We have measured the initial propagation velocity of the sperm-induced Ca2+ wave in the egg of Xenopus laevis and have compared it with the initial propagation velocities of the inositol triphosphate (IP3)-induced and Ca2+-induced Ca2+ waves. The initial mean propagation velocity of the sperm-induced wave (13μm/s) is very similar to that of the IP3-induced waves (12.3μm/s) and two times faster than the mean Ca2+-induced wave velocity (6.6μm/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 IP3, a description for Ca2+ dynamics in the endoplasmic reticulum, and a highly concentrated endoplasmic reticulum in the egg cortex. We conclude that models incorporating sperm-induced IP3 generation fit the data best and those involving the influx of either Ca2+ or a diffusible sperm factor fit the data poorly. This independence from Ca2+ influx is also supported by electrophysiological data indicating that Ca2+ influx is not needed to maintain open Cl− channels that generate the fertilization potential.

Over the past several decades, progress in the measurement of rates and interactions of molecular and cellular processes has initiated a revolution in our understanding of dynamic phenomena in cells. Spikes or bursts of plasma membrane electrical activity or intracellular signaling via receptors, second messengers, or other networked biochemical pathways in single cells, or more complex processes that involve small clusters of cells, organelles, or groups of neurons, are examples of the complex behaviors that we know take place on the cellular scale. The vast amount of quantitative information uncovered in recent years, leveraged by the intricate mechanisms already shown to exist, results in an array of possibilities that makes it quite hard to evaluate new hypotheses on an intuitive basis. Using mathematical analysis and computer simulation we can show that some seemingly reasonable hypotheses are not possible. Analysis and simulations that confirm that a given hypothesis is reasonable can often result in quantitative predictions for further experimental exploration. Rapid advances in computer hardware and software technology combined with pioneering work giving structure to the interface between mathematics and biology have put the ability to test hypotheses and evaluate mechanisms with simulations within the reach of all cell biologists and neuroscientists.

One of the hallmarks of cellular processes is their complexity. For example, in Chapter 3 we described a detailed model for the SERCA pump that might require 11 ODEs and 22 kinetic constants for its analysis. Similarly, the Hodgkin-Huxley model, which includes only three currents in the squid giant axon, involves 4 differential equations and information about three voltage gated currents. As complex as these processes are, they do not begin to represent the true complexity of cellular processes like muscle contraction or insulin secretion, which depend on the coupling of numerous dynamic components. In Chapter 5 we describe some examples of “whole-cell” modeling that attempt to deal with these larger issues.

In this chapter we shall discuss a variety of intracellular Ca2+ wave phenomena, but always from the perspective that the distance scales of interest are large enough that Ca2+ transport is well-modeled by conservation equations based on a continuum description of matter (recall Chapter 7). Although recent experimental and theoretical work suggests that the macroscopic behavior of propagating Ca2+ waves (e.g., wave speed) may depend in subtle ways on the density and distribution of intracellular Ca2+ release channels, we postpone consideration of intracellular heterogeneities such as clusters of Ca2+ release channels until later in the chapter. This makes sense because both the mathematics and simulation methods used to study nonlinear wave propagation in homogeneous media are simpler than the heterogeneous case. This simplicity should facilitate the development of intuition regarding nonlinear wave propagation. Throughout the chapter a recurring theme will be the manner in which Ca2+ buffers, through their important association with free Ca2+, can influence wave phenomena dependent on diffusion. The chapter concludes with calculations of localized Ca2+ elevations due to intracellular Ca2+ release, i.e., Ca2+ “puffs” or “sparks,” elementary events that sum to produce Ca2+ waves.

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.

During development, mammalian retinal ganglion cells (RGCs) go through marked ontogenetic changes with respect to their excitable membrane properties. Voltage-clamp studies conducted in our laboratory have shown that the amplitude, voltage-dependence and kinetics of activation and inactivation (where present) of Na(+), K(+) and Ca(2+) conductances all exhibit developmental changes during a time when the firing patterns of mammalian ganglion cells shift from being transient to being predominantly sustained in nature. In order to better understand the contribution of each conductance to the generation of spikes and spiking patterns, we have developed a model based on our experimental data. For simplicity, we have initially used experimental data obtained from postnatal ganglion cells. At this age the ontogenetic changes observed in the characteristics of the various ionic currents are complete. Utilizing the methods adopted by Hodgkin and Huxley for the giant squid axon, we have determined rate equations for the activation and inactivation properties of the I(A), I(K dr), I(Na), I(Ca L), I(Ca N), and I(leak) currents in postnatal cat RGCs. Combining these with a simplified model of the calcium-activated potassium current (I(KCa)), we have solved and analysed the resulting differential equations. While spikes and spiking patterns resembling experimental data could be obtained from a model in which [Ca(2+)i] was averaged across the whole cell, more accurate simulations were obtained when the diffusion of intracellular Ca(2+) was modeled spatially. The resulting spatial calcium gradients were more effective in gating I(KCa), and our simulations more accurately matched the recorded amplitude and shape of individual spikes as well as the frequency of maintained discharges observed in mammalian postnatal RGCs.

