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The circadian clock, a fundamental biological regulator, governs essential cellular processes in health and disease. Circadian-based therapeutic strategies are increasingly gaining recognition as promising avenues. Aligning drug administration with the circadian rhythm can enhance treatment efficacy and minimize side effects. Yet, uncovering the optimal treatment timings remains challenging, limiting their widespread adoption. In this work, we introduce a high-throughput approach integrating live-imaging and data analysis techniques to deep-phenotype cancer cell models, evaluating their circadian rhythms, growth, and drug responses. We devise a streamlined process for profiling drug sensitivities across different times of the day, identifying optimal treatment windows and responsive cell types and drug combinations. Finally, we implement multiple computational tools to uncover cellular and genetic factors shaping time-of-day drug sensitivity. Our versatile approach is adaptable to various biological models, facilitating its broad application and relevance. Ultimately, this research leverages circadian rhythms to optimize anti-cancer drug treatments, promising improved outcomes and transformative treatment strategies., (© 2024. The Author(s).)

Circadian clocks are internal timing devices that have evolved as an adaption to the omnipresent natural 24 h rhythmicity of daylight intensity. Properties of the circadian system are photoperiod dependent. The phase of entrainment varies systematically with season. Plastic photoperiod-dependent re-arrangements in the mammalian circadian core pacemaker yield an internal representation of season. Output pathways of the circadian clock regulate photoperiodic responses such as flowering time in plants or hibernation in mammals. Here, we review the concepts of seasonal entrainment and photoperiodic encoding. We introduce conceptual phase oscillator models as their high level of abstraction, but, yet, intuitive interpretation of underlying parameters allows for a straightforward analysis of principles that determine entrainment characteristics. Results from this class of models are related and discussed in the context of more complex conceptual amplitude-phase oscillators as well as contextual molecular models that take into account organism, tissue, and cell-type-specific details., (© 2023. The Author(s).)

Three parameters are important to characterize a circadian and in general any biological clock: period, phase and amplitude. While circadian periods have been shown to correlate with entrainment phases, and clock amplitude influences the phase response of an oscillator to pulse-like zeitgeber signals, the co-modulations of amplitude and periods, which we term twist , have not been studied in detail. In this paper we define two concepts: parametric twist refers to amplitude-period correlations arising in ensembles of self-sustained, limit cycle clocks in the absence of external inputs, and phase space twist refers to the co-modulation of an individual clock's amplitude and period in response to external zeitgebers. Our findings show that twist influences the interaction of oscillators with the environment, facilitating entrainment, speeding upfastening recovery to pulse-like perturbations or modifying the response of an individual clock to coupling. This theoretical framework might be applied to understand the emerging properties of other oscillating systems., Competing Interests: No competing interests were disclosed., (Copyright: © 2024 del Olmo M et al.)

The importance of RNA-binding proteins (RBPs) for plant responses to environmental stimuli and development is well documented. Insights into the portfolio of RNAs they recognize, however, clearly lack behind the understanding gathered in non-plant model organisms. Here, we characterize binding of the circadian clock-regulated Arabidopsis thaliana GLYCINE-RICH RNA-BINDING PROTEIN 7 (AtGRP7) to its target transcripts. We identified novel RNA targets from individual-nucleotide resolution UV crosslinking and immunoprecipitation (iCLIP) data using an improved bioinformatics pipeline that will be broadly applicable to plant RBP iCLIP data. 2705 transcripts with binding sites were identified in plants expressing AtGRP7-GFP that were not recovered in plants expressing an RNA-binding dead variant or GFP alone. A conserved RNA motif enriched in uridine residues was identified at the AtGRP7 binding sites. NMR titrations confirmed the preference of AtGRP7 for RNAs with a central U-rich motif. Among the bound RNAs, circadian clock-regulated transcripts were overrepresented. Peak abundance of the LHCB1.1 transcript encoding a chlorophyll-binding protein was reduced in plants overexpressing AtGRP7 whereas it was elevated in atgrp7 mutants, indicating that LHCB1.1 was regulated by AtGRP7 in a dose-dependent manner. In plants overexpressing AtGRP7, the LHCB1.1 half-life was shorter compared to wild-type plants whereas in atgrp7 mutant plants, the half-life was significantly longer. Thus, AtGRP7 modulates circadian oscillations of its in vivo binding target LHCB1.1 by affecting RNA stability., (© 2023 The Authors. The Plant Journal published by Society for Experimental Biology and John Wiley & Sons Ltd.)

