Circadian clock models

Figure 24.7 Simulation of circadian clock model for varying values of [1.0 (solid), 1.1 (dashed), 1.5 (dash-dot), 2.0 (dotted), 4.0 (asterisk)].

Figure 24.8 Simulation of circadian clock model for entraining signal with period of 20 h.

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. 

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). 

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. 

When the circadian clock says Act Now, it initiates a process that includes the activation of neuromodulatory cells in the brain stem. These cells, strategically located in the noradrenergic locus coeruleus and the serotonergic raphe, constitute what Vernon Mountcastic called a brain within the brain, because their activity (in waking) or inactivity (in REM sleep) make such a difference to the brain s mode of operation. So important are they that they must be accorded a key role in any model that attempts to account for spontaneous and drug-induced alterations in consciousness. 

Fig. 10.2 Waves through cell cycle phases in absence (a, b) or presence (c, d) of entrainment by the circadian clock. The variability of durations for all cell cycle phases is equal to 0% (left column) or 15% (right column). The curves, generated by numerical simulations of the cell cycle automaton model, show the proportions of cells in Cl, S, G2 or M phase as a function of time, for days 10-13. The time step used for simulations is equal to 1 min. The duration of the cell cycle before or in the absence of entrainment is 22 h. The successive phases of the cell cycle have the following mean durations G1 (9 h),...

To determine the effect of circadian rhythms on anticancer drug administration, it is important to incorporate the link between the circadian clock and the cell cycle. Entrainment by the circadian clock can be included in the automaton model by considering that the protein Weel undergoes circadian variation, because the circadian clock proteins CLOCK and BMAL1 induce the expression of the Weel gene (see Fig. 10.1b) [3-5]. Weel is a kinase that phosphorylates and thereby inactivates the protein kinase cdc2 (also known as the cyclin-dependent kinase Cdkl) that controls the transition G2/M and, consequently, the onset of mitosis. 

To clarify the reason why different circadian schedules of 5-FU delivery have distinct cytotoxic effects, we used the cell cycle automaton model to determine the time evolution of the fraction of cells in S phase in response to different patterns of circadian drug administration, for a cell cycle variability of 15%. The results, shown in Fig. 10.5, correspond to the case considered in Fig. 10.4, namely, entrainment of a 22-h cell cycle by the circadian clock. The data for Fig. 10.5a clearly indicate why the circadian schedule with a peak at 4 a.m. is the least toxic. The reason is that the fraction of cells in S phase is then precisely in antiphase with the circadian profile of 5-FU. Since 5-FU only affects cells in the S phase, the circadian delivery of the anticancer drug in this case kills but a negligible amount of cells. 

The results presented here point to the interest of measuring, both in normal and tumor cell populations, parameters such as the duration of the cell cycle phases and their variability, as well as the presence or absence of entrainment by the circadian clock. As shown by the results obtained with the cell cycle automaton model, these data are crucial for using the model to predict the differential outcome of various anticancer drug delivery schedules on normal and tumor cell populations. In a sub-... 

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