A mechanism for robust circadian timekeeping via stoichiometric balance

A dual-feedback loop model of the mammalian circadian clock for multi-input control of circadian phase

The molecular circadian clock is driven by interlocked transcriptional-translational feedback loops, producing oscillations in the expressions of genes and proteins to coordinate the timing of biological processes throughout the body. Modeling this system gives insight into the underlying processes driving oscillations in an activator-repressor architecture and allows us to make predictions about how to manipulate these oscillations. The knockdown or upregulation of different cellular components using small molecules can disrupt these rhythms, causing a phase shift, and we aim to determine the dosing of such molecules with a model-based control strategy. Mathematical models allow us to predict the phase response of the circadian clock to these interventions and time them appropriately but only if the model has enough physiological detail to describe these responses while maintaining enough simplicity for online optimization. We build a control-relevant, physiologically-based model of the two main feedback loops of the mammalian molecular clock, which provides sufficient detail to consider multi-input control. Our model captures experimentally observed peak to trough ratios, relative abundances, and phase differences in the model species, and we independently validate this model by showing that the in silico model reproduces much of the behavior that is observed in vitro under genetic knockout conditions. Because our model produces valid phase responses, it can be used in a model predictive control algorithm to determine inputs to shift phase. Our model allows us to consider multi-input control through small molecules that act on both feedback loops, and we find that changes to the parameters of the negative feedback loop are much stronger inputs for shifting phase. The strongest inputs predicted by this model provide targets for new experimental small molecules and suggest that the function of the positive feedback loop is to stabilize the oscillations while linking the circadian system to other clock-controlled processes.

Wake-sleep cycles are severely disrupted by diseases affecting cytoplasmic homeostasis

Circadian rhythms including wake-sleep cycles are driven by molecular time cues generated by a self-sustaining transcriptional negative feedback loop. Among all clock proteins, PERIOD (PER) is considered the pacemaker protein because its rhythm of accumulation and nuclear entry generates the timing and duration of feedback inhibition. Here we provide a new understanding of how robust PER rhythms are generated: the collective action of interacting PER molecules, not a random mass action of individual molecules, allows compensation of spatial and temporal differences (or “noise”) of individual molecules. We also show that the collective PER rhythm requires healthy cytoplasmic trafficking, and that circadian sleep disorders can arise in such conditions as obesity, aging, and neurodegenerative disorders in which the cytoplasm becomes congested.

The circadian clock is based on a transcriptional feedback loop with an essential time delay before feedback inhibition. Previous work has shown that PERIOD (PER) proteins generate circadian time cues through rhythmic nuclear accumulation of the inhibitor complex and subsequent interaction with the activator complex in the feedback loop. Although this temporal manifestation of the feedback inhibition is the direct consequence of PER’s cytoplasmic trafficking before nuclear entry, how this spatial regulation of the pacemaker affects circadian timing has been largely unexplored. Here we show that circadian rhythms, including wake-sleep cycles, are lengthened and severely unstable if the cytoplasmic trafficking of PER is disrupted by any disease condition that leads to increased congestion in the cytoplasm. Furthermore, we found that the time delay and robustness in the circadian clock are seamlessly generated by delayed and collective phosphorylation of PER molecules, followed by synchronous nuclear entry. These results provide clear mechanistic insight into why circadian and sleep disorders arise in such clinical conditions as metabolic and neurodegenerative diseases and aging, in which the cytoplasm is congested.

An Optimal Time for Treatment-Predicting Circadian Time by Machine Learning and Mathematical Modelling

Tailoring medical interventions to a particular patient and pathology has been termed personalized medicine. The outcome of cancer treatments is improved when the intervention is timed in accordance with the patient's internal time. Yet, one challenge of personalized medicine is how to consider the biological time of the patient. Prerequisite for this so-called chronotherapy is an accurate characterization of the internal circadian time of the patient. As an alternative to time-consuming measurements in a sleep-laboratory, recent studies in chronobiology predict circadian time by applying machine learning approaches and mathematical modelling to easier accessible observables such as gene expression. Embedding these results into the mathematical dynamics between clock and cancer in mammals, we review the precision of predictions and the potential usage with respect to cancer treatment and discuss whether the patient's internal time and circadian observables, may provide an additional indication for individualized treatment timing. Besides the health improvement, timing treatment may imply financial advantages, by ameliorating side effects of treatments, thus reducing costs. Summarizing the advances of recent years, this review brings together the current clinical standard for measuring biological time, the general assessment of circadian rhythmicity, the usage of rhythmic variables to predict biological time and models of circadian rhythmicity.

Misuse of the Michaelis-Menten rate law for protein interaction networks and its remedy

For over a century, the Michaelis-Menten (MM) rate law has been used to describe the rates of enzyme-catalyzed reactions and gene expression. Despite the ubiquity of the MM rate law, it accurately captures the dynamics of underlying biochemical reactions only so long as it is applied under the right condition, namely, that the substrate is in large excess over the enzyme-substrate complex. Unfortunately, in circumstances where its validity condition is not satisfied, especially so in protein interaction networks, the MM rate law has frequently been misused. In this review, we illustrate how inappropriate use of the MM rate law distorts the dynamics of the system, provides mistaken estimates of parameter values, and makes false predictions of dynamical features such as ultrasensitivity, bistability, and oscillations. We describe how these problems can be resolved with a slightly modified form of the MM rate law, based on the total quasi-steady state approximation (tQSSA). Furthermore, we show that the tQSSA can be used for accurate stochastic simulations at a lower computational cost than using the full set of mass-action rate laws. This review describes how to use quasi-steady state approximations in the right context, to prevent drawing erroneous conclusions from in silico simulations.

Basal leakage in oscillation: Coupled transcriptional and translational control using feed-forward loops

The circadian clock is a complex system that plays many important roles in most organisms. Previously, many mathematical models have been used to sharpen our understanding of the Arabidopsis clock, which brought to light the roles of each transcriptional and post-translational regulations. However, the presence of both regulations, instead of either transcription or post-translation, raised curiosity of whether the combination of these two regulations is important for the clock’s system. In this study, we built a series of simplified oscillators with different regulations to study the importance of post-translational regulation (specifically, 26S proteasome degradation) in the clock system. We found that a simple transcriptional-based oscillator can already generate sustained oscillation, but the oscillation can be easily destroyed in the presence of transcriptional leakage. Coupling post-translational control with transcriptional-based oscillator in a feed-forward loop will greatly improve the robustness of the oscillator in the presence of basal leakage. Using these general models, we were able to replicate the increased variability observed in the E3 ligase mutant for both plant and mammalian clocks. With this insight, we also predict a plausible regulator of several E3 ligase genes in the plant’s clock. Thus, our results provide insights into and the plausible importance in coupling transcription and post-translation controls in the clock system.

