The interplay of cis-regulatory elements rules circadian rhythms in mouse liver

Analytic solutions for the circadian oscillator characterize cycle dynamics and its robustness

Circadian clocks form a fundamental mechanism that promotes the correct behavior of many cellular and molecular processes by synchronizing them on a 24 h period. However, the circadian cycles remain difficult to describe mathematically. To overcome this problem, we first propose a segmentation of the circadian cycle into eight stages based on the levels of expression of the core clock components CLOCK:BMAL1, REV-ERB and PER:CRY. This cycle segmentation is next characterized through a piecewise affine model, whose analytical study allows us to propose an Algorithm to generate biologically-consistent circadian oscillators. Our study provides a characterization of the cycle dynamics in terms of four fundamental threshold parameters and one scaling parameter, shows robustness of the circadian system and its period, and identifies critical points for correct cycle progression.

Impacts of the feedback loop between sense-antisense RNAs in regulating circadian rhythms

Antisense transcripts are a unique group of non-coding RNAs and play regulatory roles in a variety of biological processes, including circadian rhythms. Per2AS is an antisense transcript to the sense core clock gene Period2 (Per2) in mouse and its expression is rhythmic and antiphasic to Per2. To understand the impact of Per2AS-Per2 interaction, we developed a new mathematical model that mechanistically described the mutually repressive relationship between Per2 and Per2AS. This mutual repression can regulate both amplitude and period of circadian oscillation by affecting a negative feedback regulation of Per2. Simulations from this model also fit with experimental observations that could not be fully explained by our previous model. Our revised model can not only serve as a foundation to build more detailed models to better understand the impact of Per2AS-Per2 interaction in the future, but also be used to analyze other sense-antisense RNA pairs that mutually repress each other.

Quantification of circadian rhythms in mammalian lung tissue snapshot data

Healthy mammalian cells have a circadian clock, a gene regulatory network that allows them to schedule their physiological processes to optimal times of the day. When healthy cells turn into cancer cells, the circadian clock often becomes cancer specifically disturbed, so there is an interest in the extraction of circadian features from gene expression data of cancer. This is challenging, as clinical gene expression samples of cancer are snapshot-like and the circadian clock is best examined using gene expression time series. In this study, we obtained lists of intersecting circadian genes in public gene expression time series data of lung tissue of mouse and baboon. We base our circadian gene lists on correlations of gene expression levels of circadian genes, which are closely associated to the phase differences between them. Combining circadian gene expression patterns of diurnal and nocturnal species of different ages provides circadian genes that are also important in healthy and cancerous human lung tissue. We tested the quality of the representation of the circadian clock in our gene lists by PCA-based reconstructions of the circadian times of the mouse and baboon samples. Then we assigned potential circadian times to the human lung tissue samples and find an intact circadian clock in the healthy human lung tissue, but an altered, weak clock in the adjacent cancerous lung tissue.

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.

A mathematical model for the role of dopamine-D2 self-regulation in the production of ultradian rhythms

Many self-motivated and goal-directed behaviours display highly flexible, approximately 4 hour ultradian (shorter than a day) oscillations. Despite lacking direct correspondence to physical cycles in the environment, these ultradian rhythms may be involved in optimizing functional interactions with the environment and reflect intrinsic neural dynamics. Current evidence supports a role of mesostriatal dopamine (DA) in the expression and propagation of ultradian rhythmicity, however, the biochemical processes underpinning these oscillations remain to be identified. Here, we use a mathematical model to investigate D2 autoreceptor-dependent DA self-regulation as the source of ultradian behavioural rhythms. DA concentration at the midbrain-striatal synapses is governed through a dual-negative feedback-loop structure, which naturally gives rise to rhythmicity. This model shows the propensity of striatal DA to produce an ultradian oscillation characterized by a flexible period that is highly sensitive to parameter variations. Circadian (approximately 24 hour) regulation consolidates the ultradian oscillations and alters their response to the phase-dependent, rapid-resetting effect of a transient excitatory stimulus. Within a circadian framework, the ultradian rhythm orchestrates behavioural activity and enhances responsiveness to an external stimulus. This suggests a role for the circadian-ultradian timekeeping hierarchy in governing organized behaviour and shaping daily experience through coordinating the motivation to engage in recurring, albeit not highly predictable events, such as social interactions.

Network switches and their role in circadian clocks

Circadian rhythms are generated by complex interactions among genes and proteins. Self-sustained ∼24 h oscillations require negative feedback loops and sufficiently strong nonlinearities that are the product of molecular and network switches. Here, we review common mechanisms to obtain switch-like behavior, including cooperativity, antagonistic enzymes, multisite phosphorylation, positive feedback, and sequestration. We discuss how network switches play a crucial role as essential components in cellular circadian clocks, serving as integral parts of transcription-translation feedback loops that form the basis of circadian rhythm generation. The design principles of network switches and circadian clocks are illustrated by representative mathematical models that include bistable systems and negative feedback loops combined with Hill functions. This work underscores the importance of negative feedback loops and network switches as essential design principles for biological oscillations, emphasizing how an understanding of theoretical concepts can provide insights into the mechanisms generating biological rhythms.

