Quantification of circadian rhythms in single cells

Discontinuous phase transition switching induced by a power-law function between dynamical parameters in coupled oscillators

In biological or physical systems, the intrinsic properties of oscillators are heterogeneous and correlated. These two characteristics have been empirically validated and have garnered attention in theoretical studies. In this paper, we propose a power-law function existed between the dynamical parameters of the coupled oscillators, which can control discontinuous phase transition switching. Unlike the special designs for the coupling terms, this generalized function within the dynamical term reveals another path for generating the first-order phase transitions. The power-law relationship between dynamic characteristics is reasonable, as observed in empirical studies, such as long-term tremor activity during volcanic eruptions and ion channel characteristics of the Xenopus expression system. Our work expands the conditions that used to be strict for the occurrence of the first-order phase transitions and deepens our understanding of the impact of correlation between intrinsic parameters on phase transitions. We explain the reason why the continuous phase transition switches to the discontinuous phase transition when the control parameter is at a critical value.

Coupling allows robust mammalian redox circadian rhythms despite heterogeneity and noise

Circadian clocks are endogenous oscillators present in almost all cells that drive daily rhythms in physiology and behavior. There are two mechanisms that have been proposed to explain how circadian rhythms are generated in mammalian cells: through a transcription-translation feedback loop (TTFL) and based on oxidation/reduction reactions, both of which are intrinsically stochastic and heterogeneous at the single cell level. In order to explore the emerging properties of stochastic and heterogeneous redox oscillators, we simplify a recently developed kinetic model of redox oscillations to an amplitude-phase oscillator with 'twist' (period-amplitude correlation) and subject to Gaussian noise. We show that noise and heterogeneity alone lead to fast desynchronization, and that coupling between noisy oscillators can establish robust and synchronized rhythms with amplitude expansions and tuning of the period due to twist. Coupling a network of redox oscillators to a simple model of the TTFL also contributes to synchronization, large amplitudes and fine-tuning of the period for appropriate interaction strengths. These results provide insights into how the circadian clock compensates randomness from intracellular sources and highlight the importance of noise, heterogeneity and coupling in the context of circadian oscillators.

Noise facilitates entrainment of a population of uncoupled limit cycle oscillators

Many biological oscillators share two properties: they are subject to stochastic fluctuations (noise) and they must reliably adjust their period to changing environmental conditions (entrainment). While noise seems to distort the ability of single oscillators to entrain, in populations of uncoupled oscillators noise allows population-level entrainment for a wider range of input amplitudes and periods. Here, we investigate how this effect depends on the noise intensity and the number of oscillators in the population. We have found that, if a population consists of a sufficient number of oscillators, increasing noise intensity leads to faster entrainment after a phase change of the input signal (jet lag) and increases sensitivity to low-amplitude input signals.

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.

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.

A spatio-temporal model to reveal oscillator phenotypes in molecular clocks: Parameter estimation elucidates circadian gene transcription dynamics in single-cells

We propose a stochastic distributed delay model together with a Markov random field prior and a measurement model for bioluminescence-reporting to analyse spatio-temporal gene expression in intact networks of cells. The model describes the oscillating time evolution of molecular mRNA counts through a negative transcriptional-translational feedback loop encoded in a chemical Langevin equation with a probabilistic delay distribution. The model is extended spatially by means of a multiplicative random effects model with a first order Markov random field prior distribution. Our methodology effectively separates intrinsic molecular noise, measurement noise, and extrinsic noise and phenotypic variation driving cell heterogeneity, while being amenable to parameter identification and inference. Based on the single-cell model we propose a novel computational stability analysis that allows us to infer two key characteristics, namely the robustness of the oscillations, i.e. whether the reaction network exhibits sustained or damped oscillations, and the profile of the regulation, i.e. whether the inhibition occurs over time in a more distributed versus a more direct manner, which affects the cells' ability to phase-shift to new schedules. We show how insight into the spatio-temporal characteristics of the circadian feedback loop in the suprachiasmatic nucleus (SCN) can be gained by applying the methodology to bioluminescence-reported expression of the circadian core clock gene Cry1 across mouse SCN tissue. We find that while (almost) all SCN neurons exhibit robust cell-autonomous oscillations, the parameters that are associated with the regulatory transcription profile give rise to a spatial division of the tissue between the central region whose oscillations are resilient to perturbation in the sense that they maintain a high degree of synchronicity, and the dorsal region which appears to phase shift in a more diversified way as a response to large perturbations and thus could be more amenable to entrainment.

Cut the noise or couple up: Coordinating circadian and synthetic clocks

Circadian clocks are important to much of life on Earth and are of inherent interest to humanity, implicated in fields ranging from agriculture and ecology to developmental biology and medicine. New techniques show that it is not simply the presence of clocks, but coordination between them that is critical for complex physiological processes across the kingdoms of life. Recent years have also seen impressive advances in synthetic biology to the point where parallels can be drawn between synthetic biological and circadian oscillators. This review will emphasize theoretical and experimental studies that have revealed a fascinating dichotomy of coupling and heterogeneity among circadian clocks. We will also consolidate the fields of chronobiology and synthetic biology, discussing key design principles of their respective oscillators.

A stochastic oscillator model simulates the entrainment of vertebrate cellular clocks by light

The circadian clock is a cellular mechanism that synchronizes various biological processes with respect to the time of the day. While much progress has been made characterizing the molecular mechanisms underlying this clock, it is less clear how external light cues influence the dynamics of the core clock mechanism and thereby entrain it with the light-dark cycle. Zebrafish-derived cell cultures possess clocks that are directly light-entrainable, thus providing an attractive laboratory model for circadian entrainment. Here, we have developed a stochastic oscillator model of the zebrafish circadian clock, which accounts for the core clock negative feedback loop, light input, and the proliferation of single-cell oscillator noise into population-level luminescence recordings. The model accurately predicts the entrainment dynamics observed in bioluminescent clock reporter assays upon exposure to a wide range of lighting conditions. Furthermore, we have applied the model to obtain refitted parameter sets for cell cultures exposed to a variety of pharmacological treatments and predict changes in single-cell oscillator parameters. Our work paves the way for model-based, large-scale screens for genetic or pharmacologically-induced modifications to the entrainment of circadian clock function.

