted lines). Both Demer et al. (12) and Charles et al. (11)
............*..........
-I
I
2
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Fig. 5. Model simulations. A: position of wave front (defined to be where [Ca2+] = 300 nM) as a function of time, for 4 different values for K Wave front was measured in ceZZs 1-4 as labeled in Fig. 2 and is plotted as distance from stimulated cell. All data in A-D are taken from these same 4 cells. Rising portions of curves correspond to intracellular waves, whereas flat portions correspond to intercellular delay. As y increases, the intercellular wave speed increases, due mainly to a decrease in intercellular delay. Jagged nature of curves is caused by low spatial resolution in numerical simulations. Higher spatial resolution would have required inordinate amounts of computer time. B: intracellular wave speed decreases with distance from stimulated cell but is not very sensitive toK C: arrival time of wave. Dotted line is data from endothelial cells (12). D: intercellular delay increases with distance from stimulated cell. Dotted lines (connecting solid symbols) are experimental data from glioma cells for 2 different levels of gap junction expression ( 11).
2 2 $
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0.6
2
0.5
10
20
30
40
0
10
20 Time (s)
30
40
A; 0.4 u, 03.
Fig. 7. Time courses of [Ca2+] from labeled cells in Fig. 2. A: model simulations using Eq. 4 for action of IP3 on its receptor. Notice how wave stops abruptly at ceZZ 6 but has a less distinct boundary at ceZZ 4, which shows a partial response. B: model simulations using Eq. 5 for action of IP3 on its receptor. Wave peaks decrease more gradually, as is seen in glioma cells (Fig. 8).
show that the peak of the wave is also an exponentially decreasing function of distance from the stimulated cell. The present model can reproduce this result by the use of a different expression for k(P). As we discussed above, there is some evidence that the action of IP3 on its receptor is not cooperative, and the expression for k should perhaps be P
k, + P
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amount of IP3 added and obtained the same results. We are not aware of any measurements of these values in airway epithelial cell cultures. By combining data from a number of different cell types, we have previously estimated the intercellular flux coefficient to be 0.08 km/s (28), considerably lower than the value needed to explain the wave. However, it is not clear that this number is valid for epithelial cell cultures. Because of the technical difficulties of measuring [IP,] as a function of time, it has so far not been possible to measure how much IP3 is produced by the stimulated cell. Thus the IP3 sensitivity ratio in these cultures is also unknown. Evidence from other cell types indicates that the IP3 receptor is extremely sensitive to IP3, saturating at - 20 nM (21). Difficulties with this value are discussed above, but if it is appropriate for epithelial cells, the model predicts that the concentration of IP3 in the stimulated cell needs to only reach - 3 PM for there to be sufficient IP3 to propagate the wave. Thus the stimulated cell does not have to produce large amounts of IP3, as long as the IP3 receptors are very sensitive to IP3. One alternative to the hypothesis presented here is that IP3 is regenerated as it moves through the culture. There is evidence that an increased concentration of Ca2+ increases the rate of production of IP3, by increasing the activity of phospholipase C (modeled in Ref. 22). However, this alternative does not appear to be consistent with the observation that intracellular oscillations do not initiate intercellular waves. If a raised [Ca2+] produces IP3, then each oscillation would be expected to initiate an intercellular wave by the production of IP3 and its subsequent movement through gap junctions. On the other hand, there is little evidence that IP3 can be regenerated in a manner that is independent of Ca2+, and, in any case, such a model presents severe theoretical difficulties. Regeneration of IP3 on its own cannot be sufficient to propagate an intercellular wave, for, if it were, the wave would not stop but would propagate through the entire culture. We have not yet been able to 1.0 0.9
(5)
Use of this expression results in model waves that have an amplitude that decreases gradually with distance from the stimulated cell (Fig. 7B). The peak of the wave, as a function of distance from the stimulated cell, agrees well with experimental data (Fig. 8). Second, the model makes quantitative predictions for the unknown parameters, in particular the intercellular flux coefficient. Simulations also show that P,,/k, is an important ratio, where P,, is the maximum concentration of IP3 achieved by the stimulated cell. We shall call this the IP3 sensitivity ratio. From Figs. 5, 6, and 8, it can be seen that the model gives good quantitative agreement with experimental data when F = 2 pm/s and the sensitivity ratio is - 300. In the simulations shown in Fig. 6, we varied the sensitivity ratio by varying h,. We could equally have varied the total
0.4
2
3 Cell Number
4
5
Fig. 8. Model and experimental curves of wave peak as a function of distance from the stimulated cell. Experimental curves are taken from Ref. 11, and the model curves were obtained using Eq. 5 for the action of IP3 on its receptor.
