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Copyright © 1999 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 77, Issue 1, 37-44, 1 July 1999

doi:10.1016/S0006-3495(99)76870-0

Biophysical Theory and Modeling

Impact of Mitochondrial Ca²⁺ Cycling on Pattern Formation and Stability

M. Falcke^*, J.L. Hudson^#, P. Camacho^§ and J.D. Lechleiter^¶, ,  

^* Max Planck Institute for Physics of Complex Systems, 01187 Dresden, Germany
^# Department of Chemical Engineering, University of Virginia, Charlottesville, Virginia 22903-2442 USA
^§ Department of Physiology, University of Texas Health Sciences Center at San Antonio, San Antonio, Texas 78284-7756 USA
^¶ Department of Molecular Medicine, Institute of Biotechnology, University of Texas Health Sciences Center at San Antonio, San Antonio, Texas 78245-3207 USA

Address reprint requests to Dr. James D. Lechleiter, Department of Molecular Medicine, Institute of Biotechnology, University of Texas Health Sciences Center, 15355 Lambda Drive, San Antonio, TX 78245-3207. Tel.: 210-567-7252; Fax: 210-567-7247.

Cytoplasmic Ca²⁺ is a ubiquitous second messenger that regulates multiple cellular processes (Tsien and Tsien, 1990,Berridge, 1993,Fewtrell, 1993,Putney and Bird, 1993,Pozzan et al,Clapham, 1995). In many cells, hormone-stimulated IP₃ production generates waves of intracellular Ca²⁺ release, which propagate across the cell (Cornell-Bell et al,Kasai and Augustine, 1990,Lechleiter et al,Lechleiter et al; Blatter and Wier, 1992,DeLisle and Welsch, 1992,Amundson and Clapham, 1993,Eidne et al,Yao and Parker, 1994,Nathanson et al,Wang and Augustine, 1995,Thomas et al). Our initial observation of rotating spirals and expanding target patterns of Ca²⁺ waves in Xenopus laevis oocytes demonstrated that intracellular Ca²⁺ release behaves as an excitable medium (Lechleiter et al,Lechleiter et al; Lechleiter and Clapham, 1992,Camacho and Lechleiter, 1995). Excitable media are locally stable such that a small perturbation results in the direct return of the system to the steady state. However, a supra-threshold disturbance causes the system to sojourn through a large nonlinear excursion before it returns to the stable steady state (Mikhailov, 1994). Target and/or spiral patterns of intercellular Ca²⁺ release have also recently been reported in rat liver tissue, in epithelial cells, and in hippocampal slice cultures (Sanderson et al,Robb-Gaspers and Thomas, 1995,Thomas et al,Harris-White et al), suggesting that the concept of an excitable medium is applicable to Ca²⁺ signaling in multicellular systems. Spatiotemporal Ca²⁺ signals have also been recently shown to control gene expression differentially (Gu and Spitzer, 1995,Dolmetsch et al,Li et al). Other biologically excitable systems exhibiting complex spatiotemporal patterns include the spread of excitation in heart muscle, the release of extracellular cAMP during Dictyostelium discoideum aggregation, and the K⁺ ion fluxes in retinal cells (Loomis, 1979,Devreotes et al,Goroleva and Bures, 1983,Davidenko et al,Rensing, 1993,Winfree, 1993,Karma, 1994,Newman and Zahs, 1997). Thus, a fundamental understanding of the mechanisms governing pattern selection as well as the creation and stability of spiral waves has been a topic of major interest in recent years (Rensing, 1993,Winfree, 1993,Karma, 1994), given their impact on such a wide variety of biological phenomena.

