Toward a detailed computational model for the mammalian circadian clock

Systems Chronotherapeutics

Chronotherapeutics aim at treating illnesses according to the endogenous biologic rhythms, which moderate xenobiotic metabolism and cellular drug response. The molecular clocks present in individual cells involve approximately fifteen clock genes interconnected in regulatory feedback loops. They are coordinated by the suprachiasmatic nuclei, a hypothalamic pacemaker, which also adjusts the circadian rhythms to environmental cycles. As a result, many mechanisms of diseases and drug effects are controlled by the circadian timing system. Thus, the tolerability of nearly 500 medications varies by up to fivefold according to circadian scheduling, both in experimental models and/or patients. Moreover, treatment itself disrupted, maintained, or improved the circadian timing system as a function of drug timing. Improved patient outcomes on circadian-based treatments (chronotherapy) have been demonstrated in randomized clinical trials, especially for cancer and inflammatory diseases. However, recent technological advances have highlighted large interpatient differences in circadian functions resulting in significant variability in chronotherapy response. Such findings advocate for the advancement of personalized chronotherapeutics through interdisciplinary systems approaches. Thus, the combination of mathematical, statistical, technological, experimental, and clinical expertise is now shaping the development of dedicated devices and diagnostic and delivery algorithms enabling treatment individualization. In particular, multiscale systems chronopharmacology approaches currently combine mathematical modeling based on cellular and whole-body physiology to preclinical and clinical investigations toward the design of patient-tailored chronotherapies. We review recent systems research works aiming to the individualization of disease treatment, with emphasis on both cancer management and circadian timing system-resetting strategies for improving chronic disease control and patient outcomes.

Computational modeling of the cell-autonomous mammalian circadian oscillator

This review summarizes various mathematical models of cell-autonomous mammalian circadian clock. We present the basics necessary for understanding of the cell-autonomous mammalian circadian oscillator, modern experimental data essential for its reconstruction and some special problems related to the validation of mathematical circadian oscillator models. This work compares existing mathematical models of circadian oscillator and the results of the computational studies of the oscillating systems. Finally, we discuss applications of the mathematical models of mammalian circadian oscillator for solving specific problems in circadian rhythm biology.

Systems Biology-Derived Discoveries of Intrinsic Clocks

A systems approach to studying biology uses a variety of mathematical, computational, and engineering tools to holistically understand and model properties of cells, tissues, and organisms. Building from early biochemical, genetic, and physiological studies, systems biology became established through the development of genome-wide methods, high-throughput procedures, modern computational processing power, and bioinformatics. Here, we highlight a variety of systems approaches to the study of biological rhythms that occur with a 24-h period-circadian rhythms. We review how systems methods have helped to elucidate complex behaviors of the circadian clock including temperature compensation, rhythmicity, and robustness. Finally, we explain the contribution of systems biology to the transcription-translation feedback loop and posttranslational oscillator models of circadian rhythms and describe new technologies and "-omics" approaches to understand circadian timekeeping and neurophysiology.

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.

Impact of time delays on oscillatory dynamics of interlinked positive and negative feedback loops

Interlinking a positive feedback loop (PFL) with a negative feedback loop (NFL) constitutes a typical motif in genetic networks, performing various functions in cell signaling. How time delay in feedback regulation affects the dynamics of such systems still remains unclear. Here, we investigate three systems of interlinked PFL and NFL with time delays: a synthetic genetic oscillator, a three-node circuit, and a simplified single-node model. The stability of steady states and the routes to oscillation in the single-node model are analyzed in detail. The amplitude and period of oscillations vary with a pointwise periodicity over a range of time delay. Larger-amplitude oscillations can be induced when the PFL has an appropriately long delay, in comparison with the PFL with no delay or short delay; this conclusion holds true for all the three systems. We unravel the underlying mechanism for the above effects via analytical derivation under a limiting condition. We also develop a stochastic algorithm for simulating a single reaction with two delays and show that robust oscillations can be maintained by the PFL with a properly long delay in the single-node system. This work presents an effective method for constructing robust large-amplitude oscillators and interprets why similar circuit architectures are engaged in timekeeping systems such as circadian clocks.

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.

The effects of time-varying temperature on delays in genetic networks

Delays in gene networks result from the sequential nature of protein assembly. However, it is unclear how models of gene networks that use delays should be modified when considering time-dependent changes in temperature. This is important, as delay is often used in models of genetic oscillators that can be entrained by periodic fluctuations in temperature. Here, we analytically derive the time dependence of delay distributions in response to time-varying temperature changes. We find that the resulting time-varying delay is nonlinearly dependent on parameters of the time-varying temperature such as amplitude and frequency, therefore, applying an Arrhenius scaling may result in erroneous conclusions. We use these results to examine a model of a synthetic gene oscillator with temperature compensation. We show that temperature entrainment follows from the same mechanism that results in temperature compensation. Under a common Arrhenius scaling alone, the frequency of the oscillator is sensitive to changes in the mean temperature but robust to changes in the frequency of a periodically time-varying temperature. When a mechanism for temperature compensation is included in the model, however, we show that the oscillator is entrained by periodically varying temperature even when maintaining insensitivity to the mean temperature.

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.

Can a systems approach produce a better understanding of mood disorders?

BACKGROUND

One in twenty-five people suffer from a mood disorder. Current treatments are sub-optimal with poor patient response and uncertain modes-of-action. There is thus a need to better understand underlying mechanisms that determine mood, and how these go wrong in affective disorders. Systems biology approaches have yielded important biological discoveries for other complex diseases such as cancer, and their potential in affective disorders will be reviewed.

SCOPE OF REVIEW

This review will provide a general background to affective disorders, plus an outline of experimental and computational systems biology. The current application of these approaches in understanding affective disorders will be considered, and future recommendations made.

