Marko Marhl

Detecting chaos from a time series

The chaotic behaviour of a driven resonant circuit is studied directly from the experimental data. We use basic nonlinear time series analysis methods that are appropriate for undergraduate courses. Mutual information and false nearest neighbours are explained in detail, and used to obtain the best possible attractor reconstruction. For the reconstructed attractor, a determinism test is performed and the largest Lyapunov exponent is calculated. We show that the largest Lyapunov exponent is positive, which is a strong indicator for the chaotic behaviour of the system. To help the reader reproduce our results and to facilitate further applications on other experimental systems, we provide user-friendly programs with graphical interface for each implemented method on our Web page.

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.

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.

Functional connectivity in islets of Langerhans from mouse pancreas tissue slices

We propose a network representation of electrically coupled beta cells in islets of Langerhans. Beta cells are functionally connected on the basis of correlations between calcium dynamics of individual cells, obtained by means of confocal laser-scanning calcium imaging in islets from acute mouse pancreas tissue slices. Obtained functional networks are analyzed in the light of known structural and physiological properties of islets. Focusing on the temporal evolution of the network under stimulation with glucose, we show that the dynamics are more correlated under stimulation than under non-stimulated conditions and that the highest overall correlation, largely independent of Euclidean distances between cells, is observed in the activation and deactivation phases when cells are driven by the external stimulus. Moreover, we find that the range of interactions in networks during activity shows a clear dependence on the Euclidean distance, lending support to previous observations that beta cells are synchronized via calcium waves spreading throughout islets. Most interestingly, the functional connectivity patterns between beta cells exhibit small-world properties, suggesting that beta cells do not form a homogeneous geometric network but are connected in a functionally more efficient way. Presented results provide support for the existing knowledge of beta cell physiology from a network perspective and shed important new light on the functional organization of beta cell syncitia whose structural topology is probably not as trivial as believed so far.

Different types of bursting calcium oscillations in non-excitable cells

Abstract In the paper different types of bursting Ca 2+ oscillations are presented. We analyse bursting behaviour in four recent mathematical models for Ca 2+ oscillations in non-excitable cells. Separately, regular, quasi-periodic, and chaotic bursting Ca 2+ oscillations are classified into several subtypes. The classification is based on the dynamics of separated fast and slow subsystems, the so-called fast–slow burster analysis. For regular bursting Ca 2+ oscillations two types of bursting are specified: Point–Point and Point–Cycle bursting. In particular, the slow passage effect, important for the Hopf–Hopf and SubHopf–SubHopf bursting subtypes, is explained by local divergence calculated for the fast subsystem. Quasi-periodic bursting Ca 2+ oscillations can be found in only one of the four studied mathematical models and appear via a homoclinic bifurcation with a homoclinic torus structure. For chaotic bursting Ca 2+ oscillations, we found that bursting patterns resulting from the period doubling root to chaos considerably differ from those appearing via intermittency and have to be treated separately. The analysis and classification of different types of bursting Ca 2+ oscillations provides better insight into mechanisms of complex intra- and intercellular Ca 2+ signalling. This improves our understanding of several important biological phenomena in cellular signalling like complex frequency–amplitude signal encoding and synchronisation of intercellular signal transduction between coupled cells in tissue.

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.

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.

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].

Sensitivity and flexibility of regular and chaotic calcium oscillations.

Sensitivity and flexibility are important properties of biological systems. These properties are here investigated for intracellular calcium oscillations. For a particular model, we comparatively investigate sensitivity and flexibility of regular and chaotic Ca(2+) oscillations. For this model, we obtain two main results. First, sensitivity of the model system to parameter shifting does not depend on the complexity of Ca(2+) oscillations. We observe, however, that both regular and chaotic Ca(2+) oscillations are highly sensitive in regions close to bifurcation points. Second, also flexibility of Ca(2+) oscillations does not significantly depend on the type of Ca(2+) oscillations. Our results show that regular as well as chaotic Ca(2+) oscillations in the studied model are highly flexible in regimes with weak dissipation. Both results are discussed in the sense of possible biological importance.

Selective regulation of cellular processes via protein cascades acting as band‐pass filters for time‐limited oscillations

We show by mathematical modelling that a two‐level protein cascade can act as a band‐pass filter for time‐limited oscillations. The band‐pass filters are then combined into a network of three‐level signalling cascades that by filtering the frequency of time‐limited oscillations selectively switches cellular processes on and off. The physiological relevance for the selective regulation of cellular processes is demonstrated for the case of regulation by time‐limited calcium oscillations.