We have previously reported that changes in intracellular Ca²⁺ signaling are dependent on the rate of mitochondrial Ca²⁺ uptake (Jouaville et al). Increasing the rate of Ca²⁺ uptake by injection of respiratory chain substrates increases Ca²⁺ wave amplitude and velocity. Curiously, increased cytosolic Ca²⁺ sequestration increases the excitation threshold and once excited, mitochondrial Ca²⁺ uptake would be expected to decrease the peak amplitude and slow the wave velocity. In this paper we theoretically account for these seemingly paradoxical observations by incorporating the complete dynamics of mitochondrial Ca²⁺ cycling into the Tang and Othmer (TO) model of Ca²⁺ wave activity (Tang et al). Our simulations show that mitochondrial Ca²⁺ efflux is a significant determinant of pattern formation and that the cytosol can exhibit a bistable behavior.

Xenopus oocyte preparation and confocal imaging of intracellular Ca²⁺ was as previously described (Jouaville et al). Briefly, oocytes were injected with Calcium Green II (50 nl, ∼12.5μM final concentration, assuming a 1:20 dilution; Molecular Probes, Eugene, OR) 30–60min before each experiment. Images were acquired with a NORAN OZ confocal laser scanning microscope at zoom 0.7 attached to a Nikon Eclipse 200 with a 20× (0.75NA) Nikon objective lens at 1-s intervals. The confocal aperture was set at 15μm. Images were analyzed with ANALYZE software (Mayo Foundation, Rochester, MN) on a Silicon Graphics workstation. Ca²⁺ increases are reported as ΔF/F, which represents (F_peak−F_rest)/F_rest. Ca²⁺ wave activity was induced by injecting a 50-nl bolus of IP₃ (Calbiochem, San Diego, CA) of 6μM (∼300nM final). All recordings were made in the absence of extracellular Ca²⁺: 96mM NaCl, 2mM KCl, 2mM MgCl₂, 5mM Hepes (pH 7.5) (GibcoBRL, Grand Island, NY), 1mM EGTA (Sigma, St. Louis, MO).

We used a scaled version of the mathematical model for numerical calculations. Integrations were performed using a Euler forward scheme, with a spatial discretization of 0.125 and a time step of 0.0005. Onset of wave propagation was in general for a spatial discretization of 0.8. Stationary states were determined as the concentration values yielding the right-hand sides of the equations in the Appendix set equal to 0. Their stability was determined by the eigenvalues of the Jacobian matrix.

Most current models of Ca²⁺ signaling (Dupont et al,DeYoung and Keizer, 1992,Atri et al,Li and Rinzel, 1994,Tang et al) are based on the observation that at low Ca²⁺ concentrations, IP₃ and Ca²⁺ work as co-agonists leading to Ca²⁺-induced Ca²⁺ release (CICR), while at high concentrations, Ca²⁺ inhibits further Ca²⁺ release and the IP₃R becomes refractory (Iino, 1990,Parker and Ivorra, 1990,Bezprozvanny et al,Finch et al). The TO model is also based on this dual modulatory role of Ca²⁺ on IP₃-mediated Ca²⁺ release (Tang and Stephenson, 1996). We extended the TO model of Ca²⁺ signaling to incorporate the mechanisms of mitochondrial Ca²⁺ cycling by adding a third equation governing the uptake and release of Ca²⁺ by mitochondria and a corresponding term in the differential equation for cytosolic Ca²⁺ (see Appendix ). The third equation was empirically formulated, essentially based on the experimental data reviewed by Gunter and Pfeiffer, 1990. The reader is referred to a series of papers published by Magnus and Keizer for a detailed biophysical model of mitochondrial Ca²⁺ handling (Magnus and Keizer, 1997,Magnus and Keizer, 1998a,Magnus and Keizer, 1998b). An important feature of mitochondrial Ca²⁺ cycling is that Ca²⁺ uptake and efflux are distinct pathways, the latter being 10–100 times kinetically slower (Gunter and Pfeiffer, 1990). The Ca²⁺ uptake mechanism is believed to be a uniporter that facilitates the diffusion of Ca²⁺ down the electrochemical gradient across the mitochondrial membrane (Gunter and Pfeiffer, 1990). Its dependence on cytosolic Ca²⁺ is modeled as a Hill function with the higher-order kinetics associated with cooperativity and saturation (Bygrave et al,Scarpa and Graziotti, 1973). A Hill coefficient of 2 is used based on a review of experimental data (Gunter and Pfeiffer, 1990). The value for the maximum uptake velocity (V_max⁽¹⁾) were chosen according to Marinos and Billett, 1981, Marinos, 1985, and Gunter and Pfeiffer, 1990. We assume that the relevant Ca²⁺ efflux mechanism for mitochondria in Xenopus oocytes is the Na⁺/Ca²⁺ exchanger. This process was also modeled as a Hill function using a second-order Na⁺ dependence (Gunter and Pfeiffer, 1990). Consequently, the exchanger is electroneutral and the Ca²⁺ efflux does not depend on the mitochondrial membrane potential. A wide range of values, from 1 to 189μM, have been reported for the half-maximum value (K_d) (Gunter and Pfeiffer, 1990). However, physiological (Rizzuto et al,Rizzuto et al,Jouaville et al) and morphological evidence (Satoh et al) indicates that mitochondria are located in close proximity to the ER, where they experience Ca²⁺ concentrations considerably higher than those in bulk cytoplasm. Rizzuto et al,Rizzuto et al estimated that the uptake velocity of Ca²⁺ released from internal stores was an order of magnitude higher than that resulting from the average bulk concentration of Ca²⁺. We incorporated locally high Ca²⁺ concentrations into the model by multiplying the value of the Ca²⁺ concentration in the mitochondrial uptake term by a factor of 2.5, based on the data published by Rizzuto et al. This correction is equivalent to rescaling the half-maximum value of K_d to K_d/2.5. The resulting small value of K_d was essential for obtaining the results presented below.

