Magnesium maintains the length of the circadian period in Arabidopsis.

The circadian clock coordinates the physiological responses of a biological system to day and night rhythms through complex loops of transcriptional/translational regulation. It can respond to external stimuli and adjust generated circadian oscillations accordingly to maintain an endogenous period close to 24 h. However, the interaction between nutritional status and circadian rhythms in plants is poorly understood. Magnesium (Mg) is essential for numerous biological processes in plants, and its homeostasis is crucial to maintain optimal development and growth. Magnesium deficiency in young Arabidopsis thaliana seedlings increased the period of circadian oscillations of the CIRCADIAN CLOCK-ASSOCIATED 1 (CCA1) promoter (pCCA1:LUC) activity and dampened their amplitude under constant light in a dose-dependent manner. Although the circadian period increase caused by Mg deficiency was light dependent, it did not depend on active photosynthesis. Mathematical modeling of the Mg input into the circadian clock reproduced the experimental increase of the circadian period and suggested that Mg is likely to affect global transcription/translation levels rather than a single component of the circadian oscillator. Upon addition of a low dose of cycloheximide to perturb translation, the circadian period increased further under Mg deficiency, which was rescued when sufficient Mg was supplied, supporting the model's prediction. These findings suggest that sufficient Mg supply is required to support proper timekeeping in plants.

From circadian clock mechanism to sleep disorders and jet lag: Insights from a computational approach.

We present ten insights that can be gained from computational models based on molecular mechanisms for the mammalian circadian clock. These insights range from the conditions in which circadian rhythms occur spontaneously to their entrainment by the light-dark (LD) cycle and to clock-related disorders of the sleep-wake cycle. Endogenous oscillations originate spontaneously from transcription-translation feedback loops involving clock proteins such as PER, CRY, CLOCK and BMAL1. Circadian oscillations occur in a parameter domain bounded by critical values. Outside this domain the circadian network ceases to oscillate and evolves to a stable steady state. This conclusion bears on the nature of arrhythmic behavior of the circadian clock, which may not necessarily be due to mutations in clock genes. Entrainment by the LD cycle occurs in a certain range of parameter values, with a phase that depends on the endogenous period of the circadian clock. A decrease in PER phosphorylation is accompanied by a decrease in endogenous period and a phase advance of the clock; this situation accounts for the familial, advanced sleep phase syndrome (FASPS). The mirror delayed sleep phase syndrome (DSPS) can be accounted for, similarly, by an increase in PER phosphorylation and a rise in autonomous period. Failure of entrainment by the LD cycle in the model corresponds to the non-24 h sleep-wake cycle syndrome, in which the phase of the circadian clock drifts in the course of time. Quasi-periodic oscillations that develop in these conditions sometimes correspond to long-period patterns in which the circadian clock is nearly entrained for long bouts of time before its phase rapidly drifts until a new regime of quasi-entrainment is re-established. In regard to jet lag, the computational approach accounts for the two modes of re-entrainment observed after an advance or delay which correspond, respectively, to an eastward or westward flight: the clock adjusts in a direction similar (orthodromic) or opposite (antidromic) to that of the shift in the LD cycle. Computational modeling predicts that in the vicinity of the switch between orthodromic and antidromic re-entrainment the circadian clock may take a very long time to resynchronize with the LD cycle. Repetitive perturbations of the circadian clock due, for example, to chronic jet lag -a situation somewhat reminiscent of shift work- may lead to quasi-periodic or chaotic oscillations. The latter irregular oscillations can sometimes be observed in normal LD cycles, raising the question of their possible relevance to fragmented sleep patterns observed in narcolepsy. The latter condition, however, appears to originate from disorders in the orexin neural circuit, which promotes wakefulness, rather than from an irregular operation of the circadian clock.

Modeling the photoperiodic entrainment of the plant circadian clock.

