Simulation of the fertilization Ca2+ wave in Xenopus laevis eggs

Analytic solutions for calcium ion fertilisation waves on the surface of eggs

The evolution of calcium fertilisation waves on the cortex of amphibian eggs can be described by a nonlinear reaction-diffusion process on the surface of a sphere. Here, we use the nonclassical symmetry technique to find an exact analytic solution that describes the evolution of the calcium concentration. The solutions presented compare well with published experimental results. The analytic solution can be used to give insight into the processes governing the fertilisation wave, such as the flow of calcium ions from the sperm entry point. By finding a spiral solution to an approximate equation linearised near saturation, we also demonstrate how solutions with other properties may be constructed using this technique.

Changes in Ca2+ Removal Can Mask the Effects of Geometry During IP3R Mediated Ca2+ Signals

Calcium (Ca²⁺) signals are ubiquitous. Most intracellular Ca²⁺ signals involve the release of Ca²⁺ from the endoplasmic reticulum (ER) through Inositol 1,4,5-Trisphosphate Receptors (IP₃Rs). The non-uniform spatial organization of IP₃Rs and the fact that their individual openings are coupled via cytosolic Ca²⁺ are key factors for the variety of spatio-temporal distributions of the cytosolic [Ca²⁺] and the versatility of the signals. In this paper we combine experiments performed in untreated and in progesterone-treated Xenopus laevis oocytes and mathematical models to investigate how the interplay between geometry (the IP₃R spatial distribution) and dynamics (the processes that characterize the release, transport, and removal of cytosolic Ca²⁺) affects the resulting signals. Signal propagation looks more continuous and spatially uniform in treated (mature) than in untreated (immature) oocytes. This could be due to the different underlying IP₃R spatial distribution that has been observed in both cell types. The models, however, show that the rate of cytosolic Ca²⁺ removal, which is also different in both cell types, plays a key role affecting the coupling between Ca²⁺ release sites in such a way that the effect of the underlying IP₃R spatial distribution can be modified.

Insemination or phosphatidic acid induces an outwardly spiraling disk of elevated Ca2+ to produce the Ca2+ wave during Xenopus laevis fertilization

During Xenopus fertilization, the initial intracellular calcium ((Ca²⁺)_i) release at the sperm-egg binding site (hot spot) has not been described without the use of inhibitors, nor related to underlying ER structure. Without inhibitors, we now report that sperm induce an initial hot spot after sperm addition to Xenopus eggs that was ~25 µm. This area is consistent with the size of ER patches and clusters of IP3 receptors that have enhanced activity. Furthermore, we find a new mechanism for the fertilization (Ca²⁺)_i wave; instead of outward diffusion of inositol 1,4,5-trisphosphate (IP₃), we find that the wave was generated by an outward, clockwise rotation of a ~63 µm disk of elevated (Ca²⁺)_i moving very rapidly at ~65 µm/s. We also suggest a new mechanism for the acceleration of the fertilization (Ca²⁺)_i wave as the disk accelerated and was joined by other rotating disks (some rotating counterclockwise) at a time when the speed of the (Ca²⁺)_i wave increases. To examine the role of phosphatidic acid (PA) in the release of (Ca²⁺)_i during Xenopus fertilization, we find that two inhibitors of PA production delayed the appearance of fertilization hot spots by ~9-12 min but did not reduce the size of hot spots and actually accelerated the later (Ca²⁺)_i wave. Surprisingly, global addition of PA to Xenopus eggs induced localized hot spots at a time and size that was similar to those induced after sperm addition. In contrast, sperm induce a rapid (Ca²⁺)_i wave (~4 µm/s) within ~30 s after hot spot appearance, whereas hot spots induced by PA required an ~32 min to induce a very slow (~1 µm/s) (Ca²⁺)_i wave with a lower peak of (Ca²⁺)_i. Thus, PA may not be required for the initial release of (Ca²⁺)_i at the sperm-egg binding site, but mimics sperm by inducing a similarly sized localized (Ca²⁺)_i release. As compared with sperm, PA may induce a weak, slow (Ca²⁺)_i wave by slowly increasing IP3 receptor clustering. Addition of PA to Xenopus oocytes, or Ca²⁺ ionophore to either Xenopus oocytes or eggs, did not induce hot spots but a global (Ca²⁺)_i wave that rapidly moved at ~12 µm/s.

Regulation of islet glucagon secretion: Beyond calcium

The islet of Langerhans plays a key role in glucose homeostasis through regulated secretion of the hormones insulin and glucagon. Islet research has focused on the insulin-secreting β-cells, even though aberrant glucagon secretion from α-cells also contributes to the aetiology of diabetes. Despite its importance, the mechanisms controlling glucagon secretion remain controversial. Proper α-cell function requires the islet milieu, where β- and δ-cells drive and constrain α-cell dynamics. The response of glucagon to glucose is similar between isolated islets and that measured in vivo, so it appears that the glucose dependence requires only islet-intrinsic factors and not input from blood flow or the nervous system. Elevated intracellular free Ca²⁺ is needed for α-cell exocytosis, but interpreting Ca²⁺ data is tricky since it is heterogeneous among α-cells at all physiological glucose levels. Total Ca²⁺ activity in α-cells increases slightly with glucose, so Ca²⁺ may serve a permissive, rather than regulatory, role in glucagon secretion. On the other hand, cAMP is a more promising candidate for controlling glucagon secretion and is itself driven by paracrine signalling from β- and δ-cells. Another pathway, juxtacrine signalling through the α-cell EphA receptors, stimulated by β-cell ephrin ligands, leads to a tonic inhibition of glucagon secretion. We discuss potential combinations of Ca²⁺ , cAMP, paracrine and juxtacrine factors in the regulation of glucagon secretion, focusing on recent data in the literature that might unify the field towards a quantitative understanding of α-cell function.

