Calcium Wave Propagation in Networks of Endothelial Cells: Model-based Theoretical and Experimental Study

In this paper, we present a combined theoretical and experimental study of the propagation of calcium signals in multicellular structures composed of human endothelial cells. We consider multicellular structures composed of a single chain of cells as well as a chain of cells with a side branch, namely a “T” structure. In the experiments, we investigate the result of applying mechano-stimulation to induce signaling in the form of calcium waves along the chain and the effect of single and dual stimulation of the multicellular structure. The experimental results provide evidence of an effect of architecture on the propagation of calcium waves. Simulations based on a model of calcium-induced calcium release and cell-to-cell diffusion through gap junctions shows that the propagation of calcium waves is dependent upon the competition between intracellular calcium regulation and architecture-dependent intercellular diffusion.     

Velocity-curvature relationship of colliding spherical calcium waves in rat cardiac myocytes.

Colliding spherical calcium waves in enzymatically isolated rat cardiac myocytes develop new wavefronts propagating perpendicular to the original direction. When investigated by confocal laser scanning microscopy (CLSM), using the fluorescent Ca2+ indicator fluo-3 AM, "cusp"-like structures become visible that are favorably approximated by double parabolae. The time-dependent position of the vertices is used to determine propagation velocity and negative curvature of the wavefront in the region of collision. It is evident that negatively curved waves propagate faster than positively curved, single waves. Considering two perfectly equal expanding circular waves, we demonstrated that the collision of calcium waves is due to an autocatalytic process (calcium-induced calcium release), and not to a simple phenomenon of interference. Following the spatiotemporal organization in simpler chemical systems maintained under conditions far from the thermodynamic equilibrium (Belousov-Zhabotinskii reaction), the dependence of the normal velocity on the curvature of the spreading wavefront is given by a linear relation. The so-called velocity-curvature relationship makes clear that the velocity is enhanced by curvature toward the direction of forward propagation and decreased by curvature away from the direction of forward propagation (with an influence of the diffusion coefficient). Experimentally obtained velocity data of both negatively and positively curved calcium waves were approximated by orthogonal weighted regression. The negative slope of the straight line resulted in an effective diffusion coefficient of 1.2 x 10(-4) mm2/s. From the so-called critical radius, which must be exceeded to initiate a traveling calcium wave, a critical volume (with enhanced [Ca2+]i) of approximately 12 microm3 was calculated. This is almost identical to the volume that is occupied by a single calcium spark.     

Anisotropic propagation of Ca2+ waves in isolated cardiomyocytes.

Digital imaging microscopy of fluor-3 fluorescence was used to study the propagation of intracellular Ca2+ waves in isolated adult rat cardiomyocytes from 17 to 37 degrees C. Ca2+ waves spread in both transverse and longitudinal direction of a myocyte. Transverse propagation was pronounced in waves starting from a focus at the edge of a myocyte and in waves following an irregular, curved path (spiral waves). For the former type of waves, propagation velocities were determined. Both transverse and longitudinal wave components propagated at constant velocity ranging from 30 to 125 micron/s. Myocytes were anisotropic with respect to wave propagation: waves propagated faster in the longitudinal than in the transverse direction. The ratio between longitudinal and transverse velocity increased from 1.30 at 17 degrees C to 1.55 at 37 degrees C. Apparent activation energies for transverse and longitudinal wave propagation were estimated to be -20 kJ/mol, suggesting that these processes are limited by diffusion of Ca2+. Direction-dependent propagation velocities are interpreted to result from the highly ordered structure of the myocytes, especially from the anisotropic arrangement of diffusion obstacles such as myofilaments and mitochondria.     

Organization, ultrastructure, and development of midgut visceral muscle in larval Aedes aegypti

The midgut muscularis of larvae of the mosquito Aedes aegypti takes the form of a grid of longitudinal and circular muscle bands. The longitudinal and circular bands overlap at near right angles at many areas of intersection. The longitudinal bands run the length of the midgut. However, some bands of circular muscle, located in the anterior midgut, pass only partway around the gut. An unusual feature was observed at some regions where longitudinal and circular bands of muscle intersect: filaments oriented at near right angles to one another were present in the same membrane-bound fiber. These cruciform regions send contractile elements into both circular and longitudinal bands. The muscularis was fixed in a contracted state, so most of the sarcomeres are represented by complete overlap of myosin and lighter staining actin filaments. Features characteristic of supercontracting muscle, including perforated Z-lines, were seen in sarcomeres of circular muscle bands. Small invaginations resembling transverse tubules were present but a sarcoplasmic reticulum was not observed. While occasional cells that may be neurons or neurosecretory cells were observed, a network that might serve to coordinate the segmentation and peristaltic movement of the muscularis was not apparent.     