In the experiments we modeled, mitochondria were energized by injection of oxidizable substrates increasing the membrane potential by ∼30mV (Jouaville et al). This corresponds to an increase of V_max⁽¹⁾ by about a factor of 5. We present results for different values of V_max⁽¹⁾ to parallel these experiments with and without energization of the mitochondria. Figure 1A shows the effect of mitochondria Ca²⁺ cycling on pulse profiles. For high mitochondrial Ca²⁺ uptake, the simulated pulse shows two phases of Ca²⁺ decay (Figure 1A), which were also observed experimentally (Jouaville et al). First, a rapid decay phase of cytosolic Ca²⁺ caused by Ca²⁺ uptake into the ER and mitochondria occurs (Camacho and Lechleiter, 1993,Hehl et al,Herrington et al). The second, slower phase of cytosolic Ca²⁺ decay can be attributed to mitochondrial Ca²⁺ efflux. The net flux of Ca²⁺ into mitochondria is initially inward, since Ca²⁺ efflux is much slower than uptake. However, as IP₃-mediated Ca²⁺ release decreases, the net movement of Ca²⁺ changes to mitochondrial efflux. This results in a prolonged elevation of cytosolic Ca²⁺ and delays the recovery of the IP₃R from the refractory state. For comparison, the dashed line in Figure 1A shows the Ca²⁺ pulse profile without energization of the mitochondria. These simulations suggest that mitochondrial Ca²⁺ cycling may modulate Ca²⁺ signaling by prolonging the recovery time of the IP₃ receptor.

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Figure 1

(A) Comparison of simulated of Ca²⁺ pulse profiles between modified and unmodified TO model (Tang et al). The pulses travel from right to left. The solid line represents a simulation taking into account mitochondrial Ca²⁺ cycling (V_max⁽¹⁾=28μM s⁻¹) and the dashed line represents a simulation without mitochondrial Ca²⁺ cycling (V_max⁽¹⁾=0μM s⁻¹). The amplitude of both pulses was normalized using their respective maximum. (B) Dependence of velocity (dashed line) and amplitude (solid line) of periodic Ca²⁺ pulses on frequency calculated with V_max⁽¹⁾=8.0μM s⁻¹. (C) Dependence of Ca²⁺ spiral wave velocity (circles), frequency (crosses), and amplitude (triangles) on V_max⁽¹⁾ for low Ca²⁺ mitochondrial uptake (V_max⁽¹⁾<V_max,cr⁽¹⁾) normalized to the value V_max⁽¹⁾=0μM s⁻¹.

