Global parameter search reveals design principles of the mammalian circadian clock

Are circadian amplitudes and periods correlated? A new twist in the story

Three parameters are important to characterize a circadian and in general any biological clock: period, phase and amplitude. While circadian periods have been shown to correlate with entrainment phases, and clock amplitude influences the phase response of an oscillator to pulse-like zeitgeber signals, the co-modulations of amplitude and periods, which we term twist, have not been studied in detail. In this paper we define two concepts: parametric twist refers to amplitude-period correlations arising in ensembles of self-sustained, limit cycle clocks in the absence of external inputs, and phase space twist refers to the co-modulation of an individual clock's amplitude and period in response to external zeitgebers. Our findings show that twist influences the interaction of oscillators with the environment, facilitating entrainment, speeding upfastening recovery to pulse-like perturbations or modifying the response of an individual clock to coupling. This theoretical framework might be applied to understand the emerging properties of other oscillating systems.

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.

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.

Network Structure of the Master Clock Is Important for Its Primary Function

A master clock located in the suprachiasmatic nucleus (SCN) regulates the circadian rhythm of physiological and behavioral activities in mammals. The SCN has two main functions in the regulation: an endogenous clock produces the endogenous rhythmic signal in body rhythms, and a calibrator synchronizes the body rhythms to the external light-dark cycle. These two functions have been determined to depend on either the dynamic behaviors of individual neurons or the whole SCN neuronal network. In this review, we first introduce possible network structures for the SCN, as revealed by time series analysis from real experimental data. It was found that the SCN network is heterogeneous and sparse, that is, the average shortest path length is very short, some nodes are hubs with large node degrees but most nodes have small node degrees, and the average node degree of the network is small. Secondly, the effects of the SCN network structure on the SCN function are reviewed based on mathematical models of the SCN network. It was found that robust rhythms with large amplitudes, a high synchronization between SCN neurons and a large entrainment ability exists mainly in small-world and scale-free type networks, but not other types. We conclude that the SCN most probably is an efficient small-world type or scale-free type network, which drives SCN function.

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.

Critical role of deadenylation in regulating poly(A) rhythms and circadian gene expression

The mammalian circadian clock is deeply rooted in rhythmic regulation of gene expression. Rhythmic transcriptional control mediated by the circadian transcription factors is thought to be the main driver of mammalian circadian gene expression. However, mounting evidence has demonstrated the importance of rhythmic post-transcriptional controls, and it remains unclear how the transcriptional and post-transcriptional mechanisms collectively control rhythmic gene expression. In mouse liver, hundreds of genes were found to exhibit rhythmicity in poly(A) tail length, and the poly(A) rhythms are strongly correlated with the protein expression rhythms. To understand the role of rhythmic poly(A) regulation in circadian gene expression, we constructed a parsimonious model that depicts rhythmic control imposed upon basic mRNA expression and poly(A) regulation processes, including transcription, deadenylation, polyadenylation, and degradation. The model results reveal the rhythmicity in deadenylation as the strongest contributor to the rhythmicity in poly(A) tail length and the rhythmicity in the abundance of the mRNA subpopulation with long poly(A) tails (a rough proxy for mRNA translatability). In line with this finding, the model further shows that the experimentally observed distinct peak phases in the expression of deadenylases, regardless of other rhythmic controls, can robustly cluster the rhythmic mRNAs by their peak phases in poly(A) tail length and abundance of the long-tailed subpopulation. This provides a potential mechanism to synchronize the phases of target gene expression regulated by the same deadenylases. Our findings highlight the critical role of rhythmic deadenylation in regulating poly(A) rhythms and circadian gene expression.

Amplitude Effects Allow Short Jet Lags and Large Seasonal Phase Shifts in Minimal Clock Models

Mathematical models of varying complexity have helped shed light on different aspects of circadian clock function. In this work, we question whether minimal clock models (Goodwin models) are sufficient to reproduce essential phenotypes of the clock: a small phase response curve (PRC), fast jet lag, and seasonal phase shifts. Instead of building a single best model, we take an approach where we study the properties of a set of models satisfying certain constraints; here, a 1h-pulse PRC with a range of 3h and clock periods between 22h and 26h is designed. Surprisingly, almost all these randomly parameterized models showed a 4h change in phase of entrainment between long and short days and jet lag durations of three to seven days in advance and delay. Moreover, intrinsic clock period influenced jet lag duration and entrainment amplitude and phase. Fast jet lag was realized in this model by means of an interesting amplitude effect: the association between clock amplitude and clock period termed "twist." This twist allows amplitude changes to speed up and slow down clocks enabling faster shifts. These findings were robust to the addition of positive feedback to the model. In summary, the known design principles of rhythm generation - negative feedback, long delay, and switch-like inhibition (we review these in detail) - are sufficient to reproduce the essential clock phenotypes. Furthermore, amplitudes play a role in determining clock properties and must be always considered, although they are difficult to measure.

Modeling Reveals a Key Mechanism for Light-Dependent Phase Shifts of Neurospora Circadian Rhythms

Light shifts and synchronizes the phase of the circadian clock to daily environments, which is critical for maintaining the daily activities of an organism. It has been proposed that such light-dependent phase shifts are triggered by light-induced upregulation of a negative element of the core circadian clock (i.e., frq, Per1/2) in many organisms, including fungi. However, we find, using systematic mathematical modeling of the Neurospora crassa circadian clock, that the upregulation of the frq gene expression alone is unable to reproduce the observed light-dependent phase responses. Indeed, we find that the depression of the transcriptional activator white-collar-1, previously shown to be promoted by FRQ and VVD, is a key molecular mechanism for accurately simulating light-induced phase response curves for wild-type and mutant strains of Neurospora. Our findings elucidate specific molecular pathways that can be utilized to control phase resetting of circadian rhythms.

