Modelling Biological Rhythms

The seasons within: a theoretical perspective on photoperiodic entrainment and encoding

Circadian clocks are internal timing devices that have evolved as an adaption to the omnipresent natural 24 h rhythmicity of daylight intensity. Properties of the circadian system are photoperiod dependent. The phase of entrainment varies systematically with season. Plastic photoperiod-dependent re-arrangements in the mammalian circadian core pacemaker yield an internal representation of season. Output pathways of the circadian clock regulate photoperiodic responses such as flowering time in plants or hibernation in mammals. Here, we review the concepts of seasonal entrainment and photoperiodic encoding. We introduce conceptual phase oscillator models as their high level of abstraction, but, yet, intuitive interpretation of underlying parameters allows for a straightforward analysis of principles that determine entrainment characteristics. Results from this class of models are related and discussed in the context of more complex conceptual amplitude-phase oscillators as well as contextual molecular models that take into account organism, tissue, and cell-type-specific details.

The Physiological and Pharmacological Significance of the Circadian Timing of the HPA Axis: A Mathematical Modeling Approach

As a potent endogenous regulator of homeostasis, the circadian time-keeping system synchronizes internal physiology to periodic changes in the external environment to enhance survival. Adapting endogenous rhythms to the external time is accomplished hierarchically with the central pacemaker located in the suprachiasmatic nucleus (SCN) signaling the hypothalamus-pituitary-adrenal (HPA) axis to release hormones, notably cortisol, which help maintain the body's circadian rhythm. Given the essential role of HPA-releasing hormones in regulating physiological functions, including immune response, cell cycle, and energy metabolism, their daily variation is critical for the proper function of the circadian timing system. In this review, we focus on cortisol and key fundamental properties of the HPA axis and highlight their importance in controlling circadian dynamics. We demonstrate how systems-driven, mathematical modeling of the HPA axis complements experimental findings, enhances our understanding of complex physiological systems, helps predict potential mechanisms of action, and elucidates the consequences of circadian disruption. Finally, we outline the implications of circadian regulation in the context of personalized chronotherapy. Focusing on the chrono-pharmacology of synthetic glucocorticoids, we review the challenges and opportunities associated with moving toward personalized therapies that capitalize on circadian rhythms.

Cryptochrome 1 as a state variable of the circadian clockwork of the suprachiasmatic nucleus: Evidence from translational switching

The suprachiasmatic nucleus (SCN) of the hypothalamus is the principal clock driving circadian rhythms of physiology and behavior that adapt mammals to environmental cycles. Disruption of SCN-dependent rhythms compromises health, and so understanding SCN time keeping will inform management of diseases associated with modern lifestyles. SCN time keeping is a self-sustaining transcriptional/translational delayed feedback loop (TTFL), whereby negative regulators inhibit their own transcription. Formally, the SCN clock is viewed as a limit-cycle oscillator, the simplest being a trajectory of successive phases that progresses through two-dimensional space defined by two state variables mapped along their respective axes. The TTFL motif is readily compatible with limit-cycle models, and in Neurospora and Drosophila the negative regulators Frequency (FRQ) and Period (Per) have been identified as state variables of their respective TTFLs. The identity of state variables of the SCN oscillator is, however, less clear. Experimental identification of state variables requires reversible and temporally specific control over their abundance. Translational switching (ts) provides this, the expression of a protein of interest relying on the provision of a noncanonical amino acid. We show that the negative regulator Cryptochrome 1 (CRY1) fulfills criteria defining a state variable: ts-CRY1 dose-dependently and reversibly suppresses the baseline, amplitude, and period of SCN rhythms, and its acute withdrawal releases the TTFL to oscillate from a defined phase. Its effect also depends on its temporal pattern of expression, although constitutive ts-CRY1 sustained (albeit less stable) oscillations. We conclude that CRY1 has properties of a state variable, but may operate among several state variables within a multidimensional limit cycle.

Common Ground between Biological Rhythms and Forensics

Simple Summary

Biological clocks regulate the timing of numerous body functions in adaption to daily repeating cycles in the environment, such as the sleep–wake phases that are trained by the cycling changes of night and day light. The identification of a deceased victim is a critical component in a forensic investigation, but it can be significantly hampered by the condition of the dead body and the lack of personal records and documents. This review links current knowledge on the molecular mechanisms of biological rhythms to forensically relevant aspects, including the time period since death, cause of death, the use of insects for forensics, sex and age of a person, ethnic background and development. Putting these findings in context demonstrates how the analysis of molecular clock analysis could be used as tool for future personal identification in forensic investigations.

Abstract

Biological clocks set the timing for a large number of essential processes in the living human organism. After death, scientific evidence is required in forensic investigations in order to collect as much information as possible on the death circumstances and personal identifiers of the deceased victim. We summarize the associations between the molecular mechanisms of biological rhythms and forensically relevant aspects, including post-mortem interval and cause of death, entomological findings, sex, age, ethnicity and development. Given their importance during lifetime, biological rhythms could be potential tools to draw conclusions on the death circumstances and the identity of a deceased person by mechanistic investigations of the different biological clocks in a forensic context. This review puts the known effects of biological rhythms on the functions of the human organism in context with potential applications in forensic fields of interest, such as personal identification, entomology as well as the determination of the post-mortem interval and cause of death.