Impacts of the feedback loop between sense-antisense RNAs in regulating circadian rhythms

Antisense transcripts are a unique group of non-coding RNAs that are transcribed from the opposite strand of a sense coding gene in an antisense orientation. Even though they do not encode a protein, these transcripts play a regulatory role in a variety of biological processes, including circadian rhythms. We and others found an antisense transcript, Per2AS , that is transcribed from the strand opposite the sense transcript Period2 ( Per2 ) and exhibits a rhythmic and antiphasic expression pattern compared to Per2 in mouse. By assuming that Per2AS and Per2 mutually repress each other, our previous mathematical model predicted that Per2AS regulates the robustness and the amplitude of circadian rhythms. In this study, we revised our previous model and developed a new mathematical model that mechanistically described the mutually repressive relationship between Per2 and Per2AS via transcriptional interference. We found that the simulation results are largely consistent with experimental observations including the counterintuitive ones that could not be fully explained by our previous model. These results indicate that our revised model serves as a foundation to build more detailed models in the future to better understand the impact of Per2AS-Per2 interaction in the mammalian circadian clock. Our mechanistic description of Per2AS-Per2 interaction can also be extended to other mathematical models that involve sense-antisense RNA pairs that mutually repress each other.

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 SCN-HPA-Periphery Circadian Timing System: Mathematical Modeling of Clock Synchronization and the Effects of Photoperiod on Jetlag Adaptation

Synchronizing the circadian timing system (CTS) to external light/dark cycles is crucial for homeostasis maintenance and environmental adaptation. The CTS is organized hierarchically, with the central pacemaker located in the suprachiasmatic nuclei (SCN) generating coherent oscillations that are entrained to light/dark cycles. These oscillations regulate the release of glucocorticoids by the hypothalamus-pituitary-adrenal (HPA) axis, which acts as a systemic entrainer of peripheral clocks throughout the body. The SCN adjusts its network plasticity in response to variations in photoperiod, leading to changes in the rhythmic release of glucocorticoids and ultimately impacting peripheral clocks. However, the effects of photoperiod-induced variations of glucocorticoids on the synchronization of peripheral clocks are not fully understood, and the interaction between jetlag adaption and photoperiod changes is unclear. This study presents a semi-mechanistic mathematical model to investigate how the CTS responds to changes in photoperiod. Specifically, the study focuses on the entrainment properties of a system composed of the SCN, HPA axis, and peripheral clocks. The results show that high-amplitude glucocorticoid rhythms lead to a more coherent phase distribution in the periphery. In addition, our study investigates the effect of photoperiod exposure on jetlag recovery time and phase shift, proposing different interventional strategies for eastward and westward jetlag. The findings suggest that decreasing photic exposure before jetlag during eastward traveling and after jetlag during westward traveling can accelerate jetlag readaptation. The study provides insights into the mechanisms of CTS organization and potential recovery strategies for transitions between time zones and lighting zones.

CRS: a circadian rhythm score model for predicting prognosis and treatment response in cancer patients

BACKGROUND

Circadian rhythm regulates complex physiological activities in organisms. A strong link between circadian dysfunction and cancer has been identified. However, the factors of dysregulation and functional significance of circadian rhythm genes in cancer have received little attention.

METHODS

In 18 cancer types from The Cancer Genome Atlas (TCGA), the differential expression and genetic variation of 48 circadian rhythm genes (CRGs) were examined. The circadian rhythm score (CRS) model was created using the ssGSEA method, and patients were divided into high and low groups based on the CRS. The Kaplan-Meier curve was created to assess the patient survival rate. Cibersort and estimate methods were used to identify the infiltration characteristics of immune cells between different CRS subgroups. Gene Expression Omnibus (GEO) dataset is used as verification queue and model stability evaluation queue. The CRS model's ability to predict chemotherapy and immunotherapy was assessed. Wilcoxon rank-sum test was used to compare the differences of CRS among different patients. We use CRS to identify potential "clock-drugs" by the connective map method.

RESULTS

Transcriptomic and genomic analyses of 48 CRGs revealed that most core clock genes are up-regulated, while clock control genes are down-regulated. Furthermore, we show that copy number variation may affect CRGs aberrations. Based on CRS, patients can be classified into two groups with significant differences in survival and immune cell infiltration. Further studies showed that patients with low CRS were more sensitive to chemotherapy and immunotherapy. Additionally, we identified 10 compounds (e.g. flubendazole, MLN-4924, ingenol) that are positively associated with CRS, and have the potential to modulate circadian rhythms.

CONCLUSIONS

CRS can be utilized as a clinical indicator to predict patient prognosis and responsiveness to therapy, and identify potential "clock-drugs".

Data-driven modelling captures dynamics of the circadian clock of Neurospora crassa

Eukaryotic circadian clocks are based on self-sustaining, cell-autonomous oscillatory feedback loops that can synchronize with the environment via recurrent stimuli (zeitgebers) such as light. The components of biological clocks and their network interactions are becoming increasingly known, calling for a quantitative understanding of their role for clock function. However, the development of data-driven mathematical clock models has remained limited by the lack of sufficiently accurate data. Here we present a comprehensive model of the circadian clock of Neurospora crassa that describe free-running oscillations in constant darkness and entrainment in light-dark cycles. To parameterize the model, we measured high-resolution time courses of luciferase reporters of morning and evening specific clock genes in WT and a mutant strain. Fitting the model to such comprehensive data allowed estimating parameters governing circadian phase, period length and amplitude, and the response of genes to light cues. Our model suggests that functional maturation of the core clock protein Frequency causes a delay in negative feedback that is critical for generating circadian rhythms.

Circadian rhythms are endogenous autonomous clocks that emancipate daily rhythms in physiology and behavior. Lately, a large body of research has contributed to our understanding of clocks’ genetic and mechanistic basis across kingdoms of life, i.e., mammals, fungi, plants, and bacteria. Several mathematical models have made key contributions to our current understanding of the design principles of the Neurospora crassa circadian clock and conditions for self-sustained oscillations. However, previous models uncovered and described the principle properties of the clock in generic manner due to a lack of experimental data. In this study, we developed a mathematical model based on systems of differential equations to describe the core clock components and estimated model parameters from luciferase data that capture experimental observations. We demonstrate the model predictive control simulation emphasizing the importance of functional maturation of the core clock protein Frequency in generating circadian rhythms.