Co-existing feedback loops generate tissue-specific circadian rhythms

The analysis of tissue-specific data-based models of the gene regulatory network of the mammalian circadian clock reveals organ-specific synergies of feedback loops.

Gene regulatory feedback loops generate autonomous circadian rhythms in mammalian tissues. The well-studied core clock network contains many negative and positive regulations. Multiple feedback loops have been discussed as primary rhythm generators but the design principles of the core clock and differences between tissues are still under debate. Here we use global optimization techniques to fit mathematical models to circadian gene expression profiles for different mammalian tissues. It turns out that for every investigated tissue multiple model parameter sets reproduce the experimental data. We extract for all model versions the most essential feedback loops and find auto-inhibitions of period and cryptochrome genes, Bmal1–Rev-erb-α loops, and repressilator motifs as possible rhythm generators. Interestingly, the essential feedback loops differ between tissues, pointing to specific design principles within the hierarchy of mammalian tissue clocks. Self-inhibitions of Per and Cry genes are characteristic for models of suprachiasmatic nucleus clocks, whereas in liver models many loops act in synergy and are connected by a repressilator motif. Tissue-specific use of a network of co-existing synergistic feedback loops could account for functional differences between organs.

Neuronal oscillations on an ultra-slow timescale: daily rhythms in electrical activity and gene expression in the mammalian master circadian clockwork

Neuronal oscillations of the brain, such as those observed in the cortices and hippocampi of behaving animals and humans, span across wide frequency bands, from slow delta waves (0.1 Hz) to ultra-fast ripples (600 Hz). Here, we focus on ultra-slow neuronal oscillators in the hypothalamic suprachiasmatic nuclei (SCN), the master daily clock that operates on interlocking transcription-translation feedback loops to produce circadian rhythms in clock gene expression with a period of near 24 h (< 0.001 Hz). This intracellular molecular clock interacts with the cell's membrane through poorly understood mechanisms to drive the daily pattern in the electrical excitability of SCN neurons, exhibiting an up-state during the day and a down-state at night. In turn, the membrane activity feeds back to regulate the oscillatory activity of clock gene programs. In this review, we emphasise the circadian processes that drive daily electrical oscillations in SCN neurons, and highlight how mathematical modelling contributes to our increasing understanding of circadian rhythm generation, synchronisation and communication within this hypothalamic region and across other brain circuits.

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.

Transition of phase response properties and singularity in the circadian limit cycle of cultured cells

The circadian system has been regarded as a limit cycle oscillator constructed by the integrated interaction of clock genes and proteins. Here, we investigated a mammalian circadian oscillation geometrically before and after a perturbation. We detected the singular point and transition from a type 1 to type 0 phase response curve (PRC) and determined the embedding dimension to show how many variables are needed to describe the limit cycle oscillation and relaxation process after a perturbation. As a perturbation, forskolin (FK) was administered to Rat-1 cells expressing the Per2::luc gene. By broadly and finely changing the phase and strength of the perturbation, we detected the transition of the PRC from type 1 to type 0 and a possible singular transition point, the property of which agreed quite well with our numerical simulation of the noisy Goodwin model, a simple yet canonical model for the transcription-translation feedback loop of the core clock genes. Furthermore, we estimated the embedding dimension of the limit cycle before and after the perturbation. The trajectory of the limit cycle was embedded in two dimensions but with the perturbation of the state point moved out of the trajectory, the relaxation process was generally embedded in higher dimensions. The average number of embedding dimensions at each dose of FK increased as the FK dose increased but most of the relaxation process was generally embedded within four dimensions. These findings support the existence of a circadian limit cycle oscillator in mammalian cells and suggest that a small number of variables determine the relaxation process after a perturbation.

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.

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.

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.

Model-driven experimental approach reveals the complex regulatory distribution of p53 by the circadian factor Period 2

The circadian clock and cell cycle networks are interlocked on the molecular level, with the core clock loop exerting a multilevel regulatory role over cell cycle components. This is particularly relevant to the circadian factor Period 2 (Per2), which modulates the stability of the tumor suppressor p53 in unstressed cells and transcriptional activity in response to genotoxic stress. Per2 binding prevents Mdm2-mediated ubiquitination of p53 and, therefore, its degradation, and oscillations in the peaks of Per2 and p53 were expected to correspond. However, our findings showed that Per2 and p53 rhythms were significantly out-of-phase relative to each other in cell lysates and in purified cytoplasmic fractions. These seemingly conflicting experimental data motivated the use of a combined theoretical and experimental approach focusing on the role played by Per2 in dictating the phase of p53 oscillations. Systematic modeling of all possible regulatory scenarios predicted that the observed phase relationship between Per2 and p53 could be simulated if (i) p53 was more stable in the nucleus than in the cytoplasm, (ii) Per2 associates to various ubiquitinated forms of p53, and (iii) Per2 mediated p53 nuclear import. These predictions were supported by a sevenfold increase in p53's half-life in the nucleus and by in vitro binding of Per2 to the various ubiquitinated forms of p53. Last, p53's nuclear shuttling was significantly favored by ectopic expression of Per2 and reduced because of Per2 down-regulation. Our combined theoretical/mathematical approach reveals how clock regulatory nodes can be inferred from oscillating time course data.