Ocean acidification hampers sperm-egg collisions, gamete fusion, and generation of Ca2+ oscillations of a broadcast spawning bivalve, Tegillarca granosa

Although the effect of ocean acidification on fertilization success of marine organisms is increasingly well documented, the underlying mechanisms are not completely understood. The fertilization success of broadcast spawning invertebrates depends on successful sperm-egg collisions, gamete fusion, and standard generation of Ca²⁺ oscillations. Therefore, the realistic effects of future ocean pCO₂ levels on these specific aspects of fertilization of Tegillarca granosa were investigated in the present study through sperm velocity trials, fertilization kinetics model analysis, and intracellular Ca²⁺ assays, respectively. Results obtained indicated that ocean acidification significantly reduced the fertilization success of T. granosa, which could be accountable by (i) decreased sperm velocity hence reducing the probability for sperm-egg collisions; (ii) lowered probability of gamete fusion for each gamete collision event; and (iii) disrupted intracellular Ca²⁺ oscillations.

Stochastic initiation and termination of calcium-mediated triggered activity in cardiac myocytes

Cardiac myocytes normally initiate action potentials in response to a current stimulus that depolarizes the membrane above an excitation threshold. Aberrant excitation can also occur due to spontaneous calcium (Ca²⁺) release (SCR) from intracellular stores after the end of a preceding action potential. SCR drives the Na⁺/Ca²⁺ exchange current inducing a "delayed afterdepolarization" that can in turn trigger an action potential if the excitation threshold is reached. This "triggered activity" is known to cause arrhythmias, but how it is initiated and terminated is not understood. Using computer simulations of a ventricular myocyte model, we show that initiation and termination are inherently random events. We determine the probability of those events from statistical measurements of the number of beats before initiation and before termination, respectively, which follow geometric distributions. Moreover, we elucidate the origin of randomness by a statistical analysis of SCR events, which do not follow a Poisson process observed in other eukaryotic cells. Due to synchronization of Ca²⁺ releases during the action potential upstroke, waiting times of SCR events after the upstroke are narrowly distributed, whereas SCR amplitudes follow a broad normal distribution with a width determined by fluctuations in the number of independent Ca²⁺ wave foci. This distribution enables us to compute the probabilities of initiation and termination of bursts of triggered activity that are maintained by a positive feedback between the action potential upstroke and SCR. Our results establish a theoretical framework for interpreting complex and varied manifestations of triggered activity relevant to cardiac arrhythmias.

The role of IP3 receptor channel clustering in Ca2+ wave propagation during oocyte maturation

During oocyte maturation, the calcium-signaling machinery undergoes a dramatic remodeling resulting in distinctly different calcium-release patterns on all organizational scales from puffs to waves. The dynamics of the Ca(2+) release wave in mature as compared to immature oocytes are defined by a slower propagation speed and longer duration of the high Ca(2+) plateau. In this chapter, we use computational modeling to identify the changes in the signaling machinery, which contribute most significantly to the alterations observed in Ca(2+) wave propagation during Xenopus oocyte maturation. In addition to loss of store-operated calcium entry and internalization of plasma membrane pumps, we propose that spatial reorganization of the IP3 receptors in the plane of the ER membrane is a key factor for the observed signaling changes in Ca(2+) wave propagation.

Calcium signaling and meiotic exit at fertilization in Xenopus egg

Calcium is a universal messenger that mediates egg activation at fertilization in all sexually reproducing species studied. However, signaling pathways leading to calcium generation and the mechanisms of calcium-induced exit from meiotic arrest vary substantially among species. Here, we review the pathways of calcium signaling and the mechanisms of meiotic exit at fertilization in the eggs of the established developmental model, African clawed frog, Xenopus laevis. We also discuss calcium involvement in the early fertilization-induced events in Xenopus egg, such as membrane depolarization, the increase in intracellular pH, cortical granule exocytosis, cortical contraction, contraction wave, cortical rotation, reformation of the nuclear envelope, sperm chromatin decondensation and sister chromatid segregation.

Simulations of pattern dynamics for reaction-diffusion systems via SIMULINK

BACKGROUND

Investigation of the nonlinear pattern dynamics of a reaction-diffusion system almost always requires numerical solution of the system's set of defining differential equations. Traditionally, this would be done by selecting an appropriate differential equation solver from a library of such solvers, then writing computer codes (in a programming language such as C or Matlab) to access the selected solver and display the integrated results as a function of space and time. This "code-based" approach is flexible and powerful, but requires a certain level of programming sophistication. A modern alternative is to use a graphical programming interface such as Simulink to construct a data-flow diagram by assembling and linking appropriate code blocks drawn from a library. The result is a visual representation of the inter-relationships between the state variables whose output can be made completely equivalent to the code-based solution.

RESULTS

As a tutorial introduction, we first demonstrate application of the Simulink data-flow technique to the classical van der Pol nonlinear oscillator, and compare Matlab and Simulink coding approaches to solving the van der Pol ordinary differential equations. We then show how to introduce space (in one and two dimensions) by solving numerically the partial differential equations for two different reaction-diffusion systems: the well-known Brusselator chemical reactor, and a continuum model for a two-dimensional sheet of human cortex whose neurons are linked by both chemical and electrical (diffusive) synapses. We compare the relative performances of the Matlab and Simulink implementations.

CONCLUSIONS

The pattern simulations by Simulink are in good agreement with theoretical predictions. Compared with traditional coding approaches, the Simulink block-diagram paradigm reduces the time and programming burden required to implement a solution for reaction-diffusion systems of equations. Construction of the block-diagram does not require high-level programming skills, and the graphical interface lends itself to easy modification and use by non-experts.

Do calcium buffers always slow down the propagation of calcium waves?

