Model Of Calcium Oscillations Research Articles

A comparison of three models of the inositol trisphosphate receptor

The inositol (1,4,5)-trisphosphate receptor (IPR) plays a crucial role in calcium dynamics in a wide range of cell types, and is often a central feature in quantitative models of calcium oscillations and waves. We compare three mathematical models of the IPR, fitting each of them to the same data set to determine ranges for the parameter values. Each of the fits indicates that fast activation of the receptor, followed by slow inactivation, is an important feature of the model, and also that the speed of inositol trisphosphate (IP 3) binding cannot necessarily be assumed to be faster than Ca 2+ activation. In addition, the model which assumed saturating binding rates of Ca 2+ to the IPR demonstrated the best fit. However, lack of convergence in the fitting procedure indicates that responses to step increases of [Ca 2+] and [IP 3] provide insufficient data to determine the parameters unambiguously in any of the models.

Stochastic kinetics of intracellular calcium oscillations.

A stochastic model of intracellular calcium oscillations is put forward by taking into account the random opening-closing of Ca2+ channels in endoplasmic reticulum (ER) membrane. The numerical results of the stochastic model show simple and complex calcium oscillations, which accord with the experiment results.

On the encoding and decoding of calcium signals in hepatocytes

Many different agonists use calcium as a second messenger. Despite intensive research in intracellular calcium signalling it is an unsolved riddle how the different types of information represented by the different agonists, is encoded using the universal carrier calcium. It is also still not clear how the information encoded is decoded again into the intracellular specific information at the site of enzymes and genes. After the discovery of calcium oscillations, one likely mechanism is that information is encoded in the frequency, amplitude and waveform of the oscillations. This hypothesis has received some experimental support. However, the mechanism of decoding of oscillatory signals is still not known. Here, we study a mechanistic model of calcium oscillations, which is able to reproduce both spiking and bursting calcium oscillations. We use the model to study the decoding of calcium signals on the basis of co-operativity of calcium binding to various proteins. We show that this co-operativity offers a simple way to decode different calcium dynamics into different enzyme activities.

Under what conditions signal transduction pathways are highly flexible in response to external forcing? A case study on calcium oscillations

Sensitivity and flexibility are typical properties of biological systems. These properties are here investigated in a model for simple and complex intracellular calcium oscillations. In particular, the influence of external periodic forcing is studied. The main point of the study is to compare responses of the system in a chaotic regime with those obtained in a regular periodic regime. We show that the response to external signals in terms of the range of synchronization is not significantly different in regular and chaotic Ca 2+ oscillations. However, both types of oscillation are highly flexible in regimes with weak dissipation. Therefore, we conclude that dissipation of free energy is a suitable index characterizing flexibility. For biological systems this appears to be of special importance since for thermodynamic reasons, notably in view of low free energy consumption, dissipation should be minimized.

An Intracellular Calcium Oscillation Model and its Property

Intracellular Ca2+ concentration oscillation is one of thetypical nonlinear dynamical phenomena in biophysics. According to theexperimental facts that when stimulated by different agonists, thecytosolic Ca2+ oscillations of hepatocytes show nearly constantamplitude for two different oscillation-form, simple spikes and complexbursts, we build a new 4-variables model based on the two modelsproposed by Hoefer [Biophys J. 77 (1999) 1244] and Kummeret al. [Biophys. J. 79 (2000) 1188] Our model cansimulate both simple-spike and multi-spike-burst oscillations withamplitude of 0.5-0.7 µmol/L in a rather wide range of parameters.

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.

Mitochondria regulate the amplitude of simple and complex calcium oscillations

In a mathematical model for simple calcium oscillations [Biophys. Chem. 71 (1998) 125], it has been shown that mitochondria play an important role in the maintenance of constant amplitudes of cytosolic Ca 2+ oscillations. Simple plausible rate laws for Ca 2+ fluxes across the inner mitochondrial membrane have been used in this model. Here we show that it is possible to use the same rate laws as a plug-in element in other existing mathematical models and obtain the same effect on amplitude regulation. This result appears to be universal, independent of the type of model and the type of Ca 2+ oscillations. We demonstrate this on two models for spiking Ca 2+ oscillations [J. Biol. Chem. 266 (1991) 11068; Cell Calcium 14 (1993) 311] and on two recent models for bursting Ca 2+ oscillations; one of them being a receptor-operated model [Biophys. J. 79 (2000) 1188] and the other one being a store-operated model [BioSystems 57 (2000) 75].

A bifurcation analysis of two coupled calcium oscillators.

In many cell types, asynchronous or synchronous oscillations in the concentration of intracellular free calcium occur in adjacent cells that are coupled by gap junctions. Such oscillations are believed to underlie oscillatory intercellular calcium waves in some cell types, and thus it is important to understand how they occur and are modified by intercellular coupling. Using a previous model of intracellular calcium oscillations in pancreatic acinar cells, this article explores the effects of coupling two cells with a simple linear diffusion term. Depending on the concentration of a signal molecule, inositol (1,4,5)-trisphosphate, coupling two identical cells by diffusion can give rise to synchronized in-phase oscillations, as well as different-amplitude in-phase oscillations and same-amplitude antiphase oscillations. Coupling two nonidentical cells leads to more complex behaviors such as cascades of period doubling and multiply periodic solutions. This study is a first step towards understanding the role and significance of the diffusion of calcium through gap junctions in the coordination of oscillatory calcium waves in a variety of cell types. (c) 2001 American Institute of Physics.