MAJOR CONCLUSIONS

Experimental systems biology has been applied to the study of affective disorders, especially at the genome and transcriptomic levels. However, data generation has been slowed by a lack of human tissue or suitable animal models. At present, computational systems biology has only be applied to understanding affective disorders on a few occasions. These studies provide sufficient novel biological insight to motivate further use of computational biology in this field.

GENERAL SIGNIFICANCE

In common with many complex diseases much time and money has been spent on the generation of large-scale experimental datasets. The next step is to use the emerging computational approaches, predominantly developed in the field of oncology, to leverage the most biological insight from these datasets. This will lead to the critical breakthroughs required for more effective diagnosis, stratification and treatment of affective disorders.

Model-based investigation of the circadian clock and cell cycle coupling in mouse embryonic fibroblasts: Prediction of RevErb-α up-regulation during mitosis

Experimental observations have put in evidence autonomous self-sustained circadian oscillators in most mammalian cells, and proved the existence of molecular links between the circadian clock and the cell cycle. Some mathematical models have also been built to assess conditions of control of the cell cycle by the circadian clock. However, recent studies in individual NIH3T3 fibroblasts have shown an unexpected acceleration of the circadian clock together with the cell cycle when the culture medium is enriched with growth factors, and the absence of such acceleration in confluent cells. In order to explain these observations, we study a possible entrainment of the circadian clock by the cell cycle through a regulation of clock genes around the mitosis phase. We develop a computational model and a formal specification of the observed behavior to investigate the conditions of entrainment in period and phase. We show that either the selective activation of RevErb-α or the selective inhibition of Bmal1 transcription during the mitosis phase, allow us to fit the experimental data on both period and phase, while a uniform inhibition of transcription during mitosis seems incompatible with the phase data. We conclude on the arguments favoring the RevErb-α up-regulation hypothesis and on some further predictions of the model.

Clock Genes, Oscillators, and Cellular Networks in the Suprachiasmatic Nuclei

The mammalian SCN contains a biological clock that drives remarkably precise circadian rhythms in vivo and in vitro. Recent advances have revealed molecular and cellular mechanisms required for the generation of these daily rhythms and their synchronization between SCN neurons and to the environmental light cycle. This review of the evidence for a cell-autonomous circadian pacemaker within specialized neurons of the SCN focuses on 6 genes implicated within the pace making mechanism, an additional 4 genes implicated in pathways from the pacemaker, and the intercellular and intracellular mechanisms that synchronize SCN neurons to each other and to solar time.

Long-Term Constant Light Induces Constitutive Elevated Expression of mPER2 Protein in the Murine SCN: A Molecular Basis for Aschoff’s Rule?

Circadian rhythms in behavior, metabolism, and physiology are based upon transcriptional/translational feedback loops involving a core set of clock genes that interact to regulate their own expression. In mammals, the SCN is the site of a master biological clock regulating circadian locomotor rhythms. The products of the clock genes mPer1, mPer2, mCry1, and mCry2 form essential components of both negative and positive elements within the SCN oscillator. The primary aims of this study were to examine clock gene abundance under longterm LL in an attempt to provide molecular correlates of the lengthened tau and daily phase delays described by Aschoff’s rule. Wheel-running behavior was recorded frommice maintained in eitherDDor LLfor 50 days. The abundance of the clock genes mPer1, mPer2, mCry1, and mCry2 and their protein products was then examined (every ~4 h) within the SCN using in situ hybridization and immunocytochemistry. Under LL conditions, mPer1, mPer2, mCry1, and mCry2 messages remained rhythmic, although the waveform of mCry2 was altered compared to DD. In LL, mPER1, mCRY1, and mCRY2 protein levels were also rhythmic and comparable to the patterns observed in DD. However, mPER2 is elevated and constitutively expressed under LL. Thus, rhythmic expression of these clock genes is not dependent on the rhythmic production of mPER2, and the acute up-regulation of mPer1 and mPer2 described for short (nonparametric) light pulses is not sustained under LL conditions. These findings suggest that mPER2 is important for the generation of phase delays in the molecular clockwork, providing a possible molecular explanation for Aschoff’s rule: LL lengthens the circadian period by inhibiting the degeneration of mPER2, and constitutively elevated levels of mPER2 enhance the phase-delaying limb of the molecular oscillator.

Modeling the Behavior of Coupled Cellular Circadian Oscillators in the Suprachiasmatic Nucleus

The suprachiasmatic nucleus (SCN) in the hypothalamus is the site of the master circadian clock in mammals, a complex tissue composed of multiple, coupled, single-cell circadian oscillators. Mathematical modeling is now providing insights on how individual SCN cells might interact and assemble to create an integrated pacemaker that governs the circadian behavior of whole animals. In this article, we will discuss the neurobiological constraints for modeling SCN behavior, system precision, implications of cellular heterogeneity, and analysis of heterogeneously coupled oscillator networks. Mathematical approaches will be critical for better understanding intercellular interactions within the SCN.

Phase Sensitivity Analysis of Circadian Rhythm Entrainment

As a biological clock, circadian rhythms evolve to accomplish a stable (robust) entrainment to environmental cycles, of which light is the most obvious. The mechanism of photic entrainment is not known, but two models of entrainment have been proposed based on whether light has a continuous (parametric) or discrete (nonparametric) effect on the circadian pacemaker. A novel sensitivity analysis is developed to study the circadian entrainment in silico based on a limit cycle approach and applied to a model of Drosophila circadian rhythm. The comparative analyses of complete and skeleton photoperiods suggest a trade-off between the contribution of period modulation (parametric effect) and phase shift (nonparametric effect) in Drosophila circadian entrainment. The results also give suggestions for an experimental study to (in)validate the two models of entrainment.