Local positive feedback by calcium in the propagation of intracellular calcium waves.

In many types of eukaryotic cells, the activation of surface receptors leads to the production of inositol 1,4,5-trisphosphate and calcium release from intracellular stores. Calcium release can occur in complex spatial patterns, including waves of release that traverse the cytoplasm. Fluorescence video microscopy was used to view calcium waves in single mouse neuroblastoma cells. The propagation of calcium waves was slowed by buffers that bind calcium quickly, such as BAPTA, but not by a buffer with slower on-rate, EGTA. This shows that a key feedback event in wave propagation is rapid diffusion of calcium occurring locally on a scale of < 1 micron. The length-speed product of wavefronts was used to determine that calcium acting in feedback diffuses at nearly the rate expected for free diffusion in aqueous solution. In cytoplasm, which contains immobile Ca2+ buffers, this rate of diffusion occurs only in the first 0.2 ms after release, within 0.4 micron of a Ca2+ release channel mouth. Calcium diffusion from an open channel to neighboring release sites is, therefore, a rate-determining regenerative step in calcium wave propagation. The theoretical limitations of the wave front analysis are discussed.     

Correlated Diffusion of Colloidal Particles near a Liquid-Liquid Interface

Optical microscopy and multi-particle tracking are used to investigate the cross-correlated diffusion of quasi two-dimensional colloidal particles near an oil-water interface. The behaviors of the correlated diffusion along longitudinal and transverse direction are asymmetric. It is shown that the characteristic length for longitudinal and transverse correlated diffusion are particle diameter and the distance from particle center to the interface, respectively, for large particle separation . The longitudinal and transverse correlated diffusion coefficient and are independent of the colloidal area fraction when , which indicates that the hydrodynamic interactions(HIs) among the particles are dominated by HIs through the surrounding fluid for small . For high area fraction , the power law exponent for the spatial decay of begins to decrease, which suggests the HIs are more contributed from the 2D particle monolayer self for large .     

Calcium signal transmission in chick sensory neurones is diffusion based

In many cells, the cytosol is an excitable medium through which calcium waves propagate by calcium induced calcium release (CICR). Many labs. have reported CICR in neurones subsequent to calcium influx through voltage gated channels. However, these have used long depolarizations. We have imaged calcium within chick sensory neurones following 50 ms depolarizations. Calcium signals travelled rapidly throughout the cell, such that changes at the cell centre were delayed by 24 ms compared to regions 3 microm from the plasma membrane. The nuclear envelope imposed a delay of 9 ms. A simple diffusion model with few unknowns gave good fits to the measured data, indicating that passive diffusion is responsible for signal transmission in these neurones. Simulations run without indicator dye did not reveal markedly different spatiotemporal dynamics, although concentration changes were larger. Simulations of calcium changes during action potentials revealed that large calcium transients occurring in the cytosol close to the nucleus are significantly attenuated by the nuclear envelope. Our results indicate that for the brief depolarisations that neurones will experience during normal signal processing calcium signals are transmitted by passive diffusion only. Diffusion is perfectly capable of transmitting the calcium signal into the interior of nerve cell bodies, and into the nucleoplasm.     

Calcium waves in agarose gel with cell organelles: implications of the velocity curvature relationship

Calcium oscillations and waves have been observed not only in several types of living cells but also in less complex systems of isolated cell organelles. Here we report the determination of apparent Ca2+ diffusion coefficients in a novel excitable medium of agarose gel with homogeneously distributed vesicles of skeletal sarcoplasmic reticulum. Spatiotemporal calcium patterns were visualized by confocal laser scanning fluorescence microscopy. To obtain characteristic parameters of the velocity curvature relationship, namely, apparent diffusion coefficient, velocity of plane calcium waves, and critical radius, positively and negatively curved wave fronts were analyzed. It is demonstrated that gel-immobilized cell organelles reveal features of an excitable medium. Apparent Ca2+ diffusion coefficients of the in vitro system, both in the absence or in the presence of mitochondria, were found to be higher than in cardiac myocytes and lower than in unbuffered agarose gel. Plane calcium waves propagated markedly slower in the in vitro system than in rat cardiac myocytes. Whereas mitochondria significantly reduced the apparent Ca2+ diffusion coefficient of the in vitro system, propagation velocity and critical size of calcium waves were found to be nearly unchanged. These results suggest that calcium wave propagation depends on the kinetics of calcium release rather than on diffusion.     

