Transition from stochastic to deterministic behavior in calcium oscillations

Simulation and modeling is becoming more and more important when studying complex biochemical systems. Most often, ordinary differential equations are employed for this purpose. However, these are only applicable when the numbers of participating molecules in the biochemical systems are large enough to be treated as concentrations. For smaller systems, stochastic simulations on discrete particle basis are more accurate. Unfortunately, there are no general rules for determining which method should be employed for exactly which problem to get the most realistic result. Therefore, we study the transition from stochastic to deterministic behavior in a widely studied system, namely the signal transduction via calcium, especially calcium oscillations. We observe that the transition occurs within a range of particle numbers, which roughly corresponds to the number of receptors and channels in the cell, and depends heavily on the attractive properties of the phase space of the respective systems dynamics. We conclude that the attractive properties of a system, expressed, e.g., by the divergence of the system, are a good measure for determining which simulation algorithm is appropriate in terms of speed and realism.     

Non-Gaussian noise-optimized intracellular cytosolic calcium oscillations

We have numerically studied the effect of a particular kind of non-Gaussian colored noise (NGN), characterized by the deviation q from Gaussian noise (q = 1), on intracellular cytosolic calcium (Ca²⁺) oscillations. It is found that, as q is increased, the Ca²⁺ oscillation regularity increases and reaches a best performance at an optimal q, and then decreases with further increasing q, which represents the occurrence of coherence resonance, i.e., the most regular Ca²⁺ oscillations. Similar phenomena occur for different values of noise intensity and correlation time of the NGN. This phenomenon of deviation-optimized Ca²⁺ oscillations show that, external non-Gaussian noises of different types can enhance and even optimize the intracellular Ca²⁺ oscillations. This result provides new insights into the constructive roles and potential applications of non-Gaussian noises in intracellular cytosolic Ca²⁺ oscillations.     

A mesoscopic stochastic mechanism of cytosolic calcium oscillations

Based on a model of intracellular calcium (Ca(2+)) oscillation with self-modulation of inositol 1,4,5-trisphosphate signal, the mesoscopic stochastic differential equations for the intracellular Ca(2+) oscillations are theoretically derived by using the chemical Langevin equation method. The effects of the finite biochemical reaction molecule number on both simple and complex cytosolic Ca(2+) oscillations are numerically studied. In the case of simple intracellular Ca(2+) oscillation, it is found that, with the increase of molecule number, the coherence resonance or autonomous resonance phenomena can occur for some external stimulation parameter values. In the cases of complex cytosolic Ca(2+) oscillations, each extremum of concentration of cytosolic Ca(2+) oscillations corresponds to a peak in the histogram of Ca(2+) concentration, and the most probability appeared during the bursting plateau level for bursting, but at the largest minimum of Ca(2+) concentration for chaos. For quasi-periodicity, however, there are only two peaks in the histogram of Ca(2+) concentration, and the most probability is located at low concentration state.     

Birhythmicity, trirhythmicity and chaos in bursting calcium oscillations

We have analyzed various types of complex calcium oscillations. The oscillations are explained with a model based on calcium-induced calcium release (CICR). In addition to the endoplasmic reticulum as the main intracellular Ca2+ store, mitochondrial and cytosolic Ca2+ binding proteins are also taken into account. This model was previously proposed for the study of the physiological role of mitochondria and the cytosolic proteins in gene rating complex Ca2+ oscillations [1]. Here, we investigated the occurrence of different types of Ca2+ oscillations obtained by the model, i.e. simple oscillations, bursting, and chaos. In a bifurcation diagram, we have shown that all these various modes of oscillatory behavior are obtained by a change of only one model parameter, which corresponds to the physiological variability of an agonist. Bursting oscillations were studied in more detail because they express birhythmicity, trirhythmicity and chaotic behavior. Two different routes to chaos are observed in the model: in addition to the usual period doubling cascade, we also show intermittency. For the characterization of the chaotic behavior, we made use of return maps and Lyapunov exponents. The potential biological role of chaos in intracellular signaling is discussed.     

