Abstract : The proper working of the liver largely depends on the fine tuning of the level of cytosolic Ca2+ in hepatocytes. Thanks to the development of imaging techniques, our understanding of the spatio-temporal organization of intracellular Ca2+ in this -and other- cell types has much improved. Many of these signals are mediated by a rise in the level of inositol 1,4,5-trisphosphate (InsP3), a second messenger which can activate the release of Ca2+ from the endoplasmic reticulum. Besides the now well-known hepatic Ca2+ oscillations induced by hormonal stimulation, intra- and intercellular Ca2+ waves have also been observed. More recently, sub-cellular Ca2+ increases associated with the coordinated opening of a few Ca2+ channels have been reported. Given the complexity of the regulations involved in the generation of such processes and the variety of time and length scales necessary to describe those phenomena, theoretical models have been largely used to gain a precise and quantitative understanding of the dynamics of intracellular Ca2+.
Here, we review the various aspects of the spatio-temporal organization of cytosolic Ca2+ in hepatocytes from the dual point of view provided by experiments and modeling. We first focus on the description and the mechanism of intracellular Ca2+ oscillations and waves. Second, we investigate in which manner these repetitive Ca2+ increases are coordinated among a set of hepatocytes coupled by gap junctions, a phenomenon known as "intercellular Ca2+ waves". Finally, we focus on the so-called elementary Ca2+ signals induced by low InsP3 concentrations, leading to Ca2+ rises having a spatial extent of a few microns. Although these small-scale events have been mainly studied in other cell types, we theoretically infer general properties of these localized intracellular Ca2+ rises that could also apply to hepatocytes.
Abstract : Intercellular calcium signals are propagated in multicellular hepatocyte systems as well as in the intact liver. The stimulation of connected hepatocytes by glycogenolytic agonists induces reproducible sequences of intracellular calcium concentration increases, resulting in unidirectional intercellular calcium waves. Hepatocytes are characterized by a gradient of vasopressin binding sites from the periportal to perivenous areas of the cell plate in hepatic lobules. Also, coordination of calcium signals between neighboring cells requires the presence of the agonist at each cell surface as well as gap junction permeability. We present a model based on the junctional coupling of several hepatocytes differing in sensitivity to the agonist and thus in the intrinsic period of calcium oscillations. In this model, each hepatocyte displays repetitive calcium spikes with a slight phase shift with respect to neighboring cells, giving rise to a phase wave. The orientation of the apparent calcium wave is imposed by the direction of the gradient of hormonal sensitivity. Calcium spikes are coordinated by the diffusion across junctions of small amounts of inositol 1,4,5-trisphosphate (InsP3). Theoretical predictions from this model are confirmed experimentally. Thus, major physiological insights may be gained from this model for coordination and spatial orientation of intercellular signals.
Abstract : Cytosolic calcium plays a crucial role as a second messenger in cellular signalling. Various cell types, including hepatocytes, display calcium oscillations when stimulated by an extracellular signal. However, the biological relevance of this temporal organization remains unclear. In this paper we investigate theoretically the effect of calcium oscillations on a particular example of cell regulation: the phosphorylation-dephosphorylation cycle controlling the activation of glycogen phosphorylase in hepatocytes. By modelling periodic varaitions in the extracellular calcium concentration, we show that calcium oscillations reduce the threshold for the activation of the enzyme. Furthemore, as the activation of the enzyme by oscillations at a given frequency depends on the kinetics of the phosphorylation-dephosphorylation cascade, specificity can be encoded by the oscillation frequency. Finally, using a model for signal-induced calcium oscillations based on calcium-induced calcium release, we show that realistic calcium oscillations can potentiate the response to an hormonal stimulation. These results indicate that calcium oscillations could contribute to increase the efficiency and specificity of cellular signalling.