Activation and propagation of Ca(2+) release during excitation-contraction coupling in atrial myocytes

Fast two-dimensional confocal microscopy and the Ca(2+) indicator fluo-4 were used to study excitation-contraction (E-C) coupling in cat atrial myocytes which lack transverse tubules and contain both subsarcolemmal junctional (j-SR) and central nonjunctional (nj-SR) sarcoplasmic reticulum. Action potentials elicited by field stimulation induced transient increases of intracellular Ca(2+) concentration ([Ca(2+)](i)) that were highly inhomogeneous. Increases started at distinct subsarcolemmal release sites spaced approximately 2 microm apart. The amplitude and the latency of Ca(2+) release from these sites varied from beat to beat. Subsarcolemmal release fused to build a peripheral ring of elevated [Ca(2+)](i), which actively propagated to the center of the cells via Ca(2+)-induced Ca(2+) release. Resting myocytes exhibited spontaneous Ca(2+) release events, including Ca(2+) sparks and local (microscopic) or global (macroscopic) [Ca(2+)](i) waves. The microscopic [Ca(2+)](i) waves propagated in a saltatory fashion along the sarcolemma ("coupled" Ca(2+) sparks) revealing the sequential activation of Ca(2+) release sites of the j-SR. Moreover, during global [Ca(2+)](i) waves, Ca(2+) release was evident from individual nj-SR sites. Ca(2+) release sites were arranged in a regular three-dimensional grid as deduced from the functional data and shown by immunostaining of ryanodine receptor Ca(2+) release channels. The longitudinal and transverse distances between individual Ca(2+) release sites were both approximately 2 microm. Furthermore, electron microscopy revealed a continuous sarcotubular network and one peripheral coupling of j-SR with the sarcolemma per sarcomere. The results demonstrate directly that, in cat atrial myocytes, the action potential-induced whole-cell [Ca(2+)](i) transient is the spatio-temporal summation of Ca(2+) release from subsarcolemmal and central sites. First, j-SR sites are activated in a stochastic fashion by the opening of voltage-dependent sarcolemmal Ca(2+) channels. Subsequently, nj-SR sites are activated by Ca(2+)-induced Ca(2+) release propagating from the periphery.     

Anisotropic diffusion of fluorescently labeled ATP in rat cardiomyocytes determined by raster image correlation spectroscopy

A series of experimental data points to the existence of profound diffusion restrictions of ADP/ATP in rat cardiomyocytes. This assumption is required to explain the measurements of kinetics of respiration, sarcoplasmic reticulum loading with calcium, and kinetics of ATP-sensitive potassium channels. To be able to analyze and estimate the role of intracellular diffusion restrictions on bioenergetics, the intracellular diffusion coefficients of metabolites have to be determined. The aim of this work was to develop a practical method for determining diffusion coefficients in anisotropic medium and to estimate the overall diffusion coefficients of fluorescently labeled ATP in rat cardiomyocytes. For that, we have extended raster image correlation spectroscopy (RICS) protocols to be able to discriminate the anisotropy in the diffusion coefficient tensor. Using this extended protocol, we estimated diffusion coefficients of ATP labeled with the fluorescent conjugate Alexa Fluor 647 (Alexa-ATP). In the analysis, we assumed that the diffusion tensor can be described by two values: diffusion coefficient along the myofibril and that across it. The average diffusion coefficients found for Alexa-ATP were as follows: 83 +/- 14 microm(2)/s in the longitudinal and 52 +/- 16 microm(2)/s in the transverse directions (n = 8, mean +/- SD). Those values are approximately 2 (longitudinal) and approximately 3.5 (transverse) times smaller than the diffusion coefficient value estimated for the surrounding solution. Such uneven reduction of average diffusion coefficient leads to anisotropic diffusion in rat cardiomyocytes. Although the source for such anisotropy is uncertain, we speculate that it may be induced by the ordered pattern of intracellular structures in rat cardiomyocytes.     