Calcium buffers are large proteins that act as binding sites for free cytosolic calcium. Since a large fraction of cytosolic calcium is bound to calcium buffers, calcium waves are widely observed under the condition that free cytosolic calcium is heavily buffered. In addition, all physiological buffered excitable systems contain multiple buffers with different affinities. It is thus important to understand the properties of waves in excitable systems with the inclusion of buffers. There is an ongoing controversy about whether or not the addition of calcium buffers into the system always slows down the propagation of calcium waves. To solve this controversy, we incorporate the buffering effect into the generic excitable system, the FitzHugh-Nagumo model, to get the buffered FitzHugh-Nagumo model, and then to study the effect of the added buffer with large diffusivity on traveling waves of such a model in one spatial dimension. We can find a critical dissociation constant (K = K(a)) characterized by system excitability parameter a such that calcium buffers can be classified into two types: weak buffers (K ∈ (K(a), ∞)) and strong buffers (K ∈ (0, K(a))). We analytically show that the addition of weak buffers or strong buffers but with its total concentration b(0)(1) below some critical total concentration b(0,c)(1) into the system can generate a traveling wave of the resulting system which propagates faster than that of the origin system, provided that the diffusivity D1 of the added buffers is sufficiently large. Further, the magnitude of the wave speed of traveling waves of the resulting system is proportional to √D1 as D1 --> ∞. In contrast, the addition of strong buffers with the total concentration b(0)(1) > b(0,c)(1) into the system may not be able to support the formation of a biologically acceptable wave provided that the diffusivity D1 of the added buffers is sufficiently large.

Reaction-diffusion systems in intracellular molecular transport and control

Chemical reactions make cells work only if the participating chemicals are delivered to desired locations in a timely and precise fashion. Most research to date has focused on active-transport mechanisms, although passive diffusion is often equally rapid and energetically less costly. Capitalizing on these advantages, cells have developed sophisticated reaction-diffusion (RD) systems that control a wide range of cellular functions-from chemotaxis and cell division, through signaling cascades and oscillations, to cell motility. These apparently diverse systems share many common features and are "wired" according to "generic" motifs such as nonlinear kinetics, autocatalysis, and feedback loops. Understanding the operation of these complex (bio)chemical systems requires the analysis of pertinent transport-kinetic equations or, at least on a qualitative level, of the characteristic times of the constituent subprocesses. Therefore, in reviewing the manifestations of cellular RD, we also describe basic theory of reaction-diffusion phenomena.

Spatiotemporal characteristics of calcium dynamics in astrocytes

Although Ca(i)(2+) waves in networks of astrocytes in vivo are well documented, propagation in vivo is much more complex than in culture, and there is no consensus concerning the dominant roles of intercellular and extracellular messengers [inositol 1,4,5-trisphosphate (IP(3)) and adenosine-5'-triphosphate (ATP)] that mediate Ca(i)(2+) waves. Moreover, to date only simplified models that take very little account of the geometrical struture of the networks have been studied. Our aim in this paper is to develop a mathematical model based on realistic cellular morphology and network connectivity, and a computational framework for simulating the model, in order to address these issues. In the model, Ca(i) (2+) wave propagation through a network of astrocytes is driven by IP(3) diffusion between cells and ATP transport in the extracellular space. Numerical simulations of the model show that different kinetic and geometric assumptions give rise to differences in Ca(i)(2+) wave propagation patterns, as characterized by the velocity, propagation distance, time delay in propagation from one cell to another, and the evolution of Ca(2+) response patterns. The temporal Ca(i)(2+) response patterns in cells are different from one cell to another, and the Ca(i)(2+) response patterns evolve from one type to another as a Ca(i)(2+) wave propagates. In addition, the spatial patterns of Ca(i)(2+) wave propagation depend on whether IP(3), ATP, or both are mediating messengers. Finally, two different geometries that reflect the in vivo and in vitro configuration of astrocytic networks also yield distinct intracellular and extracellular kinetic patterns. The simulation results as well as the linear stability analysis of the model lead to the conclusion that Ca(i)(2+) waves in astrocyte networks are probably mediated by both intercellular IP(3) transport and nonregenerative (only the glutamate-stimulated cell releases ATP) or partially regenerative extracellular ATP signaling.

Modeling Ca2+ signaling differentiation during oocyte maturation

Ca2+ is a fundamental intracellular signal that mediates a variety of disparate physiological functions often in the same cell. Ca2+ signals span a wide range of spatial and temporal scales, which endow them with the specificity required to induce defined cellular functions. Furthermore, Ca2+ signaling is highly plastic as it is modulated dynamically during normal physiological development and under pathological conditions. However, the molecular mechanisms underlying Ca2+ signaling differentiation during cellular development remain poorly understood. Oocyte maturation in preparation for fertilization provides an exceptionally well-suited model to elucidate Ca2+ signaling regulation during cellular development. This is because a Ca2+ signal with specialized spatial and temporal dynamics is universally essential for egg activation at fertilization. Here we use mathematical modeling to define the critical determinants of Ca2+ signaling differentiation during oocyte maturation. We show that increasing IP3 receptor (IP3R) affinity replicates both elementary and global Ca2+ dynamics observed experimentally following oocyte maturation. Furthermore, our model reveals that because of the Ca2+ dependency of both SERCA and the IP3R, increased IP3R affinity shifts the system's equilibrium to a new steady state of high cytosolic Ca2+, which is essential for fertilization. Therefore our model provides unique insights into how relatively small alterations of the basic molecular mechanisms of Ca2+ signaling components can lead to dramatic alterations in the spatio-temporal properties of Ca2+ dynamics.

Are buffers boring? Uniqueness and asymptotical stability of traveling wave fronts in the buffered bistable system

Traveling waves of calcium are widely observed under the condition that the free cytosolic calcium is buffered. Thus it is of physiological interest to determine how buffers affect the properties of calcium waves. Here we summarise and extend previous results on the existence, uniqueness and stability of traveling wave solutions of the buffered bistable equation, which is the simplest possible model of the upstroke of a calcium wave. Taken together, the results show that immobile buffers do not change the existence, uniqueness or stability of the traveling wave, while mobile buffers can eliminate a traveling wave. However, if a wave exists in the latter case, it remains unique and stable.

