Hierarchical organization of calcium signals in hepatocytes: from experiments to models

Regulatory disturbances in the dynamical signaling systems of C a 2 + and NO in fibroblasts cause fibrotic disorders

Studying the calcium dynamics within a fibroblast cell individually has provided only a restricted understanding of its functions. However, research efforts focusing on systems biology approaches for such investigations have been largely neglected by researchers until now. Fibroblast cells rely on signaling from calcium ( C a 2 + ) and nitric oxide (NO) to maintain their physiological functions and structural stability. Various studies have demonstrated the correlation between NO and the control of C a 2 + dynamics in cells. However, there is currently no existing model to assess the disruptions caused by various factors in regulatory dynamics, potentially resulting in diverse fibrotic disorders. A mathematical model has been developed to investigate the effects of changes in parameters such as buffer, receptor, sarcoplasmic endoplasmic reticulum C a 2 + -ATPase (SERCA) pump, and source influx on the regulation and dysregulation of spatiotemporal calcium and NO dynamics in fibroblast cells. This model is based on a system of reaction-diffusion equations, and numerical simulations are conducted using the finite element method. Disturbances in key processes related to calcium and nitric oxide, including source influx, buffer mechanism, SERCA pump, and inositol trisphosphate ( I P 3 ) receptor, may contribute to deregulation in the calcium and NO dynamics within fibroblasts. The findings also provide new insights into the extent and severity of disorders resulting from alterations in various parameters, potentially leading to deregulation and the development of fibrotic disease.

Numerical Analysis of Compound Biochemical Calcium Oscillations Process in Hepatocyte Cells

The hepatocyte cells regulate the wide range of liver function by moderating cellular activities such as lipid, protein metabolism, carbohydrate, and interact with other cells for proliferation and maintenance. In hepatocyte cells, the concentration of calcium uptake is quite extensive from various agonists such as activeG α ${G_\alpha}$ subunit, active phospholipase C, free calcium in the cytosol, and endoplasmic reticulum. The overproduction and degradation of calcium signals can cause homeostasis, liver inflammation, and liver diseases. The spatiotemporal behavior of calcium oscillation reveals the physiological role of these cellular entities in understanding the process of production and degradation. No computational attempt has been registered to date on the compound calcium regulation of these cellular entities including the memory of cells. Hence, the authors proposed a fractional order compartmental model that systematically simulates the exchange of calcium intake in cellular entities. The nonlinear equations of the rate of changes in the activeG α ${G_\alpha}$ subunit, active phospholipase C, free calcium in the cytosol, and endoplasmic reticulum are coupled to form a nonlinear fractional order initial value problem. The existence and uniqueness, stability analysis of the model is performed that validate the theoretical results and explore the dynamic behaviour of calcium oscillation in each compartment.

Pacing intracellular Ca2+ signals in exocrine acinar cells

An increase in intracellular [Ca2+ ] in exocrine acinar cells resident in the salivary glands or pancreas is a fundamental event that drives fluid secretion and exocytosis of proteins. Stimulation with secretagogues initiates Ca2+ signals with precise spatiotemporal properties thought to be important for driving physiological output. Both in vitro, in acutely isolated acini, and in vivo, in animals expressing genetically encoded indicators, individual cells appear specialized to initiate Ca2+ signals upon stimulation. Furthermore, these signals appear to spread to neighbouring cells. These properties are present in the absence of a conventional pacemaker mechanism dependent on the cyclical activation of Ca2+ -dependent or Ca2+ -conducting plasma membrane ion channels. In this article, we propose a model for 'pacing' intracellular Ca2+ signals in acinar cells based on the enhanced sensitivity of a subpopulation of individual cells and the intercellular diffusion through gap junctions of inositol 1,4,5-trisphosphate and Ca2+ to neighbouring cells.

Impact of Interdependent Ca2+ and IP3 Dynamics On ATP Regulation in A Fibroblast Model

The vital participation of Ca2+ in human organ functions such as muscular contractions, heartbeat, brain functionality, skeletal activity, etc, motivated the scientists to thoroughly research the mechanisms of calcium (Ca2+) signalling in distinct human cells. Ca2+, inositol triphosphate (IP3), and adenosine triphosphate (ATP) play important roles in cell signaling and physiological processes. ATP and its derivatives are hypothesized to be important in the pathogenic process that leads to fibrotic illnesses like fibrosis. Fluctuations in Ca2+ and IP3 in a fibroblast cell influence ATP production. To date, no evidence of coupled Ca2+ and IP3 mechanics regulating ATP generation in a fibroblast cell during fibrotic disease has been found. The current work suggests an integrated mechanism for Ca2+ and IP3 dynamics in a fibroblast cell that regulates ATP generation. Simulation has been carried out using the finite element approach. The mechanics of interdependent systems findings vary dramatically from the results of basic independent system mechanics and give fresh information about the two systems' activities. The numerical results provide new insights into the impacts of disturbances in source influx, the serca pump, and buffers on interdependent Ca2+ and IP3 dynamics and ATP synthesis in a fibroblast cell. According to the findings of this study, fibrotic disorders cannot be attributed solely to disruptions in the processes of calcium signaling mechanics but also to disruptions in IP3 regulation mechanisms affecting the regulation of calcium in the fibroblast cell and ATP release.

Study of disorders in regulatory spatiotemporal neurodynamics of calcium and nitric oxide

Experimental studies have reported the dependence of nitric oxide (NO) on the regulation of neuronal calcium ([Ca2+]) dynamics in neurons. But, there is no model available to estimate the disorders caused by various parameters in their regulatory dynamics leading to various neuronal disorders. A mathematical model to analyze the impacts due to alterations in various parameters like buffer, ryanodine receptor, serca pump, source influx, etc. leading to regulation and dysregulation of the spatiotemporal calcium and NO dynamics in neuron cells is constructed using a system of reaction-diffusion equations. The numerical simulation is performed with the finite element approach. The disturbances in the different constitutive processes of [Ca2+] and nitric oxide including source influx, buffer mechanism, ryanodine receptor, serca pump, IP3 receptor, etc. can be responsible for the dysregulation in the [Ca2+] and NO dynamics in neurons. Also, the results reveal novel information about the magnitude and intensity of disorders in response to a range of alterations in various parameters of this neuronal dynamics, which can cause dysregulation leading to neuronal diseases like Parkinson's, cerebral ischemia, trauma, etc.

