A model of the cell-autonomous mammalian circadian clock

Computational modeling establishes mechanotransduction as a potent modulator of the mammalian circadian clock

Mechanotransduction, which is the integration of mechanical signals from the external environment of a cell to changes in intracellular signaling, governs many cellular functions. Recent studies have shown that the mechanical state of the cell is also coupled to the cellular circadian clock. To investigate possible interactions between circadian rhythms and cellular mechanotransduction, we have developed a computational model that integrates the two pathways. We postulated that translocation of the transcriptional regulators MRTF (herein referring to both MRTF-A and MRTF-B), YAP and TAZ (also known as YAP1 and WWTR1, respectively; collectively denoted YAP/TAZ) into the nucleus leads to altered expression of circadian proteins. Simulations from our model predict that lower levels of cytoskeletal activity are associated with longer circadian oscillation periods and higher oscillation amplitudes, which is consistent with recent experimental observations. Furthermore, accumulation of YAP/TAZ and MRTF in the nucleus causes circadian oscillations to decay in our model. These effects hold both at the single-cell level and within a population-level framework. Finally, we investigated the effects of mutations in YAP or lamin A, the latter of which result in a class of diseases known as laminopathies. In silico, oscillations in circadian proteins are substantially weaker in populations of cells with mutations in YAP or lamin A, suggesting that defects in mechanotransduction can disrupt the circadian clock in certain disease states; however, reducing substrate stiffness in the model restores normal oscillatory behavior, suggesting a possible compensatory mechanism. Thus, our study identifies that mechanotransduction could be a potent modulatory cue for cellular clocks and that this crosstalk can be leveraged to rescue the circadian clock in disease states.

Patterns of change in regulatory modules of chemical reaction systems induced by network modification

Cellular functions are realized through the dynamics of chemical reaction networks formed by thousands of chemical reactions. Numerical studies have empirically demonstrated that small differences in network structures among species or tissues can cause substantial changes in dynamics. However, a general principle for behavior changes in response to network structure modifications is not known. The chemical reaction system possesses substructures called buffering structures, which are characterized by a certain topological index being zero. It was proven that the steady-state response to modulation of reaction parameters inside a buffering structure is localized in the buffering structure. In this study, we developed a method to systematically identify the loss or creation of buffering structures induced by the addition of a single degradation reaction from network structure alone. This makes it possible to predict the qualitative and macroscopic changes in regulation that will be caused by the network modification. This method was applied to two reaction systems: the central metabolic system and the mitogen-activated protein kinases signal transduction system. Our method enables identification of reactions that are important for biological functions in living systems.

Peripheral clock gene oscillations are perturbed in neonatal and adult rat offspring raised under adverse limited bedding conditions

Circadian (24-h) rhythms in the suprachiasmatic nucleus (SCN) are established in utero in rodents, but rhythmicity of peripheral circadian clocks appears later in postnatal development. Since peripheral oscillators can be influenced by maternal feeding and behavior, we investigated whether exposure to the adverse environmental conditions of limited bedding (LB) during postnatal life would alter rhythmicity in the SCN, adrenal gland and liver in neonatal (postnatal day PND10), juvenile (PND28) and adult rats. We also examined locomotor activity in adults. Limited bedding increased nursing time and slightly increased fragmentation of maternal behavior. Exposure to LB reduced the amplitude of Per2 in the SCN on PND10. Adrenal clock gene expression (Bmal1, Per2, Cry1, Rev-erbα, Dbp) and corticosterone secretion were rhythmic at all ages in NB offspring, whereas rhythmicity of Bmal1, Cry1 and corticosterone was abolished in neonatal LB pups. Circadian gene expression in the adrenal and liver was well established by PND28. In adults, liver expression of several circadian genes was increased at specific daytimes by LB and the microstructure of locomotor behavior was altered. Thus, changes in maternal care and behavior might provide important signals to the maturing peripheral oscillators and modify, in particular their output functions in the long-term.

Modeling the Effects of Napping and Non-napping Patterns of Light Exposure on the Human Circadian Oscillator

In early childhood, consolidation of sleep from a biphasic to a monophasic sleep-wake pattern, that is, the transition from sleeping during an afternoon nap and at night to sleeping only during the night, represents a major developmental milestone. Reduced napping behavior is associated with an advance in the timing of the circadian system; however, it is unknown if this advance represents a standard response of the circadian clock to altered patterns of light exposure or if it additionally reflects features of the developing circadian system. Using a mathematical model of the human circadian pacemaker, we investigated the impact of napping and non-napping patterns of light exposure on entrained circadian phases. Simulated light schedules were based on published data from 20 children (34.2 ± 2.0 months) with habitual napping or non-napping sleep patterns (15 nappers). We found the model predicted different circadian phases for napping and non-napping light patterns: both the decrease in afternoon light during the nap and the increase in evening light associated with napping toddlers' later bedtimes contributed to the observed circadian phase difference produced between napping and non-napping light schedules. We systematically quantified the effects on phase shifting of nap duration, timing, and light intensity, finding larger phase delays occurred for longer and earlier naps. In addition, we simulated phase response curves to a 1-h light pulse and 1-h dark pulse to predict phase and intensity dependence of these changes in light exposure. We found the light pulse produced larger shifts compared with the dark pulse, and we analyzed the model dynamics to identify the features contributing to this asymmetry. These findings suggest that napping status affects circadian timing due to altered patterns of light exposure, with the dynamics of the circadian clock and light processing mediating the effects of the dark pulse associated with a daytime nap.

