Comparison of amplitude recovery dynamics of two limit cycle oscillator models of the human circadian pacemaker

At an organism level, the mammalian circadian pacemaker is a two-dimensional system. For these two dimensions, phase (relative timing) and amplitude of the circadian pacemaker are commonly used. Both the phase and the amplitude (A) of the human circadian pacemaker can be observed within multiple physiological measures--including plasma cortisol, plasma melatonin, and core body temperature (CBT)--all of which are also used as markers of the circadian system. Although most previous work has concentrated on changes in phase of the circadian system, critically timed light exposure can significantly reduce the amplitude of the pacemaker. The rate at which the amplitude recovers to its equilibrium level after reduction can have physiological significance. Two mathematical models that describe the phase and amplitude dynamics of the pacemaker have been reported. These models are essentially equivalent in predictions of phase and in predictions of amplitude recovery for small changes from an equilibrium value (A = 1), but are markedly different in the prediction of recovery rates when A < 0.6. To determine which dynamic model best describes the amplitude recovery observed in experimental data; both models were fit to CBT data using a maximum likelihood procedure and compared using Akaike's Information Criterion (AIC). For all subjects, the model with the lower recovery rate provided a better fit to data in terms of AIC, supporting evidence that the amplitude recovery of the endogenous pacemaker is slow at low amplitudes. Experiments derived from model predictions are proposed to test the influence of low amplitude recovery on the physiological and neurobehavioral functions.     

Phase organization of circadian oscillators in extended gate and oscillator models

The suprachiasmatic nuclei (SCN) control daily oscillations in physiology and behavior. The gate-oscillator model captures functional heterogeneity in SCN and has been successful in reproducing many features of SCN. This paper investigates the mechanism of phase organization in the gate-oscillator model and finds that only stable fixed points of the phase transition function are essential to phase organization. This obvious finding forms the basis for understanding the complex phase distribution in the gate-oscillator scheme. Extending the model with a dead zone of the phase transition function and the propagation delay of the gate signal which may represent the spatial structure of SCN, the author discusses how some features of experimentally reported phase distribution, such as the existence of anti-phase neurons and fixed phase difference between neurons, could be understood in the framework of the gate-oscillator model. The extended model shows clearly the way in which the interplay between the single-cell property and the property of the network organization influence the phase distribution of SCN neurons.     

Population dynamics of clock-controlled biological species: models and why circadian rhythms are circadian

The endolymph flow inside the semicircular ducts is analytically investigated by considering a system of two hydrodynamically interconnected ducts. Rotation of this system adds an amount of motion (momentum) to parts of it. This results in an endolymph flow in generally all vestibular parts. The "external impulses" are the impulses which emerge by rotation of exclusively a particular vestibular part. The real impulses can be calculated from a set of equations which contain the external impulses. Analytical expressions are derived for the initial velocities in the ducts and for the maximum endolymph displacements. These formulae contain the external impulses and the ratios of: (1) the radii of crus commune and ducts (gamma), (2) the lengths of crus commune and ducts (lambda). It was proven that an interconnected system composed of two ducts, and also a system composed of two such semicircular duct systems, behaves as a pure rotation transducer (like a single duct does), also when it is rotated excentrically. Duct systems with polygonal and circular geometries were used to evaluate whether an optimal value of lambda would exist (gamma was already considered elsewhere). Optimum values of lambda in a range of about 0.10-0.52 were found. This rather wide range of values agrees with values from measurements. Optimization of an interconnected duct system appeared to be equal to optimization of a system composed of separate ducts.     

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.     

Multiple photopigments entrain the Mammalian circadian oscillator

Circadian rhythms are entrained to the natural day:night cycle. Melanopsin expressed in retinal ganglion cells partially accounts for circadian photoentrainment. Dkhissi-Benyahya et al. demonstrate that medium wavelength opsin (MW-opsin) also plays an important role in the process. Furthermore, they develop a model explaining wavelength-dependent photoentrainment by melanopsin and MW-opsin.     

Nature of KaiB-KaiC binding in the cyanobacterial circadian oscillator

In the cyanobacteria Synechococcus elongatus and Thermosynechococcus elongatus, the KaiA, KaiB and KaiC proteins in the presence of ATP generate a post-translational oscillator (PTO) that can be reconstituted in vitro. KaiC is the result of a gene duplication and resembles a double doughnut with N-terminal CI and C-terminal CII hexameric rings. Six ATPs are bound between subunits in both the CI and CII ring. CI harbors ATPase activity, and CII catalyzes phosphorylation and dephosphorylation at T432 and S431 with a ca. 24-h period. KaiA stimulates KaiC phosphorylation, and KaiB promotes KaiC subunit exchange and sequesters KaiA on the KaiB-KaiC interface in the final stage of the clock cycle. Studies of the PTO protein-protein interactions are convergent in terms of KaiA binding to CII but have led to two opposing models of the KaiB-KaiC interaction. Electron microscopy (EM) and small angle X-ray scattering (SAXS), together with native PAGE using full-length proteins and separate CI and CII rings, are consistent with binding of KaiB to CII. Conversely, NMR together with gel filtration chromatography and denatured PAGE using monomeric CI and CII domains support KaiB binding to CI. To resolve the existing controversy, we studied complexes between KaiB and gold-labeled, full-length KaiC with negative stain EM. The EM data clearly demonstrate that KaiB contacts the CII ring. Together with the outcomes of previous analyses, our work establishes that only CII participates in interactions with KaiA and KaiB as well as with the His kinase SasA involved in the clock output pathway.     