Cycle dynamics and synchronization in a coupled network of peripheral circadian clocks

The intercellular interactions between peripheral circadian clocks, located in tissues and organs other than the suprachiasmatic nuclei of the hypothalamus, are still very poorly understood. We propose a theoretical and computational study of the coupling between two or more clocks, using a calibrated, reduced model of the circadian clock to describe some synchronization properties between peripheral cellular clocks. Based on a piecewise linearization of the dynamics of the mutual CLOCK:BMAL1/PER:CRY inactivation term, we suggest a segmentation of the circadian cycle into six stages, to help analyse different types of synchronization between two clocks, including single stage duration, total period and maximal amplitudes. Finally, our model reproduces some recent experimental results on the effects of different regimes of time-restricted feeding in liver circadian clocks of mice.

Synergies of Multiple Zeitgebers Tune Entrainment

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

Ultradian rhythms of AKT phosphorylation and gene expression emerge in the absence of the circadian clock components Per1 and Per2

Rhythmicity of biological processes can be elicited either in response to environmental cycles or driven by endogenous oscillators. In mammals, the circadian clock drives about 24-hour rhythms of multitude metabolic and physiological processes in anticipation to environmental daily oscillations. Also at the intersection of environment and metabolism is the protein kinase—AKT. It conveys extracellular signals, primarily feeding-related signals, to regulate various key cellular functions. Previous studies in mice identified rhythmicity in AKT activation (pAKT) with elevated levels in the fed state. However, it is still unknown whether rhythmic AKT activation can be driven through intrinsic mechanisms. Here, we inspected temporal changes in pAKT levels both in cultured cells and animal models. In cultured cells, pAKT levels showed circadian oscillations similar to those observed in livers of wild-type mice under free-running conditions. Unexpectedly, in livers of Per1,2^−/− but not of Bmal1^−/− mice we detected ultradian (about 16 hours) oscillations of pAKT levels. Importantly, the liver transcriptome of Per1,2^−/− mice also showed ultradian rhythms, corresponding to pAKT rhythmicity and consisting of AKT-related genes and regulators. Overall, our findings reveal ultradian rhythms in liver gene expression and AKT phosphorylation that emerge in the absence of environmental rhythms and Per1,2^−/− genes.

This study reveals ultradian (16-hour) rhythms in the activation of the protein kinase AKT in the livers of mice, accompanied by corresponding downstream changes in gene expression. Intriguingly, these oscillations emerge in the absence of rhythmic environmental cues and in mice lacking the circadian clock proteins Per1 and Per2.

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

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

Mathematical modeling of mammalian circadian clocks affecting drug and disease responses

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

Circadian Misalignment and Metabolic Disorders: A Story of Twisted Clocks

Simple Summary

In mammals, many physiological processes follow a 24 h rhythmic pattern. These rhythms are governed by a complex network of circadian clocks, which perceives external time cues (notably light and nutrients) and adjusts the timing of metabolic and physiological functions to allow a proper adaptation of the organism to the daily changes in the environmental conditions. Circadian rhythms originate at the cellular level through a transcriptional–translational regulatory network involving a handful of clock genes. In this review, we show how adverse effects caused by ill-timed feeding or jet lag can lead to a dysregulation of this genetic clockwork, which in turn results in altered metabolic regulation and possibly in diseases. We also show how computational modeling can complement experimental observations to understand the design of the clockwork and the onset of metabolic disorders.

Abstract

Biological clocks are cell-autonomous oscillators that can be entrained by periodic environmental cues. This allows organisms to anticipate predictable daily environmental changes and, thereby, to partition physiological processes into appropriate phases with respect to these changing external conditions. Nowadays our 24/7 society challenges this delicate equilibrium. Indeed, many studies suggest that perturbations such as chronic jet lag, ill-timed eating patterns, or shift work increase the susceptibility to cardiometabolic disorders, diabetes, and cancers. However the underlying mechanisms are still poorly understood. A deeper understanding of this complex, dynamic system requires a global holistic approach for which mathematical modeling can be highly beneficial. In this review, we summarize several experimental works pertaining to the effect of adverse conditions on clock gene expression and on physiology, and we show how computational models can bring interesting insights into the links between circadian misalignment and metabolic diseases.

Incorporating age and delay into models for biophysical systems

In many biological systems, chemical reactions or changes in a physical state are assumed to occur instantaneously. For describing the dynamics of those systems, Markov models that require exponentially distributed inter-event times have been used widely. However, some biophysical processes such as gene transcription and translation are known to have a significant gap between the initiation and the completion of the processes, which renders the usual assumption of exponential distribution untenable. In this paper, we consider relaxing this assumption by incorporating age-dependent random time delays (distributed according to a given probability distribution) into the system dynamics. We do so by constructing a measure-valued Markov process on a more abstract state space, which allows us to keep track of the 'ages' of molecules participating in a chemical reaction. We study the large-volume limit of such age-structured systems. We show that, when appropriately scaled, the stochastic system can be approximated by a system of partial differential equations (PDEs) in the large-volume limit, as opposed to ordinary differential equations (ODEs) in the classical theory. We show how the limiting PDE system can be used for the purpose of further model reductions and for devising efficient simulation algorithms. In order to describe the ideas, we use a simple transcription process as a running example. We, however, note that the methods developed in this paper apply to a wide class of biophysical systems.

