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.

	DESCRIPTION OF THE SYSTEM

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

(1)

in which v_p is the rate of Ins(1,4,5)P₃ synthesis and  is the first-order constant denoting the rate of Ins(1,4,5)P₃ degradation. D_A stands for the diffusion coefficient of Ins(1,4,5)P₃ in the cytosol, the value of which has been measured in Xenopus oocytes (1). The two spatial coordinates are denoted x and y, and t is time. In the rest of the system (i.e., everywhere except region 1 in Fig. 1), Ins(1,4,5)P₃ is assumed only to diffuse, i.e., v_p =  = 0. The gradient in Ins(1,4,5)P₃ concentration is expected to create a gradient in excitability that will favor the occurrence of a spiral wave if the Ca²⁺ front possesses a free extremity (i.e., if a circular front has been broken); that the system exhibits differences in refractory times depending on the locus considered will indeed prevent the broken wave from reforming a concentric wave as it expends in this large system.

View larger version (182K): [in this window] [in a new window]   Fig. 1.   Typical geometry of system used to study mechanism of spiral Ca2+ wave initiation in Xenopus oocytes. Only 2 spatial dimensions are considered. Outer square represents a 250 × 250-µm portion of oocyte. Smaller inner square (region 1) is region in which D-myo-inositol 1,4,5-trisphosphate [Ins(1,4,5)P3] synthesis and metabolism occur. Larger inner square (region 2) possesses a higher density of Ins(1,4,5)P3 receptor than rest of system. For numerical integration, system is discretized in 270 × 270 grid points. In that frame, region 1 has a side of 30 grid points and region 2 has a side of 44 grid points. Top left corner of region 1 is grid point with coordinates (100, 80); top left corner of region 2 is grid point (137, 115). DCa2+ and DIP3, diffusion coefficients for Ca2+ and Ins(1,4,5)P3, respectively, in cytosol.

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

(2)

in which, as in previous studies (9, 10), V_M3 stands for the maximal rate of Ca²⁺ release and K_R and K_A are the threshold constants for release and activation, respectively. K_D is the half-saturation constant of the Ins(1,4,5)P₃ receptor, and  is an adimensional number that allows for a possible increase in the density of Ins(1,4,5)P₃R. Y and Z are the intraluminal and cytosolic Ca²⁺ concentrations, respectively. In the system schematized in Fig. 1,  = 1 everywhere except in region 2, in which  = 3.

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.

	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 (v_p) 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).

View larger version (98K): [in this window] [in a new window]   Fig. 2.   Numerical simulation of a spiral Ca2+ wave in a system that represents a portion of a Xenopus oocyte and has geometry shown in Fig. 1. Panels at top show how these waves first arise; panels at bottom represent the more regular spatiotemporal pattern, which is stable at least up to 300 s. Time (t) = 0 corresponds to time at which velocity of Ins(1,4,5)P3 synthesis (vp) in region 1 (smaller square, see Fig. 1) increased from 0 to 8 µM · s1. Initial conditions are at random for Ins(1,4,5)P3 and Ca2+. Color scale is linear between 0 (white) and 1.5 µM (black). Results were obtained by numerical integration of system defined in Ref. 10, with Eqs. 1 and 2, and with following parameter values: sum of basal and stimulated influx of Ca2+ from extracellular medium (Vin) = 2.7 µM · s1, maximal rate of Ca2+ pumping into endoplasmic reticulum (ER) (VM2) = 65 µM · s1, threshold constant for Ca2+ pumping (K2) = 1 µM, maximal rate of Ca2+ release from ER (VM3) = 600 µM · s1, threshold constant of release from ER (KR) = 2 µM, threshold constant of activation (KA) = 0.88 µM, passive flux of Ca2+ from cytosol to external medium (k) and from ER to cytosol (kf) = 10 and 1 s1, half-saturation constant of Ins(1,4,5)P3 (KD) = 1 µM, and n = m = 2 and p = 4, where n, m, and p are Hill coefficients for Ca2+ pumping, release, and activation of release, respectively. In region 1 (see Fig. 1) vp = 8 µM · s1 and the first-order constant denoting rate of Ins(1,4,5)P3 degradation () = 1 µM · s1, whereas both quantities are 0 everywhere else. In region 2 (larger square, see Fig. 1) the adimensional number that allows for a possible increase in density of Ins(1,4,5)P3R () = 3, whereas  = 1 everywhere else.

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.

View larger version (35K): [in this window] [in a new window]   Fig. 3.   Temporal evolution of local Ca2+ concentration in a grid point, whose coordinates are (180, 135), located in region of high Ins(1,4,5)P3R density (A) and of a grid point, with coordinates (120, 135), in bulk of system (B). Ca2+ dynamics are much more complex in region of high receptor density, which acts as a high-frequency pacemaker surrounded by an excitable system. Equations, parameters, and configuration are same as in Fig. 2.

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 v_p 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, v_p 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 (v_p =  = 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)

View larger version (91K): [in this window] [in a new window]   Fig. 4.   Numerical simulation of a spiral Ca2+ wave initiated by injection or photorelease of a poorly metabolizable analog of Ins(1,4,5)P3. Counterclockwise rotation of spiral wave is due to fact that Ins(1,4,5)P3 is diffusing from right side of block of higher receptor density, whereas in Fig. 2 it was diffusing from left side. At left, locations of these 2 regions are indicated [more central and larger square: higher density of Ins(1,4,5)P3R; smaller square: region in which level of Ins(1,4,5)P3 is initially (at t = 0) assumed to be at a high level of 22 µM]. Equations and parameters are same as in Fig. 2.

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).

	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 (V_M3 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.