Two fundamental characteristics of an excitable medium are the dependence of the wave velocity (v) on the curvature of the wavefront and on the frequency (f) of periodic wave trains (Dockery et al). The latter is called the dispersion relation v(f). Simulations of repetitive Ca²⁺ waves at various levels of stimulation exhibited the expected dependence of wave velocity on frequency (Figure 1B) observed in many other excitable systems (Mikhailov, 1994). As the frequency of the Ca²⁺ waves increases, both the velocity and amplitude decrease.

In Xenopus oocytes, stable spirals are observed under conditions of low mitochondrial Ca²⁺ uptake (Lechleiter et al,Lechleiter et al; Lechleiter and Clapham, 1992,Camacho and Lechleiter, 1995). Our model simulations also indicate that spiral patterns are stable under these conditions. The formation of spiral wave patterns is due to the curvature-velocity relationship (Dockery et al); spirals originate at the free end of propagating waves, where the wavefront curvature is largest, resulting in lower velocity and the formation of the trademark curling pattern. With increasing V_max⁽¹⁾, a decrease in rotational frequency is accompanied by a small increase in wave velocity and a small decrease in wave amplitude (Figure 1C); this behavior only holds for small values of V_max⁽¹⁾, below a critical value to be described in the following paragraph.

When mitochondria are energized in Xenopus oocytes, spiral wave patterns become unstable, disappear, and do not reform. This model predicts, in agreement with these experiments, that spirals cease to exist at a certain critical value of mitochondrial Ca²⁺ uptake (V_max,cr⁽¹⁾=14μM s⁻¹). Above this value, it is found that waves emitted from pacemakers form the pattern. Examples of these waves, as observed in experiments and simulation, are shown in Figure 2AB, and at the website http://www.mpipks-dresden.mpg.de/(falcke/thesite.html. A simulation was also carried out corresponding to an experiment (Jouaville et al) in which pyruvate/malate was injected into an oocyte. In this simulation, V_max⁽¹⁾ was increased. The changing Ca²⁺ concentration at a point is shown in Figure 2C. In two-dimensional simulations, the Ca²⁺ oscillations associated with spiral waves cease as the system moves to a new steady state. This is followed by the dominance of waves emitted from a pacemaker.

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Figure 2

Comparison of two-dimensional simulated Ca²⁺ waves with empirically obtained data in Xenopus oocytes. (A) Image of Ca²⁺ wave activity initiated by IP₃ in an oocyte in which mitochondria are energized by injection of pyruvate/malate (10mM final concentration). Experimental conditions and protocols were as previously described (Jouaville et al). (B) Simulation of Ca²⁺ pacemaker activity from a focus located in the upper left corner of the figure. This simulation takes into account high mitochondrial Ca²⁺ uptake conditions (V_max⁽¹⁾=8.0μM s⁻¹) for an area of 700×700μm². (C) Local time series of Ca²⁺ concentration for the simulation shown in (B). At the time indicated by the arrow (t=150 s), mitochondrial Ca²⁺ uptake was increased (V_max⁽¹⁾=16.0μM s⁻¹) to model an experimental injection of pyruvate/malate (Jouaville et al). After a transient, the spirals disappeared and the pacemaker activity dominated pattern formation. In this high mitochondrial activity mode, increases in amplitude and period are observed. The velocity of wave propagation also increases from 14μms⁻¹ to 23μm s⁻¹. See also http://www.mpipks-dresden.mpg.de/(falcke/thesite.html for the complete simulation.

