On mathematical modeling of circadian rhythms, performance, and alertness

Mathematical models of neurobehavioral performance and alertness have both basic science and practical applications. These models can be especially useful in predicting the effect of different sleep-wake schedules on human neurobehavioral objective performance and subjective alertness under many conditions. Several relevant models currently exist in the literature. In principle, the development and refinement of any mathematical model should be based on an explicit modeling methodology, such as the Box modeling paradigm, that formally defines the model structure and calculates the set of parameters. While most mathematical models of neurobehavioral performance and alertness include homeostatic, circadian, and sleep inertia components and their interactions, there may be fundamental differences in the equations included in these models. In part, these may be due to differences in the assumptions of the underlying physiology. Because the choice of model equations can have a dramatic influence on the results, it is necessary to consider these differences in assumptions when examining the results from a model and when comparing results across models. This article presents principles of mathematical modeling and examples of how such procedures can be applied to the development and refinement of mathematical models of neurobehavioral performance and alertness. This article also presents several methods of testing and comparing these models, suggests different uses of the models, and discusses problems with current models.     

Development and validation of computational models for mammalian circadian oscillators

Circadian rhythms are endogenous rhythms with a cycle length of approximately 24 h. Rhythmic production of specific proteins within pacemaker structures is the basis for these physiological and behavioral rhythms. Prior work on mathematical modeling of molecular circadian oscillators has focused on the fruit fly, Drosophila melanogaster. Recently, great advances have been made in our understanding of the molecular basis of circadian rhythms in mammals. Mathematical models of the mammalian circadian oscillator are needed to piece together diverse data, predict experimental results, and help us understand the clock as a whole. Our objectives are to develop mathematical models of the mammalian circadian oscillator, generate and test predictions from these models, gather information on the parameters needed for model development, integrate the molecular model with an existing model of the influence of light and rhythmicity on human performance, and make models available in BioSpice so that they can be easily used by the general community. Two new mammalian models have been developed, and experimental data are summarized. These studies have the potential to lead to new strategies for resetting the circadian clock. Manipulations of the circadian clock can be used to optimize performance by promoting alertness and physiological synchronization.     

PERIOD-TIMELESS interval timer may require an additional feedback loop

In this study we present a detailed, mechanism-based mathematical framework of Drosophila circadian rhythms. This framework facilitates a more systematic approach to understanding circadian rhythms using a comprehensive representation of the network underlying this phenomenon. The possible mechanisms underlying the cytoplasmic "interval timer" created by PERIOD-TIMELESS association are investigated, suggesting a novel positive feedback regulatory structure. Incorporation of this additional feedback into a full circadian model produced results that are consistent with previous experimental observations of wild-type protein profiles and numerous mutant phenotypes.     

Addition of a non-photic component to a light-based mathematical model of the human circadian pacemaker

Mathematical models have become vital to the study of many biological processes in humans due to the complexity of the physiological mechanisms underlying these processes and systems. While our current mathematical representation of the human circadian pacemaker has proven useful in many experimental situations, it uses as input only a direct effect of light on the circadian pacemaker. Although light (a photic stimulus) has been shown to be the primary synchronizer of the circadian pacemaker across a number of species, studies in both animals and humans have confirmed the existence of non-photic effects that also contribute to phase shifting and entrainment. We modified our light-based circadian mathematical model to reflect evidence from these studies that the sleep-wake cycle and/or associated behaviors have a non-photic effect on the circadian pacemaker. In our representation, the sleep-wake cycle and its associated behaviors provides a non-photic drive on the circadian pacemaker that acts both independently and concomitantly with light stimuli. Further experiments are required to validate fully our model and to understand the exact effect of the sleep-wake cycle as a non-photic stimulus for the human circadian pacemaker.     

The circadian timekeeping system of Drosophila

Daily rhythms in behavior, physiology and metabolism are controlled by endogenous circadian clocks. At the heart of these clocks is a circadian oscillator that keeps circadian time, is entrained by environmental cues such as light and activates rhythmic outputs at the appropriate time of day. Genetic and molecular analyses in Drosophila have revealed important insights into the molecules and mechanisms underlying circadian oscillator function in all organisms. In this review I will describe the intracellular feedback loops that form the core of the Drosophila circadian oscillator and consider how they are entrained by environmental light cycles, where they operate within the fly and how they are thought to control overt rhythms in physiology and behavior. I will also discuss where work remains to be done to give a comprehensive picture of the circadian clock in Drosophila and likely many other organisms.     

PERIOD–TIMELESS Interval Timer May Require an Additional Feedback Loop

In this study we present a detailed, mechanism-based mathematical framework of Drosophila circadian rhythms. This framework facilitates a more systematic approach to understanding circadian rhythms using a comprehensive representation of the network underlying this phenomenon. The possible mechanisms underlying the cytoplasmic “interval timer” created by PERIOD–TIMELESS association are investigated, suggesting a novel positive feedback regulatory structure. Incorporation of this additional feedback into a full circadian model produced results that are consistent with previous experimental observations of wild-type protein profiles and numerous mutant phenotypes.     

