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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 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, fastening 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.

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.

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.

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.

A defining property of circadian clocks is temperature compensation, characterized by the resilience of circadian free-running periods against changes in environmental temperature. As an underlying mechanism, the balance or critical reaction hypothesis have been proposed. While the former supposes a temperature-dependent balancing of reactions with opposite effects on circadian period, the latter assumes an insensitivity of certain critical period determining regulations upon temperature changes. 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, abolishes circadian temperature compensation in 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 knock-down cells and their dependency on temperature. Analyzing differential responses upon temperature changes in wild type and CPSF6 knockdown cells reveals candidate genes underlying circadian temperature compensation. We identify that eukaryotic translation initiation factor 2 subunit 1 (EIF2S1) is among these candidates. EIF2S1 is known as a master regulator of cellular stress responses that additionally regulates circadian rhythms. We show that knockdown of EIF2S1 furthermore impairs temperature compensation, suggesting that the role of CPSF6 in temperature compensation may be mediated by its regulation of EIF2S1.

Autonomously oscillating circadian clocks resonate with daily environmental (zeitgeber) rhythms to organize physiology around the solar day. While 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.Abstract Figure

Methods for the quantification of rhythmic biological signals have been essential for the discovery of function and design of biological oscillators. Advances in live measurements have allowed recordings of unprecedented resolution revealing a new world of complex heterogeneous oscillations with multiple noisy non-stationary features. However, our understand- ing of the underlying mechanisms regulating these oscillations has been lagging behind, partially due to the lack of simple tools to reliably quantify these complex non-stationary features. With this challenge in mind, we have developed pyBOAT, a Python-based fully automatic stand-alone software that integrates multiple steps of non-stationary oscillatory time series analysis into an easy-to-use graphical user interface. pyBOAT implements continuous wavelet analysis which is specifically designed to reveal time-dependent features. In this work we illustrate the advantages of our tool by analyzing complex non-stationary time-series profiles. Our approach integrates data-visualization, optimized sinc-filter detrending, amplitude envelope removal and a subsequent continuous-wavelet based time- frequency analysis. Finally, using analytical considerations and numerical simulations we discuss unexpected pitfalls in commonly used smoothing and detrending operations.

Methods for the quantification of rhythmic biological signals have been essential for the discovery of function and design of biological oscillators. Advances in live measurements have allowed recordings of unprecedented resolution revealing a new world of complex heterogeneous oscillations with multiple noisy non-stationary features. However, our understanding of the underlying mechanisms regulating these oscillations has been lagging behind, partially due to the lack of simple tools to reliably quantify these complex non-stationary features. With this challenge in mind, we have developed pyBOAT, a Python-based fully automatic stand-alone software that integrates multiple steps of non-stationary oscillatory time series analysis into an easy-to-use graphical user interface. pyBOAT implements continuous wavelet analysis which is specifically designed to reveal time-dependent features. In this work we illustrate the advantages of our tool by analyzing complex non-stationary time-series profiles. Our approach integrates data-visualization, optimized sinc-filter detrending, amplitude envelope removal and a subsequent continuous-wavelet based time-frequency analysis. Finally, using analytical considerations and numerical simulations we discuss unexpected pitfalls in commonly used smoothing and detrending operations.

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.

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 tissu...

Methods for the quantification of rhythmic biological signals have been essential for the discovery of function and design of biological oscillators. Advances in live measurements have allowed recordings of unprecedented resolution revealing a new world of complex heterogeneous oscillations with multiple noisy time-dependent (non-stationary) features. However, our understanding of the underlying mechanisms regulating these oscillations has been lagging behind, partially due to the lack of tools to reliably quantify features as they change in time. With this challenge in mind, we have developed pyBOAT, a framework that integrates multiple optimal steps of non-stationary oscillatory time series analysis into an open source fully automatic stand-alone software with an easy-to-use graphical user interface. pyBOAT combines data-visualization, optimized sinc-filter detrending, amplitude envelope removal and a subsequent continuous-wavelet based time-frequency analysis. Using synthetic and real-world rhythmic data, we show that our method successfully identifies all time-dependent properties even at high levels of noise. In addition, we discuss how widely used smoothing and detrending operations can lead to unexpected artifacts and interpretation bias.

Understanding entrainment of circadian rhythms is a central goal of chronobiology. Many factors, such as period, amplitude, Zeitgeber strength, and day-length, govern entrainment ranges and the phase of entrainment. Using global optimization, we derive conceptual models with just three free parameters (period, amplitude, relaxation rate) that reproduce known phenotypic features of vertebrate clocks: relatively small phase response curves (PRCs), fast re-entrainment after jet-lag, and seasonal variability to track light onset or offset. Since optimization found multiple sets of model parameters, we can study this model ensemble to gain insight into the underlying design principles. We find that amplitudes control the size of PRCs, that fast relaxation supports short jet-lag, and that specific periods allow reasonable seasonal phase shifts. Arnold onions of representative models visualize strong dependencies of entrainment on periods, relative Zeitgeber strength, and photoperiod.

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 jetlag 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 one-hour pulse PRC with a range of three hours and clock periods between 22h and 26h. Surprisingly, almost all these randomly parameterized models showed a four hour change in phase of entrainment between long and short days and jetlag durations of three to seven days in advance and delay. Moreover, intrinsic clock period influenced jetlag duration and entrainment amplitude and phase. Fast jetlag 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.

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., Author summary Circadian clocks are endogenous pacemakers that generate gene expression oscillations with a period of approximately 24h. They enable an organism to anticipate daily changes in light and temperature and allow to align physiological properties to the most beneficial time around the solar day. The suprachiasmatic nucleus (SCN) of the hypothalamus is the master circadian pacemaker in mammals that coordinates peripheral clocks throughout the body and even encodes seasons. Gene expression oscillations of circadian clock genes in this master pacemaker have been shown to dissociate after perturbations of the system such as light pulses and jet-lag. The underlying mechanism remains unknown. We show that this dissociation may occur even within a single cell of the pacemaker. Data-driven mathematical modeling suggests that the dissociation relies upon a weak interaction between interlocked gene-regulatory feedback loops. Differential responses of these feedback loops to light perturbations is consistent with the concept of morning and evening oscillators.

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.

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.

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., The suprachiasmatic nucleus (SCN) has been thought of as the master circadian clock, but peripheral circadian clocks do exist. Here, the authors show that the choroid plexus displays oscillations more robust than the SCN and that can be described as a Poincaré oscillator with negative twist.