Circadian clocks are endogenous oscillators present in almost all cells that drive daily rhythms in physiology and behavior. There are two mechanisms that have been proposed to explain how circadian rhythms are generated in mammalian cells: through a transcription-translation feedback loop (TTFL) and based on oxidation/reduction reactions, both of which are intrinsically stochastic and heterogeneous at the single cell level. In order to explore the emerging properties of stochastic and heterogeneous redox oscillators, we simplify a recently developed kinetic model of redox oscillations to an amplitude-phase oscillator with 'twist' (period-amplitude correlation) and subject to Gaussian noise. We show that noise and heterogeneity alone lead to fast desynchronization, and that coupling between noisy oscillators can establish robust and synchronized rhythms with amplitude expansions and tuning of the period due to twist. Coupling a network of redox oscillators to a simple model of the TTFL also contributes to synchronization, large amplitudes and fine-tuning of the period for appropriate interaction strengths. These results provide insights into how the circadian clock compensates randomness from intracellular sources and highlight the importance of noise, heterogeneity and coupling in the context of circadian oscillators., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2024 The Author(s).)

The synchronization of multiple oscillators serves as the central mechanism for maintaining stable circadian rhythms in physiology and behavior. Aging and disease can disrupt synchronization, leading to changes in the periodicity of circadian activities. While our understanding of the circadian clock under synchronization has advanced significantly, less is known about its behavior outside synchronization, which can also fall within a predictable domain. These states not only impact the stability of the rhythms but also modulate the period length. In C57BL/6 mice, aging, diseases, and removal of peripheral circadian oscillators often result in lengthened behavioral circadian periods. Here, we show that these changes can be explained by a surprisingly simple mathematical relationship: the frequency is the reciprocal of the period, and its distribution becomes skewed when the period distribution is symmetric. The synchronized frequency of a population in the skewed distribution and the macroscopic frequency of combined oscillators differ, accounting for some of the atypical circadian period outputs observed in networks without synchronization. Building on this finding, we investigate the dynamics of circadian outputs in the context of aging and disease, where synchronization is weakened., Competing Interests: The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest., (Copyright © 2023 Myung, Hong, Schmal, Vitet and Wu.)

A defining property of circadian clocks is temperature compensation, characterized by the resilience of their near 24-hour free-running periods against changes in environmental temperature within the physiological range. While temperature compensation is evolutionary conserved across different taxa of life and has been studied within many model organisms, its molecular underpinnings remain elusive. Posttranscriptional regulations such as temperature-sensitive alternative splicing or phosphorylation have been described as underlying reactions. Here, we show that knockdown of cleavage and polyadenylation specificity factor subunit 6 (CPSF6), a key regulator of 3'-end cleavage and polyadenylation, significantly alters circadian temperature compensation in human U-2 OS cells. We apply a combination of 3'-end-RNA-seq and mass spectrometry-based proteomics to globally quantify changes in 3' UTR length as well as gene and protein expression between wild-type and CPSF6 knockdown cells and their dependency on temperature. Since changes in temperature compensation behavior should be reflected in alterations of temperature responses within one or all of the 3 regulatory layers, we statistically assess differential responses upon changes in ambient temperature between wild-type and CPSF6 knockdown cells. By this means, we reveal candidate genes underlying circadian temperature compensation, including eukaryotic translation initiation factor 2 subunit 1 (EIF2S1)., Competing Interests: The authors have declared that no competing interests exist., (Copyright: © 2023 Schmal et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.)