Collective timekeeping among cells of the master circadian clock

The suprachiasmatic nucleus (SCN) of the anterior hypothalamus is the master circadian clock that coordinates daily rhythms in behavior and physiology in mammals. Like other hypothalamic nuclei, the SCN displays an impressive array of distinct cell types characterized by differences in neurotransmitter and neuropeptide expression. Individual SCN neurons and glia are able to display self-sustained circadian rhythms in cellular function that are regulated at the molecular level by a 24h transcriptional-translational feedback loop. Remarkably, SCN cells are able to harmonize with one another to sustain coherent rhythms at the tissue level. Mechanisms of cellular communication in the SCN network are not completely understood, but recent progress has provided insight into the functional roles of several SCN signaling factors. This review discusses SCN organization, how intercellular communication is critical for maintaining network function, and the signaling mechanisms that play a role in this process. Despite recent progress, our understanding of SCN circuitry and coupling is far from complete. Further work is needed to map SCN circuitry fully and define the signaling mechanisms that allow for collective timekeeping in the SCN network.

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.

The circadian rhythm induced by the heterogeneous network structure of the suprachiasmatic nucleus

In mammals, the master clock is located in the suprachiasmatic nucleus (SCN), which is composed of about 20 000 nonidentical neuronal oscillators expressing different intrinsic periods. These neurons are coupled through neurotransmitters to form a network consisting of two subgroups, i.e., a ventrolateral (VL) subgroup and a dorsomedial (DM) subgroup. The VL contains about 25% SCN neurons that receive photic input from the retina, and the DM comprises the remaining 75% SCN neurons which are coupled to the VL. The synapses from the VL to the DM are evidently denser than that from the DM to the VL, in which the VL dominates the DM. Therefore, the SCN is a heterogeneous network where the neurons of the VL are linked with a large number of SCN neurons. In the present study, we mimicked the SCN network based on Goodwin model considering four types of networks including an all-to-all network, a Newman-Watts (NW) small world network, an Erdös-Rényi (ER) random network, and a Barabási-Albert (BA) scale free network. We found that the circadian rhythm was induced in the BA, ER, and NW networks, while the circadian rhythm was absent in the all-to-all network with weak cellular coupling, where the amplitude of the circadian rhythm is largest in the BA network which is most heterogeneous in the network structure. Our finding provides an alternative explanation for the induction or enhancement of circadian rhythm by the heterogeneity of the network structure.

Impact of dispersed coupling strength on the free running periods of circadian rhythms

The dominant endogenous clock, named the suprachiasmatic nucleus (SCN), regulates circadian rhythms of behavioral and physiological activity in mammals. One of the main characteristics of the SCN is that the animal maintains a circadian rhythm with a period close to 24 h in the absence of a daily light-dark cycle (called the free running period). The free running period varies among species due to heterogeneity of the SCN network. Previous studies have shown that the heterogeneity in cellular coupling as well as in intrinsic neuronal periods shortens the free running period. Furthermore, as derived from experiments, one neuron's coupling strength is negatively associated with its period. It is unknown what the effects of this association between coupling strength and period are on the free running period and how the heterogeneity in coupling strength influences this free running period. In the present study we found that in the presence of a negative relationship between one neuron's coupling strength and its period, surprisingly, the dispersion of coupling strengths increases the free running period. Our present finding may shed new light on the understanding of the heterogeneous SCN network and provides an alternative explanation for the diversity of free running periods between species.

Heterogeneity induces rhythms of weakly coupled circadian neurons

The main clock located in the suprachiasmatic nucleus (SCN) regulates circadian rhythms in mammals. The SCN is composed of approximately twenty thousand heterogeneous self-oscillating neurons, that have intrinsic periods varying from 22 h to 28 h. They are coupled through neurotransmitters and neuropeptides to form a network and output a uniform periodic rhythm. Previous studies found that the heterogeneity of the neurons leads to attenuation of the circadian rhythm with strong cellular coupling. In the present study, we investigate the heterogeneity of the neurons and of the network in the condition of constant darkness. Interestingly, we found that the heterogeneity of weakly coupled neurons enables them to oscillate and strengthen the circadian rhythm. In addition, we found that the period of the SCN network increases with the increase of the degree of heterogeneity. As the network heterogeneity does not change the dynamics of the rhythm, our study shows that the heterogeneity of the neurons is vitally important for rhythm generation in weakly coupled systems, such as the SCN, and it provides a new method to strengthen the circadian rhythm, as well as an alternative explanation for differences in free running periods between species in the absence of the daily cycle.

Noise Induces Oscillation and Synchronization of the Circadian Neurons

The principle clock of mammals, named suprachiasmatic nucleus (SCN), coordinates the circadian rhythms of behavioral and physiological activity to the external 24 h light-dark cycle. In the absence of the daily cycle, the SCN acts as an endogenous clock that regulates the ~24 h rhythm of activity. Experimental and theoretical studies usually take the light-dark cycle as a main external influence, and often ignore light pollution as an external influence. However, in modern society, the light pollution such as induced by electrical lighting influences the circadian clock. In the present study, we examined the effect of external noise (light pollution) on the collective behavior of coupled circadian oscillators under constant darkness using a Goodwin model. We found that the external noise plays distinct roles in the network behavior of neurons for weak or strong coupling between the neurons. In the case of strong coupling, the noise reduces the synchronization and the period of the SCN network. Interestingly, in the case of weak coupling, the noise induces a circadian rhythm in the SCN network which is absent in noise-free condition. In addition, the noise increases the synchronization and decreases the period of the SCN network. Our findings may shed new light on the impact of the external noise on the collective behavior of SCN neurons.

The relationship between stochastic and deterministic quasi-steady state approximations

Background

The quasi steady-state approximation (QSSA) is frequently used to reduce deterministic models of biochemical networks. The resulting equations provide a simplified description of the network in terms of non-elementary reaction functions (e.g. Hill functions). Such deterministic reductions are frequently a basis for heuristic stochastic models in which non-elementary reaction functions are used to define reaction propensities. Despite their popularity, it remains unclear when such stochastic reductions are valid. It is frequently assumed that the stochastic reduction can be trusted whenever its deterministic counterpart is accurate. However, a number of recent examples show that this is not necessarily the case.

