Unité de Chronobiologie Théorique, Faculté des Sciences, Université Libre de Bruxelles, B-1050 Brussels, Belgium
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ABSTRACT |
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Spiral waves of intracellular Ca2+ have often been observed in Xenopus oocytes. Such waves can be accounted for by most realistic models for Ca2+ oscillations taking diffusion of cytosolic Ca2+ into account, but their initiation requires rather demanding and unphysiological initial conditions. Here, it is shown by means of numerical simulations that these spiral Ca2+ waves naturally arise if the cytoplasm is assumed to be heterogeneous both at the level of the synthesis and metabolism of D-myo-inositol 1,4,5-trisphosphate [Ins(1,4,5)P3] and at the level of the distribution of the Ins(1,4,5)P3 receptors. In such conditions, a spiral can be initiated in the simulations after an increase in Ins(1,4,5)P3 concentration, with the direction of rotation being determined by the position of the region of high receptor density with respect to the locus of Ins(1,4,5)P3 production.
oscillations; inositol 1,4,5-trisphosphate; spatiotemporal pattern
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INTRODUCTION |
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OSCILLATIONS AND WAVES OF cytosolic Ca2+ have been observed in a large variety of cell types after stimulation by an extracellular agonist (3, 23). These oscillations occur through the periodic exchange of Ca2+ between the cytosol and the internal stores (the sarcoplasmic or endoplasmic reticulum). Release of Ca2+ from these stores is triggered by inositol 1,4,5-trisphosphate [Ins(1,4,5)P3] synthesized by phospholipase C (PLC) in response to external stimulation. The Ins(1,4,5)P3 receptor [Ins(1,4,5)P3R] behaves as a Ca2+ channel. Moreoever, the release of Ca2+ through this channel is activated by cytosolic Ca2+ itself (4, 11). The period of oscillations and the velocity of Ca2+ wave propagation greatly depend on the cell type. The shape of the waves can also vary; in particular, immature Xenopus oocytes expressing muscarinic acetylcholine receptor subtypes can display circular, planar, and spiral Ca2+ waves (16).
Extensive experimental and theoretical work has been carried out to uncover the mechanisms underlying Ca2+ oscillations (3, 7, 20-23). After experimental results, in most models the autocatalytic regulation called Ca2+-induced Ca2+ release (CICR), by which Ca2+ activates its own release from internal stores through the Ins(1,4,5)P3R, is at the core of the oscillatory mechanism, although a mechanism based on the cross-activation of Ins(1,4,5)P3 synthesis by Ca2+ is also plausible (18). CICR can also explain the spatial propagation of planar and circular fronts resembling those observed experimentally, when the diffusion of Ca2+ inside the cell is considered. Moreover, numerous features about these waves, such as their shape, rate of propagation, or the effect of Ca2+ buffers, can be accounted for by considering detailed properties of the intracellular Ca2+ dynamics (5a, 9, 15). Numerical simulations have shown that these models can also reproduce spiral Ca2+ waves. However, in the literature, these spirals have been initiated with a rather arbitrary choice of initial conditions, which are often both exacting and unrealistic from a physiological point of view (2, 12, 15, 19).
In a previous study based on numerical simulations (10), it has been shown that the initiation of the spiral Ca2+ waves observed in cardiac cells after overloading the stores can be explained by the spatial heterogeneity created by the nucleus (17). Such an assumption does not hold in Xenopus oocytes. These cells are indeed much larger than myocytes (1 mm in diameter vs. 100 µm in length); a small obstacle like a nucleus, behaving as a barrier to the propagation of excitation, is thus not able to break concentric waves to create spirals. In the present study based on numerical simulations, we propose a simple way by which spiral Ca2+ waves could be initiated in Xenopus oocytes.