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g
0 B
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construct a model, involving the regeneration of IP3, that propagates a wave for a limited distance depending on the magnitude of the stimulus. In summary, although the present model cannot give a definite answer as to the mechanism by which intercellular Ca2+ waves are propagated, it does give a number of quantitative predictions that can be used to test the passive-diffusion hypothesis. If further experimental work confirms our predictions, it will lend strong support to the passive-diffusion hypothesis, but as yet the question remains open. APPENDIX
of the Numerical
Method
where
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13 = C$?Ax/(D + X&X). u-1
This leads to the expression
- (I + pw,
for the second derivative approximation at i = 0 with a similar expression at i = 1. In matrix form we denote the second-order difference operator with these modifications near the boundary by A. Discretization in time. An explicit time discretization for a diffusive problem requires very small time steps to be stable. Therefore an implicit time discretization is used for the diffusion terms. However, the reaction terms do not lead to a severe time step restriction and so may be handled explicitly. We use the following semi-implicit technique
iI 2.gA)
UnAun-l
Aun-2
We believe the methods used to solve this model numerically will be of interest to other researchers in the field and so we provide some of the details. For ease of presentation, we consider the following one-dimensional problem dU
a2U
-=D-+r at
dX2
where u can be thought of as the concentration of Ca2+ or IPQ and r represents the reaction terms. We consider a membrane at x = 0 where the flux is proportional to the difference in values of u across the membrane dU
D- ax (0-)
= D u
=
dX
where O+ and O- denote the values on the right and left sides, respectively, of the membrane. Discretization in space. Away from the membrane, the term d2u/dx2 can be approximated in the standard way. However, special care must be taken near the membrane. We approximate the solution on a staggered grid with spacing AX, i.e., we consider approximations = u[(i
ui
- 1/2)Ax]
Thus the point of discontinuity (the membrane) is not a point of the grid: & is the approximation AX/~ to the left and U1 to the right. Using difference formulas we would approximate the second derivative at i = 0 by u-,
- 2u,
but U1 is the value of the solution through a discontinuity. What we desire is to replace U1 by UF, the smooth extension of the left side solution through the membrane. Similarly, we would use UC in the difference expression for the second derivative at i = 1. To eliminate the U* values in terms of known values, we apply second-order approximations to the boundary conditions UF - u(j Ax
u1=D
These equations
ug Ax
+
At
4
--Rn-1
-
2
pn-2
b43)
where the superscript n denotes approximations at time nAt, and where R is the vector of reaction terms. This time stepping is called second-order semi-implicit backward differentiation, and the resulting solutions are second order accurate in space and time (3). Multigrid solution. To implement the implicit time stepping we must solve a system of equations of the form BU=K at each time step, where B = I - 2At/3 A, K is known, and U is an unknown vector. The matrix B is positive definite and so a pointwise iterative technique like Gausssymmetric, Seidel will converge. However, the iterations lead to poor error reduction for low wave number modes, especially when AX is small. A very efficient algorithm results when the low wave number error is recursively corrected by relaxation on successively coarser grids. This type of method is called a multigrid method and is described in Ref. 8. In our case, special care must be taken near the membrane when transferring corrections from coarse to fine grids. Linear interpolation using the appropriate starred values must be used. This procedure is similar in spirit to that proposed in Ref. 1 for diffusion equations with discontinuous coefficients. More than adequate accuracy for the full two-dimensional computations is obtained using simple V-cycles (8) and two sweeps of Gauss-Seidel relaxation at each grid level.
