From stochasticity to determinism in the collective dynamics of diffusively coupled cells

Role of time delay on intracellular calcium dynamics driven by non-Gaussian noises

Effect of time delay (τ) on intracellular calcium dynamics with non-Gaussian noises in transmission processes of intracellular Ca²⁺ is studied by means of second-order stochastic Runge-Kutta type algorithm. By simulating and analyzing time series, normalized autocorrelation function, and characteristic correlation time of cytosolic and calcium store’s Ca²⁺ concentration, the results exhibit: (i) intracellular calcium dynamics’s time coherence disappears and stability strengthens as τ → 0.1s; (ii) for the case of τ < 0.1s, the normalized autocorrelation functions of cytosolic and calcium store’s Ca²⁺ concentration show damped motion when τ is very short, but they trend to a level line as τ → 0.1s, and for the case of τ > 0.1s, they show different variation as τ increases, the former changes from underdamped motion to a level line, but the latter changes from damped motion to underdamped motion; and (iii) at the moderate value of time delay, reverse resonance occurs both in cytosol and calcium store.

Fundamental properties of Ca2+ signals

BACKGROUND

Ca2+ is a ubiquitous and versatile second messenger that transmits information through changes of the cytosolic Ca2+ concentration. Recent investigations changed basic ideas on the dynamic character of Ca2+ signals and challenge traditional ideas on information transmission.

SCOPE OF REVIEW

We present recent findings on key characteristics of the cytosolic Ca2+ dynamics and theoretical concepts that explain the wide range of experimentally observed Ca2+ signals. Further, we relate properties of the dynamical regulation of the cytosolic Ca2+ concentration to ideas about information transmission by stochastic signals.

MAJOR CONCLUSIONS

We demonstrate the importance of the hierarchal arrangement of Ca2+ release sites on the emergence of cellular Ca2+ spikes. Stochastic Ca2+ signals are functionally robust and adaptive to changing environmental conditions. Fluctuations of interspike intervals (ISIs) and the moment relation derived from ISI distributions contain information on the channel cluster open probability and on pathway properties.

GENERAL SIGNIFICANCE

Robust and reliable signal transduction pathways that entail Ca2+ dynamics are essential for eukaryotic organisms. Moreover, we expect that the design of a stochastic mechanism which provides robustness and adaptivity will be found also in other biological systems. Ca2+ dynamics demonstrate that the fluctuations of cellular signals contain information on molecular behavior. This article is part of a Special Issue entitled Biochemical, biophysical and genetic approaches to intracellular calcium signaling.

Stochastic fire-diffuse-fire model with realistic cluster dynamics

Living organisms use waves that propagate through excitable media to transport information. Ca2+ waves are a paradigmatic example of this type of processes. A large hierarchy of Ca2+ signals that range from localized release events to global waves has been observed in Xenopus laevis oocytes. In these cells, Ca2+ release occurs trough inositol 1,4,5-trisphosphate receptors (IP3Rs) which are organized in clusters of channels located on the membrane of the endoplasmic reticulum. In this article we construct a stochastic model for a cluster of IP3R 's that replicates the experimental observations reported in [D. Fraiman, Biophys. J. 90, 3897 (2006)]. We then couple this phenomenological cluster model with a reaction-diffusion equation, so as to have a discrete stochastic model for calcium dynamics. The model we propose describes the transition regimes between isolated release and steadily propagating waves as the IP3 concentration is increased.

Importance of cell variability for calcium signaling in rat airway myocytes

Calcium signaling controls several essential physiological functions in different cell types. Hence, it is not surprising that different aspects of Ca(2+) dynamics are in the focus of in-depth and extensive investigations. Efforts concentrate on the development of proper theoretical models that would provide a unified description of Ca(2+) signaling. Remarkably, experimentally recorded Ca(2+) signals exhibit a rather large diversity, which can be observed irrespective of the cell type, measuring techniques, or the nature of the signal. Our goal in the present study therefore is to present a theoretical explanation for the variability observed in experiments, whereby we focus on caffeine-induced Ca(2+) responses in isolated airway myocytes. By employing a stochastic model, we first test whether the observed variability can be attributed to intrinsic fluctuations that are a common feature of biochemical reactions that govern Ca(2+) signalization. We find that stochastic effects, within ranges that correspond to actual conditions in the cell, are far too modest to explain the large diversity observed in experimental data. Foremost, we reveal that only cell variability in theoretical modeling can appropriately describe the observed diversity in single-cell responses.