Modeling Active Non-Markovian Oscillations

Modeling noisy oscillations of active systems is one of the current challenges in physics and biology. Because the physical mechanisms of such processes are often difficult to identify, we propose a linear stochastic model driven by a non-Markovian bistable noise that is capable of generating self-sustained periodic oscillation. We derive analytical predictions for most relevant dynamical and thermodynamic properties of the model. This minimal model turns out to describe accurately bistablelike oscillatory motion of hair bundles in bullfrog sacculus, extracted from experimental data. Based on and in agreement with these data, we estimate the power required to sustain such active oscillations to be of the order of 100  k_{B}T per oscillation cycle.

Mathematical modeling of mammalian circadian clocks affecting drug and disease responses

To align with daily environmental changes, most physiological processes in mammals exhibit a time-of-day rhythmicity. This circadian control of physiology is intrinsically driven by a cell-autonomous clock gene network present in almost all cells of the body that drives rhythmic expression of genes that regulate numerous molecular and cellular processes. Accordingly, many aspects of pharmacology and toxicology also oscillate in a time-of-day manner giving rise to diverse effects on pharmacokinetics and pharmacodynamics. Genome-wide studies and mathematical modeling are available tools that have significantly improved our understanding of these nonlinear aspects of physiology and therapeutics. In this manuscript current literature and our prior work on the model-based approaches that have been used to explore circadian genomic systems of mammals are reviewed. Such basic understanding and having an integrative approach may provide new strategies for chronotherapeutic drug treatments and yield new insights for the restoration of the circadian system when altered by diseases.

Volterra-series approach to stochastic nonlinear dynamics: Linear response of the Van der Pol oscillator driven by white noise

The Van der Pol equation is a paradigmatic model of relaxation oscillations. This remarkable nonlinear phenomenon of self-sustained oscillatory motion underlies important rhythmic processes in nature and electrical engineering. Relaxation oscillations in a real system are usually coupled to environmental noise, which further enriches their dynamics, but makes theoretical analysis of such systems and determination of the equation parameter values a difficult task. In a companion paper we have proposed an analytic approach to a similar problem for another classical nonlinear model-the bistable Duffing oscillator. Here we extend our techniques to the case of the Van der Pol equation driven by white noise. We analyze the statistics of solutions and propose a method to estimate parameter values from the oscillator's time series. We use experimental data of active oscillations in a biophysical system to demonstrate how our method applies to real observations and can be generalized for more complex models.

Off the Clock: From Circadian Disruption to Metabolic Disease

Circadian timekeeping allows appropriate temporal regulation of an organism’s internal metabolism to anticipate and respond to recurrent daily changes in the environment. Evidence from animal genetic models and from humans under circadian misalignment (such as shift work or jet lag) shows that disruption of circadian rhythms contributes to the development of obesity and metabolic disease. Inappropriate timing of food intake and high-fat feeding also lead to disruptions of the temporal coordination of metabolism and physiology and subsequently promote its pathogenesis. This review illustrates the impact of genetically or environmentally induced molecular clock disruption (at the level of the brain and peripheral tissues) and the interplay between the circadian system and metabolic processes. Here, we discuss some mechanisms responsible for diet-induced circadian desynchrony and consider the impact of nutritional cues in inter-organ communication, with a particular focus on the communication between peripheral organs and brain. Finally, we discuss the relay of environmental information by signal-dependent transcription factors to adjust the timing of gene oscillations. Collectively, a better knowledge of the mechanisms by which the circadian clock function can be compromised will lead to novel preventive and therapeutic strategies for obesity and other metabolic disorders arising from circadian desynchrony.

Sleep Modelling across Physiological Levels

Sleep and circadian rhythms are regulated across multiple functional, spatial and temporal levels: from genes to networks of coupled neurons and glial cells, to large scale brain dynamics and behaviour. The dynamics at each of these levels are complex and the interaction between the levels is even more so, so research have mostly focused on interactions within the levels to understand the underlying mechanisms—the so-called reductionist approach. Mathematical models were developed to test theories of sleep regulation and guide new experiments at each of these levels and have become an integral part of the field. The advantage of modelling, however, is that it allows us to simulate and test the dynamics of complex biological systems and thus provides a tool to investigate the connections between the different levels and study the system as a whole. In this paper I review key models of sleep developed at different physiological levels and discuss the potential for an integrated systems biology approach for sleep regulation across these levels. I also highlight the necessity of building mechanistic connections between models of sleep and circadian rhythms across these levels.