Abstract : Examples of pulsatile signalling abound in intercellular communication, suggesting that this phenomenon represents a major function of biological rhythms. Pulsatile signals can be encoded in terms of their frequency and prove more efficient than monotonous ones whenever constant stimulation induces desensitization of target cells. We address the main examples of frequency encoding of pulsatility, besides those of neuronal nature. Considered in turn are cAMP oscillations in the slime mould Dictyostelium discoideum, the pulsatile secretion of hormones such as gonadotropin-releasing hormone or growth hormone, intracellular Ca2+ oscillations, and circadian rhythms. Models based on receptor desensitization show the possibility of optimizing cellular responses to cAMP signals in Dictyostelium or to pulsatile hormonal stimulation. The models indicate how the optimal duration of the pulsatile signal and the optimal interval between successive pulses vary as a function of the rates or receptor desensitization and resensitization and of the maximum ligand level during stimulation. The frequency encoding of intracellular Ca2+ oscillations appears to rely on another molecular mechanism. Models based on protein phosphorylation by a Ca2+ -calmodulin activated kinase show that the mean level of phosphorylated protein increases with the frequency of calcium spikes--which itself rises with the degree of stimulation--and that distinct levels of different phosphorylated proteins can be reached for a Ca2+ signal of given frequency.
Abstract: We examine theoretical models proposed for the molecular mechanism of circadian rhythms in Drosophila. The models are based on the negative feedback exerted by a complex between the PER and TIM proteins on the expression of the per and tim genes. We show that a similar model can account for circadian oscillations in Neurospora, where the protein FRQ negatively regulates the expression of the frq gene. The effect of light on the circadian rhythms is included by considering that it elicits a rise in the rate of TIM degradation in Drosophila, whereas in Neurospora it enhances the rate of frq transcription. The models account for the occurrence of sustained circadian oscillations in continuous darkness in Drosophila and Neurospora, and in continuous light in the latter organism. Numerical simulations further indicate that the periodic forcing of circadian oscillations by light-dark cycles can result either in the entrainment to the external periodicity or in aperiodic oscillations (i.e. chaos), depending on the magnitude of the periodic changes in the light-controlled parameter.
Résumé: Nous examinons des modèles théoriques proposés pour le mécanisme moléculaire des rythmes circadiens chez la drosophile. Ces modèles sont fondés sur la rétroaction négative exercée par un complexe entre les protéines PER et TIM sur l'expression des gènes per et tim. Nous montrons qu'un modèle similaire peut rendre compte des oscillations circadiennes chez Neurospora, organisme chez lequel la protéine FRQ exerce un contrôle négatif sur la transcription du gène frq. L'effet d'une perturbation lumineuse sur les rythmes circadiens est inclus dans les modèles en considérant que la lumière accroît la vitesse de dégradation de la protéine TIM chez la drosophile et la vitesse de transcription du gène frq chez Neurospora. Les modèles rendent compte de l'existence d'oscillations circadiennes entretenues en obscurité constante chez la drosophile et Neurospora, et en lumière constante chez ce dernier organisme. Les simulations numériques montrent que le forçage périodique des rythmes circadiens par des cycles lumière-obscurité peuvent résulter, selon l'amplitude des changements du paramètre de contrôle induits par la lumière, soit en l'entraînement des oscillations à la période externe, soit en des oscillations apériodiques, c'est-à-dire du chaos.
Abstract : Circadian rhythms occur in nearly all living organisms with a period close to 24 h. These rhythms constitute an important class of biological oscillators which present the characteristic of being naturally subjected to forcing by light-dark (LD) cycles. In order to investigate the conditions in which such a forcing might lead to chaos, we consider a model for a circadian limit cycle oscillator and assess its dynamic behavior when a light-sensitive parameter is periodically forced by LD cycles. We determine as a function of the forcing period and of the amplitude of the light-induced changes in the light-sensitive parameter the occurrence of various modes of dynamic behavior such as quasi-periodicity, entrainment, period-doubling and chaos. The type of oscillatory behavior markedly depends on the forcing waveform; thus the domain of entrainment grows at the expense of the domain of chaos as the forcing function progressively goes from a square wave to a sine wave. Also studied is the dependence of the phase of periodic or aperiodic oscillations on the amplitude of the light-induced changes in the control parameter. The results are discussed with respect to the main physiological role of circadian rhythms which is to allow organisms to adapt to their periodically varying environment by entrainment to the natural LD cycle.