When Ca2+ is released from internal stores in living cells, the resulting wave of increased concentration can travel without deformation (continuous propagation) or with burst-like behavior (saltatory propagation). We analyze the "fire-diffuse-fire" model in order to illuminate the differences between these two modes of propagation. We show that the Ca2+ release wave in immature Xenopus oocytes and cardiac myocytes is saltatory, whereas the fertilization wave in the mature oocyte is continuous.

Punctate releases of Ca2+, called Ca2+ sparks, originate at the regular array of t-tubules in cardiac myocytes and skeletal muscle. During Ca2+ overload sparks serve as sites for the initiation and propagation of Ca2+ waves in myocytes. Computer simulations of spark-mediated waves are performed with model release sites that reproduce the adaptive Ca2+ release observed for the ryanodine receptor. The speed of these waves is proportional to the diffusion constant of Ca2+, D, rather than D, as is true for reaction-diffusion equations in a continuous excitable medium. A simplified "fire-diffuse-fire" model that mimics the properties of Ca2+-induced Ca2+ release (CICR) from isolated sites is used to explain this saltatory mode of wave propagation. Saltatory and continuous wave propagation can be differentiated by the temperature and Ca2+ buffer dependence of wave speed.

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.

The elementary events of excitation-contraction coupling in heart muscle are Ca2+ sparks, which arise from one or more ryanodine receptors in the sarcoplasmic reticulum (SR). Here a simple numerical model is constructed to explore Ca2+ spark formation, detection, and interpretation in cardiac myocytes. This model includes Ca2+ release, cytosolic diffusion, resequestration by SR Ca2+-ATPases, and the association and dissociation of Ca2+ with endogenous Ca2+-binding sites and a diffusible indicator dye (fluo-3). Simulations in a homogeneous, isotropic cytosol reproduce the brightness and the time course of a typical cardiac Ca2+ spark, but underestimate its spatial size (∼1.1μm vs. ∼2.0μm). Back-calculating [Ca2+]i by assuming equilibrium with indicator fails to provide a good estimate of the free Ca2+ concentration even when using blur-free fluorescence data. A parameter sensitivity study reveals that the mobility, kinetics, and concentration of the indicator are essential determinants of the shape of Ca2+ sparks, whereas the stationary buffers and pumps are less influential. Using a geometrically more complex version of the model, we show that the asymmetric shape of Ca2+ sparks is better explained by anisotropic diffusion of Ca2+ ions and indicator dye rather than by subsarcomeric inhomogeneities of the Ca2+ buffer and transport system. In addition, we examine the contribution of off-center confocal sampling to the variance of spark statistics.

The concentration of Ca2+ in intracellular stores is an important factor in many aspects of Ca2+ signaling, including the generation of Ca2+ spikes, oscillations and waves, control of mitochondrial respiration, and activation of store-operated Ca2+ channels. Here we describe a consistent method for estimating the content of stores, based on the release of stored Ca2+ by thapsigargin (TG) or ionomycin (IO). Once released from stores, Ca2+ elevates [Ca2+]i transiently before it is pumped across the plasma membrane. If the dependence of the pump rate on [Ca2+]i is known, then the kinetics and amplitude of the Ca2+ transient allows the total amount of releasable Ca2+ to be estimated. We develop this quantitative approach and validate its use in human T cells, in which the Ca2+ clearance rate is an approximately linear function of [Ca2+]i. Our results support the assumption that the ER Ca2+ leak in resting T cells is unregulated, i.e. its rate is proportional to luminal [Ca2+]. The characteristic time constant for basal Ca2+ release is 110-140 s, comparable to that for activation of Ca2+ release-activated Ca2+ (CRAC) channels by TG and consistent with the dependence of ICRAC on store depletion. This method for estimating store content may be useful for quantifying the overlap between functionally distinct stores and for defining the relation between store content and cellular responses.

Oscillations in cytosolic Ca2+ concentrations in living cells are often a manifestation of propagating waves of Ca2+. Numerical simulations with a realistic model of inositol 1,4,5-trisphosphate (IP3)-induced Ca2+ wave trains lead to wave speeds that increase linearly at long times when (a) IP3 levels are in the range for Ca2+ oscillations, (b) a gradient of phase is established by either an initial ramp or pulse of IP3, and (c) IP3 concentrations asymptotically become uniform. We explore this phenomenon with analytical and numerical methods using a simple two-variable reduction of the De Young-Keizer model of the IP3 receptor that includes the influence of Ca2+ buffers. For concentrations of IP3 in the oscillatory regime, numerical solution of the resulting reaction diffusion equations produces nonlinear wave trains that shows the same asymptotic growth of wave speed. Due to buffering, diffusion of Ca2+ is quite slow and, as previously noted, these waves occur without appreciable bulk movement of Ca2+. Thus, following Neu and Murray, we explore the behavior of these waves using an asymptotic expansion based on the small size of the buffered diffusion constant for Ca2+. We find that the gradient in phase of the wave obeys Burgers' equation asymptotically in time. This result is used to explain the linear increase of the wave speed observed in the simulations.