Mathematical modeling of circadian rhythms

Circadian rhythms are endogenous ~24-hr oscillations usually entrained to daily environmental cycles of light/dark. Many biological processes and physiological functions including mammalian body temperature, the cell cycle, sleep/wake cycles, neurobehavioral performance, and a wide range of diseases including metabolic, cardiovascular, and psychiatric disorders are impacted by these rhythms. Circadian clocks are present within individual cells and at tissue and organismal levels as emergent properties from the interaction of cellular oscillators. Mathematical models of circadian rhythms have been proposed to provide a better understanding of and to predict aspects of this complex physiological system. These models can be used to: (a) manipulate the system in silico with specificity that cannot be easily achieved using in vivo and in vitro experimental methods and at lower cost, (b) resolve apparently contradictory empirical results, (c) generate hypotheses, (d) design new experiments, and (e) to design interventions for altering circadian rhythms. Mathematical models differ in structure, the underlying assumptions, the number of parameters and variables, and constraints on variables. Models representing circadian rhythms at different physiologic scales and in different species are reviewed to promote understanding of these models and facilitate their use. This article is categorized under: Physiology > Mammalian Physiology in Health and Disease Models of Systems Properties and Processes > Organ, Tissue, and Physiological Models.

Protein sequestration versus Hill-type repression in circadian clock models

Circadian (∼24 h) clocks are self-sustained endogenous oscillators with which organisms keep track of daily and seasonal time. Circadian clocks frequently rely on interlocked transcriptional-translational feedback loops to generate rhythms that are robust against intrinsic and extrinsic perturbations. To investigate the dynamics and mechanisms of the intracellular feedback loops in circadian clocks, a number of mathematical models have been developed. The majority of the models use Hill functions to describe transcriptional repression in a way that is similar to the Goodwin model. Recently, a new class of models with protein sequestration-based repression has been introduced. Here, the author discusses how this new class of models differs dramatically from those based on Hill-type repression in several fundamental aspects: conditions for rhythm generation, robust network designs and the periods of coupled oscillators. Consistently, these fundamental properties of circadian clocks also differ among Neurospora, Drosophila, and mammals depending on their key transcriptional repression mechanisms (Hill-type repression or protein sequestration). Based on both theoretical and experimental studies, this review highlights the importance of careful modelling of transcriptional repression mechanisms in molecular circadian clocks.

Entrainment of a cellular circadian oscillator by light in the presence of molecular noise

In this paper, we consider a stochastic molecular circadian oscillator described by a sequence of biological reactions and its deterministic kinetics governed by a system of ordinary differential equations in the limit of large numbers of molecules. The oscillations in the model are generated by negative feedback regulation of a gene. The focus of this paper is the entrainment of the oscillator by a periodic light signal that affects the maximal transcription rate of the gene. We introduce two scalings of the model parameters that provide independent control over the natural frequency of the oscillator and the relative noise level. We study entrainment in two ways: by visualizing the stochastic limit cycle in various projections of the discrete phase space of the system and by evaluating the maximum of the normalized cross correlation of the light signal with the number of protein molecules in the cell. The visualization method ignores the phase of the oscillator, and we find in this way that entrainment has a subtle organizing effect on the limit cycle as a whole. The cross correlation results reveal an interval of natural frequencies of the oscillator surrounding the frequency of the light signal within which maximal entrainment occurs with rather sharp drops in entrainment at the edges of this interval. The width of the interval of maximal entrainment increases with the amplitude of the light signal. These statements are applicable both to the stochastic oscillator and to its deterministic limit, but the results are most clear-cut in the deterministic case and degrade from there as the relative noise level increases.

Night Shift Work and Risk of Breast Cancer

PURPOSE OF REVIEW

Night work is increasingly common and a necessity in certain sectors of the modern 24-h society. The embedded exposure to light-at-night, which suppresses the nocturnal hormone melatonin with oncostatic properties and circadian disruption, i.e., misalignment between internal and external night and between cells and organs, are suggested as main mechanisms involved in carcinogenesis. In 2007, the International Agency for Research on Cancer (IARC) classified shift work that involves circadian disruption as probably carcinogenic to humans based on limited evidence from eight epidemiologic studies on breast cancer, in addition to sufficient evidence from animal experiments. The aim of this review is a critical update of the IARC evaluation, including subsequent and the most recent epidemiologic evidence on breast cancer risk after night work.

RECENT FINDINGS

After 2007, in total nine new case-control studies, one case-cohort study, and eight cohort studies are published, which triples the number of studies. Further, two previous cohorts have been updated with extended follow-up. The assessment of night shift work is different in all of the 26 existing studies. There is some evidence that high number of consecutive night shifts has impact on the extent of circadian disruption, and thereby increased breast cancer risk, but this information is missing in almost all cohort studies. This in combination with short-term follow-up of aging cohorts may explain why some cohort studies may have null findings. The more recent case-control studies have contributed interesting results concerning breast cancer subtypes in relation to both menopausal status and different hormonal subtypes. The large differences in definitions of both exposure and outcome may contribute to the observed heterogeneity of results from studies of night work and breast cancer, which overall points in the direction of an increased breast cancer risk, in particular after over 20 years of night shifts. Overall, there is a tendency of increased risk of breast cancer either after over 20 years of night shift or after shorter periods with many consecutive shifts. More epidemiologic research using standardized definitions of night work metrics and breast cancer subtypes as well as other cancers is needed in order to improve the epidemiologic evidence in combination with animal models of night work. Also, evidence-based preventive interventions are needed.