For circadian clocks, several current models had successfully captured the essential dynamic behavior of the clock system mainly with transcriptional regulation. Previous studies have shown that the 26S proteasome degradation controls are important in maintaining the stability of circadian rhythms. However, how the loss-of-function or over-expression mutant of this targeted degradations lead to unstable oscillation is still unclear. In this work, we investigate the importance of coupled transcriptional and post-translational feedback loop in the circadian oscillator. With general models our study indicate that the unstable behavior of degradation mutants could be caused by the increase in the basal level of the clock genes. We found that coupling a non-linear degradation control into this transcriptional based oscillator using feed-forward loop improves the robustness of the oscillator. Using this finding, we further predict some plausible regulators of Arabidopsis’s E3 ligase protein such as COP1 and SINAT5. Hence, our results provide insights on the importance of coupling transcription and post-translation controls in the clock system.

A systems biology approach to discovering pathway signaling dysregulation in metastasis

Total metastatic burden is the primary cause of death for many cancer patients. While the process of metastasis has been studied widely, much remains to be understood. Moreover, few agents have been developed that specifically target the major steps of the metastatic cascade. Many individual genes and pathways have been implicated in metastasis but a holistic view of how these interact and cooperate to regulate and execute the process remains somewhat rudimentary. It is unclear whether all of the signaling features that regulate and execute metastasis are yet fully understood. Novel features of a complex system such as metastasis can often be discovered by taking a systems-based approach. We introduce the concepts of systems modeling and define some of the central challenges facing the application of a multidisciplinary systems-based approach to understanding metastasis and finding actionable targets therein. These challenges include appreciating the unique properties of the high-dimensional omics data often used for modeling, limitations in knowledge of the system (metastasis), tumor heterogeneity and sampling bias, and some of the issues key to understanding critical features of molecular signaling in the context of metastasis. We also provide a brief introduction to integrative modeling that focuses on both the nodes and edges of molecular signaling networks. Finally, we offer some observations on future directions as they relate to developing a systems-based model of the metastatic cascade.

Coupling protocol of interlocked feedback oscillators in circadian clocks

Circadian rhythms (approx. 24 h) show the robustness of key oscillatory features such as phase, period and amplitude against external and internal variations. The robustness of Drosophila circadian clocks can be generated by interlocked transcriptional-translational feedback loops, where two negative feedback loops are coupled through mutual activations. The mechanisms by which such coupling protocols have survived out of many possible protocols remain to be revealed. To address this question, we investigated two distinct coupling protocols: activator-coupled oscillators (ACO) and repressor-coupled oscillators (RCO). We focused on the two coupling parameters: coupling dissociation constant and coupling time-delay. Interestingly, the ACO was able to produce anti-phase or morning-evening cycles, whereas the RCO produced in-phase ones. Deterministic and stochastic analyses demonstrated that the anti-phase ACO provided greater fluctuations in amplitude not only with respect to changes in coupling parameters but also to random parameter perturbations than the in-phase RCO. Moreover, the ACO deteriorated the entrainability to the day-night master clock, whereas the RCO produced high entrainability. Considering that the real, interlocked feedback loops have evolved as the ACO, instead of the RCO, we first proposed a hypothesis that the morning-evening or anti-phase cycle is more essential for Drosophila than achieving robustness and entrainability.

Bayesian modeling reveals metabolite-dependent ultrasensitivity in the cyanobacterial circadian clock

Mathematical models can enable a predictive understanding of mechanism in cell biology by quantitatively describing complex networks of interactions, but such models are often poorly constrained by available data. Owing to its relative biochemical simplicity, the core circadian oscillator in Synechococcus elongatus has become a prototypical system for studying how collective dynamics emerge from molecular interactions. The oscillator consists of only three proteins, KaiA, KaiB, and KaiC, and near-24-h cycles of KaiC phosphorylation can be reconstituted in vitro. Here, we formulate a molecularly detailed but mechanistically naive model of the KaiA-KaiC subsystem and fit it directly to experimental data within a Bayesian parameter estimation framework. Analysis of the fits consistently reveals an ultrasensitive response for KaiC phosphorylation as a function of KaiA concentration, which we confirm experimentally. This ultrasensitivity primarily results from the differential affinity of KaiA for competing nucleotide-bound states of KaiC. We argue that the ultrasensitive stimulus-response relation likely plays an important role in metabolic compensation by suppressing premature phosphorylation at nighttime.

Chemical perturbations reveal that RUVBL2 regulates the circadian phase in mammals

Transcriptional regulation lies at the core of the circadian clockwork, but how the clock-related transcription machinery controls the circadian phase is not understood. Here, we show both in human cells and in mice that RuvB-like ATPase 2 (RUVBL2) interacts with other known clock proteins on chromatin to regulate the circadian phase. Pharmacological perturbation of RUVBL2 with the adenosine analog compound cordycepin resulted in a rapid-onset 12-hour clock phase-shift phenotype at human cell, mouse tissue, and whole-animal live imaging levels. Using simple peripheral injection treatment, we found that cordycepin penetrated the blood-brain barrier and caused rapid entrainment of the circadian phase, facilitating reduced duration of recovery in a mouse jet-lag model. We solved a crystal structure for human RUVBL2 in complex with a physiological metabolite of cordycepin, and biochemical assays showed that cordycepin treatment caused disassembly of an interaction between RUVBL2 and the core clock component BMAL1. Moreover, we showed with spike-in ChIP-seq analysis and binding assays that cordycepin treatment caused disassembly of the circadian super-complex, which normally resides at E-box chromatin loci such as PER1, PER2, DBP, and NR1D1. Mathematical modeling supported that the observed type 0 phase shifts resulted from derepression of E-box clock gene transcription.