Complex calcium oscillations and the role of mitochondria and cytosolic proteins

Intracellular calcium oscillations, which are oscillatory changes of cytosolic calcium concentration in response to agonist stimulation, are experimentally well observed in various living cells. Simple calcium oscillations represent the most common pattern and many mathematical models have been published to describe this type of oscillation. On the other hand, relatively few theoretical studies have been proposed to give an explanation of complex intracellular calcium oscillations, such as bursting and chaos. In this paper, we develop a new possible mechanism for complex calcium oscillations based on the interplay between three calcium stores in the cell: the endoplasmic reticulum (ER), mitochondria and cytosolic proteins. The majority ( approximately 80%) of calcium released from the ER is first very quickly sequestered by mitochondria. Afterwards, a much slower release of calcium from the mitochondria serves as the calcium supply for the intermediate calcium exchanges between the ER and the cytosolic proteins causing bursting calcium oscillations. Depending on the permeability of the ER channels and on the kinetic properties of calcium binding to the cytosolic proteins, different patterns of complex calcium oscillations appear. With our model, we are able to explain simple calcium oscillations, bursting and chaos. Chaos is also observed for calcium oscillations in the bursting mode.     

Pharmacologically induced calcium oscillations protect neurons from increases in cytosolic calcium after trauma

Increases in cytosolic calcium ([Ca(2+)](i)) following mechanical injury are often considered a major contributing factor to the cellular sequelae in traumatic brain injury (TBI). However, very little is known on how developmental changes may affect the calcium signaling in mechanically injured neurons. One key feature in the developing brain that may directly impact its sensitivity to stretch is the reduced inhibition which results in spontaneous [Ca(2+)](i) oscillations. In this study, we examined the mechanism of stretch-induced [Ca(2+)](i) transients in 18-days in vitro (DIV) neurons exhibiting bicuculline-induced [Ca(2+)](i) oscillations. We used an in vitro model of mechanical trauma to apply a defined uniaxial strain to cultured cortical neurons and used increases in [Ca(2+)](i) as a measure of the neuronal response to the stretch insult. We found that stretch-induced increases in [Ca(2+)](i) in 18-DIV neurons were inhibited by pretreatment with either the NMDA receptor antagonist, APV [D(-)-2-Amino-5-phosphonopentanoic acid], or by depolymerizing the actin cytoskeleton prior to stretch. Blocking synaptic NMDA receptors prior to stretch significantly attenuated most of the [Ca(2+)](i) transient. In comparison, cultures with pharmacologically induced [Ca(2+)](i) oscillations showed a substantially reduced [Ca(2+)](i) peak after stretch. We provide evidence showing that a contributing factor to this mechanical desensitization from induced [Ca(2+)](i) oscillations is the PKC-mediated uncoupling of NMDA receptors (NMDARs) from spectrin, an actin-associated protein, thereby rendering neurons insensitive to stretch. These results provide novel insights into how the [Ca(2+)](i) response to stretch is initiated, and how reduced inhibition - a feature of the developing brain - may affect the sensitivity of the immature brain to trauma.     

Embedding a grammatical description in deterministic chaos: an experiment in recurrent neural learning

We consider the modeling process of a biological agent by combining the concepts of neuroinformatics and deterministic chaos. We assume that an agent observes a target process as a stochastic symbolic process, which is restricted by grammatical constraints. Our main hypothesis is that an agent would learn the target model by reconstructing an equivalent quasi-stochastic process on its deterministic neural dynamics. We employed a recurrent neural network (RNN), which is regarded as an adjustable deterministic dynamical system. Then, we conducted an experiment to observe how the RNN learns to reconstruct the target process, represented by a stochastic finite state machine in the simulation. The result revealed the capability of the RNN to evolve, by means of learning, toward chaos, which is able to mimic a target's stochastic process. We precisely analyzed the evolutionary process as well as the internal representation of the neural dynamics obtained. This analysis enabled us to clarify an interesting mechanism of the self-organization of chaos by means of neural learning, and also showed how grammar can be embedded in the evolved deterministic chaos.     

Agonist-induced cytosolic calcium oscillations originate from a specific locus in single hepatocytes

Digital imaging fluorescence microscopy of fura-2-loaded hepatocytes in primary culture has been used to examine the changes of cytosolic free Ca2+ ([Ca2+]i) in response to receptor activation by alpha 1-adrenergic agonists and vasopressin at the subcellular level. Agonist-induced Ca2+ oscillations did not occur synchronously within the cell but originated from a specific region adjacent to the cell membrane and then propagated throughout the rest of the cell, with each oscillation within a series originating from the same locus. Furthermore, hormones acting through different receptors produced Ca2+ waves with similar rates of progress (20-25 microns.s-1) which originated from the same subcellular locus. For a given cell, the rate of progress and amplitude of the Ca2+ waves were independent of applied agonist concentration and were unaffected by depletion of extracellular Ca2+. The kinetics of Ca2+ increase at different points within the cell indicated that the Ca2+ waves were not driven by diffusion but were characteristic of a self-propagating mechanism. Significantly, when cells were treated with A1F-4 to directly activate the G-protein which couples receptor occupancy to [Ca2+]i mobilization, the origin and kinetics of the Ca2+ waves were identical to those observed with hormonal stimulation. It is proposed that the spatial organization of the intracellular Ca2+ release mechanisms may have significance in the regulation of the asymmetric metabolic functions of hepatocytes and other functionally polarized cells.     