A bidomain threshold model of propagating calcium waves

We present a bidomain fire-diffuse-fire model that facilitates mathematical analysis of propagating waves of elevated intracellular calcium (Ca(2+)) in living cells. Modeling Ca(2+) release as a threshold process allows the explicit construction of traveling wave solutions to probe the dependence of Ca(2+) wave speed on physiologically important parameters such as the threshold for Ca(2+) release from the endoplasmic reticulum (ER) to the cytosol, the rate of Ca(2+) resequestration from the cytosol to the ER, and the total [Ca(2+)] (cytosolic plus ER). Interestingly, linear stability analysis of the bidomain fire-diffuse-fire model predicts the onset of dynamic wave instabilities leading to the emergence of Ca(2+) waves that propagate in a back-and-forth manner. Numerical simulations are used to confirm the presence of these so-called 'tango waves' and the dependence of Ca(2+) wave speed on the total [Ca(2+)].

Calcium and other ion dynamics during gamete maturation and fertilization

Ion currents and cytosolic free calcium ([Ca(2+)](i)) elevations are crucial events in triggering the complex machinery involved in both gamete maturation and fertilization. Oocyte maturation is triggered by hormone signaling which causes ion currents and [Ca(2+)](i) increase. Extracellular calcium seems to be required for meiosis progression since: (i) calcium depletion in the maturation medium severely affects oocyte developmental competence; (ii) the activity of plasma membrane L-type Ca(2+) currents decreases during maturation; (iii) the exposure to verapamil, a specific Ca(2+) channel blocker, decreases in vitro maturation efficiency. In spermatozoa, maturation initiates inside the epididymis and ends in the female genital tract. During their journey through the female reproductive tract, sperm undergo a dramatic selection and capacitation achieving fertilization competence. Adhesion to the tubal epithelium extends sperm life through depression of [Ca(2+)](i) until capacitation signals trigger an [Ca(2+)](i) elevation followed by sperm release. At fertilization, egg-sperm interaction evokes well-described transient and almost simultaneous events: i.e., fertilization current, a change in resting potential, and an increase in free [Ca(2+)](i) concentration. These events, termed oocyte activation, are the direct consequence of sperm interaction via either activation of a receptor or entry of a sperm factor. The latter hypothesis has been recently supported by the discovery of PCLzeta, a sperm-specific isozyme triggering a dramatic [Ca(2+)](i) increase via inositol 1,4,5-trisphosphate (IP(3)) production. The course of ion currents and [Ca(2+)](i) transients during maturation and fertilization plays a pivotal role in correct embryo development.

Simulations of intracellular calcium release dynamics in response to a high-intensity, ultrashort electric pulse

Numerical simulations for electrically induced, intracellular calcium release from the endoplasmic reticulum are reported. A two-step model is used for self-consistency. Distributed electrical circuit representation coupled with the Smoluchowski equation yields the ER membrane nanoporation for calcium outflow based on a numerical simulation. This is combined with the continuum Li-Rinzel model and drift diffusion for calcium dynamics. Our results are shown to be in agreement with reported calcium release data. A modest increase (rough doubling) of the cellular calcium is predicted in the absence of extra-cellular calcium. In particular, the applied field of 15 kV/cm with 60 ns pulse duration makes for a strong comparison. No oscillations are predicted and the net recovery period of about 5 min are both in agreement with published experimental results. A quantitative explanation for the lack of such oscillatory behavior, based on the density dependent calcium fluxes, is also provided.

A kinetic Monte Carlo simulation study of inositol 1,4,5-trisphosphate receptor (IP3R) calcium release channel

Most of the previously theoretical studies about the stochastic nature of the IP3R calcium release channel gating use the chemical master equation (CME) approach. Because of the limitations of this approach we have used a stochastic simulation algorithm (SSA) presented by Gillespie. A single subunit of De Young-Keizer (DYK) model was simulated using Gillespie algorithm. The model has been considered in its complete form with eight states. We investigate the conditions which affect the open state of the model. Calcium concentrations were the subject of fluctuation in the previous works while in this study the population of the states is the subject of stochastic fluctuations. We found out that decreasing open probability is a function of Ca(2+) concentration in fast time domain, while in slow time domain it is a function of IP3 concentration. Studying the population of each state shows a time dependent reaction pattern in fast and medium time domains (10(-4) and 10(-3)s). In this pattern the state of X(010) has a determinative role in selecting the open state path. Also, intensity and frequency of fluctuations and Ca(2+) inhibitions have been studied. The results indicate that Gillespie algorithm can be a better choice for studying such systems, without using any approximation or elimination while having acceptable accuracy. In comparison with the chemical master equation, Gillespie algorithm is also provides a wide area for studying biological systems from other points of view.

The asymptotic behavior of solutions of the buffered bistable system

In this paper, we study a model for calcium buffering with bistable nonlinearity. We present some results on the stability of equilibrium states and show that there exists a threshold phenomenon in our model. In comparing with the model without buffers, we see that stationary buffers cannot destroy the asymptotic stability of the associated equilibrium states and the threshold phenomenon. Moreover, we also investigate the propagation property of solutions with initial data being a disturbance of one of the stable states which is confined to a half-line. We show that the more stable state will eventually dominate the whole dynamics and that the speed of this propagation (or invading process) is positive.

Signal transduction pathways leading to Ca2+ release in a vertebrate model system: Lessons from Xenopus eggs

At fertilization, eggs unite with sperm to initiate developmental programs that give rise to development of the embryo. Defining the molecular mechanism of this fundamental process at the beginning of life has been a key question in cell and developmental biology. In this review, we examine sperm-induced signal transduction events that lead to release of intracellular Ca(2+), a pivotal trigger of developmental activation, during fertilization in Xenopus laevis. Recent data demonstrate that metabolism of inositol 1,4,5-trisphosphate (IP(3)), a second messenger for Ca(2+) release, is carefully regulated and involves phospholipase C (PLC) and the tyrosine kinase Src. Roles of other potential regulators in this pathway, such as phosphatidylinositol 3-kinase, heterotrimeric GTP-binding protein, phospholipase D (PLD) and phosphatidic acid (PA) are also discussed. Finally, we address roles of egg lipid/membrane microdomains or 'rafts' as a platform for the sperm-egg membrane interaction and subsequent signaling events of egg activation.

