A model of the cell-autonomous mammalian circadian clock

Coherency of circadian rhythms in the SCN is governed by the interplay of two coupling factors

Circadian clocks are autonomous oscillators driving daily rhythms in physiology and behavior. In mammals, a network of coupled neurons in the suprachiasmatic nucleus (SCN) is entrained to environmental light-dark cycles and orchestrates the timing of peripheral organs. In each neuron, transcriptional feedbacks generate noisy oscillations. Coupling mediated by neuropeptides such as VIP and AVP lends precision and robustness to circadian rhythms. The detailed coupling mechanisms between SCN neurons are debated. We analyze organotypic SCN slices from neonatal and adult mice in wild-type and multiple knockout conditions. Different degrees of rhythmicity are quantified by pixel-level analysis of bioluminescence data. We use empirical orthogonal functions (EOFs) to characterize spatio-temporal patterns. Simulations of coupled stochastic single cell oscillators can reproduce the diversity of observed patterns. Our combination of data analysis and modeling provides deeper insight into the enormous complexity of the data: (1) Neonatal slices are typically stronger oscillators than adult slices pointing to developmental changes of coupling. (2) Wild-type slices are completely synchronized and exhibit specific spatio-temporal patterns of phases. (3) Some slices of Cry double knockouts obey impaired synchrony that can lead to co–existing rhythms (“splitting”). (4) The loss of VIP-coupling leads to desynchronized rhythms with few residual local clusters. Additional information was extracted from co–culturing slices with rhythmic neonatal wild-type SCNs. These co–culturing experiments were simulated using external forcing terms representing VIP and AVP signaling. The rescue of rhythmicity via co–culturing lead to surprising results, since a cocktail of AVP-antagonists improved synchrony. Our modeling suggests that these counter-intuitive observations are pointing to an antagonistic action of VIP and AVP coupling. Our systematic theoretical and experimental study shows that dual coupling mechanisms can explain the astonishing complexity of spatio-temporal patterns in SCN slices.

The mammalian circadian clock is orchestrated by a network of coupled neurons. Brain slice preparations allow the analysis of coupling mechanisms mediated by neuropeptides. From bioluminescence recordings, we extract single cell characteristics such as period, amplitude and damping rate. Our data-based stochastic network model involves local coupling between cells and additional external forcing. Available experimental data guide our simulations with two distinct coupling and forcing mechanisms representing the neuropeptides VIP and AVP. We compare our simulations with experiments from neonatal and adult wild-type brain slices and multiple knockouts. Furthermore, we study co–culturing of slices with synchronized neonatal wild-type slices. The extreme complexity of the spatio-temporal patterns is quantified using empirical orthogonal functions (EOFs). The experimental reduction of AVP coupling leads to surprising observations. In double knockouts, inhibition of AVP signaling can improve synchrony, whereas, in triple knockouts, coherency is reduced. Our network modeling shows that these counter-intuitive observations can be explained by an antagonistic action of VIP and AVP signaling. The agreement of experiments and simulations suggests that quite complex spatio-temporal patterns can appear as emergent properties of oscillator networks with dual coupling mechanisms.

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.

Initial state perturbations as a validation method for data-driven fuzzy models of cellular networks

Background

Data-driven methods that automatically learn relations between attributes from given data are a popular tool for building mathematical models in computational biology. Since measurements are prone to errors, approaches dealing with uncertain data are especially suitable for this task. Fuzzy models are one such approach, but they contain a large amount of parameters and are thus susceptible to over-fitting. Validation methods that help detect over-fitting are therefore needed to eliminate inaccurate models.

Results

We propose a method to enlarge the validation datasets on which a fuzzy dynamic model of a cellular network can be tested. We apply our method to two data-driven dynamic models of the MAPK signalling pathway and two models of the mammalian circadian clock. We show that random initial state perturbations can drastically increase the mean error of predictions of an inaccurate computational model, while keeping errors of predictions of accurate models small.

Conclusions

With the improvement of validation methods, fuzzy models are becoming more accurate and are thus likely to gain new applications. This field of research is promising not only because fuzzy models can cope with uncertainty, but also because their run time is short compared to conventional modelling methods that are nowadays used in systems biology.