The circadian clock is an endogenous 24 hour rhythm that helps organisms anticipate and adapt to daily and seasonal variations in environment, such as the day/night cycle or changing temperatures. The plant clock is a complex network of transcription factors that regulate each other, forming interlocked feedback loops. Most of its components are light-regulated in some way, making the system highly sensitive to changes in light conditions. Here, we explore the mechanisms by which the plant clock adapts to changing day length. We first present some experimental data illustrating the variety of behaviors found in seedlings exposed to external day/night cycles different from 24h. We then use a mathematical model to characterize the response of the clock to a wide range of external cycle lengths and photoperiods. We show the existence of several domains of periodic entrainment with different ratios between the external cycle length and the period of the clock, and the presence of quasiperiodic and chaotic behaviors outside of the entrainment range. We simulate knockout mutants with impaired clock function and theoretical variants with diminished light sensitivity to highlight the role of a complex network and multiple light inputs in keeping the clock entrained over a wide range of conditions.

A Compact Model for the Complex Plant Circadian Clock.

The circadian clock is an endogenous timekeeper that allows organisms to anticipate and adapt to the daily variations of their environment. The plant clock is an intricate network of interlocked feedback loops, in which transcription factors regulate each other to generate oscillations with expression peaks at specific times of the day. Over the last decade, mathematical modeling approaches have been used to understand the inner workings of the clock in the model plant Arabidopsis thaliana. Those efforts have produced a number of models of ever increasing complexity. Here, we present an alternative model that combines a low number of equations and parameters, similar to the very earliest models, with the complex network structure found in more recent ones. This simple model describes the temporal evolution of the abundance of eight clock gene mRNA/protein and captures key features of the clock on a qualitative level, namely the entrained and free-running behaviors of the wild type clock, as well as the defects found in knockout mutants (such as altered free-running periods, lack of entrainment, or changes in the expression of other clock genes). Additionally, our model produces complex responses to various light cues, such as extreme photoperiods and non-24 h environmental cycles, and can describe the control of hypocotyl growth by the clock. Our model constitutes a useful tool to probe dynamical properties of the core clock as well as clock-dependent processes.

Modeling and simulating the Arabidopsis thaliana circadian clock using XPP-AUTO.

Circadian clocks are endogenous timekeepers that produce oscillations with a period of about one day. Their rhythmicity originates from complex gene regulatory networks at the cellular level. In the last decades, computational models have been proven to be a powerful tool in order to understand the dynamics and design principles of the complex regulatory circuitries underlying the circadian clocks of different organisms. We present the process of model development using a small and simplified two-gene regulatory network of the Arabidopsis circadian clock. Subsequently, we discuss important numerical techniques to analyze such a mathematical model using XPP-AUTO. We show how to solve deterministic and stochastic ordinary differential equations and how to compute bifurcation diagrams or simulate phase-shift experiments. We finally discuss the contributions of modeling to the understanding and dissection of the Arabidopsis circadian system.

Critical phase shifts slow down circadian clock recovery: implications for jet lag.

Advancing or delaying the light-dark (LD) cycle perturbs the circadian clock, which eventually recovers its original phase with respect to the new LD cycle. Readjustment of the clock occurs by shifting its phase in the same (orthodromic re-entrainment) or opposite direction (antidromic re-entrainment) as the shift in the LD cycle. To investigate circadian clock recovery after phase shifts of the LD cycle we use a detailed computational model previously proposed for the cellular regulatory network underlying the mammalian circadian clock. The model predicts the existence of a sharp threshold separating orthodromic from antidromic re-entrainment. In the vicinity of this threshold, resynchronization of the clock after a phase shift markedly slows down. The type of re-entrainment, the position of the threshold and the time required for resynchronization depend on multiple factors such as the autonomous period of the clock, the direction and magnitude of the phase shift, the clock biochemical kinetic parameters, and light intensity. Partitioning the phase shift into a series of smaller phases shifts decreases the impact on the recovery of the circadian clock. We use the phase response curve to predict the location of the threshold separating orthodromic and antidromic re-entrainment after advanced or delayed phase shifts of the LD cycle. The marked increase in recovery times predicted near the threshold could be responsible for the most severe disturbances of the human circadian clock associated with jet lag.

[Circadian rhythms and systems biology]. / Biologie des systèmes et rythmes cellulaires.