A Systems Biology Approach Identifies Hidden Regulatory Connections Between the Circadian and Cell-Cycle Checkpoints

Circadian rhythms form a self-sustaining, endogenous, time-keeping system that allows organisms to anticipate daily environmental changes. The core of the clock network consists of interlocking transcriptional-translational feedback loops that ensures that metabolic, behavioral, and physiological processes run on a 24 h timescale. The hierarchical nature of the clock manifests itself in multiple points of control on the daily cell division cycle, which relies on synthesis, degradation, and post-translational modification for progression. This relationship is particularly important for understanding the role of clock components in sensing stress conditions and triggering checkpoint signals that stop cell cycle progression. A case in point is the interplay among the circadian factor PERIOD2 (PER2), the tumor suppressor p53, and the oncogenic mouse double minute-2 homolog protein (MDM2), which is the p53's negative regulator. Under unstressed conditions, PER2 and p53 form a stable complex in the cytosol and, along with MDM2, a trimeric complex in the nucleus. Association of PER2 to the C-terminus end of p53 prevents MDM2-mediated ubiquitylation and degradation of p53 as well as p53's transcriptional activation. Remarkably, when not bound to p53, PER2 acts as substrate for the E3-ligase activity of MDM2; thus, PER2 is degraded in a phosphorylation-independent fashion. Unexpectedly, the phase relationship between PER2 and p53 are opposite; however, a systematic modeling approach, inferred from the oscillatory time course data of PER2 and p53, aided in identifying additional regulatory scenarios that explained, a priori, seemingly conflicting experimental data. Therefore, we advocate for a combined experimental/mathematical approach to elucidating multilevel regulatory cellular processes.

A microfluidic approach for experimentally modelling the intercellular coupling system of a mammalian circadian clock at single-cell level

In mammals, it is believed that the intercellular coupling mechanism between neurons in the suprachiasmatic nucleus (SCN) confers robustness and distinguishes the central clock from peripheral circadian oscillators. Current in vitro culturing methods used in Petri dishes to study intercellular coupling by exogenous factors invariably cause perturbations, such as simple media changes. Here, we design a microfluidic device to quantitatively study the intercellular coupling mechanism of circadian clock at the single cell level, and demonstrate that vasoactive intestinal peptide (VIP) induced coupling in clock mutant Cry1-/- mouse adult fibroblasts engineered to express the VIP receptor, VPAC2, is sufficient to synchronize and maintain robust circadian oscillations. Our study provides a proof-of-concept platform to reconstitute the intercellular coupling system of the central clock using uncoupled, single fibroblast cells in vitro, to mimic SCN slice cultures ex vivo and mouse behavior in vivo phenotypically. Such a versatile microfluidic platform may greatly facilitate the studies of intercellular regulation networks, and provide new insights into the coupling mechanisms of the circadian clock.

Clocks in the Wild: Entrainment to Natural Light

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

MYC-Associated Factor MAX is a Regulator of the Circadian Clock

The circadian transcriptional network is based on a competition between transcriptional activator and repressor complexes regulating the rhythmic expression of clock-controlled genes. We show here that the MYC-associated factor X, MAX, plays a repressive role in this network and operates through a MYC-independent binding to E-box-containing regulatory regions within the promoters of circadian BMAL1 targets. We further show that this "clock" function of MAX is required for maintaining a proper circadian rhythm and that MAX and BMAL1 contribute to two temporally alternating transcriptional complexes on clock-regulated promoters. We also identified MAX network transcriptional repressor, MNT, as a fundamental partner of MAX-mediated circadian regulation. Collectively, our data indicate that MAX regulates clock gene expression and contributes to keeping the balance between positive and negative elements of the molecular clock machinery.

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.

Systems Level Understanding of Circadian Integration with Cell Physiology

The mammalian circadian clock regulates a wide variety of physiological and behavioral processes. In turn, its disruption is associated with sleep deficiency, metabolic syndrome, neurological and psychiatric disorders, and cancer. At the turn of the century, the circadian clock was determined to be regulated by a transcriptional negative feedback mechanism composed of a dozen core clock genes. More recently, large-scale genomic studies have expanded the clock into a complex network composed of thousands of gene outputs and inputs. A major task of circadian research is to utilize systems biological approaches to uncover the governing principles underlying cellular oscillatory behavior and advance understanding of biological functions at the genomic level with spatiotemporal resolution. This review focuses on the genes and pathways that provide inputs to the circadian clock. Several emerging examples include AMP-activated protein kinase AMPK, nutrient/energy sensor mTOR, NAD⁺-dependent deacetylase SIRT1, hypoxia-inducible factor HIF1α, oxidative stress-inducible factor NRF2, and the proinflammatory factor NF-κB. Among others that continue to be revealed, these input pathways reflect the extensive interplay between the clock and cell physiology through the regulation of core clock genes and proteins. While the scope of this crosstalk is well-recognized, precise molecular links are scarce, and the underlying regulatory mechanisms are not well understood. Future research must leverage genetic and genomic tools and technologies, network analysis, and computational modeling to characterize additional modifiers and input pathways. This systems-based framework promises to advance understanding of the circadian timekeeping system and may enable the enhancement of circadian functions through related input pathways.

Casein kinase 1 dynamics underlie substrate selectivity and the PER2 circadian phosphoswitch

Post-translational control of PERIOD stability by Casein Kinase 1δ and ε (CK1) plays a key regulatory role in metazoan circadian rhythms. Despite the deep evolutionary conservation of CK1 in eukaryotes, little is known about its regulation and the factors that influence substrate selectivity on functionally antagonistic sites in PERIOD that directly control circadian period. Here we describe a molecular switch involving a highly conserved anion binding site in CK1. This switch controls conformation of the kinase activation loop and determines which sites on mammalian PER2 are preferentially phosphorylated, thereby directly regulating PER2 stability. Integrated experimental and computational studies shed light on the allosteric linkage between two anion binding sites that dynamically regulate kinase activity. We show that period-altering kinase mutations from humans to Drosophila differentially modulate this activation loop switch to elicit predictable changes in PER2 stability, providing a foundation to understand and further manipulate CK1 regulation of circadian rhythms.

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.

A data-driven phase and isostable reduced modeling framework for oscillatory dynamical systems

Phase-amplitude reduction is of growing interest as a strategy for the reduction and analysis of oscillatory dynamical systems. Augmentation of the widely studied phase reduction with amplitude coordinates can be used to characterize transient behavior in directions transverse to a limit cycle to give a richer description of the dynamical behavior. Various definitions for amplitude coordinates have been suggested, but none are particularly well suited for implementation in experimental systems where output recordings are readily available but the underlying equations are typically unknown. In this work, a reduction framework is developed for inferring a phase-amplitude reduced model using only the observed model output from an arbitrarily high-dimensional system. This framework employs a proper orthogonal reduction strategy to identify important features of the transient decay of solutions to the limit cycle. These features are explicitly related to previously developed phase and isostable coordinates and used to define so-called data-driven phase and isostable coordinates that are valid in the entire basin of attraction of a limit cycle. The utility of this reduction strategy is illustrated in examples related to neural physiology and is used to implement an optimal control strategy that would otherwise be computationally intractable. The proposed data-driven phase and isostable coordinate system and associated reduced modeling framework represent a useful tool for the study of nonlinear dynamical systems in situations where the underlying dynamical equations are unknown and in particularly high-dimensional or complicated numerical systems for which standard phase-amplitude reduction techniques are not computationally feasible.

