A qualitative model of a motor circadian rhythm

In Nature it is possible to observe diverse rhythms. Because of their adaptive characteristics, the circadian rhythms are of major importance and have been the subject of numerous experimental and theoretical studies. In this article, we give a presentation of the main results we have obtained about the motor circadian rhythm along some years of collaboration between biologists and mathematicians. We present a mathematical model simulating changes in frequency, synchronization and amplitude of the circadian oscillation during two developmental stages of the crayfish, namely, the juvenile and the adult stages. We report also some work in progress on the simulation of the phase response curve and on a simplified model of the rhythm.     

Chlamydomonas reinhardtii as a unicellular model for circadian rhythm analysis

Chlamydomonas reinhardtii has been used as an experimental model organism for circadian rhythm research for more than 30 yr. Some of the physiological rhythms of this alga are well established, and several clock mutants have been isolated. The cloning of clock genes from these mutant strains by positional cloning is under way and should give new insights into the mechanism of the circadian clock. In a spectacular space experiment, the question of the existence of an endogenous clock vs. an exogenous mechanism has been studied in this organism. With the emergence of molecular analysis of circadian rhythms in plants in 1985, a circadian gene expression pattern of several nuclear and chloroplast genes was detected. Evidence is now accumulating that shows circadian control at the translational level. In addition, the gating of the cell cycle by the circadian clock has been analyzed. This review focuses on the different aspects of circadian rhythm research in C. reinhardtii over the past 30 yr. The suitability of Chlamydomonas as a model system in chronobiology research and the adaptive significance of the observed rhythms will be discussed.     

Chlamydomonas reinhardtii as a unicellular model for circadian rhythm analysis

Chlamydomonas reinhardtii has been used as an experimental model organism for circadian rhythm research for more than 30 yr. Some of the physiological rhythms of this alga are well established, and several clock mutants have been isolated. The cloning of clock genes from these mutant strains by positional cloning is under way and should give new insights into the mechanism of the circadian clock. In a spectacular space experiment, the question of the existence of an endogenous clock vs. an exogenous mechanism has been studied in this organism. With the emergence of molecular analysis of circadian rhythms in plants in 1985, a circadian gene expression pattern of several nuclear and chloroplast genes was detected. Evidence is now accumulating that shows circadian control at the translational level. In addition, the gating of the cell cycle by the circadian clock has been analyzed. This review focuses on the different aspects of circadian rhythm research in C. reinhardtii over the past 30 yr. The suitability of Chlamydomonas as a model system in chronobiology research and the adaptive significance of the observed rhythms will be discussed.     

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.     

A qualitative mathematical model of the ontogeny of a circadian rhythm in crayfish

Based on experimental work on the ontogeny of the electroretinogram circadian rhythm in crayfish, we present a mathematical model simulating changes in both frequency and amplitude of the electroretinogram oscillation during several developmental stages until shortly before the adult age. Simultaneously, we propose a hypothetical oscillation in the hormonal release whose frequency is imposed on the electroretinogram oscillation. The model consists of two coupled nonlinear oscillators in which a dynamical response is obtained mainly through an Andronov-Hopf bifurcation. Through the construction of the model, a biological hypothesis about the essential elements underlying the ERG circadian rhythm and their interrelations is formulated and discussed.     

Neural network implementation of a three-phase model of respiratory rhythm generation

A mathematical model of the central neural mechanisms of respiratory rhythm generation is developed. This model assumes that the respiratory cycle consists of three phases: inspiration, post-inspiration, and expiration. Five respiratory neuronal groups are included: inspiratory, late-inspiratory, post-inspiratory, expiratory, and early-inspiratory neurons. Proposed interconnections among these groups are based substantially on previous physiological findings. The model produces a stable limit cycle and generally reproduces the features of the firing patterns of the 5 neuronal groups. When simulated feedback from pulmonary stretch receptors is made to excite late-inspiratory neurons and inhibit early-inspiratory neurons, the model quantitatively reproduces previous observations of the expiratory-prolonging effects of pulses and steps of vagal afferent activity presented in expiration. In addition the model reproduces expected respiratory cycle timing and amplitude responses to change of chemical drive both in the absence and in the presence of simulated stretch receptor feedback. These results demonstrate the feasibility of generating the respiratory rhythm with a simple neural network based on observed respiratory neuronal groups. Other neuronal groups not included in the model may be more important for shaping the waveforms than for generating the basic oscillation.     

Congenital anophthalmia: a circadian rhythm study

A circadian rhythm of heart rate and respiratory rate was seen at 1, 8, and 12 months of age in an infant born without ocular tissue, which supports the possibility that the time cues were nonphotic. No melatonin circadian rhythm was detected at any age up to 9 years of age, and this is most likely associated with the anophthalmia and lack of photic input to the suprachiasmatic nucleus. Usually circadian organization is present after the neonatal period and approaches adult levels with development.     

Mathematical model of stabilization of circadian rhythm in cell energy metabolism

A mathematical model for circadian self-oscillation in the carbohydrate branch of energy metabolism (CEM) was analysed. The self-oscillations are due to the reciprocal regulation of the activities of 6-phosphofructokinase and fructose-1,6-bisphosphatase by fructose-1,6-bisphosphate. The circadian period was shown to be insensitive to metabolic disturbances because of the presence in CEM of negative feedback mechanisms regulating the activities of the key enzymes 6-phosphofructokinase, fructose-1,6-bisphosphatase, pyruvate kinase and phosphoenolpyruvate carboxykinase. It has been also shown that such mechanisms are largely synergistic in their action.     