Minimal model for intracellular calcium oscillations and electrical bursting in melanotrope cells of Xenopus laevis.

A minimal model is presented to explain changes in frequency, shape, and amplitude of Ca2+ oscillations in the neuroendocrine melanotrope cell of Xenopus Laevis. It describes the cell as a plasma membrane oscillator with influx of extracellular Ca2+ via voltage-gated Ca2+ channels in the plasma membrane. The Ca2+ oscillations in the Xenopus melanotrope show specific features that cannot be explained by previous models for electrically bursting cells using one set of parameters. The model assumes a KCa-channel with slow Ca2+-dependent gating kinetics that initiates and terminates the bursts. The slow kinetics of this channel cause an activation of the Kca-channel with a phase shift relative to the intracellular Ca2+ concentration. The phase shift, together with the presence of a Na+ channel that has a lower threshold than the Ca2+ channel, generate the characteristic features of the Ca2+ oscillations in the Xenopus melanotrope cell.

Switching from Simple to Complex Oscillations in Calcium Signaling

We present a new model for calcium oscillations based on experiments in hepatocytes. The model considers feedback inhibition on the initial agonist receptor complex by calcium and activated phospholipase C, as well as receptor type-dependent self-enhanced behavior of the activated G α subunit. It is able to show simple periodic oscillations and periodic bursting, and it is the first model to display chaotic bursting in response to agonist stimulations. Moreover, our model offers a possible explanation for the differences in dynamic behavior observed in response to different agonists in hepatocytes.

NONLINEAR PROCESSES OF INTRACELLULAR CALCIUM SIGNALING AS A TARGET FOR THE INFLUENCE OF EXTREMELY LOW-FREQUENCY FIELDS

Theoretical analysis of peculiarities of reception of weak extremely low-frequency periodic signals by calcium-dependent intracellular regulatory systems was performed on the reduced “minimal” model for calcium oscillations suggested by Goldbeter et al. (Proc. Natl. Acad. Sci. USA 87, 1461–1465, 1990). The model considered the following calcium-dependent processes: the rise in intracellular free calcium concentration ([Ca2+]i) due to calcium ionophore A23187 action on a cell, activation of the Ca2+ entry through calcium channels in the plasma membrane by the initial rise in [Ca2+]i, and the Ca2+ release from intracellular stores by the calcium-induced calcium release mechanism. Calcium channels of plasma membrane were chosen as a target for the modulating signal and an additive noise influence in the model. An increase in [Ca2+]i under the influence of the modulating signal was demonstrated to depend not only on the amplitude and frequency of this signal, but also on the phase of the signal with respect t...

Augmented platelet calcium uptake in response to serotonin stimulation in patients with major depression measured using Mn 2+ influx and 45Ca 2+ uptake

There is an augmented platelet intracellular calcium response to serotonin stimulation in major depression. The role that calcium influx has in this process is not known. The objective of this study was to determine platelet calcium influx in response to serotonin by two methods, Mn 2+ influx and 45Ca 2+ uptake, in order to observe if the uptake response to serotonin was augmented in major depression by comparing the response to normal controls. The use of the two methods of calcium influx showed that serotonin stimulates calcium uptake into platelets. Furthermore, patients with major depression have significantly augmented platelet calcium uptake in response to serotonin. The interesting finding was that calcium uptake into platelets is biphasic, occurring immediately and after five minutes. These results may support the two pool model for calcium oscillations within cells whereby extracellular calcium is needed for intracellular calcium release, and for replenishment of depleted stores once intracellular calcium is released.

Model of Intercellular Calcium Oscillations in Hepatocytes: Synchronization of Heterogeneous Cells

Hepatocytes respond with repetitive cytosolic calcium spikes to stimulation by vasopressin and noradrenalin. In the intact liver, calcium oscillations occur in a synchronized fashion as periodic waves across whole liver lobules, but the mechanism of intercellular coupling remains unclear. Recently, it has been shown that individual hepatocytes can have very different intrinsic oscillation frequencies but become phase-locked when coupled by gap junctions. We investigate the gap junction hypothesis for intercellular synchronization by means of a mathematical model. It is shown that junctional calcium fluxes are effective in synchronizing calcium oscillations in coupled hepatocytes. An experimentally testable estimate is given for the junctional coupling coefficient required; it mainly depends on the degree of heterogeneity between cells. Intercellular synchronization by junctional calcium diffusion may occur also in other cell types exhibiting calcium-activated calcium release through InsP 3 receptors, if the gap junctional coupling is strong enough and the InsP 3 receptors are sufficiently sensitized by InsP 3.

Agonist-dependent phosphorylation of the inositol 1,4,5-trisphosphate receptor: A possible mechanism for agonist-specific calcium oscillations in pancreatic acinar cells.