Amplitude response and singularity behavior of circadian clock to external stimuli

Amplitude changes caused by environmental cues are universal in the circadian clock and associated with various diseases. Singularity behavior, characterized by the disruption of circadian rhythms due to critical stimuli, has been observed across various species. Several mathematical models of the circadian clock have replicated this phenomenon. A comprehensive understanding of the amplitude response remains elusive due to experimental limitations. In this study, we address this question by utilizing a simple normal form model that accurately fits previous experimental data, thereby presenting a general mechanism. We employ a geometric framework to illustrate the dynamics in different stimuli of light-induced transcription (LIT) and light-induced degradation (LID), highlighting the core role of invisible instability in amplitude response. Our model systematically elucidates how stimulus mode, phase, and strength determine amplitude responses. The results show that external stimuli induce alterations in both the amplitudes of individual oscillators and the synchronization among oscillators, collectively influencing the overall amplitude response. While experimental methods impose constraints resulting in limited outcomes under specific conditions, our model provides a comprehensive and three-dimensional mechanistic explanation. A comparison with existing experimental findings demonstrates the consistency of our proposed mechanism. Considering the response direction, the framework enables the identification of phases that lead to increased circadian amplitude. Based on this mechanism derived from the framework, stimulus strategies for resetting circadian rhythms with reduced side effects could be designed. Our results demonstrate that the framework has great potential for understanding and applying stimulus responses in the circadian clock and other limit cycle oscillations.

Controlling complex dynamical systems based on the structure of the networks

Progress of molecular biology resulted in the accumulation of information on biomolecular interactions, which are complex enough to be termed as networks. Dynamical behavior generated by complex network systems is considered to be the origin of the biological functions. One of the largest missions in modern life science is to obtain logical understanding for the dynamics of complex systems based on experimentally identified networks. However, a network does not provide sufficient information to specify dynamics explicitly, i.e. it lacks information of mathematical formulae of functions or parameter values. One has to develop mathematical models under assumptions of functions and parameter values to know the detail of dynamics of network systems. In this review, on the other hand, we introduce our own mathematical theory to understand the behavior of biological systems from the information of regulatory networks alone. Using the theory, important aspects of dynamical properties can be extracted from networks. Namely, key factors for observing/controlling the whole dynamical system are determined from network structure alone. We also show an application of the theory to a real biological system, a gene regulatory network for cell-fate specification in ascidian. We demonstrate that the system was completely controllable by experimental manipulations of the key factors identified by the theory from the information of network alone. This review article is an extended version of the Japanese article, Controlling Cell-Fate Specification System Based on a Mathematical Theory of Network Dynamics, published in SEIBUTSU BUTSURI Vol. 60, p. 349-351 (2020).

Dynamic modelling predicts lactate and IL-1β as interventional targets in metabolic-inflammation-clock regulatory loop in glioma

In an attempt to understand the role of dysregulated circadian rhythm in glioma, our recent findings highlighted the existence of a feed-forward loop between tumour metabolite lactate, pro-inflammatory cytokine IL-1β and circadian CLOCK. To further elucidate the implication of this complex interplay, we developed a mathematical model that quantitatively describes this lactate dehydrogenase A (LDHA)-IL-1β-CLOCK/BMAL1 circuit and predicts potential therapeutic targets. The model was calibrated on quantitative western blotting data in two glioma cell lines in response to either lactate inhibition or IL-1β stimulation. The calibrated model described the experimental data well and most of the parameters were identifiable, thus the model was predictive. Sensitivity analysis identified IL-1β and LDHA as potential intervention points. Mathematical models described here can be useful to understand the complex interrelationship between metabolism, inflammation and circadian rhythm, and in designing effective therapeutic strategies. Our findings underscore the importance of including the circadian clock when developing pharmacological approaches that target aberrant tumour metabolism and inflammation. Insight box  The complex interplay of metabolism-inflammation-circadian rhythm in tumours is not well understood. Our recent findings provided evidence of a feed-forward loop between tumour metabolite lactate, pro-inflammatory cytokine IL-1β and circadian CLOCK/BMAL1 in glioma. To elucidate the implication of this complex interplay, we developed a mathematical model that quantitatively describes this LDHA-IL-1β-CLOCK/BMAL1 circuit and integrates experimental data to predict potential therapeutic targets. The study employed a multi-start optimization strategy and profile likelihood estimations for parameter estimation and assessing identifiability. The simulations are in reasonable agreement with the experimental data. Sensitivity analysis found LDHA and IL-1β as potential therapeutic points. Mathematical models described here can provide insights to understand the complex interrelationship between metabolism, inflammation and circadian rhythm, and in identifying effective therapeutic targets.