A model of propagating calcium-induced calcium release mediated by calcium diffusion

The effect of sudden local fluctuations of the free sarcoplasmic [Ca++]i in cardiac cells on calcium release and calcium uptake by the sarcoplasmic reticulum (SR) was calculated with the aid of a simplified model of SR calcium handling. The model was used to evaluate whether propagation of calcium transients and the range of propagation velocities observed experimentally (0.05-15 mm s(-1)) could be predicted. Calcium fluctuations propagate by virtue of focal calcium release from the SR, diffusion through the cytosol (which is modulated by binding to troponin and calmodulin and sequestration by the SR), and subsequently induce calcium release from adjacent release sites of the SR. The minimal and maximal velocities derived from the simulation were 0.09 and 15 mm s(-1) respectively. The method of solution involved writing the diffusion equation as a difference equation in the spatial coordinates. Thus, coupled ordinary differential equations in time with banded coefficients were generated. The coupled equations were solved using Gear's sixth order predictor-corrector algorithm for stiff equations with reflective boundaries. The most important determinants of the velocity of propagation of the calcium waves were the diastolic [Ca++]i, the rate of rise of the release, and the amount of calcium released from the SR. The results are consistent with the assumptions that calcium loading causes an increase in intracellular calcium and calcium in the SR, and an increase in the amount and rate of calcium released. These two effects combine to increase the propagation velocity at higher levels of calcium loading.     

Receptors, sparks and waves in a fire-diffuse-fire framework for calcium release

Calcium ions are an important second messenger in living cells. Indeed calcium signals in the form of waves have been the subject of much recent experimental interest. It is now well established that these waves are composed of elementary stochastic release events (calcium puffs or sparks) from spatially localised calcium stores. The aim of this paper is to analyse how the stochastic nature of individual receptors within these stores combines to create stochastic behaviour on long time-scales that may ultimately lead to waves of activity in a spatially extended cell model. Techniques from asymptotic analysis and stochastic phase–plane analysis are used to show that a large cluster of receptor channels leads to a release probability with a sigmoidal dependence on calcium density. This release probability is incorporated into a computationally inexpensive model of calcium release based upon a stochastic generalisation of the fire-diffuse-fire (FDF) threshold model. Numerical simulations of the model in one and two dimensions (with stores arranged on both regular and disordered lattices) illustrate that stochastic calcium release leads to the spontaneous production of calcium sparks that may merge to form saltatory waves. Illustrations of spreading circular waves, spirals and more irregular waves are presented. Furthermore, receptor noise is shown to generate a form of array enhanced coherence resonance whereby all calcium stores release periodically and simultaneously.     

Effects of peristaltic and longitudinal wave motion of the channel wall on movement of micro-organisms: application to spermatozoa transport

A mathematical model is presented to study the motion of the spermatozoa in the cervical canal by considering the transverse waves along its tail and the transverse and longitudinal motions of the cervical wall. In an attempt to control fertility by reducing the speed of sperm, the transverse waves have been considered in the direction opposite to the motion of the spermatozoa. It has been shown that by having appropriate transverse wave motion and longitudinal velocity, the sperm may not be able to move towards the oviduct even if it could continue to have its own propelling velocity. A particular case of the motion of a thin plane sheet in a channel under peristaltic motion of its walls has also been obtained and studied.     

Transient Mitochondrial Depolarizations Reflect Focal Sarcoplasmic Reticular Calcium Release in Single Rat Cardiomyocytes

Digital imaging of mitochondrial potential in single rat cardiomyocytes revealed transient depolarizations of mitochondria discretely localized within the cell, a phenomenon that we shall call “flicker.” These events were usually highly localized and could be restricted to single mitochondria, but they could also be more widely distributed within the cell. Contractile waves, either spontaneous or in response to depolarization with 50 mM K⁺, were associated with propagating waves of mitochondrial depolarization, suggesting that propagating calcium waves are associated with mitochondrial calcium uptake and consequent depolarization. Here we demonstrate that the mitochondrial flicker was directly related to the focal release of calcium from sarcoplasmic reticular (SR) calcium stores and consequent uptake of calcium by local mitochondria. Thus, the events were dramatically reduced by (a) depletion of SR calcium stores after long-term incubation in EGTA or thapsigargin (500 nM); (b) buffering intracellular calcium using BAPTA-AM loading; (c) blockade of SR calcium release with ryanodine (30 μM); and (d) blockade of mitochondrial calcium uptake by microinjection of diaminopentane pentammine cobalt (DAPPAC), a novel inhibitor of the mitochondrial calcium uniporter. These observations demonstrate that focal SR calcium release results in calcium microdomains sufficient to promote local mitochondrial calcium uptake, suggesting a tight coupling of calcium signaling between SR release sites and nearby mitochondria.     