Simple Kinetic Models in Molecular Chronobiology

Circadian rhythms are constituted by a complex dynamical system with intertwined feedback loops, molecular switches, and self-sustained oscillations. Mathematical modeling supports understanding available heterogeneous kinetic data, highlights basic mechanisms, and can guide experimental research. Here, we introduce the basic steps from a biological question to simple models providing insight into gene-regulatory mechanisms. We illustrate the general approach by three examples: modeling decay processes, clock-controlled genes, and self-sustained oscillations.

Multiple random phosphorylations in clock proteins provide long delays and switches

Theory predicts that self-sustained oscillations require robust delays and nonlinearities (ultrasensitivity). Delayed negative feedback loops with switch-like inhibition of transcription constitute the core of eukaryotic circadian clocks. The kinetics of core clock proteins such as PER2 in mammals and FRQ in Neurospora crassa is governed by multiple phosphorylations. We investigate how multiple, slow and random phosphorylations control delay and molecular switches. We model phosphorylations of intrinsically disordered clock proteins (IDPs) using conceptual models of sequential and distributive phosphorylations. Our models help to understand the underlying mechanisms leading to delays and ultrasensitivity. The model shows temporal and steady state switches for the free kinase and the phosphoprotein. We show that random phosphorylations and sequestration mechanisms allow high Hill coefficients required for self-sustained oscillations.

Methods detecting rhythmic gene expression are biologically relevant only for strong signal

The nycthemeral transcriptome embodies all genes displaying a rhythmic variation of their mRNAs periodically every 24 hours, including but not restricted to circadian genes. In this study, we show that the nycthemeral rhythmicity at the gene expression level is biologically functional and that this functionality is more conserved between orthologous genes than between random genes. We used this conservation of the rhythmic expression to assess the ability of seven methods (ARSER, Lomb Scargle, RAIN, JTK, empirical-JTK, GeneCycle, and meta2d) to detect rhythmic signal in gene expression. We have contrasted them to a naive method, not based on rhythmic parameters. By taking into account the tissue-specificity of rhythmic gene expression and different species comparisons, we show that no method is strongly favored. The results show that these methods designed for rhythm detection, in addition to having quite similar performances, are consistent only among genes with a strong rhythm signal. Rhythmic genes defined with a standard p-value threshold of 0.01 for instance, could include genes whose rhythmicity is biologically irrelevant. Although these results were dependent on the datasets used and the evolutionary distance between the species compared, we call for caution about the results of studies reporting or using large sets of rhythmic genes. Furthermore, given the analysis of the behaviors of the methods on real and randomized data, we recommend using primarily ARS, empJTK, or GeneCycle, which verify expectations of a classical distribution of p-values. Experimental design should also take into account the circumstances under which the methods seem more efficient, such as giving priority to biological replicates over the number of time-points, or to the number of time-points over the quality of the technique (microarray vs RNAseq). GeneCycle, and to a lesser extent empirical-JTK, might be the most robust method when applied to weakly informative datasets. Finally, our analyzes suggest that rhythmic genes are mainly highly expressed genes.

To be active, genes have to be transcribed to RNA. For some genes, the transcription rate follows a circadian rhythm with a periodicity of approximately 24 hours; we call these genes “rhythmic”. In this study, we compared methods designed to detect rhythmic genes in gene expression data. The data are measures of the number of RNA molecules for each gene, given at several time-points, usually spaced 2 to 4 hours, over one or several periods of 24 hours. There are many such methods, but it is not known which ones work best to detect genes whose rhythmic expression is biologically functional. We compared these methods using a reference group of evolutionarily conserved rhythmic genes. We compared data from baboon, mouse, rat, zebrafish, fly, and mosquitoes. Surprisingly, no method was particularly effective. Furthermore, we found that only very strong rhythmic signals were relevant with each method. More precisely, when we use a usual cut-off to define rhythmic genes, the group of genes considered as rhythmic contains many genes whose rhythmicity cannot be confirmed to be biologically relevant. We also show that rhythmic genes mainly contain highly expressed genes. Finally, based on our results, we provide recommendations on which methods to use and how, and suggestions for future experimental designs.

Filtering and inference for stochastic oscillators with distributed delays

Motivation

The time evolution of molecular species involved in biochemical reaction networks often arises from complex stochastic processes involving many species and reaction events. Inference for such systems is profoundly challenged by the relative sparseness of experimental data, as measurements are often limited to a small subset of the participating species measured at discrete time points. The need for model reduction can be realistically achieved for oscillatory dynamics resulting from negative translational and transcriptional feedback loops by the introduction of probabilistic time-delays. Although this approach yields a simplified model, inference is challenging and subject to ongoing research. The linear noise approximation (LNA) has recently been proposed to address such systems in stochastic form and will be exploited here.

Results

We develop a novel filtering approach for the LNA in stochastic systems with distributed delays, which allows the parameter values and unobserved states of a stochastic negative feedback model to be inferred from univariate time-series data. The performance of the methods is tested for simulated data. Results are obtained for real data when the model is fitted to imaging data on Cry1, a key gene involved in the mammalian central circadian clock, observed via a luciferase reporter construct in a mouse suprachiasmatic nucleus.

Availability and implementation

Programmes are written in MATLAB and Statistics Toolbox Release 2016 b, The MathWorks, Inc., Natick, Massachusetts, USA. Sample code and Cry1 data are available on GitHub https://github.com/scalderazzo/FLNADD.

Supplementary information

Supplementary data are available at Bioinformatics online.

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.

