Abstract : Receptor-mediated production of inositol 1,4,5-trisphosphate (InsP3) initiates Ca2+ release and is responsible for cytosolic Ca2+ oscillations. InsP3 oscillations have also been observed in some cells. One of the enzymes controlling InsP3 catabolism, the InsP3 3-kinase, is stimulated by Ca2+; this regulation is presumably part of the reason for InsP3 oscillations that have been observed in some cells. Here, we investigate the possible role of Ca2+-activated InsP3 catabolism on the characteristics of the InsP3-induced Ca2+ oscillations. Numerical simulations show that if it is assumed that the Ca2+-independent InsP3 catabolism is predominant, Ca2+ oscillations remain qualitatively unchanged although the relative amplitude of the oscillations in InsP3 concentrations becomes minimal. We tested this prediction in hepatocytes by masking the Ca2+-dependent InsP3 catabolism by 3-kinase through the injection of massive amounts of InsP3 5-phosphatase, which is not stimulated by Ca2+. We find that in such injected hepatocytes, Ca2+ oscillations generated by modest agonist levels are suppressed, presumably because of the decreased dose in InsP3, but that at higher doses of agonist, oscillations reappear, with characteristics similar to those of untreated cells at low agonist doses. Altogether, these results suggest that oscillations in InsP3 concentration due to Ca2+-stimulated InsP3 catabolism do not play a major role for the oscillations in Ca2+ concentration.
Abstract : The rules that govern the activation and autophosphorylation of the multifunctional Ca2+ -calmodulin kinase II (CaMKII) by Ca2+ and calmodulin (CaM) are thought to underlie its ability to decode Ca2+ oscillations and to control multiple cellular functions. We propose a simple biophysical model for the activation of *CaMKII by Ca2+ and calmodulin. The model describes the transition of the subunits of the kinase between their different possible states (inactive, bound to Ca2+ -CaM, phosphorylated at Thr286, trapped and autonomous). All transitions are described by classical kinetic equations except for the autophosphorylation step, which is modeled in an empirical manner. The model quantitatively reproduces the experimentally demonstrated frequency sensitivity of CaMKII [Science 279 (1998) 227]. We further use the model to investigate the role of several characterized features of the kinase - as well as some that are not easily attainable by experiments - in its frequency-dependent responses. In cellular microdomains, CaMKII is expected to sense very brief Ca2+ spikes; our simulations under such conditions reveal that the enzyme response is tuned to optimal frequencies. This prediction is then confirmed by experimental data. This novel and simple model should help in understanding the rules that govern CaMKII regulation, as well as those involved in decoding intracellular Ca2+ signals.
Abstract : Intracellular Ca2+ oscillations most often appear as regular spikes. However, complex Ca2+ oscillations in the form of bursting or chaos have also been observed. These oscillations most probably originate from the interplay between multiple regulatory mechanisms. From an experimental point of view, the mechansims underlying these types of Ca2+ oscilations remain poorly uderstood. In this review, we first briefly review the various types of regulatory mechanims known to regulate Ca2+ oscillations, together with the related theoretical models. We show that the interplay between the activation of an InsP3-metabolizing enzyme (InsP3 3-kinase) by Ca2+ and Ca2+-induced Ca2+ release is a plausible explanation for at least some complex Ca2+ oscillations. This regulation indeed provides a mechanism for the self-modulation of the InsP3 signal, thus leading to a phenomenon analogous to the periodic forcing of an autonomous oscillator. We also briefly assess the role of luminal Ca2+ in the onset of Ca2+ oscillations, as well as the link between bistability and Ca2+ oscillations.
Abstract : Circadian rhythms are endogenous rhythms with a cycle length of approximately 24 h. Rhythmic production of specific proteins within pacemaker structures is the basis for these physiological and behavioral rhythms. Prior work on mathematical modeling of molecular circadian oscillators has focused on the fruit fly, Drosophila melanogaster. Recently, great advances have been made in our understanding of the molecular basis of circadian rhythms in mammals. Mathematical models of the mammalian circadian oscillator are needed to piece together diverse data, predict experimental results, and help us understand the clock as a whole. Our objectives are to develop mathematical models of the mammalian circadian oscillator, generate and test predictions from these models, gather information on the parameters needed for model development, integrate the molecular model with an existing model of the influence of light and rhythmicity on human performance, and make models available in BioSpice so that they can be easily used by the general community. Two new mammalian models have been developed, and experimental data are summarized. These studies have the potential to lead to new strategies for resetting the circadian clock. Manipulations of the circadian clock can be used to optimize performance by promoting alertness and physiological synchronization.