Results

Here we explain the origin of these discrepancies, and demonstrate a clear relationship between the accuracy of the deterministic and the stochastic QSSA for examples widely used in biological systems. With an analysis of a two-state promoter model, and numerical simulations for a variety of other models, we find that the stochastic QSSA is accurate whenever its deterministic counterpart provides an accurate approximation over a range of initial conditions which cover the likely fluctuations from the quasi steady-state (QSS). We conjecture that this relationship provides a simple and computationally inexpensive way to test the accuracy of reduced stochastic models using deterministic simulations.

Conclusions

The stochastic QSSA is one of the most popular multi-scale stochastic simulation methods. While the use of QSSA, and the resulting non-elementary functions has been justified in the deterministic case, it is not clear when their stochastic counterparts are accurate. In this study, we show how the accuracy of the stochastic QSSA can be tested using their deterministic counterparts providing a concrete method to test when non-elementary rate functions can be used in stochastic simulations.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-015-0218-3) contains supplementary material, which is available to authorized users.

Circadian systems biology: When time matters

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

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

The proportion of light-responsive neurons determines the limit cycle properties of the suprachiasmatic nucleus

In mammals, the central clock in the suprachiasmatic nucleus (SCN) controls physiological and behavioral circadian rhythms and is entrained to the external light-dark cycle. The ability of the SCN to entrain can be measured by exposing the animal to a light-dark cycle with a duration that deviates from 24 h (T-cycles); a wider entrainment range reflects a higher ability to entrain. The neurons of the SCN are either light responsive or light unresponsive and are mutually synchronized. The coupling and synchronization between individual SCN neurons and between groups of neurons within the SCN influence the SCN's ability to entrain. Some studies suggest that enhanced coupling decreases the entrainment range, whereas others suggest that enhanced coupling increases the entrainment range. The latter results are surprising, as they are not consistent with the prevalent assumption that the SCN is a limit cycle oscillator that has larger phase shifts when the amplitude is smaller. Here, we used the Poincaré and Goodwin models to test entrainment properties using various proportions of neurons that are responsive to an external stimulus. If all neurons receive external input, the SCN shows limit cycle behavior in all conditions. If all neurons do not receive light input, we found that the entrainment range of the SCN was positively related to coupling strength when coupling was weak. When coupling strength was stronger and above a critical value, the entrainment range was negatively correlated with coupling strength. The results obtained from our simulations were confirmed by analytical studies. Thus, the limit cycle behavior of the SCN appears to be critically dependent on the coupling strength among the neurons and the proportion of neurons that respond to the entraining stimulus.

LEMS: a language for expressing complex biological models in concise and hierarchical form and its use in underpinning NeuroML 2

Computational models are increasingly important for studying complex neurophysiological systems. As scientific tools, it is essential that such models can be reproduced and critically evaluated by a range of scientists. However, published models are currently implemented using a diverse set of modeling approaches, simulation tools, and computer languages making them inaccessible and difficult to reproduce. Models also typically contain concepts that are tightly linked to domain-specific simulators, or depend on knowledge that is described exclusively in text-based documentation. To address these issues we have developed a compact, hierarchical, XML-based language called LEMS (Low Entropy Model Specification), that can define the structure and dynamics of a wide range of biological models in a fully machine readable format. We describe how LEMS underpins the latest version of NeuroML and show that this framework can define models of ion channels, synapses, neurons and networks. Unit handling, often a source of error when reusing models, is built into the core of the language by specifying physical quantities in models in terms of the base dimensions. We show how LEMS, together with the open source Java and Python based libraries we have developed, facilitates the generation of scripts for multiple neuronal simulators and provides a route for simulator free code generation. We establish that LEMS can be used to define models from systems biology and map them to neuroscience-domain specific simulators, enabling models to be shared between these traditionally separate disciplines. LEMS and NeuroML 2 provide a new, comprehensive framework for defining computational models of neuronal and other biological systems in a machine readable format, making them more reproducible and increasing the transparency and accessibility of their underlying structure and properties.

Biophysical properties of presynaptic short-term plasticity in hippocampal neurons: insights from electrophysiology, imaging and mechanistic models

Hippocampal neurons show different types of short-term plasticity (STP). Some exhibit facilitation of their synaptic responses and others depression. In this review we discuss presynaptic biophysical properties behind heterogeneity in STP in hippocampal neurons such as alterations in vesicle priming and docking, fusion, neurotransmitter filling and vesicle replenishment. We look into what types of information electrophysiology, imaging and mechanistic models have given about the time scales and relative impact of the different properties on STP with an emphasis on the use of mechanistic models as complementary tools to experimental procedures. Taken together this tells us that it is possible for a multitude of different mechanisms to underlie the same STP pattern, even though some are more important in specific cases, and that mechanistic models can be used to integrate the biophysical properties to see which mechanisms are more important in specific cases of STP.