Cancer Clocks Out for Lunch: Disruption of Circadian Rhythm and Metabolic Oscillation in Cancer

Circadian rhythms are 24-h oscillations present in most eukaryotes and many prokaryotes that synchronize activity to the day-night cycle. They are an essential feature of organismal and cell physiology that coordinate many of the metabolic, biosynthetic, and signal transduction pathways studied in biology. The molecular mechanism of circadian rhythm is controlled both by signal transduction and gene transcription as well as by metabolic feedback. The role of circadian rhythm in cancer cell development and survival is still not well understood, but as will be discussed in this Review, accumulated research suggests that circadian rhythm may be altered or disrupted in many human cancers downstream of common oncogenic alterations. Thus, a complete understanding of the genetic and metabolic alterations in cancer must take potential circadian rhythm perturbations into account, as this disruption itself will influence how gene expression and metabolism are altered in the cancer cell compared to its non-transformed neighbor. It will be important to better understand these circadian changes in both normal and cancer cell physiology to potentially design treatment modalities to exploit this insight.

Investigations of the CLOCK and BMAL1 Proteins Binding to DNA: A Molecular Dynamics Simulation Study

The circadian locomotor output cycles kaput (CLOCK), and brain and muscle ARNT-like 1 (BMAL1) proteins are important transcriptional factors of the endogenous circadian clock. The CLOCK and BMAL1 proteins can regulate the transcription-translation activities of the clock-related genes through the DNA binding. The hetero-/homo-dimerization and DNA combination of the CLOCK and BMAL1 proteins play a key role in the positive and negative transcriptional feedback processes. In the present work, we constructed a series of binary and ternary models for the bHLH/bHLH-PAS domains of the CLOCK and BMAL1 proteins, and the DNA molecule, and carried out molecular dynamics simulations, free energy calculations and conformational analysis to explore the interaction properties of the CLOCK and BMAL1 proteins with DNA. The results show that the bHLH domains of CLOCK and BMAL1 can favorably form the heterodimer of the bHLH domains of CLOCK and BMAL1 and the homodimer of the bHLH domains of BMAL1. And both dimers could respectively bind to DNA at its H1-H1 interface. The DNA bindings of the H1 helices in the hetero- and homo-bHLH dimers present the rectangular and diagonal binding modes, respectively. Due to the function of the α-helical forceps in these dimers, the tight gripping of the H1 helices to the major groove of DNA would cause the decrease of interactions at the H1-H2 interfaces in the CLOCK and BMAL1 proteins. The additional PAS domains in the CLOCK and BMAL1 proteins affect insignificantly the interactions of the CLOCK and BMAL1 proteins with the DNA molecule due to the flexible and long loop linkers located at the middle of the PAS and bHLH domains. The present work theoretically explains the interaction mechanisms of the bHLH domains of the CLOCK and BMAL1 proteins with DNA.

Modelling the dual role of Per phosphorylation and its effect on the period and phase of the mammalian circadian clock

Circadian clocks are regulated at the post-translational level by a variety of processes among which protein phosphorylation plays a prominent, although complex, role. Thus, the phosphorylation of different sites on the clock protein PER by casein kinase I (CKI) can lead to opposite effects on the stability of the protein and on the period of circadian oscillations. Here the authors extend a computational model previously proposed for the mammalian circadian clock by incorporating two distinct phosphorylations of PER by CKI. On the basis of experimental observations the authors consider that phosphorylation at one site (denoted here PER-P1) enhances the rate of degradation of the protein and decreases the period, while phosphorylation at another site (PER-P2) stabilises the protein, enhances the transcription of the Per gene, and increases the period. The model also incorporates an additional phosphorylation of PER by the Glycogen Synthase Kinase 3 (GSK3). The authors show that the extended model incorporating the antagonistic effects of PER phosphorylations by CKI can account for observations pertaining to (i) the decrease in period in the Tau mutant, because of an increase in phosphorylation by CKI leading to PER-P1, and (ii) the familial advanced sleep phase syndrome (FASPS) in which the period is shortened and the phase of the oscillations is advanced when the rate of phosphorylation leading to PER-P2 is decreased. The model further accounts for the increase in period observed in the presence of CKI inhibitors that decrease the rate of phosphorylation leading to both PER-P1 and PER-P2. A similar increase in period results from inhibition of GSK3. [Includes supplementary material].

Using computational models to discover and understand mechanisms

Areas of biology such as cell and molecular biology have been dominated by research directed at constructing mechanistic explanations that identify parts and operations that when organized appropriately are responsible for the various phenomena they investigate. Increasingly the mechanisms hypothesized involve non-sequential organization of non-linear operations and so exceed the ability of researchers to mentally rehearse their behavior. Accordingly, scientists rely on tools of computational modeling and dynamical systems theory in advancing dynamic mechanistic explanations. Using circadian rhythm research as an exemplar, this paper explores the variety of roles computational modeling is playing. They serve not just to determine whether the mechanism will produce the desired behavior, but in the discovery process of hypothesizing mechanisms and in understanding why proposed mechanisms behave as they do.

Phase-selective entrainment of nonlinear oscillator ensembles

The ability to organize and finely manipulate the hierarchy and timing of dynamic processes is important for understanding and influencing brain functions, sleep and metabolic cycles, and many other natural phenomena. However, establishing spatiotemporal structures in biological oscillator ensembles is a challenging task that requires controlling large collections of complex nonlinear dynamical units. In this report, we present a method to design entrainment signals that create stable phase patterns in ensembles of heterogeneous nonlinear oscillators without using state feedback information. We demonstrate the approach using experiments with electrochemical reactions on multielectrode arrays, in which we selectively assign ensemble subgroups into spatiotemporal patterns with multiple phase clusters. The experimentally confirmed mechanism elucidates the connection between the phases and natural frequencies of a collection of dynamical elements, the spatial and temporal information that is encoded within this ensemble, and how external signals can be used to retrieve this information.