Spontaneous and chemoattractant-induced oscillations of cytosolic free calcium in single adherent human neutrophils

Studies with fluorescent Ca2+ indicators in large populations of neutrophils in suspension reveal a stable base line followed by a rapid agonist-induced elevation of cytosolic free calcium, [Ca2+]i, concomitant with other parameters of cellular activation. To study the role of adhesion in cell activation, we monitored [Ca2+]i in single neutrophils adhered to albumin-coated or fibronectin-coated glass coverslips before and after stimulation with the chemotactic peptide N-formyl-L-methionyl-L-leucyl-L-phenylalanine (fMLP). Human neutrophils loaded with 2 microM fura 2/AM were allowed to adhere to coverslips for 15-20 min at 37 degrees C. [Ca2+]i was monitored with a dual excitation microfluorimeter with a time resolution of 200 ms. Statistical analysis was performed using an algorithm allowing to detect significant [Ca2+]i peaks. 54% of the cells showed spontaneous [Ca2+]i oscillations. The amplitude of these [Ca2+]i peaks averaged 77 +/- 10 nM above basal levels (mean value of 110 +/- 20 nM), and their mean duration was 28 +/- 5 s; periods of [Ca2+]i bursts could last up to 15 min. In "silent" cells exhibiting a stable [Ca2+]i base line without spontaneous oscillations, low concentrations of fMLP (10(-10)-10(-9) M) could induce sustained [Ca2+]i oscillations. By contrast, higher agonist concentrations (10(-6) M) induced a single [Ca2+]i transient followed by a stable base line. 47% of the cells showing spontaneous [Ca2+]i oscillations did not respond to fMLP. Spontaneous [Ca2+]i oscillations depended on the continuous presence of extracellular Ca2+. Therefore: (i) spontaneous oscillations of [Ca2+]i occur in neutrophils adherent to various substrata; (ii) these oscillations do not preclude and can be dissociated from the response to fMLP; (iii) neutrophil functions might be controlled by [Ca2+]i oscillations rather than by sustained alterations of [Ca2+]i.     

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.     

Auxin causes oscillations of cytosolic free calcium and pH inZea mays coleoptiles

In epidermal cells of maize (Zea mays L.) coleoptiles, cytosolic pH (pH_c), cytosolic free calcium, membrane potential and changes thereof were monitored continuously and simultaneously (pH_c/, _m, Ca²⁺/ _m) using double-barrelled ion-sensitive microelectrodes. In the resting cells the cytosolic pH was 7.3–7.5 and the concentration of free calcium was 119±24 nM. One-micromolar indole-3-acetic acid (IAA), added to the external medium at pH 6.0 triggered oscillations in _m, pH_c and free calcium with a period of 20 to 30 min. Acidification of the cytosolic pH increased the cytosolic free calcium. The _m oscillations are attributed to changes in activity of the H⁺-extrusion pump at the plasmalemma, triggered off by pH and controlled by pH regulation (pH oscillation). The origin of the pH_c and Ca²⁺ changes remains unclear, but is possibly caused by auxin-receptor-induced lipid breakdown and subsequent second-messenger formation. It is suggested that the observed cytosolic pH and Ca²⁺ changes are intrinsically interrelated, and it is concluded that this onset of regulatory processes through the phytohormone IAA is indicative of calcium and protons mediating early auxin action in maize coleoptiles. It is further concluded that the double-barrelled ion-sensitive microelectrode is an invaluable tool for investigating in-vivo hormone action in plant tissues.Abbreviations and symbols FC fusicoccin - IAA indole-3-acetic acid - Mes 2-(N-morpholino)ethanesulfonic acid - pH_c cytosolic pH - Tris 2-amino-2-(hydroxymethyl)-1,3-propanediol - _m membrane potential difference (mV)     

Controlling chaos in ecology: From deterministic to individual-based models

The possibility of chaos control in biological systems has been stimulated by recent advances in the study of heart and brain tissue dynamics. More recently, some authors have conjectured that such a method might be applied to population dynamics and even play a nontrivial evolutionary role in ecology. In this paper we explore this idea by means of both mathematical and individual-based simulation models. Because of the intrinsic noise linked to individual behavior, controlling a noisy system becomes more difficult but, as shown here, it is a feasible task allowed to be experimentally tested.     

