A theoretical study on seasonality

Circadian clock period length is not consistently linked to chronotype in a wild songbird

Circadian clock properties vary between individuals and relate to variation in entrained timing in captivity. How this variation translates into behavioural differences in natural settings, however, is poorly understood. Here, we tested in great tits whether variation in the free-running period length (tau) under constant dim light (LL) was linked to the phase angle of the entrained rhythm ("chronotype") in captivity and in the wild, as recently indicated in our study species. We also assessed links between tau and the timing of first activity onset and offset under LL relative to the last experienced light-dark (LD) cycle. We kept 66 great tits, caught in two winters, in LL for 14 days and subsequently released them with a radio transmitter back to the wild, where their activity and body temperature rhythms were tracked for 1 to 22 days. For a subset of birds, chronotype was also recorded in the lab before release. Neither wild nor lab chronotypes were related to tau. We also found no correlation between lab and wild chronotypes. However, the first onset in LL had a positive relationship with tau, but only in males. Our results demonstrate that links between tau and phase of entrainment, postulated on theoretical grounds, may not consistently hold under natural conditions, possibly due to strong masking. This calls for more holistic research on how the many components of the circadian system interact with the environment to shape timing in the wild. Wild birds showed chronotypes in the field that were unlinked to their circadian period length tau measured in captivity. In males only, the first onset of activity after exposure to constant dim light did correlate with tau. Our study emphasises the need to investigate clocks in the real world, including a need to better understand masking.

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.

Weak synchronization can alter circadian period length: implications for aging and disease conditions

The synchronization of multiple oscillators serves as the central mechanism for maintaining stable circadian rhythms in physiology and behavior. Aging and disease can disrupt synchronization, leading to changes in the periodicity of circadian activities. While our understanding of the circadian clock under synchronization has advanced significantly, less is known about its behavior outside synchronization, which can also fall within a predictable domain. These states not only impact the stability of the rhythms but also modulate the period length. In C57BL/6 mice, aging, diseases, and removal of peripheral circadian oscillators often result in lengthened behavioral circadian periods. Here, we show that these changes can be explained by a surprisingly simple mathematical relationship: the frequency is the reciprocal of the period, and its distribution becomes skewed when the period distribution is symmetric. The synchronized frequency of a population in the skewed distribution and the macroscopic frequency of combined oscillators differ, accounting for some of the atypical circadian period outputs observed in networks without synchronization. Building on this finding, we investigate the dynamics of circadian outputs in the context of aging and disease, where synchronization is weakened.

Limits of entrainment of circadian neuronal networks

Circadian rhythmicity lies at the center of various important physiological and behavioral processes in mammals, such as sleep, metabolism, homeostasis, mood changes, and more. Misalignment of intrinsic neuronal oscillations with the external day-night cycle can disrupt such processes and lead to numerous disorders. In this work, we computationally determine the limits of circadian synchronization to external light signals of different frequency, duty cycle, and simulated amplitude. Instead of modeling circadian dynamics with generic oscillator models (e.g., Kuramoto-type), we use a detailed computational neuroscience model, which integrates biomolecular dynamics, neuronal electrophysiology, and network effects. This allows us to investigate the effect of small drug molecules, such as Longdaysin, and connect our results with experimental findings. To combat the high dimensionality of such a detailed model, we employ a matrix-free approach, while our entire algorithmic pipeline enables numerical continuation and construction of bifurcation diagrams using only direct simulation. We, thus, computationally explore the effect of heterogeneity in the circadian neuronal network, as well as the effect of the corrective therapeutic intervention of Longdaysin. Last, we employ unsupervised learning to construct a data-driven embedding space for representing neuronal heterogeneity.

Red Light Resets the Expression Pattern, Phase, and Period of the Circadian Clock in Plants: A Computational Approach

Simple Summary

Progress in computational biology has provided a comprehensive understanding of the dynamics of the plant circadian clock. Previously proposed models of the plant circadian clock have intended to model its entrainment using white-light/dark cycles. However, these models have failed to take into account the effect of light quality on circadian rhythms, which has been experimentally observed. In this work, we developed a computational approach to characterizing the effects of light quality on plant circadian rhythms. The results demonstrated that red light can reset the expression patterns, phases, and periods of clock component genes. The circadian period, amplitude, and phase can be co-optimized for high-quality and efficient breeding.