Macroscopic Models for Human Circadian Rhythms

Mathematical models have a long and influential history in the study of human circadian rhythms. Accurate predictive models for the human circadian light response have been used to study the impact of a host of light exposures on the circadian system. However, generally, these models do not account for the physiological basis of these rhythms. We illustrate a new paradigm for deriving models of the human circadian light response. Beginning from a high-dimensional model of the circadian neural network, we systematically derive low-dimensional models using an approach motivated by experimental measurements of circadian neurons. This systematic reduction allows for the variables and parameters of the derived model to be interpreted in a physiological context. We fit and validate the resulting models to a library of experimental measurements. Finally, we compare model predictions for experimental measurements of light levels and discuss the differences between our model’s predictions and previous models. Our modeling paradigm allows for the integration of experimental measurements across the single-cell, tissue, and behavioral scales, thereby enabling the development of accurate low-dimensional models for human circadian rhythms.

Robust synchronization of the cell cycle and the circadian clock through bidirectional coupling

The cell cycle and the circadian clock represent major cellular rhythms, which appear to be coupled. Thus the circadian factor BMAL1 controls the level of cell cycle proteins such as Cyclin E and WEE1, the latter of which inhibits the kinase CDK1 that governs the G2/M transition. In reverse the cell cycle impinges on the circadian clock through direct control by CDK1 of REV-ERBα, which negatively regulates BMAL1. These observations provide evidence for bidirectional coupling of the cell cycle and the circadian clock. By merging detailed models for the two networks in mammalian cells, we previously showed that unidirectional coupling to the circadian clock can entrain the cell cycle to 24 or 48 h, depending on the cell cycle autonomous period, while complex oscillations occur when entrainment fails. Here we show that the reverse unidirectional coupling via phosphorylation of REV-ERBα or via mitotic inhibition of transcription, both controlled by CDK1, can elicit entrainment of the circadian clock by the cell cycle. We then determine the effect of bidirectional coupling of the cell cycle and circadian clock as a function of their relative coupling strengths. In contrast to unidirectional coupling, bidirectional coupling markedly reduces the likelihood of complex oscillations. While the two rhythms oscillate independently as long as both couplings are weak, one rhythm entrains the other if one of the couplings dominates. If the couplings in both directions become stronger and of comparable magnitude, the two rhythms synchronize, generally at an intermediate period within the range defined by the two autonomous periods prior to coupling. More surprisingly, synchronization may also occur at a period slightly below or above this range, while in some conditions the synchronization period can even be much longer. Two or even three modes of synchronization may sometimes coexist, yielding examples of birhythmicity or trirhythmicity. Because synchronization readily occurs in the form of simple periodic oscillations over a wide range of coupling strengths and in the presence of multiple connections between the two oscillatory networks, the results indicate that bidirectional coupling favours the robust synchronization of the cell cycle and the circadian clock.

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.

Multi-scale modeling of the circadian modulation of learning and memory

We propose a multi-scale model to explain the time-of-day effects on learning and memory. We specifically model the circadian variation of hippocampus (HC) dependent long-term potentiation (LTP), depression (LTD), and the fear conditioning paradigm in amygdala. The model we built has both Goodwin type circadian gene regulatory network (GRN) and the conductance model of Morris-Lecar (ML) type to explain the spontaneous firing patterns (SFR) in suprachiasmatic nucleus (SCN). In the conductance model, we also include N-Methyl-D-aspartic acid receptor (NMDAR) to study the circadian dependent changes in LTP/LTD in hippocampus and include both NMDAR and α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid receptor (AMPAR) dynamics to explain the circadian modulation of fear conditioning paradigm in memory acquisition, recall, and extinction as seen in amygdala. Our multi-scale model captures the essential dynamics seen in the experiments and strongly supports the circadian time-of-the-day effects on learning and memory.

Systems approach reveals photosensitivity and PER2 level as determinants of clock-modulator efficacy

In mammals, the master circadian clock synchronizes daily rhythms of physiology and behavior with the day-night cycle. Failure of synchrony, which increases the risk for numerous chronic diseases, can be treated by phase adjustment of the circadian clock pharmacologically, for example, with melatonin, or a CK1δ/ε inhibitor. Here, using in silico experiments with a systems pharmacology model describing molecular interactions, and pharmacokinetic and behavioral experiments in cynomolgus monkeys, we find that the circadian phase delay caused by CK1δ/ε inhibition is more strongly attenuated by light in diurnal monkeys and humans than in nocturnal mice, which are common preclinical models. Furthermore, the effect of CK1δ/ε inhibition strongly depends on endogenous PER2 protein levels, which differs depending on both the molecular cause of the circadian disruption and the patient's lighting environment. To circumvent such large interindividual variations, we developed an adaptive chronotherapeutics to identify precise dosing regimens that could restore normal circadian phase under different conditions. Our results reveal the importance of photosensitivity in the clinical efficacy of clock-modulating drugs, and enable precision medicine for circadian disruption.

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.

Systems Biology Approaches and Precision Oral Health: A Circadian Clock Perspective

A vast majority of the pathophysiological and metabolic processes in humans are temporally controlled by a master circadian clock located centrally in the hypothalamic suprachiasmatic nucleus of the brain, as well as by specialized peripheral oscillators located in other body tissues. This circadian clock system generates a rhythmical diurnal transcriptional-translational cycle in clock genes and protein expression and activities regulating numerous downstream target genes. Clock genes as key regulators of physiological function and dysfunction of the circadian clock have been linked to various diseases and multiple morbidities. Emerging omics technologies permits largescale multi-dimensional investigations of the molecular landscape of a given disease and the comprehensive characterization of its underlying cellular components (e.g., proteins, genes, lipids, metabolites), their mechanism of actions, functional networks and regulatory systems. Ultimately, they can be used to better understand disease and interpatient heterogeneity, individual profile, identify personalized targetable key molecules and pathways, discover novel biomarkers and genetic alterations, which collectively can allow for a better patient stratification into clinically relevant subgroups to improve disease prediction and prevention, early diagnostic, clinical outcomes, therapeutic benefits, patient's quality of life and survival. The use of “omics” technologies has allowed for recent breakthroughs in several scientific domains, including in the field of circadian clock biology. Although studies have explored the role of clock genes using circadiOmics (which integrates circadian omics, such as genomics, transcriptomics, proteomics and metabolomics) in human disease, no such studies have investigated the implications of circadian disruption in oral, head and neck pathologies using multi-omics approaches and linking the omics data to patient-specific circadian profiles. There is a burgeoning body of evidence that circadian clock controls the development and homeostasis of oral and maxillofacial structures, such as salivary glands, teeth and oral epithelium. Hence, in the current era of precision medicine and dentistry and patient-centered health care, it is becoming evident that a multi-omics approach is needed to improve our understanding of the role of circadian clock-controlled key players in the regulation of head and neck pathologies. This review discusses current knowledge on the role of the circadian clock and the contribution of omics-based approaches toward a novel precision health era for diagnosing and treating head and neck pathologies, with an emphasis on oral, head and neck cancer and Sjögren's syndrome.