Limits on reconstruction of dynamics in networks

An observability condition number is defined for physical systems modeled by network dynamics. Assuming that the dynamical equations of the network are known and a noisy trajectory is observed at a subset of the nodes, we calculate the expected distance to the nearest correct trajectory as a function of the observation noise level and discuss how it varies over the unobserved nodes of the network. When the condition number is sufficiently large, reconstructing the trajectory from observations from the subset will be infeasible. This knowledge can be used to choose an optimal subset from which to observe a network.

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.

A Meta-Analysis Characterizing Stem-Like Gene Expression in the Suprachiasmatic Nucleus and Its Circadian Clock

Cells expressing proteins characteristic of stem cells and progenitor cells are present in the suprachiasmatic nucleus (SCN) of the adult mammalian hypothalamus. Any relationship between this distinctive feature and the master circadian clock of the SCN is unclear. Considering the lack of obvious neurogenesis in the adult SCN relative to the hippocampus and other structures that provide neurons and glia, it is possible that the SCN has partially differentiated cells that can provide neural circuit plasticity rather than ongoing neurogenesis. To test this possibility, available databases and publications were explored to identify highly expressed genes in the mouse SCN that also have known or suspected roles in cell differentiation, maintenance of stem-like states, or cell-cell interactions found in adult and embryonic stem cells and cancer stem cells. The SCN was found to have numerous genes associated with stem cell maintenance and increased motility from which we selected 25 of the most relevant genes. Over ninety percent of these stem-like genes were expressed at higher levels in the SCN than in other brain areas. Further analysis of this gene set could provide a greater understanding of how adjustments in cell contacts alter period and phase relationships of circadian rhythms. Circadian timing and its role in cancer, sleep, and metabolic disorders are likely influenced by genes selected in this study.

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

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

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.

Locally Optimal Control of Complex Networks

It has recently been shown that the minimum energy solution of the control problem for a linear system produces a control trajectory that is nonlocal. An issue then arises when the dynamics represents a linearization of the underlying nonlinear dynamics of the system where the linearization is only valid in a local region of the state space. Here we provide a solution to the problem of optimally controlling a linearized system by deriving a time-varying set that represents all possible control trajectories parametrized by time and energy. As long as the control action terminus is defined within this set, the control trajectory is guaranteed to be local. If the desired terminus of the control action is far from the initial state, a series of local control actions can be performed in series, relinearizing the dynamics at each new position.

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

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

Dissipative structures and biological rhythms

Sustained oscillations abound in biological systems. They occur at all levels of biological organization over a wide range of periods, from a fraction of a second to years, and with a variety of underlying mechanisms. They control major physiological functions, and their dysfunction is associated with a variety of physiological disorders. The goal of this review is (i) to give an overview of the main rhythms observed at the cellular and supracellular levels, (ii) to briefly describe how the study of biological rhythms unfolded in the course of time, in parallel with studies on chemical oscillations, (iii) to present the major roles of biological rhythms in the control of physiological functions, and (iv) the pathologies associated with the alteration, disappearance, or spurious occurrence of biological rhythms. Two tables present the main examples of cellular and supracellular rhythms ordered according to their period, and their role in physiology and pathophysiology. Among the rhythms discussed are neural and cardiac rhythms, metabolic oscillations such as those occurring in glycolysis in yeast, intracellular Ca⁺⁺ oscillations, cyclic AMP oscillations in Dictyostelium amoebae, the segmentation clock that controls somitogenesis, pulsatile hormone secretion, circadian rhythms which occur in all eukaryotes and some bacteria with a period close to 24 h, the oscillatory dynamics of the enzymatic network driving the cell cycle, and oscillations in transcription factors such as NF-ΚB and tumor suppressors such as p53. Ilya Prigogine's concept of dissipative structures applies to temporal oscillations and allows us to unify within a common framework the various rhythms observed at different levels of biological organization, regardless of their period and underlying mechanism.

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.