Systematic analysis of negative and positive feedback loops for robustness and temperature compensation in circadian rhythms

Temperature compensation and robustness to biological noise are two key characteristics of the circadian clock. These features allow the circadian pacemaker to maintain a steady oscillation in a wide range of environmental conditions. The presence of a time-delayed negative feedback loop in the regulatory network generates autonomous circadian oscillations in eukaryotic systems. In comparison, the circadian clock of cyanobacteria is controlled by a strong positive feedback loop. Positive feedback loops with substrate depletion can also generate oscillations, inspiring other circadian clock models. What makes a circadian oscillatory network robust to extrinsic noise is unclear. We investigated four basic circadian oscillators with negative, positive, and combinations of positive and negative feedback loops to explore network features necessary for circadian clock resilience. We discovered that the negative feedback loop system performs the best in compensating temperature changes. We also show that a positive feedback loop can reduce extrinsic noise in periods of circadian oscillators, while intrinsic noise is reduced by negative feedback loops.

Level of constitutively expressed BMAL1 affects the robustness of circadian oscillations

The circadian rhythm is a biological oscillation of physiological activities with a period of approximately 24 h, that is driven by a cell-autonomous oscillator called the circadian clock. The current model of the mammalian circadian clock is based on a transcriptional-translational negative feedback loop in which the protein products of clock genes accumulate in a circadian manner and repress their own transcription. However, several studies have revealed that constitutively expressed clock genes can maintain circadian oscillations. To understand the underlying mechanism, we expressed Bmal1 in Bmal1-disrupted cells using a doxycycline-inducible promoter and monitored Bmal1 and Per2 promoter activity using luciferase reporters. Although the levels of BMAL1 and other clock proteins, REV-ERBα and CLOCK, showed no obvious rhythmicity, robust circadian oscillation in Bmal1 and Per2 promoter activities with the correct phase relationship was observed, which proceeded in a doxycycline-concentration-dependent manner. We applied transient response analysis to the Bmal1 promoter activity in the presence of various doxycycline concentrations. Based on the obtained transfer functions, we suggest that, at least in our experimental system, BMAL1 is not directly involved in the oscillatory process, but modulates the oscillation robustness by regulating basal clock gene promoter activity.

Circadian Rhythm Regulated by Tumor Suppressor p53 and Time Delay in Unstressed Cells

Circadian function and p53 network are interconnected on the molecular level, but the dynamics induced by the interaction between the circadian factor Per2 and the tumor suppressor p53 remains poorly understood. Here, we constructed an integrative model composed of a circadian clock module and a p53-Mdm2 feedback module to study the dynamics of p53-Per2 network in unstressed cells. As expected, the model can accurately predict the circadian rhythm, which is consistent with diverse experimental observations. In addition, using a combination of theoretical analysis and numerical simulation, the results demonstrated that p53 expression enhances the phase advance of circadian rhythm and reduces the robustness of circadian rhythm. Furthermore, the time delay required for the transcription and translation of Per2 protein induces oscillations by undergoing a supercritical Hopf bifurcation, and improves the robustness of circadian rhythm. In summary, this work shows that the p53-Per2 interaction and the time delay are two essential factors for circadian functions.

A Phenomenological Mouse Circadian Pacemaker Model

Mathematical models have been used extensively in chronobiology to explore characteristics of biological clocks. In particular, for human circadian studies, the Kronauer model has been modified multiple times to describe rhythm production and responses to sensory input. This phenomenological model comprises a single set of parameters which can simulate circadian responses in humans under a variety of environmental conditions. However, corresponding models for nocturnal rodents commonly used in circadian rhythm studies are not available and may require new parameter values for different species and even strains. Moreover, due to a considerable variation in experimental data collected from mice of the same strain, within and across laboratories, a range of valid parameters is essential. This study develops a Kronauer-like model for mice by re-fitting relevant parameters to published phase response curve and period data using total least squares. Local parameter sensitivity analysis and parameter distributions determine the parameter ranges that give a near-identical model and data distribution of periods. However, the model required further parameter adjustments to match characteristics of other mouse strains, implying that the model itself detects changes in the core processes of rhythm generation and control. The model is a useful tool to understand and interpret future mouse circadian clock experiments.

Time-keeping and decision-making in living cells: Part I

To survive and reproduce, a cell must process information from its environment and its own internal state and respond accordingly, in terms of metabolic activity, gene expression, movement, growth, division and differentiation. These signal–response decisions are made by complex networks of interacting genes and proteins, which function as biochemical switches and clocks, and other recognizable information-processing circuitry. This theme issue of Interface Focus (in two parts) brings together articles on time-keeping and decision-making in living cells—work that uses precise mathematical modelling of underlying molecular regulatory networks to understand important features of cell physiology. Part I focuses on time-keeping: mechanisms and dynamics of biological oscillators and modes of synchronization and entrainment of oscillators, with special attention to circadian clocks.