Molecular mechanisms that regulate the coupled period of the mammalian circadian clock

In mammals, most cells in the brain and peripheral tissues generate circadian (∼24 h) rhythms autonomously. These self-sustained rhythms are coordinated and entrained by a master circadian clock in the suprachiasmatic nucleus (SCN). Within the SCN, the individual rhythms of each neuron are synchronized through intercellular signaling. One important feature of SCN is that the synchronized period is close to the population mean of cells' intrinsic periods. In this way, the synchronized period of the SCN stays close to the periods of cells in peripheral tissues. This is important because the SCN must entrain cells throughout the body. However, the mechanism that drives the period of the coupled SCN cells to the population mean is not known. We use mathematical modeling and analysis to show that the mechanism of transcription repression in the intracellular feedback loop plays a pivotal role in regulating the coupled period. Specifically, we use phase response curve analysis to show that the coupled period within the SCN stays near the population mean if transcriptional repression occurs via protein sequestration. In contrast, the coupled period is far from the mean if repression occurs through highly nonlinear Hill-type regulation (e.g., oligomer- or phosphorylation-based repression), as widely assumed in previous mathematical models. Furthermore, we find that the timescale of intercellular coupling needs to be fast compared to that of intracellular feedback to maintain the mean period. These findings reveal the important relationship between the intracellular transcriptional feedback loop and intercellular coupling. This relationship explains why transcriptional repression appears to occur via protein sequestration in multicellular organisms, mammals, and Drosophila, in contrast with the phosphorylation-based repression in unicellular organisms and syncytia. That is, transition to protein sequestration is essential for synchronizing multiple cells with a period close to the population mean (∼24 h).

Socially synchronized circadian oscillators

Daily rhythms of physiology and behaviour are governed by an endogenous timekeeping mechanism (a circadian 'clock'). The alternation of environmental light and darkness synchronizes (entrains) these rhythms to the natural day-night cycle, and underlying mechanisms have been investigated using singly housed animals in the laboratory. But, most species ordinarily would not live out their lives in such seclusion; in their natural habitats, they interact with other individuals, and some live in colonies with highly developed social structures requiring temporal synchronization. Social cues may thus be critical to the adaptive function of the circadian system, but elucidating their role and the responsible mechanisms has proven elusive. Here, we highlight three model systems that are now being applied to understanding the biology of socially synchronized circadian oscillators: the fruitfly, with its powerful array of molecular genetic tools; the honeybee, with its complex natural society and clear division of labour; and, at a different level of biological organization, the rodent suprachiasmatic nucleus, site of the brain's circadian clock, with its network of mutually coupled single-cell oscillators. Analyses at the 'group' level of circadian organization will likely generate a more complex, but ultimately more comprehensive, view of clocks and rhythms and their contribution to fitness in nature.

Entrainment range of nonidentical circadian oscillators by a light-dark cycle

The suprachiasmatic nucleus (SCN) is a principal circadian clock in mammals, which controls physiological and behavioral daily rhythms. The SCN has two main features: Maintaining a rhythmic cycle of approximately 24 h in the absence of a light-dark cycle (free-running period) and the ability to entrain to external light-dark cycles. Both free-running period and range of entrainment vary from one species to another. To understand this phenomenon, we investigated the diversity of a free-running period by the distribution of coupling strengths in our previous work [Phys. Rev. E 80, 030904(R) (2009)]. In this paper we numerically found that the dispersion of intrinsic periods among SCN neurons influence the entrainment range of the SCN, but has little influence on the free-running periods under constant darkness. This indicates that the dispersion of coupling strengths determines the diversity in free-running periods, while the dispersion of intrinsic periods determines the diversity in the entrainment range. A theoretical analysis based on two coupled neurons is presented to explain the results of numerical simulations.

A circadian clock-regulated toggle switch explains AtGRP7 and AtGRP8 oscillations in Arabidopsis thaliana

The circadian clock controls many physiological processes in higher plants and causes a large fraction of the genome to be expressed with a 24h rhythm. The transcripts encoding the RNA-binding proteins AtGRP7 (Arabidopsis thaliana Glycine Rich Protein 7) and AtGRP8 oscillate with evening peaks. The circadian clock components CCA1 and LHY negatively affect AtGRP7 expression at the level of transcription. AtGRP7 and AtGRP8, in turn, negatively auto-regulate and reciprocally cross-regulate post-transcriptionally: high protein levels promote the generation of an alternative splice form that is rapidly degraded. This clock-regulated feedback loop has been proposed to act as a molecular slave oscillator in clock output. While mathematical models describing the circadian core oscillator in Arabidopsis thaliana were introduced recently, we propose here the first model of a circadian slave oscillator. We define the slave oscillator in terms of ordinary differential equations and identify the model's parameters by an optimization procedure based on experimental results. The model successfully reproduces the pertinent experimental findings such as waveforms, phases, and half-lives of the time-dependent concentrations. Furthermore, we obtain insights into possible mechanisms underlying the observed experimental dynamics: the negative auto-regulation and reciprocal cross-regulation via alternative splicing could be responsible for the sharply peaking waveforms of the AtGRP7 and AtGRP8 mRNA. Moreover, our results suggest that the AtGRP8 transcript oscillations are subordinated to those of AtGRP7 due to a higher impact of AtGRP7 protein on alternative splicing of its own and of the AtGRP8 pre-mRNA compared to the impact of AtGRP8 protein. Importantly, a bifurcation analysis provides theoretical evidence that the slave oscillator could be a toggle switch, arising from the reciprocal cross-regulation at the post-transcriptional level. In view of this, transcriptional repression of AtGRP7 and AtGRP8 by LHY and CCA1 induces oscillations of the toggle switch, leading to the observed high-amplitude oscillations of AtGRP7 mRNA.

The circadian clock organizes the day in the life of a plant by causing 24h rhythms in gene expression. For example, the core clockwork of the model plant Arabidopsis thaliana causes the transcripts encoding the RNA-binding proteins AtGRP7 and AtGRP8 to undergo high amplitude oscillations with a peak at the end of the day. AtGRP7 and AtGRP8, in turn, negatively auto-regulate and reciprocally cross-regulate their own expression by causing alternative splicing of their pre-mRNAs, followed by rapid degradation of the alternatively spliced transcripts. This has led to the suggestion that they represent molecular slave oscillators downstream of the core clock. Using a mathematical model we obtain insights into possible mechanisms underlying the experimentally observed dynamics, e.g. a higher impact of AtGRP7 protein compared to the impact of AtGRP8 protein on the alternative splicing explains the experimentally observed phases of their transcript. Previously, components that reciprocally repress their own transcription (double negative loops) have been shown to potentially act as a toggle switch between two states. We provide theoretical evidence that the slave oscillator could be a bistable toggle switch as well, operating at the post-transcriptional level.