Calcium signalling in hepatic metabolism: Health and diseases

The flexibility between the wide array of hepatic functions relies on calcium (Ca2+) signalling. Indeed, Ca2+ is implicated in the control of many intracellular functions as well as intercellular communication. Thus, hepatocytes adapt their Ca2+ signalling depending on their nutritional and hormonal environment, leading to opposite cellular functions, such as glucose storage or synthesis. Interestingly, hepatic metabolic diseases, such as obesity, type 2 diabetes and non-alcoholic fatty liver diseases, are associated with impaired Ca2+ signalling. Here, we present the hepatocytes' toolkit for Ca2+ signalling, complete with regulation systems and signalling pathways activated by nutrients and hormones. We further discuss the current knowledge on the molecular mechanisms leading to alterations of Ca2+ signalling in hepatic metabolic diseases, and review the literature on the clinical impact of Ca2+-targeting therapeutics.

Simulation of biochemical dynamics of C a 2 + and P L C in fibroblast cell

Calcium dynamics is not only responsible for maintaining the framework and functions of the cell but also plays a role in the dynamics of other biochemical systems in the cell. Phospholipase C-γ l (P L C ) has a crucial role in the function of fibroblast cells. Experiments have shown thatP L C andC a 2 + have interdependent dynamics in fibroblast cells. However, no reaction-diffusion model exists for the two-way feedback system dynamics ofC a 2 + andP L C in fibroblasts till date. The computational model is designed to investigate the impact of variations in several processes, such as theS E R C A pump, buffer process, source inflow, etc., on the system dynamics ofC a 2 + andP L C in fibroblast cells. The computational findings are obtained using finite element techniques, and the consequences of dysregulation in various processes on the spatiotemporal calcium andP L C dynamics in fibroblasts are investigated. The results lead to the conclusion that the effects of buffer, source influx, diffusion, andS E R C A pump can cause fluctuations in the dynamics ofC a 2 + andP L C in fibroblasts. Disruptions in these constitutive processes can result in changes in the dynamics of calcium andP L C . Thus, the current model provides new/novel information regarding the precise dysregulatory constitutive systems that regulate calcium andP L C kinetics, such as source inflow, diffusion,S E R C A , and buffer, can be responsible for excessive calcium andP L C concentrations leading to fibrotic illnesses such as cancer and fibrosis.

Effect of disturbances in neuronal calcium and IP3 dynamics on β-amyloid production and degradation

Overproduction and accumulation of β-amyloid and its improper clearance can cause neurotoxicity leading to Alzheimer's disease. The production and degradation of β-amyloid depend on the calcium ([Ca2+]) and IP3 dynamics in the nerve cells. Thus, there is a need to understand the impacts of disturbances in the processes of [Ca2+] and IP3 dynamics on β-amyloid production and its degradation. Here, a model is proposed to investigate the role of [Ca2+] and IP3 dynamics on β-amyloid production and degradation. The problem is formulated in terms of the initial boundary value problem involving the system of two reaction-diffusion equations respectively for [Ca2+] and IP3 in the nerve cell. The solution is obtained by employing the Finite element approach. The numerical results are used to analyze the impact of various mechanisms of calcium and IP3 dynamics on β-amyloid production and degradation in a neuron cell. The results indicate that disturbances in any of the constitutive processes of interdependent calcium and IP3 dynamics like source influx, buffering, serca pump, and IP3 dynamics, etc. can cause dynamic changes in β-amyloid production and degradation, which in turn can be the cause of neurotoxicity and neuronal disorders like Alzheimer's disease. Thus, the relationships obtained by the proposed model among various mechanisms can be useful in addressing the challenges of identifying specific constitutive processes causing neuronal disorders like Alzheimer's disease, etc., and developing the framework for their diagnosis and treatment.

A Tale of two receptors

Calcium (Ca²⁺) oscillations in hepatocytes have a wide dynamic range. In particular, recent experimental evidence shows that agonist stimulation of the P2Y family of receptors leads to qualitatively diverse Ca²⁺ oscillations. We present a new model of Ca²⁺ oscillations in hepatocytes based on these experiments to investigate the mechanisms controlling P2Y-activated Ca²⁺ oscillations. The model accounts for Ca²⁺ regulation of the IP₃ receptor (IP₃R), the positive feedback from Ca²⁺ on phospholipase C (PLC) and the P2Y receptor phosphorylation by protein kinase C (PKC). Furthermore, PKC is shown to control multiple cellular substrates. Utilising the model, we suggest the activity and intensity of PLC and PKC necessary to explain the qualitatively diverse Ca²⁺ oscillations in response to P2Y receptor activation.

Calcium Oscillatory Patterns and Oocyte Activation During Fertilization: a Possible Mechanism for Total Fertilization Failure (TFF) in Human In Vitro Fertilization?

This paper reviews the effects of calcium oscillatory patterns in oocytes and early embryo development. Total fertilization failure (TFF) is the failure of fertilization in all oocytes in a human IVF cycle, even after treatment with intracytoplasmic sperm injection (ICSI). It is not well understood and currently attributed to oocyte activation deficiency. Calcium signaling is important in oocyte activation events. Calcium oscillations, in particular, have been reported in animal and human oocytes after fertilization. Abnormal calcium oscillations after fertilization may be the principal mechanism for TFF. While studies also establish strong associations between abnormal calcium oscillatory patterns and suboptimal developmental outcomes, critical basic parameters and their mechanism of action have yet to be identified. Empirical use of artificial oocyte activation (AOA) methods has shown initial success in helping patients overcome TFF. The AOA methods attempt to raise calcium levels after fertilization, but the efficacy and safety of these AOA methods are still in early stages of addressing TFF. Additional information about calcium oscillatory patterns and the effects of AOA in human ART may allow the prevention of TFF or allow treatment of TFF patients effectively and safely.

