View - Research

132 downloads 79 Views 6MB Size Report
Ca2+ release in the form of regenerative circular and spiral waves of Ca2+ ...... Biologi- cally, the foci for Ca2+ release may provide the cell with a mechanism for  ...
Cell, Vol. 69, 263-294,

April 17, 1992, Copyright

0 1992 by Cell Press

Molecular Mechanisms of Intracellular Excitability in X. laevis Oocytes James D. Lechleiter’ and David E. Clapham Department of Pharmacology Mayo Foundation Rochester, Minnesota 55905

Summary Following receptor activation in Xenopus ooctyes, spiral waves of intracellular Ca2+ release were observed. We have identified key molecular elements in the pathway that give rise to Ca*+ excitability. The patterns of Ca*+ release produced by GTP-r-S and by inositol 1,4,5-trisphosphate (IPI) are indistinguishable from receptor-induced Ca*+ patterns. The regenerative Ca*+ activity is critically dependent on the presence of IPs and on the concentration of intracellular Ca2+, but is independent of extracellular Ca2+. Broad regions of the intracellular milieu can be synchronously excited to initiate Ca2+ waves and produce pulsating foci of Ca2+ release. By testing the temperature dependence of wavefront propagation, we provide evidence for an underlying process limited by diffusion, consistent with the elementary theory of excitable media. We propose a model for intracellular Ca2+ signaling in which wave propagation is controlled by IP3-mediated Ca2+ release from internal stores, but is modulated by the cytoplasmic concentration and diffusion of Ca2+. Introduction Intracellular Ca’+ release is a familiar convergence point for many receptor-induced cell signals, controlling processes ranging from secretion and heart rate to transcription and cell division (Berridge, 1987; Berridge and Irvine, 1989). However, the molecular mechanism(s) controlling Ca*+ release are incompletely understood, and still less is known about how specific cellular instructions are encoded. Previous work has suggested that cell signals are encoded in periodic Ca2+ changes (oscillations), which are highly organized both temporally and spatially (Berridge et al., 1988; DuPont and Goldbeter, 1989; Berridge, 1990; Petersen and Wakui, 1990; Tsien and Tsien, 1990; Berridge and Moreton, 1991; Cuthbertson and Chay, 1991; Meyer, 1991; Tsunoda, 1991). Using confocal imaging techniques, we recently described the process of receptor-induced Ca2+ release in Xenopus laevis oocytes (Lechleiteret al., 1991a, 1991 b). We found that these large cells exhibited complex spatial and temporal patterns of Ca2+ release in the form of regenerative circular and spiral waves of Ca2+ release. We proposed that these complex patterns were generated by an underlying excitable medium composed of Ca2+ release processes. *Present address: Department of Neurosciences, ginia School of Medicine, Charlottesville, Virginia

University 22908-0002.

of Vir-

Calcium

Excitability is a property common to other chemical and biological systems, including the Belousov-Zhabotinsky (BZ) reaction, aggregating slime mold Dictyostelium discoideum, as well as in the electrical activity in cardiac and neural cells (Zaikin and Zhabotinsky, 1970; Allesie et al., 1973; Devreotes et al., 1983; Goroleva and Bures, 1983; Winfree, 1987). Any process that Undt?rgOeE a large excursion away from and then back to steady-state, when perturbed by a suprathreshold stimulus, may be considered excitable. An excitable medium is defined as a population of excitatory processes, coupled byacommon stimulatory signal through diffusion (Winfree, 1990). In such a medium, suprathreshold stimuli are propagated from one excitatory process to the neighboring excitatory processes by the coupling signal, creating waves of excitation. Small subthreshold perturbationsawayfrom steady-state, on the other hand, are quickly damped out. Using excitability as a model for Ca2+ release, we previously obtained estimates for a refractory period of excitability, the minimal area necessary to initiate wave propagation (critical radius), and a diffusion constant of the excitatory signal suggesting that Ca2+ itself was the coupling stimulatory signal for receptor-induced Ca2+ release in Xenopus oocytes (Lechleiter et al., 1991b). The first objective of the work presented here was to identify, at the molecular level, the key elements involved in Ca2+ excitability. To accomplish this, we directly injected second messengers into the oocytes and released caged compounds by UV laser scanning (Bliton et al., 1992) thus bypassing receptor activation and directly manipulating the Ca2+ release machinery. The second objective of this work was to test some of the predictions of our model for intracellular Ca’+ release. To this end, we artificially induced synchronous excitation of Ca*+ stores in a band defined by the caged release of inositol 1,4,5-trisphosphate (IPs) with UV laser scanning. We also examined the effects of temperature dependence of Ca2+ wave propagation. The combined results of these experiments indicate the predominant roles of IP3 receptor release sites and Ca2+ as key regulators of the Ca*+ release process in oocytes. Results G Protein-Mediated Calcium Excitability Complex spatiotemporal patterns of Ca2+ release are clearly present in hormone receptor-mediated signaling in Xenopus oocytes. We have interpreted these data as evidence for an excitable medium composed of the collection of excitable processes of Ca2+ release. Theoretically, such a medium is composed of a homogenous distribution of excitable processes (Winfree, 1990). However, the evidence for this model of Ca*+ excitability could be affected by a complex spatial distribution of receptors. We wanted to examine the dependence of pattern formation on receptor distribution, by bypassing receptor-induced Caz+ release and directly activating G protein-mediated signal

