Computer simulation of a cytosolic calcium oscillator

Effect of delay in transportation of extracellular glucose into cardiomyocytes under diabetic condition: a study through mathematical model

A four-dimensional model was built to mimic the cross-talk among plasma glucose, plasma insulin, intracellular glucose and cytoplasmic calcium of a cardiomyocyte. A time delay was considered to represent the time required for performing various cellular mechanisms between activation of insulin receptor and subsequent glucose entry from extracellular region into intracellular region of a cardiac cell. We analysed the delay-induced model and deciphered conditions for stability and bifurcation. Extensive numerical computations were performed to validate the analytical results and give further insights. Sensitivity study of the system parameters using LHS-PRCC method reveals that some rate parameters, which represent the input of plasma glucose, absorption of glucose by noncardiac cells and insulin production, are sensitive and may cause significant change in the system dynamics. It was observed that the time taken for transportation of extracellular glucose into the cell through GLUT4 plays an important role in maintaining physiological oscillations of the state variables. Parameter recalibration exercise showed that reduced input rate of glucose in the blood plasma or an alteration in transportation delay may be used for therapeutic targets in diabetic-like condition for maintaining normal cardiac function.

Ca2+ signaling in T lymphocytes: the interplay of the endoplasmic reticulum, mitochondria, membrane potential, and CRAC channels on transcription factor activation

T cell receptor stimulation initiates a cascade of reactions that cause an increase in intracellular calcium (Ca²⁺) concentration mediated through inositol 1,4,5-trisphosphate (IP₃). To understand the basic mechanisms by which the immune response in T cells is activated, it is useful to understand the signaling pathways that contain important targets for drugs in a quantitative fashion. A computational model helps us to understand how the selected elements in the pathways interact with each other, and which component plays the crucial role in systems. We have developed a mathematical model to explore the mechanism for controlling transcription factor activity, which regulates gene expression, by the modulation of calcium signaling triggered during T cell activation. The model simulates the activation and modulation of Ca²⁺ release-activated Ca²⁺ (CRAC) channels by mitochondrial dynamics and depletion of endoplasmic reticulum (ER) store, and also includes membrane potential in T-cells. The model simulates the experimental finding that increases in Ca²⁺ current enhances the activation of transcription factors and the Ca²⁺ influx through CRAC is also essential for the NFAT and NFκB activation. The model also suggests that plasma membrane Ca²⁺-ATPase (PMCA) controls a majority of the extrusion of Ca²⁺ and modulates the activation of CRAC channels. Furthermore, the model simulations explain how the complex interaction of the endoplasmic reticulum, membrane potential, mitochondria, and ion channels such as CRAC channels control T cell activation.

Restoring calcium homeostasis in diabetic cardiomyocytes: an investigation through mathematical modelling

Regulated calcium flux from sarcoplasmic reticulum could be a possible therapeutic strategy in diabetic cardiomyocyte problem.

The Phase Lag between Agonist-Induced Oscillatory Ca2+ and IP3 Signals Does Not Imply Causality (December 2015)

Activated phospholipase C (PLC*) generates 1,4,5-triphosphate (IP₃) and diacylglycerol (DAG) from phosphatidyl inositol (PIP₂). The DAG remains in the plasma membrane and co-activates conventional protein kinase C (PKC) with Ca²⁺. We have developed a mathematical model for the activation of the Ca²⁺-dependent PKC and its negative feedback on phospholipase C (PLC) and coupled it to the De Young-Keizer model for IP₃ mediated Ca²⁺ oscillations. The model describes the cascade of reactions for the translocation of PKC to plasma membrane, and simulates activation of Ca²⁺ and diacylglycerol (DAG) oscillations. The model demonstrates that oscillations in Ca²⁺ and DAG are possible with or without a positive Ca²⁺ feedback on phospholipase C consistent with experiment. In many experimental studies, the timing of the peaks of the Ca²⁺ and IP₃ oscillations have been used to suggest causality, i.e. that the IP₃ oscillations cause the Ca²⁺ oscillations. The model is used to explore this question. To this end, the positive and negative feedback between Ca²⁺ and IP₃ production are modulated, resulting in changes to the phase lag between the peaks in [Ca²⁺]_cyt and [IP]_cyt. The model simulates a possible experimental protocol that can be used to differentiate whether or not the positive feedback of Ca²⁺ on PLC is needed for the oscillations.

Model analysis of nonlinear modification of neutrophil calcium homeostasis under the influence of modulated electromagnetic radiation of extremely high frequencies

The problem of resonance effects of electromagnetic radiation (EMR) on biological objects remained unsolved till now. Previously we demonstrated that low-intensity amplitude-modulated EMR of extremely high frequencies (EHF) modified the activity of mouse neutrophils in the synergistic reaction of calcium ionophore A23187 and phorbol ester PMA. The EHF EMR influence on the neutrophils was significant at the carrier frequencies of radiation within a narrow range of 41.8-42.05 GHz and at the modulation frequency of 1 Hz. The purpose of the work was the analysis of frequency-dependent modification of intracellular free calcium concentration ([Ca(2+)](i)) by modulated EHF EMR on the basis of a special model for [Ca(2+)](i) oscillations in the neutrophils. The calcium channels of plasma membrane were chosen as the action target of external modulation in the model. The computer simulation demonstrated the rise in [Ca(2+)](i) at the influence of the external field with a threshold dependence on the modulation amplitude. The effect depended heavily on a sequence of delivery of the chemical and electromagnetic stimuli. The narrow-band rise in [Ca(2+)](i) had a phase-frequency dependence. With the modulation amplitudes exceeding the threshold value, the rise in [Ca(2+)](i) of more than 50% of the initial level was observed at the frequency of about 1 Hz and in the phase range of 0.3-2.5 radians. The results of the model analysis are in good correspondence with the experimental data obtained before, namely, with the resonance modification of the neutrophil activity at the modulation frequency of 1 Hz and with the presence of the effect only at high concentrations of calcium ionophore.