Gap junctions coordinate the propagation of glycogenolysis induced by norepinephrine in the pineal gland

Chemical and electrical synapses are the two major communication systems that permit cell-to-cell communication within the nervous system. Although most studies are focused on chemical synapses (glutamate, γ-aminobutyric acid, and other neurotransmitters), clearly both types of synapses interact and cooperate to allow the coordination of several cell functions within the nervous system. The pineal gland has limited independent axonal innervation and not every cell has access to nerve terminals. Thus, additional communication systems, such as gap junctions, have been postulated to coordinate metabolism and signaling. Using acutely isolated glands and dissociated cells, we found that gap junctions spread glycogenolytic signals from cells containing adrenoreceptors to the entire gland lacking these receptors. Our data using glycogen and lactate quantification, electrical stimulation, and high-performance liquid chromatography with electrochemical detection, demonstrate that gap junctional communication between cells of the rat pineal gland allows cell-to-cell propagation of norepinephrine-induced signal that promotes glycogenolysis throughout the entire gland. Thus, the interplay of both synapses is essential for coordinating glycogen metabolism and lactate production in the pineal gland.

The expression pattern of calcium signaling-related genes during smoltification of Salmo salar in productive conditions

Variations in the mRNA expression of hepatic and muscle genes that are related to calcium signaling were analyzed by real-time qPCR in farmed Atlantic salmon (Salmo salar L. 1758) to determine changes in expression between parr and smolt stages. These organs were selected due to their close relationship with calcium signaling and metabolism (e.g., glycolysis, oxidative phosphorylation, muscle contraction). Differential expression between smolt and parr specimens and between organs was observed. Compared to parr specimens, smolts exhibited upregulated expression of the calcitonin receptor precursor, calcitonin receptor, calcitonin isoform, parathyroid hormone, and calmodulin in the liver. This pattern was inverse in muscle, with the exception of calmodulin, which was significantly upregulated in smolts compared to parr. Additionally, plasma calcium was decreased in the smolt condition. This study is the first to characterize the expression pattern of calcium signaling-related genes in the liver and muscle of parr and smolt S. salar. However, further functional studies are required to obtain a wider understanding about the physiological changes that accompany the productive conditions during smoltification.

On the phase space structure of IP3 induced Ca2+ signalling and concepts for predictive modeling

The correspondence between mathematical structures and experimental systems is the basis of the generalizability of results found with specific systems and is the basis of the predictive power of theoretical physics. While physicists have confidence in this correspondence, it is less recognized in cellular biophysics. On the one hand, the complex organization of cellular dynamics involving a plethora of interacting molecules and the basic observation of cell variability seem to question its possibility. The practical difficulties of deriving the equations describing cellular behaviour from first principles support these doubts. On the other hand, ignoring such a correspondence would severely limit the possibility of predictive quantitative theory in biophysics. Additionally, the existence of functional modules (like pathways) across cell types suggests also the existence of mathematical structures with comparable universality. Only a few cellular systems have been sufficiently investigated in a variety of cell types to follow up these basic questions. IP₃ induced Ca²⁺signalling is one of them, and the mathematical structure corresponding to it is subject of ongoing discussion. We review the system's general properties observed in a variety of cell types. They are captured by a reaction diffusion system. We discuss the phase space structure of its local dynamics. The spiking regime corresponds to noisy excitability. Models focussing on different aspects can be derived starting from this phase space structure. We discuss how the initial assumptions on the set of stochastic variables and phase space structure shape the predictions of parameter dependencies of the mathematical models resulting from the derivation.

Computational properties of mitochondria in T cell activation and fate

In this article, we review how mitochondrial Ca²⁺ transport (mitochondrial Ca²⁺ uptake and Na⁺/Ca²⁺ exchange) is involved in T cell biology, including activation and differentiation through shaping cellular Ca²⁺ signals. Based on recent observations, we propose that the Ca²⁺ crosstalk between mitochondria, endoplasmic reticulum and cytoplasm may form a proportional–integral–derivative (PID) controller. This PID mechanism (which is well known in engineering) could be responsible for computing cellular decisions. In addition, we point out the importance of analogue and digital signal processing in T cell life and implication of mitochondrial Ca²⁺ transport in this process.

FCS experiments to quantify Ca2+ diffusion and its interaction with buffers

Ca²⁺ signals are ubiquitous. One of the key factors for their versatility is the variety of spatio-temporal distributions that the cytosolic Ca²⁺ can display. In most cell types Ca²⁺ signals not only depend on Ca²⁺ entry from the extracellular medium but also on Ca²⁺ release from internal stores, a process which is in turn regulated by cytosolic Ca²⁺ itself. The rate at which Ca²⁺ is transported, the fraction that is trapped by intracellular buffers, and with what kinetics are thus key features that affect the time and spatial range of action of Ca²⁺ signals. The quantification of Ca²⁺ diffusion in intact cells is quite challenging because the transport rates that can be inferred using optical techniques are intricately related to the interaction of Ca²⁺ with the dye that is used for its observation and with the cellular buffers. In this paper, we introduce an approach that uses Fluorescence Correlation Spectroscopy (FCS) experiments performed at different conditions that in principle allows the quantification of Ca²⁺ diffusion and of its reaction rates with unobservable (non-fluorescent) Ca²⁺ buffers. To this end, we develop the necessary theory to interpret the experimental results and then apply it to FCS experiments performed in a set of solutions containing Ca²⁺, a single wavelength Ca²⁺ dye, and a non-fluorescent Ca²⁺ buffer. We show that a judicious choice of the experimental conditions and an adequate interpretation of the fitting parameters can be combined to extract information on the free diffusion coefficient of Ca²⁺ and of some of the properties of the unobservable buffer. We think that this approach can be applied to other situations, particularly to experiments performed in intact cells.

