Revisiting spontaneous internal desynchrony using a quantitative model of sleep physiology

Early attempts to characterize free-running human circadian rhythms generated three notable results: 1) observed circadian periods of 25 hours (considerably longer than the now established 24.1- to 24.2-hour average intrinsic circadian period) with sleep delayed to later circadian phases than during entrainment; 2) spontaneous internal desynchrony of circadian rhythms and sleep/wake cycles--the former with an approximately 24.9-hour period, and the latter with a longer (28-68 hour) or shorter (12-20 hour) period; and 3) bicircadian (48-50 hour) sleep/wake cycles. All three results are reproduced by Kronauer et al.'s (1982) coupled oscillator model, but the physiological basis for that phenomenological model is unclear. We use a physiologically based model of hypothalamic and brain stem nuclei to investigate alternative physiological mechanisms that could underlie internal desynchrony. We demonstrate that experimental observations can be reproduced by changes in two pathways: promotion of orexinergic (Orx) wake signals, and attenuation of the circadian signal reaching hypothalamic nuclei. We reason that delayed sleep is indicative of an additional wake-promoting drive, which may be of behavioral origin, associated with removal of daily schedules and instructions given to participants. We model this by increasing Orx tone during wake, which reproduces the observed period lengthening and delayed sleep. Weakening circadian input to the ventrolateral preoptic nucleus (possibly mediated by the dorsomedial hypothalamus) causes desynchrony, with observed sleep/wake cycle period determined by degree of Orx up-regulation. During desynchrony, sleep/wake cycles are driven by sleep homeostasis, yet sleep bout length maintains circadian phase dependence. The model predicts sleep episodes are shortest when started near the temperature minimum, consistent with experimental findings. The model also correctly predicts that it is possible to transition to bicircadian rhythms from either a synchronized or desynchronized state. Our findings suggest that feedback from behavioral choices to physiology could play an important role in spontaneous internal desynchrony.     

A model for the enhancement of fitness in cyanobacteria based on resonance of a circadian oscillator with the external light-dark cycle

Fitness enhancement based on resonating circadian clocks has recently been demonstrated in cyanobacteria [Ouyang et al. (1998). Proc. Natl Acad. Sci. U.S.A.95, 8660-8664]. Thus, the competition between two cyanobacterial strains differing by the free-running period (FRP) of their circadian oscillations leads to the dominance of one or the other of the two strains, depending on the period of the external light-dark (LD) cycle. The successful strain is generally that which has an FRP closest to the period of the LD cycle. Of key importance for the resonance phenomenon are observations which indicate that the phase angle between the circadian oscillator and the LD cycle depends both on the latter cycle's length and on the FRP. We account for these experimental observations by means of a theoretical model which takes into account (i) cell growth, (ii) secretion of a putative cell growth inhibitor, and (iii) the existence of a cellular, light-sensitive circadian oscillator controlling growth as well as inhibitor secretion. Building on a previous analysis in which the phase angle was considered as a freely adjustable parameter [Roussel et al. (2000). J. theor. Biol.205, 321-340], we incorporate into the model a light-sensitive version of the van der Pol oscillator to represent explicitly the cellular circadian oscillator. In this way, the model automatically generates a phase angle between the circadian oscillator and the LD cycle which depends on the characteristic FRP of the strain and varies continuously with the period of the LD cycle. The model provides an explanation for the results of competition experiments between strains of different FRPs subjected to entrainment by LD cycles of different periods. The model further shows how the dominance of one strain over another in LD cycles can be reconciled with the observation that two strains characterized by different FRPs nevertheless display the same growth kinetics in continuous light or in LD cycles when present alone in the medium. Theoretical predictions are made as to how the outcome of competition depends on the initial proportions and on the FRPs of the different strains. We also determine the effect of the photoperiod and extend the analysis to the case of a competition between three cyanobacterial strains.     

Amplitude of circadian oscillations entrained by 24-h light-dark cycles

An intriguing property of circadian clocks is that their free-running period is not exactly 24h. Using models for circadian rhythms in Neurospora and Drosophila, we determine how the entrainment of these rhythms is affected by the free-running period and by the amplitude of the external light-dark cycle. We first consider the model for Neurospora, in which light acts by inducing the expression of a clock gene. We show that the amplitude of the oscillations of the clock protein entrained by light-dark cycles is maximized when the free-running period is smaller than 24h. Moreover, if the amplitude of the light-dark cycle is very strong, complex oscillations occur when the free-running period is close to 24h. In the model for circadian rhythms in Drosophila, light acts by enhancing the degradation of a clock protein. We show that while the amplitude of circadian oscillations entrained by light-dark cycles is also maximized if the free-running period is smaller than 24h, the range of entrainment is centered around 24h in this model. We discuss the physiological relevance of these results in regard to the setting of the free-running period of the circadian clock.     