Organizing and manipulating dynamic processes is important to understand and influence many natural phenomena. Here, the authors present a method to design entrainment signals that create stable phase patterns in heterogeneous nonlinear oscillators, and verify it in electrochemical reactions.

Tuning the phase of circadian entrainment

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

Inhibitory and excitatory networks balance cell coupling in the suprachiasmatic nucleus: A modeling approach

Neuronal coupling contributes to circadian rhythms formation in the suprachiasmatic nucleus (SCN). While the neurotransmitter vasoactive intestinal polypeptide (VIP) is considered essential for synchronizing the oscillations of individual neurons, γ-aminobutyric acid (GABA) does not have a clear functional role despite being highly concentrated in the SCN. While most studies have examined the role of either GABA or VIP, our mathematical modeling approach explored their interplay on networks of SCN neurons. Tuning the parameters that control the release of GABA and VIP enabled us to optimize network synchrony, which was achieved at a peak firing rate during the subjective day of about 7Hz. Furthermore, VIP and GABA modulation could adjust network rhythm amplitude and period without sacrificing synchrony. We also performed simulations of SCN networks to phase shifts during 12h:12h light-dark cycles and showed that GABA networks reduced the average time for the SCN model to re-synchronize. We hypothesized that VIP and GABA balance cell coupling in the SCN to promote synchronization of heterogeneous oscillators while allowing flexibility for adjustment to environmental changes.

Inverse problem studies of biochemical systems with structure identification of S-systems by embedding training functions in a genetic algorithm

An efficient inverse problem approach for parameter estimation, state and structure identification from dynamic data by embedding training functions in a genetic algorithm methodology (ETFGA) is proposed for nonlinear dynamical biosystems using S-system canonical models. Use of multiple shooting and decomposition approach as training functions has been shown for handling of noisy datasets and computational efficiency in studying the inverse problem. The advantages of the methodology are brought out systematically by studying it for three biochemical model systems of interest. By studying a small-scale gene regulatory system described by a S-system model, the first example demonstrates the use of ETFGA for the multifold aims of the inverse problem. The estimation of a large number of parameters with simultaneous state and network identification is shown by training a generalized S-system canonical model with noisy datasets. The results of this study bring out the superior performance of ETFGA on comparison with other metaheuristic approaches. The second example studies the regulation of cAMP oscillations in Dictyostelium cells now assuming limited availability of noisy data. Here, flexibility of the approach to incorporate partial system information in the identification process is shown and its effect on accuracy and predictive ability of the estimated model are studied. The third example studies the phenomenological toy model of the regulation of circadian oscillations in Drosophila that follows rate laws different from S-system power-law. For the limited noisy data, using a priori information about properties of the system, we could estimate an alternate S-system model that showed robust oscillatory behavior with predictive abilities.

Kernel Architecture of the Genetic Circuitry of the Arabidopsis Circadian System

A wide range of organisms features molecular machines, circadian clocks, which generate endogenous oscillations with ~24 h periodicity and thereby synchronize biological processes to diurnal environmental fluctuations. Recently, it has become clear that plants harbor more complex gene regulatory circuits within the core circadian clocks than other organisms, inspiring a fundamental question: are all these regulatory interactions between clock genes equally crucial for the establishment and maintenance of circadian rhythms? Our mechanistic simulation for Arabidopsis thaliana demonstrates that at least half of the total regulatory interactions must be present to express the circadian molecular profiles observed in wild-type plants. A set of those essential interactions is called herein a kernel of the circadian system. The kernel structure unbiasedly reveals four interlocked negative feedback loops contributing to circadian rhythms, and three feedback loops among them drive the autonomous oscillation itself. Strikingly, the kernel structure, as well as the whole clock circuitry, is overwhelmingly composed of inhibitory, rather than activating, interactions between genes. We found that this tendency underlies plant circadian molecular profiles which often exhibit sharply-shaped, cuspidate waveforms. Through the generation of these cuspidate profiles, inhibitory interactions may facilitate the global coordination of temporally-distant clock events that are markedly peaked at very specific times of day. Our systematic approach resulting in experimentally-testable predictions provides insights into a design principle of biological clockwork, with implications for synthetic biology.

A tunable artificial circadian clock in clock-defective mice

Self-sustaining oscillations are essential for diverse physiological functions such as the cell cycle, insulin secretion and circadian rhythms. Synthetic oscillators using biochemical feedback circuits have been generated in cell culture. These synthetic systems provide important insight into design principles for biological oscillators, but have limited similarity to physiological pathways. Here we report the generation of an artificial, mammalian circadian clock in vivo, capable of generating robust, tunable circadian rhythms. In mice deficient in Per1 and Per2 genes (thus lacking circadian rhythms), we artificially generate PER2 rhythms and restore circadian sleep/wake cycles with an inducible Per2 transgene. Our artificial clock is tunable as the period and phase of the rhythms can be modulated predictably. This feature, and other design principles of our work, might enhance the study and treatment of circadian dysfunction and broader aspects of physiology involving biological oscillators.

Circadian rhythms are central to health and disease and there is renewed interest in chronotherapy. Here, the authors present a mouse with an artificial circadian clock that can be pharmacologically tuned, providing a tool for future studies of circadian biology and therapy.