Abstract

Recent research in the fields of biochemistry and molecular biology has shown that different light qualities have extremely different effects on plant development, and optimizing light quality conditions can speed up plant growth. Clock-regulated red-light signaling, can enhance hypocotyl elongation, and increase seedling height and flower and fruit productivity. In order to investigate the effect of red light on circadian clocks in plants, a novel computational model was established. The expression profiles of the circadian element CCA1 from previous related studies were used to fit the model. The simulation results were validated by the expression patterns of CCA1 in Arabidopsis, including wild types and mutants, and by the phase shifts of CCA1 after red-light pulse. The model was used to further explore the complex responses to various photoperiods, such as the natural white-light/dark cycles, red/white/dark cycles, and extreme 24 h photoperiods. These results demonstrated that red light can reset the expression pattern, period, and phase of the circadian clock. Finally, we identified the dependence of phase shifts on the length of red-light pulse and the minimum red-light pulse length required for producing an observable phase shift. This work provides a promising computational approach to investigating the response of the circadian clock to other light qualities.

The emergence of polyglot entrainment responses to periodic inputs in vicinities of Hopf bifurcations in slow-fast systems

Several distinct entrainment patterns can occur in the FitzHugh-Nagumo (FHN) model under external periodic forcing. Investigating the FHN model under different types of periodic forcing reveals the existence of multiple disconnected 1:1 entrainment segments for constant, low enough values of the input amplitude when the unforced system is in the vicinity of a Hopf bifurcation. This entrainment structure is termed polyglot to distinguish it from the single 1:1 entrainment region (monoglot) structure typically observed in Arnold tongue diagrams. The emergence of polyglot entrainment is then explained using phase-plane analysis and other dynamical system tools. Entrainment results are investigated for other slow-fast systems of neuronal, circadian, and glycolytic oscillations. Exploring these models, we found that polyglot entrainment structure (multiple 1:1 regions) is observed when the unforced system is in the vicinity of a Hopf bifurcation and the Hopf point is located near a knee of a cubic-like nullcline.

Synergies of Multiple Zeitgebers Tune Entrainment

Circadian rhythms are biological rhythms with a period close to 24 h. They become entrained to the Earth's solar day via different periodic cues, so-called zeitgebers. The entrainment of circadian rhythms to a single zeitgeber was investigated in many mathematical clock models of different levels of complexity, ranging from the Poincaré oscillator and the Goodwin model to biologically more detailed models of multiple transcriptional translational feedback loops. However, circadian rhythms are exposed to multiple coexisting zeitgebers in nature. Therefore, we study synergistic effects of two coexisting zeitgebers on different components of the circadian clock. We investigate the induction of period genes by light together with modulations of nuclear receptor activities by drugs and metabolism. Our results show that the entrainment of a circadian rhythm to two coexisting zeitgebers depends strongly on the phase difference between the two zeitgebers. Synergistic interactions of zeitgebers can strengthen diurnal rhythms to reduce detrimental effects of shift-work and jet lag. Medical treatment strategies which aim for stable circadian rhythms should consider interactions of multiple zeitgebers.

Light-induced synchronization of the SCN coupled oscillators and implications for entraining the HPA axis