Figure 3A shows the transient state of a spiral wave tip when Ca²⁺ uptake exceeds the critical value (V_max⁽¹⁾>V_max,cr⁽¹⁾). When the tip bends in the early stage of spiral formation, another small amplitude wave emerges from the back of the wave at the highest curvature (indicated by white arrow). Mitochondrial Ca²⁺ efflux is responsible for this secondary wave, which in turn is responsible for prolonging the refractory state of the IP₃R and preventing spiral formation. Although efflux plays a fundamental role in the destabilization of the spiral core, it is not the sole determinant. Planar waves exist at frequencies higher than those at which spiral waves occur. This indicates that wavefront curvature also contributes to spiral core instability. Near the spiral tip, where the wavefront curvature is the highest, Ca²⁺ efflux is focused. This focal increase in Ca²⁺ further prolongs the refractory period of IP₃Rs. Thus, both curvature and mitochondrial efflux are responsible for the generation of the secondary wave which forces the tip outward, thereby preventing spiral pattern formation (Figure 3B). This phenomenon was experimentally observed in the oocyte after energization as shown in Figure 3CD. The free end of a Ca²⁺ wave is forced outward by a secondary Ca²⁺ wave and the spiral fails to form.

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Figure 3

Stability and formation of the spiral wave core. (A) Simulated development of the free end of a spiral at high mitochondrial Ca²⁺ uptake (V_max⁽¹⁾=20μM s⁻¹) for an area of 500×500μm². When the curvature becomes too high, a secondary wave appears (white arrow) and prevents further development of the spiral by forcing the tip outward. (B) State of the spiral core 7s after the simulation shown in (A). (C) Experimental image of an unstable spiral core in an oocyte. Ca²⁺ wave activity was initiated by simultaneous injection of IP₃ (6μM final concentration) and pyruvate/malate (10mM final). Notice that the free end of the Ca²⁺ wave fails to form a spiral pattern. (D) Image of the experimental Ca²⁺ wave shown in (C) collected 3s later. Note that the tip of the free end was forced outward by the unstable core, as simulated in (B). (E) Bistability of Ca²⁺ wave activity in Xenopus oocytes with energized mitochondria. IP₃-induced Ca²⁺ wave activity observed in an oocyte preinjected with pyruvate/malate (10mM final concentration). The direction of wave propagation is indicated by the white arrow. The fluorescence intensity (ΔF/F) of the white frame is plotted in (F).

When periodic wave patterns of different frequencies are present in a medium, they compete for space. As time goes on, the pattern with the highest frequency generally gains spatial control of the field (Mikhailov, 1994). We recently showed that spiral waves dominate pacemakers in Xenopus oocytes (Lechleiter, 1998); this indicates that the former have higher frequencies. Thus, it is only after the spirals disappeared above V_max,cr⁽¹⁾—when mitochondria are energized—that the lower-frequency pacemakers can govern the pattern formation in the oocyte. The dependence of the amplitude, frequency, and velocity of waves in oocytes on the state of energization of mitochondria can now be readily explained. Energization results in a wave pattern dominated by slow pacemakers. The smaller frequency leads to an increase in wave amplitude and velocity, according to the dispersion relation.

The local dynamics of our model yield three stationary states, each with different concentrations of cytosolic Ca²⁺. At low mitochondrial Ca²⁺ uptake (small V_max⁽¹⁾), only the state with the lowest cytosolic Ca²⁺ is stable and the cytosol behaves as an excitable system. Our calculations indicate that the system becomes bistable at the uptake value V_max,b⁽¹⁾=9.6μM s⁻¹ (i.e., V_max,b⁽¹⁾<V_max,cr⁽¹⁾). At this point, the stationary state with the highest cytosolic Ca²⁺ concentration is stabilized by increased mitochondrial Ca²⁺ cycling. The system now has two stable stationary states. When both states exist at adjacent locations, the interface moves so that the volume occupied by one of the states grows at the expense of the other (see Mikhailov, 1994 for bistable systems in general). This moving interface is called a front and the state, which loses volume, is termed metastable. Whether the system switches by a front from low to high cytosolic Ca²⁺, or vice versa, depends on the degree of mitochondrial energization. In most of the bistable region that we consider, the state of high cytosolic Ca²⁺ is metastable. In our bistable system, both waves (pulses) and fronts occur and below V_max,cr⁽¹⁾ spirals form. Above V_max,cr⁽¹⁾, the region of high Ca²⁺ can expand if it is surrounded by a wave, even though it is the metastable state. Thus, a front of transition from low to high Ca²⁺ can occur in this parameter range if it immediately follows a wave. This occurs when the unstable spiral core expands (Figure 3A–D). If the wave leading the front is extinguished by collision with another wave, the front reverses its direction of motion. Another way that this patch of high Ca²⁺ in Figure 3B disappears is that a pacemaker inside it starts a front that returns the region to a state of low Ca²⁺. Finally, this creates a pattern in which the waves emitted by pacemakers become the dominant structure of the bistable system http://www.mpipks-dresden.mpg.de/~falcke/thesite.html).