Mathematical modeling of circadian rhythms

Circadian rhythms are endogenous ~24-hr oscillations usually entrained to daily environmental cycles of light/dark. Many biological processes and physiological functions including mammalian body temperature, the cell cycle, sleep/wake cycles, neurobehavioral performance, and a wide range of diseases including metabolic, cardiovascular, and psychiatric disorders are impacted by these rhythms. Circadian clocks are present within individual cells and at tissue and organismal levels as emergent properties from the interaction of cellular oscillators. Mathematical models of circadian rhythms have been proposed to provide a better understanding of and to predict aspects of this complex physiological system. These models can be used to: (a) manipulate the system in silico with specificity that cannot be easily achieved using in vivo and in vitro experimental methods and at lower cost, (b) resolve apparently contradictory empirical results, (c) generate hypotheses, (d) design new experiments, and (e) to design interventions for altering circadian rhythms. Mathematical models differ in structure, the underlying assumptions, the number of parameters and variables, and constraints on variables. Models representing circadian rhythms at different physiologic scales and in different species are reviewed to promote understanding of these models and facilitate their use. This article is categorized under: Physiology > Mammalian Physiology in Health and Disease Models of Systems Properties and Processes > Organ, Tissue, and Physiological Models.

Computational modeling and analysis of the impacts of sleep deprivation on glucose stimulated insulin secretion

Circadian clock is an exquisite internal biological clock functioning in all living organisms. Lifestyle changes such as shift work or frequent travelling might result in malfunctioning of the central and consequently the peripheral clocks leading to different metabolic disorders. Disruptions in β cell clock have been found to be a potential reason behind β cell failure that makes a person prone towards developing type 2 diabetes (T2DM). In this study, a Petri net model for β cell circadian clock has been developed, followed by analysis of the negative impacts of sleep deprivation conditions on the process of glucose stimulated insulin secretion (GSIS) through misalignment of circadian clock. The analysis of structural properties of the Petri net model reveals robustness of the circadian system. The simulation results predict that sleep loss negatively affects the expression of circadian genes which eventually leads to impaired GSIS and β cell failure. These results suggest that sleep/wake cycle is a vital contributor for the entrainment of the circadian clock and normal functioning of β cell.

A saturated reaction in repressor synthesis creates a daytime dead zone in circadian clocks

Negative feedback loops (NFLs) for circadian clocks include light-responsive reactions that allow the clocks to shift their phase depending on the timing of light signals. Phase response curves (PRCs) for light signals in various organisms include a time interval called a dead zone where light signals cause no phase shift during daytime. Although the importance of the dead zone for robust light entrainment is known, how the dead zone arises from the biochemical reactions in an NFL underlying circadian gene expression rhythms remains unclear. In addition, the observation that the light-responsive reactions in the NFL vary between organisms raises the question as to whether the mechanism for dead zone formation is common or distinct between different organisms. Here we reveal by mathematical modeling that the saturation of a biochemical reaction in repressor synthesis in an NFL is a common mechanism of daytime dead zone generation. If light signals increase the degradation of a repressor protein, as in Drosophila, the saturation of repressor mRNA transcription nullifies the effect of light signals, generating a dead zone. In contrast, if light signals induce the transcription of repressor mRNA, as in mammals, the saturation of repressor translation can generate a dead zone by cancelling the influence of excess amount of mRNA induced by light signals. Each of these saturated reactions is located next to the light-responsive reaction in the NFL, suggesting a design principle for daytime dead zone generation.

Light-entrainable circadian clocks form behavioral and physiological rhythms in organisms. The light-entrainment properties of these clocks have been studied by measuring phase shifts caused by light pulses administered at different times. The phase response curves of various organisms include a time window called the dead zone where the phase of the clock does not respond to light pulses. However, the mechanism underlying the dead zone generation remains unclear. We show that the saturation of biochemical reactions in feedback loops for circadian oscillations generates a dead zone. The proposed mechanism is generic, as it functions in different models of the circadian clocks and biochemical oscillators. Our analysis indicates that light-entrainment properties are determined by biochemical reactions at the single-cell level.

Modeling the dynamic behavior of biochemical regulatory networks

Strategies for modeling the complex dynamical behavior of gene/protein regulatory networks have evolved over the last 50 years as both the knowledge of these molecular control systems and the power of computing resources have increased. Here, we review a number of common modeling approaches, including Boolean (logical) models, systems of piecewise-linear or fully non-linear ordinary differential equations, and stochastic models (including hybrid deterministic/stochastic approaches). We discuss the pro's and con's of each approach, to help novice modelers choose a modeling strategy suitable to their problem, based on the type and bounty of available experimental information. We illustrate different modeling strategies in terms of some abstract network motifs, and in the specific context of cell cycle regulation.

Waveforms of molecular oscillations reveal circadian timekeeping mechanisms

Circadian clocks play a pivotal role in orchestrating numerous physiological and developmental events. Waveform shapes of the oscillations of protein abundances can be informative about the underlying biochemical processes of circadian clocks. We derive a mathematical framework where waveforms do reveal hidden biochemical mechanisms of circadian timekeeping. We find that the cost of synthesizing proteins with particular waveforms can be substantially reduced by rhythmic protein half-lives over time, as supported by previous plant and mammalian data, as well as our own seedling experiment. We also find that previously enigmatic, cyclic expression of positive arm components within the mammalian and insect clocks allows both a broad range of peak time differences between protein waveforms and the symmetries of the waveforms about the peak times. Such various peak-time differences may facilitate tissue-specific or developmental stage-specific multicellular processes. Our waveform-guided approach can be extended to various biological oscillators, including cell-cycle and synthetic genetic oscillators.

Hang-Hyun Jo et al. derive a mathematical framework for analyzing circadian clock waveforms. Using data from plants and animals, they find that waveforms of clock protein profiles provide important information about the biochemical mechanisms of circadian timekeeping.