The suprachiasmatic nucleus (SCN) synchronizes the physiological rhythms to the external light-dark cycle and tunes the dynamics of circadian rhythms to photoperiod fluctuations. Changes in the neuronal network topologies are suggested to cause adaptation of the SCN in different photoperiods, resulting in the broader phase distribution of neuron activities in long photoperiods (LP) compared to short photoperiods (SP). Regulated by the SCN output, the level of glucocorticoids is elevated in short photoperiod, which is associated with peak disease incidence. The underlying coupling mechanisms of the SCN and the interplay between the SCN and the HPA axis have yet to be fully elucidated. In this work, we propose a mathematical model including a multiple-cellular SCN compartment and the HPA axis to investigate the properties of the circadian timing system under photoperiod changes. Our model predicts that the probability-dependent network is more energy-efficient than the distance-dependent network. Coupling the SCN network by intra-subpopulation and inter-subpopulation forces, we identified the negative correlation between robustness and plasticity of the oscillatory network. The HPA rhythms were predicted to be strongly entrained to the SCN rhythms with a pro-inflammatory high-amplitude glucocorticoid profile under SP. The fast temporal topology switch of the SCN network was predicted to enhance synchronization when the synchronization is not complete. These synchronization and circadian dynamics alterations might govern the seasonal variation of disease incidence and its symptom severity.

Mathematical Modeling in Circadian Rhythmicity

Circadian clocks are autonomous systems able to oscillate in a self-sustained manner in the absence of external cues, although such Zeitgebers are typically present. At the cellular level, the molecular clockwork consists of a complex network of interlocked feedback loops. This chapter discusses self-sustained circadian oscillators in the context of nonlinear dynamics theory. We suggest basic steps that can help in constructing a mathematical model and introduce how self-sustained generations can be modeled using ordinary differential equations. Moreover, we discuss how coupled oscillators synchronize among themselves or entrain to periodic signals. The development of mathematical models over the last years has helped to understand such complex network systems and to highlight the basic building blocks in which oscillating systems are built upon. We argue that, through theoretical predictions, the use of simple models can guide experimental research and is thus suitable to model biological systems qualitatively.

Motif structure for the four subgroups within the suprachiasmatic nuclei affects its entrainment ability

Circadian rhythms of physiological and behavioral activities are regulated by a central clock. This clock is located in the bilaterally symmetrical suprachiasmatic nucleus (SCN) of mammals. Each nucleus contains a light-sensitive group of neurons, named the ventrolateral (VL) part, with the rest of the neurons being insensitive to light, named the dorsomedial (DM) group. While the coupling between the VL and DM subgroups have been investigated quite well, the communication among the four subgroups across the nuclei did not get a lot of attention. In this article, we theoretically analyzed seven motiflike connection patterns to investigate the network of the two nuclei of the SCN as a whole in relation to the function of the SCN. We investigated the entrainment ability of the SCN and found that the entrainment range is larger in the motifs containing a link between the two VL parts across the nuclei, but it is smaller in the motifs that contain a link between the two DM parts across the nuclei. The SCN may strengthen or weaken connections between the left and right nucleus to accomodate changes in external conditions, such as resynchronization after a jet lag, adjustment to photoperiod or for the aging SCN.

Principles underlying the complex dynamics of temperature entrainment by a circadian clock

Autonomously oscillating circadian clocks resonate with daily environmental (zeitgeber) rhythms to organize physiology around the solar day. Although entrainment properties and mechanisms have been studied widely and in great detail for light-dark cycles, entrainment to daily temperature rhythms remains poorly understood despite that they are potent zeitgebers. Here we investigate the entrainment of the chronobiological model organism Neurospora crassa, subject to thermocycles of different periods and fractions of warm versus cold phases, mimicking seasonal variations. Depending on the properties of these thermocycles, regularly entrained rhythms, period-doubling (frequency demultiplication) but also irregular aperiodic behavior occurs. We demonstrate that the complex nonlinear phenomena of experimentally observed entrainment dynamics can be understood by molecular mathematical modeling.

Light entrainment of the SCN circadian clock and implications for personalized alterations of corticosterone rhythms in shift work and jet lag