Transcription-based circadian mechanism controls the duration of molecular clock states in response to signaling inputs

The molecular oscillator of the mammalian circadian clock consists in a dynamical network of genes and proteins whose main regulatory mechanisms occur at the transcriptional level. From a dynamical point of view, the mechanisms leading to an oscillatory solution with an orderly protein peak expression and a clear day/night phase distinction remain unclear. Our goal is to identify the essential interactions needed to generate phase opposition between the activating CLOCK:BMAL1 and the repressing PER:CRY complexes and to better distinguish these two main clock molecular phases relating to rest/activity and fast/feeding cycles. To do this, we develop a transcription-based mathematical model centered on linear combinations of the clock controlled elements (CCEs): E-box, R-box and D-box. Each CCE is responsive to activators and repressors. After model calibration with single-cell data, we explore entrainment and period tuning via interplay with metabolism. Variation of the PER degradation rate γ_p, relating to the tau mutation, results in asymmetric changes in the duration of the different clock molecular phases. Time spent at the state of high PER/PER:CRY decreases with γ_p, while time spent at the state of high BMAL1 and CRY1, both proteins with activity in promoting insulin sensitivity, remains constant. This result suggests a possible mechanism behind the altered metabolism of tau mutation animals. Furthermore, we expose the clock system to two regulatory inputs, one relating to the fast/feeding cycle and the other to the light-dependent synchronization signaling. We observe the phase difference between these signals to also affect the relative duration of molecular clock states. Simulated circadian misalignment, known to correlate with insulin resistance, leads to decreased duration of BMAL1 expression. Our results reveal a possible mechanism for clock-controlled metabolic homeostasis, whereby the circadian clock controls the relative duration of different molecular (and metabolic) states in response to signaling inputs.

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

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

Translational thresholds in a core circadian clock model

Organisms have evolved in a daily cyclic environment, developing circadian cell-autonomous clocks that temporally organize a wide range of biological processes. Translation is a highly regulated process mainly associated with the activity of microRNAs (miRNAs) at the translation initiation step that impacts on the molecular circadian clock dynamics. Recently, a molecular titration mechanism was proposed to explain the interactions between some miRNAs and their target mRNAs; new evidence also indicates that regulation by miRNA is a nonlinear process such that there is a threshold level of target mRNA below which protein production is drastically repressed. These observations led us to use a theoretical model of the circadian molecular clock to study the effect of miRNA-mediated translational thresholds on the molecular clock dynamics. We model the translational threshold by introducing a phenomenological Hill equation for the kinetics of PER translation and show how the parameters associated with translation kinetics affect the period, amplitude, and time delays between clock mRNA and clock protein expression. We show that our results are useful for analyzing experiments related to the translational regulation of negative elements of transcriptional-translational feedback loops. We also provide new elements for thinking about the translational threshold as a mechanism that favors the emergence of circadian rhythmicity, the tuning of the period-delay relationship and the cell capacity to control the protein oscillation amplitude with almost negligible changes in the mRNA amplitudes.

Bayesian inference of distributed time delay in transcriptional and translational regulation

MOTIVATION

Advances in experimental and imaging techniques have allowed for unprecedented insights into the dynamical processes within individual cells. However, many facets of intracellular dynamics remain hidden, or can be measured only indirectly. This makes it challenging to reconstruct the regulatory networks that govern the biochemical processes underlying various cell functions. Current estimation techniques for inferring reaction rates frequently rely on marginalization over unobserved processes and states. Even in simple systems this approach can be computationally challenging, and can lead to large uncertainties and lack of robustness in parameter estimates. Therefore we will require alternative approaches to efficiently uncover the interactions in complex biochemical networks.

RESULTS

We propose a Bayesian inference framework based on replacing uninteresting or unobserved reactions with time delays. Although the resulting models are non-Markovian, recent results on stochastic systems with random delays allow us to rigorously obtain expressions for the likelihoods of model parameters. In turn, this allows us to extend MCMC methods to efficiently estimate reaction rates, and delay distribution parameters, from single-cell assays. We illustrate the advantages, and potential pitfalls, of the approach using a birth-death model with both synthetic and experimental data, and show that we can robustly infer model parameters using a relatively small number of measurements. We demonstrate how to do so even when only the relative molecule count within the cell is measured, as in the case of fluorescence microscopy.

AVAILABILITY AND IMPLEMENTATION

Accompanying code in R is available at https://github.com/cbskust/DDE_BD.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

An Inactivation Switch Enables Rhythms in a Neurospora Clock Model

Autonomous endogenous time-keeping is ubiquitous across many living organisms, known as the circadian clock when it has a period of about 24 h. Interestingly, the fundamental design principle with a network of interconnected negative and positive feedback loops is conserved through evolution, although the molecular components differ. Filamentous fungus Neurospora crassa is a well-established chrono-genetics model organism to investigate the underlying mechanisms. The core negative feedback loop of the clock of Neurospora is composed of the transcription activator White Collar Complex (WCC) (heterodimer of WC1 and WC2) and the inhibitory element called FFC complex, which is made of FRQ (Frequency protein), FRH (Frequency interacting RNA Helicase) and CK1a (Casein kinase 1a). While exploring their temporal dynamics, we investigate how limit cycle oscillations arise and how molecular switches support self-sustained rhythms. We develop a mathematical model of 10 variables with 26 parameters to understand the interactions and feedback among WC1 and FFC elements in nuclear and cytoplasmic compartments. We performed control and bifurcation analysis to show that our novel model produces robust oscillations with a wild-type period of 22.5 h. Our model reveals a switch between WC1-induced transcription and FFC-assisted inactivation of WC1. Using the new model, we also study the possible mechanisms of glucose compensation. A fairly simple model with just three nonlinearities helps to elucidate clock dynamics, revealing a mechanism of rhythms' production. The model can further be utilized to study entrainment and temperature compensation.