Stochastic entrainment of a stochastic oscillator

In this work, we consider a stochastic oscillator described by a discrete-state continuous-time Markov chain, in which the states are arranged in a circle, and there is a constant probability per unit time of jumping from one state to the next in a specified direction around the circle. At each of a sequence of equally spaced times, the oscillator has a specified probability of being reset to a particular state. The focus of this work is the entrainment of the oscillator by this periodic but stochastic stimulus. We consider a distinguished limit, in which (i) the number of states of the oscillator approaches infinity, as does the probability per unit time of jumping from one state to the next, so that the natural mean period of the oscillator remains constant, (ii) the resetting probability approaches zero, and (iii) the period of the resetting signal approaches a multiple, by a ratio of small integers, of the natural mean period of the oscillator. In this distinguished limit, we use analytic and numerical methods to study the extent to which entrainment occurs.

On the Inference of Functional Circadian Networks Using Granger Causality

Being able to infer one way direct connections in an oscillatory network such as the suprachiastmatic nucleus (SCN) of the mammalian brain using time series data is difficult but crucial to understanding network dynamics. Although techniques have been developed for inferring networks from time series data, there have been no attempts to adapt these techniques to infer directional connections in oscillatory time series, while accurately distinguishing between direct and indirect connections. In this paper an adaptation of Granger Causality is proposed that allows for inference of circadian networks and oscillatory networks in general called Adaptive Frequency Granger Causality (AFGC). Additionally, an extension of this method is proposed to infer networks with large numbers of cells called LASSO AFGC. The method was validated using simulated data from several different networks. For the smaller networks the method was able to identify all one way direct connections without identifying connections that were not present. For larger networks of up to twenty cells the method shows excellent performance in identifying true and false connections; this is quantified by an area-under-the-curve (AUC) 96.88%. We note that this method like other Granger Causality-based methods, is based on the detection of high frequency signals propagating between cell traces. Thus it requires a relatively high sampling rate and a network that can propagate high frequency signals.

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.

Circadian Gating of the Mammalian Cell Cycle Restriction Point: A Mathematical Analysis

A critical decision in the mammalian cell cycle is whether to pass through the restriction point (R-point) or enter the cell cycle. In this letter, we modeled the decision-making system of the mammalian cell cycle entry and the simulated circadian regulation of the R-point driven by external epithelial growth factor (EGF) patterns. Our conceptual model replicated key signaling behaviors observed experimentally, suggesting that the proposed network captured the essential system features. The model revealed the dramatic importance of the EGF dynamics on promoting cell proliferation, showed that the EGF signal duration was more important than the signal strength for driving cells past the R-point, and suggested that the loss of circadian control of the cell cycle entry could be associated with cancer development.

Distinct roles for GABA across multiple timescales in mammalian circadian timekeeping

The suprachiasmatic nuclei (SCN), the central circadian pacemakers in mammals, comprise a multiscale neuronal system that times daily events. We use recent advances in graphics processing unit computing to generate a multiscale model for the SCN that resolves cellular electrical activity down to the timescale of individual action potentials and the intracellular molecular events that generate circadian rhythms. We use the model to study the role of the neurotransmitter GABA in synchronizing circadian rhythms among individual SCN neurons, a topic of much debate in the circadian community. The model predicts that GABA signaling has two components: phasic (fast) and tonic (slow). Phasic GABA postsynaptic currents are released after action potentials, and can both increase or decrease firing rate, depending on their timing in the interspike interval, a modeling hypothesis we experimentally validate; this allows flexibility in the timing of circadian output signals. Phasic GABA, however, does not significantly affect molecular timekeeping. The tonic GABA signal is released when cells become very excited and depolarized; it changes the excitability of neurons in the network, can shift molecular rhythms, and affects SCN synchrony. We measure which neurons are excited or inhibited by GABA across the day and find GABA-excited neurons are synchronized by-and GABA-inhibited neurons repelled from-this tonic GABA signal, which modulates the synchrony in the SCN provided by other signaling molecules. Our mathematical model also provides an important tool for circadian research, and a model computational system for the many multiscale projects currently studying brain function.

Predicted Role of NAD Utilization in the Control of Circadian Rhythms during DNA Damage Response

The circadian clock is a set of regulatory steps that oscillate with a period of approximately 24 hours influencing many biological processes. These oscillations are robust to external stresses, and in the case of genotoxic stress (i.e. DNA damage), the circadian clock responds through phase shifting with primarily phase advancements. The effect of DNA damage on the circadian clock and the mechanism through which this effect operates remains to be thoroughly investigated. Here we build an in silico model to examine damage-induced circadian phase shifts by investigating a possible mechanism linking circadian rhythms to metabolism. The proposed model involves two DNA damage response proteins, SIRT1 and PARP1, that are each consumers of nicotinamide adenine dinucleotide (NAD), a metabolite involved in oxidation-reduction reactions and in ATP synthesis. This model builds on two key findings: 1) that SIRT1 (a protein deacetylase) is involved in both the positive (i.e. transcriptional activation) and negative (i.e. transcriptional repression) arms of the circadian regulation and 2) that PARP1 is a major consumer of NAD during the DNA damage response. In our simulations, we observe that increased PARP1 activity may be able to trigger SIRT1-induced circadian phase advancements by decreasing SIRT1 activity through competition for NAD supplies. We show how this competitive inhibition may operate through protein acetylation in conjunction with phosphorylation, consistent with reported observations. These findings suggest a possible mechanism through which multiple perturbations, each dominant during different points of the circadian cycle, may result in the phase advancement of the circadian clock seen during DNA damage.

Many physiological processes are regulated by the circadian clock, and we are continuing to learn about the role of the circadian clock in disease. Research in recent years has begun to shed light on the feedback mechanisms that exist between circadian regulation and other processes, including metabolism and the response to DNA damage. A challenge has been to understand the dynamic nature of the protein interactions of these processes, which often involve protein modification as a means of communicating cellular states, such as damaged DNA. Here we have devised a model that simulates an alteration of the circadian clock that is observed during DNA damage response. A novel aspect of this model is the inclusion of SIRT1, a protein that regulates core circadian proteins through modification and helps to repress gene expression. SIRT1 is dependent on a metabolite regulated by the circadian clock and is depleted during DNA damage. In conjunction with a second form of protein modification, our results suggest that multiple forms of protein modification may contribute to the experimentally observed alterations to circadian function.