Prevalence of stochasticity in experimentally observed responses of pancreatic acinar cells to acetylcholine

Calcium ions play an important role in both intra- and intercellular signaling. In pancreatic acinar cells intracellular Ca(2+) regulates exocytotic secretion and fluid secretion. In this paper we study the typical experimental traces of Ca(2+) responses in pancreatic acinar cells obtained in response to the physiological agonist acetylcholine. To determine whether they are stochastic or deterministic in nature, we analyze the traces with methods of nonlinear time series analysis. In particular, by performing surrogate data tests and employing a determinism test for short time series, we show that the responses of pancreatic acinar cells to acetylcholine are stochastic with only faintly expressed deterministic features. Presented results thus corroborate the notion that mathematical models should take stochasticity explicitly into account when describing intra- and intercellular processes, and that indeed further efforts should be directed toward this subject.

Effectors of the frequency of calcium oscillations in HEK-293 cells: wavelet analysis and a computer model

Oscillations of the intracellular concentration of Ca(2+) in cultured HEK-293 cells, which heterologously expressed the calcium-sensing receptor, were recorded with the fluorophore Fura-2 using fluorescence microscopy. HEK-293 cells are extremely sensitive to small perturbations in extracellular calcium concentrations. Resting cells were attached to cover slips and perifused with saline solution containing physiologically relevant extracellular Ca(2+) concentrations in the range 0.5-5 mM. Acquired digitized images of the cells showed oscillatory fluctuations in the intracellular Ca(2+) concentration over the time course, and were processed as a function of the change in Fura-2 excitation ratio and frequency at 12-37 degrees C. Newly developed data processing techniques with wavelet analysis were used to estimate the frequency at which the rectified sinusoidal oscillations occurred; we estimated ~4 min(-1) under normal conditions. Temperature variations revealed an Arrhenius relationship in oscillation frequency. A critical Ca(2+) concentration of ~2 mM was estimated, below which oscillations did not occur. These data were used to develop a kinetic model of the system that was simulated using Mathematica; kinetic parameter values were adjusted to match the experimentally observed oscillations of intracellular Ca(2+) concentration as a function of extracellular Ca(2+) concentration, and temperature; and from these, limit cycles were obtained and control coefficients were estimated for all parameters.

Establishing the stochastic nature of intracellular calcium oscillations from experimental data

Calcium has been established as a key messenger in both intra- and intercellular signaling. Experimentally observed intracellular calcium responses to different agonists show a variety of behaviors from simple spiking to complex oscillatory regimes. Here we study typical experimental traces of calcium oscillations in hepatocytes obtained in response to phenylephrine and ATP. The traces were analyzed with methods of nonlinear time series analysis in order to determine the stochastic/deterministic nature of the intracellular calcium oscillations. Despite the fact that the oscillations appear, visually, to be deterministic yet perturbed by noise, our analyses provide strong evidence that the measured calcium traces in hepatocytes are prevalently of stochastic nature. In particular, bursting calcium oscillations are temporally correlated Gaussian series distorted by a monotonic, instantaneous, time-independent function, whilst the spiking behavior appears to have a dynamical nonlinear component whereby the overall determinism level is still low. The biological importance of this finding is discussed in relation to the mechanisms incorporated in mathematical models as well as the role of stochasticity and determinism at cellular and tissue levels which resemble typical statistical and thermodynamic effects in physics.

Spatial coherence resonance in excitable biochemical media induced by internal noise

We show that in a spatially extended excitable medium, presently modelled with diffusively coupled FitzHugh-Nagumo neurons, internal stochasticity is able to extract a characteristic spatial frequency of waves on the spatial grid. Internal noise is introduced via a stochastic simulation method and is the only agent acting on the system. Remarkably, the spatial periodicity is best pronounced at an intermediate level of internal stochasticity. Thus, the reported phenomenon is an observation of internal noise spatial coherence resonance in excitable biochemical media.