Phase-locked signals elucidate circuit architecture of an oscillatory pathway

This paper introduces the concept of phase-locking analysis of oscillatory cellular signaling systems to elucidate biochemical circuit architecture. Phase-locking is a physical phenomenon that refers to a response mode in which system output is synchronized to a periodic stimulus; in some instances, the number of responses can be fewer than the number of inputs, indicative of skipped beats. While the observation of phase-locking alone is largely independent of detailed mechanism, we find that the properties of phase-locking are useful for discriminating circuit architectures because they reflect not only the activation but also the recovery characteristics of biochemical circuits. Here, this principle is demonstrated for analysis of a G-protein coupled receptor system, the M3 muscarinic receptor-calcium signaling pathway, using microfluidic-mediated periodic chemical stimulation of the M3 receptor with carbachol and real-time imaging of resulting calcium transients. Using this approach we uncovered the potential importance of basal IP3 production, a finding that has important implications on calcium response fidelity to periodic stimulation. Based upon our analysis, we also negated the notion that the Gq-PLC interaction is switch-like, which has a strong influence upon how extracellular signals are filtered and interpreted downstream. Phase-locking analysis is a new and useful tool for model revision and mechanism elucidation; the method complements conventional genetic and chemical tools for analysis of cellular signaling circuitry and should be broadly applicable to other oscillatory pathways.

Key to robust discernment of cell circuit architecture is to have as many distinct response features as possible for comparison and evaluation. One under-appreciated characteristic of oscillatory circuits is that under periodic stimulation, these systems will exhibit responses synchronized to this stimulatory input, a phenomenon termed phase-locking. We demonstrate that phase-locked response characteristics vary noticeably depending on circuit activation and recovery properties; these response characteristics thereby provide a unique set of criteria for oscillatory circuit architecture analysis. The concept is validated through experiments on an oscillatory calcium pathway in mammalian cells; the experimental setup allowed us to explore, for the first time, the properties of chemically induced phase-locking of intracellular signals. Observations of this phenomenon were then used to test the predictions of several existing mathematical models of calcium signaling. Most of the models we evaluated were unable to match all our experimental observations, suggesting that current models are missing mechanistic elements in the context of calcium signaling for the cell type and receptor/stimulant tested. The observations of phase-locking further led us to identify one simple mechanistic modification that would account for all the experimental observations. The techniques and methodology presented should be broadly applicable to a variety of biological oscillators.

Complex intracellular calcium oscillations. A theoretical exploration of possible mechanisms

Intracellular Ca(2+) oscillations are commonly observed in a large number of cell types in response to stimulation by an extracellular agonist. In most cell types the mechanism of regular spiking is well understood and models based on Ca(2+)-induced Ca(2+) release (CICR) can account for many experimental observations. However, cells do not always exhibit simple Ca(2+) oscillations. In response to given agonists, some cells show more complex behaviour in the form of bursting, i.e. trains of Ca(2+) spikes separated by silent phases. Here we develop several theoretical models, based on physiologically plausible assumptions, that could account for complex intracellular Ca(2+) oscillations. The models are all based on one- or two-pool models based on CICR. We extend these models by (i) considering the inhibition of the Ca(2+)-release channel on a unique intracellular store at high cytosolic Ca(2+) concentrations, (ii) taking into account the Ca(2+)-activated degradation of inositol 1,4,5-trisphosphate (IP(3)), or (iii) considering explicity the evolution of the Ca(2+) concentration in two different pools, one sensitive and the other one insensitive to IP(3). Besides simple periodic oscillations, these three models can all account for more complex oscillatory behaviour in the form of bursting. Moreover, the model that takes the kinetics of IP(3) into account shows chaotic behaviour.

A method for determining the dependence of calcium oscillations on inositol trisphosphate oscillations

In some cell types, oscillations in the concentration of free intracellular calcium ([Ca2+]) are accompanied by oscillations in the concentration of inositol 1,4,5-trisphosphate ([IP3]). However, in most cell types it is still an open question as to whether oscillations in [IP3] are necessary for Ca2+ oscillations in vivo, or whether they merely follow passively. Using a wide range of models, we show that the response to an artificially applied pulse of IP3 can be used to distinguish between these two cases. Hence, we show that muscarinic receptor-mediated, long-period Ca2+ oscillations in pancreatic acinar cells depend on [IP3] oscillations, whereas short-period Ca2+ oscillations in airway smooth muscle do not.

