An Unstable Singularity Underlies Stochastic Phasing of the Circadian Clock in Individual Cyanobacterial Cells

When lowering temperature, the in vivo circadian clock in cyanobacteria follows and surpasses the in vitro protein clock trough the Hopf bifurcation

The in vivo circadian clock in single cyanobacteria is studied here by time-lapse fluorescence microscopy when the temperature is lowered below 25°C. We first disentangle the circadian clock behavior from the bacterial cold shock response by identifying a sequence of "death steps" based on cellular indicators. By analyzing only "alive" traces, we show that the dynamic response of individual oscillatory traces to a step-down temperature signal is described by a simple Stuart-Landau oscillator model. The same dynamical analysis applied to in vitro data (KaiC phosphorylation level following a temperature step-down) allows for extracting and comparing both clock's responses to a temperature step down. It appears, therefore, that both oscillators go through a similar supercritical Hopf bifurcation. Finally, to quantitatively describe the temperature dependence of the resulting in vivo and in vitro Stuart-Landau parameters [Formula: see text] and [Formula: see text], we propose two simplified analytical models: temperature-dependent positive feedback or time-delayed negative feedback that is temperature compensated. Our results provide strong constraints for future models by revealing a specific time scale for transitory regimes in the cyanobacterial circadian system and its temperature dependence.

Environmental and molecular noise buffering by the cyanobacterial clock in individual cells

Circadian clocks enable organisms to anticipate daily cycles, while being robust to molecular and environmental noise. Here, we show how the clock of the cyanobacterium Synechococcus elongatus PCC 7942 buffers genetic and environmental perturbations through its core KaiABC phosphorylation loop. We first characterise single-cell clock dynamics in clock mutants using a microfluidics device that allows precise control of the microenvironment. We find that known clock regulators are dispensable for clock robustness, whilst perturbations of the core clock reveal that the wild type operates at a noise optimum that we can reproduce in a stochastic model of just the core phosphorylation loop. We then examine how the clock responds to noisy environments, including natural light conditions. The model accurately predicts how the clock filters out environmental noise, including fast light fluctuations, to keep time while remaining responsive to environmental shifts. Our findings illustrate how a simple clock network can exhibit complex noise filtering properties, advancing our understanding of how biological circuits can perform accurately in natural environments.

The cyanobacterial circadian clock couples to pulsatile processes using pulse amplitude modulation

Cellular processes are dynamic and often oscillatory, requiring precise coordination for optimal cell function.1,2,3,4,5,6,7 How distinct oscillatory processes can couple within a single cell remains an open question. Here, we use the cyanobacterial circadian clock8,9 as a model system to explore the coupling of oscillatory and pulsatile gene circuits. The cyanobacterial circadian clock generates 24-h oscillations in downstream targets10,11,12,13,14,15 to time processes across the day/night cycle.9,16,17,18,19,20,21,22 This timing is partly mediated by the clock's modulation of the activity of alternative sigma factors,14,23,24,25 which direct RNA polymerase to specific promoters.26 Using single-cell time-lapse microscopy and modeling, we find that the clock modulates the amplitude of expression pulses of the alternative sigma factor RpoD4, which occurs only at cell division. This pulse amplitude modulation (PAM), analogous to AM regulation in radio transmission,27 allows the clock to robustly generate a 24-h rhythm in rpoD4 expression despite rpoD4's pulsing frequency being non-circadian. By modulating cell division rates, we find that, as predicted by our model, PAM regulation generates the same 24-h period in rpoD4 pulse amplitude over a range of rpoD4 pulse frequencies. Furthermore, we identify a functional significance of rpoD4 expression levels: deletion of rpoD4 results in smaller cell sizes, whereas an increase in rpoD4 expression leads to larger cell sizes in a dose-dependent manner. Thus, our work reveals a link between the cell cycle, clock, and RpoD4 in cyanobacteria and suggests that PAM regulation can be a general mechanism for biological clocks to robustly modulate pulsatile downstream processes.

Insights into a clock's fidelity through vesicular encapsulation

The single-celled cyanobacterium, Synechococcus elongatus , generates circadian rhythms with exceptional fidelity and synchrony despite their femtoliter volumes. Here, we explore the mechanistic aspects of this fidelity, by reconstituting the KaiABC post-translational oscillator (PTO) in cell-mimetic giant vesicles (GUVs) under well-defined conditions in vitro . PTO proteins were encapsulated with a coefficient of variation that closely matched protein variations observed in live cells. Using fluorescently labeled KaiB and confocal microscopy, we were able to measure circadian rhythms generated by thousands of encapsulated PTOs at the single-vesicle level for several days as a function of protein concentration and GUV size. We find that PTO fidelity decreased with decreasing levels of encapsulated PTO proteins and in smaller GUVs. We also observed that in encapsulated PTOs, a significant fraction of KaiB localized to GUV membranes like it does in cyanobacteria. A mathematical model that uses empirical bulk concentration and stoichiometry limitations suggests that cyanobacteria overcome challenges to fidelity by expressing high levels of PTO proteins along with the CikA and SasA proteins, which buffer stochastic variations in the levels of KaiA and KaiB, respectively. Further, the model suggests that the transcription-translation feedback loop (TTFL) contributes at most a small percentage to the overall fidelity of the cyanobacterial circadian clock under constant conditions but is essential for maintaining phase synchrony. Our results are the first experimental demonstration of populations of synthetic cells that can autonomously keep circadian time. Additionally, the approach of using bulk relationships to understand complex phenomena in cell-like systems could be useful for understanding other collective behavior important in biology, such as liquid-liquid phase separation.