The choroid plexus is an important circadian clock component

Mammalian circadian clocks have a hierarchical organization, governed by the suprachiasmatic nucleus (SCN) in the hypothalamus. The brain itself contains multiple loci that maintain autonomous circadian rhythmicity, but the contribution of the non-SCN clocks to this hierarchy remains unclear. We examine circadian oscillations of clock gene expression in various brain loci and discovered that in mouse, robust, higher amplitude, relatively faster oscillations occur in the choroid plexus (CP) compared to the SCN. Our computational analysis and modeling show that the CP achieves these properties by synchronization of “twist” circadian oscillators via gap-junctional connections. Using an in vitro tissue coculture model and in vivo targeted deletion of the Bmal1 gene to silence the CP circadian clock, we demonstrate that the CP clock adjusts the SCN clock likely via circulation of cerebrospinal fluid, thus finely tuning behavioral circadian rhythms.

The suprachiasmatic nucleus (SCN) has been thought of as the master circadian clock, but peripheral circadian clocks do exist. Here, the authors show that the choroid plexus displays oscillations more robust than the SCN and that can be described as a Poincaré oscillator with negative twist.

Measuring Relative Coupling Strength in Circadian Systems

Modern imaging techniques allow the monitoring of circadian rhythms of single cells. Coupling between these single cellular circadian oscillators can generate coherent periodic signals on the tissue level that subsequently orchestrate physiological outputs. The strength of coupling in such systems of oscillators is often unclear. In particular, effects on coupling strength by varying cell densities, by knockouts, and by inhibitor applications are debated. In this study, we suggest to quantify the relative coupling strength via analyzing period, phase, and amplitude distributions in ensembles of individual circadian oscillators. Simulations of different oscillator networks show that period and phase distributions become narrower with increasing coupling strength. Moreover, amplitudes can increase due to resonance effects. Variances of periods and phases decay monotonically with coupling strength, and can serve therefore as measures of relative coupling strength. Our theoretical predictions are confirmed by studying recently published experimental data from PERIOD2 expression in slices of the suprachiasmatic nucleus during and after the application of tetrodotoxin (TTX). On analyzing the corresponding period, phase, and amplitude distributions, we can show that treatment with TTX can be associated with a reduced coupling strength in the system of coupled oscillators. Analysis of an oscillator network derived directly from the data confirms our conclusions. We suggest that our approach is also applicable to quantify coupling in fibroblast cultures and hepatocyte networks, and for social synchronization of circadian rhythmicity in rodents, flies, and bees.

Entrainment Maps

Circadian oscillators found across a variety of species are subject to periodic external light-dark forcing. Entrainment to light-dark cycles enables the circadian system to align biological functions with appropriate times of day or night. Phase response curves (PRCs) have been used for decades to gain valuable insights into entrainment; however, PRCs may not accurately describe entrainment to photoperiods with substantial amounts of both light and dark due to their reliance on a single limit cycle attractor. We have developed a new tool, called an entrainment map, that overcomes this limitation of PRCs and can assess whether, and at what phase, a circadian oscillator entrains to external forcing with any photoperiod. This is a 1-dimensional map that we construct for 3 different mathematical models of circadian clocks. Using the map, we are able to determine conditions for existence and stability of phase-locked solutions. In addition, we consider the dependence on various parameters such as the photoperiod and intensity of the external light as well as the mismatch in intrinsic oscillator frequency with the light-dark cycle. We show that the entrainment map yields more accurate predictions for phase locking than methods based on the PRC. The map is also ideally suited to calculate the amount of time required to achieve entrainment as a function of initial conditions and the bifurcations of stable and unstable periodic solutions that lead to loss of entrainment.

Deconstructing Circadian Rhythmicity with Models and Manipulations

A master brain clock, localized to the hypothalamic suprachiasmatic nucleus (SCN), coordinates daily rhythms of physiology and behavior. Within the SCN, interconnected individual neurons are oscillators that, as an ensemble, function to send a coherent timing signal to the brain and body. However, individually, these neurons display different amplitudes, periods, and phases of oscillation. The dynamic properties of the SCN have been characterized over several spatial levels of analysis, from proteins to cells to tissues, and over several temporal ranges, from milliseconds to weeks. Modeling tools guide empirical research in this complex and multiscale spatiotemporal environment. Given that the SCN is a prototypical example of oscillating neural systems, principles of its organization hold promise as general prototypes of rhythms in other frequencies.