At very high energization of the mitochondria (V_max⁽¹⁾>16.4μM s⁻¹), the simulations show that fronts from low to high cytosolic Ca²⁺ continue to exist outside the spiral core. This indicates that the region of high cytosolic Ca²⁺ emerging from the spiral instability continues to expand even if the leading pulse becomes annihilated. Experimental evidence for such a transition in oocytes is shown in Figure 3E and F. Fronts from high to low Ca²⁺ cease to exist at V_max⁽¹⁾=23μM s⁻¹, i.e., for 16.4μM s⁻¹<V_max⁽¹⁾=23μM s⁻¹ fronts in both directions and waves co-exist. Which waveform arises depends on the situation initiating it. If a front is initiated at V_max⁽¹⁾>23μM s⁻¹, the resting Ca²⁺ concentration in the oocyte is predicted to switch to the stationary state with high cytosolic Ca²⁺, and wave activity stops.

The mechanism of mitochondrial-induced spiral instability described above suggests that spirals could be recovered by increasing cytosolic Ca²⁺ removal. Experimental studies in Xenopus oocytes show that overexpression of Ca²⁺-ATPases permits spiral wave formation even in the presence of energized mitochondria (Figure 4A). We simulated increased SERCA expression in our model by factoring an increase of 10% in the density of SERCAs. Calculations were performed assuming high mitochondrial Ca²⁺ uptake, where spiral wave formation is unstable. Consistent with our experimental observations, the simulation shows that an increase in SERCA density restores spiral formation (Figure 4B). This observation also supports the mechanism proposed above, in which pattern instability is attributed to increased mitochondrial Ca²⁺ cycling.

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Figure 4

(A) Overexpression of SERCA 2b rescues IP₃-induced spiral pattern formation of Ca²⁺ waves in an oocyte preinjected with pyruvate/malate. (B) Simulation of a spiral Ca²⁺ wave at high mitochondrial Ca²⁺ uptake (V_max⁽¹⁾=20μM s⁻¹) and increased SERCA density, which was simulated by factoring into the calculation a 10% higher maximum Ca²⁺ pump rate (P_max^r=5.9μM s⁻¹).

In summary, we have resolved an experimental paradox on the effects of energization of mitochondria on Ca²⁺ wave activity in Xenopus oocytes. Namely, that increased mitochondrial Ca²⁺ sequestration leads to increased Ca²⁺ wave amplitude and velocity (Jouaville et al). This outcome is theoretically predicted when both mitochondrial Ca²⁺ uptake and Ca²⁺ efflux are incorporated into the TO model of Ca²⁺ activity. Our simulations also demonstrate that mitochondria play a critical role in pattern formation and stability. We find that increased mitochondrial Ca²⁺ efflux destabilizes the formation of spirals. Furthermore, energization of mitochondria creates a bistable cytosol in which resting Ca²⁺ concentrations can stably exist at levels higher than normal. Interestingly, the sperm-induced Ca²⁺ wave in Xenopus eggs has also recently been modeled as a bistable system (Fontanilla and Nuccitelli, 1998,Wagner et al). Taken together, these model simulations provide a new understanding of the role played by mitochondrial Ca²⁺ cycling in pattern formation and in controlling intracellular Ca²⁺ levels.