Evaluation of multidrug cancer chronotherapy based on cell cycle model under influences of circadian clock

The intracellular circadian clock mechanisms are known to affect various substantial cellular machinery such as cell cycle progression, inflammatory response, apoptosis, and DNA repair. Cancer growth in various tissues is still under circadian control, which may be at least partly underlain by the survived connections between the intracellular machinery and the clock. Considering such findings, chronotherapy has been applied to cancer treatments, in which anti-cancer drugs are administered in scheduled circadian times so as to resolve the trade-off between damages against the normal and cancer cells. However, any effective administration strategy has not yet been established especially in a quantitative sense. In this study, we develop an automaton model of cell division cycle interacting with circadian clock and suffering from a probability of cell death. A cancer cell is modeled by shortening/ lengthening the cell division interval and a transition to motility state under starving condition. Population proliferating dynamics in 3D space are simulated under the diffusion of nutrient factor and the anti-cancer drugs from a vessel. The simulation results show that the drug administration schedule could differentiate the damages against proliferation of normal and cancer cells. This implies the existence of optimal timing for the drug administration, which could provide an efficient strategy of chronotherapeutic treatment of cancer.

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

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

Hypothesis driven single cell dual oscillator mathematical model of circadian rhythms

Molecular mechanisms responsible for 24 h circadian oscillations, entrainment to external cues, encoding of day length and the time-of-day effects have been well studied experimentally. However, it is still debated from the molecular network point of view whether each cell in suprachiasmatic nuclei harbors two molecular oscillators, where one tracks dawn and the other tracks dusk activities. A single cell dual morning and evening oscillator was proposed by Daan et al., based on the molecular network that has two sets of similar non-redundant per1/cry1 and per2/cry2 circadian genes and each can independently maintain their endogenous oscillations. Understanding of dual oscillator dynamics in a single cell at molecular level may provide insight about the circadian mechanisms that encodes day length variations and its response to external zeitgebers. We present here a realistic dual oscillator model of circadian rhythms based on the series of hypotheses proposed by Daan et al., in which they conjectured that the circadian genes per1/cry1 track dawn while per2/cry2 tracks dusk and they together constitute the morning and evening oscillators (dual oscillator). Their hypothesis also provides explanations about the encoding of day length in terms of molecular mechanisms of per/cry expression. We frame a minimal mathematical model with the assumption that per1 acts a morning oscillator and per2 acts as an evening oscillator and to support and interpret this assumption we fit the model to the experimental data of per1/per2 circadian temporal dynamics, phase response curves (PRC's), and entrainment phenomena under various light-dark conditions. We also capture different patterns of splitting phenomena by coupling two single cell dual oscillators with neuropeptides vasoactive intestinal polypeptide (VIP) and arginine vasopressin (AVP) as the coupling agents and provide interpretation for the occurrence of splitting in terms of ME oscillators, though they are not required to explain the morning and evening oscillators. The proposed dual oscillator model based on Daan's hypothesis supports per1 and per2 playing the role of morning and evening oscillators respectively and this may be the first step towards the understanding of the core molecular mechanism responsible for encoding the day length.

Computational modeling of the cell-autonomous mammalian circadian oscillator

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

Systems Biology-Derived Discoveries of Intrinsic Clocks

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

Comparative Analysis of Vertebrate Diurnal/Circadian Transcriptomes

From photosynthetic bacteria to mammals, the circadian clock evolved to track diurnal rhythms and enable organisms to anticipate daily recurring changes such as temperature and light. It orchestrates a broad spectrum of physiology such as the sleep/wake and eating/fasting cycles. While we have made tremendous advances in our understanding of the molecular details of the circadian clock mechanism and how it is synchronized with the environment, we still have rudimentary knowledge regarding its connection to help regulate diurnal physiology. One potential reason is the sheer size of the output network. Diurnal/circadian transcriptomic studies are reporting that around 10% of the expressed genome is rhythmically controlled. Zebrafish is an important model system for the study of the core circadian mechanism in vertebrate. As Zebrafish share more than 70% of its genes with human, it could also be an additional model in addition to rodent for exploring the diurnal/circadian output with potential for translational relevance. Here we performed comparative diurnal/circadian transcriptome analysis with established mouse liver and other tissue datasets. First, by combining liver tissue sampling in a 48h time series, transcription profiling using oligonucleotide arrays and bioinformatics analysis, we profiled rhythmic transcripts and identified 2609 rhythmic genes. The comparative analysis revealed interesting features of the output network regarding number of rhythmic genes, proportion of tissue specific genes and the extent of transcription factor family expression. Undoubtedly, the Zebrafish model system will help identify new vertebrate outputs and their regulators and provides leads for further characterization of the diurnal cis-regulatory network.