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.

Entrainment of the suprachiasmatic nucleus network by a light-dark cycle

The synchronization of biological activity with the alternation of day and night (circadian rhythm) is performed in the brain by a group of neurons, constituting the suprachiasmatic nucleus (SCN). The SCN is divided into two subgroups of oscillating cells: the ventrolateral (VL) neurons, which are exposed to light (photic signal), and the dorsomedial (DM) neurons, which are coupled to the VL cells. When the coupling between these neurons is strong enough, the system synchronizes with the photic period. Upon increasing the cell coupling, the entrainment of the DM cells has been recently shown to occur via a very sharp (jumping) transition when the period of the photic input is larger than the intrinsic period of the cells. Here, we characterize this transition with a simple realistic model. We show that two bifurcations possibly lead to the disappearance of the endogenous mode. Using a mean-field model, we show that the jumping transition results from a supercritical Hopf-like bifurcation. This finding implies that both the period and strength of the stimulating photic signal, and the relative fraction of cells in the VL and DM compartments, are crucial in determining the synchronization of the system.

(Re)inventing the circadian feedback loop

For 20 years, researchers have thought that circadian clocks are defined by feedback loops of transcription and translation. The rediscovery of posttranslational circadian oscillators in diverse organisms forces us to rethink this paradigm. Meanwhile, the original "basic" feedback loops of canonical circadian clocks have swelled to include dozens of additional proteins acting in interlocked loops. We review several self-sustained clock mechanisms and propose that minimum requirements for diurnal timekeeping might be simpler than those of actual free-running circadian oscillators. Thus, complex mechanisms of circadian timekeeping might have evolved from random connections between unrelated feedback loops with independent but limited time-telling capability.

Photic desynchronization of two subgroups of circadian oscillators in a network model of the suprachiasmatic nucleus with dispersed coupling strengths

The suprachiasmatic nucleus (SCN) is the master circadian clock in mammals and is composed of thousands of neuronal oscillators expressing different intrinsic periods. These oscillators form a coupled network with a free-running period around 24 h in constant darkness and entrainable to the external light-dark cycle (T cycle). Coupling plays an important role in setting the period of the network and its range of entrainment. Experiments in rats have shown that two subgroups of oscillators within the SCN, a ventrolateral (VL) subgroup that receives photic input and a dorsomedial (DM) subgroup that is coupled to VL, can be desynchronized under a short (22-h) T cycle, with VL entrained to the cycle and DM free-running. We use a modified Goodwin model to understand how entrainment of the subgroups to short (22-h) and long (26-h) T cycles is influenced by light intensity, the proportion of neurons that receives photic input, and coupling heterogeneity. We find that the model's critical value for the proportion of photically-sensitive neurons is in accord with actual experimental estimates, while the model's inclusion of dispersed coupling can account for the experimental observation that VL and DM desynchronize more readily under the 22-h than under the 26-h T cycle. Heterogeneous intercellular coupling within the SCN is likely central to the generation of complex behavioral patterns.

Persistent cell-autonomous circadian oscillations in fibroblasts revealed by six-week single-cell imaging of PER2::LUC bioluminescence

Biological oscillators naturally exhibit stochastic fluctuations in period and amplitude due to the random nature of molecular reactions. Accurately measuring the precision of noisy oscillators and the heterogeneity in period and strength of rhythmicity across a population of cells requires single-cell recordings of sufficient length to fully represent the variability of oscillations. We found persistent, independent circadian oscillations of clock gene expression in 6-week-long bioluminescence recordings of 80 primary fibroblast cells dissociated from PER2::LUC mice and kept in vitro for 6 months. Due to the stochastic nature of rhythmicity, the proportion of cells appearing rhythmic increases with the length of interval examined, with 100% of cells found to be rhythmic when using 3-week windows. Mean period and amplitude are remarkably stable throughout the 6-week recordings, with precision improving over time. For individual cells, precision of period and amplitude are correlated with cell size and rhythm amplitude, but not with period, and period exhibits much less cycle-to-cycle variability (CV 7.3%) than does amplitude (CV 37%). The time series are long enough to distinguish stochastic fluctuations within each cell from differences among cells, and we conclude that the cells do exhibit significant heterogeneity in period and strength of rhythmicity, which we measure using a novel statistical metric. Furthermore, stochastic modeling suggests that these single-cell clocks operate near a Hopf bifurcation, such that intrinsic noise enhances the oscillations by minimizing period variability and sustaining amplitude.

Effect of network architecture on synchronization and entrainment properties of the circadian oscillations in the suprachiasmatic nucleus

In mammals, the suprachiasmatic nucleus (SCN) of the hypothalamus constitutes the central circadian pacemaker. The SCN receives light signals from the retina and controls peripheral circadian clocks (located in the cortex, the pineal gland, the liver, the kidney, the heart, etc.). This hierarchical organization of the circadian system ensures the proper timing of physiological processes. In each SCN neuron, interconnected transcriptional and translational feedback loops enable the circadian expression of the clock genes. Although all the neurons have the same genotype, the oscillations of individual cells are highly heterogeneous in dispersed cell culture: many cells present damped oscillations and the period of the oscillations varies from cell to cell. In addition, the neurotransmitters that ensure the intercellular coupling, and thereby the synchronization of the cellular rhythms, differ between the two main regions of the SCN. In this work, a mathematical model that accounts for this heterogeneous organization of the SCN is presented and used to study the implication of the SCN network topology on synchronization and entrainment properties. The results show that oscillations with larger amplitude can be obtained with scale-free networks, in contrast to random and local connections. Networks with the small-world property such as the scale-free networks used in this work can adapt faster to a delay or advance in the light/dark cycle (jet lag). Interestingly a certain level of cellular heterogeneity is not detrimental to synchronization performances, but on the contrary helps resynchronization after jet lag. When coupling two networks with different topologies that mimic the two regions of the SCN, efficient filtering of pulse-like perturbations in the entrainment pattern is observed. These results suggest that the complex and heterogeneous architecture of the SCN decreases the sensitivity of the network to short entrainment perturbations while, at the same time, improving its adaptation abilities to long term changes.