Amplitude response and singularity behavior of circadian clock to external stimuli

Amplitude changes caused by environmental cues are universal in the circadian clock and associated with various diseases. Singularity behavior, characterized by the disruption of circadian rhythms due to critical stimuli, has been observed across various species. Several mathematical models of the circadian clock have replicated this phenomenon. A comprehensive understanding of the amplitude response remains elusive due to experimental limitations. In this study, we address this question by utilizing a simple normal form model that accurately fits previous experimental data, thereby presenting a general mechanism. We employ a geometric framework to illustrate the dynamics in different stimuli of light-induced transcription (LIT) and light-induced degradation (LID), highlighting the core role of invisible instability in amplitude response. Our model systematically elucidates how stimulus mode, phase, and strength determine amplitude responses. The results show that external stimuli induce alterations in both the amplitudes of individual oscillators and the synchronization among oscillators, collectively influencing the overall amplitude response. While experimental methods impose constraints resulting in limited outcomes under specific conditions, our model provides a comprehensive and three-dimensional mechanistic explanation. A comparison with existing experimental findings demonstrates the consistency of our proposed mechanism. Considering the response direction, the framework enables the identification of phases that lead to increased circadian amplitude. Based on this mechanism derived from the framework, stimulus strategies for resetting circadian rhythms with reduced side effects could be designed. Our results demonstrate that the framework has great potential for understanding and applying stimulus responses in the circadian clock and other limit cycle oscillations.

Principles, mechanisms and functions of entrainment in biological oscillators

Entrainment is a phenomenon in which two oscillators interact with each other, typically through physical or chemical means, to synchronize their oscillations. This phenomenon occurs in biology to coordinate processes from the molecular to organismal scale. Biological oscillators can be entrained within a single cell, between cells or to an external input. Using six illustrative examples of entrainable biological oscillators, we discuss the distinctions between entrainment and synchrony and explore features that contribute to a system's propensity to entrain. Entrainment can either enhance or reduce the heterogeneity of oscillations within a cell population, and we provide examples and mechanisms of each case. Finally, we discuss the known functions of entrainment and discuss potential functions from an evolutionary perspective.

Circa-SCOPE: high-throughput live single-cell imaging method for analysis of circadian clock resetting

Circadian clocks are self-sustained and cell-autonomous oscillators. They respond to various extracellular cues depending on the time-of-day and the signal intensity. Phase Transition Curves (PTCs) are instrumental in uncovering the full repertoire of responses to a given signal. However, the current methodologies for reconstructing PTCs are low-throughput, laborious, and resource- and time-consuming. We report here the development of an efficient and high throughput assay, dubbed Circadian Single-Cell Oscillators PTC Extraction (Circa-SCOPE) for generating high-resolution PTCs. This methodology relies on continuous monitoring of single-cell oscillations to reconstruct a full PTC from a single culture, upon a one-time intervention. Using Circa-SCOPE, we characterize the effects of various pharmacological and blood-borne resetting cues, at high temporal resolution and a wide concentration range. Thus, Circa-SCOPE is a powerful tool for comprehensive analysis and screening for circadian clocks' resetting cues, and can be valuable for basic as well as translational research.

Robust, coherent, and synchronized circadian clock-controlled oscillations along Anabaena filaments

Circadian clocks display remarkable reliability despite significant stochasticity in biomolecular reactions. We study the dynamics of a circadian clock-controlled gene at the individual cell level in Anabaena sp. PCC 7120, a multicellular filamentous cyanobacterium. We found significant synchronization and spatial coherence along filaments, clock coupling due to cell-cell communication, and gating of the cell cycle. Furthermore, we observed low-amplitude circadian oscillatory transcription of kai genes encoding the post-transcriptional core oscillatory circuit and high-amplitude oscillations of rpaA coding for the master regulator transducing the core clock output. Transcriptional oscillations of rpaA suggest an additional level of regulation. A stochastic one-dimensional toy model of coupled clock cores and their phosphorylation states shows that demographic noise can seed stochastic oscillations outside the region where deterministic limit cycles with circadian periods occur. The model reproduces the observed spatio-temporal coherence along filaments and provides a robust description of coupled circadian clocks in a multicellular organism.

Origin of exponential growth in nonlinear reaction networks

Exponentially growing systems are prevalent in nature, spanning all scales from biochemical reaction networks in single cells to food webs of ecosystems. How exponential growth emerges in nonlinear systems is mathematically unclear. Here, we describe a general theoretical framework that reveals underlying principles of long-term growth: scalability of flux functions and ergodicity of the rescaled systems. Our theory shows that nonlinear fluxes can generate not only balanced growth but also oscillatory or chaotic growth modalities, explaining nonequilibrium dynamics observed in cell cycles and ecosystems. Our mathematical framework is broadly useful in predicting long-term growth rates from natural and synthetic networks, analyzing the effects of system noise and perturbations, validating empirical and phenomenological laws on growth rate, and studying autocatalysis and network evolution.