Regulating the Suprachiasmatic Nucleus (SCN) Circadian Clockwork: Interplay between Cell-Autonomous and Circuit-Level Mechanisms

The suprachiasmatic nucleus (SCN) is the principal circadian clock of the brain, directing daily cycles of behavior and physiology. SCN neurons contain a cell-autonomous transcription-based clockwork but, in turn, circuit-level interactions synchronize the 20,000 or so SCN neurons into a robust and coherent daily timer. Synchronization requires neuropeptide signaling, regulated by a reciprocal interdependence between the molecular clockwork and rhythmic electrical activity, which in turn depends on a daytime Na⁺ drive and nighttime K⁺ drag. Recent studies exploiting intersectional genetics have started to identify the pacemaking roles of particular neuronal groups in the SCN. They support the idea that timekeeping involves nonlinear and hierarchical computations that create and incorporate timing information through the interactions between key groups of neurons within the SCN circuit. The field is now poised to elucidate these computations, their underlying cellular mechanisms, and how the SCN clock interacts with subordinate circadian clocks across the brain to determine the timing and efficiency of the sleep-wake cycle, and how perturbations of this coherence contribute to neurological and psychiatric illness.

Feedback Loops of the Mammalian Circadian Clock Constitute Repressilator

Mammals evolved an endogenous timing system to coordinate their physiology and behaviour to the 24h period of the solar day. While it is well accepted that circadian rhythms are generated by intracellular transcriptional feedback loops, it is still debated which network motifs are necessary and sufficient for generating self-sustained oscillations. Here, we systematically explore a data-based circadian oscillator model with multiple negative and positive feedback loops and identify a series of three subsequent inhibitions known as "repressilator" as a core element of the mammalian circadian oscillator. The central role of the repressilator motif is consistent with time-resolved ChIP-seq experiments of circadian clock transcription factors and loss of rhythmicity in core clock gene knockouts.

A systems theoretic approach to analysis and control of mammalian circadian dynamics

The mammalian circadian clock is a complex multi-scale, multivariable biological control system. In the past two decades, methods from systems engineering have led to numerous insights into the architecture and functionality of this system. In this review, we examine the mammalian circadian system through a process systems lens. We present a mathematical framework for examining the cellular circadian oscillator, and show recent extensions for understanding population-scale dynamics. We provide an overview of the routes by which the circadian system can be systemically manipulated, and present in silico proof of concept results for phase resetting of the clock via model predictive control.

BioNSi: A Discrete Biological Network Simulator Tool

Modeling and simulation of biological networks is an effective and widely used research methodology. The Biological Network Simulator (BioNSi) is a tool for modeling biological networks and simulating their discrete-time dynamics, implemented as a Cytoscape App. BioNSi includes a visual representation of the network that enables researchers to construct, set the parameters, and observe network behavior under various conditions. To construct a network instance in BioNSi, only partial, qualitative biological data suffices. The tool is aimed for use by experimental biologists and requires no prior computational or mathematical expertise. BioNSi is freely available at http://bionsi.wix.com/bionsi , where a complete user guide and a step-by-step manual can also be found.

Circadian Clock Genes Modulate Human Bone Marrow Mesenchymal Stem Cell Differentiation, Migration and Cell Cycle

Many of the components that regulate the circadian clock have been identified in organisms and humans. The influence of circadian rhythm (CR) on the regulation of stem cells biology began to be evaluated. However, little is known on the role of CR on human mesenchymal stem cell (hMSCs) properties. The objective of this study was to investigate the influence of CR on the differentiation capacities of bone marrow hMSCs, as well as the regulation of cell cycle and migration capabilities. To that, we used both a chemical approach with a GSK-3β specific inhibitor (2’E,3’Z-6-bromoindirubin-3’-oxime, BIO) and a knockdown of CLOCK and PER2, two of the main genes involved in CR regulation. In these experimental conditions, a dramatic inhibition of adipocyte differentiation was observed, while osteoblastic differentiation capacities were not modified. In addition, cell migration was decreased in PER2^-/- cells. Lastly, downregulation of circadian clock genes induced a modification of the hMSCs cell cycle phase distribution, which was shown to be related to a change of the cyclin expression profile. Taken together, these data showed that CR plays a role in the regulation of hMSCs differentiation and division, and likely represent key factor in maintaining hMSCs properties.