In order to adapt to their cycling environment, virtually all living organisms have developed an internal timer, the circadian clock. In mammals, the circadian pacemaker is composed of about 20,000 neurons, called the suprachiasmatic nucleus (SCN) located in the hypothalamus. The SCN receives light signals from the retina and controls peripheral circadian clocks to ensure the proper timing of physiological processes. In each SCN neuron, a genetic regulatory network enables the circadian expression of the clock genes, but individual dynamics are highly heterogeneous in dispersed cell culture: many cells present damped oscillations and the period of the oscillations varies from cell to cell. In addition, the neurotransmitters that ensure the intercellular coupling, and thereby the synchronization of the cellular rhythms, differ between the two main regions of the SCN. We present here a mathematical model that accounts for this heterogeneous organization of the SCN and study the implication of the network topology on synchronization and entrainment properties. Our results show that cellular heterogeneity may help the resynchronization after jet lag and suggest that the complex architecture of the SCN decreases the sensitivity of the network to short entrainment perturbations while, at the same time, improving its adaptation abilities to long term changes.

Modeling circadian clocks: Roles, advantages, and limitations

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

Modeling circadian clocks: From equations to oscillations

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

Circadian regulation of intracellular G-protein signalling mediates intercellular synchrony and rhythmicity in the suprachiasmatic nucleus

Synchronous oscillations of thousands of cellular clocks in the suprachiasmatic nucleus (SCN), the circadian centre, are coordinated by precisely timed cell–cell communication, the principle of which is largely unknown. Here we show that the amount of RGS16 (regulator of G protein signalling 16), a protein known to inactivate Gαi, increases at a selective circadian time to allow time-dependent activation of intracellular cyclic AMP signalling in the SCN. Gene ablation of Rgs16 leads to the loss of circadian production of cAMP and as a result lengthens circadian period of behavioural rhythm. The temporally precise regulation of the cAMP signal by clock-controlled RGS16 is needed for the dorsomedial SCN to maintain a normal phase-relationship to the ventrolateral SCN. Thus, RGS16-dependent temporal regulation of intracellular G protein signalling coordinates the intercellular synchrony of SCN pacemaker neurons and thereby defines the 24 h rhythm in behaviour.

Circadian rhythm is controlled by the suprachiasmatic nucleus and the mechanisms that control the rhythm are largely undiscovered. In this study, a G protein regulator, RGS16, is shown to be involved in the production of cyclic AMP that is required for the suprachiasmatic nucleus to maintain rhythm

Mechanism of phase splitting in two coupled groups of suprachiasmatic-nucleus neurons

The phase-splitting behavior of coupled suprachiasmatic-nucleus neurons has been observed in many mammals, and its mechanism is still not completely understood. Based on our previous work [C. Gu, J. Wang, and Z. Liu, Phys. Rev. E 80, 030904(R) (2009)] on the free-running periods of neurons in the suprachiasmatic nucleus, we present here a modified Goodwin oscillator model to explain the mechanism of phase splitting. In contrast to the previous phase model, the modified Goodwin oscillator model contains the information on both the phase and amplitude and, thus, can show more features than the purely phase model, including all three behaviors of synchronization, phase splitting, and amplitude death and the distributed periodicity in the regions of synchronization and phase splitting, etc. An analytic phase model is extracted from the modified Goodwin oscillator model to explain the dependence of periodicity on the parameters. Moreover, both the modified Goodwin oscillator model and the analytic phase model show that the ensemble frequency can be enhanced or reduced by the time delay.

Coupling governs entrainment range of circadian clocks

Circadian clock oscillator properties that are crucial for synchronization with the environment (entrainment) are studied in experiment and theory.

The ratio between stimulus (zeitgeber) strength and oscillator amplitude, and the rigidity of the oscillatory system (relaxation rate upon perturbation) determine entrainment properties. Coupling among oscillators affects both qualities resulting in increased amplitude and rigidity.

Uncoupled lung clocks entrain to extreme zeitgeber cycles, whereas the coupled oscillator system in the suprachiasmatic nucleus (SCN) does not; however, when coupling in the SCN is inhibited, larger ranges of entrainment can be achieved.

Daily rhythms in physiology, metabolism and behavior are controlled by an endogenous circadian timing system, which has evolved to synchronize an organism to periodically recurring environmental conditions, such as light–dark or temperature cycles. In mammals, the circadian system relies on cell-autonomous oscillators residing in almost every cell of the body. Cells of the SCN in the anterior hypothalamus are able to generate precise, long-lasting self-sustained circadian oscillations, which drive most rhythmic behavioral and physiological outputs, and which are believed to originate from the fact that the SCN tissue consists of tightly coupled cells (Aton and Herzog, 2005). In contrast, peripheral oscillators, such as lung tissue, exhibit seemingly damped and usually less precise oscillations, which are thought to be brought about by the lack of intercellular coupling.