Computational and experimental insights into the circadian effects of SIRT1

The circadian clock orchestrates 24-h rhythms in physiology in most living organisms. At the molecular level, the dogma is that circadian oscillations are based on a negative transcriptional feedback loop. Recent studies found the NAD+-dependent histone deacetylase, SIRT1, directly regulates acetylation status of clock components and influences circadian amplitude in cells. While Nakahata et al. [Nakahata Y, Kaluzova M (2008) Cell 134:329-340] reported that loss of SIRT1 increases amplitude through BMAL1 acetylation, Asher et al. [Asher G, Gatfield D (2008) Cell 134:317-328] reported that loss of SIRT1 decreases amplitude through an increase in acetylated PER2. To address this SIRT1 paradox, we developed a circadian enzymatic model. Predictions from this model and experimental validation strongly align with the findings of Asher et al., with PER2 as the primary target of SIRT1. Further, the model suggested SIRT1 influences BMAL1 expression through actions on PGC1α. We validated this finding experimentally. Thus, our computational and experimental approaches suggest SIRT1 positively regulates clock function through actions on PER2 and PGC1α.

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.

Protein sequestration versus Hill-type repression in circadian clock models

Circadian (∼24 h) clocks are self-sustained endogenous oscillators with which organisms keep track of daily and seasonal time. Circadian clocks frequently rely on interlocked transcriptional-translational feedback loops to generate rhythms that are robust against intrinsic and extrinsic perturbations. To investigate the dynamics and mechanisms of the intracellular feedback loops in circadian clocks, a number of mathematical models have been developed. The majority of the models use Hill functions to describe transcriptional repression in a way that is similar to the Goodwin model. Recently, a new class of models with protein sequestration-based repression has been introduced. Here, the author discusses how this new class of models differs dramatically from those based on Hill-type repression in several fundamental aspects: conditions for rhythm generation, robust network designs and the periods of coupled oscillators. Consistently, these fundamental properties of circadian clocks also differ among Neurospora, Drosophila, and mammals depending on their key transcriptional repression mechanisms (Hill-type repression or protein sequestration). Based on both theoretical and experimental studies, this review highlights the importance of careful modelling of transcriptional repression mechanisms in molecular circadian clocks.

Entrainment of a cellular circadian oscillator by light in the presence of molecular noise

In this paper, we consider a stochastic molecular circadian oscillator described by a sequence of biological reactions and its deterministic kinetics governed by a system of ordinary differential equations in the limit of large numbers of molecules. The oscillations in the model are generated by negative feedback regulation of a gene. The focus of this paper is the entrainment of the oscillator by a periodic light signal that affects the maximal transcription rate of the gene. We introduce two scalings of the model parameters that provide independent control over the natural frequency of the oscillator and the relative noise level. We study entrainment in two ways: by visualizing the stochastic limit cycle in various projections of the discrete phase space of the system and by evaluating the maximum of the normalized cross correlation of the light signal with the number of protein molecules in the cell. The visualization method ignores the phase of the oscillator, and we find in this way that entrainment has a subtle organizing effect on the limit cycle as a whole. The cross correlation results reveal an interval of natural frequencies of the oscillator surrounding the frequency of the light signal within which maximal entrainment occurs with rather sharp drops in entrainment at the edges of this interval. The width of the interval of maximal entrainment increases with the amplitude of the light signal. These statements are applicable both to the stochastic oscillator and to its deterministic limit, but the results are most clear-cut in the deterministic case and degrade from there as the relative noise level increases.

Fundamental Characteristics of Single-Cell Aging in Diploid Yeast

Single-cell-level experimentation can elucidate key biological insights about cellular aging that are masked in population-level studies. However, the extensive time requirement of tracking single cells has historically prevented their long-term longitudinal observation. Using a microfluidic device that automates microscopic monitoring of diploid Saccharomyces cerevisiae cells throughout their replicative lifespan, here we report the fundamental characteristics of single-cell aging for diploid yeast. We find that proteins with short versus long half-lives exhibit distinct dynamics as cells age and that the intercellular gene expression noise increases during aging, whereas the intracellular noise stays unchanged. A stochastic model provides quantitative mechanistic insights into the observed noise dynamics and sheds light on the age-dependent intracellular noise differences between diploid and haploid yeast. Our work elucidates how a set of canonical phenotypes dynamically change while the host cells are aging in real time, providing essential insights for a comprehensive understanding on and control of lifespan at the single-cell level.

•

A microfluidic device facilitates longitudinal observation of aging diploid yeast

•

Proteins with short versus long half-lives exhibit distinct dynamics as cells age

•

Intercellular gene expression noise increases during replicative aging

•

Unlike haploid yeast, intracellular noise is unchanged during aging in diploid yeast

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

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

Macroscopic models for networks of coupled biological oscillators

The study of synchronization of coupled biological oscillators is fundamental to many areas of biology including neuroscience, cardiac dynamics, and circadian rhythms. Mathematical models of these systems may involve hundreds of variables in thousands of individual cells resulting in an extremely high-dimensional description of the system. This often contrasts with the low-dimensional dynamics exhibited on the collective or macroscopic scale for these systems. We introduce a macroscopic reduction for networks of coupled oscillators motivated by an elegant structure we find in experimental measurements of circadian protein expression and several mathematical models for coupled biological oscillators. The observed structure in the collective amplitude of the oscillator population differs from the well-known Ott-Antonsen ansatz, but its emergence can be characterized through a simple argument depending only on general phase-locking behavior in coupled oscillator systems. We further demonstrate its emergence in networks of noisy heterogeneous oscillators with complex network connectivity. Applying this structure, we derive low-dimensional macroscopic models for oscillator population activity. This approach allows for the incorporation of cellular-level experimental data into the macroscopic model whose parameters and variables can then be directly associated with tissue- or organism-level properties, thereby elucidating the core properties driving the collective behavior of the system.

Modeling transcriptional co-regulation of mammalian circadian clock

The circadian clock is a self-sustaining oscillator that has a period of about 24 hours at the molecular level. The oscillator is a transcription-translation feedback loop system composed of several genes. In this paper, a scalar nonlinear differential equation with two delays, modeling the transcriptional co-regulation in mammalian circadian clock, is proposed and analyzed. Sufficient conditions are established for the asymptotic stability of the unique nontrivial positive equilibrium point of the model by studying an exponential polynomial characteristic equation with delay-dependent coefficients. The existence of the Hopf bifurcations can be also obtained. Numerical simulations of the model with proper parameter values coincide with the theoretical result.