Theoretical approaches for the dynamics of complex biological systems from information of networks

Modern biology has provided many examples of large networks describing the interactions between multiple species of bio-molecules. It is believed that the dynamics of molecular activities based on such networks are the origin of biological functions. On the other hand, we have a limited understanding for dynamics of molecular activity based on networks. To overcome this problem, we have developed two structural theories, by which the important aspects of the dynamical properties of the system are determined only from information on the network structure, without assuming other quantitative details. The first theory, named Linkage Logic, determines a subset of molecules in regulatory networks, by which any long-term dynamical behavior of the whole system can be identified/controlled. The second theory, named Structural Sensitivity Analysis, determines the sensitivity responses of the steady state of chemical reaction networks to perturbations of the reaction rate. The first and second theories investigate the dynamical properties of regulatory and reaction networks, respectively. The first theory targets the attractors of the regulatory network systems, whereas the second theory applies only to the steady states of the reaction network systems, but predicts their detailed behavior. To demonstrate the utility of our methods several biological network systems, and show they are practically useful to analyze behaviors of biological systems.

Circadian systems biology: When time matters

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

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

A mammalian circadian clock model incorporating daytime expression elements

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

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

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

Modeling circadian clock-cell cycle interaction effects on cell population growth rates

The circadian clock and the cell cycle are two tightly coupled oscillators. Recent analytical studies have shown counter-intuitive effects of circadian gating of the cell cycle on growth rates of proliferating cells which cannot be explained by a molecular model or a population model alone. In this work, we present a combined molecular-population model that studies how coupling the circadian clock to the cell cycle, through the protein WEE1, affects a proliferating cell population. We show that the cell cycle can entrain to the circadian clock with different rational period ratios and characterize multiple domains of entrainment. We show that coupling increases the growth rate for autonomous periods of the cell cycle around 24 h and above 48 h. We study the effect of mutation of circadian genes on the growth rate of cells and show that disruption of the circadian clock can lead to abnormal proliferation. Particularly, we show that Cry 1, Cry 2 mutations decrease the growth rate of cells, Per 2 mutation enhances it and Bmal 1 knockout increases it for autonomous periods of the cell cycle less than 21 h and decreases it elsewhere. Combining a molecular model to a population model offers new insight on the influence of the circadian clock on the growth of a cell population. This can help chronotherapy which takes benefits of physiological rhythms to improve anti-cancer efficacy and tolerance to drugs by administering treatments at a specific time of the day.

Brain clock driven by neuropeptides and second messengers

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

An intensity ratio of interlocking loops determines circadian period length

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

Bayesian parameter inference and model selection by population annealing in systems biology

Parameter inference and model selection are very important for mathematical modeling in systems biology. Bayesian statistics can be used to conduct both parameter inference and model selection. Especially, the framework named approximate Bayesian computation is often used for parameter inference and model selection in systems biology. However, Monte Carlo methods needs to be used to compute Bayesian posterior distributions. In addition, the posterior distributions of parameters are sometimes almost uniform or very similar to their prior distributions. In such cases, it is difficult to choose one specific value of parameter with high credibility as the representative value of the distribution. To overcome the problems, we introduced one of the population Monte Carlo algorithms, population annealing. Although population annealing is usually used in statistical mechanics, we showed that population annealing can be used to compute Bayesian posterior distributions in the approximate Bayesian computation framework. To deal with un-identifiability of the representative values of parameters, we proposed to run the simulations with the parameter ensemble sampled from the posterior distribution, named “posterior parameter ensemble”. We showed that population annealing is an efficient and convenient algorithm to generate posterior parameter ensemble. We also showed that the simulations with the posterior parameter ensemble can, not only reproduce the data used for parameter inference, but also capture and predict the data which was not used for parameter inference. Lastly, we introduced the marginal likelihood in the approximate Bayesian computation framework for Bayesian model selection. We showed that population annealing enables us to compute the marginal likelihood in the approximate Bayesian computation framework and conduct model selection depending on the Bayes factor.

