The circadian oscillator analysed at the single-transcript level

Network switches and their role in circadian clocks

Circadian rhythms are generated by complex interactions among genes and proteins. Self-sustained ∼24 h oscillations require negative feedback loops and sufficiently strong nonlinearities that are the product of molecular and network switches. Here, we review common mechanisms to obtain switch-like behavior, including cooperativity, antagonistic enzymes, multisite phosphorylation, positive feedback, and sequestration. We discuss how network switches play a crucial role as essential components in cellular circadian clocks, serving as integral parts of transcription-translation feedback loops that form the basis of circadian rhythm generation. The design principles of network switches and circadian clocks are illustrated by representative mathematical models that include bistable systems and negative feedback loops combined with Hill functions. This work underscores the importance of negative feedback loops and network switches as essential design principles for biological oscillations, emphasizing how an understanding of theoretical concepts can provide insights into the mechanisms generating biological rhythms.

Coupling allows robust mammalian redox circadian rhythms despite heterogeneity and noise

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.

An improved rhythmicity analysis method using Gaussian Processes detects cell-density dependent circadian oscillations in stem cells

MOTIVATION

Detecting oscillations in time series remains a challenging problem even after decades of research. In chronobiology, rhythms (for instance in gene expression, eclosion, egg-laying, and feeding) tend to be low amplitude, display large variations amongst replicates, and often exhibit varying peak-to-peak distances (non-stationarity). Most currently available rhythm detection methods are not specifically designed to handle such datasets, and are also limited by their use of P-values in detecting oscillations.

RESULTS

We introduce a new method, ODeGP (Oscillation Detection using Gaussian Processes), which combines Gaussian Process regression and Bayesian inference to incorporate measurement errors, non-uniformly sampled data, and a recently developed non-stationary kernel to improve detection of oscillations. By using Bayes factors, ODeGP models both the null (non-rhythmic) and the alternative (rhythmic) hypotheses, thus providing an advantage over P-values. Using synthetic datasets, we first demonstrate that ODeGP almost always outperforms eight commonly used methods in detecting stationary as well as non-stationary symmetric oscillations. Next, by analyzing existing qPCR datasets, we demonstrate that our method is more sensitive compared to the existing methods at detecting weak and noisy oscillations. Finally, we generate new qPCR data on mouse embryonic stem cells. Surprisingly, we discover using ODeGP that increasing cell-density results in rapid generation of oscillations in the Bmal1 gene, thus highlighting our method's ability to discover unexpected and new patterns. In its current implementation, ODeGP is meant only for analyzing single or a few time-trajectories, not genome-wide datasets.

AVAILABILITY AND IMPLEMENTATION

ODeGP is available at https://github.com/Shaonlab/ODeGP.

An improved rhythmicity analysis method using Gaussian Processes detects cell-density dependent circadian oscillations in stem cells

Detecting oscillations in time series remains a challenging problem even after decades of research. In chronobiology, rhythms in time series (for instance gene expression, eclosion, egg-laying and feeding) datasets tend to be low amplitude, display large variations amongst replicates, and often exhibit varying peak-to-peak distances (non-stationarity). Most currently available rhythm detection methods are not specifically designed to handle such datasets. Here we introduce a new method, ODeGP ( O scillation De tection using G aussian P rocesses), which combines Gaussian Process (GP) regression with Bayesian inference to provide a flexible approach to the problem. Besides naturally incorporating measurement errors and non-uniformly sampled data, ODeGP uses a recently developed kernel to improve detection of non-stationary waveforms. An additional advantage is that by using Bayes factors instead of p-values, ODeGP models both the null (non-rhythmic) and the alternative (rhythmic) hypotheses. Using a variety of synthetic datasets we first demonstrate that ODeGP almost always outperforms eight commonly used methods in detecting stationary as well as non-stationary oscillations. Next, on analyzing existing qPCR datasets that exhibit low amplitude and noisy oscillations, we demonstrate that our method is more sensitive compared to the existing methods at detecting weak oscillations. Finally, we generate new qPCR time-series datasets on pluripotent mouse embryonic stem cells, which are expected to exhibit no oscillations of the core circadian clock genes. Surprisingly, we discover using ODeGP that increasing cell density can result in the rapid generation of oscillations in the Bmal1 gene, thus highlighting our method’s ability to discover unexpected patterns. In its current implementation, ODeGP (available as an R package) is meant only for analyzing single or a few time-trajectories, not genome-wide datasets.