Identifying a stochastic clock network with light entrainment for single cells of Neurospora crassa

Stochastic networks for the clock were identified by ensemble methods using genetic algorithms that captured the amplitude and period variation in single cell oscillators of Neurospora crassa. The genetic algorithms were at least an order of magnitude faster than ensemble methods using parallel tempering and appeared to provide a globally optimum solution from a random start in the initial guess of model parameters (i.e., rate constants and initial counts of molecules in a cell). The resulting goodness of fit [Formula: see text] was roughly halved versus solutions produced by ensemble methods using parallel tempering, and the resulting [Formula: see text] per data point was only [Formula: see text] = 2,708.05/953 = 2.84. The fitted model ensemble was robust to variation in proxies for "cell size". The fitted neutral models without cellular communication between single cells isolated by microfluidics provided evidence for only one Stochastic Resonance at one common level of stochastic intracellular noise across days from 6 to 36 h of light/dark (L/D) or in a D/D experiment. When the light-driven phase synchronization was strong as measured by the Kuramoto (K), there was degradation in the single cell oscillations away from the stochastic resonance. The rate constants for the stochastic clock network are consistent with those determined on a macroscopic scale of 10⁷ cells.

Damped circadian oscillation in the absence of KaiA in Synechococcus

Proteins KaiA, KaiB and KaiC constitute a biochemical circadian oscillator in the cyanobacterium Synechococcus elongatus. It has been reported kaiA inactivation completely abolishes circadian oscillations. However, we show here that kaiBC promoter activity exhibits a damped, low-amplitude oscillation with a period of approximately 24 h in kaiA-inactivated strains. The damped rhythm resonates with external cycles with a period of 24–26 h, indicating that its natural frequency is similar to that of the circadian clock. Double-mutation experiments reveal that kaiC, kaiB, and sasA (encoding a KaiC-binding histidine kinase) are all required for the damped oscillation. Further analysis suggests that the kaiA-less damped transcriptional rhythm requires KaiB-KaiC complex formation and the transcription-translation feedback loop, but not the KaiC phosphorylation cycle. Our results provide insights into mechanisms that could potentially underlie the diurnal/circadian behaviors observed in other bacterial species that possess kaiB and kaiC homologues but lack a kaiA homologue.

Proteins KaiA, KaiB and KaiC constitute a biochemical circadian oscillator in Synechococcus cyanobacteria. Here, Kawamoto et al. show that kaiBC promoter activity exhibits a damped, low-amplitude circadian oscillation in the absence of KaiA, which could explain the circadian rhythms observed in other bacteria that lack a kaiA homologue.

Chemical modulation of circadian rhythms and assessment of cellular behavior via indirubin and derivatives

Circadian rhythms are critical regulators of many physiological and behavioral functions. The use and abilities of small molecules to affect oscillations have recently received significant attention. These manipulations can be reversible and tunable, and have been used to study various biological mechanisms and molecular properties. Here, we outline procedures for assessment of cellular circadian changes following treatment with small molecules, using luminescent reporters. We describe reporter generation, luminometry experiments, and data analysis. Protocols for studies of accompanying effects on cells, including motility, viability, and anchorage-independent proliferation assays are also presented. As examples, we use indirubin-3'-oxime and two derivatives, 5-iodo-indirubin-3'-oxime and 5-sulfonic acid-indirubin-3'-oxime. In this case study, we analyze effects of these compounds on Bmal1 and Per2 (positive and negative core circadian elements) oscillations and provide step-by-step protocols for data analysis, including removal of trends from raw data, period estimations, and statistical analysis. The reader is provided with detailed protocols, and guidance regarding selection of and alternative approaches.

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.

ECHO: an application for detection and analysis of oscillators identifies metabolic regulation on genome-wide circadian output

MOTIVATION

Time courses utilizing genome scale data are a common approach to identifying the biological pathways that are controlled by the circadian clock, an important regulator of organismal fitness. However, the methods used to detect circadian oscillations in these datasets are not able to accommodate changes in the amplitude of the oscillations over time, leading to an underestimation of the impact of the clock on biological systems.

RESULTS

We have created a program to efficaciously identify oscillations in large-scale datasets, called the Extended Circadian Harmonic Oscillator application, or ECHO. ECHO utilizes an extended solution of the fixed amplitude oscillator that incorporates the amplitude change coefficient. Employing synthetic datasets, we determined that ECHO outperforms existing methods in detecting rhythms with decreasing oscillation amplitudes and in recovering phase shift. Rhythms with changing amplitudes identified from published biological datasets revealed distinct functions from those oscillations that were harmonic, suggesting purposeful biologic regulation to create this subtype of circadian rhythms.

AVAILABILITY AND IMPLEMENTATION

ECHO's full interface is available at https://github.com/delosh653/ECHO. An R package for this functionality, echo.find, can be downloaded at https://CRAN.R-project.org/package=echo.find.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

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.

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.

Coherency of circadian rhythms in the SCN is governed by the interplay of two coupling factors

Circadian clocks are autonomous oscillators driving daily rhythms in physiology and behavior. In mammals, a network of coupled neurons in the suprachiasmatic nucleus (SCN) is entrained to environmental light-dark cycles and orchestrates the timing of peripheral organs. In each neuron, transcriptional feedbacks generate noisy oscillations. Coupling mediated by neuropeptides such as VIP and AVP lends precision and robustness to circadian rhythms. The detailed coupling mechanisms between SCN neurons are debated. We analyze organotypic SCN slices from neonatal and adult mice in wild-type and multiple knockout conditions. Different degrees of rhythmicity are quantified by pixel-level analysis of bioluminescence data. We use empirical orthogonal functions (EOFs) to characterize spatio-temporal patterns. Simulations of coupled stochastic single cell oscillators can reproduce the diversity of observed patterns. Our combination of data analysis and modeling provides deeper insight into the enormous complexity of the data: (1) Neonatal slices are typically stronger oscillators than adult slices pointing to developmental changes of coupling. (2) Wild-type slices are completely synchronized and exhibit specific spatio-temporal patterns of phases. (3) Some slices of Cry double knockouts obey impaired synchrony that can lead to co–existing rhythms (“splitting”). (4) The loss of VIP-coupling leads to desynchronized rhythms with few residual local clusters. Additional information was extracted from co–culturing slices with rhythmic neonatal wild-type SCNs. These co–culturing experiments were simulated using external forcing terms representing VIP and AVP signaling. The rescue of rhythmicity via co–culturing lead to surprising results, since a cocktail of AVP-antagonists improved synchrony. Our modeling suggests that these counter-intuitive observations are pointing to an antagonistic action of VIP and AVP coupling. Our systematic theoretical and experimental study shows that dual coupling mechanisms can explain the astonishing complexity of spatio-temporal patterns in SCN slices.