The suprachiasmatic nucleus (SCN) functions as the central pacemaker aligning physiological and behavioral oscillations to day/night (activity/inactivity) transitions. The light signal entrains the molecular clock of the photo-sensitive ventrolateral (VL) core of the SCN which in turn entrains the dorsomedial (DM) shell via the neurotransmitter vasoactive intestinal polypeptide (VIP). The shell converts the VIP rhythmic signals to circadian oscillations of arginine vasopressin (AVP), which eventually act as a neurotransmitter signal entraining the hypothalamic–pituitary–adrenal (HPA) axis, leading to robust circadian secretion of glucocorticoids. In this work, we discuss a semi-mechanistic mathematical model that reflects the essential hierarchical structure of the photic signal transduction from the SCN to the HPA axis. By incorporating the interactions across the core, the shell, and the HPA axis, we investigate how these coupled systems synchronize leading to robust circadian oscillations. Our model predicts the existence of personalized synchronization strategies that enable the maintenance of homeostatic rhythms while allowing for differential responses to transient and permanent light schedule changes. We simulated different behavioral situations leading to perturbed rhythmicity, performed a detailed computational analysis of the dynamic response of the system under varying light schedules, and determined that (1) significant interindividual diversity and flexibility characterize adaptation to varying light schedules; (2) an individual’s tolerances to jet lag and alternating shift work are positively correlated, while the tolerances to jet lag and transient shift work are negatively correlated, which indicates trade-offs in an individual’s ability to maintain physiological rhythmicity; (3) weak light sensitivity leads to the reduction of circadian flexibility, implying that light therapy can be a potential approach to address shift work and jet lag related disorders. Finally, we developed a map of the impact of the synchronization within the SCN and between the SCN and the HPA axis as it relates to the emergence of circadian flexibility.

Naturalistic Intensities of Light at Night: A Review of the Potent Effects of Very Dim Light on Circadian Responses and Considerations for Translational Research

In this review, we discuss the remarkable potency and potential applications of a form of light that is often overlooked in a circadian context: naturalistic levels of dim light at night (nLAN), equivalent to intensities produced by the moon and stars. It is often assumed that such low levels of light do not produce circadian responses typically associated with brighter light levels. A solid understanding of the impacts of very low light levels is complicated further by the broad use of the somewhat ambiguous term “dim light,” which has been used to describe light levels ranging seven orders of magnitude. Here, we lay out the argument that nLAN exerts potent circadian effects on numerous mammalian species, and that given conservation of anatomy and function, the efficacy of light in this range in humans warrants further investigation. We also provide recommendations for the field of chronobiological research, including minimum requirements for the measurement and reporting of light, standardization of terminology (specifically as it pertains to “dim” light), and ideas for reconsidering old data and designing new studies.

Conceptual Models of Entrainment, Jet Lag, and Seasonality

Understanding entrainment of circadian rhythms is a central goal of chronobiology. Many factors, such as period, amplitude, Zeitgeber strength, and daylength, govern entrainment ranges and phases of entrainment. We have tested whether simple amplitude-phase models can provide insight into the control of entrainment phases. Using global optimization, we derived conceptual models with just three free parameters (period, amplitude, and relaxation rate) that reproduce known phenotypic features of vertebrate clocks: phase response curves (PRCs) with relatively small phase shifts, fast re-entrainment after jet lag, and seasonal variability to track light onset or offset. Since optimization found multiple sets of model parameters, we could study this model ensemble to gain insight into the underlying design principles. We found complex associations between model parameters and entrainment features. Arnold onions of representative models visualize strong dependencies of entrainment on periods, relative Zeitgeber strength, and photoperiods. Our results support the use of oscillator theory as a framework for understanding the entrainment of circadian clocks.

Clocks in the Wild: Entrainment to Natural Light

Entrainment denotes a process of coordinating the internal circadian clock to external rhythmic time-cues (Zeitgeber), mainly light. It is facilitated by stronger Zeitgeber signals and smaller period differences between the internal clock and the external Zeitgeber. The phase of entrainment ψ is a result of this process on the side of the circadian clock. On Earth, the period of the day-night cycle is fixed to 24 h, while the periods of circadian clocks distribute widely due to natural variation within and between species. The strength and duration of light depend locally on season and geographic latitude. Therefore, entrainment characteristics of a circadian clock vary under a local light environment and distribute along geoecological settings. Using conceptual models of circadian clocks, we investigate how local conditions of natural light shape global patterning of entrainment through seasons. This clock-side entrainment paradigm enables us to predict systematic changes in the global distribution of chronotypes.