Circadian systems biology: When time matters

The circadian clock is a powerful endogenous timing system, which allows organisms to fine-tune their physiology and behaviour to the geophysical time. The interplay of a distinct set of core-clock genes and proteins generates oscillations in expression of output target genes which temporally regulate numerous molecular and cellular processes. The study of the circadian timing at the organismal as well as at the cellular level outlines the field of chronobiology, which has been highly interdisciplinary ever since its origins. The development of high-throughput approaches enables the study of the clock at a systems level. In addition to experimental approaches, computational clock models exist which allow the analysis of rhythmic properties of the clock network. Such mathematical models aid mechanistic understanding and can be used to predict outcomes of distinct perturbations in clock components, thereby generating new hypotheses regarding the putative function of particular clock genes. Perturbations in the circadian timing system are linked to numerous molecular dysfunctions and may result in severe pathologies including cancer. A comprehensive knowledge regarding the mechanistic of the circadian system is crucial to develop new procedures to investigate pathologies associated with a deregulated clock.

In this manuscript we review the combination of experimental methodologies, bioinformatics and theoretical models that have been essential to explore this remarkable timing-system. Such an integrative and interdisciplinary approach may provide new strategies with regard to chronotherapeutic treatment and new insights concerning the restoration of the circadian timing in clock-associated diseases.

A theoretical study on seasonality

In addition to being endogenous, a circadian system must be able to communicate with the outside world and align its rhythmicity to the environment. As a result of such alignment, external Zeitgebers can entrain the circadian system. Entrainment expresses itself in coinciding periods of the circadian oscillator and the Zeitgeber and a stationary phase difference between them. The range of period mismatches between the circadian system and the Zeitgeber that Zeitgeber can overcome to entrain the oscillator is called an entrainment range. The width of the entrainment range usually increases with increasing Zeitgeber strength, resulting in a wedge-like Arnold tongue. This classical view of entrainment does not account for the effects of photoperiod on entrainment. Zeitgebers with extremely small or large photoperiods are intuitively closer to constant environments than equinoctial Zeitgebers and hence are expected to produce a narrower entrainment range. In this paper, we present theoretical results on entrainment under different photoperiods. We find that in the photoperiod-detuning parameter plane, the entrainment zone is shaped in the form of a skewed onion. The bottom and upper points of the onion are given by the free-running periods in DD and LL, respectively. The widest entrainment range is found near photoperiods of 50%. Within the onion, we calculated the entrainment phase that varies over a range of 12 h. The results of our theoretical study explain the experimentally observed behavior of the entrainment phase in dependence on the photoperiod.

Effects of different per translational kinetics on the dynamics of a core circadian clock model

Living beings display self-sustained daily rhythms in multiple biological processes, which persist in the absence of external cues since they are generated by endogenous circadian clocks. The period (per) gene is a central player within the core molecular mechanism for keeping circadian time in most animals. Recently, the modulation PER translation has been reported, both in mammals and flies, suggesting that translational regulation of clock components is important for the proper clock gene expression and molecular clock performance. Because translational regulation ultimately implies changes in the kinetics of translation and, therefore, in the circadian clock dynamics, we sought to study how and to what extent the molecular clock dynamics is affected by the kinetics of PER translation. With this objective, we used a minimal mathematical model of the molecular circadian clock to qualitatively characterize the dynamical changes derived from kinetically different PER translational mechanisms. We found that the emergence of self-sustained oscillations with characteristic period, amplitude, and phase lag (time delays) between per mRNA and protein expression depends on the kinetic parameters related to PER translation. Interestingly, under certain conditions, a PER translation mechanism with saturable kinetics introduces longer time delays than a mechanism ruled by a first-order kinetics. In addition, the kinetic laws of PER translation significantly changed the sensitivity of our model to parameters related to the synthesis and degradation of per mRNA and PER degradation. Lastly, we found a set of parameters, with realistic values, for which our model reproduces some experimental results reported recently for Drosophila melanogaster and we present some predictions derived from our analysis.

Inference on periodicity of circadian time series

Estimation of the period length of time-course data from cyclical biological processes, such as those driven by the circadian pacemaker, is crucial for inferring the properties of the biological clock found in many living organisms. We propose a methodology for period estimation based on spectrum resampling (SR) techniques. Simulation studies show that SR is superior and more robust to non-sinusoidal and noisy cycles than a currently used routine based on Fourier approximations. In addition, a simple fit to the oscillations using linear least squares is available, together with a non-parametric test for detecting changes in period length which allows for period estimates with different variances, as frequently encountered in practice. The proposed methods are motivated by and applied to various data examples from chronobiology.

Modeling spatiotemporal dynamics of bacterial populations

Quantitative modeling of spatiotemporal dynamics of cells facilitates understanding and engineering of biological systems. Using a synthetic bacterial ecosystem as a workbench, we present the approach to mathematically simulate the spatiotemporal population dynamics of the ecosystem. A description of ecosystem's genetic construction and model development is firstly given. Parameter estimation and computational approach for the derived partial differential equations (PDEs) are then given. Spatiotemporal pattern formation is computed by numerically solving the PDE model. Biodiversity of the ecosystem and its impacts by cellular seeding distance and motility are computed according to the cell distribution patterns.