MYOD1 functions as a clock amplifier as well as a critical co-factor for downstream circadian gene expression in muscle

In the present study we show that the master myogenic regulatory factor, MYOD1, is a positive modulator of molecular clock amplitude and functions with the core clock factors for expression of clock-controlled genes in skeletal muscle. We demonstrate that MYOD1 directly regulates the expression and circadian amplitude of the positive core clock factor Bmal1. We identify a non-canonical E-box element in Bmal1 and demonstrate that is required for full MYOD1-responsiveness. Bimolecular fluorescence complementation assays demonstrate that MYOD1 colocalizes with both BMAL1 and CLOCK throughout myonuclei. We demonstrate that MYOD1 and BMAL1:CLOCK work in a synergistic fashion through a tandem E-box to regulate the expression and amplitude of the muscle specific clock-controlled gene, Titin-cap (Tcap). In conclusion, these findings reveal mechanistic roles for the muscle specific transcription factor MYOD1 in the regulation of molecular clock amplitude as well as synergistic regulation of clock-controlled genes in skeletal muscle.

Oxysterols and Gastrointestinal Cancers Around the Clock

This review focuses on the role of oxidized sterols in three major gastrointestinal cancers (hepatocellular carcinoma, pancreatic, and colon cancer) and how the circadian clock affects the carcinogenesis by regulating the lipid metabolism and beyond. While each field of research (cancer, oxysterols, and circadian clock) is well-studied within their specialty, little is known about the intertwining mechanisms and how these influence the disease etiology in each cancer type. Oxysterols are involved in pathology of these cancers, but final conclusions about their protective or damaging effects are elusive, since the effect depends on the type of oxysterol, concentration, and the cell type. Oxysterol concentrations, the expression of key regulators liver X receptors (LXR), farnesoid X receptor (FXR), and oxysterol-binding proteins (OSBP) family are modulated in tumors and plasma of cancer patients, exposing these proteins and selected oxysterols as new potential biomarkers and drug targets. Evidence about how cholesterol/oxysterol pathways are intertwined with circadian clock is building. Identified key contact points are different forms of retinoic acid receptor related orphan receptors (ROR) and LXRs. RORs and LXRs are both regulated by sterols/oxysterols and the circadian clock and in return also regulate the same pathways, representing a complex interplay between sterol metabolism and the clock. With this in mind, in addition to classical therapies to modulate cholesterol in gastrointestinal cancers, such as the statin therapy, the time is ripe also for therapies where time and duration of the drug application is taken as an important factor for successful therapies. The final goal is the personalized approach with chronotherapy for disease management and treatment in order to increase the positive drug effects.

Reducing the complexity of mathematical models for the plant circadian clock by distributed delays

A major bottleneck in the modelling of biological networks is the parameter explosion problem - the exponential increase in the number of parameters that need to be optimised to data as the size of the model increases. Here, we address this problem in the context of the plant circadian clock by applying the method of distributed delays. We show that using this approach, the system architecture can be simplified efficiently - reducing the number of parameters - whilst still preserving the core mechanistic dynamics of the gene regulatory network. Compared to models with discrete time-delays, which are governed by functional differential equations, the distributed delay models can be converted into sets of equivalent ordinary differential equations, enabling the use of standard methods for numerical integration, and for stability and bifurcation analyses. We demonstrate the efficiency of our modelling approach by applying it to three exemplar mathematical models of the Arabidopsis circadian clock of varying complexity, obtaining significant reductions in complexity in each case. Moreover, we revise one of the most up-to-date Arabidopsis models, updating the regulation of the PRR9 and PRR7 genes by LHY in accordance with recent experimental data. The revised model more accurately reproduces the LHY-induction experiments of core clock genes, compared with the original model. Our work thus shows that the method of distributed delays facilitates the optimisation and reformulation of genetic network models.

Deciphering the Dynamics of Interlocked Feedback Loops in a Model of the Mammalian Circadian Clock

Mathematical models of fundamental biological processes play an important role in consolidating theory and experiments, especially if they are systematically developed, thoroughly characterized, and well tested by experimental data. In this work, we report a detailed bifurcation analysis of a mathematical model of the mammalian circadian clock network developed by Relógio et al., noteworthy for its consistency with available data. Using one- and two-parameter bifurcation diagrams, we explore how oscillations in the model depend on the expression levels of its constituent genes and the activities of their encoded proteins. These bifurcation diagrams allow us to decipher the dynamics of interlocked feedback loops by parametric variation of genes and proteins in the model. Among other results, we find that REV-ERB, a member of a subfamily of orphan nuclear receptors, plays a critical role in the intertwined dynamics of Relógio's model. The bifurcation diagrams reported here can be used for predicting how the core clock network responds-in terms of period, amplitude and phases of oscillations-to different perturbations.

Modeling clock-related metabolic syndrome due to conflicting light and food cues

Most organisms possess a light- and food- entrainable circadian clock system enabling their adaptation to daily environmental changes in sunlight and food availability. The mammalian circadian system is composed of multiple clocks throughout the body. These local clocks are entrained by nutrient, neural, endocrine and temperature cues and drive diverse physiological functions including metabolism. In particular, the clock of the pancreatic β cell rhythmically regulates the transcription of genes involved in glucose-stimulated insulin secretion. Perturbations of this fine-tuned oscillatory network increase the susceptibility to diseases. Besides chronic jet lag and shift work, common perturbations are ill-timed eating patterns which can lead to metabolic troubles (such as hypoinsulinemia). We have built a mathematical model describing the clock-dependent pancreatic regulation of glucose homeostasis in rodents. After calibrating the model using experimental data, we have investigated the effect of restricting food access to the normal rest phase. Our simulations show that the conflict between the light-dark cycle and the feeding-fasting cycle creates a differential phase shift in the expression of core clock genes (consistent with experimental observations). Our model further predicts that this induces a non-concomitance between nutrient cues and clock-controlled cues driving metabolic outputs which results in hypoinsulinemia, hyperglycemia as well as in a loss of food anticipation.