The effect of repeated pulses of light at the same time on period responses of the rat circadian pacemaker

A consequence of simple velocity-based models is that, in response to light pulses, the circadian period should adjust inversely to phase. In addition, because of the interaction of circadian period and phase response, earlier circadian period changes should modify later circadian period changes. The literature contains few mentions of response curves of circadian period responses following light pulses. Rats were exposed to four pulses of light (60 minutes, 1000 lux) at the same circadian time, a minimum of 26 days apart; we assessed period responses and possible bias in the period-response curve. Modulation of circadian period following light-induced phase responses was examined by assessing the period of running wheel activity onset. Phase and circadian period were not consistently found to share an inverse relationship. Moreover, biases in initial period tended to be increased by the experimental protocol regardless of circadian time of pulse. Rats with a short initial (high-velocity) period had a lengthened period, while rats with a long initial period (low velocity) tended to have a reduce period. However, rats with a long initial period were phase delay biased, not phase advance biased. These results do not support a simple velocity model of the pacemaker. (Chronobiology International, 18(2), 187-201, 2001)     

Bifurcations in a mathematical model for circadian oscillations of clock genes

Circadian oscillations with a period of about 24h are observed in nearly all living organisms as conspicuous biological rhythms. In this paper, we investigate various kinds of bifurcation phenomena produced in a circadian oscillator model of Drosophila. In Drosophila, it is known that circadian oscillations in the levels of two proteins, PER and TIM, result from the negative feedback exerted by a PER-TIM complex on the expression of the per and tim genes that code for the two proteins. For studying circadian oscillations of proteins in Drosophila, a mathematical model has been proposed. The model cannot only account for regular circadian oscillations in environmental conditions such as constant darkness, but also give rise to more complex oscillatory phenomena including chaos and birhythmicity. By calculating bifurcations using Kawakami's method, we obtain detailed bifurcation diagrams related to stable and unstable invariant sets, and identify parameter regions in which the model generates complex oscillations as well as regular circadian oscillations. Moreover, we study bifurcations observed in the model incorporating the effect on a light-dark (LD) cycle and show that the waveform of the periodic variation in the light-induced parameter has a marked influence on the global bifurcation structure or the type of dynamic behavior resulting from the forcing term of the circadian oscillator by the LD cycles.     

Real-time monitoring of chloroplast gene expression by a luciferase reporter: evidence for nuclear regulation of chloroplast circadian period

Chloroplast-encoded genes, like nucleus-encoded genes, exhibit circadian expression. How the circadian clock exerts its control over chloroplast gene expression, however, is poorly understood. To facilitate the study of chloroplast circadian gene expression, we developed a codon-optimized firefly luciferase gene for the chloroplast of Chlamydomonas reinhardtii as a real-time bioluminescence reporter and introduced it into the chloroplast genome. The bioluminescence of the reporter strain correlated well with the circadian expression pattern of the introduced gene and satisfied all three criteria for circadian rhythms. Moreover, the period of the rhythm was lengthened in per mutants, which are phototactic rhythm mutants carrying a long-period gene in their nuclear genome. These results demonstrate that chloroplast gene expression rhythm is a bona fide circadian rhythm and that the nucleus-encoded circadian oscillator determines the period length of the chloroplast rhythm. Our reporter strains can serve as a powerful tool not only for analysis of the circadian regulation mechanisms of chloroplast gene expression but also for a genetic approach to the molecular oscillator of the algal circadian clock.     

Properties of a feedback model for the circadian clock of Hemideina thoracica (Orthoptera; Stenopelmatidae)

The single oscillator feedback model describing the circadian system of the nocturnal insect, Hemideina thoracica, (Gander and Lewis, 1979) has been developed and refined by restricting the range of parameter values which successfully simulate the known behaviour of this insect's circadian system. The model accounts for Aschoffs Law for nocturnal animals, and makes new predictions on the combined effects of temperature and constant light on the period of the free-running rhythm, which were verified experimentally. These simulations also indicate that the same general feedback model can be used to describe the circadian systems of other organisms.     