Abstract : We analyze the dynamics of a bienzymatic system consisting of isocitrate dehydrogenase (IDH, EC. 184.108.40.206), which transforms NADP+ into NADPH, and of diaphorase (DIA, EC 220.127.116.11), which catalyzes the reverse reaction. Experimental evidence as well as a theoretical model showed the possibility of a coexistence between two stable steady states in this reaction system [G.M. Guidi et al., Biophys. J. 74 (1998) 1229-1240], owing to the regulatory properties of IDH. Here we extend this analysis by considering the behavior of the model proposed for the IDH-DIA bienzymatic system in conditions where the system is open to an influx of its substrates isocitrate and NADP+ and to an efflux of all metabolic species. The analysis indicates that in addition to different modes of bistability (including mushrooms and isolas), sustained oscillations can be observed in such conditions. These results point to the isocitrate dehydrogenase reaction coupled to diaphorase as a suitable candidate for further experimental and theoretical studies of bistability and oscillations in biochemical systems. The results obtained in this particular bienzymatic system bear on other enzymatic systems possessing a cyclical nature, which are known to play significant roles in a variety of metabolic and cellular regulatory processes.
Abstract : The hair follicle cycle successively goes through the anagen, catagen, telogen, and latency phases, which correspond, respectively, to hair growth, arrest, shedding, and absence before a new anagen phase is initiated. Experimental observations collected over a period of 14 years in a group of 10 male volunteers, alopecic and nonalopecic, allowed us to determine the characteristics of scalp hair follicle cycles. On the basis of these observations, we propose a follicular automaton model to simulate the dynamics of human hair cycles. The automaton model is defined by a set of rules that govern the stochastic transitions of each follicle between the successive states anagen, telogen, and latency, and the subsequent return to anagen. The transitions occur independently for each follicle, after time intervals given stochastically by a distribution characterized by a mean and a variance. The follicular automaton model accounts both for the dynamical transitions observed in a single follicle and for the behavior of an ensemble of independently cycling follicles. Thus, the model successfully reproduces the evolution of the fractions of follicle populations in each of the three phases, which fluctuate around steady-state or slowly drifting values. We apply the follicular automaton model to the study of spatial patterns of follicular growth that result from a spatially heterogeneous distribution of parameters such as the mean duration of anagen phase. When considering that follicles die or miniaturize after going through a critical number of successive cycles, the model can reproduce the evolution to hair patterns similar to well known types of diffuse or androgenetic alopecia.
Abstract : Thanks to genetic and biochemical advances on the molecular mechanism of circadian rhythms in Drosophila, theoretical models closely related to experimental observations can be considered for the regulatory mechanism of the circadian clock in this organism. Modeling is based on the autoregulatory negative feedback exerted by a complex between the PER and TIM proteins on the expression of the per and tim genes. The model predicts the occurrence of sustained circadian oscillations in continuous darkness. When incorporating light-induced TIM degradation, the model accounts for damping of oscillations in constant light, entrainment of the rhythm by light-dark cycles of varying period or photoperiod, and phase shifting by light pulses. The model further provides a molecular dynamical explanation for the permanent or transient suppression of circadian rhythmicity triggered in a variety of organisms by a critical pulse of light. Finally, the model shows that to produce a robust rhythm the various clock genes must be expressed at the appropriate levels since sustained oscillations only occur in a precise range of parameter values.
Abstract : We examine theoretical models proposed for the molecular mechanism of circadian rhythms in Drosophila. In a recent experimental study, Ouyang et al. (1998, Proc. Natl. Acad. Sci. U.S.A. 95, 8660-8664) have shown that, in direct competition, cyanobacterial strains whose circadian clocks have free-running periods (FRPs) which match the period of an imposed light/dark (LD) cycle exclude strains whose FRPs are out of resonance with the LD cycle. These differences in competitive fitness are observed despite the lack of measurable differences in monoculture growth rates between the strains. Here we show that the experimental results are consistent with a mathematical model in which cells rhythmically produce a metabolic inhibitor to which they display a sensitivity modulated by their circadian rhythm. We argue that models in which there is a circadian modulation of nutrient uptake kinetics cannot account for the results of these experiments. We discuss possible experiments to further characterize this phenomenon. The experimental protocol we propose can be used to distinguish between mutual inhibition and substrate depletion as underlying causes of the competitive advantage of circadian resonance.