The mammalian circadian clock is orchestrated by a network of coupled neurons. Brain slice preparations allow the analysis of coupling mechanisms mediated by neuropeptides. From bioluminescence recordings, we extract single cell characteristics such as period, amplitude and damping rate. Our data-based stochastic network model involves local coupling between cells and additional external forcing. Available experimental data guide our simulations with two distinct coupling and forcing mechanisms representing the neuropeptides VIP and AVP. We compare our simulations with experiments from neonatal and adult wild-type brain slices and multiple knockouts. Furthermore, we study co–culturing of slices with synchronized neonatal wild-type slices. The extreme complexity of the spatio-temporal patterns is quantified using empirical orthogonal functions (EOFs). The experimental reduction of AVP coupling leads to surprising observations. In double knockouts, inhibition of AVP signaling can improve synchrony, whereas, in triple knockouts, coherency is reduced. Our network modeling shows that these counter-intuitive observations can be explained by an antagonistic action of VIP and AVP signaling. The agreement of experiments and simulations suggests that quite complex spatio-temporal patterns can appear as emergent properties of oscillator networks with dual coupling mechanisms.

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.

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.

Modeling-Based Investigation of the Effect of Noise in Cellular Systems

Noise is pervasive in cellular biology and inevitably affects the dynamics of cellular processes. Biological systems have developed regulatory mechanisms to ensure robustness with respect to noise or to take advantage of stochasticity. We review here, through a couple of selected examples, some insights on possible robustness factors and constructive roles of noise provided by computational modeling. In particular, we focus on (1) factors that likely contribute to the robustness of oscillatory processes such as the circadian clocks and the cell cycle, (2) how reliable coding/decoding of calcium-mediated signaling could be achieved in presence of noise and, in some cases, enhanced through stochastic resonance, and (3) how embryonic cell differentiation processes can exploit stochasticity to create heterogeneity in a population of identical cells.

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.

Biological information systems: Evolution as cognition-based information management

An alternative biological synthesis is presented that conceptualizes evolutionary biology as an epiphenomenon of integrated self-referential information management. Since all biological information has inherent ambiguity, the systematic assessment of information is required by living organisms to maintain self-identity and homeostatic equipoise in confrontation with environmental challenges. Through their self-referential attachment to information space, cells are the cornerstone of biological action. That individualized assessment of information space permits self-referential, self-organizing niche construction. That deployment of information and its subsequent selection enacted the dominant stable unicellular informational architectures whose biological expressions are the prokaryotic, archaeal, and eukaryotic unicellular forms. Multicellularity represents the collective appraisal of equivocal environmental information through a shared information space. This concerted action can be viewed as systematized information management to improve information quality for the maintenance of preferred homeostatic boundaries among the varied participants. When reiterated in successive scales, this same collaborative exchange of information yields macroscopic organisms as obligatory multicellular holobionts. Cognition-Based Evolution (CBE) upholds that assessment of information precedes biological action, and the deployment of information through integrative self-referential niche construction and natural cellular engineering antecedes selection. Therefore, evolutionary biology can be framed as a complex reciprocating interactome that consists of the assessment, communication, deployment and management of information by self-referential organisms at multiple scales in continuous confrontation with environmental stresses.

Differences in intrinsic amplitudes of neuronal oscillators improve synchronization in the suprachiasmatic nucleus

In mammals, a main clock located in the suprachiasmatic nucleus (SCN) regulates the ∼24 h rhythms of behavioral and physiological activities exposed to a natural 24 light-dark cycle or even under constant darkness. The rhythms originate from self-sustained oscillations of the SCN neurons, which differ in both intrinsic periods and intrinsic amplitudes. The intrinsic periods and the intrinsic amplitudes were found to be bound to specific regions in the previous experiments. In particular, neurons of smaller amplitudes and larger periods are located in a ventrolateral part, and neurons of larger amplitudes and smaller periods are in a dorsomedial part. In the present study, we examined the effects of the differences in the intrinsic frequencies and the differences in the intrinsic amplitudes of neuronal oscillators on the synchronization, respectively. We found that the differences in the intrinsic frequencies weaken the synchronization, whereas the differences in the intrinsic amplitudes strengthen the synchronization. Our finding may shed light on the effects of the heterogenous properties of individual neurons on the collective behaviors of the SCN network and provide a way to enhance the synchronization.

Potential contribution of tandem circadian enhancers to nonlinear oscillations in clock gene expression

Limit-cycle oscillations require the presence of nonlinear processes. Although mathematical studies have long suggested that multiple nonlinear processes are required for autonomous circadian oscillation in clock gene expression, the underlying mechanism remains controversial. Here we show experimentally that cell-autonomous circadian transcription of a mammalian clock gene requires a functionally interdependent tandem E-box motif; the lack of either of the two E-boxes results in arrhythmic transcription. Although previous studies indicated the role of the tandem motifs in increasing circadian amplitude, enhancing amplitude does not explain the mechanism for limit-cycle oscillations in transcription. In this study, mathematical analysis suggests that the interdependent behavior of enhancer elements including not only E-boxes but also ROR response elements might contribute to limit-cycle oscillations by increasing transcriptional nonlinearity. As expected, introduction of the interdependence of circadian enhancer elements into mathematical models resulted in autonomous transcriptional oscillation with low Hill coefficients. Together these findings suggest that interdependent tandem enhancer motifs on multiple clock genes might cooperatively enhance nonlinearity in the whole circadian feedback system, which would lead to limit-cycle oscillations in clock gene expression.