Mathematical analysis of robustness of oscillations in models of the mammalian circadian clock

Circadian rhythms in a wide range of organisms are mediated by molecular mechanisms based on transcription-translation feedback. In this paper, we use bifurcation theory to explore mathematical models of genetic oscillators, based on Kim & Forger’s interpretation of the circadian clock in mammals. At the core of their models is a negative feedback loop whereby PER proteins (PER1 and PER2) bind to and inhibit their transcriptional activator, BMAL1. For oscillations to occur, the dissociation constant of the PER:BMAL1 complex, K^d, must be ≤ 0.04 nM, which is orders of magnitude smaller than a reasonable expectation of 1–10 nM for this protein complex. We relax this constraint by two modifications to Kim & Forger’s ‘single negative feedback’ (SNF) model: first, by introducing a multistep reaction chain for posttranscriptional modifications of Per mRNA and posttranslational phosphorylations of PER, and second, by replacing the first-order rate law for degradation of PER in the nucleus by a Michaelis-Menten rate law. These modifications increase the maximum allowable K^d to ~2 nM. In a third modification, we consider an alternative rate law for gene transcription to resolve an unrealistically large rate of Per2 transcription at very low concentrations of BMAL1. Additionally, we studied extensions of the SNF model to include a second negative feedback loop (involving REV-ERB) and a supplementary positive feedback loop (involving ROR). Contrary to Kim & Forger’s observations of these extended models, we find that, with our modifications, the supplementary positive feedback loop makes the oscillations more robust than observed in the models with one or two negative feedback loops. However, all three models are similarly robust when accounting for circadian rhythms (~24 h period) with K^d ≥ 1 nM. Our results provide testable predictions for future experimental studies.

The circadian rhythm aligns bodily functions to the day/night cycle and is important for our health. The rhythm originates from an intracellular molecular clock mechanism that mediates rhythmic gene expression. It is long understood that transcriptional negative feedback with sufficient time delay is key to generating circadian oscillations. However, some of the most widely cited mathematical models for the circadian clock suffer from problems of parameter ‘fragilities’. That is, sustained oscillations are possible only for physically unrealistic parameter values. A recent model by Kim & Forger nicely incorporates the inhibitory binding of PER proteins to their transcription activator BMAL1, but oscillations in the Kim-Forger model require a binding affinity between PER and BMAL1 that is orders of magnitude larger than observed binding affinities of protein complexes. To rectify this problem, we make several physiologically credible modifications to the Kim-Forger model, which allow oscillations to occur with more realistic binding affinities. The modified model is further extended to explore the potential roles of supplementary feedback loops in the mammalian clock mechanism. Ultimately, accurate models of the circadian clock will provide better predictive tools for chronotherapy and chrono-pharmacology studies.

Functional observability and target state estimation in large-scale networks

Observing the states of a network is fundamental to our ability to explore and control the dynamics of complex natural, social, and technological systems. The problem of determining whether the system is observable has been addressed by network control researchers over the past decade. Progress on the further problem of actually designing and implementing efficient algorithms to infer the states from limited measurements has been hampered by the high dimensionality of large-scale networks. Noting that often only a small number of state variables in a network are essential for control, intervention, and monitoring purposes, this work develops a graph-based theory and highly scalable methods that achieve accurate estimation of target variables of network systems with minimal sensing and computational resources.

The quantitative understanding and precise control of complex dynamical systems can only be achieved by observing their internal states via measurement and/or estimation. In large-scale dynamical networks, it is often difficult or physically impossible to have enough sensor nodes to make the system fully observable. Even if the system is in principle observable, high dimensionality poses fundamental limits on the computational tractability and performance of a full-state observer. To overcome the curse of dimensionality, we instead require the system to be functionally observable, meaning that a targeted subset of state variables can be reconstructed from the available measurements. Here, we develop a graph-based theory of functional observability, which leads to highly scalable algorithms to 1) determine the minimal set of required sensors and 2) design the corresponding state observer of minimum order. Compared with the full-state observer, the proposed functional observer achieves the same estimation quality with substantially less sensing and fewer computational resources, making it suitable for large-scale networks. We apply the proposed methods to the detection of cyberattacks in power grids from limited phase measurement data and the inference of the prevalence rate of infection during an epidemic under limited testing conditions. The applications demonstrate that the functional observer can significantly scale up our ability to explore otherwise inaccessible dynamical processes on complex networks.