Dichlorodiphenylchloroethylene elevates cytosolic calcium concentrations and oscillations in primary cultures of human granulosa-lutein cells

1,1-Dichloro-2,2-bis(p-chlorophenyl)ethylene (DDE), a metabolite of DDT (1,1-dichlorodiphenyltrichloroethane), is a persistent hormonally active environmental toxicant that has been found in human serum and follicular fluid. The objective of this study was to determine whether DDE can alter free calcium ion concentrations in the cytosol ([Ca(2+)](cyt)) of human granulosa cells. Changes in [Ca(2+)](cyt) in single cells loaded with Fura-2 were studied using a dynamic digital Ca(2+) imaging system. At a concentration of 100 ng/ml, DDE stimulated small elevations of [Ca(2+)](cyt) accompanied by Ca(2+) oscillations. At 1 microg DDE/ml, there was a biphasic Ca(2+) response with marked elevations of [Ca(2+)](cyt) over time. In Ca(2+)-free medium, cells showed an initial small elevation of [Ca(2+)](cyt), which was magnified after addition of Ca(2+) to the medium. Washing the cells after DDE treatment failed to remove the elevated [Ca(2+)](cyt) and oscillations, both of which were eliminated by addition of EGTA. ATP also induced [Ca(2+)](cyt) elevations and oscillations, and these effects were potentiated when DDE was added. FSH induced transient [Ca(2+)](cyt) elevations, whereas hCG caused a prolonged elevation and marked oscillations in [Ca(2+)](cyt). These results suggest that DDE at concentrations normally found in human tissues induces elevations in [Ca(2+)](cyt) in granulosa-lutein cells. Our data therefore highlight a novel mechanism through which DDE can alter endocrine homeostasis and possibly act as an endocrine toxicant.     

GnRH-induced cytosolic calcium oscillations in pituitary gonadotrophs: phase resetting by membrane depolarization.

Cultured rat pituitary gonadotrophs under whole-cell voltage clamp conditions respond to the hypothalamic hormone GnRH with synchronized oscillatory changes in both cytosolic Ca2+ concentration ([Ca2+]i) and [Ca2+]i-activated, apamin-sensitive K+ current (IK(Ca)). We found, and report here for the first time, that in GnRH-stimulated cells a brief depolarizing pulse can elicit a transient [Ca2+]i rise similar to the endogenous cycle. Furthermore, Ca2+ entry during a single depolarizing pulse was found to shift the phase of subsequent endogenous [Ca2+]i oscillations, which thereafter continue to occur at their previous frequency before the pulse. Application of two consecutive depolarizing pulses showed that the size of the [Ca2+]i rise evoked by the second pulse depended on the time lapsed between two consecutive pulses, indicating that each endogenous or evoked [Ca2+]i rise cycle leaves the Ca2+ release mechanism of the gonadotroph in a refractory state. Recovery from this condition can be described by an exponential function of the time lapsed between the pulses (time constant of ca. 1 s). We propose that the underlying mechanism in both refractoriness after endogenous cycles and phase resetting by a brief pulse of Ca2+ entry involves the InsP3 receptor-channel molecule presumed to be located on the cytosolic aspect of the endoplasmic reticulum membrane.     

Model for external influences on cellular signal transduction pathways including cytosolic calcium oscillations

Experiments on the effects of extremely-low-frequency (ELF) electric and magnetic fields on cells of the immune system, T-lymphocytes in particular, suggest that the external field interacts with the cell at the level of intracellular signal transduction pathways. These are directly connected with changes in the calcium-signaling processes of the cell. Based on these findings, a theoretical model for receptor-controlled cytosolic calcium oscillations and for external influences on the signal transduction pathway is presented. We discuss the possibility that the external field acts on the kinetics of the signal transduction between the activated receptors at the cell membrane and the G-proteins. It is shown that, depending on the specific combination of cell internal biochemical and external physical parameters, entirely different responses of the cell can occur. We compare the effects of a coherent (periodic) modulation and of incoherent perturbations (noise). The model and the calculations are based on the theory of self-sustained, nonlinear oscillators. It is argued that these systems form an ideal basis for information-encoding processes in biological systems. © 1995 Wiley-Liss, Inc.     