Identifying stochastic oscillations in single-cell live imaging time series using Gaussian processes

Multiple biological processes are driven by oscillatory gene expression at different time scales. Pulsatile dynamics are thought to be widespread, and single-cell live imaging of gene expression has lead to a surge of dynamic, possibly oscillatory, data for different gene networks. However, the regulation of gene expression at the level of an individual cell involves reactions between finite numbers of molecules, and this can result in inherent randomness in expression dynamics, which blurs the boundaries between aperiodic fluctuations and noisy oscillators. This underlies a new challenge to the experimentalist because neither intuition nor pre-existing methods work well for identifying oscillatory activity in noisy biological time series. Thus, there is an acute need for an objective statistical method for classifying whether an experimentally derived noisy time series is periodic. Here, we present a new data analysis method that combines mechanistic stochastic modelling with the powerful methods of non-parametric regression with Gaussian processes. Our method can distinguish oscillatory gene expression from random fluctuations of non-oscillatory expression in single-cell time series, despite peak-to-peak variability in period and amplitude of single-cell oscillations. We show that our method outperforms the Lomb-Scargle periodogram in successfully classifying cells as oscillatory or non-oscillatory in data simulated from a simple genetic oscillator model and in experimental data. Analysis of bioluminescent live-cell imaging shows a significantly greater number of oscillatory cells when luciferase is driven by a Hes1 promoter (10/19), which has previously been reported to oscillate, than the constitutive MoMuLV 5’ LTR (MMLV) promoter (0/25). The method can be applied to data from any gene network to both quantify the proportion of oscillating cells within a population and to measure the period and quality of oscillations. It is publicly available as a MATLAB package.

Technological advances now allow us to observe gene expression in real-time at a single-cell level. In a wide variety of biological contexts this new data has revealed that gene expression is highly dynamic and possibly oscillatory. It is thought that periodic gene expression may be useful for keeping track of time and space, as well as transmitting information about signalling cues. Classifying a time series as periodic from single cell data is difficult because it is necessary to distinguish whether peaks and troughs are generated from an underlying oscillator or whether they are aperiodic fluctuations. To this end, we present a novel tool to classify live-cell data as oscillatory or non-oscillatory that accounts for inherent biological noise. We first demonstrate that the method outperforms a competing scheme in classifying computationally simulated single-cell data, and we subsequently analyse live-cell imaging time series. Our method is able to successfully detect oscillations in a known genetic oscillator, but it classifies data from a constitutively expressed gene as aperiodic. The method forms a basis for discovering new gene expression oscillators and quantifying how oscillatory activity alters in response to changes in cell fate and environmental or genetic perturbations.

Dissociation between two subgroups of the suprachiasmatic nucleus affected by the number of damped oscillated neurons

In mammals, the main clock located in the suprachiasmatic nucleus (SCN) of the brain synchronizes the body rhythms to the environmental light-dark cycle. The SCN is composed of about 2×10^{4} neurons which can be classified into three oscillatory phenotypes: self-sustained oscillators, damped oscillators, and arrhythmic neurons. Exposed to an artificial external light-dark cycle with a period of 22h instead of 24h, two subgroups of the SCN can become desynchronized (dissociated). The ventrolateral (VL) subgroup receives photic input and is entrained to the external cycle and a dorsomedial (DM) subgroup oscillates with its endogenous (i.e., free running) period and is synchronized to the external light-dark cycle through coupling from the VL. In the present study, we examined the effects of damped oscillatory neurons on the dissociation between VL and DM under an external 22h cycle. We found that, with increasing numbers of damped oscillatory neurons located in the VL, the dissociation between the VL and DM emerges, but if these neurons are increasingly present in the DM the dissociation disappears. Hence, the damped oscillatory neurons in different subregions of the SCN play distinct roles in the dissociation between the two subregions of the SCN. This shows that synchrony between SCN subregions is affected by the number of damped oscillatory neurons and the location of these cells. We suggest that more knowledge on the number and the location of these cells may explain why some species do show a dissociation between the subregions and others do not, as the distribution of oscillatory types of neurons offers a plausible and novel candidate mechanism to explain heterogeneity.

Excitability in the p53 network mediates robust signaling with tunable activation thresholds in single cells

Cellular signaling systems precisely transmit information in the presence of molecular noise while retaining flexibility to accommodate the needs of individual cells. To understand design principles underlying such versatile signaling, we analyzed the response of the tumor suppressor p53 to varying levels of DNA damage in hundreds of individual cells and observed a switch between distinct signaling modes characterized by isolated pulses and sustained oscillations of p53 accumulation. Guided by dynamic systems theory we show that this requires an excitable network structure comprising positive feedback and provide experimental evidence for its molecular identity. The resulting data-driven model reproduced all features of measured signaling responses and is sufficient to explain their heterogeneity in individual cells. We present evidence that heterogeneity in the levels of the feedback regulator Wip1 sets cell-specific thresholds for p53 activation, providing means to modulate its response through interacting signaling pathways. Our results demonstrate how excitable signaling networks can provide high specificity, sensitivity and robustness while retaining unique possibilities to adjust their function to the physiology of individual cells.

Accurate predictions of population-level changes in sequence and structural properties of HIV-1 Env using a volatility-controlled diffusion model

The envelope glycoproteins (Envs) of HIV-1 continuously evolve in the host by random mutations and recombination events. The resulting diversity of Env variants circulating in the population and their continuing diversification process limit the efficacy of AIDS vaccines. We examined the historic changes in Env sequence and structural features (measured by integrity of epitopes on the Env trimer) in a geographically defined population in the United States. As expected, many Env features were relatively conserved during the 1980s. From this state, some features diversified whereas others remained conserved across the years. We sought to identify “clues” to predict the observed historic diversification patterns. Comparison of viruses that cocirculate in patients at any given time revealed that each feature of Env (sequence or structural) exists at a defined level of variance. The in-host variance of each feature is highly conserved among individuals but can vary between different HIV-1 clades. We designate this property “volatility” and apply it to model evolution of features as a linear diffusion process that progresses with increasing genetic distance. Volatilities of different features are highly correlated with their divergence in longitudinally monitored patients. Volatilities of features also correlate highly with their population-level diversification. Using volatility indices measured from a small number of patient samples, we accurately predict the population diversity that developed for each feature over the course of 30 years. Amino acid variants that evolved at key antigenic sites are also predicted well. Therefore, small “fluctuations” in feature values measured in isolated patient samples accurately describe their potential for population-level diversification. These tools will likely contribute to the design of population-targeted AIDS vaccines by effectively capturing the diversity of currently circulating strains and addressing properties of variants expected to appear in the future.

HIV-1 is the causative agent of the global AIDS pandemic. The envelope glycoproteins (Envs) of HIV-1 constitute a primary target for antibody-based vaccines. However, the diversity of Envs in the population limits the potential efficacy of this approach. Accurate estimates of the range of variants that currently infect patients and those expected to appear in the future will likely contribute to the design of population-targeted immunogens. We found that different properties (features) of Env have different propensities for small “fluctuations” in their values among viruses that infect patients at any given time point. This propensity of each feature for in-host variance, which we designate “volatility”, is conserved among patients. We apply this parameter to model the evolution of features (in patients and population) as a diffusion process driven by their “diffusion coefficients” (volatilities). Using volatilities measured from a few patient samples from the 1980s, we accurately predict properties of viruses that evolved in the population over the course of 30 years. The diffusion-based model described here efficiently captures evolution of phenotypes in biological systems controlled by a dominant random component.