Timing of circadian genes in mammalian tissues

Circadian clocks are endogenous oscillators driving daily rhythms in physiology. The cell-autonomous clock is governed by an interlocked network of transcriptional feedback loops. Hundreds of clock-controlled genes (CCGs) regulate tissue specific functions. Transcriptome studies reveal that different organs (e.g. liver, heart, adrenal gland) feature substantially varying sets of CCGs with different peak phase distributions. To study the phase variability of CCGs in mammalian peripheral tissues, we develop a core clock model for mouse liver and adrenal gland based on expression profiles and known cis-regulatory sites. ‘Modulation factors’ associated with E-boxes, ROR-elements, and D-boxes can explain variable rhythms of CCGs, which is demonstrated for differential regulation of cytochromes P450 and 12 h harmonics. By varying model parameters we explore how tissue-specific peak phase distributions can be generated. The central role of E-boxes and ROR-elements is confirmed by analysing ChIP-seq data of BMAL1 and REV-ERB transcription factors.

It is not the parts, but how they interact that determines the behaviour of circadian rhythms across scales and organisms

Biological rhythms, generated by feedback loops containing interacting genes, proteins and/or cells, time physiological processes in many organisms. While many of the components of the systems that generate biological rhythms have been identified, much less is known about the details of their interactions. Using examples from the circadian (daily) clock in three organisms, Neurospora, Drosophila and mouse, we show, with mathematical models of varying complexity, how interactions among (i) promoter sites, (ii) proteins forming complexes, and (iii) cells can have a drastic effect on timekeeping. Inspired by the identification of many transcription factors, for example as involved in the Neurospora circadian clock, that can both activate and repress, we show how these multiple actions can cause complex oscillatory patterns in a transcription–translation feedback loop (TTFL). Inspired by the timekeeping complex formed by the NMO–PER–TIM–SGG complex that regulates the negative TTFL in the Drosophila circadian clock, we show how the mechanism of complex formation can determine the prevalence of oscillations in a TTFL. Finally, we note that most mathematical models of intracellular clocks model a single cell, but compare with experimental data from collections of cells. We find that refitting the most detailed model of the mammalian circadian clock, so that the coupling between cells matches experimental data, yields different dynamics and makes an interesting prediction that also matches experimental data: individual cells are bistable, and network coupling removes this bistability and causes the network to be more robust to external perturbations. Taken together, we propose that the interactions between components in biological timekeeping systems are carefully tuned towards proper function. We also show how timekeeping can be controlled by novel mechanisms at different levels of organization.

Cooperation of luteinizing hormone signaling pathways in preovulatory avian follicles regulates circadian clock expression in granulosa cell

Ovulation in birds is triggered by a surge of luteinizing hormone (LH), and the ovulatory cycle is affected by the circadian rhythms of clock genes transcription levels in follicles. The influence of LH signaling cascades action on circadian clock genes was investigated using granulosa cells of preovulatory follicles from Roman hens cultured in a serum-free system. The expression of core oscillators (Bmal1, Clock, Cry1, Per2, and Rev-erbβ), clock-controlled gene (Star), Egr-1 and LHr was measured by quantitative real-time PCR. Significant changes in clock genes transcription levels were observed in control groups over 24 h, indicating that cell-autonomous rhythms exist in granulosa cells. Intriguingly, the transcript levels of clock genes increased with LH treatment during 24 h of culture; they peaked 4 h in advance of controls and second but weaker oscillations were also observed. It appeared that LH changed the cell-autonomous rhythm and cycle time of clock genes. To further investigate the LH signaling cascades, inhibitors of cyclic adenosine monophosphate (cAMP), p38 mitogen-activated protein kinases (p38MAPK) and extracellular signal-regulated kinases 1 and 2 (ERK1/2) pathways were used. The transcript levels of clock genes were suppressed by blocking cAMP, but increased with similar expression patterns by blocking the p38MPAK and ERK1/2 pathways over 24 h. Thus, the influence of LH signaling cascades in chicken ovulation is mediated by the cAMP pathway and also involves the p38MAPK and ERK1/2 pathways.

