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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.

DNA-damaging chemotherapeutics are widely used in cancer treatments, but for solid tumors they often leave a residual tumor-cell population. Here we investigated how cellular states might affect the response of individual cells in a clonal population to cisplatin, a DNA-damaging chemotherapeutic agent. Using a live-cell reporter of cell cycle phase and long-term imaging, we monitored single-cell proliferation before, at the time of, and after treatment. We found that in response to cisplatin, cells either arrested or died, and the ratio of these outcomes depended on the dose. While we found that the cell cycle phase at the time of cisplatin addition was not predictive of outcome, the proliferative history of the cell was: highly proliferative cells were more likely to arrest than to die, whereas slowly proliferating cells showed a higher probability of death. Information theory analysis revealed that the dose of cisplatin had the greatest influence on the cells’ decisions to arrest or die, and that the proliferation status interacted with the cisplatin dose to further guide this decision. These results show an unexpected effect of proliferation status in regulating responses to cisplatin and suggest that slowly proliferating cells within tumors may be acutely vulnerable to chemotherapy.

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.

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.

The current model of the mammalian circadian oscillator is predominantly based on data from genetics and biochemistry experiments, while the cell biology of circadian clocks is still in its infancy. Here, we describe a new strategy for the efficient generation of knock-in reporter cell lines using CRISPR technology that is particularly useful for lowly or transiently expressed genes, such as those coding for circadian clock proteins. We generated single and double knock-in cells with endogenously expressed PER2 and CRY1 fused to fluorescent proteins, which allowed to simultaneously monitor the dynamics of CRY1 and PER2 proteins in live single cells. Both proteins are highly rhythmic in the nucleus of human cells with PER2 showing a much higher amplitude than CRY1. Surprisingly, CRY1 protein is nuclear at all circadian times indicating the absence of circadian gating of nuclear import. Furthermore, in the nucleus of individual cells CRY1 abundance rhythms are phase-delayed (∼5 hours), and CRY1 levels are much higher (>6 times) compared to PER2 questioning the current model of the circadian oscillator. Our knock-in strategy will allow the generation of additional single, double or triple knock-in cells for circadian clock proteins, which should greatly advance our understanding about the cell biology of circadian clocks.

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.

Most species evolved a circadian clock to adapt to the 24 h period of the solar day. In mammals, these clocks generate endogenous rhythms by regulatory gene networks in almost every cell. A pacemaker, the suprachiasmatic nucleus (SCN) as the master clock, receives environmental input and orchestrates peripheral organs via sympathetic enervation, temperature and humoral factors. However, the mechanisms by which this synchronization is achieved are largely unknown. In order to elucidate paradigms of environmental information transfer within the circadian network, we address the following questions: How is environmental information perceived by different circadian networks? Do different circadian networks vary in their responses to a given signal, and, if so, do the differences depend on inherent circadian properties? Which part of the signal (onset, offset, duration, strength) is relevant for the responses? To address these questions, we combine experimental data from cultured single cells and organotypic slices with mathematical models of circadian oscillators and find that temperature signals have a strong impact on circadian rhythms, depending on the specific circadian properties of the clock cells.

Autonomous circadian clocks drive daily rhythms in physiology and behaviour. A network of coupled neurons, the suprachiasmatic nucleus (SCN), serves as a robust self-sustained circadian pacemaker. Synchronization of this timer to the environmental light-dark cycle is crucial for an organism's fitness. In a recent theoretical and experimental study it was shown that coupling governs the entrainment range of circadian clocks. We apply the theory of coupled oscillators to analyse how diffusive and mean-field coupling affects the entrainment range of interacting cells. Mean-field coupling leads to amplitude expansion of weak oscillators and, as a result, reduces the entrainment range. We also show that coupling determines the rigidity of the synchronized SCN network, i.e. the relaxation rates upon perturbation. %(Floquet exponents). Our simulations and analytical calculations using generic oscillator models help to elucidate how coupling determines the entrainment of the SCN. Our theoretical framework helps to interpret experimental data.