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DESCRIPTION OF THE SYSTEM |
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The propagation of concentric Ca2+ waves has been extensively simulated by considering the diffusion of cytosolic Ca2+ in the various models initially developed to account for Ca2+ oscillations in homogeneous conditions (2, 9, 12, 15). Among these models, the one based on a phenomenological description of CICR is particularly well adapted for the study of Ca2+ waves, as it contains only two variables; a detailed description of this model, which is used in the present numerical study to simulate the Ca2+ dynamics in Xenopus oocytes, can be found elsewhere (8, 9, 12).
Spiral Ca2+ waves generally arise from the asymmetric breaking of concentric waves. In a cell as large as the Xenopus oocyte, the asymmetry could arise from the existence of a gradient in Ins(1,4,5)P3 concentration due to a spatially restricted synthesis of the latter messenger. The substrate of PLC for Ins(1,4,5)P3 synthesis is indeed located in the plasma membrane (3); moreover, the Ins(1,4,5)P3 5-phosphatase, the main enzyme responsible for Ins(1,4,5)P3 metabolism, is mainly present on the cell surface (6). Thus, in our two-dimensional system designed to represent a portion of a Xenopus oocyte, it has been assumed that Ins(1,4,5)P3 synthesis and metabolism only occur in a small region (region 1 on Fig. 1) that is arbitrarily chosen as a square having a side of 27.8 µm. In this region, the time evolution of Ins(1,4,5)P3 concentration (A) is given by
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(1) |
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The breakage of the Ins(1,4,5)P3-induced Ca2+ wave can be provoked by some heterogeneity in the cytoplasm. On the basis of the assumption that the Ca2+-releasing mechanisms are heterogeneously distributed in the cytoplasm, region 2 in Fig. 1 is supposed to possess a higher density of Ins(1,4,5)P3 receptor; this region is a square with 40.7-µm sides. From a quantitative point of view, the distinctive feature of this area is that the maximal velocity of Ca2+ release from the internal stores has a larger value than in the rest of the system. The rate of release (V3) now takes the form
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(2) |
The full system explicitly considers the evolution of Ins(1,4,5)P3 concentration and of both intravesicular and cytosolic Ca2+ concentrations. Diffusion of intravesicular Ca2+ is not taken into account. A computer program was developed to numerically integrate these coupled partial derivative equations, using a variable time step Gear integration method. The dimension of the Cartesian grid used to simulate Ca2+ and Ins(1,4,5)P3 diffusion is 0.926 µm. The Laplacian is discretized using the finite difference method. No flux boundary conditions are used. This system of 270 × 270 × 3 differential equations is solved on Silicon Graphics R10000 workstation.
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RESULTS |
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Numerical integration of the system defined in Ref. 10, in the geometry
shown in Fig. 1, gives rise to spiral
Ca2+ waves. Such time-dependent,
spatial structures of Ca2+ are
shown in Fig. 2; the three
panels at top show the rather complex
behavior that first arises when the rate of
Ins(1,4,5)P3 synthesis (vp)
is increased up to 8 µM · s1.
The Ca2+ front is not circular
because the regions close to the locus of
Ins(1,4,5)P3
synthesis are more excitable than the bulk of the system. After a
transient period, the duration of which depends on the initial
conditions, a more regular spiral
Ca2+ wave becomes visible and
keeps on rotating clockwise. However, the spatiotemporal
Ca2+ pattern in the region
possessing a higher density of
Ins(1,4,5)P3R (region
2 in Fig. 1), which contains the tip
of the spiral, remains irregular. The average wavelength of the
Ca2+ spiral is on the order of 130 µm, and the rotation time is slightly larger than 2 s; thus the
wavelength is in good agreement with experimental observations, whereas
the period is too short by a factor of two (12).
