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(alpha) 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.     

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 shape hormonally induced cytoplasmic calcium oscillations and modulate exocytosis

Pituitary gonadotropes transduce hormonal input into cytoplasmic calcium ([Ca(2+)](cyt)) oscillations that drive rhythmic exocytosis of gonadotropins. Using Calcium Green-1 and rhod-2 as optical measures of cytoplasmic and mitochondrial free Ca(2+), we show that mitochondria sequester Ca(2+) and tune the frequency of [Ca(2+)](cyt) oscillations in rat gonadotropes. Mitochondria accumulated Ca(2+) rapidly and in phase with elevations of [Ca(2+)](cyt) after GnRH stimulation or membrane depolarization. Inhibiting mitochondrial Ca(2+) uptake by the protonophore CCCP reduced the frequency of GnRH-induced [Ca(2+)](cyt) oscillations or, occasionally, stopped them. Much of the Ca(2+) that entered mitochondria is bound by intramitochondrial Ca(2+) buffering systems. The mitochondrial Ca(2+) binding ratio may be dynamic because [Ca(2+)](mit) appeared to reach a plateau as mitochondrial Ca(2+) accumulation continued. Entry of Ca(2+) into mitochondria was associated with a small drop in the mitochondrial membrane potential. Ca(2+) was extruded from mitochondria more slowly than it entered, and much of this efflux could be blocked by CGP-37157, a selective inhibitor of mitochondrial Na(+)-Ca(2+) exchange. Plasma membrane capacitance changes in response to depolarizing voltage trains were increased when CCCP was added, showing that mitochondria lower the local [Ca(2+)](cyt) near sites that trigger exocytosis. Thus, we demonstrate a central role for mitochondria in a significant physiological response.     

Intracellular calcium oscillations regulate ciliary beat frequency of airway epithelial cells

The effect of ATP-induced Ca2+ oscillations on ciliary activity was examined in airway epithelial cells by simultaneously measuring the ciliary beat frequency (CBF) and the intracellular Ca2+ concentration ([Ca2+]i) near the base of the cilia. Exposure to extracellular ATP (ATPo) induces a rapid and large increase in both [Ca2+]i and CBF, followed by oscillations in [Ca2+]i and a sustained elevation in CBF. After each Ca2+ oscillation, the [Ca2+]i returned to near basal values. By contrast, the CBF remained elevated during these Ca2+ oscillations, although each Ca2+ oscillation induced small variations in CBF. During Ca2+ oscillations, increases in CBF closely followed the rising phase of increases in [Ca2+]i, but declines in CBF lagged behind declines in [Ca2+]i. Higher frequency Ca2+ oscillations reduced variations in CBF, producing a stable and sustained elevation in CBF. The maximal CBF was induced by Ca2+ oscillations and was 15% greater than the CBF induced by the substantially larger initial [Ca2+]i increase. These data demonstrate that the rate of CBF is not directly dependent on the absolute [Ca2+]i, but is dependent on the differential changes in [Ca2+]i and suggest that CBF in airway epithelial cells is regulated by frequency-modulated Ca2+ signaling.     

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.     

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.     

Mitochondria and calcium signaling

The kinetic properties for the uptake, storage and release of Ca2+ from isolated mitochondria accurately predict the behaviour of the organelles within the intact cell. While the steady-state cycling of Ca2+ across the inner membrane between independent uptake and efflux pathways seems at first sight to be symmetrical, the distinctive kinetics of the uniporter, which is highly dependent on external free Ca2+ concentration and the efflux pathway, whose activity is clamped over a wide range of total matrix Ca2+ by the solubility of the calcium phosphate complex provide a mechanism whereby mitochondria reversibly sequester transient elevations in cytoplasmic Ca2+. Under non-stimulated conditions, the same transport processes can regulate matrix Ca2+ concentrations and hence citric acid cycle activity.     

Sustained oscillations in glycolysis: an experimental and theoretical study of chaotic and complex periodic behavior and of quenching of simple oscillations

We report sustained oscillations in glycolysis conducted in an open system (a continuous-flow, stirred tank reactor; CSTR) with inflow of yeast extract as well as glucose. Depending on the operating conditions, we observe simple or complex periodic oscillations or chaos. We report the response of the system to instantaneous additions of small amounts of several substrates as functions of the amount added and the phase of the addition. We simulate oscillations and perturbations by a kinetic model based on the mechanism of glycolysis in a CSTR. We find that the response to particular perturbations forms an efficient tool for elucidating the mechanism of biochemical oscillations.     