Inositol 1,4,5-trisphosphate (IP3)-induced Ca2+ mobilization leads to depletion of the endoplasmic reticulum (ER) and an increase in Ca2+ entry. We show here for the gonadotroph, an excitable endocrine cell, that sensing of ER Ca2+ content can occur without the Ca2+ release-activated Ca2+ current (Icrac), but rather through the coupling of IP3-induced Ca2+ oscillations to plasma membrane voltage spikes that gate Ca2+ entry. Thus we demonstrate that capacitative Ca2+ entry is accomplished through Ca(2+)-controlled Ca2+ entry. We develop a comprehensive model, with parameter values constrained by available experimental data, to simulate the spatiotemporal behavior of agonist-induced Ca2+ signals in both the cytosol and ER lumen of gonadotrophs. The model combines two previously developed models, one for ER-mediated Ca2+ oscillations and another for plasma membrane potential-driven Ca2+ oscillations. Simulations show agreement with existing experimental records of store content, cytosolic Ca2+ concentration ([Ca2+]i), and electrical activity, and make a variety of new, experimentally testable predictions. In particular, computations with the model suggest that [Ca2+]i in the vicinity of the plasma membrane acts as a messenger for ER content via Ca(2+)-activated K+ channels and Ca2+ pumps in the plasma membrane. We conclude that, in excitable cells that do not express Icrac, [Ca2+]i profiles provide a sensitive mechanism for regulating net calcium flux through the plasma membrane during both store depletion and refilling.

A simplified mechanism that mimics "adaptation" of the ryanodine receptor (RyR) has been developed and its significance for Ca2+(-)induced Ca2+ release and Ca2+ oscillations investigated. For parameters that reproduce experimental data for the RyR from cardiac cells, adaptation of the RyR in combination with sarco/endoplasmic reticulum Ca2+ ATPase Ca2+ pumps in the internal stores can give rise to either low [Cai2+] steady states or Ca2+ oscillations coexisting with unphysiologically high [Cai2+] steady states. In this closed-cell-type model rapid, adaptation-dependent Ca2+ oscillations occur only in limited ranges of parameters. In the presence of Ca2+ influx and efflux from outside the cell (open-cell model) Ca2+ oscillations occur for a wide range of physiological parameter values and have a period that is determined by the rate of Ca2+ refilling of the stores. Although the rate of adaptation of the RyR has a role in determining the shape and the period of the Ca2+ spike, it is not essential for their existence. This is in marked contrast with what is observed for the inositol 1,4,5-trisphosphate receptor for which the biphasic activation and inhibition of its activity by Ca2+ are sufficient to produce oscillations. Results for this model are compared with those based on Ca2+(-)induced Ca2+ release alone in the bullfrog sympathetic neuron. This kinetic model should be suitable for analyzing phenomena associated with "Ca2+ sparks," including their merger into Ca2+ waves in cardiac myocytes.

Using ideas from statistical nonequilibrium thermodynamics we develop a spatially nonlocal theory of the influence of diffusion on rapid chemical reactions. The new approach generalizes the hydrodynamic (local) theory by including elementary events that influence both diffusion and reaction on molecular length scales and eliminates problems in the local theory that occur at short times. Examples of the nonlocal approach are given for several reaction schemes, including a simple reversible reaction A+B■C, and single species reactions of the form A+A→products. For low densities, the present theory is equivalent to the usual Smoluchowski method. In this regime, an intermediate version of the theory, including only nonlocal effects due to reaction, is compared to the full nonlocal theory and to the earlier hydrodynamic level theory and is shown to produce results for the time dependence of the radial distribution function that are similar at low density to the full theory. The application of the new approach at higher densities and to lower dimensions is described briefly.

Using a recent generalization of fluctuation theory that includes spatially nonlocal effects, we examine the influence of nonlocal chemical reactivities on rapid chemical reactions in two dimensions. We focus attention of the irreversible reaction A + A --> products at low density, exploring the time dependence of the radial distribution of pairs of reactants, its asymptotic time dependence, and steady states in the presence of inputs. As in the case of three dimensions, we find that the nonlocal theory removes inherent problems with the local theory that occur at short times, gives the expected logarithmic divergence in the absence of inputs, but has well-behaved steady state properties in the presence of inputs. Thus, the nonlocal theory appears to provide a systematic method for treating rapid reactions in membranes and on surfaces at higher densities.