Light and the human circadian clock

The circadian clock can only reliably fulfil its function if it is stably entrained. Most clocks use the light-dark cycle as environmental signal (zeitgeber) for this active synchronisation. How we think about clock function and entrainment has been strongly influenced by the early concepts of the field's pioneers, and the astonishing finding that circadian rhythms continue a self-sustained oscillation in constant conditions has become central to our understanding of entrainment.Here, we argue that we have to rethink these initial circadian dogmas to fully understand the circadian programme and how it entrains. Light is also the prominent zeitgeber for the human clock, as has been shown experimentally in the laboratory and in large-scale epidemiological studies in real life, and we hypothesise that social zeitgebers act through light entrainment via behavioural feedback loops (zeitnehmer). We show that human entrainment can be investigated in detail outside of the laboratory, by using the many 'experimental' conditions provided by the real world, such as daylight savings time, the 'forced synchrony' imposed by the introduction of time zones, or the fact that humans increasingly create their own light environment. The conditions of human entrainment have changed drastically over the past 100 years and have led to an increasing discrepancy between biological and social time (social jetlag). The increasing evidence that social jetlag has detrimental consequences for health suggests that shift-work is only an extreme form of circadian misalignment, and that the majority of the population in the industrialised world suffers from a similarly 'forced synchrony'.

The effects of combining serotonin reuptake inhibition and 5-HT7 receptor blockade on circadian rhythm regulation in rodents

Disruption of circadian rhythms may lead to mood disorders. The present study investigated the potential therapeutic utility of combining a 5-HT7 antagonist with a selective serotonin (5-HT) reuptake inhibitor (SSRI), the standard of care in depression, on circadian rhythm regulation. In tissue explants of the suprachiasmatic nucleus (SCN) from PER2::LUC mice genetically modified to report changes in the expression of a key clock protein, the period length of PER2 bioluminescence was shortened in the presence of AS19, a 5-HT7 partial agonist. This reduction was blocked by SB269970, a selective 5-HT7 antagonist. The SSRI, escitalopram, had no effect alone on period length, but a combination with SB269970, yielded significant increases. Dosed in vivo, escitalopram had little impact on the occurrence of activity onsets in rats given access to running wheels, whether the drug was given acutely or sub-chronically. However, preceding the escitalopram treatment with a single acute dose of SB269970 produced robust phase delays, in keeping with the in vitro explant data. Taken together, these findings suggest that the combination of an SSRI and a 5-HT7 receptor antagonist has a greater impact on circadian rhythms than that observed with either agent alone, and that such a multimodal approach may be of therapeutic value in treating patients with poor clock function.

Tissue-Specific Interaction of Per1/2 and Dec2 in the Regulation of Fibroblast Circadian Rhythms

In mammals, the molecular circadian clockwork is comprised of interlocked transcriptional-translational feedback loops (TTLs). Three Period ( Per1-3) and 2 Dec ( Dec1/2) genes interact in regulating the activity of the transcriptional activators CLOCK/NPAS2 and BMAL1. While deletion of Per1 and Per2 in mice results in behavioral arrhythmicity, Dec deletion has less dramatic effects on activity rhythms, affecting primarily phase of entrainment and free-running period. In intact animals, clock gene mutant phenotypes are often masked due to intercellular coupling mechanisms that stabilize cellular rhythms. Therefore, to study Per/Dec genetic interaction at the cellular level, we isolated fibroblasts from different tissues of Per1, Per2, and Dec2 single and double mutant mice. We show that in the cellular TTL, Pers and Dec2 act in a principally synergistic way, but tissue-specific differences in this interaction are seen. A rescue of rhythmicity in Per2 mutant cells after additional deletion of Dec2 was observed, indicating that in the absence of Per2, DEC2 destabilizes TTL function. Rhythm power in Per1/Dec2 and Per2/Dec2 double mutants was strongly reduced, suggesting that interaction of Dec2 with both Per genes is important for stabilizing clock period. Contrary to what was observed for behavior, nonsynergistic effects of Dec2 and Per1/2 mutations were observed on cellular clock phase regulation that do not correlate with period effects. Our data reveal cell type-specific interactions of Per1/2 and Dec2 in the regulation of period, phase, and rhythm sustainment, emphasizing the differential organization of the mammalian clock machinery in different tissues.

Modeling circadian clocks: Roles, advantages, and limitations

Circadian rhythms are endogenous oscillations characterized by a period of about 24h. They constitute the biological rhythms with the longest period known to be generated at the molecular level. The abundance of genetic information and the complexity of the molecular circuitry make circadian clocks a system of choice for theoretical studies. Many mathematical models have been proposed to understand the molecular regulatory mechanisms that underly these circadian oscillations and to account for their dynamic properties (temperature compensation, entrainment by light dark cycles, phase shifts by light pulses, rhythm splitting, robustness to molecular noise, intercellular synchronization). The roles and advantages of modeling are discussed and illustrated using a variety of selected examples. This survey will lead to the proposal of an integrated view of the circadian system in which various aspects (interlocked feedback loops, inter-cellular coupling, and stochasticity) should be considered together to understand the design and the dynamics of circadian clocks. Some limitations of these models are commented and challenges for the future identified.