The endogenous circadian timing system has evolved to synchronize an organism to periodically recurring environmental conditions. Those external time cues are called Zeitgebers. When entrained by a Zeitgeber, the intrinsic oscillator adopts a fixed phase relation ψ to the Zeitgeber. Here, we systematically study how the phase of entrainment depends on clock and Zeitgeber properties. We combine numerical simulations of amplitude-phase models with predictions from analytically tractable models. In this way we derive relations between the phase of entrainment ψ to the mismatch between the endogenous and Zeitgeber period, the Zeitgeber strength, and the range of entrainment. A core result is the "180° rule" asserting that the phase ψ varies over a range of about 180° within the entrainment range. The 180° rule implies that clocks with a narrow entrainment range ("strong oscillators") exhibit quite flexible entrainment phases. We argue that this high sensitivity of the entrainment phase contributes to the wide range of human chronotypes.

The suprachiasmatic nucleus (SCN) coordinates via multiple outputs physiological and behavioural circadian rhythms. The SCN is composed of a heterogeneous network of coupled oscillators that entrain to the daily light–dark cycles. Outside the physiological entrainment range, rich locomotor patterns of desynchronized rhythms are observed. Previous studies interpreted these results as the output of different SCN neural subpopulations. We find, however, that even a single periodically driven oscillator can induce such complex desynchronized locomotor patterns. Using signal analysis, we show how the observed patterns can be consistently clustered into two generic oscillatory interaction groups: modulation and superposition. In seven of 17 rats undergoing forced desynchronization, we find a theoretically predicted third spectral component. Combining signal analysis with the theory of coupled oscillators, we provide a framework for the study of circadian desynchronization.