Diverse development and higher sensitivity of the circadian clocks to changes in maternal-feeding regime in a rat model of cardio-metabolic disease

Spontaneously hypertensive rats (SHR) develop cardiovascular and metabolic pathology in adulthood when their circadian system exhibits significant aberrances compared with healthy control rats. This study was aimed to elucidate how the SHR circadian system develops during ontogenesis and to assess its sensitivity to changes in maternal-feeding regime. Analysis of ontogenesis of clock gene expression rhythms in the suprachiasmatic nuclei, liver and colon revealed significant differences in SHR compared with Wistar rats. In the suprachiasmatic nuclei of the hypothalamus (SCN) and liver, the development of a high-amplitude expression rhythm selectively for Bmal1 was delayed compared with Wistar rat. The atypical development of the SHR circadian clocks during postnatal ontogenesis might arise from differences in maternal behavior between SHR and Wistar rats that were detected soon after delivery. It may also arise from higher sensitivity of the circadian clocks in the SHR SCN, liver and colon to maternal behavior related to changes in the feeding regime because in contrast to Wistar rat, the SHR SCN and peripheral clocks during the prenatal period and the hepatic clock during the early postnatal period were phase shifted due to exposure of mothers to a restricted feeding regime. The maternal restricted feeding regime shifted the clocks despite the fact that the mothers were maintained under the light/dark cycle. Our findings of the diverse development and higher sensitivity of the developing circadian system of SHR to maternal cues might result from previously demonstrated differences in the SHR circadian genotype and may potentially contribute to cardiovascular and metabolic diseases, which the animal model spontaneously develops.     

Revised limit cycle oscillator model of human circadian pacemaker

In 1990, Kronauer proposed a mathematical model of the effects of light on the human circadian pacemaker. This study presents several refinements to Kronauer's original model of the pacemaker that enable it to predict more accurately the experimental results from a number of different studies of the effects of the intensity, timing, and duration of light stimuli on the human circadian pacemaker. These refinements include the following: The van der Pol oscillator from Kronauer's model has been replaced with a higher order limit cycle oscillator so that the system's amplitude recovery is slower near the singularity and faster near the limit cycle; the phase and amplitude of the circadian rhythm in sensitivity to light from Kronauer's model has been refined so that the peak sensitivity to light on the limit cycle now occurs approximately 4 h before the core body temperature minimum (CBTmin) and is three times as great as the minimum sensitivity on the limit cycle; the critical phase (at which type 1 phase response curves [PRCs] can be distinguished from type 0 PRCs) that occurs at CBT,n now corresponds to 0.8 h after the minimum of x (x(min) in this refined model rather than to the exact timing of x(min) as in Kronauer's model; a direct effect of light on circadian period was incorporated into the model such that as light intensity increases, the period decreases, which is in accordance with Aschoff's rule.     

Internal noise-driven circadian oscillator in Drosophila

An internal noise-driven oscillator was studied in a two-variable Drosophila model, where both positive feedback and negative feedback are crucial to the circadian oscillations. It is shown that internal noise could sustain reliable oscillations for the parameter which produces a stable steady state in the deterministic system. The noise-sustained oscillations are interpreted by using phase plane analysis. The period of such oscillations fluctuates slightly around the period of deterministic oscillations and the coherence of oscillations becomes the best at an optimal internal noise intensity, indicating the occurrence of intrinsic coherence resonance. In addition, in the oscillatory region, the coherence of noisy circadian oscillations is suppressed by the internal noise, but the period is hardly affected, demonstrating the robustness of the Drosophila model for circadian rhythms to the intrinsic noise.     

The period length of fibroblast circadian gene expression varies widely among human individuals

Mammalian circadian behavior is governed by a central clock in the suprachiasmatic nucleus of the brain hypothalamus, and its intrinsic period length is believed to affect the phase of daily activities. Measurement of this period length, normally accomplished by prolonged subject observation, is difficult and costly in humans. Because a circadian clock similar to that of the suprachiasmatic nucleus is present in most cell types, we were able to engineer a lentiviral circadian reporter that permits characterization of circadian rhythms in single skin biopsies. Using it, we have determined the period lengths of 19 human individuals. The average value from all subjects, 24.5 h, closely matches average values for human circadian physiology obtained in studies in which circadian period was assessed in the absence of the confounding effects of light input and sleep–wake cycle feedback. Nevertheless, the distribution of period lengths measured from biopsies from different individuals was wider than those reported for circadian physiology. A similar trend was observed when comparing wheel-running behavior with fibroblast period length in mouse strains containing circadian gene disruptions. In mice, inter-individual differences in fibroblast period length correlated with the period of running-wheel activity; in humans, fibroblasts from different individuals showed widely variant circadian periods. Given its robustness, the presented procedure should permit quantitative trait mapping of human period length.     