Cdl 284

Figure 1. G Ca% Activity

Protein-Induced

Regenerative

(A and B) Spatial patterns of Ca2+ release for single optical slices recorded at 340 and 190 s, respectively. (C and D) Stereo view of the spatiotemporal pattern of Ca% release. Caz+ activity was rendered as a volume by sequentially stacking 550 images, captured at 1 s intervals, of a single optical slice in the x-y plane of the oocyte. By presenting a volume of Caz+ activity, it is possible to display the temporal changes of Ca2+ release in the space of a single figure, compared with the space required for the sequential display of hundreds of frames as in (A) and (B) (see text). A brightest-pixel algorithm was also used for presentation. This routine displaysonly the brightest pixel along theviewers line of sight and adds depth to the twodimensional image (Lechleiter et al., 1991a). Resting Cap+ isshown in blue, Ca2;’ increases in green, and purple indicates time when oocyte was scanned with UV light. Color scale intensity is shown in (E), where the absolute intensity of the central pixel is plotted; only 100 of a total of 255 intensity levels are shown. The z-axis is labeled in seconds.

transduction. Xenopus oocytes were simultaneously injected (50 nl) with the Ca2+ dye indicator flue-3 (25 nl of 1 mM; ~25 uM final concentration) and caged GTPr-S (25 nl of 66 mM; ~1.65 mM final concentration). After a 2030 min equilibration period, a single optical slice (760 x 760 x 40 urn) near the plasma membrane surface of the oocyte was confocally imaged at 1 s intervals (Figure 1). The bath solution contained less than 10 nM free Ca2+ to exclude extracellular Ca2+ as a significant source for intracellular Ca2+ release. From 26 to 30 s, the optical slice was laser scanned with ultraviolet (UV) light (shown in purple) to release GTP-r-S uniformly throughout the imaging plane. The integrated dwell time of the UV laser was less than 10 tus at any one location. A focal increase in Ca*+ was immediately apparent at 31 s. From this focus, a wave of Ca’+ irregularly propagated, in bursts, across the imaging plane. The Ca2+ concentration remained high and fairly constant throughout the image until 130-150 s, where it began to decline. At this time, several distinct regions distributed throughout the oocyte began to exhibit regenerative focal activity, initiating waves of Ca2+ release, propagation, and annihilation. This period of time is the regenerative phase of Ca’+ activity. The particular region that showed Ca* release immediately after UV scanning also developed into a prominent pulsating focus of Ca2+ release, producing circular patterns of wave propagation (Figure 1B). The ends of some incomplete arcs of Ca2+ developed into spiral waves (Figure 1A; see below). The complete temporal change in Ca2+ was rendered as a stereo volume by stacking 550 sequential images of the same optical slice, captured at 1 s intervals (Figures IC and 1 D). In these volumes of Ca2+ activity, time is represented by the z-axis. This form of presentation compresses over time Caw activity that is normally shown sequentially in hundreds of frames, as in Figures 1A and 16, into the space of a single figure. Interestingly, the peak

