Albert Goldbeter

Computational approaches to cellular rhythms

Oscillations arise in genetic and metabolic networks as a result of various modes of cellular regulation. In view of the large number of variables involved and of the complexity of feedback processes that generate oscillations, mathematical models and numerical simulations are needed to fully grasp the molecular mechanisms and functions of biological rhythms. Models are also necessary to comprehend the transition from simple to complex oscillatory behaviour and to delineate the conditions under which they arise. Examples ranging from calcium oscillations to pulsatile intercellular communication and circadian rhythms illustrate how computational biology contributes to clarify the molecular and dynamical bases of cellular rhythms.

Toward a detailed computational model for the mammalian circadian clock

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-Erb α 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-Erbα 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, whereas the lack of entrainment by light-dark 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.

A model for circadian oscillations in the Drosophila period protein (PER)

The mechanism of circadian oscillations in the period protein (PER) in Drosophila is investigated by means of a theoretical model. Taking into account recent experimental observations, the model for the circadian clock is based on multiple phosphorylation of PER and on the negative feedback exerted by PER on the transcription of the period (per) gene. This minimal biochemical model provides a molecular basis for circadian oscillations of the limit cycle type. During oscillations, the peak in per mRNA precedes by several hours the peak in total PER protein. The results support the view that multiple PER phosphorylation introduces times delays which strengthen the capability of negative feedback to produce oscillations. The analysis shows that the rhythm only occurs in a range bounded by two critical values of the maximum rate of PER degradation. A similar result is obtained with respect to the rate of PER transport into the nucleus. The results suggest a tentative explanation for the altered period of per mutants, in terms of variations in the rate of PER degradation.

Robustness of circadian rhythms with respect to molecular noise

We use a core molecular model capable of generating circadian rhythms to assess the robustness of circadian oscillations with respect to molecular noise. The model is based on the negative feedback exerted by a regulatory protein on the expression of its gene. Such a negative regulatory mechanism underlies circadian oscillations of the PER protein in Drosophila and of the FRQ protein in Neurospora. The model incorporates gene transcription into mRNA, translation of mRNA into protein, reversible phosphorylation leading to degradation of the regulatory protein, transport of the latter into the nucleus, and repression of gene expression by the nuclear form of the protein. To assess the effect of molecular noise, we perform stochastic simulations after decomposing the deterministic model into elementary reaction steps. The oscillations predicted by the stochastic simulations agree with those obtained with the deterministic version of the model. We show that robust circadian oscillations can occur already with a limited number of mRNA and protein molecules, in the range of tens and hundreds, respectively. Entrainment by light/dark cycles and cooperativity in repression enhance the robustness of circadian oscillations with respect to molecular noise.

A Model for Circadian Rhythms in Drosophila Incorporating the Formation of a Complex between the PER and TIM Proteins

The authors present a model for circadian oscillations of the Period (PER) and Timeless (TIM) proteins in Drosophila. The model for the circadian clock is based on multiple phosphorylation of PER and TIM and on the negative feedback exerted by a nuclear PER-TIM complex on the transcription of the perand tim genes. Periodic behavior occurs in a large domain of parameter space in the form of limit cycle oscillations. These sustained oscillations occur in conditions corresponding to continuous darkness or to entrainment by light-dark cycles and are in good agreement with experimental observations on the temporal variations of PER and TIM and of per and tim mRNAs. Birhythmicity (coexistence of two periodic regimes) and aperiodic oscillations (chaos) occur in a restricted range of parameter values. The results are compared to the predictions of a model based on the sole regulation by PER. Both the formation of a complex between PER and TIM and protein phosphorylation are found to favor oscillatory behavior. Determining how the period depends on several key parameters allows us to test possible molecular explanations proposed for the altered period in the perl and pers mutants. The extended model further allows the construction of phase-response curves based on the light-induced triggering of TIM degradation. These curves, established as a function of both the duration and magnitude of the effect of a light pulse, match the phase-response curves obtained experimentally in the wild type and pers mutant of Drosophila.

Dissipative Structures for an Allosteric Model: Application to Glycolytic Oscillations

An allosteric model of an open monosubstrate enzyme reaction is analyzed for the case where the enzyme, containing two protomers, is activated by the product. It is shown that this system can lead to instabilities beyond which a new state organized in time or in space (dissipative structure) can be reached. The conditions for both types of instabilities are presented and the occurrence of a temporal structure, consisting of a limit cycle behavior, is determined numerically as a function of the important parameters involved in the system. Sustained oscillations in the product and substrate concentrations are shown to occur for acceptable values of the allosteric and kinetic constants; moreover, they seem to be favored by substrate activation. The model is applied to phosphofructokinase, which is the enzyme chiefly responsible for glycolytic oscillations and which presents the same pattern of regulation as the allosteric enzyme appearing in the model. A qualitative and quantitative agreement is obtained with the experimental observations concerning glycolytic self-oscillations.