Abstract : L'étude de l'horloge circadienne illustre comment la modélisation aide à réduire la complexité des mécanismes moléculaires à l'origine des rythmes biologiques.
Abstract : Circadian rhythms, characterized by a period of about 24h, are generated in nearly all living organisms by the negative autoregulation of clock gene expression. Deterministic models based on this genetic regulation account for circadian oscillations in constant environmental conditions (e.g., in constant darkness) and for entrainment of these rhythms by light-dark cycles. When the number of clock mRNA and protein molecules is low, it is necessary to resort to stochastic simulations to assess the influence of molecular noise on circadian oscillations. Indeed, it is possible that the autoregulatory mechanism of gene expression might not produce stable rhythms due to fluctuations if the number of molecules involved in the clock mechanism remains too low. We have compared the deterministic and stochastic approaches for a model based on the negative autoregulation of a clock gene. We show by means of stochastic simulations that robust circadian oscillations can already occur when the maximum number of mRNA and protein molecules is of the order of a few tens or hundreds, respectively. Furthermore, the results indicate that the cooperativity characterizing the repression of the transcription process strenghtens the robustness of circadian rhythms and that entrainment by light-dark cycles stabilizes the phase of the oscillations.
Résumé : Les rythmes caractérisés par une période proche de 24 h sont générés, chez quasiment tous les organismes vivants, par l'autorégulation négative de l'expression de gènes de l'horloge circadienne. Les modèles déterministes fondés sur cette régulation génétique permettent de rendre compte de ces oscillations circadiennes dans des conditions environnementales constantes (en obscurité constante par exemple) et de l'entraînement de ces rythmes par des cycles lumière-obscurité. Lorsque le nombre de molécules d'ARNm et de protéines est faible, il est nécessaire de recourir à des simulations stochastiques pour déterminer l'influence du bruit moléculaire sur les oscillations circadiennes. Il est en effet possible que le mécanisme d'autorégulation de l'expression génétique ne puisse produire de rythme stable, en raison des fluctuations, si le nombre de molécules impliquées dans le mécanisme de l'horloge est peu élevé. Nous avons effectué la comparaison entre approches déterministe et stochastique pour un modèle fondé sur l'autorégulation négative d'un gène de l'horloge circadienne. Après avoir développé le modèle déterministe en étapes élémentaires, nous montrons, à l'aide de simulations numériques, que des oscillations circadiennes robustes peuvent se produire déjà pour un nombre de molécules de l'ordre de quelques dizaines d'ARNm et quelques centaines de protéines. En outre, les résultats indiquent que la coopérativité caractérisant la répression de la transcription augmente la robustesse des rythmes tandis que l'entraînement par des cycles lumière-obscurité stabilise la phase des oscillations circadiennes.
Abstract : Circadian rhythms are endogenous oscillations that occur with a period close to 24 h in nearly all living organisms. These rhythms originate from the negative autoregulation of gene expression. Deterministic models based on such genetic regulatory processes account for the occurrence of circadian rhythms in constant environmental conditions (e.g., constant darkness), for entrainment of these rhythms by light-dark cycles, and for their phase-shifting by light pulses. When the numbers of protein and mRNA molecules involved in the oscillations are small, as may occur in cellular conditions, it becomes necessary to resort to stochastic simulations to assess the influence of molecular noise on circadian oscillations. We address the effect of molecular noise by considering the stochastic version of a deterministic model previously proposed for circadian oscillations of the PER and TIM proteins and their mRNAs in Drosophila. The model is based on repression of the per and tim genes by a complex between the PER and TIM proteins. Numerical simulations of the stochastic version of the model are performed by means of the Gillespie method. The predictions of the stochastic approach compare well with those of the deterministic model with respect both to sustained oscillations of the limit cycle type and to the influence of the proximity from a bifurcation point beyond which the system evolves to a stable steady state. Stochastic simulations indicate that robust circadian oscillations can emerge at the cellular level even when the maximum numbers of mRNA and protein molecules involved in the oscillations are of the order of only a few tens or hundreds. The stochastic model also reproduces the evolution to a strange attractor in conditions where the deterministic PERTIM model admits chaotic behaviour. The difference between periodic oscillations of the limit cycle type and aperiodic oscillations (i.e. chaos) persists in the presence of molecular noise, as shown by means of Poincaré sections. The progressive obliteration of periodicity observed as the number of molecules decreases can thus be distinguished from the aperiodicity originating from chaotic dynamics. As long as the numbers of molecules involved in the oscillations remain sufficiently large (of the order of a few tens or hundreds, or more), stochastic models therefore provide good agreement with the predictions of the deterministic model for circadian rhythms.