amplitudes for individual Ca2+ waves occurred during the regenerative phase of C$+ activity. This is demonstrated in Figure 1 E, where the intensity of a single pixel, located in the center of the imaging field, is plotted for the entire sequence of images. As the cytoplasmic concentrations continued to drop toward resting levels, the number of pulsating foci decreased. Prior focal regions still supported wave propagation, but the reduction in activity caused fewer annihilations and permitted longer, unbroken arcs of C3+ waves. The waves continued to decrease in frequency over time but were still present 20 min after the initial release of GTP-Y-S (data not shown). The spatiotemporal patterns of this and 7 of 6 other oocytes induced by uncaged GTP+S were qualitatively similar to receptor-induced patterns. In the one exception, an initial Cap+ wave was observed, but no regenerative activity occurred in its wake. We quantitated this similarity by examining the effects of curvature on wavefront velocities, since increasing curvature is predicted to result in faster propagation speeds in an excitable medium (Zykov, 1980; Keener, 1986). We restricted our analysis to circular patterns of pulsating Ca2+ release that were less than 4 s old. At greater times, wavefront edges were often less distinct, due to the slowing of wavefront velocity by partially recovered excitable regions. This dependence of propagation speed on the frequency of activity is referred to as the dispersion effect (Miller and Rinzel, 1981; Dockery et al., 1988). Ca*+ patterns were then processed by sequentially subtracting consecutive images, producing active Ca2+ wavefronts of 1 s duration. In Figure 2A, an expanding circular wavefront is presented. For clarity, only the wavefront edge plus one-fourth of the previous edge is shown in thelefthandcolumn. Fromthisfigure,it isalreadyapparent that larger velocities (note the difference in radii between wavefront edges) are associated with the larger curvatures at 3 and 4 s, compared with 1 and 2 s. A plot of curvature

Intracellular 285

Calcium

Excitability

hormone receptor-mediated parameters for Ca2+ waves (Lechleiter et al., 1991 b). In terms of Ca2+ release, it will be interesting to determine whether these parameters are conserved in other cell types or if Ca2+ activity is scaled by cell size. We concluded from these experiments that Ca2+ excitability induced by receptors and that induced by uncaged GTP-r-S were indistinguishable, and that the excitable processes of Ca*+ release were distal to receptor activation. We focused the next series of experiments on the role of IPs in Ca2+ excitability, since receptor G protein coupling likely stimulated CaZ+ release through phosphatidyl inositol turnover (Berridge and Irvine, 1989).

- 40

B

-z

- 30 2 25 -20 A .t - 10 58 >

GTP-y-S I

I

I

I

-.04

703

-.02

-.Ol

curvature Figure 2. Curvature Induced Ca2+ Waves (A) Expanding only wavefront Images were above resting (B) Curvature each indicated from (A).

Effect

(LO

0

(pm-l)

on Wavefront

Propagation

for GTP--r-S-

circular wave pattern shown at O-4 s. Left column shows edges plus one-fourth of the previous wavefront edge. sequentially subtracted to define the wavefront edge Ca% levels. versus velocity plot of four expanding circular waves, by different symbols. Closed circles are data plotted

versus velocity from four expanding Ca*+ wavefronts is shown in Figure 28. Here the difference in radii was used as an estimate of velocity, and the inverse of the radii midpoints was used as an estimate of the curvature. In agreement with theory, we observed an approximately linear relationship between increasing curvature and increasing velocity. A linear regression fit of these data gave a mean planar velocity (zero curvature) of 28 urn/s, a mean critical radius of 14.2 urn (zero velocity), and a mean diffusion constant of the excitatory signal (line slope) of 3.97 x lO-6 cm2/s. These estimates were quantitatively similar to