Entrainment range of the suprachiasmatic nucleus affected by the difference in the neuronal amplitudes between the light-sensitive and light-insensitive regions

Mammals not only can be synchronized to the natural 24-h light-dark cycle, but also to a cycle with a non-24-h period. The range of the period of the external cycle, for which the animals can be entrained to, is called the entrainment range, which differs among species. The entrainment range as a characteristic of the animal is determined by the main circadian clock, i.e., the suprachiasmatic nucleus (SCN) in the brain. The SCN is composed of ∼10000 heterogeneous neurons, which can be divided into two subgroups, i.e., the ventrolateral subgroup (VL) directly receiving the light information from the retina and relaying the information to the dorsomedial subgroup (DM). Among the SCN neurons, the amplitudes are different; however, it is unclear that the amplitude is related to the location of the neurons in experiments. In the present study, we examined the effect of the difference in the neuronal amplitude between the VL and the DM on the entrainment range of the SCN, based on a mathematical model, i.e., the Poincaré model, which is used to describe the circadian clock. We find that the maximal entrainment range is obtained when the difference is equal to a critical point. If the difference of the amplitudes of the VL neurons to the amplitudes of the DM neurons is smaller than a critical point, with the increase of the difference, the entrainment range of the SCN increases, while if the difference is larger than the critical point, the entrainment range decreases with the increase of the difference. Our finding may give a potential explanation for the diversity of the entrainment range among species.

Regulating the Suprachiasmatic Nucleus (SCN) Circadian Clockwork: Interplay between Cell-Autonomous and Circuit-Level Mechanisms

The suprachiasmatic nucleus (SCN) is the principal circadian clock of the brain, directing daily cycles of behavior and physiology. SCN neurons contain a cell-autonomous transcription-based clockwork but, in turn, circuit-level interactions synchronize the 20,000 or so SCN neurons into a robust and coherent daily timer. Synchronization requires neuropeptide signaling, regulated by a reciprocal interdependence between the molecular clockwork and rhythmic electrical activity, which in turn depends on a daytime Na⁺ drive and nighttime K⁺ drag. Recent studies exploiting intersectional genetics have started to identify the pacemaking roles of particular neuronal groups in the SCN. They support the idea that timekeeping involves nonlinear and hierarchical computations that create and incorporate timing information through the interactions between key groups of neurons within the SCN circuit. The field is now poised to elucidate these computations, their underlying cellular mechanisms, and how the SCN clock interacts with subordinate circadian clocks across the brain to determine the timing and efficiency of the sleep-wake cycle, and how perturbations of this coherence contribute to neurological and psychiatric illness.

Feedback Loops of the Mammalian Circadian Clock Constitute Repressilator

Mammals evolved an endogenous timing system to coordinate their physiology and behaviour to the 24h period of the solar day. While it is well accepted that circadian rhythms are generated by intracellular transcriptional feedback loops, it is still debated which network motifs are necessary and sufficient for generating self-sustained oscillations. Here, we systematically explore a data-based circadian oscillator model with multiple negative and positive feedback loops and identify a series of three subsequent inhibitions known as "repressilator" as a core element of the mammalian circadian oscillator. The central role of the repressilator motif is consistent with time-resolved ChIP-seq experiments of circadian clock transcription factors and loss of rhythmicity in core clock gene knockouts.

The effects of non-self-sustained oscillators on the en-trainment ability of the suprachiasmatic nucleus

In mammals, the circadian rhythms of behavioral and physiological activities are regulated by an endogenous clock located in the suprachiasmatic nucleus (SCN). The SCN is composed of ~20,000 neurons, of which some are capable of self-sustained oscillations, while the others do not oscillate in a self-sustainable manner, but show arrhythmic patterns or damped oscillations. Thus far, the effects of these non-self-sustained oscillatory neurons are not fully explored. Here, we examined how the proportion of the non-self-sustained oscillators affects the free running period under constant darkness and the ability to entrain to the light-dark cycle. We find that the proportion does not affect the free running period, but plays a significant role in the range of entrainment. We also find that its effect on the entrainment range depends on the region where the non-self-sustained oscillators are located. If the non-self-sustained oscillatory neurons are situated in the light-sensitive subregion, the entrainment range narrows when the proportion increases. If they are situated in the light-insensitive subregion, however, the entrainment range broadens with the increase of the proportion. We suggest that the heterogeneity within the light-sensitive and light-insensitive subregions of the SCN has important consequences for how the clock works.

Synchronizing stochastic circadian oscillators in single cells of Neurospora crassa

The synchronization of stochastic coupled oscillators is a central problem in physics and an emerging problem in biology, particularly in the context of circadian rhythms. Most measurements on the biological clock are made at the macroscopic level of millions of cells. Here measurements are made on the oscillators in single cells of the model fungal system, Neurospora crassa, with droplet microfluidics and the use of a fluorescent recorder hooked up to a promoter on a clock controlled gene-2 (ccg-2). The oscillators of individual cells are stochastic with a period near 21 hours (h), and using a stochastic clock network ensemble fitted by Markov Chain Monte Carlo implemented on general-purpose graphical processing units (or GPGPUs) we estimated that >94% of the variation in ccg-2 expression was stochastic (as opposed to experimental error). To overcome this stochasticity at the macroscopic level, cells must synchronize their oscillators. Using a classic measure of similarity in cell trajectories within droplets, the intraclass correlation (ICC), the synchronization surface ICC is measured on >25,000 cells as a function of the number of neighboring cells within a droplet and of time. The synchronization surface provides evidence that cells communicate, and synchronization varies with genotype.

A systems theoretic approach to analysis and control of mammalian circadian dynamics

The mammalian circadian clock is a complex multi-scale, multivariable biological control system. In the past two decades, methods from systems engineering have led to numerous insights into the architecture and functionality of this system. In this review, we examine the mammalian circadian system through a process systems lens. We present a mathematical framework for examining the cellular circadian oscillator, and show recent extensions for understanding population-scale dynamics. We provide an overview of the routes by which the circadian system can be systemically manipulated, and present in silico proof of concept results for phase resetting of the clock via model predictive control.