Several epidemiological studies have suggested that obesity complicated with insulin resistance and type 2 diabetes exerts deleterious effects on the skeleton. While obesity coexists with estrogen deficiency in postmenopausal women, their combined effects on the skeleton are poorly studied. Thus, we investigated the impact of high-fat diet (HFD) on bone and metabolism of ovariectomized (OVX) female mice (C57BL/6J). OVX or sham operated mice were fed either HFD (60%fat) or normal diet (10%fat) for 12 weeks. HFD-OVX group exhibited pronounced increase in body weight (~86% in HFD and ~122% in HFD-OVX, p < 0.0005) and impaired glucose tolerance. Bone microCT-scanning revealed a pronounced decrease in trabecular bone volume/total volume (BV/TV) (-15.6 ± 0.48% in HFD and -37.5 ± 0.235% in HFD-OVX, p < 0.005) and expansion of bone marrow adipose tissue (BMAT; +60.7 ± 9.9% in HFD vs. +79.5 ± 5.86% in HFD-OVX, p < 0.005). Mechanistically, HFD-OVX treatment led to upregulation of genes markers of senescence, bone resorption, adipogenesis, inflammation, downregulation of gene markers of bone formation and bone development. Similarly, HFD-OVX treatment resulted in significant changes in bone tissue levels of purine/pyrimidine and Glutamate metabolisms, known to play a regulatory role in bone metabolism. Obesity and estrogen deficiency exert combined deleterious effects on bone resulting in accelerated cellular senescence, expansion of BMAT and impaired bone formation leading to decreased bone mass. Our results suggest that obesity may increase bone fragility in postmenopausal women., (© 2022 The Authors. Aging Cell published by Anatomical Society and John Wiley & Sons Ltd.)

Experiments that compare rhythmic properties across different genetic alterations and entrainment conditions underlie some of the most important breakthroughs in circadian biology. A robust estimation of the rhythmic properties of the circadian signals goes hand in hand with these discoveries. Widely applied traditional signal analysis methods such as fitting cosine functions or Fourier transformations rely on the assumption that oscillation periods do not change over time. However, novel high-resolution recording techniques have shown that, most commonly, circadian signals exhibit time-dependent changes of periods and amplitudes which cannot be captured with the traditional approaches. In this chapter we introduce a method to determine time-dependent properties of oscillatory signals, using the novel open-source Python-based Biological Oscillations Analysis Toolkit (pyBOAT). We show with examples how to detect rhythms, compute and interpret high-resolution time-dependent spectral results, analyze the main oscillatory component, and to subsequently determine these main components' time-dependent instantaneous period, amplitude, and phase. We introduce step-by-step how such an analysis can be done by means of the easy-to-use point-and-click graphical user interface (GUI) provided by pyBOAT or executed within a Python programming environment. Concepts are explained using simulated signals as well as experimentally obtained time series., (© 2022. The Author(s).)

Autonomously oscillating circadian clocks resonate with daily environmental (zeitgeber) rhythms to organize physiology around the solar day. Although entrainment properties and mechanisms have been studied widely and in great detail for light-dark cycles, entrainment to daily temperature rhythms remains poorly understood despite that they are potent zeitgebers. Here we investigate the entrainment of the chronobiological model organism Neurospora crassa , subject to thermocycles of different periods and fractions of warm versus cold phases, mimicking seasonal variations. Depending on the properties of these thermocycles, regularly entrained rhythms, period-doubling (frequency demultiplication) but also irregular aperiodic behavior occurs. We demonstrate that the complex nonlinear phenomena of experimentally observed entrainment dynamics can be understood by molecular mathematical modeling., Competing Interests: The authors declare no competing interests., (© 2021 The Authors.)

Clock genes Cry1 and Cry2, inhibitory components of core molecular feedback loop, are regarded as critical molecules for the circadian rhythm generation in mammals. A double knockout of Cry1 and Cry2 abolishes the circadian behavioral rhythm in adult mice under constant darkness. However, robust circadian rhythms in PER2::LUC expression are detected in the cultured suprachiasmatic nucleus (SCN) of Cry1/Cry2 deficient neonatal mice and restored in adult SCN by co-culture with wild-type neonatal SCN. These findings led us to postulate the compensatory molecule(s) for Cry1/Cry2 deficiency in circadian rhythm generation. We examined the roles of Chrono and Dec1/Dec2 proteins, the suppressors of Per(s) transcription similar to CRY(s). Unexpectedly, knockout of Chrono or Dec1/Dec2 in the Cry1/Cry2 deficient mice did not abolish but decoupled the coherent circadian rhythm into three different periodicities or significantly shortened the circadian period in neonatal SCN. DNA microarray analysis for the SCN of Cry1/Cry2 deficient mice revealed substantial increases in Per(s), Chrono and Dec(s) expression, indicating disinhibition of the transactivation by BMAL1/CLOCK. Here, we conclude that Chrono and Dec1/Dec2 do not compensate for absence of CRY1/CRY2 in the circadian rhythm generation but contribute to the coherent circadian rhythm expression in the neonatal mouse SCN most likely through integration of cellular circadian rhythms., (© 2021. The Author(s).)