lPsDependent Ca*+ Excitability To examine the role of inositol triphosphates in Ca2+ excitability, preinjected oocytes (~50 uM flue3, -final concentration) were again injected 20-30 min later with either IPs (50 nl of 20 uM, ~1 uM final) or the nonhydrolyzable analog, inositol 1,4,5trisphosphorothioate(lP&) (also ml uM final), and then imaged confocally as described above. In short, the resultant spatiotemporal patterns of Ca*+ release induced by both IPs (9 of 12 oocytes; cf. Figure 3) and IP& (15 of 16 oocytes; cf. Figure 4) were similar to receptor- and G protein-induced Ca*+ activity. The first observation was generally a Ca*+ wave that propagated throughout the imaging plane, producing an elevated cytoplasmic Ca2+ concentration in its wake (cf. Figure 38). Second, the cytosolic Ca2+ decreased and regenerative Cd+ activity developed (Figures 3A, 3C, 3D, and 4B-4D; note the differences in time scale). The magnitude of individual wavefronts and the number of pulsating foci initially increased, but gradually, over a period of minutes, the trend reversed, with the number and frequency of pulsating foci decreasing. The magnitudes of the remaining Ca*+ wavefronts were not significantly smaller than peak amplitude (< 10%) for either IPa- or IP&-induced regenerative activity. This suggests that the metabolism of IPs is not responsible for the cessation of Ca*+ activity. To investigate further whether IP3-induced regenerative activity was similar to receptor- and G protein-mediated activity, the dependence of wavefront velocity on curvature was determined. Analysis of four expanding circular wavefronts from an oocyte injected with IP& yielded a mean planar velocity of 29.5 urn/s, a mean critical radius of 9.8 urn, and a mean diffusion constant for the excitatory signal of 2.94 x 1O+ cm2/s. The similarity between these estimates and those reported above suggested that Ca2+ activity was dominated by IP3-induced Ca2+ release and distal steps in regenerative release rather than prior steps in signal transduction. Stimulation of Ca2+ release, whether by an irreversibly activated G protein or by a nonhydrolyzable analog of IPa, did not result in gross alteration of the CaZ+ release pathway, since the measured activity was indistinguishable from both receptor-mediated and (directly mediated) IPs injections. Generation of Spiral Ca*+ Spiral waves are examples tions that can be generated ure 5A). The formation of

Waves of the complex pattern formaby an excitable medium (Figthis complex pattern can be

Cell 286

Figure tivity

L. IR-induced

Regenerative

Ca2+ Ac-

(A and B) Spatial patterns of Ca2+ release recorded just prior (270 s) to the addition of caffeine (280 s) and during the initial wave of Ca*+ (95 s). (C and D) Stereo view of the spatiotemporal pattern of Ca2+ release. Ca% activity was rendered as in Figure 1. Gray scale intensity is shown in (D), where the intensity of the central pixel is plotted.

attributed to two properties of excitability when applied to intracellular Ca*+ release. The first is a finite refractory period subsequent to excitation. This period is equated to the time when the underlying Ca*+ processes are unable to release Ca2+. Refractory periods result in the annihilation of Ca*+ wavefronts when they impinge on recently excited areas. Refractory regions recover and support ensuing excitation and Ca*+ wave propagation. The second property is a regenerative stimulus occurring at multiple foci, equated here to the pulsatile release of Ca*+ at multiple foci. Individually, a suprathreshold stimulus would create circular patterns of Ca*+ release. However, when multiple foci are active, the multitude of collisions create incomplete arcs of Ca*+ release and, more importantly, create multiple regions with different degrees of recovering excitability. Since the speed of propagation is dependent on the extent of recovery (dispersion effect), differentially recovered regions can dictate the direction of wavefront propagation and may therefore regulate the direction of the Ca*+ signal within the cell. The sequence of Ca*+ images in Figure 5B show the key features in the development of a spiral wave. A single primary focus, centrally located at frames 3, 15, 27, and 39 s, initiated Cd+ waves at set intervals (&every 6 s). In total, eight circular waves were initiated prior to frame 40 s, but for presentation every other 6 s period has been omitted. Critical to the development of the spiral wave was the nearly concurrent activity of multiple foci along a line to the lower right of the primary focal site (~5 o’clock, frame 4 s). These sites interrupted the symmetrical circular wavefront and created an incomplete arc of Ca*+ release (frame 13 s). More importantly, these sites created a lane of recovering excitability, which directed the advancing end of the wavefront arc. Frequently, propagating Ca*+ arcs are blocked at refractory regions, but on occasion (frame 14 s), the propagating end is directed back around itself. At frames 15 and 16 s, the second pulse of the primary pulsating focus annihilated the turning Ca*+ arc. By frame 27 s, however, the end of the Ca2+ arc had turned faster than the primary