Spectrofluorimetric and image recordings of spontaneous and lectin-induced cytosolic calcium oscillations in Jurkat T cells

Intracellular variations in Ca2+ concentrations have been measured in single Jurkat T lymphocyte variants (77 6.8 and E6.1) using Fura-2 as a probe. Under basal conditions, the cytosolic Ca2+ level is stable but some cells show spontaneous Ca2+ oscillations (frequency, 0.30 +/- 0.06 Hz). These oscillations are sensitive to the external concentration of Ca2+ since they can no longer be observed when the bathing solution is replaced (superfusion) with a Ca(2+)-free medium or when a Ca2+ chelator (EGTA) is added. Various changes in the cytosolic concentration of Ca2+ ([Ca2+]i) can be observed when the cells are exposed to the mitogenic lectin phytohemagglutinin (PHA, 80 nM). For instance, in the case of non-oscillating cells, the lectin induces either a rapid increase in [Ca2+]i that is followed by a sustained response (plateau) or it triggers Ca2+ spikes. In the case of experiments done in Ca(2+)-free medium, only the initial spike was observed. In the case of spontaneously oscillating cells, PHA induces a rapid increase in [Ca2+]i that is followed by a plateau where oscillations are absent. In every case, the PHA-dependent Ca2+ response is abrogated in a Ca(2+)-free medium. Computer simulations based on the model of Goldbeter et al. [27] show that the various Ca2+ responses of Jurkat cells are related to the cytosolic level of free Ca2+. Video imaging analyses show that the cellular Ca2+ responses are not homogeneous whether the observations are made in spontaneously oscillating Jurkat cells or when they are exposed to PHA.     

Injection of inositol trisphosphorothioate into Limulus ventral photoreceptors causes oscillations of free cytosolic calcium

Limulus ventral photoreceptors contain calcium stores sensitive to release by D-myo-inositol 1,4,5 trisphosphate (InsP3) and a calcium-activated conductance that depolarizes the cell. Mechanisms that terminate the response to InsP3 were investigated using nonmetabolizable DL-myo-inositol 1,4,5 trisphosphorothioate (InsPS3). An injection of 1 mM InsPS3 into a photoreceptor's light-sensitive lobe caused an initial elevation of cytosolic free calcium ion concentration (Cai) and a depolarization lasting only 1-2 s. A period of densensitization followed, during which injections of InsPS3 were ineffective. As sensitivity recovered, oscillations of membrane potential began, continuing for many minutes with a frequency of 0.07-0.3 Hz. The activity of InsPS3 probably results from the D-stereoisomer, since L-InsP3 was much less effective than InsP3. Injections of 1 mM InsP3 caused an initial depolarization and a period of densensitization similar to that caused by 1 mM InsPS3, but no sustained oscillations of membrane potential. The initial response to InsPS3 or InsP3 may therefore be terminated by densensitization, rather than by metabolism. Metabolism of InsP3 may prevent oscillations of membrane potential after sensitivity has recovered. The InsPS3-induced oscillations of membrane potential accompanied oscillations of Cai and were abolished by injection of ethyleneglycol-bis (beta-aminoethyl ether)-N,N'-tetraacetic acid. Removal of extracellular calcium reduced the frequency of oscillation but not its amplitude. Under voltage clamp, oscillations of inward current were observed. These results indicate that periodic bursts of calcium release underly the oscillations of membrane potential. After each burst, the sensitivity of the cell to injected InsP3 was greatly reduced, recovering during the interburst interval. The oscillations may, therefore, result in part from a periodic variation in sensitivity to a constant concentration of InsPS3. Prior injection of calcium inhibited depolarization by InsPS3, suggesting that feedback inhibition of InsPS3-induced calcium release by elevated Cai may mediate desensitization between bursts and after injections of InsPS3.     

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 deterministic approach to survival statistics

Survival functions of the form p(t) = exp[–(t)], > 0 can be generated by deterministic nonlinear, asymptotically stable (chaotic) dynamical systems. These systems thus provide an alternative to stochastic interpretations of failure time data. We use this approach to analyze cancer patient survival statistics. In this manner we are able to obtain fresh insights into the implications of negative and positive clinical trials.