The mammalian circadian clock is well-known to be important for our sleep-wake cycles, as well as other daily rhythms such as temperature regulation, hormone release or feeding-fasting cycles. Under normal conditions, these daily cyclic events follow 24 h limit cycle oscillations, but under some circumstances, more complex nonlinear phenomena, such as the emergence of chaos, or the splitting of physiological dynamics into oscillations with two different periods, can be observed. These nonlinear events have been described at the organismic and tissue level, but whether they occur at the cellular level is still unknown. Our results show that period-doubling, chaos and splitting appear in different models of the mammalian circadian clock with interlocked feedback loops and in the absence of external forcing. We find that changes in the degradation of clock genes and proteins greatly alter the dynamics of the system and can induce complex nonlinear events. Our findings highlight the role of degradation rates in determining the oscillatory behaviour of clock components, and can contribute to the understanding of molecular mechanisms of circadian dysregulation.

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., (Copyright © 2020 The Author(s). Published by Elsevier Ltd.. All rights reserved.)

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., (Copyright © 2020 Tokuda, Schmal, Ananthasubramaniam and Herzel.)

Entrainment denotes a process of coordinating the internal circadian clock to external rhythmic time-cues (Zeitgeber), mainly light. It is facilitated by stronger Zeitgeber signals and smaller period differences between the internal clock and the external Zeitgeber. The phase of entrainment ψ is a result of this process on the side of the circadian clock. On Earth, the period of the day-night cycle is fixed to 24 h, while the periods of circadian clocks distribute widely due to natural variation within and between species. The strength and duration of light depend locally on season and geographic latitude. Therefore, entrainment characteristics of a circadian clock vary under a local light environment and distribute along geoecological settings. Using conceptual models of circadian clocks, we investigate how local conditions of natural light shape global patterning of entrainment through seasons. This clock-side entrainment paradigm enables us to predict systematic changes in the global distribution of chronotypes., (Copyright © 2020 Schmal, Herzel and Myung.)

Circadian rhythms are generated by interlocked transcriptional-translational negative feedback loops (TTFLs), the molecular process implemented within a cell. The contributions, weighting and balancing between the multiple feedback loops remain debated. Dissociated, free-running dynamics in the expression of distinct clock genes has been described in recent experimental studies that applied various perturbations such as slice preparations, light pulses, jet-lag, and culture medium exchange. In this paper, we provide evidence that this "presumably transient" dissociation of circadian gene expression oscillations may occur at the single-cell level. Conceptual and detailed mechanistic mathematical modeling suggests that such dissociation is due to a weak interaction between multiple feedback loops present within a single cell. The dissociable loops provide insights into underlying mechanisms and general design principles of the molecular circadian clock., Competing Interests: The authors have declared that no competing interests exist.

Mammalian circadian clocks have a hierarchical organization, governed by the suprachiasmatic nucleus (SCN) in the hypothalamus. The brain itself contains multiple loci that maintain autonomous circadian rhythmicity, but the contribution of the non-SCN clocks to this hierarchy remains unclear. We examine circadian oscillations of clock gene expression in various brain loci and discovered that in mouse, robust, higher amplitude, relatively faster oscillations occur in the choroid plexus (CP) compared to the SCN. Our computational analysis and modeling show that the CP achieves these properties by synchronization of "twist" circadian oscillators via gap-junctional connections. Using an in vitro tissue coculture model and in vivo targeted deletion of the Bmal1 gene to silence the CP circadian clock, we demonstrate that the CP clock adjusts the SCN clock likely via circulation of cerebrospinal fluid, thus finely tuning behavioral circadian rhythms.