focus, initiating a spiral wavefront. Once initiated, the curvature effect of an excitable medium can maintain spiral wavefront propagation. This effect states that a convex (negative curvature) wavefront propagates more slowly than a planar wave, which in turn propagates more slowly than a concave wavefront (positive curvature; Zykov, 1980; Keener, 1966). Synchronous Excitation of Ca*+ Release Fundamental to our model of intracellular Ca*+ release are the individual excitable processes collectively referred to as the excitable medium. By definition, every process within this medium is excitable, and any small collection of excitable processes (>the critical radius; cf. Zykov, 1960; Keener, 1966; Foerster et al., 1989) is capable of initiating a Ca*+ wave, provided the region is not refractory. These characteristics provided a critical test for our model and predicted that a preselected region, when synchronously excited, would initiate a propagating Ca*+ wave. In agreement with theory, we found that extensive regions of the oocyte were capable of initiating Ca*+ waves, but that some regions were more easily excitable. We also found that under conditions of basal Ca*+ activity, the “hot spots” initiated repetitive pulses of Ca*+ release. Oocytes were initially injected with flue-3 (~25 FM final) and caged IPs (~170 uM final) and imaged confocally as described above. A restricted band of caged IPs (~50 x 760 urn) was then released using a laser scanned UV band (pass lasting 1130 second; see Experimental Procedures). Under these conditions, Ca*+ release was observed and propagated from the band into the adjacent regions of the oocyte. However, propagation was slow (300 nM, as previously reported (Lechleiter et al., 1991a). Image acquisitions were also recorded in the zero Cap+ solution given above. UV Laser Scanning Simultaneous UV and visible wavelength confocal scanning was possible due to a recent modification in our confocal microscope (Bliton et al., 1992). The duration of a scan was determined by a shutter placed only in the path of the UV laserbeam (Coherent lnova 90 argon laser with UV optics supplying 100 mW total power distributed at lines 334, 351, and 363 nm). Thus, 1130 s duration refers to the period of time when the oocyte is simultaneously scanned with UV and visible wavelengths of light. Acknowledgments We thank Steven Girard for helpful discussions on the theoretical treatment of excitability, Chris Bliton for her work in confocal UV scanning, and Dr.% Patricia Camacho and Ernest Peralta for their careful critiques of this manuscript. This work was supported by the American

Intracellular 293

Calcium

Excitability

Heart Association (J. D. L. and D. E. C.), by the Whitaker Foundation (D. E. C.), and by NIH (D. E. C). The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked “advertisemenr” in accordance with 18 USC Section 1734 solely to indicate this fact. Received

December

5, 1991; revised

January

Allesie, M. A., Bonke, F. I. M., and Schopman. F. J. G. (1973). movement in rabbit atrial muscle as a mechanism of tachycardia. Res. 33, 54-82.

Kaplan, J. H., and Ellis-Davies, G. C. R. (1988). Photolabile chelators for the rapid photorelease of divalent cations. Proc. Natl. Acad. Sci. USA 85, 6571-6575.

trisphosphate and diacylglycerol: two Annu. Rev. Biochem. 56, 159-193.

Berridge, 9586.

M. J. (1990).

oscillations.

Berridge, signalling.

M. J., and Irvine, R. F. (1989). Nature 347, 197-205.

J. Biol. Chem.

R. B. (1991). Calcium

H. (1987). Activation of a generated step increases neurons. Proc. Natl. Acad.

lino, M. (1990). Biphasic Ca*+ dependence of inositol 1,4,5-trisphosphate-induced Ca release in smooth muscle cells of the guinea pig Taenia caeci. J. Gen. Physiol. 95, 1103-l 122.

M. J. (1987). lnositol second messengers.

Berridge, M. J., and Moreton. Curr. Biol. 7, 296-297.