Modelling of simple and complex calcium oscillations. From single-cell responses to intercellular signalling

This review provides a comparative overview of recent developments in the modelling of cellular calcium oscillations. A large variety of mathematical models have been developed for this wide-spread phenomenon in intra- and intercellular signalling. From these, a general model is extracted that involves six types of concentration variables: inositol 1,4,5-trisphosphate (IP3), cytoplasmic, endoplasmic reticulum and mitochondrial calcium, the occupied binding sites of calcium buffers, and the fraction of active IP3 receptor calcium release channels. Using this framework, the models of calcium oscillations can be classified into 'minimal' models containing two variables and 'extended' models of three and more variables. Three types of minimal models are identified that are all based on calcium-induced calcium release (CICR), but differ with respect to the mechanisms limiting CICR. Extended models include IP3--calcium cross-coupling, calcium sequestration by mitochondria, the detailed gating kinetics of the IP3 receptor, and the dynamics of G-protein activation. In addition to generating regular oscillations, such models can describe bursting and chaotic calcium dynamics. The earlier hypothesis that information in calcium oscillations is encoded mainly by their frequency is nowadays modified in that some effect is attributed to amplitude encoding or temporal encoding. This point is discussed with reference to the analysis of the local and global bifurcations by which calcium oscillations can arise. Moreover, the question of how calcium binding proteins can sense and transform oscillatory signals is addressed. Recently, potential mechanisms leading to the coordination of oscillations in coupled cells have been investigated by mathematical modelling. For this, the general modelling framework is extended to include cytoplasmic and gap-junctional diffusion of IP3 and calcium, and specific models are compared. Various suggestions concerning the physiological significance of oscillatory behaviour in intra- and intercellular signalling are discussed. The article is concluded with a discussion of obstacles and prospects.

A cellular automaton model of cellular signal transduction

On the basis of cellular automata models, a software specifically tailored to model biochemical reactions involved in cellular signal transduction was implemented on a personal computer. Recent data regarding desensitization processes in mouse Leydig cells are used to simulate the underlying reactions of signal transduction. Pretreatment of real Leydig cells with different molecules results in a modification of the signal transduction cascade leading to a diminished response of the cells during subsequent stimulations. The model is capable of simulating the complex behavior of this intracellular second messenger production in a qualitative and semi-quantitative way. The results indicate that quantitative simulations on a molecular level will be possible once appropriate computer hardware is available. The simulations and results of the cellular automaton presented are easily described and comprehended, which make it a useful tool that will facilitate research in cellular signal transduction and other fields covering complex reaction networks.

Simulations of the effects of inositol 1,4,5-trisphosphate 3-kinase and 5-phosphatase activities on Ca2+ oscillations

Inositol 1,4,5-trisphosphate (Ins-1,4,5-P3) is responsible for Ca2+ mobilization in response to external stimulation in many cell types. The latter phenomenon often occurs as repetitive Ca2+ spikes. In this study, the effect of the two Ins-1,4,5-P3 metabolizing enzymes (Ins-1,4,5-P3 3-kinase and 5-phosphatase) on the temporal pattern of Ca2+ oscillations has been investigated. On the basis of the well-documented Ins-1,4,5-P3 3-kinase stimulation by the Ca2+/calmodulin complex and of the experimentally-determined kinetic characteristics of these enzymes, we predict that 5-phosphatase primarily controls the levels of Ins-1,4,5-P3 and, thereby, the occurrence and frequency of Ca2+ oscillations. Consequently, the model reproduces the experimental observation performed in Chinese hamster ovary cells that 5-phosphatase overexpression has a much more pronounced effect on the pattern of Ca2+ oscillations than 3-kinase overexpression. We also investigated, in more detail, under which conditions a similar effect could be observed in other cell types expressing various Ins-1,4,5-P3 3-kinase activities.

Isoprenylated human brain type I inositol 1,4,5-trisphosphate 5-phosphatase controls Ca2+ oscillations induced by ATP in Chinese hamster ovary cells

D-myo-Inositol 1,4,5-trisphosphate (InsP3) 5-phosphatase and 3-kinase are thought to be critical regulatory enzymes in the control of InsP3 and Ca2+ signaling. In brain and many other cells, type I InsP3 5-phosphatase is the major phosphatase that dephosphorylates InsP3 and D-myo-inositol 1,3,4,5-tetrakisphosphate. The type I 5-phosphatase appears to be associated with the particulate fraction of cell homogenates. Molecular cloning of the human brain enzyme identifies a C-terminal farnesylation site CVVQ. Post-translational modification of this enzyme promotes membrane interactions and changes in specific activity. We have now compared the cytosolic Ca2+ ([Ca2+]i) responses induced by ATP, thapsigargin, and ionomycin in Chinese hamster ovary (CHO-K1) cells transfected with the intact InsP3 5-phosphatase and with a mutant in which the C-terminal cysteine cannot be farnesylated. [Ca2+]i was also measured in cells transfected with an InsP3 3-kinase construct encoding the A isoform. The Ca2+ oscillations detected in the presence of 1 microM ATP in control cells were totally lost in 87.5% of intact (farnesylated) InsP3 5-phosphatase-transfected cells, while such a loss occurred in only 1.1% of the mutant InsP3 5-phosphatase-transfected cells. All cells overexpressing the InsP3 3-kinase also responded with an oscillatory pattern. However, in contrast to control cells, the [Ca2+]i returned to base-line levels in between a couple of oscillations. The [Ca2+]i responses to thapsigargin and ionomycin were identical for all cells. The four cell clones compared in this study also behaved similarly with respect to capacitative Ca2+ entry. In permeabilized cells, no differences in extent of InsP3-induced Ca2+ release nor in the threshold for InsP3 action were observed among the four clones and no differences in the expression levels of the various InsP3 receptor isoforms could be shown between the clones. Our data support the contention that the ATP-induced increase in InsP3 concentration in transfected CHO-K1 cells is essentially restricted to the site of its production near the plasma membrane, where it can be metabolized by the type I InsP3 5-phosphatase. This enzyme directly controls the [Ca2+]i response and the Ca2+ oscillations in intact cells.