Modeling Strengthens Molecular Link between Circadian Polymorphisms and Major Mood Disorders

Anxiety and other mood disorders, such as major depressive disorder (MDD) and seasonal affective disorder (SAD), affect nearly one-fifth of the global population and disproportionately affect young adults. Individuals affected by mood disorders are frequently plagued by sleep and circadian problems, and recent genetic studies provide ample support for the association of circadian and sleep syndromes with depression and anxiety. Mathematical modeling has been crucial in understanding some of the essential features of the mammalian circadian clock and is now a vital tool for dissecting how circadian genes regulate the molecular mechanisms that influence mood. Here, we model the effect of five clock gene polymorphisms, previously linked to mood disorders, on circadian gene expression and, ultimately, on the period length and amplitude of the clock, two parameters that dictate the phase, or alignment, of the clock relative to the environment. We then test whether these gene variants are associated with circadian phenotypes (Horne-Ostberg Morningness-Eveningness scores) and well-established measures of depression (Beck Depression Inventory) and anxiety (State-Trait Anxiety Inventory) in a population of undergraduates ( n = 546). In this population, we find significant allelic and/or genotypic associations between CRY2 and two PER3 variants and diurnal preference. The PER3 length polymorphism (rs57875989) was significantly associated with depression in this sample, and individuals homozygous for the PER3 single nucleotide polymorphism (SNP) (rs228697) reported significantly higher anxiety. Our simple model satisfies available experimental knockdown conditions as well as existing data on clock polymorphisms associated with mood. In addition, our model enables us to predict circadian phenotypes (e.g., altered period length, amplitude) associated with mood disorders in order to identify critical effects of clock gene mutations on CRY/BMAL binding and to predict that the intronic SNPs studied represent gain-of-function mutations, causing increased transcription rate. Given the user-friendly structure of our model, we anticipate that it will be useful for further study of the relationships among clock polymorphisms, circadian misalignment, and mood disorders.

Long negative feedback loop enhances period tunability of biological oscillators

Oscillatory phenomena play a major role in organisms. In some biological oscillations such as cell cycles and heartbeats, the period can be tuned without significant changes in the amplitude. This property is called (period) tunability, one of the prominent features of biological oscillations. However, how biological oscillators produce tunable oscillations remains largely unexplored. We tackle this question using computational experiments. It has been reported that positive-plus-negative feedback oscillators produce tunable oscillations through the hysteresis-based mechanism. First, in this study, we confirmed that positive-plus-negative feedback oscillators generate tunable oscillations. Second, we found that tunability is positively correlated with the dynamic range of oscillations. Third, we showed that long negative feedback oscillators without any additional positive feedback loops can produce tunable oscillations. Finally, we computationally demonstrated that by lengthening the negative feedback loop, the Repressilator, known as a non-tunable synthetic gene oscillator, can be converted into a tunable oscillator. This work provides synthetic biologists with clues to design tunable gene oscillators.

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

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

Modeling the interactions of sense and antisense Period transcripts in the mammalian circadian clock network

In recent years, it has become increasingly apparent that antisense transcription plays an important role in the regulation of gene expression. The circadian clock is no exception: an antisense transcript of the mammalian core-clock gene PERIOD2 (PER2), which we shall refer to as Per2AS RNA, oscillates with a circadian period and a nearly 12 h phase shift from the peak expression of Per2 mRNA. In this paper, we ask whether Per2AS plays a regulatory role in the mammalian circadian clock by studying in silico the potential effects of interactions between Per2 and Per2AS RNAs on circadian rhythms. Based on the antiphasic expression pattern, we consider two hypotheses about how Per2 and Per2AS mutually interfere with each other's expression. In our pre-transcriptional model, the transcription of Per2AS RNA from the non-coding strand represses the transcription of Per2 mRNA from the coding strand and vice versa. In our post-transcriptional model, Per2 and Per2AS transcripts form a double-stranded RNA duplex, which is rapidly degraded. To study these two possible mechanisms, we have added terms describing our alternative hypotheses to a published mathematical model of the molecular regulatory network of the mammalian circadian clock. Our pre-transcriptional model predicts that transcriptional interference between Per2 and Per2AS can generate alternative modes of circadian oscillations, which we characterize in terms of the amplitude and phase of oscillation of core clock genes. In our post-transcriptional model, Per2/Per2AS duplex formation dampens the circadian rhythm. In a model that combines pre- and post-transcriptional controls, the period, amplitude and phase of circadian proteins exhibit non-monotonic dependencies on the rate of expression of Per2AS. All three models provide potential explanations of the observed antiphasic, circadian oscillations of Per2 and Per2AS RNAs. They make discordant predictions that can be tested experimentally in order to distinguish among these alternative hypotheses.

Evidence for widespread dysregulation of circadian clock progression in human cancer

The ubiquitous daily rhythms in mammalian physiology are guided by progression of the circadian clock. In mice, systemic disruption of the clock can promote tumor growth. In vitro, multiple oncogenes can disrupt the clock. However, due to the difficulties of studying circadian rhythms in solid tissues in humans, whether the clock is disrupted within human tumors has remained unknown. We sought to determine the state of the circadian clock in human cancer using publicly available transcriptome data. We developed a method, called the clock correlation distance (CCD), to infer circadian clock progression in a group of samples based on the co-expression of 12 clock genes. Our method can be applied to modestly sized datasets in which samples are not labeled with time of day and coverage of the circadian cycle is incomplete. We used the method to define a signature of clock gene co-expression in healthy mouse organs, then validated the signature in healthy human tissues. By then comparing human tumor and non-tumor samples from twenty datasets of a range of cancer types, we discovered that clock gene co-expression in tumors is consistently perturbed. Subsequent analysis of data from clock gene knockouts in mice suggested that perturbed clock gene co-expression in human cancer is not caused solely by the inactivation of clock genes. Furthermore, focusing on lung cancer, we found that human lung tumors showed systematic changes in expression in a large set of genes previously inferred to be rhythmic in healthy lung. Our findings suggest that clock progression is dysregulated in many solid human cancers and that this dysregulation could have broad effects on circadian physiology within tumors. In addition, our approach opens the door to using publicly available data to infer circadian clock progression in a multitude of human phenotypes.