Modeling the interrelations between the calcium oscillations and ER membrane potential oscillations

A refined electrochemical model accounting for intracellular calcium oscillations and their interrelations with oscillations of the potential difference across the membrane of the endoplasmic reticulum (ER) or other intracellular calcium stores is established. The ATP dependent uptake of Ca2+ from the cytosol into the ER, the Ca2+ release from the ER through channels following a calcium-induced calcium release mechanism, and a potential-dependent Ca2+ leak flux out of the ER are included in the model and described by plausible rate laws. The binding of calcium to specific proteins such as calmodulin is taken into account. The quasi-electroneutrality condition allows us to express the transmembrane potential in terms of the concentrations of cytosolic calcium and free binding sites on proteins, which are the two independent variables of the model. We include monovalent ions in the model, because they make up a considerable portion in the balance of electroneutrality. As the permeability of the endoplasmic membrane for these ions is much higher than that for calcium ions, we assume the former to be in Nernst equilibrium. A stability analysis of the steady-state solutions (which are unique or multiple depending on parameter values) is carried out and the Hopf bifurcation leading from stable steady states to self-sustained oscillations is analysed with the help of appropriate mathematical techniques. The oscillations obtained by numerical integration exhibit the typical spike-like shape found in experiments and reasonable values of frequency and amplitude. The model describes the process of switching between stationary and pulsatile regimes as well as changes in oscillation frequency upon parameter changes. It turns out that calcium oscillations can arise without a permanent influx of calcium into the cell, when a calcium-buffering system such as calmodulin is included.     

Chaotic oscillations in a simple collapsible-tube model.

A steady flow through a segment of externally pressurized, collapsible tube can become unstable to a wide variety of self-excited oscillations of the internal flow and tube walls. A simple, one-dimensional model of the conventional laboratory apparatus, which has been shown previously to predict steady flows and multiple modes of oscillation, is investigated numerically here. Large amplitude oscillations are shown to have a relaxation structure, and the nonlinear interaction between different modes is shown to give rise to quasiperiodic and apparently aperiodic behavior. These predictions are shown to compare favorably with experimental observations.     

Inositol trisphosphate and calcium oscillations

InsP3 has two important functions in generating Ca2+ oscillations. It releases Ca2+ from the internal store and it can contribute to Ca2+ entry. A hypothesis has been developed to describe a mechanism for Ca2+ oscillations with particular emphasis on the way agonist concentration regulates oscillator frequency. The main idea is that the InsP3 receptors are sensitized to release Ca2+ periodically by cyclical fluctuations of Ca2+ within the lumen of the endoplasmic reticulum. Each time a pulse of Ca2+ is released, the luminal level of Ca2+ declines and has to be replenished before the InsP3 receptors are resensitized to deliver the next pulse of Ca2+. It is this loading of the internal store that explains why frequency is sensitive to external Ca2+ and may also account for how variations in agonist concentration are translated into changes in oscillation frequency. Variations in agonist-induced entry of external Ca2+, which can occur through different mechanisms, determine the variable rates of store loading responsible for adjusting the sensitivity of the InsP3 receptors to produce the periodic pulses of Ca2+. The Ca2+ oscillator is an effective analogue-to-digital converter in that variations in the concentration of the external stimulus are translated into a change in oscillator frequency.     

Mitochondria as biosensors of calcium microdomains

The notion that the agonist-dependent increases in intracellular Ca2+ concentration, on ubiquitous signalling mechanism, occur with a tightly regulated spatio-temporal pattern has become an established concept in modern cell biology. As a consequence, the concept is emerging that the recruitment of specific intracellular targets and effector system mechanisms depends on exposure to local [Ca2+] that differs substantially from the mean [Ca2+]. A striking example is provided by mitochondria, intracellular organelles that have been overlooked for a long time in the field of calcium signalling due to the low affinity of their Ca(2+)-uptake pathways. We will summarize here some of the evidence indicating that these organelles actively participate in Ca2+ homeostasis in physiological conditions (with consequences not only for the control of their function, but also for the modulation of the complexity of calcium signals) because they have the capability to respond to microdomains of high [Ca2+] transiently generated in their proximity by the opening of Ca2+ channels.     