Quantal release, incremental detection, and long-period Ca2+ oscillations in a model based on regulatory Ca2+-binding sites along the permeation pathway

Quantal release, incremental detection, and oscillations are three types of Ca2+ responses that can be obtained in different conditions, after stimulation of the intracellular Ca2+ stores by submaximum concentrations of inositol 1,4,5-triphosphate (InsP3). All three phenomena are thought to occur through the regulatory properties of the InsP3 receptor/Ca2+ channel. In the present study, we perform further analysis of the model (Swillens et al., 1994, Proc. Natl. Acad. Sci. USA. 91:10074-10078) previously proposed for transient InsP3-induced Ca2+ release, based on the bell-shaped dependence of the InsP3 receptor activity on the Ca2+ level and on the existence of an intermediate Ca2+ domain located around the mouth of the channel. We show that Ca2+ oscillations also arise in the latter model. Conditions for the occurrence of the various behaviors are investigated. Numerical simulations also show that the existence of an intermediate Ca2+ domain can markedly increase the period of oscillations. Periods on the order of 1 min can indeed be accounted for by the model when one assigns realistic values to the kinetic constants of the InsP3 receptor, which, in the absence of a domain, lead to oscillations with periods of a few seconds. Finally, theoretical support in favor of a positive cooperativity in the regulation of the InsP3 receptor by Ca2+ is presented.

Crosstalk between cellular morphology and calcium oscillation patterns. Insights from a stochastic computer model

Agonist-induced oscillations in the concentration of intracellular free calcium ([Ca2+]i) display a wide variety of temporal and spatial patterns. In non-excitable cells, typical oscillatory patterns are somewhat cell-type specific and range from frequency-encoded, repetitive Ca2+ spikes to oscillations that are more sinusoidal in shape. Although the response of a cell population, even to the same stimulus, is often extremely heterogeneous, the response of the same cell to successive exposures can be remarkably similar. We propose that such "Ca2+ fingerprints' can be a consequence of cell-specific morphological properties. The hypothesis is tested by means of a stochastic computer simulation of a two-dimensional model for oscillatory Ca2+ waves which encompasses the basic elements of the two-pool oscillator introduced by Goldbeter et al. (Goldbeter A., Dupont G., Berridge M.J. Minimal model for signal-induced Ca(2+)-oscillations and for their frequency encoding through protein phosphorylation. Proc Natl Acad Sci USA 1990; 87: 1461-1465). In the framework of our extended spatiotemporal model, single cells can display various oscillation patterns which depend on the agonist dose, Ca2+ diffusibility, and several morphological parameters. These are, for example, size and shape of the cell and the cell nucleus, the amount and distribution of Ca2+ stores, and the subcellular location of the inositol(1,4,5)-trisphosphate-generating apparatus.

InsP3-induced Ca2+ excitability of the endoplasmic reticulum

Oscillations in intracellular Ca2+ can be induced by a variety of cellular signalling processes (Woods et al., 1986; Berridge 1988; Jacob et al., 1988) and appear to play a role in secretion (Stojilković et al., 1994), fertilization (Miyazaki et al., 1993), and smooth muscle contraction (Iino and Tsukioka, 1994). Recently, great progress has been made in understanding the mechanisms involved in a particular class of Ca2+ oscillation, associated with the second messenger inositol 1,4,5-trisphosphate (InsP3) (Berridge, 1993). Working in concert with intracellular Ca2+, InsP3 controls Ca2+ release via the InsP3 receptor in the endoplasmic reticulum (ER) (Berridge and Irvine, 1989). The IP3 receptor is regulated by its coagonists InsP3 and Ca2+, which both activate and inhibit Ca2+ release (Finch et al., 1991; Bezprozvanny et al., 1991; De Young and Keizer, 1992). These processes, together with the periodic activation of Ca2+ uptake into the ER, have been identified as key features in the mechanism of InsP3-induced Ca2+ oscillations in pituitary gonadotrophs (Li et al., 1994), Xenopus laevis oocytes (Lechleiter and Clapham, 1992; Atri et al., 1993), and other cell types (Keizer and De Young, 1993). Earlier discussions and models of InsP3-induced Ca2+ oscillations focused on the nature and number of internal releasable pools of Ca2+ (Goldbeter et al., 1990; Swillens and Mercan, 1990; Somogyi and Stucki, 1991), the importance of oscillations in InsP3 (Meyer and Stryer, 1988), and other issues not based on detailed experimental findings in specific cells types.