Modern imaging techniques allow the monitoring of circadian rhythms of single cells. Coupling between these single cellular circadian oscillators can generate coherent periodic signals on the tissue level that subsequently orchestrate physiological outputs. The strength of coupling in such systems of oscillators is often unclear. In particular, effects on coupling strength by varying cell densities, by knockouts, and by inhibitor applications are debated. In this study, we suggest to quantify the relative coupling strength via analyzing period, phase, and amplitude distributions in ensembles of individual circadian oscillators. Simulations of different oscillator networks show that period and phase distributions become narrower with increasing coupling strength. Moreover, amplitudes can increase due to resonance effects. Variances of periods and phases decay monotonically with coupling strength, and can serve therefore as measures of relative coupling strength. Our theoretical predictions are confirmed by studying recently published experimental data from PERIOD2 expression in slices of the suprachiasmatic nucleus during and after the application of tetrodotoxin (TTX). On analyzing the corresponding period, phase, and amplitude distributions, we can show that treatment with TTX can be associated with a reduced coupling strength in the system of coupled oscillators. Analysis of an oscillator network derived directly from the data confirms our conclusions. We suggest that our approach is also applicable to quantify coupling in fibroblast cultures and hepatocyte networks, and for social synchronization of circadian rhythmicity in rodents, flies, and bees.

Motivation: Neural activities of the brain occur through the formation of spatio-temporal patterns. In recent years, macroscopic neural imaging techniques have produced a large body of data on these patterned activities, yet a numerical measure of spatio-temporal coherence has often been reduced to the global order parameter, which does not uncover the degree of spatial correlation. Here, we propose to use the spatial autocorrelation measure Moran's I, which can be applied to capture dynamic signatures of spatial organization. We demonstrate the application of this technique to collective cellular circadian clock activities measured in the small network of the suprachiasmatic nucleus (SCN) in the hypothalamus., Results: We found that Moran's I is a practical quantitative measure of the degree of spatial coherence in neural imaging data. Initially developed with a geographical context in mind, Moran's I accounts for the spatial organization of any interacting units. Moran's I can be modified in accordance with the characteristic length scale of a neural activity pattern. It allows a quantification of statistical significance levels for the observed patterns. We describe the technique applied to synthetic datasets and various experimental imaging time-series from cultured SCN explants. It is demonstrated that major characteristics of the collective state can be described by Moran's I and the traditional Kuramoto order parameter R in a complementary fashion., Availability and Implementation: Python 2.7 code of illustrative examples can be found in the Supplementary Material., Contact: christoph.schmal@charite.de or grigory.bordyugov@hu-berlin.de., Supplementary Information: Supplementary data are available at Bioinformatics online., (© The Author(s) 2017. Published by Oxford University Press.)

Lymphocytes circulate through lymph nodes (LN) in search for antigen in what is believed to be a continuous process. Here, we show that lymphocyte migration through lymph nodes and lymph occurred in a non-continuous, circadian manner. Lymphocyte homing to lymph nodes peaked at night onset, with cells leaving the tissue during the day. This resulted in strong oscillations in lymphocyte cellularity in lymph nodes and efferent lymphatic fluid. Using lineage-specific genetic ablation of circadian clock function, we demonstrated this to be dependent on rhythmic expression of promigratory factors on lymphocytes. Dendritic cell numbers peaked in phase with lymphocytes, with diurnal oscillations being present in disease severity after immunization to induce experimental autoimmune encephalomyelitis (EAE). These rhythms were abolished by genetic disruption of T cell clocks, demonstrating a circadian regulation of lymphocyte migration through lymph nodes with time-of-day of immunization being critical for adaptive immune responses weeks later., (Copyright © 2017 The Author(s). Published by Elsevier Inc. All rights reserved.)

Background: The analysis of circadian leaf movement rhythms is a simple yet effective method to study effects of treatments or gene mutations on the circadian clock of plants. Currently, leaf movements are analysed using time lapse photography and subsequent bioinformatics analyses of leaf movements. Programs that are used for this purpose either are able to perform one function (i.e. leaf tip detection or rhythm analysis) or their function is limited to specific computational environments. We developed a leaf movement analysis tool-PALMA-that works in command line and combines image extraction with rhythm analysis using Fast Fourier transformation and non-linear least squares fitting., Results: We validated PALMA in both simulated time series and in experiments using the known short period mutant sensitivity to red light reduced 1 ( srr1 - 1 ). We compared PALMA with two established leaf movement analysis tools and found it to perform equally well. Finally, we tested the effect of reduced iron conditions on the leaf movement rhythms of wild type plants. Here, we found that PALMA successfully detected period lengthening under reduced iron conditions., Conclusions: PALMA correctly estimated the period of both simulated and real-life leaf movement experiments. As a platform-independent console-program that unites both functions needed for the analysis of circadian leaf movements it is a valid alternative to existing leaf movement analysis tools.