Mitochondria and calcium signaling in embryonic development

Calcium (Ca²⁺) is a simple but critical signal for controlling various cellular processes and is especially important in fertilization and embryonic development. The dynamic change of cellular Ca²⁺ concentration and homeostasis are tightly regulated. Cellular Ca²⁺ increases by way of Ca²⁺ influx from extracellular medium and Ca²⁺ release from cellular stores of the endoplasmic reticulum (ER) and sarcoplasmic reticulum (SR). The elevated Ca²⁺ is subsequently sequestered by expelling it out of the cell or by pumping back to the ER/SR. Mitochondria function as a power house for energy production via oxidative phosphorylation in most eukaryotes. In addition to this well-known function, mitochondria are also recognized to regulate Ca²⁺ homeostasis through different mechanisms. Although critical roles of Ca²⁺ signaling in fertilization and embryonic development are known, the involvement of mitochondria in these processes are not fully understood. This review is focused on the role of mitochondrial respiratory chain complex I in the regulation of Ca²⁺ signaling pathway and gene expression in embryonic development, especially on the new findings in the cardiac development of Xenopus embryos. The data demonstrate that mitochondria modulate Ca²⁺ signaling and the Ca²⁺-dependent NFAT pathway and its target gene which are essential for embryonic heart development.     

Modelling the transition from simple to complex Ca2+oscillations in pancreatic acinar cells

A mathematical model is proposed which systematically investigates complex calcium oscillations in pancreatic acinar cells. This model is based on calcium-induced calcium release via inositol trisphosphate receptors (IPR) and ryanodine receptors (RyR) and includes calcium modulation of inositol (1,4,5) trisphosphate (IP₃) levels through feedback regulation of degradation and production. In our model, the apical and the basal regions are separated by a region containing mitochondria, which is capable of restricting Ca²⁺ responses to the apical region. We were able to reproduce the observed oscillatory patterns, from baseline spikes to sinusoidal oscillations. The model predicts that calcium-dependent production and degradation of IP₃ is a key mechanism for complex calcium oscillations in pancreatic acinar cells. A partial bifurcation analysis is performed which explores the dynamic behaviour of the model in both apical and basal regions.     

Mitochondria as sensors and regulators of calcium signalling

During the past two decades calcium (Ca(2+)) accumulation in energized mitochondria has emerged as a biological process of utmost physiological relevance. Mitochondrial Ca(2+) uptake was shown to control intracellular Ca(2+) signalling, cell metabolism, cell survival and other cell-type specific functions by buffering cytosolic Ca(2+) levels and regulating mitochondrial effectors. Recently, the identity of mitochondrial Ca(2+) transporters has been revealed, opening new perspectives for investigation and molecular intervention.     

Frequency decoding of calcium oscillations

Background

Calcium (Ca^2 +) oscillations are ubiquitous signals present in all cells that provide efficient means to transmit intracellular biological information. Either spontaneously or upon receptor ligand binding, the otherwise stable cytosolic Ca^2 + concentration starts to oscillate. The resulting specific oscillatory pattern is interpreted by intracellular downstream effectors that subsequently activate different cellular processes. This signal transduction can occur through frequency modulation (FM) or amplitude modulation (AM), much similar to a radio signal. The decoding of the oscillatory signal is typically performed by enzymes with multiple Ca^2 + binding residues that diversely can regulate its total phosphorylation, thereby activating cellular program. To date, NFAT, NF-κB, CaMKII, MAPK and calpain have been reported to have frequency decoding properties.

Scope of review

The basic principles and recent discoveries reporting frequency decoding of FM Ca^2 + oscillations are reviewed here.

Major conclusions

A limited number of cellular frequency decoding molecules of Ca^2 + oscillations have yet been reported. Interestingly, their responsiveness to Ca^2 + oscillatory frequencies shows little overlap, suggesting their specific roles in cells.

General significance

Frequency modulation of Ca^2 + oscillations provides an efficient means to differentiate biological responses in the cell, both in health and in disease. Thus, it is crucial to identify and characterize all cellular frequency decoding molecules to understand how cells control important cell programs.