Chaos in intracellular Ca2+ oscillations in a new model for non-excitable cells

A new model for intracellular Ca2+ oscillations is presented. The new model reinterprets two previous models, the ICC and CICR mechanisms, and incorporates the bell-shaped dependence of Ca2+ release on cytosolic [Ca2+]. Complex oscillations and chaos are found with this new model, confirming experimental observations of complex oscillations. A rich bifurcation sequence is found for the model as the stimulation due to agonist (R) is varied, including a period doubling route to chaos and a period-adding sequence of mixed-mode states.

Properties of intracellular Ca2+ waves generated by a model based on Ca(2+)-induced Ca2+ release

Cytosolic Ca2+ waves occur in a number of cell types either spontaneously or after stimulation by hormones, neurotransmitters, or treatments promoting Ca2+ influx into the cells. These waves can be broadly classified into two types. Waves of type 1, observed in cardiac myocytes or Xenopus oocytes, correspond to the propagation of sharp bands of Ca2+ throughout the cell at a rate that is high enough to permit the simultaneous propagation of several fronts in a given cells. Waves of type 2, observed in hepatocytes, endothelial cells, or various kinds of eggs, correspond to the progressive elevation of cytosolic Ca2+ throughout the cell, followed by its quasi-homogeneous return down to basal levels. Here we analyze the propagation of these different types of intracellular Ca2+ waves in a model based on Ca(2+)-induced Ca2+ release (CICR). The model accounts for transient or sustained waves of type 1 or 2, depending on the size of the cell and on the values of the kinetic parameters that measure Ca2+ exchange between the cytosol, the extracellular medium, and intracellular stores. Two versions of the model based on CICR are considered. The first version involves two distinct Ca2+ pools sensitive to inositol 1,4,5-trisphosphate (IP3) and Ca2+, respectively, whereas the second version involves a single pool sensitive both to Ca2+ and IP3 behaving as co-agonists for Ca2+ release. Intracellular Ca2+ waves occur in the two versions of the model based on CICR, but fail to propagate in the one-pool model at subthreshold levels of IP3. For waves of type 1, we investigate the effect of the spatial distribution of Ca(2+)-sensitive Ca2+ stores within the cytosol, and show that the wave fails to propagate when the distance between the stores exceeds a critical value on the order of a few microns. We also determine how the period and velocity of the waves are affected by changes in parameters measuring stimulation, Ca2+ influx into the cell, or Ca2+ pumping into the stores. For waves of type 2, the numerical analysis indicates that the best qualitative agreement with experimental observations is obtained for phase waves. Finally, conditions are obtained for the occurrence of "echo" waves that are sometimes observed in the experiments.

A membrane potential model with counterions for cytosolic calcium oscillations

An initial model has been proposed to describe a mechanism for cytosolic calcium oscillations [Jafri MS. Vajda S. Pasik P. Gillo B. (1992) A membrane model for cytosolic calcium oscillations: a study using Xenopus oocytes. Biophys. J., 63, 235-246]. In this paper we extend our original model to include the effects of counterion movement into the ER in response to calcium release. This produces smoother oscillations over a wider parameter range. We have lowered the endoplasmic reticulum (ER) intraluminal free calcium concentration and shown that the oscillations can occur at lower ER membrane potentials, consistent with physiological values. The improved model is then tested with two representative paradigms that are currently under investigation by many researchers. The model predicts that the reduction of the ER calcium pump (Ca-ATPase) rate can cause the termination of cytosolic calcium oscillations in an active cell, and induce oscillations in a resting cell. This result is consistent with experiments with thapsigargin, a Ca-ATPase activity inhibitor. In addition, we simulate the latency period for the response to the application of agonist and offer a plausible explanation for it. Our mathematical model is currently the only model that formulates the contributions of calcium binding proteins, ER membrane potential, ER counterion movements, and distinct calcium pump populations, and describes their effects on cytosolic calcium oscillations.

Quantitative analysis of cytosolic free calcium oscillations in neutrophils by mathematical modelling

Mathematical models are often used to elucidate mechanisms behind cytosolic Ca2+ oscillations. We have evaluated the use of mathematical modelling to analyse and quantify Ca2+ signal patterns, in single, adherent human neutrophils (PMN) after stimulation by the bacterial peptide N-formyl-methionyl-leucyl-phenylalanine (fMLP). The cells were loaded with Fura-2 and fluctuations in cytosolic Ca2+ recorded with a video based digital imaging system. A new indirect intracellular calibration method was introduced to avoid the uncertainty in obtaining an equilibrium between the extracellular and intracellular calcium concentrations. Two different approaches to mathematical modelling were used. First, we applied a sensitivity analysis with a two-pool model by assuming an optimal situation using reliable a priori estimates of all structural parameters (e.g. Hill coefficients and dissociation constants). We found that the a priori estimates of the other 5 more variable parameters must lie within the range of 25-400% of the postulated true parameter values to be reliable in a parameter estimation method. Small changes (less than 5%) in those variable parameter values induced very different types of signal patterns which may have some relevance in evaluating a possible functional significance to the oscillatory signals. Second, we employed a one-pool, non oscillatory model integrated with a power spectrum method as a tool to quantify the dose dependency between fMLP (1-1000 nM) and parameters describing the biphasic process of calcium signalling and parameters describing only the oscillatory components. We conclude that the frequency of the observed oscillations assembled around one characteristic frequency independent of fMLP concentration, and sinusoidal oscillations were observed most frequently in PMN stimulated to a moderate peak [Ca2+]i level.