Abstract : Msn2 and Msn4 are two related transcriptional activators that mediate a general response to stress in yeast Saccharomyces cerevisiae by eliciting the expression of specific sets of genes. In response to stress or nutritional limitation, Msn2 and Msn4 migrate from the cytoplasm to the nucleus. Using GFP-tagged constructs and high-resolution time-lapse video microscopy on single cells, we show that light emitted by the microscope also triggers this migration. Unexpectedly, the population of Msn2 or Msn4 molecules shuttles repetitively into and out of the nucleus with a periodicity of a few minutes. A large heterogeneity in the oscillatory response to stress is observed between individual cells. This periodic behavior, which can be induced by various types of stress, at intermediate stress levels, is not dependent upon protein synthesis and persists when the DNA-binding domain of Msn2 is removed. The cAMP-PKA pathway controls the sensitivity of the oscillatory nucleocytoplasmic shuttling. In the absence of PKA, Msn4 continues to oscillate while Msn2 is maintained in the nucleus. We show that a computational model based on the possibility that Msn2 and Msn4 participate in autoregulatory loops controlling their subcellular localization can account for the oscillatory behavior of the two transcription factors.
Abstract : Drosophila is by far the most advanced model to understand the complex biochemical interactions upon which circadian clocks rely. Most of the genes that have been characterized so far were isolated through genetic screens using the locomotor activity rhythms of the adults as a circadian output. In addition, new techniques are available to deregulate gene expression in specific cells, allowing to analyze the growing number of developmental genes that also play a role as clock genes. However, one of the major challenges in circadian biology remains to properly interpret complex behavioral data and use them to fuel molecular models. This review tries to describe the problems that clockwatchers have to face when using Drosophila activity rhythms to understand the multiple facets of circadian function.
Abstract : We present a computational model for the mammalian circadian clock based on the intertwined positive and negative regulatory loops involving the Per, Cry, Bmal1, Clock, and Rev-Erba genes. In agreement with experimental observations, the model can give rise to sustained circadian oscillations in continuous darkness, characterized by an antiphase relationship between Per/Cry/Rev-Erba and Bmal1 mRNAs. Sustained oscillations correspond to the rhythms autonomously generated by suprachiasmatic nuclei. For other parameter values, damped oscillations can also be obtained in the model. These oscillations, which transform into sustained oscillations when coupled to a periodic signal, correspond to rhythms produced by peripheral tissues. When incorporating the light-induced expression of the Per gene, the model accounts for entrainment of the oscillations by light-dark cycles. Simulations show that the phase of the oscillations can then vary by several hours with relatively minor changes in parameter values. Such a lability of the phase could account for physiological disorders related to circadian rhythms in humans, such as advanced or delayed sleep phase syndrome, while the lack of entrainment by lightdark cycles can be related to the non-24h sleep-wake syndrome. The model uncovers the possible existence of multiple sources of oscillatory behavior. Thus, in conditions where the indirect negative autoregulation of Per and Cry expression is inoperative the model indicates the possibility that sustained oscillations might still arise from the negative autoregulation of Bmal1 expression.
Abstract : Two new theoretical models show how negative feedback loops incorporating a time delay can account for a variety of transcriptional oscillations, such as the mechanism of the segmentation clock in zebrafish and a number of recently identified transcriptional oscillators.