A Mathematical Model to Characterize the Role of Light Adaptation in Mammalian Circadian Clock

In response to a light stimulus, the mammalian circadian clock first dramatically increases the expression of Per1 mRNA, and then drops to a baseline even when light persists. This phenomenon is known as light adaptation, which has been experimentally proven to be related to the CRTC1-SIK1 pathway in suprachiasmatic nucleus (SCN). However, the role of this light adaptation in the circadian rhythm remains to be elucidated. To reveal the in-depth function of light adaptation and the underlying dynamics, we proposed a mathematical model for the CRTC1-SIK1 network and coupled it to a mammalian circadian model. The simulation result proved that the light adaptation is achieved by the self-inhibition of the CRTC1/CREB complex. Also, consistently with experimental observations, this adaptation mechanism can limit the phase response to short-term light stimulus, and it also restricts the rate of the phase shift in a jet lag protocol to avoid overly rapid re-entrainment. More importantly, this light adaptation is predicted to prevent the singularity behavior in the cell population, which represents the abolishment of circadian rhythmicity due to desynchronization of oscillating cells. Furthermore, it has been shown to provide refractoriness to successive stimuli with short gap. Therefore, we concluded that the light adaptation generated by the CRTC1-SIK1 pathway in the SCN provides a robust mechanism, allowing the circadian system to maintain homeostasis in the presence of light perturbations. These results not only give new insights into the dynamics of light adaptation from a computational perspective but also lead us to formulate hypotheses about the related physiological significance.

The Goodwin Oscillator and its Legacy

In the 1960's Brian Goodwin published a couple of mathematical models showing how feedback inhibition can lead to oscillations and discussed possible implications of this behaviour for the physiology of the cell. He also presented key ideas about the rich dynamics that may result from the coupling between such biochemical oscillators. Goodwin's work motivated a series of theoretical investigations aiming at identifying minimal mechanisms to generate limit cycle oscillations and deciphering design principles of biological oscillators. The three-variable Goodwin model (adapted by Griffith) can be seen as a core model for a large class of biological systems, ranging from ultradian to circadian clocks. We summarize here main ideas and results brought by Goodwin and review a couple of modeling works directly or indirectly inspired by Goodwin's findings.

A mathematical model of the circadian clock and drug pharmacology to optimize irinotecan administration timing in colorectal cancer

Scheduling anticancer drug administration over 24 h may critically impact treatment success in a patient-specific manner. Here, we address personalization of treatment timing using a novel mathematical model of irinotecan cellular pharmacokinetics and -dynamics linked to a representation of the core clock and predict treatment toxicity in a colorectal cancer (CRC) cellular model. The mathematical model is fitted to three different scenarios: mouse liver, where the drug metabolism mainly occurs, and two human colorectal cancer cell lines representing an in vitro experimental system for human colorectal cancer progression. Our model successfully recapitulates quantitative circadian datasets of mRNA and protein expression together with timing-dependent irinotecan cytotoxicity data. The model also discriminates time-dependent toxicity between the different cells, suggesting that treatment can be optimized according to their cellular clock. Our results show that the time-dependent degradation of the protein mediating irinotecan activation, as well as an oscillation in the death rate may play an important role in the circadian variations of drug toxicity. In the future, this model can be used to support personalized treatment scheduling by predicting optimal drug timing based on the patient's gene expression profile.

Exploiting circadian memory to hasten recovery from circadian misalignment

Recent years have seen a sustained interest in the development of circadian reentrainment strategies to limit the deleterious effects of jet lag. Due to the dynamical complexity of many circadian models, phase-based model reduction techniques are often an imperative first step in the analysis. However, amplitude coordinates that capture lingering effects (i.e., memory) from past inputs are often neglected. In this work, we focus on these amplitude coordinates using an operational phase and an isostable coordinate framework in the context of the development of jet-lag amelioration strategies. By accounting for the influence of circadian memory, we identify a latent phase shift that can prime one's circadian cycle to reentrain more rapidly to an expected time-zone shift. A subsequent optimal control problem is proposed that balances the trade-off between control effort and the resulting latent phase shift. Data-driven model identification techniques for the inference of necessary reduced order, phase-amplitude-based models are considered in situations where the underlying model equations are unknown, and numerical results are illustrated in both a simple planar model and in a coupled population of circadian oscillators.

Circadian Effects of Drug Responses

Circadian rhythms describe physiological systems that repeat themselves with a cycle of approximately 24 h. Our understanding of the cellular and molecular origins of these oscillations has improved dramatically, allowing us to appreciate the significant role these oscillations play in maintaining physiological homeostasis. Circadian rhythms allow living organisms to predict and efficiently respond to a dynamically changing environment, set by repetitive day/night cycles. Since circadian rhythms underlie almost every aspect of human physiology, it is unsurprising that they also influence the response of a living organism to disease, stress, and therapeutics. Therefore, not only do the mechanisms that maintain health and disrupt homeostasis depend on our internal circadian clock, but also the way drugs are perceived and function depends on these physiological rhythms. We present a holistic view of the therapeutic process, discussing components such as disease state, pharmacokinetics, and pharmacodynamics, as well as adverse reactions that are critically affected by circadian rhythms. We outline challenges and opportunities in moving toward personalized medicine approaches that explore and capitalize on circadian rhythms for the benefit of the patient.