Characterization of the effect of the mitochondrial protein Hint2 on intracellular Ca(2+) dynamics

Hint2, one of the five members of the superfamily of the histidine triad AMP-lysine hydrolase proteins, is expressed in mitochondria of various cell types. In human adrenocarcinoma cells, Hint2 modulates Ca(2+) handling by mitochondria. As Hint2 is highly expressed in hepatocytes, we investigated if this protein affects Ca(2+) dynamics in this cell type. We found that in hepatocytes isolated from Hint2(-/-) mice, the frequency of Ca(2+) oscillations induced by 1 μM noradrenaline was 150% higher than in the wild-type. Using spectrophotometry, we analyzed the rates of Ca(2+) pumping in suspensions of mitochondria prepared from hepatocytes of either wild-type or Hint2(-/-) mice; we found that Hint2 accelerates Ca(2+) pumping into mitochondria. We then resorted to computational modeling to elucidate the possible molecular target of Hint2 that could explain both observations. On the basis of a detailed model for mitochondrial metabolism proposed in another study, we identified the respiratory chain as the most probable target of Hint2. We then used the model to predict that the absence of Hint2 leads to a premature opening of the mitochondrial permeability transition pore in response to repetitive additions of Ca(2+) in suspensions of mitochondria. This prediction was then confirmed experimentally.

Modeling the intracellular organization of calcium signaling

Calcium (Ca²⁺) is a key signaling ion that plays a fundamental role in many cellular processes in most types of tissues and organisms. The versatility of this signaling pathway is remarkable. Depending on the cell type and the stimulus, intracellular Ca²⁺ increases can last over different periods, as short spikes or more sustained signals. From a spatial point of view, they can be localized or invade the whole cell. Such a richness of behaviors is possible thanks to numerous exchange processes with the external medium or internal Ca²⁺ pools, mainly the endoplasmic or sarcoplasmic reticulum and mitochondria. These fluxes are also highly regulated. In order to get an accurate description of the spatiotemporal organization of Ca²⁺ signaling, it is useful to resort to modeling. Thus, each flux can be described by an appropriate kinetic expression. Ca²⁺ dynamics in a given cell type can then be simulated by a modular approach, consisting of the assembly of computational descriptions of the appropriate fluxes and regulations. Modeling can also be used to get insight into the mechanisms of decoding of the Ca²⁺ signals responsible for cellular responses. Cells can use frequency or amplitude coding, as well as take profit of Ca²⁺ oscillations to increase their sensitivity to small average Ca²⁺ increases.

Calcium signaling in the liver

Intracellular free Ca(2+) ([Ca(2+)]i) is a highly versatile second messenger that regulates a wide range of functions in every type of cell and tissue. To achieve this versatility, the Ca(2+) signaling system operates in a variety of ways to regulate cellular processes that function over a wide dynamic range. This is particularly well exemplified for Ca(2+) signals in the liver, which modulate diverse and specialized functions such as bile secretion, glucose metabolism, cell proliferation, and apoptosis. These Ca(2+) signals are organized to control distinct cellular processes through tight spatial and temporal coordination of [Ca(2+)]i signals, both within and between cells. This article will review the machinery responsible for the formation of Ca(2+) signals in the liver, the types of subcellular, cellular, and intercellular signals that occur, the physiological role of Ca(2+) signaling in the liver, and the role of Ca(2+) signaling in liver disease.

Cell-type-specific modelling of intracellular calcium signalling: a urothelial cell model

Calcium signalling plays a central role in regulating a wide variety of cell processes. A number of calcium signalling models exist in the literature that are capable of reproducing a variety of experimentally observed calcium transients. These models have been used to examine in more detail the mechanisms underlying calcium transients, but very rarely has a model been directly linked to a particular cell type and experimentally verified. It is important to show that this can be achieved within the general theoretical framework adopted by these models. Here, we develop a framework designed specifically for modelling cytosolic calcium transients in urothelial cells. Where possible, we draw upon existing calcium signalling models, integrating descriptions of components known to be important in this cell type from a number of studies in the literature. We then add descriptions of several additional pathways that play a specific role in urothelial cell signalling, including an explicit ionic influx term and an active pumping mechanism that drives the cytosolic calcium concentration to a target equilibrium. The resulting one-pool model of endoplasmic reticulum (ER)-dependent calcium signalling relates the cytosolic, extracellular and ER calcium concentrations and can generate a wide range of calcium transients, including spikes, bursts, oscillations and sustained elevations in the cytosolic calcium concentration. Using single-variate robustness and multivariate sensitivity analyses, we quantify how varying each of the parameters of the model leads to changes in key features of the calcium transient, such as initial peak amplitude and the frequency of bursting or spiking, and in the transitions between bursting- and plateau-dominated modes. We also show that, novel to our urothelial cell model, the ionic and purinergic P2Y pathways make distinct contributions to the calcium transient. We then validate the model using human bladder epithelial cells grown in monolayer cell culture and show that the model robustly captures the key features of the experimental data in a way that is not possible using more generic calcium models from the literature.

Hierarchic stochastic modelling applied to intracellular Ca(2+) signals

Important biological processes like cell signalling and gene expression have noisy components and are very complex at the same time. Mathematical analysis of such systems has often been limited to the study of isolated subsystems, or approximations are used that are difficult to justify. Here we extend a recently published method (Thurley and Falcke, PNAS 2011) which is formulated in observable system configurations instead of molecular transitions. This reduces the number of system states by several orders of magnitude and avoids fitting of kinetic parameters. The method is applied to Ca(2+) signalling. Ca(2+) is a ubiquitous second messenger transmitting information by stochastic sequences of concentration spikes, which arise by coupling of subcellular Ca(2+) release events (puffs). We derive analytical expressions for a mechanistic Ca(2+) model, based on recent data from live cell imaging, and calculate Ca(2+) spike statistics in dependence on cellular parameters like stimulus strength or number of Ca(2+) channels. The new approach substantiates a generic Ca(2+) model, which is a very convenient way to simulate Ca(2+) spike sequences with correct spiking statistics.

Intercellular Ca(2+) waves: mechanisms and function

Intercellular calcium (Ca(2+)) waves (ICWs) represent the propagation of increases in intracellular Ca(2+) through a syncytium of cells and appear to be a fundamental mechanism for coordinating multicellular responses. ICWs occur in a wide diversity of cells and have been extensively studied in vitro. More recent studies focus on ICWs in vivo. ICWs are triggered by a variety of stimuli and involve the release of Ca(2+) from internal stores. The propagation of ICWs predominately involves cell communication with internal messengers moving via gap junctions or extracellular messengers mediating paracrine signaling. ICWs appear to be important in both normal physiology as well as pathophysiological processes in a variety of organs and tissues including brain, liver, retina, cochlea, and vascular tissue. We review here the mechanisms of initiation and propagation of ICWs, the key intra- and extracellular messengers (inositol 1,4,5-trisphosphate and ATP) mediating ICWs, and the proposed physiological functions of ICWs.