One-pool model for Ca2+ oscillations involving Ca2+ and inositol 1,4,5-trisphosphate as co-agonists for Ca2+ release

Experimental observations indicate that Ca(2+)-induced Ca2+ release (CICR) may underlie Ca2+ oscillations in a variety of cells. In its original version, a theoretical model for signal-induced Ca2+ oscillations based on CICR assumed the existence of two types of pools, one sensitive to inositol 1,4,5-trisphosphate (IP3) and the other one sensitive to Ca2+. Recent experiments indicate that Ca2+ channels may sometimes be sensitive to both IP3 and Ca2+. Such a regulation may be viewed as Ca(2+)-sensitized IP3-induced Ca2+ release or, alternatively, as a form of IP3-sensitized CICR. We show that sustained oscillations can still occur in a one-pool model, provided that the same Ca2+ channels are sensitive to both Ca2+ and IP3 behaving as co-agonists. This model and the two-pool model based on CICR both account for a number of experimental observations but differ in some respects. Thus, while in the two-pool model the latency and period of Ca2+ oscillations are of the same order of magnitude and correlate in a roughly linear manner, latency in the one-pool model is always brief and remains much shorter than the period of oscillations. Moreover, the first Ca2+ spike is much larger than the following ones in the one-pool model. These distinctive properties might provide an explanation for the differences in Ca2+ oscillations observed in various cell types.

Calcium wave propagation by calcium-induced calcium release: an unusual excitable system

We discuss in detail the behaviour of a model, proposed by Goldbeter et al. (1990. Proc. natn. Acad. Sci. 87, 1461-1465), for intracellular calcium wave propagation by calcium-induced calcium release, focusing our attention on excitability and the propagation of waves in one spatial dimension. The model with no diffusion behaves like a generic excitable system, and threshold behaviour, excitability and oscillations can be understood within this general framework. However, when diffusion is included, the model no longer behaves like a generic excitable system; the fast and slow variables are not distinct and previous results on excitable systems do not necessarily apply. We consider a piecewise linear simplification of the model, and construct travelling pulse and periodic plane wave solutions to the simplified model. The analogous behaviour in the full model is studied numerically. Goldbeter's model for calcium-induced calcium release is an excitable system of a type not previously studied in detail.

Hepatocyte growth factor induces calcium mobilization and inositol phosphate production in rat hepatocytes

The effects of hepatocyte growth factor (HGF) on intracellular Ca2+ mobilization were studied using fura-2-loaded single rat hepatocytes. Hepatocytes microperfused with different amounts of HGF responded with a rapid concentration-dependent rise in the cytosolic free Ca2+ concentration with a maximum increase of 142% at 80 ng/ml of HGF. The lag period of the Ca2+ response was decreased with increasing HGF concentrations, being 64 +/- 12 s, 42 +/- 6 s, and 14 +/- 2 s, respectively, with 8, 20, and 80 ng/ml of HGF. The detailed pattern of Ca2+ transients, however, was variable. Out of 16 cells tested using 20 ng/ml of HGF, 68% showed sustained oscillatory responses, whereas other cells showed a sustained increase in the cytosolic-free Ca2+ upon exposure to HGF, which was dependent on the presence of extracellular Ca2+. HGF also induced Ca2+ entry across the plasma membrane. Mobilization of Ca2+ by HGF was accompanied by a rapid accumulation of inositol 1,4,5-trisphosphate (Ins 1,4,5-P3). The effects of HGF and epidermal growth factor (EGF) were comparable and partly additive for Ins 1,4,5-P3 production and for the sustained phase of Ca2+ mobilization. Preincubation of cells with 10 microM of genistein to inhibit protein tyrosine kinases abolished the HGF-induced Ca2+ response and also inhibited HGF-induced Ins 1,4,5-P3 production in rat liver cells. These data indicate that early events in the signal transduction pathways mediated by HGF and EGF have in common the requirements for tyrosine kinase activity, Ins 1,4,5-P3 production, and Ca2+ mobilization.

A membrane model for cytosolic calcium oscillations. A study using Xenopus oocytes

Cytosolic calcium oscillations occur in a wide variety of cells and are involved in different cellular functions. We describe these calcium oscillations by a mathematical model based on the putative electrophysiological properties of the endoplasmic reticulum (ER) membrane. The salient features of our membrane model are calcium-dependent calcium channels and calcium pumps in the ER membrane, constant entry of calcium into the cytosol, calcium dependent removal from the cytosol, and buffering by cytoplasmic calcium binding proteins. Numerical integration of the model allows us to study the fluctuations in the cytosolic calcium concentration, the ER membrane potential, and the concentration of free calcium binding sites on a calcium binding protein. The model demonstrates the physiological features necessary for calcium oscillations and suggests that the level of calcium flux into the cytosol controls the frequency and amplitude of oscillations. The model also suggests that the level of buffering affects the frequency and amplitude of the oscillations. The model is supported by experiments indirectly measuring cytosolic calcium by calcium-induced chloride currents in Xenopus oocytes as well as cytosolic calcium oscillations observed in other preparations.