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.     

The mammalian circadian system: models and physiology

Mammalian circadian rhythms have been studied in great detail using primarily two different methods. One method is usually referred to as the formal analysis of rhythms. Its goal is to describe the properties of both rhythms and their underlying mechanisms, and it aims at the development of adequate mathematical models of the circadian system. The other method is the physiological analysis of the mechanisms that generate and entrain rhythms. Its goal is the identification of the anatomical components of the circadian system and the elucidation at a cellular and molecular level of how these components work. This paper reviews how the formal analysis of circadian systems, primarily in rodents, set the agenda for physiological studies, and the degree to which this agenda has been fulfilled. It then discusses how physiological analyses of the system have helped to redefine issues such as the nature and identity of the pacemaker, the nature of the entrainment process, the roles of photic and nonphotic cues, and the role of feedback in the circadian system. The continued commerce between these two approaches has led to a sophisticated appreciation of the complexities and subtleties of circadian organization in mammals. The further integration of formal and physiological analyses remains a challenging goal for the future.     

The cost of circadian desynchrony: Evidence,insights and open questions

Coordinated daily rhythms are evident in most aspects of our physiology, driven by internal timing systems known as circadian clocks. Our understanding of how biological clocks are built and function has grown exponentially over the past 20 years. With this has come an appreciation that disruption of the clock contributes to the pathophysiology of numerous diseases, from metabolic disease to neurological disorders to cancer. However, it remains to be determined whether it is the disruption of our rhythmic physiology per se (loss of timing itself), or altered functioning of individual clock components that drive pathology. Here, we review the importance of circadian rhythms in terms of how we (and other organisms) relate to the external environment, but also in relation to how internal physiological processes are coordinated and synchronized. These issues are of increasing importance as many aspects of modern life put us in conflict with our internal clockwork.

     

An Example of Circular Statistics in Chronobiological Studies: Analysis of Polymorphism in the Emergence Rhythms of Schistosoma Mansoni Cercariae

In the not too distant past, it was common belief that rhythms in the physical environment were the driving force, to which organisms responded passively, for the observed daily rhythms in measurable physiological and behavioral variables. The demonstration that this was not the case, but that both plants and animals possess accurate endogenous time-measuring machinery (i.e., circadian clocks) contributed to heightening interest in the study of circadian biological rhythms. In the last few decades, flourishing studies have demonstrated that most organisms have at least one internal circadian timekeeping device that oscillates with a period close to that of the astronomical day (i.e., 24h). To date, many of the physiological mechanisms underlying the control of circadian rhythmicity have been described, while the improvement of molecular biology techniques has permitted extraordinary advancements in our knowledge of the molecular components involved in the machinery underlying the functioning of circadian clocks in many different organisms, man included. In this review, we attempt to summarize our current understanding of the genetic and molecular biology of circadian clocks in cyanobacteria, fungi, insects, and mammals. (Chronobiology International, 17(4), 433–451, 2000)     

Circadian clocks: what makes them tick?

In the not too distant past, it was common belief that rhythms in the physical environment were the driving force, to which organisms responded passively, for the observed daily rhythms in measurable physiological and behavioral variables. The demonstration that this was not the case, but that both plants and animals possess accurate endogenous time-measuring machinery (i.e., circadian clocks) contributed to heightening interest in the study of circadian biological rhythms. In the last few decades, flourishing studies have demonstrated that most organisms have at least one internal circadian timekeeping device that oscillates with a period close to that of the astronomical day (i.e., 24h). To date, many of the physiological mechanisms underlying the control of circadian rhythmicity have been described, while the improvement of molecular biology techniques has permitted extraordinary advancements in our knowledge of the molecular components involved in the machinery underlying the functioning of circadian clocks in many different organisms, man included. In this review, we attempt to summarize our current understanding of the genetic and molecular biology of circadian clocks in cyanobacteria, fungi, insects, and mammals. (Chronobiology International, 17(4), 433-451, 2000)     

On analyzing circadian rhythms data using nonlinear mixed models with harmonic terms

Wang, Ke, and Brown (2003, Biometrics59, 804-812) developed a smoothing-based approach for modeling circadian rhythms with random effects. Their approach is flexible in that fixed and random covariates can affect both the amplitude and phase shift of a nonparametrically smoothed periodic function. In motivating their approach, Wang et al. stated that a simple sinusoidal function is too restrictive. In addition, they stated that "although adding harmonics can improve the fit, it is difficult to decide how many harmonics to include in the model, and the results are difficult to interpret." We disagree with the notion that harmonic models cannot be a useful tool in modeling longitudinal circadian rhythm data. In this note, we show how nonlinear mixed models with harmonic terms allow for a simple and flexible alternative to Wang et al.'s approach. We show how to choose the number of harmonics using penalized likelihood to flexibly model circadian rhythms and to estimate the effect of covariates on the rhythms. We fit harmonic models to the cortisol circadian rhythm data presented by Wang et al. to illustrate our approach. Furthermore, we evaluate the properties of our procedure with a small simulation study. The proposed parametric approach provides an alternative to Wang et al.'s semiparametric approach and has the added advantage of being easy to implement in most statistical software packages.     