Identifying a stochastic clock network with light entrainment for single cells of Neurospora crassa

Stochastic networks for the clock were identified by ensemble methods using genetic algorithms that captured the amplitude and period variation in single cell oscillators of Neurospora crassa. The genetic algorithms were at least an order of magnitude faster than ensemble methods using parallel tempering and appeared to provide a globally optimum solution from a random start in the initial guess of model parameters (i.e., rate constants and initial counts of molecules in a cell). The resulting goodness of fit [Formula: see text] was roughly halved versus solutions produced by ensemble methods using parallel tempering, and the resulting [Formula: see text] per data point was only [Formula: see text] = 2,708.05/953 = 2.84. The fitted model ensemble was robust to variation in proxies for "cell size". The fitted neutral models without cellular communication between single cells isolated by microfluidics provided evidence for only one Stochastic Resonance at one common level of stochastic intracellular noise across days from 6 to 36 h of light/dark (L/D) or in a D/D experiment. When the light-driven phase synchronization was strong as measured by the Kuramoto (K), there was degradation in the single cell oscillations away from the stochastic resonance. The rate constants for the stochastic clock network are consistent with those determined on a macroscopic scale of 10⁷ cells.

For Whom the Clock Ticks: Clinical Chronobiology for Infectious Diseases

The host defense against pathogens varies among individuals. Among the factors influencing host response, those associated with circadian disruptions are emerging. These latter depend on molecular clocks, which control the two partners of host defense: microbes and immune system. There is some evidence that infections are closely related to circadian rhythms in terms of susceptibility, clinical presentation and severity. In this review, we overview what is known about circadian rhythms in infectious diseases and update the knowledge about circadian rhythms in immune system, pathogens and vectors. This heuristic approach opens a new fascinating field of time-based personalized treatment of infected patients.

The Kai-Protein Clock-Keeping Track of Cyanobacteria's Daily Life

Life has adapted to Earth's day-night cycle with the evolution of endogenous biological clocks. Whereas these circadian rhythms typically involve extensive transcription-translation feedback in higher organisms, cyanobacteria have a circadian clock, which functions primarily as a protein-based post-translational oscillator. Known as the Kai system, it consists of three proteins KaiA, KaiB, and KaiC. In this chapter, we provide a detailed structural overview of the Kai components and how they interact to produce circadian rhythms of global gene expression in cyanobacterial cells. We discuss how the circadian oscillation is coupled to gene expression, intertwined with transcription-translation feedback mechanisms, and entrained by input from the environment. We discuss the use of mathematical models and summarize insights into the cyanobacterial circadian clock from theoretical studies. The molecular details of the Kai system are well documented for the model cyanobacterium Synechococcus elongatus, but many less understood varieties of the Kai system exist across the highly diverse phylum of Cyanobacteria. Several species contain multiple kai-gene copies, while others like marine Prochlorococcus strains have a reduced kaiBC-only system, lacking kaiA. We highlight recent findings on the genomic distribution of kai genes in Bacteria and Archaea and finally discuss hypotheses on the evolution of the Kai system.

Computational modelling unravels the precise clockwork of cyanobacteria

Precisely timing the regulation of gene expression by anticipating recurring environmental changes is a fundamental part of global gene regulation. Circadian clocks are one form of this regulation, which is found in both eukaryotes and prokaryotes, providing a fitness advantage for these organisms. Whereas many different eukaryotic groups harbour circadian clocks, cyanobacteria are the only known oxygenic phototrophic prokaryotes to regulate large parts of their genes in a circadian fashion. A decade of intensive research on the mechanisms and functionality using computational and mathematical approaches in addition to the detailed biochemical and biophysical understanding make this the best understood circadian clock. Here, we summarize the findings and insights into various parts of the cyanobacterial circadian clock made by mathematical modelling. These findings have implications for eukaryotic circadian research as well as synthetic biology harnessing the power and efficiency of global gene regulation.

Biophysical clocks face a trade-off between internal and external noise resistance

Many organisms use free running circadian clocks to anticipate the day night cycle. However, others organisms use simple stimulus-response strategies ('hourglass clocks') and it is not clear when such strategies are sufficient or even preferable to free running clocks. Here, we find that free running clocks, such as those found in the cyanobacterium Synechococcus elongatus and humans, can efficiently project out light intensity fluctuations due to weather patterns ('external noise') by exploiting their limit cycle attractor. However, such limit cycles are necessarily vulnerable to 'internal noise'. Hence, at sufficiently high internal noise, point attractor-based 'hourglass' clocks, such as those found in a smaller cyanobacterium with low protein copy number, Prochlorococcus marinus, can outperform free running clocks. By interpolating between these two regimes in a diverse range of oscillators drawn from across biology, we demonstrate biochemical clock architectures that are best suited to different relative strengths of external and internal noise.