Gurney, A. M., Tsien, R. Y., and Lester, potassium current by rapid photochemically of intracellular calcium in rat sympathetic Sci. USA 84,3496-3500.

Circus Circ.

Berridge. interacting

lnositol

N. A., and Bures, J. (1983). Spiral waves of spreading dein the isolated chicken retina. J. Neurobiol. 74, 353-363.

Harootunian, A. T., Kao, J. P. Y., Paranjape, S., and Tsien, R. Y. (1991). Generation of calcium oscillations in fibroblasts by positive feedback between calcium and IPJ. Science 257, 75-78.

29, 1992.

References

Calcium

Goroleva, pression

265,9583-

phosphates

and cell

wavesand

spirals.

Berridge, M. J., Cobbold, P. H.. and Cuthbertson, K. S. R. (1988). Spatial and temporal aspects of cell signalling. Phil. Trans. Roy. Sot. (Lond.) B 320, 325-343. Bezprozvanny, I., Watras, J., and Ehrlich, B. E. (1991). Bell-shaped calcium-response curves of Ins(1 ,4,5)P3- and calcium-gated channels from endoplasmic reticulum of cerebellum. Nature 357, 751-754. Bliton, C., Lechleiter, J., and Clapham, D. (1992). Optical modifications enabling simultaneous ultraviolet and visible confocal microscopy. J. Microsc., in press.

Keener, J. P. (1986). A geometrical theory for spiral waves media. SIAM J. Appl. Math. 46, 1039-1056.

in excitable

Kubota, H. Y., Yoshimoto, Y., Yoneda, M., and Hiramota, Y. (1987). Free calcium wave upon activation in Xenopus eggs. Dev. Biol. 179, 129-136. Lechleiter, J., Girard, S., Clapham, D., and Peralta, E. (1991 a). Subcellular patterns of calcium release determined by G protein-specfic residues of muscarinic receptors. Nature 350, 505-508. Lechleiter, J., Girard, S., Peralta, E., and Clapham, D. (1991 b). Spiral calcium wave propagation and annihilation in Xenopus laevis oocytes. Science 252, 123-l 26. Meyer, T. (1991). 675-678.

Cell signaling

by second

messenger

waves.

Cell 64,

Miledi, R., and Parker, I. (1984). Chloride current induced by injection of calcium into Xenopus oocytes. J. Physiol. 357, 173-183.

Busa, W. B., and Nuccitelli, R. (1985). An elevated free cytosolic Ca2+ wave follows fertilization in eggs of the frog, Xenopus laevis. J. Cell Biol. 700, 1325-1329.

Miller, R. N., and Rinzel, J. (1981). The dependence of impulse propagation speed on firing frequency, dispersion, for the Hodgkin-Huxley model. Biophys. J. 34, 227-259.

Busa, W. B., Ferguson, J. E., Joseph, S. K., Williamson, Nuccitelli, R. (1985). Activation of frog (Xenopus/aevis)eggs trisphosphate. 1. Characterization of Ca*+ release from stores. J. Cell Biol. 707, 677-682.

Missiaen, L., Taylor, C. W., and Berridge, M. J. (1991). Spontaneous calcium release from inositol trisphosphate-sensitive calcium stores. Nature 352, 241-244.

J. R., and by inositol intracellular

Cuthbertson, K. S. Ft., and Chay, T. R. (1991). Modelling receptorcontrolled intracellular calcium oscillators, Cell Calcium 72, 97-109. DeLisle, S., Krause, K., Denning, G., Potter, B. V. L., and Welsh, M. J. (1990). Effect of inositol trisphosphate and calcium on oscillating elevations of intracellular calcium in Xenopus oocytes. J. Biol. Chem. 265, 11726-11730. Devreotes, P. N., Potel, M. J., and Mackay, S. A. (1983). Quantitative analysis of cyclic AMP waves mediating aggregation in Dictyostelium discoideum. Dev. Biol. 96, 405-415. Dockery, traveling 177-191.