Oscillations and waves of cytosolic calcium: insights from theoretical models

Oscillations in cytosolic Ca2+ occur in a wide variety of cells, either spontaneously or as a result of external stimulation. This process is often accompanied by intracellular Ca2+ waves. A number of theoretical models have been proposed to account for the periodic generation and spatial propagation of Ca2+ signals. These models are reviewed and their predictions compared with experimental observations. Models for Ca2+ oscillations can be distinguished according to whether or not they rely on the concomitant, periodic variation in inositol 1,4,5-trisphosphate. Such a variation, however, is not required in models based on Ca(2+)-induced Ca2+ release. When Ca2+ diffusion is incorporated into these models, propagating waves of cytosolic Ca2+ arise, with profiles and rates comparable to those seen in the experiments.

[Modeling in biology. Structured analysis of intracellular calcium oscillations in electrically non-excitable cells]

In this paper a systematic approach to the mathematical modeling of intracellular Ca2+ oscillations is introduced. After a structured analysis a stochastic model of the system is derived which is numerically tractable by means of a stochastic simulation. A critical discussion of theoretical models for Ca2+ oscillations reveals that not all of the proposed mechanisms are consistent with experimental data. In addition, a model for oscillatory calcium waves is presented. Uncovering these mechanisms facilitates the design of anti-mitotic drugs interfering with Ca2+ metabolism.

Two roles of Ca2+ in agonist stimulated Ca2+ oscillations

We propose a mechanism for agonist-stimulated Ca2+ oscillations that involves two roles for cytosolic Ca2+: (a) inhibition of inositol-1,4,5-trisphosphate (IP3) stimulated Ca2+ release from the endoplasmic reticulum (ER) and (b) stimulation of the production of IP3 through its action on phospholipase C (PLC), via a Gq protein related mechanism. Relying on quantitative experiments by Parker, I., and I. Ivorra (1990. Proc. Natl. Acad. Sci. USA. 87:260-264) on the inhibition of Ca2+ release from the ER using caged-IP3, we develop a kinetic model of inhibition that allows us to simulate closely their experiments. The model assumes that the ER IP3 receptor is a tetramer of independent subunits that can bind both Ca2+ and IP3. Upon incorporation of the action of Ca2+ on PLC that leads to production of IP3, we observe in-phase-oscillations of Ca2+ and IP3 at intermediate values of agonist stimulation. The oscillations occur on a time scale of 10-20 s, which is comparable to the time scale for inhibition in Xenopus oocytes. Analysis of the mechanism shows that Ca(2+)-inhibition of IP3-stimulated Ca2+ release from the ER is an essential step in the mechanism. We also find that the effect of Ca2+ on PLC can lead to an indirect increase of cytosolic Ca2+, superficially resembling "Ca(2+)-induced Ca(2+)-release." The mechanism that we propose appears to be consistent with recent experiments on REF52 cells by Harootunian, A. T., J. P. Y. Kao, S. Paranjape, and R. Y. Tsien. (1991. Science [Wash. DC]. 251:75-78.) and we propose additional experiments to help test its underlying assumptions.

Acetylcholine activates non-selective cation and chloride conductances in canine and guinea-pig tracheal myocytes

1. Membrane currents activated by acetylcholine (ACh) were investigated in isolated canine and guinea-pig tracheal myocytes using the nystatin perforated patch configuration of whole-cell recording. ACh caused depolarization accompanied by a membrane conductance increase. 2. When cells were held under voltage clamp (holding potential, Vh = -60 mV), ACh elicited inward current (IACh) of up to 3900 pA, with a reversal potential (Erev) of approximately -20 mV. 3. Removal of extracellular Na+ (Na+o) reduced but did not eliminate IACh. IACh remaining in the absence of Na+ reversed direction close to the predicted equilibrium potential for Cl-. Erev shifted 32 +/- 4 mV per 10-fold change of [Cl-]i. Increasing external [K+] caused Erev to shift in the positive direction. These results suggest that ACh activated chloride and non-selective cation conductances. 4. In the absence of Na+o, the Cl- channel blockers SITS or niflumic acid reversibly antagonized IACh. 5. Caffeine and ryanodine elicited currents both in the presence and absence of Na+o; these currents had a reversal potential similar to that of IACh. Caffeine applied before ACh occluded the response to ACh. 6. We also observed two types of spontaneous membrane currents. Spontaneous transient outward currents (STOCs) may represent Ca(2+)-activated K+ currents. Spontaneous inward currents were also observed which were reduced in magnitude (but not eliminated) by removal of Na+o and reversed direction at approximately the Cl- equilibrium potential. The spontaneous inward currents and STOCs were coincident and were reversibly suppressed by ACh. 7. ACh elicited contractions of cells under voltage clamp at -60 mV, an effect also observed in the absence of extracellular Ca2+ or when IACh was reduced by omission of Na+o and exposure to Cl- channel blockers. The number of cells which did contract in response to ACh decreased, however, when the concentration of internal Cl- decreased. 8. All effects of ACh on contraction and membrane currents were antagonized by atropine. 9. We conclude that activation of muscarinic receptors in mammalian tracheal myocytes causes release of Ca2+ from intracellular stores and subsequent activation of Cl- and non-selective cation conductances. This is the first direct demonstration of these conductances in tracheal smooth muscle cells. Activation of these conductances does not appear to be required for contraction. However, regulation of cytosolic Cl- levels may be important for release and uptake of Ca2+ from internal stores.