+ u,
Ax2
D
+ pu,
Ax2
q+ = 7 (
can be solved giving + (1 - NJ,
u; 2
ur -
+ u() 2
J. Sneyd was supported by National Institute of General Medical Sciences Grant GM-48682, B. Wetton was supported by a Canadian National Sciences and Engineering Research Council grant, A. Charles was supported by a Veterans Affairs and Career Development Award, and M. Sanderson was supported by National Heart, Lung, and Blood Institute Grant HL-49288 and the Smokeless Tobacco Research Council. Address for reprint requests: J. Sneyd, Dept. of Mathematics, Univ. of Canterbury, Private Bag 4880, Christchurch, New Zealand. Received
13 March
1994; accepted in final form 21 November
1994.
REFERENCES 1. Alcouffe, R. E., A. Brand& J. E. Dendy, and J. W. Painter. The multi-grid method for the diffusion equation with strongly discontinuous coefficients. SIAdM J. Sci. Stat. Comput. 2: 430454,198l.
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Some Details
CALCIUM
INTERCELLULAR
WAVES
Cl545
18. Keizer, J., and G. DeYoung. Simplification of a realistic model of IPS-induced Ca2+ oscillations. J. Theor. Biol. 166: 431-442, 1994. 19. Li, Y.-X., and J. Rinzel. Equations for InsPz receptor-mediated [Ca2+] oscillations derived from a detailed kinetic model: a Hodgkin-Huxley like formalism. J. Theor. Biol. 166: 461-473, 1994. 20. Lytton, J., M. Westlin, S. E. Burk, G. E. Shull, and D. H. MacLennan. Functional comparisons between isoforms of the sarcoplamic or endoplasmic reticulum family of calcium pumps. J. Biol. Chem. 267: 14483-14489,1992. 21. Meyer, T., D. Holowka, and L. Stryer. Highly cooperative opening of calcium channels by inositol 1,4,5-trisphosphate. Science Wash. DC 240: 653-656,1988. 22. Meyer, T., and L. Stryer. Molecular model for receptorstimulated calcium spiking. Proc. Natl. Acad. Sci. USA 85: 5051-5055,1988. I., and I. Ivorra. Inhibition by Ca2+ of inositol trisphos23. Parker, phate-mediated Ca 2+ liberation: a possible mechanism for oscillatory release of Ca 2+. Proc. Natl. Acad. Sci. USA 87: 260-264, 1990. J. B., S. W. Sernett, S. DeLisle, P. M. Snyder, M. J. 24. Parys, Welsh, and K. P. Campbell. Isolation, characterization, and localization of the inositol 1,4,5-trisphosphate receptor protein in Xenopus laevis oocytes. J. Biol. Chem. 267: 18776-18782, 1992. R. M., M. Poitras, G. Boulay, and G. 25. Ribeiro-do-Valle, Guillemette. The important discrepancy between the apparent affinity observed in Ca2+ mobilization studies and the Kd measured in binding studies is a consequence of the quanta1 process by which inositol 1,4,5-trisphosphate releases Ca2+ from bovine adrenal cortex microsomes. Cell Calcium 15: 79-88, 1994. M. J., A. C. Charles, S. Boitano, and E. R. 26. Sanderson, Dirksen. Mechanisms and function of intercellular calcium signaling. Mol. Cell. Endocrinol. 98: 173-187, 1994. 27. Sanderson, M. J., A. C. Charles, and E. R. Dirksen. Mechanical stimulation and intercellular communication increases intracellular Ca2+ in epithelial cells. Cell Regul. 1: 585-596, 1990. 28. Sneyd, J., A. C. Charles, and M. J. Sanderson. A model for the propagation of intercellular calcium waves. Am. J. Physiol. 266 (CelZ Physiol. 35): C293-C302, 1994. 29. Storey, D. J., S. B. Shears, C. J. Kirk, and R. H. Michell. Stepwise enzymatic dephosphorylation of inositol 1,4,5trisphosphate to inositol in liver. Nature Lond. 312: 374-376, 1984. S. S.-H., A. A. Alonsi, and S. H. Thompson. The 30. Wang, lifetime of inositol 1,4,5-trisphosphate in single cells. J. Gen. Physiol. 105: 149-171, 1995. 31. Yao, Y., and I. Parker. Potentiation of inositol trisphosphateinduced Ca2+ mobilization in Xenopus oocytes by cytosolic Ca2+. J. Phwiol. Lond. 458: 319-338, 1992.
