Abstract : Nearly all living organisms display circadian oscillations characterized by a period close to 24 h. 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 core deterministic model previously proposed for circadian oscillations of the PER protein and its mRNA in Drosophila. The model is based on cooperative repression of the per gene by the PER protein. 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 to both sustained oscillations of the limit cycle type and the influence of the proximity from a bifurcation point below 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 simulations also reproduce the evolution toward a strange attractor in conditions where an extended version of the deterministic model admits chaotic behavior. These results show how regulatory feedback processes at the cellular level allow the emergence of a coherent biological rhythm out of molecular noise.
Abstract : Fertilization triggers repetitive waves of cytosolic Ca(2+) in the egg of many species. The mechanism involved in the generation of Ca(2+) waves has been studied in much detail in mature ascidian eggs, by raising artificially the level of inositol 1,4,5-trisphosphate [Ins(1,4,5)P(3)] or of its poorly metabolizable analogue, glycero-myo-phosphatidylinositol 4,5-bisphosphate [gPtdIns(4,5)P(2)]. Here, we use this strategy and the experimental results it provides to develop a realistic theoretical model for repetitive Ca(2+) wave generation and propagation in mature eggs. The model takes into account the heterogeneous spatial distribution of the endoplasmic reticulum. Our results corroborate the hypothesis that Ca(2+) wave pacemakers are associated with cortical accumulations of endoplasmic reticulum. The model is first tested and validated by the adequate match between its theoretical predictions and the observed effects of localized injections of massive amounts of Ins(1,4,5)P(3) analogues. In a second step, we use the model to make some propositions about the possible characteristics of the sperm factor. We find that to account for the spatial characteristics of the first series of Ca(2+) waves seen at fertilization in ascidian eggs, it has to be assumed that, if the sperm factor is a phospholipase C, it is Ca(2+)-sensitive and highly diffusible. Although the actual state of knowledge does not allow us to explain the observed relocalization of the Ca(2+) wave pacemaker site, the model corroborates the assumption that PtdIns(4,5)P(2), the substrate for phospholipase C is distributed over the entire egg. We also predict that the dose of sperm factor injected into the egg should modulate the temporal characteristics of the first, long-lasting fertilization wave.
Abstract : Most living organisms have developed the capability of generating autonomously sustained oscillations with a period close to 24 h. The mechanism responsible for these circadian rhythms relies on the negative regulation exerted by a protein on the expression of its own gene. Deterministic models for circadian rhythms account for the occurrence of autonomous oscillations of the limit cycle type, for their entrainment by lightdark cycles, and for their phase shifting by light pulses. Such models, however, do not take into consideration the molecular fluctuations which arise when the number of molecules involved in the regulatory mechanism is low. Here we resort to a stochastic description of a core model for circadian rhythms to study the emergence of coherent oscillations in gene expression in the presence of molecular noise. We show that despite the "bar code" pattern of gene activation, robust circadian oscillations can be observed. Simulations of the deterministic, fully developed version of the circadian model indicate, however, that sustained oscillations only emerge above a critical value of the rate constants characterizing the reversible binding of repressor to the gene, while below this value the system evolves towards an excitable steady state. This explains why, depending on whether or not the critical value of these rate constants is exceeded, stochastic simulations of the model produce coherent oscillations or very noisy oscillations with a highly variable period.
Abstract : During development, Myxococcus xanthus cells produce a series of spatial patterns by coordinating their motion through a contact-dependent signal, the C-signal. C-signaling modulates the frequency at which cells reverse their gliding direction. It does this by interacting with the Frz system (a homolog of the Escherichia coli chemosensory system) via a cascade of covalent modifications. Here we show that introducing a negative feedback into this cascade results in oscillatory behavior of the signaling circuit. The model explains several aspects of M. xanthus behavior during development, including the nonrandom distribution of reversal times, and the differences in response of the reversal frequency to both moderate and high levels of C-signaling at different developmental stages. We also propose experiments to test the model.
Abstract : We extend the study of a computational model recently proposed for the mammalian circadian clock (Proc. Natl Acad. Sci. USA 100 (2003) 7051). The model, based on the intertwined positive and negative regulatory loops involving the Per, Cry, Bmal1, and Clock genes, can give rise to sustained circadian oscillations in conditions of continuous darkness. These limit cycle oscillations correspond to circadian rhythms autonomously generated by suprachiasmatic nuclei and by some peripheral tissues. By using different sets of parameter values producing circadian oscillations, we compare the effect of the various parameters and show that both the occurrence and the period of the oscillations are generally most sensitive to parameters related to synthesis or degradation of Bmal1 mRNA and BMAL1 protein. The mechanism of circadian oscillations relies on the formation of an inactive complex between PER and CRY and the activators CLOCK and BMAL1 that enhance Per and Cry expression. Bifurcation diagrams and computer simulations nevertheless indicate the possible existence of a second source of oscillatory behavior. Thus, sustained oscillations might arise from the sole negative autoregulation of Bmal1 expression. This second oscillatory mechanism may not be functional in physiological conditions, and its period need not necessarily be circadian. When incorporating the light-induced expression of the Per gene, the model accounts for entrainment of the oscillations by lightdark (LD) cycles. Long-term suppression of circadian oscillations by a single light pulse can occur in the model when a stable steady state coexists with a stable limit cycle. The phase of the oscillations upon entrainment in LD critically depends on the parameters that govern the level of CRY protein. Small changes in the parameters governing CRY levels can shift the peak in Per mRNA from the L to the D phase, or can prevent entrainment. The results are discussed in relation to physiological disorders of the sleep-wake cycle linked to perturbations of the human circadian clock, such as the familial advanced sleep phase syndrome or the non-24 h sleep-wake syndrome.