On the Adaptive Significance of Circadian Clocks for Their Owners

Circadian rhythms are believed to be an evolutionary adaptation to daily environmental cycles resulting from Earth's rotation about its axis. A trait evolved through a process of natural selection is considered as adaptation; therefore, rigorous demonstration of adaptation requires evidence suggesting evolution of a trait by natural selection. Like any other adaptive trait, circadian rhythms are believed to be advantageous to living beings through some perceived function. Circadian rhythms are thought to confer advantage to their owners through scheduling of biological functions at appropriate time of daily environmental cycle (extrinsic advantage), coordination of internal physiology (intrinsic advantage), and through their role in responses to seasonal changes. So far, the adaptive value of circadian rhythms has been tested in several studies and evidence indeed suggests that they confer advantage to their owners. In this review, we have discussed the background for development of the framework currently used to test the hypothesis of adaptive significance of circadian rhythms. Critical examination of evidence reveals that there are several lacunae in our understanding of circadian rhythms as adaptation. Although it is well known that demonstrating a given trait as adaptation (or setting the necessary criteria) is not a trivial task, here we recommend some of the basic criteria and suggest the nature of evidence required to comprehensively understand circadian rhythms as adaptation. Thus, we hope to create some awareness that may benefit future studies in this direction. (Author correspondence: vsharma@jncasr.ac.in or vksharmas@gmail.com)     

Acute exposure to 2G phase shifts the rat circadian timing system

The circadian timing system (CTS) provides internal and external temporal coordination of an animal's physiology and behavior. In mammals, the generation and coordination of these circadian rhythms is controlled by a neural pacemaker, the suprachiasmatic nucleus (SCN), located within the hypothalamus. The pacemaker is synchronized to the 24 hour day by time cues (zeitgebers) such as the light/dark cycle. When an animal is exposed to an environment without time cues, the circadian rhythms maintain internal temporal coordination but exhibit a "free-running" condition in which the period length is determined by the internal pacemaker. Maintenance of internal and external temporal coordination are critical for normal physiological and psychological function in human and non-human primates. Exposure to altered gravitational environments has been shown to affect the amplitude, mean, and timing of circadian rhythms in species ranging from unicellular organisms to man. However, it has not been determined whether altered gravitational fields have a direct effect on the neural pacemaker, or affect peripheral physiological systems that express these circadian parameters. In previous studies, the ability of a stimulus to phase shift circadian rhythms was used to determine whether a stimulus has a direct effect on the neural pacemaker. The present experiment was performed in order to determine whether acute exposure to a hyperdynamic field could phase shift circadian rhythms.     

Intrinsic period and light intensity determine the phase relationship between melatonin and sleep in humans

The internal circadian clock and sleep-wake homeostasis regulate the timing of human brain function, physiology, and behavior so that wakefulness and its associated functions are optimal during the solar day and that sleep and its related functions are optimal at night. The maintenance of a normal phase relationship between the internal circadian clock, sleep-wake homeostasis, and the light-dark cycle is crucial for optimal neurobehavioral and physiological function. Here, the authors show that the phase relationship between these factors-the phase angle of entrainment (psi)-is strongly determined by the intrinsic period (tau) of the master circadian clock and the strength of the circadian synchronizer. Melatonin was used as a marker of internal biological time, and circadian period was estimated during a forced desynchrony protocol. The authors observed relationships between the phase angle of entrainment and intrinsic period after exposure to scheduled habitual wakefulness-sleep light-dark cycle conditions inside and outside of the laboratory. Individuals with shorter circadian periods initiated sleep and awakened at a later biological time than did individuals with longer circadian periods. The authors also observed that light exposure history influenced the phase angle of entrainment such that phase angle was shorter following exposure to a moderate bright light (approximately 450 lux)-dark/wakefulness-sleep schedule for 5 days than exposure to the equivalent of an indoor daytime light (approximately 150 lux)-dark/wakefulness-sleep schedule for 2 days. These findings demonstrate that neurobiological and environmental factors interact to regulate the phase angle of entrainment in humans. This finding has important implications for understanding physiological organization by the brain's master circadian clock and may have implications for understanding mechanisms underlying circadian sleep disorders.