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The complexity of the Ca2+ dynamics in the region with a higher density of Ins(1,4,5)P3R is visible by examination of the evolution of the level of cytosolic Ca2+ at a particular grid point of this region. Such a time series [grid point (180, 135)] is shown in Fig. 3A. This region acts as a high-frequency pacemaker because of the high rate of Ca2+ release from the stores in this area. Only a fraction of the Ca2+ spikes there initiated will be able to propagate in the surrounding region, which has a smaller potentiality to release Ca2+. Thus the tip of the spiral sometimes breaks and finally disappears when it encounters a refractory region characterized by a basal density of Ins(1,4,5)P3R. After some time, the new extremity of the front can bend again, thus forming a new tip. Alternatively, a new front is sometimes emitted by region 2, which is in the oscillatory regime; such a front then combines with the extremity of the large spiral, so that the global appearance of the Ca2+ wave remains the same. The regular temporal evolution of cytosolic Ca2+ in the grid point (120, 135), located in a region with a basal density of Ins(1,4,5)P3R, is shown in Fig. 3B.
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The respective locations of the regions of
Ins(1,4,5)P3
metabolism and synthesis, on the one hand, and of high
Ins(1,4,5)P3R density, on the other hand, play a crucial role in determining the
occurrence of a spiral wave. In fact, to generate a spiral, the region
of high receptor density has to be located in a steep gradient of
Ins(1,4,5)P3
concentration; as long as this condition is fulfilled, a phenomenon
that depends on various couterbalancing factors such as the positions
of the regions and the parameters vp and ,
spiral waves do not accurately depend on the geometry of the system.
For example, in a system like the one schematized in Fig. 1, the
Ca2+ wave still displays a spiral
shape when regions
1 and
2 are moved away from one another if,
at the same time,
vp is increased
(not shown). Also, the shape and dimensions of these areas can be
varied in the simulations without qualitatively affecting the
spatiotemporal dynamics of cytosolic
Ca2+. In real cells, regions of
high receptor density would certainly be distributed in a more random
fashion. The effect of randomly distributed
Ca2+-releasing sites has already
been investigated in other theoretical studies (5a, 15a). In such
conditions, the waves can become abortive at small doses of
Ins(1,4,5)P3 or
at very low density of
Ins(1,4,5)P3R; also, the front is more irregular, reflecting the inhomogeneous distribution of releasing sites. However, these studies clearly show
that the continuous approximation certainly remains a good approximation of the qualitative behavior of the wave. In this respect,
it appears that the occurence of spiral
Ca2+ waves would be little
affected by a distribution of
Ins(1,4,5)P3R that is less regular than in the present simulated system; the region
that, on average, possesses a sufficiently higher density of
Ins(1,4,5)P3R
would behave as the pacemaker site.
In experiments, Ca2+ waves are
often initiated by the injection or the photorelease of a poorly
metabolizable analog of
Ins(1,4,5)P3 into
the oocyte (16, 20). Such a situation can be simulated by considering
that the level of
Ins(1,4,5)P3 is
initially high in a well-defined region of the system that would
correspond, for example, to the part of the oocyte that has been
flashed. Moreover, it is then considered that this
Ins(1,4,5)P3 is
not metabolized or synthesized
(vp = = 0);
the initially localized high level of
Ins(1,4,5)P3
spreads because of diffusion. This system, which also generates a
gradient of
Ins(1,4,5)P3
concentration onto a region possessing a higher density of
Ins(1,4,5)P3R,
can also generate spiral Ca2+
waves. This is illustrated in Fig. 4, in
which the larger, more central square (indicated for both
t = 9 and
t = 10.25) indicates the region of
higher density of
Ins(1,4,5)P3R
(same location as region
2 in Fig. 1) and the other, smaller
square shows the portion of the oocyte in which the level of
Ins(1,4,5)P3 was
initially (i.e., at t = 0) at a higher
level. As can be seen in the frame showing the situation at
t = 10.75 s, the dynamics in the
region of higher receptor density is complex, as in Figs. 2 and 3. In this particular case, the small "semicircular" front will not propagate further away outside the block because the surrounding medium
is refractory. However, it will annihilate the part of the front that
forms the tip of the larger spiral (see Fig. 4, t = 11.75)