Pathway-level analysis of genome-wide circadian dynamics in diverse tissues in rat and mouse

A computational framework is developed to enable the characterization of genome-wide, multi-tissue circadian dynamics at the level of "functional groupings of genes" defined in the context of signaling, cellular/genetic processing and metabolic pathways in rat and mouse. Our aim is to identify how individual genes come together to generate orchestrated rhythmic patterns and how these may vary within and across tissues. We focus our analysis on four tissues (adipose, liver, lung, and muscle). A genome-wide pathway-centric analysis enables us to develop a comprehensive picture on how the observed circadian variation at the individual gene level, orchestrates functional responses at the pathway level. Such pathway-based "meta-data" analysis enables the rational integration and comparison across platforms and/or experimental designs evaluating emergent dynamics, as opposed to comparisons of individual elements. One of our key findings is that when considering the dynamics at the pathway level, a complex behavior emerges. Our work proposes that tissues tend to coordinate gene's circadian expression in a way that optimizes tissue-specific pathway activity, depending of each tissue's broader role in homeostasis.

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.

Complementary phase responses via functional differentiation of dual negative feedback loops

Multiple feedback loops are often found in gene regulations for various cellular functions. In mammalian circadian clocks, oscillations of Period1 (Per1) and Period2 (Per2) expression are caused by interacting negative feedback loops (NFLs) whose protein products with similar molecular functions repress each other. However, Per1 expression peaks earlier than Per2 in the pacemaker tissue, raising the question of whether the peak time difference reflects their different dynamical functions. Here, we address this question by analyzing phase responses of the circadian clock caused by light-induced transcription of both Per1 and Per2 mRNAs. Through mathematical analyses of dual NFLs, we show that phase advance is mainly driven by light inputs to the repressor with an earlier expression peak as Per1, whereas phase delay is driven by the other repressor with a later peak as Per2. Due to the complementary contributions to phase responses, the ratio of light-induced transcription rates between Per1 and Per2 determines the magnitude and direction of phase shifts at each time of day. Specifically, stronger Per1 light induction than Per2 results in a phase response curve (PRC) with a larger phase advance zone than delay zone as observed in rats and hamsters, whereas stronger Per2 induction causes a larger delay zone as observed in mice. Furthermore, the ratio of light-induced transcription rates required for entrainment is determined by the relation between the circadian and light-dark periods. Namely, if the autonomous period of a circadian clock is longer than the light-dark period, a larger light-induced transcription rate of Per1 than Per2 is required for entrainment, and vice versa. In short, the time difference between Per1 and Per2 expression peaks can differentiate their dynamical functions. The resultant complementary contributions to phase responses can determine entrainability of the circadian clock to the light-dark cycle.

Circadian rhythms and the HPA axis: A systems view

The circadian timing system comprises a network of time-keeping clocks distributed across a living host whose responsibility is to allocate resources and distribute functions temporally to optimize fitness. The molecular structures generating these rhythms have evolved to accommodate the rotation of the earth in an attempt to primarily match the light/dark periods during the 24-hr day. To maintain synchrony of timing across and within tissues, information from the central clock, located in the suprachiasmatic nucleus, is conveyed using systemic signals. Leading among those signals are endocrine hormones, and while the hypothalamic-pituitary-adrenal axis through the release of glucocorticoids is a major pacesetter. Interestingly, the fundamental units at the molecular and physiological scales that generate local and systemic signals share critical structural properties. These properties enable time-keeping systems to generate rhythmic signals and allow them to adopt specific properties as they interact with each other and the external environment. The purpose of this review is to provide a broad overview of these structures, discuss their functional characteristics, and describe some of their fundamental properties as these related to health and disease. This article is categorized under: Immune System Diseases > Computational Models Immune System Diseases > Biomedical Engineering.

Electrophysiological Approaches to Studying the Suprachiasmatic Nucleus

In mammals, the part of the nervous system responsible for most circadian behavior can be localized to a bilaterally paired structure in the hypothalamus known as the suprachiasmatic nucleus (SCN). Understanding the mammalian circadian system will require a detailed multilevel analysis of neural SCN circuits ex vivo and in vivo. Many of the techniques and approaches that are used for the analysis of the circuitry driving circadian oscillations in the SCN are similar to those employed in other brain regions. There is, however, one fundamental difference that needs to be taken into consideration, that is, the physiological, cell, and molecular properties of SCN neurons vary with the time of day. In this chapter, we will consider the preparations and electrophysiological techniques that we have used to analyze the SCN circuit focusing on the acute brain slice and intact, freely moving animal.

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.

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.

Mechanistic Modeling of Biochemical Systems without A Priori Parameter Values Using the Design Space Toolbox v.3.0

Mechanistic models of biochemical systems provide a rigorous description of biological phenomena. They are indispensable for making predictions and elucidating biological design principles. To date, mathematical analysis and characterization of these models encounter a bottleneck consisting of large numbers of unknown parameter values. Here, we introduce the Design Space Toolbox v.3.0 (DST3), a software implementation of the Design Space formalism enabling mechanistic modeling without requiring previous knowledge of parameter values. This is achieved by using a phenotype-centric modeling approach, in which the system is first decomposed into a series of biochemical phenotypes. Parameter values realizing phenotypes of interest are subsequently predicted. DST3 represents the most generally applicable implementation of the Design Space formalism and offers unique advantages over earlier versions. By expanding the Design Space formalism and streamlining its distribution, DST3 represents a valuable tool for elucidating biological design principles and designing novel synthetic circuits.