The properties of inositol 1,4,5-trisphosphate (IP3)-dependent intracellular calcium oscillations in pancreatic acinar cells depend crucially on the agonist used to stimulate them. Acetylcholine or carbachol (CCh) cause high-frequency (10-12-s period) calcium oscillations that are superimposed on a raised baseline, while cholecystokinin (CCK) causes long-period (>100-s period) baseline spiking. We show that physiological concentrations of CCK induce rapid phosphorylation of the IP3 receptor, which is not true of physiological concentrations of CCh. Based on this and other experimental data, we construct a mathematical model of agonist-specific intracellular calcium oscillations in pancreatic acinar cells. Model simulations agree with previous experimental work on the rates of activation and inactivation of the IP3 receptor by calcium (DuFour, J.-F., I.M. Arias, and T.J. Turner. 1997. J. Biol. Chem. 272:2675-2681), and reproduce both short-period, raised baseline oscillations, and long-period baseline spiking. The steady state open probability curve of the model IP3 receptor is an increasing function of calcium concentration, as found for type-III IP3 receptors by Hagar et al. (Hagar, R.E., A.D. Burgstahler, M.H. Nathanson, and B.E. Ehrlich. 1998. Nature. 396:81-84). We use the model to predict the effect of the removal of external calcium, and this prediction is confirmed experimentally. We also predict that, for type-III IP3 receptors, the steady state open probability curve will shift to lower calcium concentrations as the background IP3 concentration increases. We conclude that the differences between CCh- and CCK-induced calcium oscillations in pancreatic acinar cells can be explained by two principal mechanisms: (a) CCK causes more phosphorylation of the IP3 receptor than does CCh, and the phosphorylated receptor cannot pass calcium current; and (b) the rate of calcium ATPase pumping and the rate of calcium influx from the outside the cell are greater in the presence of CCh than in the presence of CCK.

Modelling oscillations of calcium and endoplasmic reticulum transmembrane potential: Role of the signalling and buffering proteins and of the size of the Ca 2+ sequestering ER subcompartments

Intracellular calcium oscillations provide a natural clock that may be of crucial importance for the timing of many cellular processes. Elucidating of the mechanisms underlying these oscillations is of particular interest. The theoretical description presented here extends existing models of calcium oscillations by allowing for two types of proteins differing in their calcium-binding properties. This model reflects experimental findings by considering both a fast calcium-binding process to low-affinity protein binding sites such as found in the N-domains of calmodulin or troponin C and a class of high-affinity calcium binding proteins with slow binding kinetics (e.g., parvalbumin or the C-domains of calmodulin and troponin C). Furthermore, recalling that calcium is mainly stored in small subcompartments of the ER, it is argued that only a small fraction of its overall volume participates in the rapid release and uptake of calcium. The effect of the size of this fraction is studied. The hypothesis saying that any electric potential difference across the ER membrane would be dissipated by the highly permeant ions is critically examined by an analytical estimation based on the electroneutrality condition and by numerical integration of the complete model equations. It is predicted theoretically that the transmembrane potential of the ER calcium stores, which is up to now virtually impossible to determine in experiment, builds up in the millivolt range at physiological concentrations of monovalent ions. The phenomenology of oscillations is studied by numerical integration. The model reproduces experimentally observed values of frequency and amplitude as well as the typical spike-like shape of oscillations. The model reveals also the time course of a shift of the bound Ca 2+ population from the low-affinity binding sites to the high-affinity binding sites.

Mitochondria as an important factor in the maintenance of constant amplitudes of cytosolic calcium oscillations

Theoretical models of intracellular calcium oscillations have hitherto focused on the endoplasmic reticulum (ER) as an internal calcium store. These models reproduced the large variability in oscillation frequency observed experimentally. In the present contribution, we extend our earlier model [Marhl et al., Biophys. Chem., 63 (1997) 221] by including, in addition to the ER, mitochondria as calcium stores. Simple plausible rate laws are used for the calcium uptake into, and release from, the mitochondria. It is demonstrated with the help of this extended model that mitochondria are likely to act in favour of frequency encoding by enabling the maintenance of fairly constant amplitudes over wide ranges of frequency.

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.

Model for receptor-controlled cytosolic calcium oscillations and for external influences on the signal pathway

The external stimulation of many cells by a hormone, for example, often leads to an oscillating cytosolic calcium concentration. This periodic behavior is now designated the cytosolic calcium oscillator. A theoretical model is presented that describes this behavior on the basis of inositol(1,4,5)trisphosphate-induced calcium oscillations. In contrast to other models only a single positive feedback loop is taken into account to obtain oscillations. The model includes important innovations compared to other approaches. It includes the contribution of extracellular calcium and its modification after the stimulation of the cell. Furthermore, the signal pathway that leads to cytosolic calcium oscillations is described in more detail than in other models. This enables investigations on the influence of additional parameters like external electromagnetic fields on the signal transduction pathway. The model and the calculations are based on the theory of nonlinear self-sustained oscillators.

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

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

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.