J. D., Keener, J. P., and Tyson, waves in the Belousov-Zhabotinskii

J. J. (1988). Dispersion of reaction. Physica D 30,

Dupont, G., and Goldbeter, A. (1989). Theorectical insights into the origin of signal-induced calcium oscillations. In Cell to Cell Signalling: From Experiments to Theoretical Models (London: Academic Press Ltd.), pp. 461-474. Endo, M. (1985). Calcium release from sarcoplasmic Topics Membr. Trans. 25, 181-230.

reticulum.

Fabiato, A., and Fabiato, F. J. (1979). Calculator programs ing the composition of the solutions containing multiple ligands used for experiments in skinned muscle cells. (Paris) 75, 483-505. Finch, E. A., Turner, coagonist of inositol Science 252,443~446.

Curr.

for computmetals and J. Physiol.

T. J.. and Goldin, S. M. (1991). Calcium as a 1,4.5-trisphosphate-induced calcium release.

Foerster, P., Muller, S. C., and Hess, 8. (1989). Critical size and curvature of wave formation in an excitable chemical medium. Proc. Natl. Acad. Sci. USA 88, 8831-6834. Girard, S., Luckhoff, A., Lechleiter, J., Sneyd, J., and Clapham, D. (1992). Two-dimensional model of calcium waves reproduces the patterns observed in Xenopus oocytes. Biophys. J., in press,

Parker, I., and Ivorra, I. (1990a). Inhibition by Ca*+ of inositol trisphosphate-mediated Ca2+ liberation: a possible mechanism for oscillatory release of Ca2+. Proc. Natl. Acad. Sci. USA 87, 260-264. Parker, I., and Ivorra, I. (1990b). Localized all-or-none tion by inositol trisphosphate. Science 250, 977-979.

calcium

libera-

Parker, I., and Ivorra, I. (1991). Caffeine inhibits inositol trisphosphatemediated liberation of intracellular calcium in Xenopus oocytes. Physiol. 433, 229-240.

J.

Petersen, 0. H., and Wakui. M. (1990). Oscillating intracellular Ca*+ signals evoked by activation of receptors linked to inositol lipid hydroly sis: mechanism of generation. J. Membr. Biol. 718, 93-105. Segel, I. H. (1976). & Sons).

Biochemical

Calculations

(New

York:

John Wiley

Smith, J. S., Imagawa, T., and Ma, J. (1988). Purified ryanodine receptor from rabbit skeletal muscle is the calcium-release channel of sarcoplasmic reticulum. J. Gen. Physiol. 92, l-26. Swann, K., and Whitaker, M (1986). The part played by inositol trisphosphate and calcium in the propagation of the fertilization wave in sea urchin eggs. J. Cell Biol. 703, 2333-2342. Thastrup, O., Cullen, P. J., and Drobak, 8. K. (1990). Thapsigargin, a tumor promoter, discharges intracellular Ca2+ stores by specific inhibition of the endoplasmic reticulum Ca2+-ATPase. Proc. Natl. Acad. Sci. USA 87,2466-2470. Tsien, R. W., and Tsien, R. Y. (1990). Calcium oscillations. Annu. Rev. Cell Biol. 6, 715-760. Tsunoda, Y. (1991). Oscillatory New Biologist 3, 3-17.

Ca*+ signaling

channels,

stores,

and its cellular

and

function.

Wakui, M., Potter, B. V. L., and Petersen, 0. H. (1989). Pulsatile intracellular calcium release does not depend on fluctuations in inositol triphosphate concentration. Nature 339, 317-320. Winfree, A. T. (1980). Excitable kinetics and excitable etry of Biological Time (New York: Springer-Verlag),

media. In Geomp, 246.

Cdl 294

Winfree, A. T. (1967). When Time sional Dynamics of Electrochemical (Princeton, New Jersey: Princeton Winfree, A. T. (1990). 43, 297-309.

Vortices

Zaikin, A. N., andzhabotinsky, agation in two-dimensional ture 225, 535-537.

Breaks Down: The Three-DimenWaves and Cardiac Arrhythmias University Press).

in motionless

media. Appl. Mech.

Rev.

A. M. (1970). Concentrationwave propliquid-phase self-oscillating systems. Na-

Zykov, V. S. (1960). Analytical evaluation of the dependence of the speed of an excitation wave in a two-dimensional excitable medium on the curvature of its front. Biophys. J. 25, 906-911.