Circadian clock oscillator properties that are crucial for synchronization with the environment (entrainment) are studied in experiment and theory. The ratio between stimulus (zeitgeber) strength and oscillator amplitude, and the rigidity of the oscillatory system (relaxation rate upon perturbation) determine entrainment properties. Coupling among oscillators affects both qualities resulting in increased amplitude and rigidity. Uncoupled lung clocks entrain to extreme zeitgeber cycles, whereas the coupled oscillator system in the suprachiasmatic nucleus (SCN) does not; however, when coupling in the SCN is inhibited, larger ranges of entrainment can be achieved., Daily rhythms in physiology, metabolism and behavior are controlled by an endogenous circadian timing system, which has evolved to synchronize an organism to periodically recurring environmental conditions, such as light–dark or temperature cycles. In mammals, the circadian system relies on cell-autonomous oscillators residing in almost every cell of the body. Cells of the SCN in the anterior hypothalamus are able to generate precise, long-lasting self-sustained circadian oscillations, which drive most rhythmic behavioral and physiological outputs, and which are believed to originate from the fact that the SCN tissue consists of tightly coupled cells (Aton and Herzog, 2005). In contrast, peripheral oscillators, such as lung tissue, exhibit seemingly damped and usually less precise oscillations, which are thought to be brought about by the lack of intercellular coupling. Precise synchronization of these rhythms within the organism, but also with the environment (so-called entrainment), is an essential part of circadian organization. Entrainment is one of the cornerstones of circadian biology (Roenneberg et al, 2003). In evolution, the phase of a rhythmic variable is selective rather than its endogenous period. Thus, the synchronization of endogenous rhythms to zeitgeber cycles of the environment (resulting in a specific phase of entrainment) is fundamental for the adaptive value of circadian clocks. In this study, we systematically investigated the properties of circadian oscillators that are essential for entrainment behavior and describe coupling as a primary determinant. As an experimental starting point of this study, we found that the circadian oscillators of lung tissue have a larger range of entrainment than SCN tissue—they readily entrained to extreme experimental temperature cycle of 20 or 28 h, whereas SCN tissue did not (Figure 4). For this purpose, we cultured SCN and lung slices derived from mice that express luciferase as fusion protein together with the clock protein PERIOD2 (Yoo et al, 2004). The detection of luciferase-driven bioluminescence allowed us to follow molecular clock gene activity in real-time over the course of several days. In theoretical analyses, we show that both the ratio of amplitude and zeitgeber strength and, importantly, inter-oscillator coupling are major determinants for entrainment. The reason for coupling being critical is twofold: (i) Coupling makes an oscillatory system more rigid, i.e., it relaxes faster in response to a perturbation, and (ii) coupling increases the amplitude of the oscillatory system. Both of these consequences of coupling lead to a smaller entrainment range, because zeitgeber stimuli affect the oscillatory system less if the relaxation is fast and the amplitude is high (Figure 1). From these theoretical considerations, we conclude that the lung clock probably constitutes a weak oscillatory system, likely because a lack in coupling leads to a slow amplitude relaxation. (Circadian amplitude is not particularly low in lung (Figure 4).) In contrast, the SCN constitutes a rigid oscillator, whereby coupling and its described consequences probably are the primary causes for this rigidity. We then tested these theoretical predictions by experimentally perturbing coupling in the SCN (with MDL and TTX; O'Neill et al, 2008; Yamaguchi et al, 2003) and find that, indeed, reducing the coupling weakens the circadian oscillatory system in the SCN, which results in an enlargement of the entrainment range (Figure 6). Why is the SCN designed to be a stronger circadian oscillator than peripheral organs? We speculate that the position of the SCN—as the tissue that conveys environmental timing information (i.e., light) to the rest of the body—makes it necessary to create a circadian clock that is robust against noisy environmental stimuli. The SCN oscillator needs to be robust enough to be protected from environmental noise, but flexible enough to fulfill its function as an entrainable clock even in extreme photoperiods (i.e., seasons). By the same token, peripheral clocks are more protected from the environmental zeitgebers due to intrinsic homeostatic mechanisms. Thus, they do not necessarily need to develop a strong oscillatory system (e.g., by strengthening the coupling), rather they need to stay flexible enough to respond to direct or indirect signals from the SCN, such as hormonal, neural, temperature or metabolic signals. Such a design ensures that only robust and persistent environmental signals trigger an SCN resetting response, while SCN signals can relatively easily be conveyed to the rest of the body. Thus, the robustness in the SCN clock likely serves as a filter for environmental noise. In summary, using a combination of simulation studies, analytical calculations and experiments, we uncovered critical features for entrainment, such as zeitgeber-to-amplitude ratio and amplitude relaxation rate. Coupling is a primary factor that governs these features explaining important differences in the design of SCN and peripheral oscillators that ensure a robust, but also flexible circadian system., Circadian clocks are endogenous oscillators driving daily rhythms in physiology and behavior. Synchronization of these timers to environmental light–dark cycles (‘entrainment') is crucial for an organism's fitness. Little is known about which oscillator qualities determine entrainment, i.e., entrainment range, phase and amplitude. In a systematic theoretical and experimental study, we uncovered these qualities for circadian oscillators in the suprachiasmatic nucleus (SCN—the master clock in mammals) and the lung (a peripheral clock): (i) the ratio between stimulus (zeitgeber) strength and oscillator amplitude and (ii) the rigidity of the oscillatory system (relaxation rate upon perturbation) determine entrainment properties. Coupling among oscillators affects both qualities resulting in increased amplitude and rigidity. These principles explain our experimental findings that lung clocks entrain to extreme zeitgeber cycles, whereas SCN clocks do not. We confirmed our theoretical predictions by showing that pharmacological inhibition of coupling in the SCN leads to larger ranges of entrainment. These differences between master and the peripheral clocks suggest that coupling-induced rigidity in the SCN filters environmental noise to create a robust circadian system.