Limit Cycle Models for Circadian Rhythms Based on Transcriptional Regulation in Drosophila and Neurospora

We examine theoretical models for circadian oscillations based on transcriptional regulation in Drosophila and Neurospora. For Drosophila, the molecular model is based on the negative feedback exerted on the expression of the per and tim genes by the complex formed between the PER and TIM proteins. For Neurospora, similarly, the model relies on the feedback exerted on the expression of the frq gene by its protein product FRQ. In both models, sustained rhythmic variations in protein and mRNA levels occur in continuous darkness, in the form of limit cycle oscillations. The effect of light on circadian rhythms is taken into account in the models by considering that it triggers degradation of the TIM protein in Drosophila, and frq transcription in Neurospora. When incorporating the control exerted by light at the molecular level, we show that the models can account for the entrainment of circadian rhythms by light-dark cycles and for the damping of the oscillations in constant light, though such damping occurs more readily in the Drosophila model. The models account for the phase shifts induced by light pulses and allow the construction of phase response curves. These compare well with experimental results obtained in Drosophila. The model for Drosophila shows that when applied at the appropriate phase, light pulses of appropriate duration and magnitude can permanently or transiently suppress circadian rhythmicity. We investigate the effects of the magnitude of light-induced changes on oscillatory behavior. Finally, we discuss the common and distinctive features of circadian oscillations in the two organisms.

A model based on receptor desensitization for cyclic AMP signaling in Dictyostelium cells

We analyze a model based on receptor modification for the cAMP signaling system that controls aggregation of the slime mold Dictyostelium discoideum after starvation. The model takes into account both the desensitization of the cAMP receptor by reversible phosphorylation and the activation of adenylate cyclase that follows binding of extracellular cAMP to the unmodified receptor. The dynamics of the signaling system is studied in terms of three variables, namely, intracellular and extracellular cAMP, and the fraction of receptor in active state. Using parameter values collected from experimental studies on cAMP signaling and receptor phosphorylation, we show that the model accounts qualitatively and, in a large measure, quantitatively for the various modes of dynamic behavior observed in the experiments: (a) autonomous oscillations of cAMP, (b) relay of suprathreshold cAMP pulses, i.e., excitability, characterized by both an absolute and a relative refractory period, and (c) adaptation to constant cAMP stimuli. A two-variable version of the model is used to demonstrate the link between excitability and oscillations by phase plane analysis. The response of the model to repetitive stimulation allows comprehension, in terms of receptor desensitization, of the role of periodic signaling in Dictyostelium and, more generally, the function of pulsatile patterns of hormone secretion.

Modeling the segmentation clock as a network of coupled oscillations in the Notch, Wnt and FGF signaling pathways

The formation of somites in the course of vertebrate segmentation is governed by an oscillator known as the segmentation clock, which is characterized by a period ranging from 30 min to a few hours depending on the organism. This oscillator permits the synchronized activation of segmentation genes in successive cohorts of cells in the presomitic mesoderm in response to a periodic signal emitted by the segmentation clock, thereby defining the future segments. Recent microarray experiments [Dequeant, M.L., Glynn, E., Gaudenz, K., Wahl, M., Chen, J., Mushegian, A., Pourquie, O., 2006. A complex oscillating network of signaling genes underlies the mouse segmentation clock. Science 314, 1595-1598] indicate that the Notch, Wnt and Fibroblast Growth Factor (FGF) signaling pathways are involved in the mechanism of the segmentation clock. By means of computational modeling, we investigate the conditions in which sustained oscillations occur in these three signaling pathways. First we show that negative feedback mediated by the Lunatic Fringe protein on intracellular Notch activation can give rise to periodic behavior in the Notch pathway. We then show that negative feedback exerted by Axin2 on the degradation of beta-catenin through formation of the Axin2 destruction complex can produce oscillations in the Wnt pathway. Likewise, negative feedback on FGF signaling mediated by the phosphatase product of the gene MKP3/Dusp6 can produce oscillatory gene expression in the FGF pathway. Coupling the Wnt, Notch and FGF oscillators through common intermediates can lead to synchronized oscillations in the three signaling pathways or to complex periodic behavior, depending on the relative periods of oscillations in the three pathways. The phase relationships between cycling genes in the three pathways depend on the nature of the coupling between the pathways and on their relative autonomous periods. The model provides a framework for analyzing the dynamics of the segmentation clock in terms of a network of oscillating modules involving the Wnt, Notch and FGF signaling pathways.

One-pool model for Ca2+ oscillations involving Ca2+ and inositol 1,4,5-trisphosphate as co-agonists for Ca2+ release

Experimental observations indicate that Ca(2+)-induced Ca2+ release (CICR) may underlie Ca2+ oscillations in a variety of cells. In its original version, a theoretical model for signal-induced Ca2+ oscillations based on CICR assumed the existence of two types of pools, one sensitive to inositol 1,4,5-trisphosphate (IP3) and the other one sensitive to Ca2+. Recent experiments indicate that Ca2+ channels may sometimes be sensitive to both IP3 and Ca2+. Such a regulation may be viewed as Ca(2+)-sensitized IP3-induced Ca2+ release or, alternatively, as a form of IP3-sensitized CICR. We show that sustained oscillations can still occur in a one-pool model, provided that the same Ca2+ channels are sensitive to both Ca2+ and IP3 behaving as co-agonists. This model and the two-pool model based on CICR both account for a number of experimental observations but differ in some respects. Thus, while in the two-pool model the latency and period of Ca2+ oscillations are of the same order of magnitude and correlate in a roughly linear manner, latency in the one-pool model is always brief and remains much shorter than the period of oscillations. Moreover, the first Ca2+ spike is much larger than the following ones in the one-pool model. These distinctive properties might provide an explanation for the differences in Ca2+ oscillations observed in various cell types.