Coupling Controls the Synchrony of Clock Cells in Development and Knockouts

In mammals, a network of coupled neurons within the hypothalamus coordinates physiological rhythms with daily changes in the environment. In each neuron, delayed negative transcriptional feedbacks generate oscillations, albeit noisy and unreliable ones. Coupling mediated by diffusible neuropeptides lends precision and robustness to circadian rhythms. The double knockout of Cryptochrome Cry turns adult mice arrhythmic. But, remarkably, double knockout neonates continue to show robust oscillation much like wild-type neonates and appear to lose rhythmicity with development. We study quantitatively dispersed neurons and brain slices from wild-type and Cry double knockout mice to understand the links between single cell rhythmicity and intercellular coupling. We quantify oscillator properties of dispersed cells using nonlinear regression and study bifurcations diagrams of network models. We find that varying just three parameters-oscillator strength, strength of coupling, and timing of coupling-can reproduce experimentally observed features. In particular, modeling reveals that minor changes in timing of coupling can destroy synchronization as observed in adult slices from knockout mice.

The synchronization of neuronal oscillators determined by the directed network structure of the suprachiasmatic nucleus under different photoperiods

The main function of the principal clock located in the suprachiasmatic nucleus (SCN) of mammals is synchronizing the body rhythms to the 24 h light-dark cycle. Additionally, the SCN is able to adapt to the photoperiod of the cycle which varies among seasons. Under the long photoperiod (LP), the synchronization degree of the SCN neurons is lower than that under the photoperiod (SP). In the present study, a potential explanation is given for this phenomenon. We propose that the asymmetrical coupling between the light-signal-sensitive part (the ventralateral part, abbreviation: VL) and the light-signal-insensitive part (the dorsalmedial part, abbreviation: DM) of the SCN plays a role in the synchronization degree, which is reflected by the ratio of the number of the directed links from the VL neurons to the DM neurons to the total links of both directions between the VL and the DM. The ratio is assumed to characterize the directed network structure under different photoperiods, which is larger under the SP and smaller under the LP. We found that with the larger ratio in the situation of the SP, the synchronization degree is higher. Our finding may shed new light on the asymmetrical coupling between the VL and the DM, and the network structure of the SCN.

Circadian mRNA expression: insights from modeling and transcriptomics

Circadian clocks synchronize organisms to the 24 h rhythms of the environment. These clocks persist under constant conditions, have their origin at the cellular level, and produce an output of rhythmic mRNA expression affecting thousands of transcripts in many mammalian cell types. Here, we review the charting of circadian output rhythms in mRNA expression, focusing on mammals. We emphasize the challenges in statistics, interpretation, and quantitative descriptions that such investigations have faced and continue to face, and outline remaining outstanding questions.

Quantifying Stochastic Noise in Cultured Circadian Reporter Cells

Stochastic noise at the cellular level has been shown to play a fundamental role in circadian oscillations, influencing how groups of cells entrain to external cues and likely serving as the mechanism by which cell-autonomous rhythms are generated. Despite this importance, few studies have investigated how clock perturbations affect stochastic noise—even as increasing numbers of high-throughput screens categorize how gene knockdowns or small molecules can change clock period and amplitude. This absence is likely due to the difficulty associated with measuring cell-autonomous stochastic noise directly, which currently requires the careful collection and processing of single-cell data. In this study, we show that the damping rate of population-level bioluminescence recordings can serve as an accurate measure of overall stochastic noise, and one that can be applied to future and existing high-throughput circadian screens. Using cell-autonomous fibroblast data, we first show directly that higher noise at the single-cell results in faster damping at the population level. Next, we show that the damping rate of cultured cells can be changed in a dose-dependent fashion by small molecule modulators, and confirm that such a change can be explained by single-cell noise using a mathematical model. We further demonstrate the insights that can be gained by applying our method to a genome-wide siRNA screen, revealing that stochastic noise is altered independently from period, amplitude, and phase. Finally, we hypothesize that the unperturbed clock is highly optimized for robust rhythms, as very few gene perturbations are capable of simultaneously increasing amplitude and lowering stochastic noise. Ultimately, this study demonstrates the importance of considering the effect of circadian perturbations on stochastic noise, particularly with regard to the development of small-molecule circadian therapeutics.

As most organisms exist in an environment that changes predictably with a 24-hour period, highly optimized genetic circuits turn on and off the production of key regulatory proteins to anticipate the day/night cycle. In humans, the demands of a modern society have required that we deviate from this evolutionarily prescribed sleep and feeding schedule, resulting in increased long-term risks of metabolic disease. There is therefore a desire to find pharmacological treatments that would restore the normal functioning of our circadian clock despite irregular behavioral schedules. One aspect of these treatments that is often overlooked in searching for candidate drugs is how these treatments might affect the accuracy of the circadian timing system. Recording the time of each cell is possible but difficult; as a result single-cell approaches cannot be scaled up to high-throughput searches. In this paper, we show that it is possible to estimate how much the noise of a system has changed by looking only at the averaged protein production of an entire population of cells. Such an approach allows us to analyze prior data from high-throughput screens, and show that the natural clock has been highly optimized to be both accurate and high amplitude.

The Goodwin model revisited: Hopf bifurcation, limit-cycle, and periodic entrainment

The three-variable Goodwin oscillator is a minimal model demonstrating the emergence of oscillations in simple biochemical feedback systems. As a prototypical oscillator, this model was extensively studied from a theoretical point of view and applied to various biological systems, including circadian clocks. Here, we reexamine this model, derive analytically the amplitude equation near the Hopf bifurcation and investigate the effect of a periodic modulation of the oscillator. In particular, we compare the entrainment performance when the free oscillator displays either self-sustained or damped oscillations. We discuss the results in the context of circadian oscillators.

Modeling and simulating the Arabidopsis thaliana circadian clock using XPP-AUTO

Circadian clocks are endogenous timekeepers that produce oscillations with a period of about one day. Their rhythmicity originates from complex gene regulatory networks at the cellular level. In the last decades, computational models have been proven to be a powerful tool in order to understand the dynamics and design principles of the complex regulatory circuitries underlying the circadian clocks of different organisms. We present the process of model development using a small and simplified two-gene regulatory network of the Arabidopsis circadian clock. Subsequently, we discuss important numerical techniques to analyze such a mathematical model using XPP-AUTO. We show how to solve deterministic and stochastic ordinary differential equations and how to compute bifurcation diagrams or simulate phase-shift experiments. We finally discuss the contributions of modeling to the understanding and dissection of the Arabidopsis circadian system.