CLOCKΔ19 mutation modifies the manner of synchrony among oscillation neurons in the suprachiasmatic nucleus

In mammals, the principal circadian oscillator exists in the hypothalamic suprachiasmatic nucleus (SCN). In the SCN, CLOCK works as an essential component of molecular circadian oscillation, and ClockΔ19 mutant mice show unique characteristics of circadian rhythms such as extended free running periods, amplitude attenuation, and high-magnitude phase-resetting responses. Here we investigated what modifications occur in the spatiotemporal organization of clock gene expression in the SCN of ClockΔ19 mutants. The cultured SCN, sampled from neonatal homozygous ClockΔ19 mice on an ICR strain comprising PERIOD2::LUCIFERASE, demonstrated that the Clock gene mutation not only extends the circadian period, but also affects the spatial phase and period distribution of circadian oscillations in the SCN. In addition, disruption of the synchronization among neurons markedly attenuated the amplitude of the circadian rhythm of individual oscillating neurons in the mutant SCN. Further, with numerical simulations based on the present studies, the findings suggested that, in the SCN of the ClockΔ19 mutant mice, stable oscillation was preserved by the interaction among oscillating neurons, and that the orderly phase and period distribution that makes a phase wave are dependent on the functionality of CLOCK.

Stability of Wake-Sleep Cycles Requires Robust Degradation of the PERIOD Protein

Robustness in biology is the stability of phenotype under diverse genetic and/or environmental perturbations. The circadian clock has remarkable stability of period and phase that-unlike other biological oscillators-is maintained over a wide range of conditions. Here, we show that the high fidelity of the circadian system stems from robust degradation of the clock protein PERIOD. We show that PERIOD degradation is regulated by a balance between ubiquitination and deubiquitination, and that disruption of this balance can destabilize the clock. In mice with a loss-of-function mutation of the E3 ligase gene β-Trcp2, the balance of PERIOD degradation is perturbed and the clock becomes dramatically unstable, presenting a unique behavioral phenotype unlike other circadian mutant animal models. We believe that our data provide a molecular explanation for how circadian phases, such as wake-sleep onset times, can become unstable in humans, and we present a unique mouse model to study human circadian disorders with unstable circadian rhythm phases.

Encoding seasonal information in a two-oscillator model of the multi-oscillator circadian clock

The suprachiasmatic nucleus (SCN) is a collection of about 10 000 neurons, each of which functions as a circadian clock with slightly different periods and phases, that work in concert with form and maintain the master circadian clock for the organism. The diversity among neurons confers on the SCN the ability to robustly encode both the 24-h light pattern as well as the seasonal time. Cluster synchronization brings the different neurons into line and reduces the large population to essentially two oscillators, coordinated by a macroscopic network motif of asymmetric repulsive-attractive coupling. We recount the steps leading to this simplification and rigorously examine the two-oscillator case by seeking an analytical solution. Through these steps, we identify physiologically relevant parameters that shape the behaviour of the SCN network and delineate its ability to store past details of seasonal variation in photoperiod.

Reentrainment of the circadian pacemaker during jet lag: East-west asymmetry and the effects of north-south travel

The normal alignment of circadian rhythms with the 24-h light-dark cycle is disrupted after rapid travel between home and destination time zones, leading to sleep problems, indigestion, and other symptoms collectively known as jet lag. Using mathematical and computational analysis, we study the process of reentrainment to the light-dark cycle of the destination time zone in a model of the human circadian pacemaker. We calculate the reentrainment time for travel between any two points on the globe at any time of the day and year. We construct one-dimensional entrainment maps to explain several properties of jet lag, such as why most people experience worse jet lag after traveling east than west. We show that this east-west asymmetry depends on the endogenous period of the traveler's circadian clock as well as daylength. Thus the critical factor is not simply whether the endogenous period is greater than or less than 24 h as is commonly assumed. We show that the unstable fixed point of an entrainment map determines whether a traveler reentrains through phase advances or phase delays, providing an understanding of the threshold that separates orthodromic and antidromic modes of reentrainment. Contrary to the conventional wisdom that jet lag only occurs after east-west travel across multiple time zones, we predict that the change in daylength encountered during north-south travel can cause jet lag even when no time zones are crossed. Our techniques could be used to provide advice to travelers on how to minimize jet lag on trips involving multiple destinations and a combination of transmeridian and translatitudinal travel.

Quantitative Systems Biology to decipher design principles of a dynamic cell cycle network: the "Maximum Allowable mammalian Trade-Off-Weight" (MAmTOW)

Network complexity is required to lend cellular processes flexibility to respond timely to a variety of dynamic signals, while simultaneously warranting robustness to protect cellular integrity against perturbations. The cell cycle serves as a paradigm for such processes; it maintains its frequency and temporal structure (although these may differ among cell types) under the former, but accelerates under the latter. Cell cycle molecules act together in time and in different cellular compartments to execute cell type-specific programs. Strikingly, the timing at which molecular switches occur is controlled by abundance and stoichiometry of multiple proteins within complexes. However, traditional methods that investigate one effector at a time are insufficient to understand how modulation of protein complex dynamics at cell cycle transitions shapes responsiveness, yet preserving robustness. To overcome this shortcoming, we propose a multidisciplinary approach to gain a systems-level understanding of quantitative cell cycle dynamics in mammalian cells from a new perspective. By suggesting advanced experimental technologies and dedicated modeling approaches, we present innovative strategies (i) to measure absolute protein concentration in vivo, and (ii) to determine how protein dosage, e.g., altered protein abundance, and spatial (de)regulation may affect timing and robustness of phase transitions. We describe a method that we name “Maximum Allowable mammalian Trade–Off–Weight” (MAmTOW), which may be realized to determine the upper limit of gene copy numbers in mammalian cells. These aspects, not covered by current systems biology approaches, are essential requirements to generate precise computational models and identify (sub)network-centered nodes underlying a plethora of pathological conditions.

Circadian Clock Model Supports Molecular Link Between PER3 and Human Anxiety

Generalized anxiety and major depression have become increasingly common in the United States, affecting 18.6 percent of the adult population. Mood disorders can be debilitating, and are often correlated with poor general health, life dissatisfaction, and the need for disability benefits due to inability to work. Recent evidence suggests that some mood disorders have a circadian component, and disruptions in circadian rhythms may even trigger the development of these disorders. However, the molecular mechanisms of this interaction are not well understood. Polymorphisms in a circadian clock-related gene, PER3, are associated with behavioral phenotypes (extreme diurnal preference in arousal and activity) and sleep/mood disorders, including seasonal affective disorder (SAD). Here we show that two PER3 mutations, a variable number tandem repeat (VNTR) allele and a single-nucleotide polymorphism (SNP), are associated with diurnal preference and higher Trait-Anxiety scores, supporting a role for PER3 in mood modulation. In addition, we explore a potential mechanism for how PER3 influences mood by utilizing a comprehensive circadian clock model that accurately predicts the changes in circadian period evident in knock-out phenotypes and individuals with PER3-related clock disorders.