Intercellular Ca2+ wave propagation through gap-junctional Ca2+ diffusion: a theoretical study

Intercellular regenerative calcium waves in systems such as the liver and the blowfly salivary gland have been hypothesized to spread through calcium-induced calcium release (CICR) and gap-junctional calcium diffusion. A simple mathematical model of this mechanism is developed. It includes CICR and calcium removal from the cytoplasm, cytoplasmic and gap-junctional calcium diffusion, and calcium buffering. For a piecewise linear approximation of the calcium kinetics, expressions in terms of the cellular parameters are derived for 1) the condition for the propagation of intercellular waves, and 2) the characteristic time of the delay of a wave encountered at the gap junctions. Intercellular propagation relies on the local excitation of CICR in the perijunctional space by gap-junctional calcium influx. This mechanism is compatible with low effective calcium diffusivity, and necessitates that CICR can be excited in every cell along the path of a wave. The gap-junctional calcium permeability required for intercellular waves in the model falls in the range of reported gap-junctional permeability values. The concentration of diffusive cytoplasmic calcium buffers and the maximal rate of CICR, in the case of inositol 1,4,5-trisphosphate (IP3) receptor calcium release channels set by the IP(3) concentration, are shown to be further determinants of wave behavior.     

Transverse Viscoelastic Extension in Nitella: II. Effects of Acid and Ions

The transverse viscoelastic extension of isolated Nitella cell walls is stimulated by acid pH and by Mg²⁺ and K⁺ ions. In the presence of 1 millimolar citrate-phosphate buffer the threshold pH in the transverse direction is 3.5, compared to 4.5 in the longitudinal direction. The relative amounts of extension stimulated by acid are comparable in the two directions at their respective thresholds. Longitudinal and transverse Mg²⁺ ion-induced extensibility begins at 10 millimolar and reaches a plateau between 10 and 100 millimolar. The threshold for K⁺ ion enhancement is near 10 millimolar in the longitudinal direction and 50 millimolar in the transverse direction. Maximum stimulation by K⁺ is obtained at 250 millimolar. At their respective maxima, Mg²⁺ and K⁺ induce equal amounts of extension. However, the relative amount of extension induced by ions is significantly less in the transverse than in the longitudinal direction. Ions and acids appear to affect different sites in the wall, inasmuch as neither treatment abolishes the effect of the other. Walls from rapidly growing cells are more sensitive to stimulation than nongrowing cells in the longitudinal direction but not in the transverse direction.     

Regional differences in cholinergic regulation of potassium current in feline esophageal circular smooth muscle

Potassium channels are important contributors to membrane excitability in smooth muscles. There are regional differences in resting membrane potential and K(+)-channel density along the length of the feline circular smooth muscle esophagus. The aim of this study was to assess responses of K(+)-channel currents to cholinergic (ACh) stimulation along the length of the feline circular smooth muscle esophageal body. Perforated patch-clamp technique assessed K(+)-channel responses to ACh stimulation in isolated smooth muscle cells from the circular muscle layer of the esophageal body at 2 (distal)- and 4-cm (proximal) sites above the lower esophageal sphincter. Western immunoblots assessed ion channel and receptor expression. ACh stimulation produced a transient increase in outward current followed by inhibition of spontaneous transient outward currents. These ACh-induced currents were abolished by blockers of large-conductance Ca(2+)-dependent K(+) channels (BK(Ca)). Distal cells demonstrated a greater peak current density in outward current than cells from the proximal region and a longer-lasting outward current increase. These responses were abolished by atropine and the specific M(3) receptor antagonist 4-DAMP but not the M(1) receptor antagonist pirenzipine or the M(2) receptor antagonist methoctramine. BK(Ca) expression along the smooth muscle esophagus was similar, but M(3) receptor expression was greater in the distal region. Therefore, ACh can differentially activate a potassium channel (BK(Ca)) current along the smooth muscle esophagus. This activation probably occurs through release of intracellular calcium via an M(3) pathway and has the potential to modulate the timing and amplitude of peristaltic contraction along the esophagus.     