Discerning mechanistically rewired biological pathways by cumulative interaction heterogeneity statistics

Changes in response of a biological pathway could be a consequence of either pathway rewiring, changed input, or a combination of both. Most pathway analysis methods are not designed for mechanistic rewiring such as regulatory element variations. This limits our understanding of biological pathway evolution. Here we present a Q-method to discern whether changed pathway response is caused by mechanistic rewiring of pathways due to evolution. The main innovation is a cumulative pathway interaction heterogeneity statistic accounting for rewiring-specific effects on the rate of change of each molecular variable across conditions. The Q-method remarkably outperformed differential-correlation based approaches on data from diverse biological processes. Strikingly, it also worked well in differentiating rewired chaotic systems, whose dynamics are notoriously difficult to predict. Applying the Q-method on transcriptome data of four yeasts, we show that pathway interaction heterogeneity for known metabolic and signaling pathways is indeed a predictor of interspecies genetic rewiring due to unbalanced TATA box-containing genes among the yeasts. The demonstrated effectiveness of the Q-method paves the way to understanding network evolution at the resolution of functional biological pathways.

Noise facilitates transcriptional control under dynamic inputs

Cells must respond sensitively to time-varying inputs in complex signaling environments. To understand how signaling networks process dynamic inputs into gene expression outputs and the role of noise in cellular information processing, we studied the immune pathway NF-κB under periodic cytokine inputs using microfluidic single-cell measurements and stochastic modeling. We find that NF-κB dynamics in fibroblasts synchronize with oscillating TNF signal and become entrained, leading to significantly increased NF-κB oscillation amplitude and mRNA output compared to non-entrained response. Simulations show that intrinsic biochemical noise in individual cells improves NF-κB oscillation and entrainment, whereas cell-to-cell variability in NF-κB natural frequency creates population robustness, together enabling entrainment over a wider range of dynamic inputs. This wide range is confirmed by experiments where entrained cells were measured under all input periods. These results indicate that synergy between oscillation and noise allows cells to achieve efficient gene expression in dynamically changing signaling environments.

A mammalian circadian clock model incorporating daytime expression elements

Models of the mammalian clock have traditionally been based around two feedback loops-the self-repression of Per/Cry by interfering with activation by BMAL/CLOCK, and the repression of Bmal/Clock by the REV-ERB proteins. Recent experimental evidence suggests that the D-box, a transcription factor binding site associated with daytime expression, plays a larger role in clock function than has previously been understood. We present a simplified clock model that highlights the role of the D-box and illustrate an approach for finding maximum-entropy ensembles of model parameters, given experimentally imposed constraints. Parameter variability can be mitigated using prior probability distributions derived from genome-wide studies of cellular kinetics. Our model reproduces predictions concerning the dual regulation of Cry1 by the D-box and Rev-ErbA/ROR response element (RRE) promoter elements and allows for ensemble-based predictions of phase response curves (PRCs). Nonphotic signals such as Neuropeptide Y (NPY) may act by promoting Cry1 expression, whereas photic signals likely act by stimulating expression from the E/E' box. Ensemble generation with parameter probability restraints reveals more about a model's behavior than a single optimal parameter set.

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

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

An introduction to systems toxicology

Systems toxicology combines novel and historical experimental data to generate increasingly complex models of the biological response to chemical exposure.

Mathematical model of the Tat-Rev regulation of HIV-1 replication in an activated cell predicts the existence of oscillatory dynamics in the synthesis of viral components

Background

The life cycle of human immunodeficiency virus type-1 (HIV-1) makes possible the realization of regulatory strategies that can lead to complex dynamical behavior of the system. We analyze the strategy which is based on two feedback mechanisms, one mediating a positive regulation of the virus replication by Tat protein via the antitermination of the genomic RNAs transcription on TAR (transactivation responsive) element of the proviral DNA and the second mechanism providing a negative regulation of the splicing of the full-length (9 kb) RNAs and incompletely spliced (4 kb) RNAs via their transport from the nucleus to the cytoplasm. Although the existence of these two regulatory feedback loops has been considered in other mathematical models, none of them examined the conditions for the emergence of complex oscillatory patterns in the intracellular dynamics of viral components.

Results

We developed a mechanistic mathematical model for the Tat-Rev mediated regulation of HIV-1 replication, which considers the activation of proviral DNA transcription, the Tat-specific antitermination of transcription on TAR-element, resulting in the synthesis of the full-length 9 kb RNA, the splicing of the 9 kb RNA down to the 4 kb RNA and the 4 kb RNA to 2 kb RNA, the transport of 2 kb mRNAs from the nucleus to the cytoplasm by the intracellular mechanisms, the multiple binding of the Rev protein to RRE (Rev Response Element) sites on 9 kb and 4 kb RNA resulting in their export to the cytoplasm and the synthesis of Tat and Rev proteins in the cytoplasm followed by their transport into the nucleus. The degradation of all viral proteins and RNAs both in the cytoplasm and the nucleus is described. The model parameters values were derived from the published literature data. The model was used to examine the dynamics of the synthesis of the viral proteins Tat and Rev, the mRNAs under the intracellular conditions specific for activated HIV-1 infected macrophages. In addition, we analyzed alternative hypotheses for the re-cycling of the Rev proteins both in the cytoplasm and the nuclear pore complex.

Conclusions

The quantitative mathematical model of the Tat-Rev regulation of HIV-1 replication predicts the existence of oscillatory dynamics which depends on the efficacy of the Tat and TAR interaction as well as on the Rev-mediated transport processes. The biological relevance of the oscillatory regimes for the HIV-1 life cycle is discussed.