Weak coupling between intracellular feedback loops explains dissociation of clock gene dynamics

Circadian rhythms are generated by interlocked transcriptional-translational negative feedback loops (TTFLs), the molecular process implemented within a cell. The contributions, weighting and balancing between the multiple feedback loops remain debated. Dissociated, free-running dynamics in the expression of distinct clock genes has been described in recent experimental studies that applied various perturbations such as slice preparations, light pulses, jet-lag, and culture medium exchange. In this paper, we provide evidence that this "presumably transient" dissociation of circadian gene expression oscillations may occur at the single-cell level. Conceptual and detailed mechanistic mathematical modeling suggests that such dissociation is due to a weak interaction between multiple feedback loops present within a single cell. The dissociable loops provide insights into underlying mechanisms and general design principles of the molecular circadian clock.

Heterogeneity in relaxation rate improves the synchronization of oscillatory neurons in a model of the SCN

The circadian rhythms in mammals, that are regulated by the suprachiasmatic nucleus (SCN) of the brain, have been observed even in the absence of a light-dark cycle. The SCN is composed of about 10 000 autonomous neuronal oscillators, which are heterogenous in many oscillatory properties, including the heterogeneity in relaxation rates. Although the relaxation rate affects the entrainability of the SCN as a whole, not much is known about the reasons why the heterogeneity in relaxation rate exists. In the present study, based on a Poincaré model, we examine whether the heterogeneity in the relaxation rate affects the synchronization of the SCN neuronal oscillators under constant darkness. Both our simulations and theoretical results show that the heterogeneity improves the synchronization. Our findings provide an alternative explanation for the existence of the heterogeneity in the SCN neurons and shed light on the effect of neuronal heterogeneity on the collective behavior of the SCN neurons.

Modeling the influence of chronopharmacological administration of synthetic glucocorticoids on the hypothalamic-pituitary-adrenal axis

Natural glucocorticoids, a class of cholesterol-derived hormones, modulate an array of metabolic, anti-inflammatory, immunosuppressive and cognitive signaling. The synthesis of natural glucocorticoids, largely cortisol in humans, is regulated by the hypothalamic-pituitary-adrenal (HPA) axis and exhibits pronounced circadian variation. Considering the central regulatory function of endogenous glucocorticoids, maintenance of the circadian activity of the HPA axis is essential to host survival and chronic disruption of such activity leads to systemic complications. There is a great deal of interest in synthetic glucocorticoids due to the immunosuppressive and anti-inflammatory properties and the development of novel dosing regimens that can minimize the disruption of endogenous activity, while still maintaining the pharmacological benefits of long-term synthetic glucocorticoid therapy. Synthetic glucocorticoids are associated with an increased risk of developing the pathological disorders related to chronic suppression of cortisol rhythmicity as a result of the potent negative feedback by synthetic glucocorticoids on the HPA axis precursors. In this study, a mathematical model was developed to explore the influence of chronopharmacological dosing of exogenous glucocorticoids on the endogenous cortisol rhythm considering intra-venous and oral dosing. Chronic daily dosing resulted in modification of the circadian rhythmicity of endogenous cortisol with the amplitude and acrophase of the altered rhythm dependent on the administration time. Simulations revealed that the circadian features of the endogenous cortisol rhythm can be preserved by proper timing of administration. The response following a single dose was not indicative of the response following long-term, repeated chronopharmacological dosing of synthetic glucocorticoids. Furthermore, simulations revealed the inductive influence of long-term treatment was only associated with low to moderate doses, while high doses generally led to suppression of endogenous activity regardless of the chronopharmacological dose. Finally, chronic daily dosing was found to alter the responsiveness of the HPA axis, such that a decrease in the amplitude of the cortisol rhythm resulted in a partial loss in the time-of-day dependent response to CRH stimulation, while an increase in the amplitude was associated with a more pronounced time-of-day dependence of the response.