The methamphetamine-sensitive circadian oscillator (MASCO) in mice

The suprachiasmatic nucleus (SCN) orchestrates synchrony among many peripheral oscillators and is required for circadian rhythms of locomotor activity and many physiological processes. However, the unique effects of methamphetamine (MAP) on circadian behavior suggest the presence of an SCN-independent, methamphetamine-sensitive circadian oscillator (MASCO). Substantial data collected using rat models show that chronic methamphetamine dramatically lengthens circadian period of locomotor activity rhythms and induces rhythms in animals lacking an SCN. However, the anatomical substrate and the molecular components of the MASCO are unknown. The response to MAP is less well studied in mice, a model that would provide the genetic tools to probe the molecular components of this extra-SCN oscillator. The authors tested the effects of chronic MAP on 2 strains of intact and SCN-lesioned mice in constant dark and constant light. Furthermore, they applied various MAP availability schedules to SCN-lesioned mice to confirm the circadian nature of the underlying oscillator. The results indicate that this oscillator has circadian properties. In intact mice, the MASCO interacts with the SCN in a manner that is strain, sex, and dose dependent. In SCN-lesioned mice, it induces robust free-running locomotor rhythmicity, which persists for up to 14 cycles after methamphetamine is withdrawn. In the future, localization of the MASCO and characterization of its underlying molecular mechanism, as well as its interactions with other oscillators in the body, will be essential to a complete understanding of the organization of the mammalian circadian system.     

Mechanics and resonance of the cyanobacterial circadian oscillator

Recent experiments elucidated the structure and function of the cyanobacterial circadian oscillator, which is driven by sunlight intensity variation and therefore by Earth's rotation. It is known that cyanobacteria appeared about 3.5 billion years ago and that Earth's rotational speed is continuously decreasing because of tidal friction. What is the effect of the continuous slowdown of Earth's rotation on the operation of the cyanobacterial oscillator? To answer this question we derived the oscillator's equation of motion directly from experimental data, coupled it with Earth's rotation and computed its natural periods and its resonance curve. The results show that there are two resonance peaks of the “cyanobacterial oscillator-rotating Earth” system, indicating that cyanobacteria used more efficiently the solar energy during the geological period in which the day length varied from about 11 to 15 h and make more efficient use of solar energy at the geological period which started with a day length of 21 h and will end at a day length of 28 h.     

A circadian oscillator model based on empirical data

A model based on the van der Pol equation has been developed to predict the pattern of adaptation of aircrew and other travellers to rapid time-zone transitions, when the exposure to light cannot be quantified. The parameters of the model include the stiffness (mu) and the intrinsic period (T0), which together define the free-running period, and the external force (F). The parameter values were estimated by using a simplex minimization technique to fit the output from the model to body temperature data from 12 individuals before, and over a 12-day period immediately after, a 10-h eastward transition between London and Sydney. Data were collected at three equally spaced points during each sleep period and at the end of four 45-min rest periods during the day. The fitting procedure enabled the parameters of the temperature rhythm to be estimated after correcting for the masking effect of sleep. The average estimates of mu (0.38 h) and T0 (24.24 h) were close to earlier estimates based on forced desynchronization experiments, and the mean free-running period, calculated from these, was 24.50 h. The mean value of the external force F (0.54) was surprisingly high, and this may reflect the strong outdoor light levels during the days in Sydney. Estimates of phase, based on the model solutions, suggested that 11 subjects adapted by a phase delay and 1 by a phase advance. However, the amplitude of the rhythms was much reduced at times when the phase was changing rapidly. Simulations using the range of the model parameters for the 12 individuals predicted that adaptation to within 1 h after a 10-h eastward transition would be achieved within between 3 and 11 days. However, since these predictions are dependent on the choice of external force, estimates may need to be more conservative in real-life situations when light exposure cannot be measured.     

The molecular clockwork of a protein-based circadian oscillator

The circadian clock of the cyanobacterium Synechococcuselongatus PCC 7942 is governed by a core oscillator consisting of the proteins KaiA, KaiB, and KaiC. Remarkably, circadian oscillations in the phosphorylation state of KaiC can be reconstituted in a test tube by mixing the three Kai proteins and adenosine triphosphate. The in vitro oscillator provides a well-defined system in which experiments can be combined with mathematical analysis to understand the mechanism of a highly robust biological oscillator. In this Review, we summarize the biochemistry of the Kai proteins and examine models that have been proposed to explain how oscillations emerge from the properties of the oscillator’s constituents.