Activation of a cGMP-sensitive calcium-dependent chloride channel may cause transition from calcium waves to whole cell oscillations in smooth muscle cells

In vitro, alpha-adrenoreceptor stimulation of rat mesenteric small arteries often leads to a rhythmic change in wall tension, i.e., vasomotion. Within the individual smooth muscle cells of the vascular wall, vasomotion is often preceded by a period of asynchronous calcium waves. Abruptly, these low-frequency waves may transform into high-frequency whole cell calcium oscillations. Simultaneously, multiple cells synchronize, leading to rhythmic generation of tension. We present a mathematical model of vascular smooth muscle cells that aims at characterizing this sudden transition. Simulations show calcium waves sweeping through the cytoplasm when the sarcoplasmic reticulum (SR) is stimulated to release calcium. A rise in cGMP leads to the experimentally observed transition from waves to whole cell calcium oscillations. At the same time, membrane potential starts to oscillate and the frequency approximately doubles. In this transition, the simulated results point to a key role for a recently discovered cGMP-sensitive calcium-dependent chloride channel. This channel depolarizes the membrane in response to calcium released from the SR. In turn, depolarization causes a uniform opening of L-type calcium channels on the cell surface, stimulating a synchronized release of SR calcium and inducing the shift from waves to whole cell oscillations. The effect of the channel is therefore to couple the processes of the SR with those of the membrane. We hypothesize that the shift in oscillatory mode and the associated onset of oscillations in membrane potential within the individual cell may underlie sudden intercellular synchronization and the appearance of vasomotion.     

Propagated disturbances of transverse potential gradient in intracellular fibrils as the source of motive forces for longitudinal transport in cells

Summary It is proposed as a working hypothesis that conformational changes propagated like waves along intracellular fibrils (tubules, microtubules, microfilaments) have an electric component,i.e., there are waves of disturbance of electric potential in the fibrils. The paper considers the unavoidable consequences of the wave. The latter is accompanied by local electric field in the boundary layer of cytoplasmic fluid. Both positively and negatively charged particles may be attracted to the fibril in certain regions of the field and, being attracted, the particle may be under the action of longitudinal component of electric force. When the force is strong enough to move the particle with wave velocity, the particle will travel smoothly along the fibril, otherwise the movement will be saltatory or of agitation type. Net electroosmotic flow in one direction in the boundary layer of fluid may be expected when the waves are propagated in series. Turbulent motion of the fluid caused by the waves may provide the basis for activated diffusion. Asymmetry of the wave may account for polar transport of this sort. The electric field transmitted along the fibril across a sieve pore in phloem may facilitate electroosmotically the flow through the pore. Quantitative requirements of the hypothesis that electric field generated by the waves may account for different aspects of longitudinal transport in cells are apparently met.     

A single-pool model for intracellular calcium oscillations and waves in the Xenopus laevis oocyte.

We construct a minimal model of cytosolic free Ca2+ oscillations based on Ca2+ release via the inositol 1,4,5-trisphosphate (IP3) receptor/Ca2+ channel (IP3R) of a single intracellular Ca2+ pool. The model relies on experimental evidence that the cytosolic free calcium concentration ([Ca2+]c) modulates the IP3R in a biphasic manner, with Ca2+ release inhibited by low and high [Ca2+]c and facilitated by intermediate [Ca2+]c, and that channel inactivation occurs on a slower time scale than activation. The model produces [Ca2+]c oscillations at constant [IP3] and reproduces a number of crucial experiments. The two-dimensional spatial model with IP3 dynamics, cytosolic diffusion of IP3 (Dp = 300 microns 2 s-1), and cytosolic diffusion of Ca2+ (Dc = 20 microns 2 s-1) produces circular, planar, and spiral waves of Ca2+ with speeds of 7-15 microns.s-1, which annihilate upon collision. Increasing extracellular [Ca2+] influx increases wave speed and baseline [Ca2+]c. A [Ca2+]c-dependent Ca2+ diffusion coefficient does not alter the qualitative behavior of the model. An important model prediction is that channel inactivation must occur on a slower time scale than activation in order for waves to propagate. The model serves to capture the essential macroscopic mechanisms that are involved in the production of intracellular Ca2+ oscillations and traveling waves in the Xenopus laevis oocyte.