The Circadian Clock Drives Mast Cell Functions in Allergic Reactions

Allergic diseases are known to vary in the severity of their symptoms throughout the day/night cycle. This rhythmicity is also observed in mast cell function and responsiveness. Mast cells are key effector cells of allergic reactions and release cytokines, chemokines, and important inflammatory mediators such as histamine, which have been shown to display diurnal variation. Recent research clarified that mast cells are controlled by their internal clock-which is regulated by a specific set of clock genes-as well as external factors such as light sensed by the suprachiasmatic nuclei, hormonal status, or diet. Here, we give an overview of the connections between circadian clock, mast cells, and allergic disease. Further work aimed at studying the role of chronotherapy/chronomedicine should take into account this rhythmic nature of not only mast cells but also the immune responses generated by mast cell signaling.

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.

Modeling circadian variability of core-clock and clock-controlled genes in four tissues of the rat

Circadian clocks, present in almost all cells of the body, are entrained to rhythmic changes in the environment (e.g. light/dark cycles). Genes responsible for this timekeeping are named core-clock genes, which through transcriptional feedback interactions mediated by transcription factor binding to Ebox/RRE/Dbox elements can generate oscillatory activity of their expression. By regulating the transcription of other clock-controlled genes (CCGs) circadian information is transmitted to tissue and organ levels. Recent studies have indicated that there is a considerable variability of clock-controlled gene expression between tissues both with respect to the circadian genes that are regulated and to their phase lags. In this work, a mathematical model was adapted to explore the dynamics of core-clock and clock-controlled genes measured in four tissues of the rat namely liver, muscle, adipose, and lung. The model efficiently described the synchronous rhythmicity of core-clock genes and further predicted that their phases are mainly regulated by Per2 and Cry1 transcriptional delays and Rev-Erba and Cry1 degradation rates. Similarly, after mining databases for potential Ebox/RRE/Dbox elements in the promoter region of clock-controlled genes, the phase variabilities of the same genes between different tissues were described. The analysis suggests that inter-tissue circadian variability of the same clock-controlled genes is an inherent component of homeostatic function and may arise due to different transcription factor activities on Ebox/RRE/Dbox elements.

Daily variation of gene expression in diverse rat tissues

Circadian information is maintained in mammalian tissues by a cell-autonomous network of transcriptional feedback loops that have evolved to optimally regulate tissue-specific functions. An analysis of daily gene expression in different tissues, as well as an evaluation of inter-tissue circadian variability, is crucial for a systems-level understanding of this transcriptional circuitry. Affymetrix gene chip measurements of liver, muscle, adipose, and lung tissues were obtained from a rich time series light/dark experiment, involving 54 normal rats sacrificed at 18 time points within the 24-hr cycle. Our analysis revealed a high degree of circadian regulation with a variable distribution of phases among the four tissues. Interestingly, only a small number of common genes maintain circadian activity in all tissues, with many of them consisting of "core-clock" components with synchronous rhythms. Our results suggest that inter-tissue circadian variability is a critical component of homeostatic body function and is mediated by diverse signaling pathways that ultimately lead to highly tissue-specific transcription regulation.

Epigenetic Basis of Circadian Rhythm Disruption in Cancer

Self-sustained and synchronized to environmental stimuli, circadian clocks are under genetic and epigenetic regulation. Recent findings have greatly increased our understanding of epigenetic plasticity governed by circadian clock. Thus, the link between circadian clock and epigenetic machinery is reciprocal. Circadian clock can affect epigenetic features including genomic DNA methylation, noncoding RNA, mainly miRNA expression, and histone modifications resulted in their 24-h rhythms. Concomitantly, these epigenetic events can directly modulate cyclic system of transcription and translation of core circadian genes and indirectly clock output genes. Significant findings interlocking circadian clock, epigenetics, and cancer have been revealed, particularly in breast, colorectal, and blood cancers. Aberrant methylation of circadian gene promoter regions and miRNA expression affected circadian gene expression, together with 24-h expression oscillation pace have been frequently observed.

A Mathematical Model of the Liver Circadian Clock Linking Feeding and Fasting Cycles to Clock Function

To maintain energy homeostasis despite variable energy supply and consumption along the diurnal cycle, the liver relies on a circadian clock synchronized to food timing. Perturbed feeding and fasting cycles have been associated with clock disruption and metabolic diseases; however, the mechanisms are unclear. To address this question, we have constructed a mathematical model of the mammalian circadian clock, incorporating the metabolic sensors SIRT1 and AMPK. The clock response to various temporal patterns of AMPK activation was simulated numerically, mimicking the effects of a normal diet, fasting, and a high-fat diet. The model reproduces the dampened clock gene expression and NAD⁺ rhythms reported for mice on a high-fat diet and predicts that this effect may be pharmacologically rescued by timed REV-ERB agonist administration. Our model thus identifies altered AMPK signaling as a mechanism leading to clock disruption and its associated metabolic effects and suggests a pharmacological approach to resetting the clock in obesity.