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

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

Timing of neuropeptide coupling determines synchrony and entrainment in the mammalian circadian clock

Robust synchronization is a critical feature of several systems including the mammalian circadian clock. The master circadian clock in mammals consists of about 20000 ‘sloppy’ neuronal oscillators within the hypothalamus that keep robust time by synchronization driven by inter-neuronal coupling. The complete understanding of this synchronization in the mammalian circadian clock and the mechanisms underlying it remain an open question. Experiments and computational studies have shown that coupling individual oscillators can achieve robust synchrony, despite heterogeneity and different network topologies. But, much less is known regarding the mechanisms and circuits involved in achieving this coupling, due to both system complexity and experimental limitations. Here, we computationally study the coupling mediated by the primary coupling neuropeptide, vasoactive intestinal peptide (VIP) and its canonical receptor, VPAC2R, using the transcriptional elements and generic mode of VIP-VPAC2R signaling. We find that synchrony is only possible if VIP (an inducer of Per expression) is released in-phase with activators of Per expression. Moreover, anti-phasic VIP release suppresses coherent rhythms by moving the network into a desynchronous state. Importantly, experimentally observed rhythms in VPAC2R have little effect on network synchronization, but can improve the amplitude of the SCN network rhythms while narrowing the network entrainment range. We further show that these findings are valid across several computational network models. Thus, we identified a general design principle to achieve robust synchronization: An activating coupling agent, such as VIP, must act in-phase with the activity of core-clock promoters. More generally, the phase of coupling is as critical as the strength of coupling from the viewpoint of synchrony and entrainment.

Synchronization among multiple oscillators is a common theme in many biological and engineered systems. Here, we look at its use by the mammalian biological clock to keep accurate time. Through biochemical interactions among a network of inaccurate neuron clocks, a strong precise clock is produced. Although we are gradually learning more about these biochemical interactions, the details still remain largely unclear. Studies, both computational and experimental, have shown that the strength of the rhythmic interaction critically decides if a system can synchronize, i.e., the interactions must be strong enough. In this work, we show that the rhythmic interaction between these neuronal clocks must be timed correctly (in the right phase) in addition to being strong enough to synchronize the network. Activating (repressing) interactions must coincide with other activators (repressors) in each neuron to achieve synchrony. Since this principle imposes certain properties on synchronizing interactions, these properties can be used to identify and understand novel interaction mechanisms. Further, these principles are applicable to interactions between cellular oscillators in other tissues and organisms.

Variants in glucose- and circadian rhythm-related genes affect the response of energy expenditure to weight-loss diets: the POUNDS LOST Trial

BACKGROUND

Circadian rhythm has been shown to be related to glucose metabolism and risk of diabetes, probably through effects on energy balance. Recent genome-wide association studies identified variants in circadian rhythm-related genes (CRY2 and MTNR1B) associated with glucose homeostasis.

OBJECTIVE

We tested whether CRY2 and MTNR1B genotypes affected changes in measures of energy expenditure in response to a weight-loss diet intervention in a 2-y randomized clinical trial, the POUNDS (Preventing Overweight Using Novel Dietary Strategies) LOST Trial.

DESIGN

The variants CRY2 rs11605924 (n = 721) and MTNR1B rs10830963 (n = 722) were genotyped in overweight or obese adults who were randomly assigned to 1 of 4 weight-loss diets that differed in their proportions of macronutrients. Respiratory quotient (RQ) and resting metabolic rate (RMR) were measured.

RESULTS

By 2 y of diet intervention, the A allele of CRY2 rs11605924 was significantly associated with a greater reduction in RQ (P = 0.03) and a greater increase in RMR and RMR/kg (both P = 0.04). The G allele of MTNR1B rs10830963 was significantly associated with a greater increase in RQ (P = 0.01) but was not related to changes in RMR and RMR/kg. In addition, we found significant gene-diet fat interactions for both CRY2 (P-interaction = 0.02) and MTNR1B (P-interaction < 0.001) in relation to 2-y changes in RQ.

CONCLUSIONS

Our data indicate that variants in the circadian-related genes CRY2 and MTNR1B may affect long-term changes in energy expenditure, and dietary fat intake may modify the genetic effects. This trial was registered at www.clinicaltrials.gov as NCT00072995.