Analytical study of robustness of a negative feedback oscillator by multiparameter sensitivity

Background

One of the distinctive features of biological oscillators such as circadian clocks and cell cycles is robustness which is the ability to resume reliable operation in the face of different types of perturbations. In the previous study, we proposed multiparameter sensitivity (MPS) as an intelligible measure for robustness to fluctuations in kinetic parameters. Analytical solutions directly connect the mechanisms and kinetic parameters to dynamic properties such as period, amplitude and their associated MPSs. Although negative feedback loops are known as common structures to biological oscillators, the analytical solutions have not been presented for a general model of negative feedback oscillators.

Results

We present the analytical expressions for the period, amplitude and their associated MPSs for a general model of negative feedback oscillators. The analytical solutions are validated by comparing them with numerical solutions. The analytical solutions explicitly show how the dynamic properties depend on the kinetic parameters. The ratio of a threshold to the amplitude has a strong impact on the period MPS. As the ratio approaches to one, the MPS increases, indicating that the period becomes more sensitive to changes in kinetic parameters. We present the first mathematical proof that the distributed time-delay mechanism contributes to making the oscillation period robust to parameter fluctuations. The MPS decreases with an increase in the feedback loop length (i.e., the number of molecular species constituting the feedback loop).

Conclusions

Since a general model of negative feedback oscillators was employed, the results shown in this paper are expected to be true for many of biological oscillators. This study strongly supports that the hypothesis that phosphorylations of clock proteins contribute to the robustness of circadian rhythms. The analytical solutions give synthetic biologists some clues to design gene oscillators with robust and desired period.

Amplitude metrics for cellular circadian bioluminescence reporters

Bioluminescence rhythms from cellular reporters have become the most common method used to quantify oscillations in circadian gene expression. These experimental systems can reveal phase and amplitude change resulting from circadian disturbances, and can be used in conjunction with mathematical models to lend further insight into the mechanistic basis of clock amplitude regulation. However, bioluminescence experiments track the mean output from thousands of noisy, uncoupled oscillators, obscuring the direct effect of a given stimulus on the genetic regulatory network. In many cases, it is unclear whether changes in amplitude are due to individual changes in gene expression level or to a change in coherence of the population. Although such systems can be modeled using explicit stochastic simulations, these models are computationally cumbersome and limit analytical insight into the mechanisms of amplitude change. We therefore develop theoretical and computational tools to approximate the mean expression level in large populations of noninteracting oscillators, and further define computationally efficient amplitude response calculations to describe phase-dependent amplitude change. At the single-cell level, a mechanistic nonlinear ordinary differential equation model is used to calculate the transient response of each cell to a perturbation, whereas population-level dynamics are captured by coupling this detailed model to a phase density function. Our analysis reveals that amplitude changes mediated at either the individual-cell or the population level can be distinguished in tissue-level bioluminescence data without the need for single-cell measurements. We demonstrate the effectiveness of the method by modeling experimental bioluminescence profiles of light-sensitive fibroblasts, reconciling the conclusions of two seemingly contradictory studies. This modeling framework allows a direct comparison between in vitro bioluminescence experiments and in silico ordinary differential equation models, and will lead to a better quantitative understanding of the factors that affect clock amplitude.

Improving the accuracy of expression data analysis in time course experiments using resampling

Background

As time series experiments in higher eukaryotes usually obtain data from different individuals collected at the different time points, a time series sample itself is not equivalent to a true biological replicate but is, rather, a combination of several biological replicates. The analysis of expression data derived from a time series sample is therefore often performed with a low number of replicates due to budget limitations or limitations in sample availability. In addition, most algorithms developed to identify specific patterns in time series dataset do not consider biological variation in samples collected at the same conditions.

Results

Using artificial time course datasets, we show that resampling considerably improves the accuracy of transcripts identified as rhythmic. In particular, the number of false positives can be greatly reduced while at the same time the number of true positives can be maintained in the range of other methods currently used to determine rhythmically expressed genes.

Conclusions

The resampling approach described here therefore increases the accuracy of time series expression data analysis and furthermore emphasizes the importance of biological replicates in identifying oscillating genes. Resampling can be used for any time series expression dataset as long as the samples are acquired from independent individuals at each time point.

Electronic supplementary material

The online version of this article (doi:10.1186/s12859-014-0352-8) contains supplementary material, which is available to authorized users.

A model for the fast synchronous oscillations of firing rate in rat suprachiasmatic nucleus neurons cultured in a multielectrode array dish

When dispersed and cultured in a multielectrode dish (MED), suprachiasmatic nucleus (SCN) neurons express fast oscillations of firing rate (FOFR; fast relative to the circadian cycle), with burst duration ∼10 min, and interburst interval varying from 20 to 60 min in different cells but remaining nevertheless rather regular in individual cells. In many cases, separate neurons in distant parts of the 1 mm recording area of a MED exhibited correlated FOFR. Neither the mechanism of FOFR nor the mechanism of their synchronization among neurons is known. Based on recent data implicating vasoactive intestinal polypeptide (VIP) as a key intercellular synchronizing agent, we built a model in which VIP acts as both a feedback regulator to generate FOFR in individual neurons, and a diffusible synchronizing agent to produce coherent electrical output of a neuronal network. In our model, VIP binding to its (VPAC₂) receptors acts through G_s G-proteins to activate adenylyl cyclase (AC), increase intracellular cAMP, and open cyclic-nucleotide-gated (CNG) cation channels, thus depolarizing the cell and generating neuronal firing to release VIP. In parallel, slowly developing homologous desensitization and internalization of VPAC₂ receptors terminates elevation of cAMP and thereby provides an interpulse silent interval. Through mathematical modeling, we show that this VIP/VPAC₂/AC/cAMP/CNG-channel mechanism is sufficient for generating reliable FOFR in single neurons. When our model for FOFR is combined with a published model of synchronization of circadian rhythms based on VIP/VPAC₂ and Per gene regulation synchronization of circadian rhythms is significantly accelerated. These results suggest that (a) auto/paracrine regulation by VIP/VPAC₂ and intracellular AC/cAMP/CNG-channels are sufficient to provide robust FOFR and synchrony among neurons in a heterogeneous network, and (b) this system may also participate in synchronization of circadian rhythms.