Analysis of precision in chemical oscillators: implications for circadian clocks

Biochemical reaction networks often exhibit spontaneous self-sustained oscillations. An example is the circadian oscillator that lies at the heart of daily rhythms in behavior and physiology in most organisms including humans. While the period of these oscillators evolved so that it resonates with the 24 h daily environmental cycles, the precision of the oscillator (quantified via the Q factor) is another relevant property of these cell-autonomous oscillators. Since this quantity can be measured in individual cells, it is of interest to better understand how this property behaves across mathematical models of these oscillators. Current theoretical schemes for computing the Q factors show limitations for both high-dimensional models and in the vicinity of Hopf bifurcations. Here, we derive low-noise approximations that lead to numerically stable schemes also in high-dimensional models. In addition, we generalize normal form reductions that are appropriate near Hopf bifurcations. Applying our approximations to two models of circadian clocks, we show that while the low-noise regime is faithfully recapitulated, increasing the level of noise leads to species-dependent precision. We emphasize that subcomponents of the oscillator gradually decouple from the core oscillator as noise increases, which allows us to identify the subnetworks responsible for robust rhythms.

Human chronotypes from a theoretical perspective

The endogenous circadian timing system has evolved to synchronize an organism to periodically recurring environmental conditions. Those external time cues are called Zeitgebers. When entrained by a Zeitgeber, the intrinsic oscillator adopts a fixed phase relation to the Zeitgeber. Here, we systematically study how the phase of entrainment depends on clock and Zeitgeber properties. We combine numerical simulations of amplitude-phase models with predictions from analytically tractable models. In this way we derive relations between the phase of entrainment to the mismatch between the endogenous and Zeitgeber period, the Zeitgeber strength, and the range of entrainment. A core result is the “180° rule” asserting that the phase varies over a range of about 180° within the entrainment range. The 180° rule implies that clocks with a narrow entrainment range (“strong oscillators”) exhibit quite flexible entrainment phases. We argue that this high sensitivity of the entrainment phase contributes to the wide range of human chronotypes.

Homeostatic and circadian contribution to EEG and molecular state variables of sleep regulation

STUDY OBJECTIVES

Besides their well-established role in circadian rhythms, our findings that the forebrain expression of the clock-genes Per2 and Dbp increases and decreases, respectively, in relation to time spent awake suggest they also play a role in the homeostatic aspect of sleep regulation. Here, we determined whether time of day modulates the effects of elevated sleep pressure on clock-gene expression. Time of day effects were assessed also for recognized electrophysiological (EEG delta power) and molecular (Homer1a) markers of sleep homeostasis.

DESIGN

EEG and qPCR data were obtained for baseline and recovery from 6-h sleep deprivation starting at ZT0, -6, -12, or -18.

SETTING

Mouse sleep laboratory.

PARTICIPANTS

Male mice.

INTERVENTIONS

Sleep deprivation.

RESULTS

The sleep-deprivation induced changes in Per2 and Dbp expression importantly varied with time of day, such that Per2 could even decrease during sleep deprivations occurring at the decreasing phase in baseline. Dbp showed similar, albeit opposite dynamics. These unexpected results could be reliably predicted assuming that these transcripts behave according to a driven damped harmonic oscillator. As expected, the sleep-wake distribution accounted for a large degree of the changes in EEG delta power and Homer1a. Nevertheless, the sleep deprivation-induced increase in delta power varied also with time of day with higher than expected levels when recovery sleep started at dark onset.

CONCLUSIONS

Per2 and delta power are widely used as exclusive state variables of the circadian and homeostatic process, respectively. Our findings demonstrate a considerable cross-talk between these two processes. As Per2 in the brain responds to both sleep loss and time of day, this molecule is well positioned to keep track of and to anticipate homeostatic sleep need.

CITATION

Curie T; Mongrain V; Dorsaz S; Mang GM; Emmenegger Y; Franken P. Homeostatic and circadian contribution to EEG and molecular state variables of sleep regulation. SLEEP 2013;36(3):311-323.

Mathematical modeling in chronobiology

Circadian clocks are autonomous oscillators entrained by external Zeitgebers such as light-dark and temperature cycles. On the cellular level, rhythms are generated by negative transcriptional feedback loops. In mammals, the suprachiasmatic nucleus (SCN) in the anterior part of the hypothalamus plays the role of the central circadian pacemaker. Coupling between individual neurons in the SCN leads to precise self-sustained oscillations even in the absence of external signals. These neuronal rhythms orchestrate the phasing of circadian oscillations in peripheral organs. Altogether, the mammalian circadian system can be regarded as a network of coupled oscillators. In order to understand the dynamic complexity of these rhythms, mathematical models successfully complement experimental investigations. Here we discuss basic ideas of modeling on three different levels (1) rhythm generation in single cells by delayed negative feedbacks, (2) synchronization of cells via external stimuli or cell-cell coupling, and (3) optimization of chronotherapy.

Weakly circadian cells improve resynchrony

The mammalian suprachiasmatic nuclei (SCN) contain thousands of neurons capable of generating near 24-h rhythms. When isolated from their network, SCN neurons exhibit a range of oscillatory phenotypes: sustained or damping oscillations, or arrhythmic patterns. The implications of this variability are unknown. Experimentally, we found that cells within SCN explants recover from pharmacologically-induced desynchrony by re-establishing rhythmicity and synchrony in waves, independent of their intrinsic circadian period We therefore hypothesized that a cell's location within the network may also critically determine its resynchronization. To test this, we employed a deterministic, mechanistic model of circadian oscillators where we could independently control cell-intrinsic and network-connectivity parameters. We found that small changes in key parameters produced the full range of oscillatory phenotypes seen in biological cells, including similar distributions of period, amplitude and ability to cycle. The model also predicted that weaker oscillators could adjust their phase more readily than stronger oscillators. Using these model cells we explored potential biological consequences of their number and placement within the network. We found that the population synchronized to a higher degree when weak oscillators were at highly connected nodes within the network. A mathematically independent phase-amplitude model reproduced these findings. Thus, small differences in cell-intrinsic parameters contribute to large changes in the oscillatory ability of a cell, but the location of weak oscillators within the network also critically shapes the degree of synchronization for the population.