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An interesting change in the Ca2+ spiral shown in Fig. 4 with respect to the one shown in Fig. 2 is that the former one rotates counterclockwise. This is due to the fact that the Ins(1,4,5)P3 is now diffusing from the right side of the obstacle, whereas in Fig. 2 it was diffusing from the left side. This result does not depend on how the gradient in Ins(1,4,5)P3 is generated [by a localized region of Ins(1,4,5)P3 synthesis and metabolism or by an initially localized increase in Ins(1,4,5)P3]. Such a counterclockwise rotation of the spiral can also be observed in the simulations in the same conditions as in Fig. 2, if the two areas indicated in Fig. 1 are moved in such a manner that region 1 becomes located to the right of region 2. Although rather intuitive from a geometrical point of view (Figs. 2 and 4 are more or less mirror images), these differences make some physiological sense because the oocyte is polarized. Moreover, this prediction could be tested experimentally by injecting boluses of Ins(1,4,5)P3 at various regions of the cell; the change of location of the pipette should in some cases induces a change in the direction of spinning of the spiral. Also interesting to mention is the fact that in Fig. 4, as in many other simulations, the spiral is only transient. Depending on the system, spirals rotating from 5 to ~25 times before their transformation into concentric waves have been observed in the simulations. Such transient Ca2+ spirals have been reported experimentally (12). This contrasts with the situation shown in Fig. 2, in which the spiral appears as a stable spatiotemporal pattern (the stability has been tested until t = 300 s).
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DISCUSSION |
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It is well known that a circular front that breaks in an asymmetric medium can initiate a spiral. The present simulations show that this concept might explain the origin of the spiral Ca2+ waves that have been observed in Xenopus oocytes. A region characterized by a higher density of Ins(1,4,5)P3R can act as a source of heterogeneity that breaks the Ca2+ wave, and the Ins(1,4,5)P3 gradient due to either spatially restricted Ins(1,4,5)P3 synthesis and metabolism or to injection of Ins(1,4,5)P3 into a localized region of the oocyte can induce asymmetry of the medium. Moreover, this mechanism of spiral Ca2+ wave initiation is rather robust with respect to changes in the values of the dynamic parameters or in the detailed configuration of the system that represents a portion of the cell. In that respect, it is reasonable to assume that a three-dimensional configuration corresponding to the spatial extension of the system schematized in Fig. 1 could generate scroll waves such as the ones occurring in oocytes.
In contrast with a previous study aimed at investigating the origin of spiral Ca2+ waves in cardiac myocytes and in which an unexcitable region is responsible for spiral wave initiation, in the present work, spiral Ca2+ waves are best initiated when the existence of a region possessing a larger potentiality to release Ca2+ is assumed. If, in contrast, region 2 (see Fig. 1) is assumed to have a lower density of Ins(1,4,5)P3 than the rest of the system, a single Ca2+ front is initiated in region 1, which is initially characterized by a high level of Ins(1,4,5)P3; when it encounters the refractory region, the front breaks and propagates on both sides of the obstacle, after which, in most cases, both parts of the wave merge again into a circular front. Other numerical studies have shown that concentric Ca2+ waves can sometimes transform into spiral ones when encountering refractory blocks; however, this mechanism is much less likely to occur in real cells, as some very precise relationships between the respective locations of the refractory block and the Ca2+ front must be fulfilled.
That the microscopic spatial arrangement of the diverse processes
involved in the Ca2+ dynamics play
an important role in determining the global aspect of the
Ca2+ waves has already been
emphasized for various phenomena. For example, it has been shown that
the saltatory nature of the Ca2+
waves seen in HeLa cells (5) might be due to the inhomogeneous distribution of the
Ins(1,4,5)P3R
throughout the cytoplasm (5a, 15a). In hepatocytes, it has been
proposed that the Ca2+ waves
always originate from a specific locus, which differs from one cell to
the other, because this region possesses a larger density of
Ins(1,4,5)P3R
(24). Accordingly, in the present simulations, the block of higher
receptor density acts as the initiation site for the
Ca2+ waves. In our system, this
region (region
2 in Fig. 1) is the only one to be in
the oscillatory regime, as the rest of the cytoplasm is in an excitable
state; such a difference is obtained by varying the local maximal
velocity of Ca2+ release
(VM3 in
Eq.