Phase response curves (PRCs) are widely used in circadian clocks, neuroscience, and heart physiology. They quantify the response of an oscillator to pulse-like perturbations. Phase response curves provide valuable information on the properties of oscillators and their synchronization. This chapter discusses biological self-sustained oscillators (circadian clock, physiological rhythms, etc.) in the context of nonlinear dynamics theory. Coupled oscillators can synchronize with different frequency ratios, can generate toroidal dynamics (superposition of independent frequencies), and may lead to deterministic chaos. These nonlinear phenomena can be analyzed with the aid of a phase transition curve, which is intimately related to the phase response curve. For illustration purposes, this chapter discusses a model of circadian oscillations based on a delayed negative feedback. In a second part, the chapter provides a step-by-step recipe to measure phase response curves. It discusses specifications of this recipe for circadian rhythms, heart rhythms, neuronal spikes, central pattern generators, and insect communication. Finally, it stresses the predictive power of measured phase response curves. PRCs can be used to quantify the coupling strength of oscillations, to classify oscillator types, and to predict the complex dynamics of periodically driven oscillations.

The circadian clock coordinates daily physiological, metabolic and behavioural rhythms. These endogenous oscillations are synchronized with external cues (‘zeitgebers’), such as daily light and temperature cycles. When the circadian clock is entrained by a zeitgeber, the phase difference ψ between the phase of a clock-controlled rhythm and the phase of the zeitgeber is of fundamental importance for the fitness of the organism. The phase of entrainment ψ depends on the mismatch between the intrinsic period τ and the zeitgeber period T and on the ratio of the zeitgeber strength to oscillator amplitude. Motivated by the intriguing complexity of empirical data and by our own experiments on temperature entrainment of mouse suprachiasmatic nucleus (SCN) slices, we present a theory on how clock and zeitgeber properties determine the phase of entrainment. The wide applicability of the theory is demonstrated using mathematical models of different complexity as well as by experimental data. Predictions of the theory are confirmed by published data on Neurospora crassa strains for different period mismatches τ − T and varying photoperiods. We apply a novel regression technique to analyse entrainment of SCN slices by temperature cycles. We find that mathematical models can explain not only the stable asymptotic phase of entrainment, but also transient phase dynamics. Our theory provides the potential to explore seasonal variations of circadian rhythms, jet lag and shift work in forthcoming studies.

Circadian clocks are internal timekeepers present in almost all organisms. Driven by a genetic network of highly conserved structure, they generate self-sustained oscillations that entrain to periodic external signals such as the 24 h light–dark cycle. Vertebrates possess multiple, functionally overlapping homologues of the core clock genes. Furthermore, vertebrate clocks entrain to a range of periods three times as narrow as that of other organisms. We asked whether genetic redundancies play a role in governing entrainment properties and analysed locomotor activity rhythms of genetically modified mice lacking one set of clock homologues. Exposing them to non-24 h light–dark cycles, we found that the mutant mice have a wider entrainment range than the wild types. Spectral analysis furthermore revealed nonlinear phenomena of periodically forced self-sustained oscillators for which the entrainment range relates inversely to oscillator amplitude. Using the forced oscillator model to explain the observed differences in entrainment range between mutant and wild-type mice, we sought to quantify the overall oscillator amplitude of their clocks from the activity rhythms and found that mutant mice have weaker circadian clocks than wild types. Our results suggest that genetic redundancy strengthens the circadian clock leading to a narrow entrainment range in vertebrates.

Entrainment, where oscillators synchronize to an external signal, is ubiquitous in nature. The transient time leading to entrainment plays a major role in many biological processes. Our goal is to unveil the specific dynamics that leads to fast entrainment. By studying a generic model, we characterize the transient time to entrainment and show how it is governed by two basic properties of an oscillator: the radial relaxation time and the phase velocity distribution around the limit cycle. Those two basic properties are inherent in every oscillator. This concept can be applied to many biological systems to predict the average transient time to entrainment or to infer properties of the underlying oscillator from the observed transients. We found that both a sinusoidal oscillator with fast radial relaxation and a spike-like oscillator with slow radial relaxation give rise to fast entrainment. As an example, we discuss the jet-lag experiments in the mammalian circadian pacemaker.