Autonomous regulation of free Ca2+ concentrations in isolated plant cell nuclei: a mathematical analysis

Experiments performed on nuclei isolated from animal or plant cells have provided evidence that the nucleus generates directly specific nucleoplasmic calcium transients in response to external stimuli. Recent data suggest that isolated plant nuclei might be considered as a closed system where the nuclear concentration of free calcium would be regulated by reversible movements between the nucleoplasm and nuclear stores. We have addressed the relevance of this hypothesis by developing a mathematical approach to simulate nucleoplasmic calcium dynamics generated under various pH and temperature conditions. Here, we show that the experimental results could be explained provided that calcium channels as well as systems transporting calcium are present on the inner nuclear membrane. The putative channels would allow the entry of calcium into the nucleoplasm whereas the elusive transporting system(s) would contribute to replenish the nuclear stores. The simple proposed model is versatile enough to explain and predict autonomous changes in free calcium in the nucleoplasm of isolated plant nuclei.

Ca2+ shuttling between endoplasmic reticulum and mitochondria underlying Ca2+ oscillations

Although many cell functions are regulated by Ca(2+) oscillations induced by a cyclic release of Ca(2+) from intracellular Ca(2+) stores, the pacemaker mechanism of Ca(2+) oscillations remains to be explained. Using green fluorescent protein-based Ca(2+) indicators that are targeted to intracellular Ca(2+) stores, the endoplasmic reticulum (ER) and mitochondria, we found that Ca(2+) shuttles between the ER and mitochondria in phase with Ca(2+) oscillations. Following agonist stimulation, Ca(2+) release from the ER generated the first Ca(2+) oscillation and loaded mitochondria with Ca(2+). Before the second Ca(2+) oscillation, Ca(2+) release from the mitochondria by means of the Na(+)/Ca(2+) exchanger caused a gradual increase in cytoplasmic Ca(2+) concentration, inducing a regenerative ER Ca(2+) release, which generated the peak of Ca(2+) oscillation and partially reloaded the mitochondria. This sequence of events was repeated until mitochondrial Ca(2+) was depleted. Thus, Ca(2+) shuttling between the ER and mitochondria may have a pacemaker role in the generation of Ca(2+) oscillations.

Modeling and analysis of calcium signaling events leading to long-term depression in cerebellar Purkinje cells

Modeling and simulation of the calcium signaling events that precede long-term depression of synaptic activity in cerebellar Purkinje cells are performed using the Virtual Cell biological modeling framework. It is found that the unusually high density and low sensitivity of inositol-1,4,5-trisphosphate receptors (IP3R) are critical to the ability of the cell to generate and localize a calcium spike in a single dendritic spine. The results also demonstrate the model's capability to simulate the supralinear calcium spike observed experimentally during coincident activation of the parallel and climbing fibers. The sensitivity of the calcium spikes to certain biological and geometrical effects is investigated as well as the mechanisms that underlie the cell's ability to generate the supralinear spike. The sensitivity of calcium release rates from the IP3R to calcium concentrations, as well as IP3 concentrations, allows the calcium spike to form. The diffusion barrier caused by the small radius of the spine neck is shown to be important, as a threshold radius is observed above which a spike cannot be formed. Additionally, the calcium buffer capacity and diffusion rates from the spine are found to be important parameters in shaping the calcium spike.

Direct stochastic simulation of Ca2+ motion in Xenopus eggs

The release of important intracellular ions has been widely modeled using two approaches, namely, (1) Fickian diffusion, in which sometimes tensorial diffusion coefficients are used to fit observed temporally varying concentrations of calcium, and (2) cellular automata, which produce a set of localized finite difference equations that result in complex global behavior. Here, we take a different approach, employing some assumed, a priori, distribution of ion-binding proteins in the cell, and some assumed biochemical capture and release characteristics to explain ionic motion, and ultimately, distribution. We study several scenarios for ion distribution, based on differences in binder action and distribution. The numbers and strengths of ion binders, spatial variation in inositol 1,4,5-triphosphate concentration, together with the escalating distribution of ionic diffusion speed, are found to be key factors leading to concavity in the Ca2+ wave shape. We also offer an explanation for geometrical effects on previously observed ion diffusion speeds in the cellular cortex of the Xenopus laevis egg during fertilization, based on an angle-of-view correction.

Cortically restricted production of IP3 leads to propagation of the fertilization Ca2+ wave along the cell surface in a model of the Xenopus egg

The fertilization Ca2+ wave in Xenopus laevis is a single, large wave of elevated free cytosolic Ca2+ concentration that emanates from the point of sperm-egg fusion and traverses the entire diameter of the egg. This phenomenon appears to involve an increase in inositol-1,4,5-trisphosphate (IP3) resulting from interaction of the sperm and egg, which then results in the activation of the endoplasmic reticulum Ca2+ release machinery. We have proposed models based on a static elevated distribution of IP3, and dynamic [IP3], however, these models have suggested that the fertilization wave passes through the center of the egg. Complementing these earlier models, we propose a more detailed model of the fertilization Ca2+ wave in Xenopus eggs to explore the hypothesis that IP3 is produced only at or near the plasma membrane. In this case, we find that the wave propagates primarily through the cortex of the egg, and that Ca2+ -induced production of IP3 at the plasma membrane allows IP3 to propagate in advance of the wave. Our model includes Ca2+ -dependent production of IP3 at the plasma membrane and IP3 degradation. Simulations in 1 dimension and axi-symmetric 3 dimensions illustrate the basic features of the wave.