Circadian rhythms are daily, near 24-h oscillations in biological processes that nearly all organisms on Earth experience. Single cells contain a molecular clock that drives circadian rhythms in physiology and, when many cells synchronize in a population, daily behaviors. We hypothesized that small differences in intrinsic cellular properties allow for a diversity of circadian periods and amplitudes across cells. We observed circadian cells and their synchrony before, during, and after limiting communication between cells and then compared their intrinsic properties to their resynchronization behavior. We found that arrhythmic, weakly oscillating, and self-sustained circadian cells rejoined the rhythmic population independent of their cell-intrinsic oscillations. Using a mechanistic computational model of circadian cells, we found that resynchronization could be enhanced by including more weak oscillators or by placing weak oscillators at more connected nodes in the network. We conclude that intrinsic properties (e.g. oscillator weakness and responsiveness) and network structure (e.g. positions of weak oscillators) can independently buffer tissue rhythms from perturbations. This reveals how cellular and network properties impose rules on systems of circadian cells that must achieve synchrony from a desynchronized state, for example during perinatal development or when forced to overcome societal constraints on sleep-wake behavior, such as working early or late shifts.

Modeling the seasonal adaptation of circadian clocks by changes in the network structure of the suprachiasmatic nucleus

The dynamics of circadian rhythms needs to be adapted to day length changes between summer and winter. It has been observed experimentally, however, that the dynamics of individual neurons of the suprachiasmatic nucleus (SCN) does not change as the seasons change. Rather, the seasonal adaptation of the circadian clock is hypothesized to be a consequence of changes in the intercellular dynamics, which leads to a phase distribution of electrical activity of SCN neurons that is narrower in winter and broader during summer. Yet to understand this complex intercellular dynamics, a more thorough understanding of the impact of the network structure formed by the SCN neurons is needed. To that effect, we propose a mathematical model for the dynamics of the SCN neuronal architecture in which the structure of the network plays a pivotal role. Using our model we show that the fraction of long-range cell-to-cell connections and the seasonal changes in the daily rhythms may be tightly related. In particular, simulations of the proposed mathematical model indicate that the fraction of long-range connections between the cells adjusts the phase distribution and consequently the length of the behavioral activity as follows: dense long-range connections during winter lead to a narrow activity phase, while rare long-range connections during summer lead to a broad activity phase. Our model is also able to account for the experimental observations indicating a larger light-induced phase-shift of the circadian clock during winter, which we show to be a consequence of higher synchronization between neurons. Our model thus provides evidence that the variations in the seasonal dynamics of circadian clocks can in part also be understood and regulated by the plasticity of the SCN network structure.

Circadian clocks drive the temporal coordination of internal biological processes, which in turn determine daily rhythms in physiology and behavior in the most diverse organisms. In mammals, the 24-hour timing clock resides in the suprachiasmatic nucleus (SCN) of the hypothalamus. The SCN is a network of interconnected neurons that serves as a robust self-sustained circadian pacemaker. The electrical activity of these neurons and their synchronization with the 24-hour cycle is established via the environmental day and night cycles. Apart from daily luminance changes, mammals are exposed to seasonal day length changes as well. Remarkably, it has been shown experimentally that the seasonal adaptations to different photoperiods are related to the modifications of the neuronal activity of the SCN due to the plasticity of the network. In our paper, by developing a mathematical model of the SCN architecture, we explore in depth the role of the structure of this important neuronal network. We show that the redistribution of the neuronal activity during winter and summer can in part be explained by structural changes of the network. Interestingly, the alterations of the electrical activity patterns can be related with small-world properties of our proposed SCN network.

Systems biology of circadian-immune interactions

There is increasing evidence that the immune system is regulated by circadian rhythms. A wide range of immune parameters, such as the number of red blood cells and peripheral blood mononuclear cells as well as the level of critical immune mediators, such as cytokines, undergo daily fluctuations. Current experimental data indicate that circadian information reaches immune tissues mainly through diurnal patterns of autonomic and endocrine rhythms. In addition, immune factors such as cytokines can also influence the phase of the circadian clock, providing bidirectional flow of circadian information between the neuroendocrine and immune systems. This network of neuroendocrine-immune interactions consists of complexly integrated molecular feedback and feedforward loops that function in synchrony in order to optimize immune response. Chronic stress can disrupt this intrinsic orchestration, as several endocrine signals of chronically stressed patients present blunted rhythmic characteristics. Reprogramming of biological rhythms has recently gained much attention as a potent method to leverage homeostatic circadian controls to ultimately improve clinical outcomes. Elucidation of the intrinsic properties of such complex systems and optimization of intervention strategies require not only an accurate identification of the signaling pathways that mediate host responses, but also a system-level description and evaluation.

Intrinsic noise analyzer: a software package for the exploration of stochastic biochemical kinetics using the system size expansion

The accepted stochastic descriptions of biochemical dynamics under well-mixed conditions are given by the Chemical Master Equation and the Stochastic Simulation Algorithm, which are equivalent. The latter is a Monte-Carlo method, which, despite enjoying broad availability in a large number of existing software packages, is computationally expensive due to the huge amounts of ensemble averaging required for obtaining accurate statistical information. The former is a set of coupled differential-difference equations for the probability of the system being in any one of the possible mesoscopic states; these equations are typically computationally intractable because of the inherently large state space. Here we introduce the software package intrinsic Noise Analyzer (iNA), which allows for systematic analysis of stochastic biochemical kinetics by means of van Kampen's system size expansion of the Chemical Master Equation. iNA is platform independent and supports the popular SBML format natively. The present implementation is the first to adopt a complementary approach that combines state-of-the-art analysis tools using the computer algebra system Ginac with traditional methods of stochastic simulation. iNA integrates two approximation methods based on the system size expansion, the Linear Noise Approximation and effective mesoscopic rate equations, which to-date have not been available to non-expert users, into an easy-to-use graphical user interface. In particular, the present methods allow for quick approximate analysis of time-dependent mean concentrations, variances, covariances and correlations coefficients, which typically outperforms stochastic simulations. These analytical tools are complemented by automated multi-core stochastic simulations with direct statistical evaluation and visualization. We showcase iNA's performance by using it to explore the stochastic properties of cooperative and non-cooperative enzyme kinetics and a gene network associated with circadian rhythms. The software iNA is freely available as executable binaries for Linux, MacOSX and Microsoft Windows, as well as the full source code under an open source license.