In addition to being endogenous, a circadian system must be able to communicate with the outside world and align its rhythmicity to the environment. As a result of such alignment, external Zeitgebers can entrain the circadian system. Entrainment expresses itself in coinciding periods of the circadian oscillator and the Zeitgeber and a stationary phase difference between them. The range of period mismatches between the circadian system and the Zeitgeber that Zeitgeber can overcome to entrain the oscillator is called an entrainment range. The width of the entrainment range usually increases with increasing Zeitgeber strength, resulting in a wedge-like Arnold tongue. This classical view of entrainment does not account for the effects of photoperiod on entrainment. Zeitgebers with extremely small or large photoperiods are intuitively closer to constant environments than equinoctial Zeitgebers and hence are expected to produce a narrower entrainment range. In this paper, we present theoretical results on entrainment under different photoperiods. We find that in the photoperiod-detuning parameter plane, the entrainment zone is shaped in the form of a skewed onion. The bottom and upper points of the onion are given by the free-running periods in DD and LL, respectively. The widest entrainment range is found near photoperiods of 50%. Within the onion, we calculated the entrainment phase that varies over a range of 12 h. The results of our theoretical study explain the experimentally observed behavior of the entrainment phase in dependence on the photoperiod.

Circadian clocks are endogenous timekeepers that produce oscillations with a period of about one day. Their rhythmicity originates from complex gene regulatory networks at the cellular level. In the last decades, computational models have been proven to be a powerful tool in order to understand the dynamics and design principles of the complex regulatory circuitries underlying the circadian clocks of different organisms. We present the process of model development using a small and simplified two-gene regulatory network of the Arabidopsis circadian clock. Subsequently, we discuss important numerical techniques to analyze such a mathematical model using XPP-AUTO. We show how to solve deterministic and stochastic ordinary differential equations and how to compute bifurcation diagrams or simulate phase-shift experiments. We finally discuss the contributions of modeling to the understanding and dissection of the Arabidopsis circadian system.

The circadian clock controls many physiological processes in higher plants and causes a large fraction of the genome to be expressed with a 24h rhythm. The transcripts encoding the RNA-binding proteins AtGRP7 (Arabidopsis thaliana Glycine Rich Protein 7) and AtGRP8 oscillate with evening peaks. The circadian clock components CCA1 and LHY negatively affect AtGRP7 expression at the level of transcription. AtGRP7 and AtGRP8, in turn, negatively auto-regulate and reciprocally cross-regulate post-transcriptionally: high protein levels promote the generation of an alternative splice form that is rapidly degraded. This clock-regulated feedback loop has been proposed to act as a molecular slave oscillator in clock output. While mathematical models describing the circadian core oscillator in Arabidopsis thaliana were introduced recently, we propose here the first model of a circadian slave oscillator. We define the slave oscillator in terms of ordinary differential equations and identify the model's parameters by an optimization procedure based on experimental results. The model successfully reproduces the pertinent experimental findings such as waveforms, phases, and half-lives of the time-dependent concentrations. Furthermore, we obtain insights into possible mechanisms underlying the observed experimental dynamics: the negative auto-regulation and reciprocal cross-regulation via alternative splicing could be responsible for the sharply peaking waveforms of the AtGRP7 and AtGRP8 mRNA. Moreover, our results suggest that the AtGRP8 transcript oscillations are subordinated to those of AtGRP7 due to a higher impact of AtGRP7 protein on alternative splicing of its own and of the AtGRP8 pre-mRNA compared to the impact of AtGRP8 protein. Importantly, a bifurcation analysis provides theoretical evidence that the slave oscillator could be a toggle switch, arising from the reciprocal cross-regulation at the post-transcriptional level. In view of this, transcriptional repression of AtGRP7 and AtGRP8 by LHY and CCA1 induces oscillations of the toggle switch, leading to the observed high-amplitude oscillations of AtGRP7 mRNA.