Simulation of calcium waves in ascidian eggs: insights into the origin of the pacemaker sites and the possible nature of the sperm factor

Fertilization triggers repetitive waves of cytosolic Ca2+ in the egg of many species. The mechanism involved in the generation of Ca2+ waves has been studied in much detail in mature ascidian eggs, by raising artificially the level of inositol 1,4,5-trisphosphate [Ins(1,4,5)P3] or of its poorly metabolizable analogue, glycero-myo-phosphatidylinositol 4,5-bisphosphate [gPtdIns(4,5)P2]. Here, we use this strategy and the experimental results it provides to develop a realistic theoretical model for repetitive Ca2+ wave generation and propagation in mature eggs. The model takes into account the heterogeneous spatial distribution of the endoplasmic reticulum. Our results corroborate the hypothesis that Ca2+ wave pacemakers are associated with cortical accumulations of endoplasmic reticulum. The model is first tested and validated by the adequate match between its theoretical predictions and the observed effects of localized injections of massive amounts of Ins(1,4,5)P3 analogues. In a second step, we use the model to make some propositions about the possible characteristics of the sperm factor. We find that to account for the spatial characteristics of the first series of Ca2+ waves seen at fertilization in ascidian eggs, it has to be assumed that, if the sperm factor is a phospholipase C, it is Ca2+-sensitive and highly diffusible. Although the actual state of knowledge does not allow us to explain the observed relocalization of the Ca2+ wave pacemaker site, the model corroborates the assumption that PtdIns(4,5)P2, the substrate for phospholipase C is distributed over the entire egg. We also predict that the dose of sperm factor injected into the egg should modulate the temporal characteristics of the first, long-lasting fertilization wave.

A wave of IP3 production accompanies the fertilization Ca2+ wave in the egg of the frog, Xenopus laevis: theoretical and experimental support

The fertilization Ca2+ wave in Xenopus laevis is a single, large wave of elevated free Ca2+ that is initiated at the point of sperm-egg fusion and traverses the entire width of the egg. This Ca2+ wave involves an increase in inositol-1,4,5-trisphosphate (IP3) resulting from the interaction of the sperm and egg, which then results in the activation of the endoplasmic reticulum Ca2+ release machinery. The extraordinarily large size of this cell (1.2 mm diameter) together with the small surface region of sperm-receptor activation makes special demands on the IP3-dependent Ca2+ mobilizing machinery. We propose a detailed model of the fertilization Ca2+ wave in Xenopus eggs that requires an accompanying wave of IP3 production. While the Ca2+ wave is initiated by a localized increase of IP3 near the site of sperm-egg fusion, the Ca2+ wave propagates via IP3 production correlated with the Ca2+ wave-possibly via Ca(2+)-mediated PLC activation. Such a Ca(2+)-mediated IP(3) production wave has not been required previously to explain the fertilization Ca2+ wave in eggs; we argue this is necessary to explain the observed IP3 dynamics in Xenopus eggs. To test our hypothesis, we have measured the IP3 levels from 20 nl "sips" of the egg cortex during wave propagation. We were unable to detect the low IP3 levels in unfertilized eggs, but after fertilization, [IP3] ranged from 175 to 430 nM at the sperm entry point and from 120 to 700 nM 90 degrees away once the Ca2+ wave passed that region about 2 min after fertilization. Prior to the Ca2+ wave reaching that region the IP3 levels were undetectable. Since significant IP3 could not diffuse to this region from the sperm entry point within 2 min, this observation is consistent with a regenerative wave of IP3 production.

Sheet excitability and nonlinear wave propagation

In the Xenopus laevis oocyte, calcium ion channels are clustered in a thin shell. Motivated by this morphology, we study a general class of reaction-diffusion systems that include most of the well-known models that support wave propagation but restricting excitability to a "sheet" of codimension 1. We find waves that undergo propagation failure with increasing diffusion coefficient and a scaling regime in which the wave speed is independent of it.

Temporal integration by calcium dynamics in a model neuron

The calculation and memory of position variables by temporal integration of velocity signals is essential for posture, the vestibulo-ocular reflex (VOR) and navigation. Integrator neurons exhibit persistent firing at multiple rates, which represent the values of memorized position variables. A widespread hypothesis is that temporal integration is the outcome of reverberating feedback loops within recurrent networks, but this hypothesis has not been proven experimentally. Here we present a single-cell model of a neural integrator. The nonlinear dynamics of calcium gives rise to propagating calcium wave-fronts along dendritic processes. The wave-front velocity is modulated by synaptic inputs such that the front location covaries with the temporal sum of its previous inputs. Calcium-dependent currents convert this information into concomitant persistent firing. Calcium dynamics in single neurons could thus be the physiological basis of the graded persistent activity and temporal integration observed in neurons during analog memory tasks.

Modeling the dependence of the period of intracellular Ca2+ waves on SERCA expression

Contrary to intuitive expectations, overexpression of sarco-endoplasmic reticulum (ER) Ca(2+) ATPases (SERCAs) in Xenopus oocytes leads to a decrease in the period and an increase in the amplitude of intracellular Ca(2+) waves. Here we examine these experimental findings by modeling Ca(2+) release using a modified Othmer-Tang-model. An increase in the period and a reduction in the amplitude of Ca(2+) wave activity are obtained when increases in SERCA density are simulated while keeping all other parameters of the model constant. However, Ca(2+) wave period can be reduced and the wave amplitude and velocity can be significantly increased when an increase in the luminal ER Ca(2+) concentration due to SERCA overexpression is incorporated into the model. Increased luminal Ca(2+) occurs because increased SERCA activity lowers cytosolic Ca(2+), which is partially replenished by Ca(2+) influx across the plasma membrane. These simulations are supported by experimental data demonstrating higher luminal Ca(2+) levels, decreased periods, increased amplitude, and increased velocity of Ca(2+) waves in response to increased SERCA density.

Simulation of waves in calcium models with 3D spherical geometry

Waves of calcium ions are present in fertilized eggs of many species. Models for pulse and tidal wave propagation have usually been studied in one or two spatial coordinates only. We examine in three spatial coordinates some established models, based on Ca(2+)-induced Ca(2+)-release from both (assumed) continuously or heterogeneously distributed stores of endoplasmic reticulum (ER) through channels activated by inositol triphosphate (IP(3)). With continuous IP(3) distribution decreasing radially towards the interior, we obtain concave pulse shapes for waves penetrating the interior. Concave waves are also recorded in systems with ER confined to distributions of small spheres (microdomains) inside the cell, which we simulate for front waves (tides) in bistable systems.