Calcium signalling and liver regeneration

After partial hepatectomy (PH) the initial mass of the organ is restored through a complex network of cellular interactions that orchestrate both proliferative and hepatoprotective signalling cascades. Among agonists involved in this network many of them drive Ca(2+) movements. During liver regeneration in the rat, hepatocyte cytosolic Ca(2+) signalling has been shown on the one hand to be deeply remodelled and on the other hand to enhance progression of hepatocytes through the cell cycle. Mechanisms through which cytosolic Ca(2+) signals impact on hepatocyte cell cycle early after PH are not completely understood, but at least they include regulation of immediate early gene transcription and ERK and CREB phosphorylation. In addition to cytosolic Ca(2+), there is also evidence that mitochondrial Ca(2+) and also nuclear Ca(2+) may be critical for the regulation of liver regeneration. Finally, Ca(2+) movements in hepatocytes, and possibly in other liver cells, not only impact hepatocyte progression in the cell cycle but more generally may regulate cellular homeostasis after PH.

BIFURCATION ANALYSIS OF CALCIUM OSCILLATIONS: TIME-SCALE SEPARATION, CANARDS, AND FREQUENCY LOWERING

The behavior of calcium oscillations near bifurcations is analyzed for three different models. For the model developed by Somogyi and Stucki [42], it is shown that the range of oscillations is bounded by supercritical and subcritical Hopf bifurcations. Near the latter, canard orbits arise, that is, quasi-harmonic oscillations with a very small amplitude grow very fast to become pulsed oscillations. The potential biological significance of this behavior is discussed. A time-scale analysis of this model is performed and an approximation formula for the oscillation period is derived. For two models that we presented earlier [30, 31], it is shown that a homoclinic bifurcation and an infinite period bifurcation, respectively, occur. These imply that the oscillation period can reach arbitrarily high values. This behavior is discussed in the light of frequency encoding, and the scaling laws of the oscillation period are given.

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.

Amplitude-encoded calcium oscillations in fish cells

The reaction of intracellular Ca(2+) to different agonist stimuli in primary hepatocytes from rainbow trout (Oncorhynchus mykiss) as well as the permanent fish cell line RTL-W1 was investigated systematically. In addition to "classical" agonists such as phenylephrine and ATP, model environmental toxicants like 4-nitrophenol and 3,4-dichloroaniline were used to elucidate possible interactions between toxic effects and Ca(2+) signaling. We report Ca(2+) oscillations in response to several stimuli in RTL-W1 cells and to a lesser extent in primary hepatocytes. Moreover, these Ca(2+) oscillations are amplitude-encoded in contrast to their mammalian counterpart. Bioinformatics and computational analysis were employed to identify key players of Ca(2+) signaling in fish and to determine likely causes for the experimentally observed differences between the Ca(2+) dynamics in fish cells compared to those in mammalian liver cells.

Using Jensen's inequality to explain the role of regular calcium oscillations in protein activation

Oscillations of cytosolic Ca(2 +) are very important for cellular signalling in excitable and non-excitable cells. The information of various extracellular stimuli is encoded into oscillating patterns of Ca(2 +) that subsequently lead to the activation of different Ca(2 +)-sensitive target proteins in the cell. The question remains, however, why this information is transmitted by means of an oscillating rather than a constant signal. Here we show that, in fact, Ca(2 +) oscillations can achieve a better activation of target proteins than a comparable constant signal with the same amount of Ca(2 +) used. For this we use Jensen's inequality that describes the relation between the function value of the average of a set of argument values and the average of the function values of the arguments from that set. We analyse the role of the cooperativity of the binding of Ca(2 +) and of zero-order ultrasensitivity, which are two properties that are often observed in experiments on the activation of Ca(2 +)-sensitive target proteins. Our results apply to arbitrary oscillation shapes and a very general decoding model, thus generalizing the observations of several previous studies. We compare our results with data from experimental studies investigating the activation of nuclear factor of activated T cells (NFAT) and Ras by oscillatory and constant signals. Although we are restricted to specific approximations due to the lack of detailed kinetic data, we find good qualitative agreement with our theoretical predictions.

From puffs to global Ca2+ signals: how molecular properties shape global signals

The universality of Ca(2+) as second messenger in living cells is achieved by a rich spectrum of spatiotemporal cellular concentration dynamics. Ca(2+) release from internal storage compartments plays a key role in shaping cytosolic Ca(2+) signals. Deciphering this signaling mechanism is essential for a deeper understanding of its physiological function and general concepts of cell signaling. Here, we review recent experimental findings demonstrating the stochasticity of Ca(2+) oscillations and its relevance for modeling Ca(2+) dynamics. The stochasticity arises by the hierarchical signal structure that carries molecular fluctuations of single channels onto the level of the cell leading to a stochastic medium as theoretically predicted. The result contradicts the current opinion of Ca(2+) being a cellular oscillator. We demonstrate that cells use array enhanced coherence resonance to form rather regular spiking signals and that the "oscillations" carry information despite the involved stochasticity. The knowledge on the underlying mechanism also allows for determination of intrinsic properties from global observations. In the second part of the paper, we briefly survey different modeling approaches with regard to the experimental results. We focus on the dependence of the standard deviation on the mean period of the oscillations. It shows that limit cycle oscillations cannot describe the experimental data and that generic models have to include the spatial aspects of Ca(2+) signaling.

Long-range interaction effects on calcium-wave propagation

In this paper, numerical simulation of calcium waves in a network of cells coupled together by a paracrine signaling is investigated. The model takes into account the long-range interaction between cells due to the action of extracellular messengers, which provide links between first-neighbor cells, but also on cells located far away from the excited cell. When considering bidirectional coupling, the long-range interaction influences neither the frequency nor the amplitude of oscillations, contrary to one-directional coupling. The long-range interaction influences the speed of propagation of Ca2+ waves in the network and induces enlargement of the transition zone before the steady regime of propagation is attained. We also investigate the long-range effects on the colonization of a given niche by a pathogenic microorganism signal on calcium wave propagation in the network.