2). In
Xenopus oocytes themselves, the
so-called "Ca2+ puffs" are
thought to originate from the opening of multiple Ins(1,4,5)P3R
gathered in clusters (20). Also, a gradient in the level of
Ins(1,4,5)P3
through the cell might explain the initiation point of the repetitive
propagating fronts (13) and could play a role in the existence of
kinematic Ca2+ waves (14). Thus
the spiral Ca2+ waves that are
frequently seen at the level of the entire
Xenopus oocyte might simply result
from the microscopic organization of the
Ca2+-releasing machinery.
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NOTE ADDED IN PROOF |
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Another plausible mechanism for spiral Ca2+ wave initiation has been recently proposed by A. McKenzie and J. Sneyd (Int. J. Bifurc. Chaos. In press). In this study, spiral waves are initiated by simulating the release of Ins(1,4,5)P3 at three different loci of the oocyte, in the absence of heterogeneity in the distribution of Ca2+ stores.
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ACKNOWLEDGEMENTS |
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I thank J. Lauzeral and J. Halloy for very fruitful discussions and A. Goldbeter for continuous support.
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FOOTNOTES |
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This work was supported by the "Actions de Recherche Concertée" Program (ARC 94-99) launched by the Division of Scientific Research, Ministry of Science and Education, French Community of Belgium.
G. Dupont is Chargé de Recherches at the Belgian Fonds National de la Recherche Scientifique.
Address for reprint requests: G. Dupont, Unité de Chronobiologie Théorique, Faculté des Sciences, Université Libre de Bruxelles CP231, B-1050 Brussels, Belgium.
Received 1 December 1997; accepted in final form 17 March 1998.
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REFERENCES |
---|
![]() ![]() ![]() ![]() ![]() ![]() |
---|
1.
Allbritton, N.,
T. Meyer,
and
L. Stryer.
Range of messenger action of calcium ion and inositol 1,4,5-trisphosphate.
Science
258:
1812-1815,
1992[Medline].
2.
Atri, A.,
J. Amundson,
D. Clapham,
and
J. Sneyd.
A single pool model for intracellular calcium oscillations and waves in the Xenopus laevis oocytes.
Biophys. J.
65:
1727-1739,
1993[Abstract].
3.
Berridge, M. J.
Inositol trisphosphate and calcium signalling.
Nature
361:
315-325,
1993[Medline].
4.
Bezprozvanny, I.,
J. Watras,
and
B. Ehrlich.
Bell-shaped calcium response curves of Ins(1,4,5)P3 and calcium-gated channels from endoplasmic reticulum of cerebellum.
Nature
351:
751-754,
1991[Medline].
5.
Bootman, M.,
E. Niggli,
M. Berridge,
and
P. Lipp.
Imaging the hierarchical nature of Ca2+ signalling in HeLa cells.
J. Physiol. (Lond.)
499:
307-314,
1997[Abstract].
5a.
Bugrim, A.,
A. Zhabotinsky,
and
I. Epstein.
Calcium waves in a model with a random spatially discrete distribution of Ca2+ release sites.
Biophys. J.
73:
2897-2906,
1997[Abstract].
6.
De Smedt, F.,
A. Boom,
X. Pesesse,
S. Schiffmann,
and
C. Erneux.
Post-translational modification of human brain type I inositol-1,4,5-trisphosphate 5-phosphatase by farnesylation.
J. Biol. Chem.
271:
10419-10424,
1996
7.
Dupont, G. Spatio-temporal organization of cytosolic
Ca2+ signals: from experimental to
theoretical aspects. Comments Theor.
Biol. In press.
8.
Dupont, G.,
and
A. Goldbeter.
One-pool model for Ca2+ oscillations involving Ca2+ and inositol 1,4,5-trisphosphate as co-agonists for Ca2+ release.