Sperm initiate a Ca2+ wave in frog eggs that is more similar to Ca2+ waves initiated by IP3 than by Ca2+

We have measured the initial propagation velocity of the sperm-induced Ca(2+) wave in the egg of Xenopus laevis and have compared it with the initial propagation velocities of the inositol triphosphate (IP(3))-induced and Ca(2+)-induced Ca(2+) waves. The initial mean propagation velocity of the sperm-induced wave (13 microm/s) is very similar to that of the IP(3)-induced waves (12.3 microm/s) and two times faster than the mean Ca(2+)-induced wave velocity (6.6 microm/s). We have generated realistic simulations of the fertilization wave in the frog egg using a computational technique based on the finite difference method. Modeling refinements presented here include equations for the production, degradation, and diffusion of IP(3), a description for Ca(2+) dynamics in the endoplasmic reticulum, and a highly concentrated endoplasmic reticulum in the egg cortex. We conclude that models incorporating sperm-induced IP(3) generation fit the data best and those involving the influx of either Ca(2+) or a diffusible sperm factor fit the data poorly. This independence from Ca(2+) influx is also supported by electrophysiological data indicating that Ca(2+) influx is not needed to maintain open Cl(-) channels that generate the fertilization potential.

Selection of intracellular calcium patterns in a model with clustered Ca2+ release channels

A two-dimensional model is proposed for intracellular Ca2+ waves, which incorporates both the discrete nature of Ca2+ release sites in the endoplasmic reticulum membrane and the stochastic dynamics of the clustered inositol 1,4,5-triphosphate (IP3) receptors. Depending on the Ca2+ diffusion coefficient and concentration of IP3, various spontaneous Ca2+ patterns, such as calcium puffs, local waves, abortive waves, global oscillation, and tide waves, can be observed. We further investigate the speed of the global waves as a function of the IP3 concentration and the Ca2+ diffusion coefficient and under what conditions the spatially averaged Ca2+ response can be described by a simple set of ordinary differential equations.

Buffers and oscillations in intracellular Ca2+ dynamics

I model the behavior of intracellular Ca(2+) release with high buffer concentrations. The model uses a spatially discrete array of channel clusters. The channel subunit dynamics is a stochastic representation of the DeYoung-Keizer model. The calculations show that the concentration profile of fast buffer around an open channel is more localized than that of slow buffers. Slow buffers allow for release of larger amounts of Ca(2+) from the endoplasmic reticulum and hence bind more Ca(2+) than fast buffers with the same dissociation constant and concentration. I find oscillation-like behavior for high slow buffer concentration and low Ca(2+) content of the endoplasmic reticulum. High concentration of slow buffer leads to oscillation-like behavior by repetitive wave nucleation for high Ca(2+) content of the endoplasmic reticulum. Localization of Ca(2+) release by slow buffer, as used in experiments, can be reproduced by the modeling approach.

On the role of stochastic channel behavior in intracellular Ca2+ dynamics

I present a stochastic model for intracellular Ca(2+) oscillations. The model starts from stochastic binding and dissociation of Ca(2+) to binding sites on a single subunit of the IP(3)-receptor channel but is capable of simulating large numbers of clusters for many oscillation periods too. I find oscillations with variable periods ranging from 17 s to 120 s and a standard deviation well in the experimentally observed range. Long period oscillations can be perceived as nucleation phenomenon and can be observed for a large variety of single channel dynamics. Their period depends on the geometric characteristics of the cluster array. Short periods are in the range of the time scale of channel dynamics. Both long and short period oscillations occur for parameters with a nonoscillatory deterministic regime.

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.

Mitochondrial modulation of intracellular Ca(2+) signaling

IP3-mediated Ca(2+) release plays a fundamental role in many cell signaling processes and has been the subject of numerous modeling studies. Only recently has the important role that mitochondria play in the dynamics of intracellular Ca(2+) signaling begun to be considered in experimental work and in computational models. Mitochondria sequester large amounts of Ca(2+) and thus have a modulatory effect on intracellular Ca(2+) signaling, and mitochondrial uptake of Ca(2+), in turn, has a regulatory effect on mitochondrial function. Here we integrate a well-established model of IP3-mediated Ca(2+) signaling with a detailed model of mitochondrial Ca(2+) handling and metabolic function. The incorporation of mitochondria results in oscillations in a bistable formulation of the IP3 model, and increasing metabolic substrate decreases the frequency of these oscillations consistent with the literature. Ca(2+) spikes from the cytosol are communicated into mitochondria and are shown to induce realistic metabolic changes. The model has been formulated using a modular approach that is easy to modify and should serve as a useful basis for the investigation of questions regarding the interaction of these two systems.

Analysis of nonlinear dynamics on arbitrary geometries with the Virtual Cell

The Virtual Cell is a modeling tool that allows biologists and theorists alike to specify and simulate cell-biophysical models on arbitrarily complex geometries. The framework combines an intuitive, front-end graphical user interface that runs in a web browser, sophisticated server-side numerical algorithms, a database for storage of models and simulation results, and flexible visualization capabilities. In this paper, we present an overview of the capabilities of the Virtual Cell, and, for the first time, the detailed mathematical formulation used as the basis for spatial computations. We also present summaries of two rather typical modeling projects, in order to illustrate the principal capabilities of the Virtual Cell. (c) 2001 American Institute of Physics.

Physiological modeling with virtual cell framework

This article describes a computational framework for cell biological modeling and simulation that is based on the mapping of experimental biochemical and electrophysiological data onto experimental images. The framework is designed to enable the construction of complex general models that encompass the general class of problems coupling reaction and diffusion.

Stochastic spreading of intracellular Ca(2+) release

We study the spreading of calcium-induced calcium release with the stochastic DeYoung-Keizer-model of the inositol 1,4,5-trisphosphate receptor channel. The model shows a transition from isolated release events to steadily propagating waves with increasing IP3 concentration. A state--stochastic backfiring--was found in the regime of steady propagation. The model can be reduced by an adiabatic elimination of the partial differential equation for the Ca(2+) concentration to a lattice of stochastic channel clusters.