How does intracellular Ca2+ oscillate: by chance or by the clock?

Ca2+ oscillations have been considered to obey deterministic dynamics for almost two decades. We show for four cell types that Ca2+ oscillations are instead a sequence of random spikes. The standard deviation of the interspike intervals (ISIs) of individual spike trains is similar to the average ISI; it increases approximately linearly with the average ISI; and consecutive ISIs are uncorrelated. Decreasing the effective diffusion coefficient of free Ca2+ using Ca2+ buffers increases the average ISI and the standard deviation in agreement with the idea that individual spikes are caused by random wave nucleation. Array-enhanced coherence resonance leads to regular Ca2+ oscillations with small standard deviation of ISIs.

Hybrid stochastic and deterministic simulations of calcium blips

Intracellular calcium release is a prime example for the role of stochastic effects in cellular systems. Recent models consist of deterministic reaction-diffusion equations coupled to stochastic transitions of calcium channels. The resulting dynamics is of multiple time and spatial scales, which complicates far-reaching computer simulations. In this article, we introduce a novel hybrid scheme that is especially tailored to accurately trace events with essential stochastic variations, while deterministic concentration variables are efficiently and accurately traced at the same time. We use finite elements to efficiently resolve the extreme spatial gradients of concentration variables close to a channel. We describe the algorithmic approach and we demonstrate its efficiency compared to conventional methods. Our single-channel model matches experimental data and results in intriguing dynamics if calcium is used as charge carrier. Random openings of the channel accumulate in bursts of calcium blips that may be central for the understanding of cellular calcium dynamics.

Connexin channel permeability to cytoplasmic molecules

Connexin channels are known to be permeable to a variety of cytoplasmic molecules. The first observation of second messenger junctional permeability, made approximately 30 years ago, sparked broad interest in gap junction channels as mediators of intercellular molecular signaling. Since then, much has been learned about the diversity of connexin channels with regard to isoform diversity, tissue and developmental distribution, modes of channel regulation, assembly, expression, biochemical modification and permeability, all of which appear to be dynamically regulated. This information has expanded the potential roles of connexin channels in development, physiology and disease, and made their elucidation much more complex--30 years ago such an orchestra of junctional dynamics was unanticipated. Only recently, however, have investigators been able to directly address, in this more complex framework, the key issue: what specific biological molecules, second messengers and others, are able to permeate the various types of connexin channels, and how well? An important related issue, given the ever-growing list of connexin-related pathologies, is how these permeabilities are altered by disease-causing connexin mutations. Together, many studies show that a variety of cytoplasmic molecules can permeate the different types of connexin channels. A few studies reveal differences in permeation by different molecules through a particular type of connexin channel, and differences in permeation by a particular molecule through different types of connexin channels. This article describes and evaluates the various methods used to obtain these data, presents an annotated compilation of the results, and discusses the findings in the context of what can be inferred about mechanism of selectivity and potential relevance to signaling. The data strongly suggest that highly specific interactions take place between connexin pores and specific biological molecular permeants, and that those interactions determine which cytoplasmic molecules can permeate and how well. At this time, the nature of those interactions is unclear. One hopes that with more detailed permeability and structural information, the specific molecular mechanisms of the selectivity can be elucidated.

Physical manipulation of calcium oscillations facilitates osteodifferentiation of human mesenchymal stem cells

The role of cytosolic calcium oscillation has long been recognized in the regulation of cellular and molecular interactions. Information embedded in calcium oscillation can provide molecular cues for cell behavior such as cell differentiation. Although calcium dynamics are versatile and likely to depend on the cell type, the calcium dynamics in human mesenchymal stem cells (hMSCs) and its role in differentiation are yet to be fully elucidated. In the present study we characterized the calcium oscillation profiles in hMSCs before and after subjecting the cells to the osteoinductive factors. Our findings indicate that the calcium spikes decreased rapidly with osteodifferentiation to a level observed in terminally differentiated human osteoblasts. In addition, the calcium oscillations appear to serve as a bidirectional signal during hMSC differentiation. While an altered calcium oscillation pattern may be an indicator for hMSC differentiation, it is also likely to be involved in directing hMSC differentiation. Treatment of hMSCs with a noninvasive electrical stimulation, for example, not only altered the calcium oscillations but also facilitated osteodifferentiation. Regulation of calcium oscillation by external physical stimulation could amplify hMSC differentiation into a tissue-specific lineage and may offer an alternate biotechnology to harness the unique properties of stem cells.

Calcium Dynamics: Spatio‐Temporal Organization from the Subcellular to the Organ Level

Many essential physiological processes are controlled by calcium. To ensure reliability and specificity, calcium signals are highly organized in time and space in the form of oscillations and waves. Interesting findings have been obtained at various scales, ranging from the stochastic opening of a single calcium channel to the intercellular calcium wave spreading through an entire organ. A detailed understanding of calcium dynamics thus requires a link between observations at different scales. It appears that some regulations such as calcium-induced calcium release or PLC activation by calcium, as well as the weak diffusibility of calcium ions play a role at all levels of organization in most cell types. To comprehend how calcium waves spread from one cell to another, specific gap-junctional coupling and paracrine signaling must also be taken into account. On the basis of a pluridisciplinar approach ranging from physics to physiology, a unified description of calcium dynamics is emerging, which could help understanding how such a small ion can mediate so many vital functions in living systems.