Spatiotemporal separation of PER and CRY posttranslational regulation in the mammalian circadian clock

Posttranslational regulation of clock proteins is an essential part of mammalian circadian rhythms, conferring sensitivity to metabolic state and offering promising targets for pharmacological control. Two such regulators, casein kinase 1 (CKI) and F-box and leucine-rich repeat protein 3 (FBXL3), modulate the stability of closely linked core clock proteins period (PER) and cryptochrome (CRY), respectively. Inhibition of either CKI or FBXL3 leads to longer periods, and their effects are independent despite targeting proteins with similar roles in clock function. A mechanistic understanding of this independence, however, has remained elusive. Our analysis of cellular circadian clock gene reporters further differentiated between the actions of CKI and FBXL3 by revealing opposite amplitude responses from each manipulation. To understand the functional relationship between the CKI-PER and FBXL3-CRY pathways, we generated robust mechanistic predictions by applying a bootstrap uncertainty analysis to multiple mathematical circadian models. Our results indicate that CKI primarily regulates the accumulating phase of the PER-CRY repressive complex by controlling the nuclear import rate, whereas FBXL3 separately regulates the duration of transcriptional repression in the nucleus. Dynamic simulations confirmed that this spatiotemporal separation is able to reproduce the independence of the two regulators in period regulation, as well as their opposite amplitude effect. As a result, this study provides further insight into the molecular clock machinery responsible for maintaining robust circadian rhythms.

Modeling Interindividual Differences in Spontaneous Internal Desynchrony Patterns

A physiologically based mathematical model of a putative sleep-wake regulatory network is used to investigate the transition from typical human sleep patterns to spontaneous internal desynchrony behavior observed under temporal isolation conditions. The model sleep-wake regulatory network describes the neurotransmitter-mediated interactions among brainstem and hypothalamic neuronal populations that participate in the transitions between wake, rapid eye movement (REM) sleep, and non-REM (NREM) sleep. Physiologically based interactions among these sleep-wake centers and the suprachiasmatic nucleus (SCN), whose activity is driven by an established circadian oscillator model, mediate circadian modulation of sleep-wake behavior. When the sleep-wake and circadian rhythms are synchronized, the model simulates stereotypically normal human sleep-wake behavior within the limits of individual variation, including typical NREM-REM cycling across the night. When effects of temporal isolation are simulated by increasing the period of the sleep-wake cycle, the model replicates spontaneous internal desynchrony with the appropriate dependence of multiple features of REM sleep on circadian phase. In temporal isolation experiments, subjects have exhibited different desynchronized sleep-wake behaviors. Our model can generate similar ranges of desynchronized behaviors by variations in the period of the sleep-wake cycle and the strength of interactions between the SCN and the sleep-wake centers. Analysis of the model suggests that similar mechanisms underlie several different desynchronized behaviors and that the phenomenon of phase trapping may be dependent on SCN modulation of REM sleep-promoting centers. These results provide predictions for physiologically plausible mechanisms underlying interindividual variations in sleep-wake behavior observed during temporal isolation experiments.

Bayesian parameter inference by Markov chain Monte Carlo with hybrid fitness measures: theory and test in apoptosis signal transduction network

When exact values of model parameters in systems biology are not available from experiments, they need to be inferred so that the resulting simulation reproduces the experimentally known phenomena. For the purpose, Bayesian statistics with Markov chain Monte Carlo (MCMC) is a useful method. Biological experiments are often performed with cell population, and the results are represented by histograms. On another front, experiments sometimes indicate the existence of a specific bifurcation pattern. In this study, to deal with both type of such experimental results and information for parameter inference, we introduced functions to evaluate fitness to both type of experimental results, named quantitative and qualitative fitness measures respectively. We formulated Bayesian formula for those hybrid fitness measures (HFM), and implemented it to MCMC (MCMC-HFM). We tested MCMC-HFM first for a kinetic toy model with a positive feedback. Inferring kinetic parameters mainly related to the positive feedback, we found that MCMC-HFM reliably infer them with both qualitative and quantitative fitness measures. Then, we applied the MCMC-HFM to an apoptosis signal transduction network previously proposed. For kinetic parameters related to implicit positive feedbacks, which are important for bistability and irreversibility of the output, the MCMC-HFM reliably inferred these kinetic parameters. In particular, some kinetic parameters that have the experimental estimates were inferred without these data and the results were consistent with the experiments. Moreover, for some parameters, the mixed use of quantitative and qualitative fitness measures narrowed down the acceptable range of parameters. Taken together, our approach could reliably infer the kinetic parameters of the target systems.