Confocal laser scanning microscopy of calcium dynamics in living cells

Confocal laser scanning microscopy (CLSM) is widely used to monitor intracellular calcium levels in living cells loaded with calcium-sensitive fluorophores. This review examines the basic advantages and limitations of CLSM in in vivo imaging analyses of calcium dynamics. The benefits of utilizing ratioed images and dextran-conjugated fluorophores are addressed, and practical aspects of handling confocal datasets are outlined. After considering some relatively new microscopical methods that can be used in conjunction with conventional CLSM, possible future applications of confocal techniques in analyses of intracellular calcium dynamics are discussed.

Comparative biology of calcium signaling during fertilization and egg activation in animals

During animal fertilizations, each oocyte or egg must produce a proper intracellular calcium signal for development to proceed normally. As a supplement to recent synopses of fertilization-induced calcium responses in mammals, this paper reviews the spatiotemporal properties of calcium signaling during fertilization and egg activation in marine invertebrates and compares these patterns with what has been reported for other animals. Based on the current database, fertilization causes most oocytes or eggs to generate multiple wavelike calcium oscillations that arise at least in part from the release of internal calcium stores sensitive to inositol 1,4,5-trisphosphate (IP3). Such calcium waves are modulated by upstream pathways involving oolemmal receptors and/or soluble sperm factors and in turn regulate calcium-sensitive targets required for subsequent development. Both "protostome" animals (e.g., mollusks, annelids, and arthropods) and "deuterostomes" (e.g., echinoderms and chordates) display fertilization-induced calcium waves, IP3-mediated calcium signaling, and the ability to use a combination of external calcium influx and internal calcium release. Such findings fail to support the dichotomy in calcium signaling modes that had previously been proposed for protostomes vs deuterostomes and instead suggest that various features of fertilization-induced calcium signals are widely shared throughout the animal kingdom.

Impact of mitochondrial Ca2+ cycling on pattern formation and stability

Energization of mitochondria significantly alters the pattern of Ca2+ wave activity mediated by activation of the inositol (1,4,5) trisphosphate (IP3) receptor (IP3R) in Xenopus oocytes. The number of pulsatile foci is reduced and spiral Ca2+ waves are no longer observed. Rather, target patterns of Ca2+ release predominate, and when fragmented, fail to form spirals. Ca2+ wave velocity, amplitude, decay time, and periodicity are also increased. We have simulated these experimental findings by supplementing an existing mathematical model with a differential equation for mitochondrial Ca2+ uptake and release. Our calculations show that mitochondrial Ca2+ efflux plays a critical role in pattern formation by prolonging the recovery time of IP3Rs from a refractory state. We also show that under conditions of high energization of mitochondria, the Ca2+ dynamics can become bistable with a second stable stationary state of high resting Ca2+ concentration.

Characterization of the sperm-induced calcium wave in Xenopus eggs using confocal microscopy

We have used confocal microscopy to examine the [Ca2+]i increase in the albino eggs of the frog Xenopus laevis after fertilization. Eggs were placed in agar wells with their animal poles downward so that fertilization occurred preferentially in the equatorial plane, and confocal microscopy was used to provide a two-dimensional optical section through the three-dimensional Ca2+ wave. These data indicate that the wave of increased [Ca2+]i traverses the entire egg and converges uniformly on the antipode. We show that ratioing two different fluorescent dyes to correct for variations in cell thickness is not a reliable technique for this very thick cell due to differential absorption with depth. Indo-1-dextran proves to be a more reliable Ca2+ indicator in this respect. Indo-1-dextran measurements indicate that the resting [Ca2+]i is not uniform throughout the egg but exhibits a 15% higher [Ca2+]i in the cortex than deep in the cytoplasm. This difference is accentuated during wave propagation and is not dependent on extracellular Ca2+. The average peak [Ca2+]i in the center of the egg as the wave propagates through it is 0.7 microM, approximately 60% of the peak cortical [Ca2+]i. The wave velocity through the center of the egg (5.7 micron/s) is slower than that in the cortex (8.9 micron/s), and both velocities vary slightly during transit. The cortical wave speed is particularly high at the beginning (15.7 micron/s) and end (17.2 micron/s) of the wave. Eggs injected with 30-80 microM of 3 kD heparin to compete with inositol-1,4,5,-trisphosphate for binding to its receptor exhibited multiple localized spots of elevated [Ca2+]i, and many of these did not initiate a wave. For those that did lead to a wave, it was usually slow moving and exhibited a reduced (60% reduction) amplitude compared with controls.

Simulation of the fertilization Ca2+ wave in Xenopus laevis eggs

In the preceding paper Fontanilla and Nuccitelli (Biophysical Journal 75:2079-2087 (1998)) present detailed measurements of the shape and speed of the fertilization Ca2+ wave in Xenopus laevis eggs. In order to help interpret their results, we develop here a computational technique based on the finite element method that allows us to carry out realistic simulations of the fertilization wave. Our simulations support the hypothesis that the physiological state of the mature egg is bistable, i.e., that its cytoplasm can accommodate two alternative physiological Ca2+ concentrations: a low concentration characteristic of the prefertilization state and a greatly elevated concentration characteristic of the state following the passage of the wave. We explore this hypothesis by assuming that the bistability is due to the release and re-uptake properties of the endoplasmic reticulum (ER) as determined by inositol trisphosphate (IP3) receptor/Ca2+ channels and sarcoendoplasmic reticulum calcium ATPase (SERCA) pumps. When combined with buffered diffusion of Ca2+ in the cytoplasm, our simulations show that inhomogeneities in the Ca2+ release properties near the plasma membrane are required to explain the temporal and spatial dependences of the shape and speed of these waves. Our results are consistent with an elevated IP3 concentration near the plasma membrane in the unfertilized egg that is augmented significantly near the site of fertilization. These gradients are essential in determining the concave shape of the Ca2+ fertilization wave front.