Models of IP3 and Ca2+ oscillations: frequency encoding and identification of underlying feedbacks

Hormones that act through the calcium-releasing messenger, inositol 1,4,5-trisphosphate (IP3), cause intracellular calcium oscillations, which have been ascribed to calcium feedbacks on the IP3 receptor. Recent studies have shown that IP3 levels oscillate together with the cytoplasmic calcium concentration. To investigate the functional significance of this phenomenon, we have developed mathematical models of the interaction of both second messengers. The models account for both positive and negative feedbacks of calcium on IP3 metabolism, mediated by calcium activation of phospholipase C and IP3 3-kinase, respectively. The coupled IP3 and calcium oscillations have a greatly expanded frequency range compared to calcium fluctuations obtained with clamped IP3. Therefore the feedbacks can be physiologically important in supporting the efficient frequency encoding of hormone concentration observed in many cell types. This action of the feedbacks depends on the turnover rate of IP3. To shape the oscillations, positive feedback requires fast IP3 turnover, whereas negative feedback requires slow IP3 turnover. The ectopic expression of an IP3 binding protein has been used to decrease the rate of IP3 turnover experimentally, resulting in a dose-dependent slowing and eventual quenching of the Ca2+ oscillations. These results are consistent with a model based on positive feedback of Ca2+ on IP3 production.

A buffering SERCA pump in models of calcium dynamics

Many cells use oscillations in calcium concentration to transmit messages. The oscillations largely result from an influx of calcium into the cytosol from the endoplasmic reticulum (ER), followed by an efflux of calcium from the cytosol back into the ER. The sarcoplasmic/endoplasmic reticulum calcium ATPase (SERCA) pump pumps calcium into the ER. It binds calcium on the cytosolic side and releases it on the ER side and in the delay between binding and release, calcium is buffered by the pump. We developed a model of a buffering SERCA pump and investigated whether including this in a model of calcium oscillations has any significant effects. We found that the oscillations produced when using the SERCA pump, which does not buffer calcium, have a larger amplitude and a slightly smaller period than when using the buffering SERCA pump. We show that the buffering SERCA pump shows adaptation to a stimulus, and we demonstrate that, by using a bidirectional SERCA pump, we are able to eliminate futile cycling of calcium between the cytosol and ER when the cell is at rest.

Is there a specific role for the plasma membrane Ca2+ -ATPase in the hepatocyte?

The plasma membrane Ca2+ -ATPase (PMCA) is responsible for the fine, long-term regulation of the cytoplasmic calcium concentration by extrusion of this cation from the cell. Although the general kinetic mechanisms for the action of both, well coordinated hydrolytic activity and calcium transport are reasonably understood in the majority of cell types, due to the complex physiologic and biochemical characteristics shown by the hepatocyte, the study of this enzyme in this cell type has become a real challenge. Here, we review the various molecular aspects known to date to be associated with liver PMCA activity, and outline the strategies to follow for establishing the role of this enzyme in the overall physiology of the hepatocyte. In this way, we first concentrate on the basic biochemical aspects of liver cell PMCA, and place an important emphasis on expression of its molecular forms to finally focus on the critical hormonal regulation of the enzyme. Although these complex aspects have been studied mainly under normal conditions, the significance of PMCA in the calcium homeostasis of an abnormal liver cell is also reviewed.

Effects of the prostaglandins PGF2alpha and PGE2 on calcium signaling in rat hepatocyte doublets

Coordination of intercellular Ca2+ signals is important for certain hepatic functions including biliary flow and glucose output. Prostaglandins, such as PGF2alpha and PGE2, may modify these hepatocyte functions by inducing Ca2+ increase, but very little is known about the organization of the Ca2+ signals induced by these agonists. We studied Ca2+ signals induced by PGF2alpha and PGE2 in fura-2 AM-loaded hepatocyte doublets. Even though both prostaglandins induced Ca2+ oscillations, neither PGF2alpha nor PGE2 induced coordinated Ca2+ oscillations in hepatocyte doublets. Gap junction permeability (GJP), assessed by fluorescence recovery after photobleaching, showed that this absence of coordination was not related to a defect in GJP. Inositol (1,4,5)trisphosphate [Ins(1,4,5)P3] assays and the increase in Ins(1,4,5)P3 receptor sensitivity to Ins(1,4,5)P3 observed in response to thimerosal suggested that the absence of coordination was a consequence of the very small quantity of Ins(1,4,5)P3 formed by these prostaglandins. Furthermore, when PGE2 and PGF2alpha were added just before norepinephrine, they favored the coordination of Ca2+ signals induced by norepinephrine. However, GJP between hepatocyte doublets was strongly inhibited by prolonged (>or=2 h) treatment with PGF2alpha, thereby preventing the coordination of Ca2+ oscillations induced by norepinephrine in these cells. Thus, depending on the time window, prostaglandins, specially PGF2alpha, may enhance or diminish the propagation of Ca2+ signals. They may therefore contribute to the fine tuning of Ca2+ wave-dependent functions, such as nerve stimulation, hormonal regulation of liver metabolism, or bile secretion, in both normal and pathogenic conditions.

Biophysical re-equilibration of Ca2+ fluxes as a simple biologically plausible explanation for complex intracellular Ca2+ release patterns

Physiological regulation of Ca(2+) release from the endoplasmic reticulum (ER) is critical for cell function. Recent direct measurements of free [Ca(2+)] inside the ER ([Ca(2+)](ER)) revealed that [Ca(2+)](ER) itself is a key regulator of ER Ca(2+) handling. However, the role of this new regulatory process in generating various patterns of Ca(2+) release remains to be elucidated in detail. Here, we incorporate the recently quantified experimental correlations between [Ca(2+)](ER) and Ca(2+) movements across the ER membrane into a mathematical model ER Ca(2+) handling. The model reproduces basic experimental dynamics of [Ca(2+)](ER). Although this was not goal in model design, the model also exhibits mechanistically unclear experimental phenomena such as "quantal" Ca(2+) release, and "store charging" by increasing resting cytosolic [Ca(2+)]. While more complex explanations cannot be ruled out, on the basis of our data we propose that "quantal release" and "store charging" could be simple re-equilibration phenomena, predicted by the recently quantified biophysical dynamics of Ca(2+) movements across the ER membrane.