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DST3 extends the Design Space formalism by identifying additional phenotypes

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Additional phenotypes arise from cycles, conservations, and metabolic imbalances

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DST3 enables mechanistic modeling without previous knowledge of parameter values

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It fully unlocks the potential of the novel phenotype-centric modeling strategy

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.

Astrocytic Modulation of Neuronal Activity in the Suprachiasmatic Nucleus: Insights from Mathematical Modeling

The suprachiasmatic nucleus (SCN) of the hypothalamus consists of a highly heterogeneous neuronal population networked together to allow precise and robust circadian timekeeping in mammals. While the critical importance of SCN neurons in regulating circadian rhythms has been extensively studied, the roles of SCN astrocytes in circadian system function are not well understood. Recent experiments have demonstrated that SCN astrocytes are circadian oscillators with the same functional clock genes as SCN neurons. Astrocytes generate rhythmic outputs that are thought to modulate neuronal activity through pre- and postsynaptic interactions. In this study, we developed an in silico multicellular model of the SCN clock to investigate the impact of astrocytes in modulating neuronal activity and affecting key clock properties such as circadian rhythmicity, period, and synchronization. The model predicted that astrocytes could alter the rhythmic activity of neurons via bidirectional interactions at tripartite synapses. Specifically, astrocyte-regulated extracellular glutamate was predicted to increase neuropeptide signaling from neurons. Consistent with experimental results, we found that astrocytes could increase the circadian period and enhance neural synchronization according to their endogenous circadian period. The impact of astrocytic modulation of circadian rhythm amplitude, period, and synchronization was predicted to be strongest when astrocytes had periods between 0 and 2 h longer than neurons. Increasing the number of neurons coupled to the astrocyte also increased its impact on period modulation and synchrony. These computational results suggest that signals that modulate astrocytic rhythms or signaling (e.g., as a function of season, age, or treatment) could cause disruptions in circadian rhythm or serve as putative therapeutic targets.

Conceptual Models of Entrainment, Jet Lag, and Seasonality

Understanding entrainment of circadian rhythms is a central goal of chronobiology. Many factors, such as period, amplitude, Zeitgeber strength, and daylength, govern entrainment ranges and phases of entrainment. We have tested whether simple amplitude-phase models can provide insight into the control of entrainment phases. Using global optimization, we derived conceptual models with just three free parameters (period, amplitude, and relaxation rate) that reproduce known phenotypic features of vertebrate clocks: phase response curves (PRCs) with relatively small phase shifts, fast re-entrainment after jet lag, and seasonal variability to track light onset or offset. Since optimization found multiple sets of model parameters, we could study this model ensemble to gain insight into the underlying design principles. We found complex associations between model parameters and entrainment features. Arnold onions of representative models visualize strong dependencies of entrainment on periods, relative Zeitgeber strength, and photoperiods. Our results support the use of oscillator theory as a framework for understanding the entrainment of circadian clocks.

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.

Quantitative live imaging of Venus::BMAL1 in a mouse model reveals complex dynamics of the master circadian clock regulator

Evolutionarily conserved circadian clocks generate 24-hour rhythms in physiology and behaviour that adapt organisms to their daily and seasonal environments. In mammals, the suprachiasmatic nucleus (SCN) of the hypothalamus is the principal co-ordinator of the cell-autonomous clocks distributed across all major tissues. The importance of robust daily rhythms is highlighted by experimental and epidemiological associations between circadian disruption and human diseases. BMAL1 (a bHLH-PAS domain-containing transcription factor) is the master positive regulator within the transcriptional-translational feedback loops (TTFLs) that cell-autonomously define circadian time. It drives transcription of the negative regulators Period and Cryptochrome alongside numerous clock output genes, and thereby powers circadian time-keeping. Because deletion of Bmal1 alone is sufficient to eliminate circadian rhythms in cells and the whole animal it has been widely used as a model for molecular disruption of circadian rhythms, revealing essential, tissue-specific roles of BMAL1 in, for example, the brain, liver and the musculoskeletal system. Moreover, BMAL1 has clock-independent functions that influence ageing and protein translation. Despite the essential role of BMAL1 in circadian time-keeping, direct measures of its intra-cellular behaviour are still lacking. To fill this knowledge-gap, we used CRISPR Cas9 to generate a mouse expressing a knock-in fluorescent fusion of endogenous BMAL1 protein (Venus::BMAL1) for quantitative live imaging in physiological settings. The Bmal1Venus mouse model enabled us to visualise and quantify the daily behaviour of this core clock factor in central (SCN) and peripheral clocks, with single-cell resolution that revealed its circadian expression, anti-phasic to negative regulators, nuclear-cytoplasmic mobility and molecular abundance.