Downloaded from http://ajpcell.physiology.org/ by 10.220.32.247 on September 3, 2017
2. Allbritton, N. L., T. Meyer, and L. Stryer. Range of messenger action of calcium ion and inositol 1,4,5trisphosphate. Science Wash. DC 258: 1812-1815,1992. 3. Ascher, U. M., S. J. Ruuth, and B. T. R. Wetton. Implicitexplicit methods for time-dependent PDEs. SIAlM J. Numer. Anal. In press. 4. Atri, A., J. Amundson, D. Clapham, and J. Sneyd. A single-pool model for intracellular calcium oscillations and waves in thexenopus laevis oocyte. Biophys. J. 65: 1727-1739, 1993. 5. Berridge, M. J. Inositol trisphosphate and calcium signalling. Nature Lond. 361: 315-325, 1993. 6. Bezprozvanny, I., J. Watras, and B. E. Ehrlich. Bell-shaped calcium-response curves of Ins( 1,4,5)P3- and calcium-gated channels from endoplasmic reticulum of cerebellum. Nature Lond. 351: 751-754,199l. 7. Boitano, S., E. R. Dirksen, and M. J. Sanderson. Intercellular propagation of calcium waves mediated by inositol trisphosphate. Science Wash. DC 258: 292-294, 1992. W. L. A MuLtigrid Tutorial. Philadelphia, PA: Sot. 8. Briggs, Industrial Appl. Math., 1987. 9. Charles, A. C., E. R. Dirksen, J. E. Merrill, and M. J. Sanderson. Mechanisms of intercellular calcium signaling in glial cells studied with dantrolene and thapsigargin. Glia 7: 134-145,1993. 10. Charles, A. C., J. E. Merrill, E. R. Dirksen, and M. J. Sanderson. Intercellular signaling in glial cells: calcium waves and oscillations in response to mechanical stimulation and glutamate. Neuron 6: 983-992, 1991. 11. Charles, A. C., C. C. G. Naus, D. Zhu, G. M. Kidder, E. R. Dirksen, and M. J. Sanderson. Intercellular calcium signaling via gap junctions in glioma cells. J. CeZZ Biol. 118: 195-201, 1992. 12. Demer, L. L., C. M. Wortham, E. R. Dirksen, and M. J. Sanderson. Mechanical stimulation induces intercellular calcium signaling in bovine aortic endothelial cells. Am. J. Physiol. 264 (Heart Circ. Physiol. 33): H2094-H2102, 1993. 13. De Young, G. W., and J. Keizer. A single pool IPs-receptor based model for agonist-stimulated Ca2+ oscillations. Proc. NatZ. Acad. Sci. USA 89: 9895-9899,1992. 14. Finch, E. A., T. J. Turner, and S. M. Goldin. Calcium as a coagonist of inositol 1,4,5-trisphosphate-induced calcium release. Science Wash. DC 252: 443-446,199l. 15. Hansen, M., S. Boitano, E. R. Dirksen, and M. Sanderson. Intercellular calcium signaling induced by extracellular ATP and mechanical stimulation in airway epithelial cells. J. Cell Sci. 106: 995-1004,1993. 16. Iino, M. Biphasic Ca 2+ dependence of inositoll,4,5-trisphosphateinduced Ca2+ release in smooth muscle cells of the guinea pig Taenia caeci. J. Gen. Physiol. 95: 1103-1122, 1990. 17. Jafri, M. S., and J. Keizer. Diffusion of IPS, but not Ca2+, is necessary for a class of IPs-induced Ca2+ waves. Proc. NatZ. Acad. Sci. USA 91: 9485-9489.1994.
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