Characterization of Ca2+ transients induced by intracellular photorelease of InsP3 in mouse ovarian oocytes

Ca2+ transients (measured with Fluo-3) were induced in single mouse ovarian oocytes by photolytic liberation of InsP3. The time course of cytosolic Ca2+ changes induced in this way is composed of distinct phases: upstroke, fast decline, slow declining plateau and fast decline to rest level. All the phases reflect mainly intracellular redistributions of the ion and not influx, since they are not strongly dependent on external Ca2+ or on changes in transmembrane potential. Often sustained Ca2+ oscillations followed the first InsP3-induced Ca2+ transient. These persisted for several minutes in the absence of external Ca2+. The initial rate of Ca2+ rise and the delay between the InsP3 stimulus and Ca2+ upstroke are correlated with the amount of liberated InsP3. A second InsP3 stimulation, applied during the plateau, causes only small Ca2+ elevations, lacking the upstroke phase. A second, full sized, transient could be elicited only after a complete return to the basal level. Vanadate, applied intracellularly, appeared to inhibit the re-uptake phase into the stores, stabilizing the plateau level. The present observations suggest that in mouse oocytes the InsP3-sensitive stores provide only a small and graded Ca2+ release which may then act as a trigger for a more substantial Ca(2+)-induced Ca2+ release (CICR) process.

Signal-induced Ca2+ oscillations: properties of a model based on Ca(2+)-induced Ca2+ release

We consider a simple, minimal model for signal-induced Ca2+ oscillations based on Ca(2+)-induced Ca2+ release. The model takes into account the existence of two pools of intracellular Ca2+, namely, one sensitive to inositol 1,4,5 trisphosphate (InsP3) whose synthesis is elicited by the stimulus, and one insensitive to InsP3. The discharge of the latter pool into the cytosol is activated by cytosolic Ca2+. Oscillations in cytosolic Ca2+ arise in this model either spontaneously or in an appropriate range of external stimulation; these oscillations do not require the concomitant, periodic variation of InsP3. The following properties of the model are reviewed and compared with experimental observations: (a) Control of the frequency of Ca2+ oscillations by the external stimulus or extracellular Ca2+; (b) correlation of latency with period of Ca2+ oscillations obtained at different levels of stimulation; (c) effect of a transient increase in InsP3; (d) phase shift and transient suppression of Ca2+ oscillations by Ca2+ pulses, and (e) propagation of Ca2+ waves. It is shown that on all these counts the model provides a simple, unified explanation for a number of experimental observations in a variety of cell types. The model based on Ca(2+)-induced Ca2+ release can be extended to incorporate variations in the level of InsP3 as well as desensitization of the InsP3 receptor; besides accounting for the phenomena described by the minimal model, the extended model might also account for the occurrence of complex Ca2+ oscillations.

Modelling receptor-controlled intracellular calcium oscillators

This paper presents mathematical models for the hepatocyte calcium oscillator which follow the concepts in a class of informal models developed to account for the striking dependence on the receptor type of several features of the calcium oscillations, in particular the shape and duration of the free calcium transients. The essence of these models is that the transients should be timed by a build-up of activated GTP-binding proteins, which, combined with positive feedback processes and perhaps with cooperative effects, leads to a sudden activation of phospholipase C (PLC), followed by negative feedback processes which switch off the calcium rise and lead to a fall in free calcium back to resting levels. These models predict pulsatile oscillations in inositol (1,4,5)P3 as well as in free calcium. We show that receptor-controlled intracellular calcium oscillators involving an unknown positive feedback pathway onto PLC and negative feedback from protein kinase C (PKC) onto G-proteins and receptors, or negative feedback by stimulation of GTPase activity can simulate many of the features of observed intracellular calcium oscillations. These oscillators exhibit a dependence of frequency on agonist concentration and a dependence of transient duration on receptor and G-protein type. We also show that a PLC-dependent GTPase activating factor (GAF) could provide explanations for some otherwise puzzling features of intracellular calcium oscillations.

Ca2+ extrusion across plasma membrane and Ca2+ uptake by intracellular stores

The aim of this review is to summarize the various systems that remove Ca2+ from the cytoplasm. We will initially focus on the Ca2+ pump and the Na(+)-Ca2+ exchanger of the plasma membrane. We will review the functional regulation of these systems and the recent progress obtained with molecular-biology techniques, which pointed to the existence of different isoforms of the Ca2+ pump. The Ca2+ pumps of the sarco(endo)plasmic reticulum will be discussed next, by summarizing the discoveries obtained with molecular-biology techniques, and by reviewing the physiological regulation of these proteins. We will finally briefly review the mitochondrial Ca(2+)-uptake mechanism.