Sleep initiation and initial sleep intensity: interactions of homeostatic and circadian mechanisms

Sleep initiation and sleep intensity in humans show a dissimilar time course. The propensity of sleep initiation (PSI), as measured by the multiple sleep latency test, remains at a relatively constant level throughout the habitual period of waking or exhibits a midafternoon peak. When waking is extended into the sleep period, PSI rises rapidly within a few hours. In contrast, sleep intensity, as measured by electroencephalographic slow-wave activity during naps, shows a gradual increase during the period of habitual waking. In the two-process model of sleep regulation, it corresponds to the rising limb of the homeostatic Process S. We propose that PSI is determined by the difference between Process S and the threshold H defining sleep onset, which is modulated by the circadian process C. In contrast to a previous version of the model, the parameters of H (amplitude, phase, skewness) differ from those of threshold L, which defines sleep termination. The present model is able to simulate the time course of PSI under baseline conditions as well as following recovery sleep after extended sleep deprivation. The simulations suggest that during the regular period of waking, a circadian process counteracts the increasing sleep propensity induced by a homeostatic process. Data obtained in the rat indicate that during the circadian period of predominant waking, a circadian process prevents a major intrusion of sleep.     

A phase dynamics model of human circadian rhythms

Nonphotic entrainment of an overt sleep-wake rhythm and a circadian pacemaker-driving temperature/melatonin rhythm suggests existence of feedback mechanisms in the human circadian system. In this study, the authors constructed a phase dynamics model that consisted of two oscillators driving temperature/melatonin and sleep-wake rhythms, and an additional oscillator generating an overt sleep-wake rhythm. The feedback mechanism was implemented by modifying couplings between the constituent oscillators according to the history of correlations between them. The model successfully simulated the behavior of human circadian rhythms in response to forced rest-activity schedules under free-run situations: the sleep-wake rhythm is reentrained with the circadian pacemaker after release from the schedule, there is a critical period for the schedule to fully entrain the sleep-wake rhythm, and the forced rest-activity schedule can entrain the circadian pacemaker with the aid of exercise. The behavior of human circadian rhythms was reproduced with variations in only a few model parameters. Because conventional models are unable to reproduce the experimental results concerned here, it was suggested that the feedback mechanisms included in this model underlie nonphotic entrainment of human circadian rhythms.     

Phase response curves of a molecular model oscillator: implications for mutual coupling of paired oscillators

Increasing evidence indicates that the accessory medulla is the circadian pacemaker controlling locomotor activity rhythms in insects. A prominent group of neurons of this neuropil shows immunoreactivity to the peptide pigment-dispersing hormone (PDH). In Drosophila melanogaster, the PDH-immunoreactive (PDH-ir) lateral neurons, which also express the clock genes period and timeless, are assumed to be circadian pacemaker cells themselves. In other insects, such as Leucophaea maderae, a subset of apparently homologue PDH-ir cells is a candidate for the circadian coupling pathway of the bilaterally symmetric clocks. Although knowledge about molecular mechanisms of the circadian clockwork is increasing rapidly, very little is known about mechanisms of circadian coupling. The authors used a computer model, based on the molecular feedback loop of the clock genes in D. melanogaster, to test the hypothesis that release of PDH is involved in the coupling between bilaterally paired oscillators. They can show that a combination of all-delay- and all-advance-type interactions between two model oscillators matches best the experimental findings on mutual pacemaker coupling in L. maderae. The model predicts that PDH affects the phosphorylation rate of clock genes and that in addition to PDH, another neuroactive substance is involved in the coupling pathway, via an all-advance type of interaction. The model suggests that PDH and light pulses, represented by two distinct classes of phase response curves, have different targets in the oscillatory feedback loop and are, therefore, likely to act in separate input pathways to the clock.     

A statistical model of the human core-temperature circadian rhythm

We formulate a statistical model of the human core-temperature circadian rhythm in which the circadian signal is modeled as a van der Pol oscillator, the thermoregulatory response is represented as a first-order autoregressive process, and the evoked effect of activity is modeled with a function specific for each circadian protocol. The new model directly links differential equation-based simulation models and harmonic regression analysis methods and permits statistical analysis of both static and dynamical properties of the circadian pacemaker from experimental data. We estimate the model parameters by using numerically efficient maximum likelihood algorithms and analyze human core-temperature data from forced desynchrony, free-run, and constant-routine protocols. By representing explicitly the dynamical effects of ambient light input to the human circadian pacemaker, the new model can estimate with high precision the correct intrinsic period of this oscillator ( approximately 24 h) from both free-run and forced desynchrony studies. Although the van der Pol model approximates well the dynamical features of the circadian pacemaker, the optimal dynamical model of the human biological clock may have a harmonic structure different from that of the van der Pol oscillator.