Cellular rhythms represent a field of choice for studies in system biology. The examples of circadian rhythms and of the cell cycle show how the experimental and modeling approaches contribute to clarify the conditions in which periodic behavior spontaneously arises in regulatory networks at the cellular level. Circadian rhythms originate from intertwined positive and negative feedback loops controlling the expression of several clock genes. Models can be used to address the dynamical bases of physiological disorders related to dysfunctions of the mammalian circadian clock. The cell cycle is driven by a network of cyclin-dependent kinases (Cdks). Modeled in the form of four modules coupled through multiple regulatory interactions, the Cdk network operates in an oscillatory manner in the presence of sufficient amounts of growth factor. For circadian rhythms and the cell cycle, as for other recently observed cellular rhythms, periodic behavior represents an emergent property of biological systems related to their regulatory structure.

Modeling the circadian clock: from molecular mechanism to physiological disorders.

Based on genetic and biochemical advances on the molecular mechanism of circadian rhythms, a computational model for the mammalian circadian clock is used to examine the dynamical bases of circadian-clock-related physiological disorders in humans. Entrainment by the light-dark cycle with a phase advance or a phase delay is associated with the Familial advanced sleep phase syndrome (FASPS) or the Delayed sleep phase syndrome (DSPS), respectively. Lack of entrainment corresponding to the occurrence of quasiperiodic oscillations with or without phase jump can be associated with the non-24 h sleep-wake syndrome. In the close vicinity of the entrainment domain, the model uncovers the possibility of infradian oscillations of very long period. Perturbation in the form of chronic jet lag, as used in mice, prevents entrainment of the circadian clock and results in chaotic or quasiperiodic oscillations. It is important to clarify the conditions for entrainment and for its failure because dysfunctions of the circadian clock may lead to physiological disorders, which pertain not only to the sleep-wake cycle but also to mood and cancer.

Modeling the mammalian circadian clock: sensitivity analysis and multiplicity of oscillatory mechanisms.

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 light-dark (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-24h sleep-wake syndrome.

Development and validation of computational models for mammalian circadian oscillators.

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.

Circadian rhythms of locomotor activity in Drosophila.

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.

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 alpha 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-Erbalpha 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.

Stochastic models for circadian rhythms: effect of molecular noise on periodic and chaotic behaviour.

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 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 PER-TIM 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.

From simple to complex oscillatory behavior in metabolic and genetic control networks.

We present an overview of mechanisms responsible for simple or complex oscillatory behavior in metabolic and genetic control networks. Besides simple periodic behavior corresponding to the evolution toward a limit cycle we consider complex modes of oscillatory behavior such as complex periodic oscillations of the bursting type and chaos. Multiple attractors are also discussed, e.g., the coexistence between a stable steady state and a stable limit cycle (hard excitation), or the coexistence between two simultaneously stable limit cycles (birhythmicity). We discuss mechanisms responsible for the transition from simple to complex oscillatory behavior by means of a number of models serving as selected examples. The models were originally proposed to account for simple periodic oscillations observed experimentally at the cellular level in a variety of biological systems. In a second stage, these models were modified to allow for complex oscillatory phenomena such as bursting, birhythmicity, or chaos. We consider successively (1) models based on enzyme regulation, proposed for glycolytic oscillations and for the control of successive phases of the cell cycle, respectively; (2) a model for intracellular Ca(2+) oscillations based on transport regulation; (3) a model for oscillations of cyclic AMP based on receptor desensitization in Dictyostelium cells; and (4) a model based on genetic regulation for circadian rhythms in Drosophila. Two main classes of mechanism leading from simple to complex oscillatory behavior are identified, namely (i) the interplay between two endogenous oscillatory mechanisms, which can take multiple forms, overt or more subtle, depending on whether the two oscillators each involve their own regulatory feedback loop or share a common feedback loop while differing by some related process, and (ii) self-modulation of the oscillator through feedback from the system's output on one of the parameters controlling oscillatory behavior. However, the latter mechanism may also be viewed as involving the interplay between two feedback processes, each of which might be capable of producing oscillations. Although our discussion primarily focuses on the case of autonomous oscillatory behavior, we also consider the case of nonautonomous complex oscillations in a model for circadian oscillations subjected to periodic forcing by a light-dark cycle and show that the occurrence of entrainment versus chaos in these conditions markedly depends on the wave form of periodic forcing. (c) 2001 American Institute of Physics.