Computational modelling unravels the precise clockwork of cyanobacteria

Precisely timing the regulation of gene expression by anticipating recurring environmental changes is a fundamental part of global gene regulation. Circadian clocks are one form of this regulation, which is found in both eukaryotes and prokaryotes, providing a fitness advantage for these organisms. Whereas many different eukaryotic groups harbour circadian clocks, cyanobacteria are the only known oxygenic phototrophic prokaryotes to regulate large parts of their genes in a circadian fashion. A decade of intensive research on the mechanisms and functionality using computational and mathematical approaches in addition to the detailed biochemical and biophysical understanding make this the best understood circadian clock. Here, we summarize the findings and insights into various parts of the cyanobacterial circadian clock made by mathematical modelling. These findings have implications for eukaryotic circadian research as well as synthetic biology harnessing the power and efficiency of global gene regulation.

Optimal entrainment of circadian clocks in the presence of noise

Circadian clocks are biochemical oscillators that allow organisms to estimate the time of the day. These oscillators are inherently noisy due to the discrete nature of the reactants and the stochastic character of their interactions. To keep these oscillators in sync with the daily day-night rhythm in the presence of noise, circadian clocks must be coupled to the dark-light cycle. In this paper, we study the entrainment of phase oscillators as a function of the intrinsic noise in the system. Using stochastic simulations, we compute the optimal coupling strength, intrinsic frequency, and shape of the phase-response curve, that maximize the mutual information between the phase of the clock and time. We show that the optimal coupling strength and intrinsic frequency increase with the noise, but that the shape of the phase-response curve varies nonmonotonically with the noise: in the low-noise regime, it features a dead zone that increases in width as the noise increases, while in the high-noise regime, the width decreases with the noise. These results arise from a tradeoff between maximizing stability-noise suppression-and maximizing linearity of the input-output, i.e., time-phase, relation. We also show that three analytic approximations-the linear-noise approximation, the phase-averaging method, and linear-response theory-accurately describe different regimes of the coupling strength and the noise.

Dependence of the entrainment on the ratio of amplitudes between two subgroups in the suprachiasmatic nucleus

Organisms can be synchronized not only to the natural 24-h light-dark cycle but also to artificial non-24-h cycles. Interestingly, when the period of the cycle is far from 24 h, organisms may show complicated behavioral patterns. For example, exposed to a 22-h light-dark cycle, in behavioral activity of rats, a phenomenon called "dissociation" emerges, i.e., one periodic component shows a 22-h period and the other shows a period close to the endogenous period of the animal (around 24 h). It has been found that these two components are regulated by two subgroups of the suprachiasmatic nucleus (SCN), respectively, with the ventrolateral part regulating the 22-h component and the dorsomedial part regulating the other component. In the present study, based on a mathematical model, we will examine how the ratio of amplitudes between these two subgroups affects the entrainment of the SCN to the external 22-h light-dark cycle. Our results show that the dissociation happens when the ratio is smaller than 1 and the maximal entrainment (synchronization) ability of the SCN to the external cycle is obtained when the ratio is larger than 1. Our finding sheds light on the dissociation between the subgroups and suggests that the heterogeneity in the amplitudes alter the entrainment ability of the SCN.

Modeling the Influence of Seasonal Differences in the HPA Axis on Synchronization of the Circadian Clock and Cell Cycle

Synchronization of biological functions to environmental signals enables organisms to anticipate and appropriately respond to daily external fluctuations and is critical to the maintenance of homeostasis. Misalignment of circadian rhythms with environmental cues is associated with adverse health outcomes. Cortisol, the downstream effector of hypothalamic-pituitary-adrenal (HPA) activity, facilitates synchronization of peripheral biological processes to the environment. Cortisol levels exhibit substantial seasonal rhythmicity, with peak levels occurring during the short-photoperiod winter months and reduced levels occurring in the long-photoperiod summer season. Seasonal changes in cortisol secretion could therefore alter its entraining capabilities, resulting in a season-dependent modification in the alignment of biological activities with the environment. We develop a mathematical model to investigate the influence of photoperiod-induced seasonal differences in the circadian rhythmicity of the HPA axis on the synchronization of the peripheral circadian clock and cell cycle in a heterogeneous cell population. Model simulations predict that the high-amplitude cortisol rhythms in winter result in the greatest entrainment of peripheral oscillators. Furthermore, simulations predict a circadian gating of the cell cycle with respect to the expression of peripheral clock genes. Seasonal differences in cortisol rhythmicity are also predicted to influence mitotic synchrony, with a high-amplitude winter rhythm resulting in the greatest synchrony and a shift in timing of the cell cycle phases, relative to summer. Our results highlight the primary interactions among the HPA axis, the peripheral circadian clock, and the cell cycle and thereby provide an improved understanding of the implications of circadian misalignment on the synchronization of peripheral regulatory processes.