Precise synchronization of these rhythms within the organism, but also with the environment (so-called entrainment), is an essential part of circadian organization. Entrainment is one of the cornerstones of circadian biology (Roenneberg et al, 2003). In evolution, the phase of a rhythmic variable is selective rather than its endogenous period. Thus, the synchronization of endogenous rhythms to zeitgeber cycles of the environment (resulting in a specific phase of entrainment) is fundamental for the adaptive value of circadian clocks. In this study, we systematically investigated the properties of circadian oscillators that are essential for entrainment behavior and describe coupling as a primary determinant.

As an experimental starting point of this study, we found that the circadian oscillators of lung tissue have a larger range of entrainment than SCN tissue—they readily entrained to extreme experimental temperature cycle of 20 or 28 h, whereas SCN tissue did not (Figure 4). For this purpose, we cultured SCN and lung slices derived from mice that express luciferase as fusion protein together with the clock protein PERIOD2 (Yoo et al, 2004). The detection of luciferase-driven bioluminescence allowed us to follow molecular clock gene activity in real-time over the course of several days.

In theoretical analyses, we show that both the ratio of amplitude and zeitgeber strength and, importantly, inter-oscillator coupling are major determinants for entrainment. The reason for coupling being critical is twofold: (i) Coupling makes an oscillatory system more rigid, i.e., it relaxes faster in response to a perturbation, and (ii) coupling increases the amplitude of the oscillatory system. Both of these consequences of coupling lead to a smaller entrainment range, because zeitgeber stimuli affect the oscillatory system less if the relaxation is fast and the amplitude is high (Figure 1). From these theoretical considerations, we conclude that the lung clock probably constitutes a weak oscillatory system, likely because a lack in coupling leads to a slow amplitude relaxation. (Circadian amplitude is not particularly low in lung (Figure 4).) In contrast, the SCN constitutes a rigid oscillator, whereby coupling and its described consequences probably are the primary causes for this rigidity. We then tested these theoretical predictions by experimentally perturbing coupling in the SCN (with MDL and TTX; O'Neill et al, 2008; Yamaguchi et al, 2003) and find that, indeed, reducing the coupling weakens the circadian oscillatory system in the SCN, which results in an enlargement of the entrainment range (Figure 6).

Why is the SCN designed to be a stronger circadian oscillator than peripheral organs? We speculate that the position of the SCN—as the tissue that conveys environmental timing information (i.e., light) to the rest of the body—makes it necessary to create a circadian clock that is robust against noisy environmental stimuli. The SCN oscillator needs to be robust enough to be protected from environmental noise, but flexible enough to fulfill its function as an entrainable clock even in extreme photoperiods (i.e., seasons). By the same token, peripheral clocks are more protected from the environmental zeitgebers due to intrinsic homeostatic mechanisms. Thus, they do not necessarily need to develop a strong oscillatory system (e.g., by strengthening the coupling), rather they need to stay flexible enough to respond to direct or indirect signals from the SCN, such as hormonal, neural, temperature or metabolic signals. Such a design ensures that only robust and persistent environmental signals trigger an SCN resetting response, while SCN signals can relatively easily be conveyed to the rest of the body. Thus, the robustness in the SCN clock likely serves as a filter for environmental noise.

In summary, using a combination of simulation studies, analytical calculations and experiments, we uncovered critical features for entrainment, such as zeitgeber-to-amplitude ratio and amplitude relaxation rate. Coupling is a primary factor that governs these features explaining important differences in the design of SCN and peripheral oscillators that ensure a robust, but also flexible circadian system.

Circadian clocks are endogenous oscillators driving daily rhythms in physiology and behavior. Synchronization of these timers to environmental light–dark cycles (‘entrainment') is crucial for an organism's fitness. Little is known about which oscillator qualities determine entrainment, i.e., entrainment range, phase and amplitude. In a systematic theoretical and experimental study, we uncovered these qualities for circadian oscillators in the suprachiasmatic nucleus (SCN—the master clock in mammals) and the lung (a peripheral clock): (i) the ratio between stimulus (zeitgeber) strength and oscillator amplitude and (ii) the rigidity of the oscillatory system (relaxation rate upon perturbation) determine entrainment properties. Coupling among oscillators affects both qualities resulting in increased amplitude and rigidity. These principles explain our experimental findings that lung clocks entrain to extreme zeitgeber cycles, whereas SCN clocks do not. We confirmed our theoretical predictions by showing that pharmacological inhibition of coupling in the SCN leads to larger ranges of entrainment. These differences between master and the peripheral clocks suggest that coupling-induced rigidity in the SCN filters environmental noise to create a robust circadian system.

Autonomous synchronization of chemically coupled synthetic oscillators

Synthetic biology has recently provided functional single-cell oscillators. With a few exceptions, however, synchronization across a population has not been achieved yet. In particular, designing a cell coupling mechanism to achieve autonomous synchronization is not straightforward since there are usually several different design alternatives. Here, we propose a method to mathematically predict autonomous synchronization properties, and to identify the network structure with the best performance, thus increasing the feasibility for a successful implementation in vivo.Our method relies on the reduction of ODE-based models for synthetic oscillators to a phase description, and the subsequent analysis of the phase model either in the spatially homogeneous or heterogeneous case. This analysis identifies three major factors determining if and when autonomous synchronization can be achieved, namely cell density, cell to cell variability, and structural design decisions. Moreover, when considering a spatially heterogeneous medium, we observe phase waves. These waves may hinder synchronization substantially, and their suppression should be considered in the design process.In contrast to previous work, we analyze the synchronization process of models of experimentally validated synthetic oscillators in mammalian cells. Alternative designs for cell-to-cell communication via a quorum sensing mechanism differ in few mechanistic details, but these differences have important implications for autonomous synchronization. Our analysis suggests that not only the periodical transcription of the protein producing the signaling molecule, but also of the receptor protein is necessary to achieve good performance.