Cell Calcium
14:
311-322,
1993[Medline].
9.
Dupont, G.,
and
A. Goldbeter.
Properties of intracellular Ca2+ waves generated by a model based on Ca2+-induced Ca2+ release.
Biophys. J.
67:
2191-2204,
1994[Abstract].
10.
Dupont, G.,
J. Pontes,
and
A. Goldbeter.
Modeling spiral Ca2+ waves in single cardiac cells: role of the spatial heterogeneity created by the nucleus.
Am. J. Physiol.
271 (Cell Physiol. 40):
C1390-C1399,
1996
11.
Finch, E.,
T. Turner,
and
S. Goldin.
Calcium as a coagonist of inositol 1,4,5-trisphosphate-induced calcium release.
Science
252:
443-446,
1991[Medline].
12.
Girard, S.,
A. Lückhoff,
J. Lechleiter,
J. Sneyd,
and
D. Clapham.
Two-dimensional model of calcium waves reproduces the patterns observed in Xenopus oocytes.
Biophys. J.
61:
509-517,
1992[Abstract].
13.
Jacob, R.
Calcium oscillations in endothelial cells.
Cell Calcium
12:
127-134,
1991[Medline].
14.
Jafri, M.,
and
J. Keizer.
Diffusion of inositol 1,4,5-trisphosphate, but not Ca2+, is necessary for a class of inositol 1,4,5-trisphosphate-induced Ca2+ waves.
Proc. Natl. Acad. Sci. USA
91:
9485-9489,
1994
15.
Jafri, M.,
and
J. Keizer.
On the roles of Ca2+ diffusion, Ca2+ buffers, and the endoplasmic reticulum in IP3-induced Ca2+ waves.
Biophys. J.
69:
2139-2153,
1995[Abstract].
15a.
Keizer, J., and G. Smith. Spark-to-wave
transition: saltatory transmission of calcium waves in cardiac
myocytes. Biophys. Chem.
In press.
16.
Lechleiter, J.,
S. Girard,
E. Peralta,
and
D. Clapham.
Spiral calcium wave propagation and annihilation in Xenopus laevis oocytes.
Science
252:
123-126,
1991[Medline].
17.
Lipp, P.,
and
E. Niggli.
Microscopic spiral waves reveal positive feedback in subcellular calcium signalling.
Biophys. J.
65:
2272-2276,
1993[Abstract].
18.
Meyer, T.,
and
L. Stryer.
Calcium spiking.
Annu. Rev. Biophys. Biophys. Chem.
20:
153-174,
1991[Medline].
19.
Othmer, H.,
and
Y. Tang.
Oscillations and waves in a model of calcium dynamics.
In: Experimental and Theoretical Advances in Biological Pattern Formation, edited by H. Othmer,
J. Murray,
and P. Maini. London: Plenum, 1993, p. 277-313.
20.
Parker, I.,
and
I. Ivorra.
Confocal microfluorimetry of Ca2+ signals evoked in Xenopus oocytes by photoreleased inositol trisphosphate.
J. Physiol. (Lond.)
461:
133-165,
1993[Abstract].
21.
Sneyd, J.,
J. Keizer,
and
M. Sanderson.
Mechanisms of calcium oscillations and waves: a quantitative analysis.
FASEB J.
9:
1463-1472,
1995
22.
Tang, Y.,
J. Stephenson,
and
H. Othmer.
Simplification and analysis of models of calcium dynamics based on IP3-sensitive calcium channel kinetics.
Biophys. J.
70:
246-263,
1996[Abstract].
23.
Thomas, A.,
G. Bird,
G. Hajnoczky,
L. Robb-Gaspers,
and
J. Putney.
Spatial and temporal aspects of calcium signalling.
FASEB J.
10:
1505-1517,
1996
24.
Thomas, A.,
D. Renard,
and
T. Rooney.
Spatial and temporal organization of calcium signalling in hepatocytes.
Cell Calcium
12:
111-127,
1991[Medline].
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