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.

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.

Reconciling periodic rhythms of large-scale biological networks by optimal control

Periodic rhythms are ubiquitous phenomena that illuminate the underlying mechanism of cyclic activities in biological systems, which can be represented by cyclic attractors of the related biological network. Disorders of periodic rhythms are detrimental to the natural behaviours of living organisms. Previous studies have shown that the state transition from one to another attractor can be accomplished by regulating external signals. However, most of these studies until now have mainly focused on point attractors while ignoring cyclic ones. The aim of this study is to investigate an approach for reconciling abnormal periodic rhythms, such as diminished circadian amplitude and phase delay, to the regular rhythms of complex biological networks. For this purpose, we formulate and solve a mixed-integer nonlinear dynamic optimization problem simultaneously to identify regulation variables and to determine optimal control strategies for state transition and adjustment of periodic rhythms. Numerical experiments are implemented in three examples including a chaotic system, a mammalian circadian rhythm system and a gastric cancer gene regulatory network. The results show that regulating a small number of biochemical molecules in the network is sufficient to successfully drive the system to the target cyclic attractor by implementing an optimal control strategy.

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.

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.

At the Interface of Lifestyle, Behavior, and Circadian Rhythms: Metabolic Implications

Nutrient metabolism is under circadian regulation. Disruption of circadian rhythms by lifestyle and behavioral choices such as work schedules, eating patterns, and social jetlag, seriously impacts metabolic homeostasis. Metabolic dysfunction due to chronic misalignment of an organism's endogenous rhythms is detrimental to health, increasing the risk of obesity, metabolic and cardiovascular disease, diabetes, and cancer. In this paper, we review literature on recent findings on the mechanisms that communicate metabolic signals to circadian clocks and vice versa, and how human behavioral changes imposed by societal and occupational demands affect the physiological networks integrating peripheral clocks and metabolism. Finally, we discuss factors possibly contributing to inter-individual variability in response to circadian changes in the context of metabolic (dys)function.

Hilbert transform-based time-series analysis of the circadian gene regulatory network

In this work, the authors propose the Hilbert transform (HT)‐based numerical method to analyse the time series of the circadian rhythms. They demonstrate the application of HT by taking both deterministic and stochastic time series that they get from the simulation of the fruit fly model Drosophila melanogaster and show how to extract the period, construct phase response curves, determine period sensitivity of the parameters to perturbations and build Arnold tongues to identify the regions of entrainment. They also derive a phase model that they numerically simulate to capture whether the circadian time series entrains to the forcing period completely (phase locking) or only partially (phase slips) or neither. They validate the phase model, and numerics with the experimental time series forced under different temperature cycles. Application of HT to the circadian time series appears to be a promising tool to extract the characteristic information about circadian rhythms.

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.

Dietary-fat effect of the rs10830963 polymorphism in MTNR1B on insulin resistance in response to 3 months weight-loss diets

BACKGROUND & AIMS

The risk allele (G) of rs10830963 in the melatonin receptor 1 B (MTNR1B) gene presents an association with obesity. We study the effect of this SNP on cardiovascular risk factors and weight loss secondary to 2hypocaloric diets.

METHODS

361 obese subjects were randomly allocated during 3 months (Diet M - high monounsaturated fat hypocaloric diet vs. Diet P - high polyunsaturated fat hypocaloric diet). Anthropometric parameters, fasting blood glucose, C-reactive protein (CRP), insulin concentration, insulin resistance (HOMA-IR), lipid profile and adipocytokines levels were measured. Genotype of MTNR1B gene polymorphism (rs10830963) was evaluated.

RESULTS

All anthropometric parameters, systolic blood pressure and leptin levels decreased in all subjects after both diets. This improvement of anthropometric parameters was higher in non G allele carriers than G allele carriers. After dietary intervention with Diet M, (CC vs. CG + GG); total cholesterol (delta: -10.4 ± 2.1mg/dl vs. -6.4 ± 1.2mg/dl: P <.05), LDL-cholesterol (delta:-7.1 ± 0.9mg/dl vs. -2.8 ± 0.8mg/dl: P <.05), insulin (delta:-3.0 ± 0.8 UI/L vs. -2.0 ± 1.0 UI/L: P<.05) and HOMA-IR (delta:-3.4 ± 1.0 units vs. -2.9 ± 0.9 units: P<.05) improved in no G allele carriers. After Diet P, in the group of subjects without G allele CC, insulin levels (delta: -2.9 ± 1.0 UI/L vs. -0.6 ± 0.2 UI/L: P <.05) and HOMA-IR (delta (CC vs. CG + GG): -0.8 ± 0.2 units vs. -0.4 ± 0.3 units: P <.05) decreased, too.

CONCLUSIONS

Our study detected a relationship of rs10830963 MTNR1B SNP with body weight loss and insulin resistance modification induced by 2different hypocaloric. Only monounsaturated enriched hypocaloric diet and in no-G allele carriers showed a significant effect on lipoproteins.

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.