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

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

Feedback loops interlocked at competitive binding sites amplify and facilitate genetic oscillations

Positive and negative feedback loops are often present in regulatory networks for genetic oscillations. Relative time scales and integration of these feedback loops are key to robust oscillations in expression levels. Using examples from the circadian clock and synthetic genetic oscillators, we study positive and negative feedback loops interlocked at competitive binding sites. In the mammalian circadian clock, a key clock gene Bmal1 is regulated by the activator ROR and the repressor REV-ERB. Conversely, Bmal1 activates both of them, forming interlocked feedback loops. Previous experiments indicate that the activator and repressor compete for the same binding sites in the Bmal1 promoter. Transcription patterns predict that ROR peaks later than REV-ERB and, moreover, the peak phase difference between them is small. Using mathematical modeling we reveal an optimal ratio of dissociation constants of an activator and a repressor for the competitive binding sites to enhance the amplitude of Bmal1 oscillations. This optimal ratio arises only when the amplitude of the repressor is larger than that of the activator. Secondly, we reveal that the preference of binding sites for an activator and a repressor depends on their relative time scales. A previous study demonstrated that noncompetitive binding sites are preferable for synthetic genetic oscillators that comprise a fast activator and a slow repressor with a large time scale separation. Here we show that when their time scales are similar, competitive binding sites are more likely to generate oscillation than noncompetitive sites. In contrast, for a slow activator and a fast repressor with a small phase difference as in Bmal1 regulation, noncompetitive binding sites are advantageous for amplifying oscillations. Our results, therefore, predict that additional mechanisms are necessary to compensate the disadvantage of the Bmal1 promoter and further facilitate amplification under the regulation by ROR and REV-ERB.

Frequency doubling in the cyanobacterial circadian clock

Organisms use circadian clocks to generate 24‐h rhythms in gene expression. However, the clock can interact with other pathways to generate shorter period oscillations. It remains unclear how these different frequencies are generated. Here, we examine this problem by studying the coupling of the clock to the alternative sigma factor sigC in the cyanobacterium Synechococcus elongatus. Using single‐cell microscopy, we find that psbAI, a key photosynthesis gene regulated by both sigC and the clock, is activated with two peaks of gene expression every circadian cycle under constant low light. This two‐peak oscillation is dependent on sigC, without which psbAI rhythms revert to one oscillatory peak per day. We also observe two circadian peaks of elongation rate, which are dependent on sigC, suggesting a role for the frequency doubling in modulating growth. We propose that the two‐peak rhythm in psbAI expression is generated by an incoherent feedforward loop between the clock, sigC and psbAI. Modelling and experiments suggest that this could be a general network motif to allow frequency doubling of outputs.

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.

Time-Dependent Gene Network Modelling by Sequential Monte Carlo

Most existing methods used for gene regulatory network modeling are dedicated to inference of steady state networks, which are prevalent over all time instants. However, gene interactions evolve over time. Information about the gene interactions in different stages of the life cycle of a cell or an organism is of high importance for biology. In the statistical graphical models literature, one can find a number of methods for studying steady-state network structures while the study of time varying networks is rather recent. A sequential Monte Carlo method, namely particle filtering (PF), provides a powerful tool for dynamic time series analysis. In this work, the PF technique is proposed for dynamic network inference and its potentials in time varying gene expression data tracking are demonstrated. The data used for validation are synthetic time series data available from the DREAM4 challenge, generated from known network topologies and obtained from transcriptional regulatory networks of S. cerevisiae. We model the gene interactions over the course of time with multivariate linear regressions where the parameters of the regressive process are changing over time.

Identifying Novel Transcriptional Regulators with Circadian Expression

Organisms adapt their physiology and behavior to the 24-h day-night cycle to which they are exposed. On a cellular level, this is regulated by intrinsic transcriptional-translational feedback loops that are important for maintaining the circadian rhythm. These loops are organized by members of the core clock network, which further regulate transcription of downstream genes, resulting in their circadian expression. Despite progress in understanding circadian gene expression, only a few players involved in circadian transcriptional regulation, including transcription factors, epigenetic regulators, and long noncoding RNAs, are known. Aiming to discover such genes, we performed a high-coverage transcriptome analysis of a circadian time course in murine fibroblast cells. In combination with a newly developed algorithm, we identified many transcription factors, epigenetic regulators, and long intergenic noncoding RNAs that are cyclically expressed. In addition, a number of these genes also showed circadian expression in mouse tissues. Furthermore, the knockdown of one such factor, Zfp28, influenced the core clock network. Mathematical modeling was able to predict putative regulator-effector interactions between the identified circadian genes and may help for investigations into the gene regulatory networks underlying circadian rhythms.

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.

Signal integration by the CYP1A1 promoter--a quantitative study

Genes involved in detoxification of foreign compounds exhibit complex spatiotemporal expression patterns in liver. Cytochrome P450 1A1 (CYP1A1), for example, is restricted to the pericentral region of liver lobules in response to the interplay between aryl hydrocarbon receptor (AhR) and Wnt/β-catenin signaling pathways. However, the mechanisms by which the two pathways orchestrate gene expression are still poorly understood. With the help of 29 mutant constructs of the human CYP1A1 promoter and a mathematical model that combines Wnt/β-catenin and AhR signaling with the statistical mechanics of the promoter, we systematically quantified the regulatory influence of different transcription factor binding sites on gene induction within the promoter. The model unveils how different binding sites cooperate and how they establish the promoter logic; it quantitatively predicts two-dimensional stimulus-response curves. Furthermore, it shows that crosstalk between Wnt/β-catenin and AhR signaling is crucial to understand the complex zonated expression patterns found in liver lobules. This study exemplifies how statistical mechanical modeling together with combinatorial reporter assays has the capacity to disentangle the promoter logic that establishes physiological gene expression patterns.

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.