Model circadian

P. Smolen, D. A. Baxter, and J. H. Byrne, Modeling circadian oscillations with interlocking positive and negative feedback loops. J. Neurosci. 21, 6644-6656 (2001). 

A common property of biological closed-loop circuits is that they exhibit remarkable robustness to disturbances and fluctuations in operating conditions. For example, the clock should maintain a nearly 24-hr period, even though the organism is exposed to temperature changes, which affect the rates of biochemical reactions. The model circadian clock can be simulated by perturbing values of the kinetic constants. The same clock simulation is evaluated for the following values of the parameter [1.0 1.1 1.5 2.0 4.0]. The period of the... 

Autocatalysis can cause sustained oscillations in batch systems. This idea originally met with skepticism. Some chemists believed that sustained oscillations would violate the second law of thermodynamics, but this is not true. Oscillating batch systems certainly exist, although they must have some external energy source or else the oscillations will eventually subside. An important example of an oscillating system is the circadian rhythm in animals. A simple model of a chemical oscillator, called the Lotka-Volterra reaction, has the assumed mechanism ... 

Hudson, R. and Distel, H. (1989) Temporal pattern of suckling in rabbit pups A model of circadian synchrony between mother and young. In S.M. Reppert (Ed.), Development of Circadian Rhythmicity and Photoperiodism in Mammals. Perinatology Press, Boston, pp. 83-102. 

Circadian variations in basal conditions Absence or presence of an effect compartment The PK model of absorption and disposition and the parameters to be estimated Inclusion or exclusion of specific patient data... 

E. Stochastic Versus Deterministic Models for Circadian Rhythms The Cell-Cycle Clock Newly Discovered Cellular Rhythms... 

Figure 1. In most examples of biological rhythms, sustained oscillations correspond to the evolution toward a hmit cycle. The limit cycle shown here was obtained in a model for circadian oscillations of the PER protein and per mRNA in Drosophila [107].

Molecular models for circadian rhythms were initially proposed [107] for circadian oscillations of the PER protein and its mRNA in Drosophila, the first organism for which detailed information on the oscillatory mechanism became available [100]. The case of circadian rhythms in Drosophila illustrates how the need to incorporate experimental advances leads to a progressive increase in the complexity of theoretical models. A first model governed by a set of five kinetic equations is shown in Fig. 3A it is based on the negative control exerted by the PER protein on the expression of the per gene [107]. Numerical simulations show that for appropriate parameter values, the steady state becomes unstable and limit cycle oscillations appear (Fig. 1). 

Further extensions of the model are required to address the dynamical consequences of these additional regulatory loops and of the indirect nature of the negative feedback on gene expression. Such extended models have been proposed for Drosophila [112, 113] and mammals [113]. The model for the circadian clock mechanism in mammals is schematized in Fig. 3C. The presence of additional mRNA and protein species, as well as of multiple complexes formed between the various clock proteins, complicates the model, which is now governed by a system of 16 or 19 kinetic equations. Sustained or damped oscillations can occur in this model for parameter values corresponding to continuous darkness. As observed in the experiments on the mammalian clock. Email mRNA oscillates in opposite phase with respect to Per and Cry mRNAs [97]. The model displays the property of entrainment by the ED cycle... 

The results obtained with the model for the mammalian circadian clock provide cues for circadian-rhythm-related sleep disorders in humans [117]. Thus permanent phase shifts in LD conditions could account for (a) the familial advanced sleep phase syndrome (FASPS) associated with PER hypopho-sphorylation [118, 119] and (b) the delayed sleep phase syndrome, which is also related to PER [120]. People affected by FASPS fall asleep around 7 30 p.m. and awake around 4 30 a.m. The duration of sleep is thus normal, but the phase is advanced by several hours. Moreover, the autonomous period measured for circadian rhythms in constant conditions is shorter [121]. The model shows that a decrease in the activity of the kinase responsible for PER phosphorylation is indeed accompanied by a reduction of the circadian period in continuous darkness and by a phase advance upon entrainment of the rhythm by the LD cycle [114]. 

For some parameter values the model for the mammalian clock fails to allow entrainment by 24-h LD cycles, regardless of the amplitude of the light-induced change in Per expression. The question arises whether there exists a syndrome corresponding to this mode of dynamic behavior predicted by the model. Indeed there exists such a syndrome, known as the non-24-h sleep-wake syndrome, in which the phase of the sleep-wake pattern continuously varies with respect to the LD cycle that is, the patient free-runs in LD conditions [117]. Disorders of the sleep-wake cycle associated with alterations in the dynamics of the circadian clock belong to the broad class of dynamical diseases [122, 123], although the term syndrome seems more appropriate for some of these conditions. 

Another common perturbation of the circadian clock is the jet lag, which results from an abmpt shift in the phase of the LD cycle to which the rhythm is naturally entrained. The molecular bases of the jet lag are currently being investigated [124]. The model for the circadian clock is being used to probe the various ways by which the clock returns to the limit cycle trajectory after a sudden shift in the phase of the LD cycle. 

Only deterministic models for cellular rhythms have been discussed so far. Do such models remain valid when the numbers of molecules involved are small, as may occur in cellular conditions Barkai and Leibler [127] stressed that in the presence of small amounts of mRNA or protein molecules, the effect of molecular noise on circadian rhythms may become significant and may compromise the emergence of coherent periodic oscillations. The way to assess the influence of molecular noise on circadian rhythms is to resort to stochastic simulations [127-129]. Stochastic simulations of the models schematized in Fig. 3A,B show that the dynamic behavior predicted by the corresponding deterministic equations remains valid as long as the maximum numbers of mRNA and protein molecules involved in the circadian clock mechanism are of the order of a few tens and hundreds, respectively [128]. In the presence of molecular noise, the trajectory in the phase space transforms into a cloud of points surrounding the deterministic limit cycle. 

D. Gonze, M. Roussel, and A. Goldbeter, A model for the enhancement of fitness in cyanobacteria based on resonance of a circadian oscillator with the external light-dark cycle. J. Theor. Biol. 214, 577-597 (2002). 

P. Ruoff, M. Vinsjevik, C. Monneijahn, and L. Rensing, The Goodwin model Simulating the effect of light pulses on the circadian sporulation rhythm of Neurospora crassa. J. Theor. Biol. 209, 29 2 (2001). 

J. C. Leloup and A. Goldbeter, A model for circadian rhythms in Drosophila incorporating the formation of a complex between the PER and TIM proteins. J. Biol. Rhythms 13, 70-87 (1998). 

H. R. Ueda, M. Hagiwara, and H. Kitano, Robust oscillations within the interlocked feedback model of Drosophila circadian rhythm. J. Theor. Biol. 210, 401 106 (2001). 

J. C. Leloup and A. Goldbeter, Toward a detailed computational model for the mammalian circadian clock. Proc. Natl. Acad. Sci. USA 100, 7051-7056 (2003). 

D. Gonze and A. Goldbeter, Entrainment versus chaos in a model for a circadian oscillator driven by light-dark cycles. J. Stat. Phys. 101, 649-663 (2000). 

Controlled chaos may also factor into the generation of rhythmic behavior in living systems. A recently proposed modeL describes the central circadian oscillator as a chaotic attractor. Limit cycle mechanisms have been previously offered to explain circadian clocks and related phenomena, but they are limited to a single stable periodic behavior. In contrast, a chaotic attractor can generate rich dynamic behavior. Attractive features of such a model include versatility of period selection as well as use of control elements of the type already well known for metabolic circuitry. 

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