Appendix A

Mathematical model of Ca²⁺ signaling incorporating mitochondrial Ca²⁺ cycling

In the TO model (Tang and Othmer, 1994,Tang et al), Ca²⁺ efflux of the endoplasmic reticulum (ER) is described by the first term of the c-dynamics (Eq. (A1)). The dependence of Ca²⁺ release on both IP₃ and Ca²⁺ is modeled by assuming one site for IP₃ binding, one activating and one inhibitory site for Ca²⁺ binding. The channel is open when IP₃ and Ca²⁺ are bound at the activating site. Binding of Ca²⁺ at the inhibitory site closes the channel. It is assumed that IP₃ is spatially and temporally constant since experimental data show that the turnover rate of IP₃ in Xenopus oocytes is of the order of minutes (Allbritton et al). Extrusion of Ca²⁺ through the activity of plasma membrane ATPases and exchangers is modeled as a small leak term. The TO model assumes that the efflux of Ca²⁺ from the ER is proportional to the difference between lumenal and cytosolic Ca²⁺ concentration. This provides an additional feedback of the ER Ca²⁺ content on cytosolic Ca²⁺ dynamics. Once Ca²⁺ is released from the ER into the cytosol, the cycle of Ca²⁺ dynamics is closed by cytosolic Ca²⁺ removal. Resequestration of Ca²⁺ into internal stores is modeled by adding a term for sarcoendoplasmatic reticulum Ca²⁺-ATPases (SERCAs) activity.

The modified TO model of Ca²⁺ signaling is given in Eqs. (A1). We present the unscaled model for clarity. Dimensionless time and space were obtained in the standard manner by scaling with epsilon and the square root of (diffusion coefficient)/epsilon.

(A1)

(A2)

(A3)

The Ca²⁺ concentrations in the cytosol, ER, and mitochondria are denoted c, c_r, and m, respectively. The ratio of the effective volume of the ER to the cytosolic is v_r=0.185. The effective volume of all mitochondria to the cytosolic volume is v_m=0.1. C_M is the total amount of Ca²⁺ in the oocyte (c+v_rc_r+v_mm) divided by the cell volume (1+v_r+v_m). C_M=1.56μM and is assumed constant (i.e., no Ca²⁺ flux occurs across the cell membrane). This allows the Ca²⁺ concentration inside the ER to be represented by c_r=((1+v_r+v_m)C_M−c−v_mm)/v_r. Consequently, c_r does not appear explicitly in the equations. P_r^leak=0.0097s⁻¹ is the leakage permeability coefficient of the ER. The fraction of inhibited IP₃R ion channels is denoted by n. P_r^chan=3.89s⁻¹ denotes the channel permeability coefficient of the IP₃R. β₀=2.96, β₁=0.12μM, and β₂=0.1μM are the dissociation constants for binding of IP₃ to the IP₃R, Ca²⁺ to the activating site and Ca²⁺ to the inhibiting site of the IP₃R, respectively. β₀ depends on the IP₃ concentration (=0.27μM, see Tang et al for details). ϵ=0.15s⁻¹ is the rate constant for the release of Ca²⁺ from the inhibiting site of the IP₃R. For the Ca²⁺-ATPases in the ER, P_r^max=5.31μMs⁻¹ is the maximum pump rate of the SERCAs and K_r=0.0296μM is the corresponding half-maximum value. D=50μm²s⁻¹ is the effective diffusion coefficient of Ca²⁺. For mitochondrial Ca²⁺ uptake, V_max⁽¹⁾ was varied according to the change of mitochondrial uptake in experiments, and the half-maximum value of mitochondrial Ca²⁺ uptake is K_d=1.5μM. For mitochondrial Ca²⁺ efflux, the maximum release velocity of the Na⁺/Ca²⁺ exchanger is V_max⁽²⁾=3μMs⁻¹ and the half-maximum value of its dependence on m is K_m=1μM. The Na⁺ concentration is 10mM and the half-maximum value of the Na⁺ concentration dependence of the Na⁺/Ca²⁺ exchanger is K_Na=5mM; ∇ denotes the nabla operator.