Zonation of the metabolic action of vasopressin in the bivascularly perfused rat liver

Predominance of the vasopressin binding capacity in the hepatic perivenous area leads to the hypothesis that the metabolic effects of the hormone should also be more pronounced in this area. Until now this question has been approached solely by experiments with isolated hepatocytes where an apparent absence of metabolic zonation was found. We have reexamined this question using the bivascularly perfused liver. In this system periportal cells can be reached in a selective manner with substrates and effectors via the hepatic artery when retrograde perfusion (hepatic vein --> portal vein) is done. The action of vasopressin (1-10 nM) on glycogenolysis, initial calcium efflux, glycolysis and oxygen uptake were measured. The results revealed that the action of vasopressin in the liver is heterogeneously distributed. Glycogenolysis stimulation and initial calcium efflux were predominant in the perivenous area, irrespective of the vasopressin concentration. Oxygen uptake was stimulated in the perivenous area; in the periportal area it ranged from inhibition at low vasopressin concentrations to stimulation at high ones. Lactate production was generally greater in the perivenous zone, whereas the opposite occurred with pyruvate production. Analysis of these and other results suggests that at least three factors are contributing to the heterogenic response of the liver parenchyma to vasopressin: a) receptor density, which tends to favour the perivenous zone; b) cell-to-cell interactions, which tend to favour situations where the perivenous zone is amply supplied with vasopressin; and c) the different response capacities of perivenous and periportal cells.

Role of voltage-dependent modulation of store Ca2+ release in synchronization of Ca2+ oscillations

Slow waves are rhythmic depolarizations that underlie mechanical activity of many smooth muscles. Slow waves result through rhythmic Ca(2+) release from intracellular Ca(2+) stores through inositol 1,4,5-trisphosphate (IP(3)) sensitive receptors and Ca(2+)-induced Ca(2+) release. Ca(2+) oscillations are transformed into membrane depolarizations by generation of a Ca(2+)-activated inward current. Importantly, the store Ca(2+) oscillations that underlie slow waves are entrained across many cells over large distances. It has been shown that IP(3) receptor-mediated Ca(2+) release is enhanced by membrane depolarization. Previous studies have implicated diffusion of Ca(2+) or the second messenger IP(3) across gap junctions in synchronization of Ca(2+) oscillations. In this study, a novel mechanism of Ca(2+) store entrainment through depolarization-induced IP(3) receptor-mediated Ca(2+) release is investigated. This mechanism is significantly different from chemical coupling-based mechanisms, as membrane potential has a coupling effect over distances several orders of magnitude greater than either diffusion of Ca(2+) or IP(3) through gap junctions. It is shown that electrical coupling acting through voltage-dependent modulation of store Ca(2+) release is able to synchronize oscillations of cells even when cells are widely separated and have different intrinsic frequencies of oscillation.

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.

Differential regulation of proteins by bursting calcium oscillations--a theoretical study

Calcium in ionic form is a second messenger connecting several input signals to several target processes in the cell. The question arises how one second messenger can transmit more than one signal simultaneously (bow-tie structure of signalling). Experimental data on calcium dynamics often show patterns of successive low-peak and high-peak oscillatory phases, known as bursting. Here, we propose that bursting calcium oscillations can perform the function of simultaneous transmission of two signals at physiological calcium concentrations, for example, by selective activation of two calcium-binding proteins. This differential regulation by periodic bursting is investigated in a theoretical model. The two proteins are assumed to be activated by calcium, and one of them is assumed to be subject to biphasic regulation due to additional inhibitory binding sites. To explore which characteristics of the complex signal could be responsible for independent regulation of low-peak activated and spike activated targets, different bursting patterns of simplified square pulses are applied. Depending on the change in the bursting pattern, one protein can be gradually activated at a constant level of the other protein's activity, or the two proteins can be activated simultaneously, or one protein can be activated while the other one is deactivated simultaneously. Thus, the two proteins can be regulated virtually independently.

Calcium oscillations in a triplet of pancreatic acinar cells

We use a mathematical model of calcium dynamics in pancreatic acinar cells to investigate calcium oscillations in a ring of three coupled cells. A connected group of cells is modeled in two different ways: 1), as coupled point oscillators, each oscillator being described by a spatially homogeneous model; and 2), as spatially distributed cells coupled along their common boundaries by gap-junctional diffusion of inositol trisphosphate and/or calcium. We show that, although the point-oscillator model gives a reasonably accurate general picture, the behavior of the spatially distributed cells cannot always be predicted from the simpler analysis; spatially distributed diffusion and cell geometry both play important roles in determining behavior. In particular, oscillations in which two cells are in synchrony, with the third phase-locked but not synchronous, appears to be more dominant in the spatially distributed model than in the point-oscillator model. In both types of model, intercellular coupling leads to a variety of synchronous, phase-locked, or asynchronous behaviors. For some parameter values there are multiple, simultaneous stable types of oscillation. We predict 1), that intercellular calcium diffusion is necessary and sufficient to coordinate the responses in neighboring cells; 2), that the function of intercellular inositol trisphosphate diffusion is to smooth out any concentration differences between the cells, thus making it easier for the diffusion of calcium to synchronize the oscillations; 3), that groups of coupled cells will tend to respond in a clumped manner, with groups of synchronized cells, rather than with regular phase-locked periodic intercellular waves; and 4), that enzyme secretion is maximized by the presence of a pacemaker cell in each cluster which drives the other cells at a frequency greater than their intrinsic frequency.

Local dissipation and coupling properties of cellular oscillators: a case study on calcium oscillations

Synchronised signal transduction between cells is crucial, since it assures fast and immutable information processing, which is vital for flawless functioning of living organisms. The question arises how to recognise the ability of a cell to be easily coupled with other cells. In the present paper, we investigate the system properties that determine best coupling abilities and assure the most efficient signal transduction between cells. A case study is done for intercellular calcium oscillations. For a particular diffusion-like coupled system of cellular oscillators, we determined the minimal gap-junctional permeability that is necessary for synchronisation of initially asynchronous oscillators. Our results show that dissipation is a crucial system property that determines the coupling ability of cellular oscillators. We found that low dissipation assures synchronisation of coupled cells already at very low gap-junctional permeability, whereas highly dissipative oscillators require much higher gap-junctional permeability in order to synchronise. The results are discussed in the sense of their biological importance for systems where the synchronous responses of cells were recognised to be indispensable for appropriate physiological functioning of the tissue.