Brain clock driven by neuropeptides and second messengers

The master circadian pacemaker in mammals is localized in a small portion of the brain called the suprachiasmatic nucleus (SCN). It is unclear how the SCN produces circadian rhythms. A common interpretation is that the SCN produces oscillations through the coupling of genetic oscillators in the neurons. The coupling is effected by a network of neuropeptides and second messengers. This network is crucial for the correct function of the SCN. However, models that study a possible oscillatory behavior of the network itself have received little attention. Here we propose and analyze a model to examine this oscillatory potential. We show that an intercellular oscillator emerges in the SCN as a result of the neuropeptide and second messenger dynamics. We find that this intercellular clock can produce circadian rhythms by itself with and without genetic clocks. We also found that the model is robust to perturbation of parameters and can be entrained by light-dark cycles.

The dynamics of NF-κB pathway regulated by circadian clock

The circadian clock regulates many physiological parameters involving immune response to infectious agents, which is mediated by activation of the transcription factor NF-κB. Thus, understanding the NF-κB dynamics regulated by circadian clocks will help in developing better therapeutics. To this end, we proposed a detailed model in the present work on the basis of understanding inflammatory response under control from circadian clocks. Our results show that the frequencies and amplitudes of the NF-κB oscillation are dependent on the strength and modes of coupling to circadian clock. This circadian control of NF-κB pathway can therefore serve as a useful mechanism in keeping the system in check and controlling inflammatory response induced by infection and other agents. The results are consistent with earlier experimental findings.

Positive feedback promotes oscillations in negative feedback loops

A simple three-component negative feedback loop is a recurring motif in biochemical oscillators. This motif oscillates as it has the three necessary ingredients for oscillations: a three-step delay, negative feedback, and nonlinearity in the loop. However, to oscillate, this motif under the common Goodwin formulation requires a high degree of cooperativity (a measure of nonlinearity) in the feedback that is biologically “unlikely.” Moreover, this recurring negative feedback motif is commonly observed augmented by positive feedback interactions. Here we show that these positive feedback interactions promote oscillation at lower degrees of cooperativity, and we can thus unify several common kinetic mechanisms that facilitate oscillations, such as self-activation and Michaelis-Menten degradation. The positive feedback loops are most beneficial when acting on the shortest lived component, where they function by balancing the lifetimes of the different components. The benefits of multiple positive feedback interactions are cumulative for a majority of situations considered, when benefits are measured by the reduction in the cooperativity required to oscillate. These positive feedback motifs also allow oscillations with longer periods than that determined by the lifetimes of the components alone. We can therefore conjecture that these positive feedback loops have evolved to facilitate oscillations at lower, kinetically achievable, degrees of cooperativity. Finally, we discuss the implications of our conclusions on the mammalian molecular clock, a system modeled extensively based on the three-component negative feedback loop.

An intensity ratio of interlocking loops determines circadian period length

Circadian clocks allow organisms to orchestrate the daily rhythms in physiology and behaviors, and disruption of circadian rhythmicity can profoundly affect fitness. The mammalian circadian oscillator consists of a negative primary feedback loop and is associated with some 'auxiliary' loops. This raises the questions of how these interlocking loops coordinate to regulate the period and maintain its robustness. Here, we focused on the REV-ERBα/Cry1 auxiliary loop, consisting of Rev-Erbα/ROR-binding elements (RORE) mediated Cry1 transcription, coordinates with the negative primary feedback loop to modulate the mammalian circadian period. The silicon simulation revealed an unexpected rule: the intensity ratio of the primary loop to the auxiliary loop is inversely related to the period length, even when post-translational feedback is fixed. Then we measured the mRNA levels from two loops in 10-mutant mice and observed the similar monotonic relationship. Additionally, our simulation and the experimental results in human osteosarcoma cells suggest that a coupling effect between the numerator and denominator of this intensity ratio ensures the robustness of circadian period and, therefore, provides an efficient means of correcting circadian disorders. This ratio rule highlights the contribution of the transcriptional architecture to the period dynamics and might be helpful in the construction of synthetic oscillators.

The bird circadian clock: insights from a computational model

The circadian timekeeping system appears more complex in birds than in mammals. In mammals, the main pacemaker is centralized in the suprachiasmatic nuclei, whereas in birds, the pacemaker involves the interplay between the pineal and hypothalamic oscillators. In order to investigate the consequence of this complex mechanism, we propose here a mathematical model for the bird circadian clock. The model is based on the internal resonance between the pineal and hypothalamic oscillators, each described by Goodwin-like equations. We show that, consistently with experimental observations, self-sustained oscillations can be generated by mutual inhibitory coupling of the 2 clocks, even if individual oscillators present damped oscillations. We study the effect of constant and periodic administrations of melatonin, which, in intact birds, acts as the coupling variable between the pineal and the hypothalamus, and compare the prediction of the model with the experiments performed in pinealectomized birds. We also assess the entrainment properties when the system is subject to light-dark cycles. Analyses of the entrainment range, resynchronization time after jet lag, and entrainment phase with respect to the photoperiod lead us to formulate hypotheses about the physiological advantage of the particular architecture of the avian circadian clock. Although minimal, our model opens promising perspectives in modeling and understanding the bird circadian clock.