Modeling circadian clocks: From equations to oscillations

Circadian rhythms are generated at the cellular level by a small but tightly regulated genetic network. In higher eukaryotes, interlocked transcriptional-translational feedback loops form the core of this network, which ensures the activation of the right genes (proteins) at the right time of the day. Understanding how such a complex molecular network can generate robust, self-sustained oscillations and accurately responds to signals from the environment (such as light and temperature) is greatly helped by mathematical modeling. In the present paper we review some mathematical models for circadian clocks, ranging from abstract, phenomenological models to the most detailed molecular models. We explain how the equations are derived, highlighting the challenges for the modelers, and how the models are analyzed. We show how to compute bifurcation diagrams, entrainment, and phase response curves. In the subsequent paper, we discuss, through a selection of examples, how modeling efforts have contributed to a better understanding of the dynamics of the circadian regulatory network.

Positive autoregulation delays the expression phase of mammalian clock gene Per2

In mammals, cellular circadian rhythms are generated by a transcriptional-translational autoregulatory network that consists of clock genes that encode transcriptional regulators. Of these clock genes, Period1 (Per1) and Period2 (Per2) are essential for sustainable circadian rhythmicity and photic entrainment. Intriguingly, Per1 and Per2 mRNAs exhibit circadian oscillations with a 4-hour phase difference, but they are similarly transactivated by CLOCK-BMAL1. In this study, we investigated the mechanism underlying the phase difference between Per1 and Per2 through a combination of mathematical simulations and molecular experiments. Mathematical analyses of a model for the mammalian circadian oscillator demonstrated that the slow synthesis and fast degradation of mRNA tend to advance the oscillation phase of mRNA expression. However, the phase difference between Per1 and Per2 was not reproduced by the model, which implemented a 1.1-fold difference in degradation rates and a 3-fold difference in CLOCK-BMAL1 mediated inductions of Per1 and Per2 as estimated in cultured mammalian cells. Thus, we hypothesized the existence of a novel transcriptional activation of Per2 by PER1/2 such that the Per2 oscillation phase was delayed. Indeed, only the Per2 promoter, but not Per1, was strongly induced by both PER1 and PER2 in the presence of CLOCK-BMAL1 in a luciferase reporter assay. Moreover, a 3-hour advance was observed in the transcriptional oscillation of the delta-Per2 reporter gene lacking cis-elements required for the induction by PER1/2. These results indicate that the Per2 positive feedback regulation is a significant factor responsible for generating the phase difference between Per1 and Per2 gene expression.

Multiscale complexity in the mammalian circadian clock

The field of systems biology studies how the interactions among individual components (e.g. genes and proteins) yield interesting and complex behavior. The circadian (daily) timekeeping system in mammals is an ideal system to study complexity because of its many biological scales (from genes to animal behavior). A wealth of data at each of these scales has recently been discovered. Within each scale, modeling can advance our understanding of challenging problems that arise in studying mammalian timekeeping. However, future work must focus on bridging the multiple spatial and temporal scales in the modeling of SCN network. Here we review recent advances, and then delve into a few areas that are promising research directions. We also discuss the flavor of modeling needed (simple or detailed) as well as new techniques that are needed to meet the challenges in modeling data across scales.

Minimum protein oscillator based on multisite phosphorylation∕dephosphorylation

The authors propose a novel minimum oscillator whereby a protein with multiple phosphorylation sites directly embedded in a negative feedback loop can exhibit oscillation. They demonstrate that if the fully phosphorylated substrate inhibits the first phosphorylation step in a cooperative manner, multisite substrates can exhibit oscillatory behaviour at the presence of a kinase and phosphatase. With a fixed number of sites, the non-linearity of the negative feedback and the substrate∕enzyme ratio must be above certain threshold values to generate undamped oscillation. There is an inverse relationship between the number of phosphorylation sites and the minimum non-linearity of the negative feedback required for oscillation; that is, the ultrasensitivity and time delay rooted in multisite phosphorylation compensate for the explicit non-linearity in the negative feedback. The period and amplitude of oscillation are mainly determined by the number of phosphorylation sites and the substrate∕enzyme ratio. The authors' results suggest that a multisite protein can be exploited for the construction of a synthetic protein oscillator featuring simplicity, robustness and tunability.

Circadian oscillators in eukaryotes

The biological clock, present in nearly all eukaryotes, has evolved such that organisms can adapt to our planet's rotation in order to anticipate the coming day or night as well as unfavorable seasons. As all modern high-precision chronometers, the biological clock uses oscillation as a timekeeping element. In this review, we describe briefly the discovery, historical development, and general properties of circadian oscillators. The issue of temperature compensation (TC) is discussed, and our present understanding of the underlying genetic and biochemical mechanisms in circadian oscillators are described with special emphasis on Neurospora crassa, mammals, and plants.