Characterizing the amplitude dynamics of the human core-temperature circadian rhythm using a stochastic-dynamic model

Two measures, amplitude and phase, have been used to describe the characteristics of the endogenous human circadian pacemaker, a biological clock located in the hypothalamus. Although many studies of change in circadian phase with respect to different stimuli have been conducted, the physiologic implications of the amplitude changes (dynamics) of the pacemaker are unknown. It is known that phase changes of the human circadian pacemaker have a significant impact on sleep timing and content, hormone secretion, subjective alertness and neurobehavioral performance. However, the changes in circadian amplitude with respect to different stimuli are less well documented. Although amplitude dynamics of the human circadian pacemaker are observed in physiological rhythms such as plasma cortisol, plasma melatonin and core temperature data, currently methods are not available to accurately characterize the amplitude dynamics from these rhythms. Of the three rhythms core temperature is the only reliable variable that can be monitored continuously in real time with a high sampling rate. To characterize the amplitude dynamics of the circadian pacemaker we propose a stochastic-dynamic model of core temperature data that contains both stochastic and dynamic characteristics. In this model the circadian component that has a dynamic characteristic is represented as a perturbation solution of the van der Pol equation and the thermoregulatory response in the data that has a stochastic characteristic is represented as a first-order autoregressive process. The model parameters are estimated using data with a maximum likelihood procedure and the goodness-of-fit measures along with the associated standard error of the estimated parameters provided inference about the amplitude dynamics of the pacemaker. Using this model we analysed core temperature data from an experiment designed to exhibit amplitude dynamics. We found that the circadian pacemaker recovers slowly to an equilibrium level following amplitude suppression. In humans this reaction to perturbation from equilibrium value has potential physiological implications.     

昼夜节律对呼吸功能的影响

昼夜节律是生物以24小时左右为基本时间单位而发生的生理、行为的周期变化。人类血压、体温、激素分泌等都存在昼夜节律,调节人类的基本活动。研究发现,人类的呼吸功能存在日间和夜间的差异。那么,呼吸是否也存在昼夜节律?呼吸的昼夜节律是否受睡眠、觉醒状态的影响?本文对上述问题进行初步阐述。     

Robust oscillations within the interlocked feedback model of Drosophila circadian rhythm

A mechanism for generating circadian rhythms has been of major interest in recent years. After the discovery of per and tim, a model with a simple feedback loop involving per and tim has been proposed. However, it is recognized that the simple feedback model cannot account for phenotypes generated by various mutants. A recent report by Glossop, Lyons & Hardin [Science286, 766 (1999)] on Drosophila suggests involvement of another feedback loop by dClk that is interlocked with per-tim feedback loop. In order to examine whether interlocked feedback loops can be a basic mechanism for circadian rhythms, a mathematical model was created and examined. Through extensive simulation and mathematical analysis, it was revealed that the interlocked feedback model accounts for the observations that are not explained by the simple feedback model. Moreover, the interlocked feedback model has robust properties in oscillations.     

Dual-scope of circadian rhythm biology

Sleep and Biological Rhythms -     

Simulation of circadian rhythm generation in the suprachiasmatic nucleus with locally coupled self-sustained oscillators

In mammals, circadian rhythms are driven by a pacemaker located in the suprachiasmatic nuclei (SCN) of the anterior hypothalamus. The firing rate of neurons within the SCN exhibits a circadian rhythm. There is evidence that individual neurons within the SCN act as circadian oscillators. Rhythm generation in the SCN was therefore modeled by a system of self-sustained oscillators. The model is composed of up to 10000 oscillatory elements arranged in a square array. Each oscillator has its own (randomly determined) intrinsic period reflecting the widely dispersed periods observed in the SCN. The model behavior was investigated mainly in the absence of synchronizing zeitgebers. Due to local coupling the oscillators synchronized and an overall rhythm emerged. This indicates that a locally coupled system is capable of integrating the output of individual clock cells with widely dispersed periods. The period of the global output (average of all oscillators) corresponded to the average of the intrinsic periods and was stable even for small amplitudes and during transients. Noise, reflecting biological fluctuations at the cellular level, distorted the global rhythm in small arrays. The period of the rhythm could be stabilized by increasing the array size, which thus increased the robustness against noise. Since different regions of the SCN have separate output pathways, the array of oscillators was subdivided into four quadrants. Sudden deviations of periodicity sometimes appeared in one quadrant, while the periods of the other quadrants were largely unaffected. This result could represent a model for splitting, which has been observed in animal experiments. In summary, the multi-oscillator model of the SCN showed a broad repertoire of dynamic patterns, revealed a stable period (even during transients) with robustness against noise, and was able to account for such a complex physiological behavior as splitting.     

昼夜节律系统与成人昼夜节律睡眠觉醒障碍

昼夜节律是存在于所有生命体中、接近24小时的内源性生物节律。昼夜节律与社会或环境节律的长期不同步,会引起睡眠、情绪等一系列变化。本文阐述了昼夜节律系统与睡眠之间的联系,重点介绍成人昼夜节律睡眠觉醒障碍疾病的临床研究成果,以期加强临床医生对该病的认识和诊治。     

Twofold effect of blue light on a circadian rhythm in Acetabularia

Abstract

The circadian chloroplast migration in Acetabularia mediterranea was monitored by continuously measuring the transmission of the cells near the apex. Under continuous red light the amplitude of the rhythm decreased rapidly within a few days. However, circadian changes of chloroplast density were still detectable even after 28 days of red light, indicating the persistence of the rhythm. When blue light was added after red light preirradiation of several days phase shifts were observed which were expressed as advances as well as delays. The period of the rhythm proved to be strongly dependent on the intensity of the continuous blue light which was given in addition to red light. Different red light intensities did not change the period. The occurrence of both effects indicates that the sensory transduction of blue light photoreception in Acetabularia works in two different ways: quanta counting processes and processes of light intensity measurement.