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.

Experimental and Mathematical Analyses Relating Circadian Period and Phase of Entrainment in Neurospora crassa

Circadian rhythms are observed in most organisms on earth and are known to play a major role in successful adaptation to the 24-h cycling environment. Circadian phenotypes are characterized by a free-running period that is observed in constant conditions and an entrained phase that is observed in cyclic conditions. Thus, the relationship between the free-running period and phase of entrainment is of interest. A popular simple rule has been that the entrained phase is the expression of the period in a cycling environment (i.e., that a short period causes an advanced phase and a long period causes a delayed phase). However, there are experimental data that are not explained by this simple relationship, and no systematic study has been done to explore all possible period-phase relationships. Here, we show the existence of stable period-phase relationships that are exceptions to this rule. First, we analyzed period-phase relationships using populations with different degrees of genome complexity. Second, we generated isogenic F1 populations by crossing 14 classical period mutants to the same female and analyzed 2 populations with a short period/delayed phase and a long period/advanced phase. Third, we generated a mathematical model to account for such variable relationships between period and phase. Our analyses support the view that the circadian period of an organism is not the only predictor of the entrained phase.

Tuning the phase of circadian entrainment

The circadian clock coordinates daily physiological, metabolic and behavioural rhythms. These endogenous oscillations are synchronized with external cues ('zeitgebers'), such as daily light and temperature cycles. When the circadian clock is entrained by a zeitgeber, the phase difference ψ between the phase of a clock-controlled rhythm and the phase of the zeitgeber is of fundamental importance for the fitness of the organism. The phase of entrainment ψ depends on the mismatch between the intrinsic period τ and the zeitgeber period T and on the ratio of the zeitgeber strength to oscillator amplitude. Motivated by the intriguing complexity of empirical data and by our own experiments on temperature entrainment of mouse suprachiasmatic nucleus (SCN) slices, we present a theory on how clock and zeitgeber properties determine the phase of entrainment. The wide applicability of the theory is demonstrated using mathematical models of different complexity as well as by experimental data. Predictions of the theory are confirmed by published data on Neurospora crassa strains for different period mismatches τ - T and varying photoperiods. We apply a novel regression technique to analyse entrainment of SCN slices by temperature cycles. We find that mathematical models can explain not only the stable asymptotic phase of entrainment, but also transient phase dynamics. Our theory provides the potential to explore seasonal variations of circadian rhythms, jet lag and shift work in forthcoming studies.

A Compact Model for the Complex Plant Circadian Clock

The circadian clock is an endogenous timekeeper that allows organisms to anticipate and adapt to the daily variations of their environment. The plant clock is an intricate network of interlocked feedback loops, in which transcription factors regulate each other to generate oscillations with expression peaks at specific times of the day. Over the last decade, mathematical modeling approaches have been used to understand the inner workings of the clock in the model plant Arabidopsis thaliana. Those efforts have produced a number of models of ever increasing complexity. Here, we present an alternative model that combines a low number of equations and parameters, similar to the very earliest models, with the complex network structure found in more recent ones. This simple model describes the temporal evolution of the abundance of eight clock gene mRNA/protein and captures key features of the clock on a qualitative level, namely the entrained and free-running behaviors of the wild type clock, as well as the defects found in knockout mutants (such as altered free-running periods, lack of entrainment, or changes in the expression of other clock genes). Additionally, our model produces complex responses to various light cues, such as extreme photoperiods and non-24 h environmental cycles, and can describe the control of hypocotyl growth by the clock. Our model constitutes a useful tool to probe dynamical properties of the core clock as well as clock-dependent processes.