Circadian control of global gene expression patterns

An internal time-keeping mechanism has been observed in almost every organism studied from archaea to humans. This circadian clock provides a competitive advantage in fitness and survival ( 18, 30, 95, 129, 137 ). Researchers have uncovered the molecular composition of this internal clock by combining enzymology, molecular biology, genetics, and modeling approaches. However, understanding the mechanistic link between the clock and output responses has been elusive. In three model organisms, Arabidopsis thaliana, Drosophila melanogaster, and Mus musculus, whole-genome expression arrays have enabled researchers to investigate how maintaining a time-keeping mechanism connects to an adaptive advantage. Here, we review the impacts transcriptomics have had on our understanding of the clock and how this molecular clock connects with system-level circadian responses. We explore the discoveries made possible by high-throughput RNA assays, the network approaches used to investigate these large transcript datasets, and potential future directions.

Entrainment Concepts Revisited

The traditional approaches to predict entrainment of circadian clocks by light are based on 2 concepts that were never successfully unified: the non-parametric approach assumes that entrainment occurs via discrete daily phase shifts while the parametric approach assumes that entrainment involves changes of the clock’s velocity. Here the authors suggest a new approach to predict and model entrainment. Unlike the traditional approaches, it does not assume a priori the mechanism of how the internal and external cycle lengths are matched (via phase shifts, velocity changes, or even other mechanisms). It is based on a circadian integrated response characteristic (CIRC) that describes how the circadian system integrates light signals at different circadian phases, without specifying exactly when and how fast its progression is affected. Light around subjective dawn compresses the internal cycle; light around subjective dusk expands it. While the phase response curve (PRC) describes the results of experiments using light stimuli (of specified duration and intensity), the CIRC reflects how the system integrates any given light profile, be it single pulses or any form of light-dark cycle (from skeleton photoperiods to natural light profiles). CIRCs are characterized by their shape (determining the extent of their dead zone) and their asymmetry (the ratio of its compressing and expanding portions). They are dimensionless (time/time), and their maximum is by definition 1. To make predictions about entrainment, the CIRC is multiplied with the light intensity/sensitivity at any given time point. Unlike the PRC and the velocity response curve, the CIRC can be assessed on the basis of entrained steady states, by modeling experimental results. The CIRC approach makes several predictions that can be tested experimentally.

Measurement of single-cell dynamics

Populations of cells are almost always heterogeneous in function and fate. To understand the plasticity of cells, it is vital to measure quantitatively and dynamically the molecular processes that underlie cell-fate decisions in single cells. Early events in cell signalling often occur within seconds of the stimulus, whereas intracellular signalling processes and transcriptional changes can take minutes or hours. By contrast, cell-fate decisions, such as whether a cell divides, differentiates or dies, can take many hours or days. Multiparameter experimental and computational methods that integrate quantitative measurement and mathematical simulation of these noisy and complex processes are required to understand the highly dynamic mechanisms that control cell plasticity and fate.

Discoveries of rhythms in human biological functions: a historical review

Though there are very early and ancient observations on the daily variation in physiological and pathophysiological functions (e.g., bronchial asthma), more detailed and scientific reports were not published until the beginning of the 17th century. The aim of this review is to bring those reports to the attention of researchers of chronobiology and chronopharmacology. The ancient books and their contents, which constitute the basis for this review, are part of the personal library collection of the author; numerous observations and reports on biologic rhythms in man are presented here for the first time. The intent of this review is to demonstrate that the fields of chronobiology and chronopharmacology are not only a new and modern branch of science, but that it stands on the shoulders of wonderful and insightful observations and explanations made by our scientific forefathers. It is the hope that the reader will enjoy the richness of the ancient reports that contribute to our present knowledge achieved through astute early biologic rhythm research.

Inhibition of casein kinase I epsilon/delta produces phase shifts in the circadian rhythms of Cynomolgus monkeys

INTRODUCTION

Circadian rhythms in mammals depend upon the cyclic oscillations of transcriptional/translational feedback loops in pacemaker cells of the suprachiasmatic nucleus. The rise and fall of clock-related proteins is a function of synthesis and degradation, the latter involving phosphorylation by casein kinase Iepsilon and delta.

OBJECTIVE

Earlier studies by our lab described the actions of a selective CKIepsilon/delta inhibitor, PF-670462, on circadian behavior in rats; the present work extended these studies to a diurnal species, Cynomolgus monkeys.

MATERIALS AND METHODS

General cage activity was used to estimate the circadian rhythms of eight telemeterized monkeys under baseline conditions and following s.c. doses of PF-670462.

RESULTS AND DISCUSSION

Consolidated bouts of activity were noted during periods of light with a repeating period length of roughly 24 h based on their onset. Reassessment in constant dim light (42 vs. 450 lx) again yielded period lengths of 24 h, in this instance revealing the animals' endogenous rhythm. PF-670462 (10-100 mg/kg s.c.) produced a dose-dependent phase delay in much the same manner as that observed previously in rats. Dosing occurred 1.5 h prior to lights-off, roughly coincident with peaking levels of PER protein, a primary substrate of CKIepsilon/delta.

CONCLUSIONS

These findings suggest that the time of dosing, when held constant in both the monkey and rat studies, produced nearly identical effects despite the subjects' diurnal or nocturnal preference. Importantly, these changes in rhythm occurred in the presence of light, revealing the drug as a powerful zeitgeber in a non-human primate and, by extension, in man.