PRD-2 mediates clock-regulated perinuclear localization of clock gene RNAs within the circadian cycle of Neurospora

The transcription-translation negative feedback loops underlying animal and fungal circadian clocks are remarkably similar in their molecular regulatory architecture and, although much is understood about their central mechanism, little is known about the spatiotemporal dynamics of the gene products involved. A common feature of these circadian oscillators is a significant temporal delay between rhythmic accumulation of clock messenger RNAs (mRNAs) encoding negative arm proteins, for example, frq in Neurospora and Per1-3 in mammals, and the appearance of the clock protein complexes assembled from the proteins they encode. Here, we report use of single-molecule RNA fluorescence in situ hybridization (smFISH) to show that the fraction of nuclei actively transcribing the clock gene frq changes in a circadian manner, and that these mRNAs cycle in abundance with fewer than five transcripts per nucleus at any time. Spatial point patterning statistics reveal that frq is spatially clustered near nuclei in a time of day-dependent manner and that clustering requires an RNA-binding protein, PRD-2 (PERIOD-2), recently shown also to bind to mRNA encoding another core clock component, casein kinase 1. An intrinsically disordered protein, PRD-2 displays behavior in vivo and in vitro consistent with participation in biomolecular condensates. These data are consistent with a role for phase-separating RNA-binding proteins in spatiotemporally organizing clock mRNAs to facilitate local translation and assembly of clock protein complexes.

Inference and uncertainty quantification of stochastic gene expression via synthetic models

Estimating uncertainty in model predictions is a central task in quantitative biology. Biological models at the single-cell level are intrinsically stochastic and nonlinear, creating formidable challenges for their statistical estimation which inevitably has to rely on approximations that trade accuracy for tractability. Despite intensive interest, a sweet spot in this trade-off has not been found yet. We propose a flexible procedure for uncertainty quantification in a wide class of reaction networks describing stochastic gene expression including those with feedback. The method is based on creating a tractable coarse-graining of the model that is learned from simulations, a synthetic model, to approximate the likelihood function. We demonstrate that synthetic models can substantially outperform state-of-the-art approaches on a number of non-trivial systems and datasets, yielding an accurate and computationally viable solution to uncertainty quantification in stochastic models of gene expression.

An effective model of endogenous clocks and external stimuli determining circadian rhythms

Circadian endogenous clocks of eukaryotic organisms are an established and rapidly developing research field. To investigate and simulate in an effective model the effect of external stimuli on such clocks and their components we developed a software framework for download and simulation. The application is useful to understand the different involved effects in a mathematical simple and effective model. This concerns the effects of Zeitgebers, feedback loops and further modifying components. We start from a known mathematical oscillator model, which is based on experimental molecular findings. This is extended with an effective framework that includes the impact of external stimuli on the circadian oscillations including high dose pharmacological treatment. In particular, the external stimuli framework defines a systematic procedure by input-output-interfaces to couple different oscillators. The framework is validated by providing phase response curves and ranges of entrainment. Furthermore, Aschoffs rule is computationally investigated. It is shown how the external stimuli framework can be used to study biological effects like points of singularity or oscillators integrating different signals at once. The mathematical framework and formalism is generic and allows to study in general the effect of external stimuli on oscillators and other biological processes. For an easy replication of each numerical experiment presented in this work and an easy implementation of the framework the corresponding Mathematica files are fully made available. They can be downloaded at the following link: https://www.biozentrum.uni-wuerzburg.de/bioinfo/computing/circadian/ .