Low-dimensional Dynamics of Two Coupled Biological Oscillators

Identification of angiotensin II-responsive circadian clock gene expression in adrenal zona glomerulosa cells and human adrenocortical H295R cells

The mammalian circadian timing system is organized in a hierarchy, with the master clock residing in the suprachiasmatic nucleus (SCN) of the hypothalamus and subsidiary peripheral clocks in peripheral tissues. Because of the diversity of peripheral tissues and cell-types in the body, the existence of autonomous clock and identification of its potential entrainment signals need to be empirically defined on a cell type-by-cell type basis. In this study, we characterized the basic circadian clock properties of the adrenal zona glomerulosa cells, or ZG cells. Using isolated adrenal explants from Per2Luc mice, dissociated ZG cells from Per2-dluc rats, and a related human adrenocortical cell line H295R, we showed that ZG cells possess genetically-encoded, self-sustained and cell-autonomous circadian clock. As to the potential entrainment signals, angiotensin II (Ang II) caused phase-dependent phase-shifts of adrenal ZG cells in cultured slices. Ang II treatment also drove initiation (or reset) of circadian clock gene expression in H295R cells with associated immediate up-regulation of PER1 and E4BP4 mRNA expression. We found that the type I Ang II receptor blocker CV11974, one of the most widely used clinical drugs for hypertensive diseases, caused attenuation of the phase resetting of H295R cells. Our in vitro data provide a basis to understand and argue for the adrenal gland ZG cells as a component of autonomous and entrainable peripheral clocks.

A combined mathematical and experimental approach reveals the drivers of time-of-day drug sensitivity in human cells

The circadian clock plays a pivotal role in regulating various aspects of cancer, influencing tumor growth and treatment responses. There are significant changes in drug efficacy and adverse effects when drugs are administered at different times of the day, underscoring the importance of considering the time of day in treatments. Despite these well-established findings, chronotherapy approaches in drug treatment have yet to fully integrate into clinical practice, largely due to the stringent clinical requirements for proving efficacy and safety, alongside the need for deeper mechanistic insights. In this study, we employ a combined mathematical and experimental approach to systematically investigate the factors influencing time-of-day drug sensitivity in human cells. Here we show how circadian and drug properties independently shape time-of-day profiles, providing valuable insights into the temporal dynamics of treatment responses. Understanding how drug efficacy fluctuates throughout the day holds immense potential for the development of personalized treatment strategies aligned with an individual's internal biological clock, revolutionizing cancer treatment by maximizing therapeutic benefits. Moreover, our framework offers a promising avenue for refining future drug screening efforts, paving the way for more effective and targeted therapies across diverse tissue types.

Unveiling the complexity of Arnold's tongues in a breathing-soliton laser

Synchronization occurs ubiquitously in nature and science. The synchronization regions generally broaden monotonically with the strength of the forcing, thereby featuring a tongue-like shape in parameter space, known as Arnold's tongue. Such a shape is universal, prevailing in many diverse synchronized systems. Theoretical studies suggest that, under strong external forcing, the shape of the synchronization regions can change substantially and even holes can appear in the solid patterns. However, experimentally accessing these abnormal regimes is quite challenging mainly because many real-world systems displaying synchronization become fragile under strong forcing. Here, we are able to observe these intriguing regimes in a breathing-soliton laser. Two types of abnormal synchronization regions are unveiled, namely, a leaf- and a ray-like shape. High-resolution control of the loss allows holes to be revealed in the synchronization regions. Our work opens the possibility to study intriguing synchronization dynamics using a simple breathing-soliton laser as a test bed.

Morphological and Electrical Properties of Proteinoid-Actin Networks

Proteinoids, or thermal proteins, are produced by heating amino acids. Proteinoids form hollow microspheres in water. The microspheres produce oscillation of electrical potential. Actin is a filament-forming protein responsible for communication, information processing and decision making in eukaryotic cells. We synthesize randomly organized networks of proteinoid microspheres spanned by actin filaments and study their morphology and electrical potential oscillatory dynamics. We analyze proteinoid-actin networks' responses to electrical stimulation. The signals come from logistic maps, the Lorenz attractor, the Rossler oscillator, and the FitzHugh-Nagumo system. We show how the networks attenuated the signals produced by these models. We demonstrate that emergent logical patterns derived from oscillatory behavior of proteinoid-actin networks show characteristics of Boolean logic gates, providing evidence for the computational ability to combine different components through architectural changes in the dynamic interface. Our experimental laboratory study paves a base for generation of proto-neural networks and implementation of neuromorphic computation with them.

Acoustic frequency comb generation on a composite diamond/silicon microcantilever in ambient air

Acoustic frequency combs (AFCs) contain equidistant coherent signals with unconventional possibilities on metrology. Previously, implementation of AFCs on mechanical microresonators with large air damping loss is difficult, which restricted their atmospheric applications. In this work, we explore the potentials of a composite diamond/silicon microcantilever for parametric generation of AFCs in ambient air. We discover that the diamond layer provides a viable route to reduce the linewidth of the primary flexural mode, yielding a 7.1-times increase of the quality factor. We develop a parametric driving scheme that enables generation of AFCs through injection locking and sequential nonlinear dynamic transitions involving subharmonic synchronization (Arnold tongue), and chaotic dynamics. Ultimately, we realize AFCs with a frequency range extending 800 kHz in the air. This work advances the understanding of AFCs and provides a viable route towards their applications in ambient air for high precision metrology.

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.

Direct single-cell observation of a key Escherichia coli cell-cycle oscillator

Initiation of DNA replication in Escherichia coli is coupled to cell size via the DnaA protein, whose activity is dependent on its nucleotide-bound state. However, the oscillations in DnaA activity have never been observed at the single-cell level. By measuring the volume-specific production rate of a reporter protein under control of a DnaA-regulated promoter, we could distinguish two distinct cell-cycle oscillators. The first, driven by both DnaA activity and SeqA repression, shows a causal relationship with cell size and divisions, similarly to initiation events. The second one, a reporter of DnaA activity alone, loses the synchrony and causality properties. Our results show that transient inhibition of gene expression by SeqA keeps the oscillation of volume-sensing DnaA activity in phase with the subsequent division event and suggest that DnaA activity peaks do not correspond directly to initiation events.

A coupled model between circadian, cell-cycle, and redox rhythms reveals their regulation of oxidative stress

Most organisms possess three biological oscillators, circadian clock, cell cycle, and redox rhythm, which are autonomous but interact each other. However, whether their interactions and autonomy are beneficial for organisms remains unclear. Here, we modeled a coupled oscillator system where each oscillator affected the phase of the other oscillators. We found that multiple types of coupling prevent a high H2O2 level in cells at M phase. Consequently, we hypothesized a high H2O2 sensitivity at the M phase and found that moderate coupling reduced cell damage due to oxidative stress by generating appropriate phase relationships between three rhythms, whereas strong coupling resulted in an elevated cell damage by increasing the average H2O2 level and disrupted the cell cycle. Furthermore, the multicellularity model revealed that phase variations among cells confer flexibility in synchronization with environments at the expense of adaptability to the optimal environment. Thus, both autonomy and synchrony among the oscillators are important for coordinating their phase relationships to minimize oxidative stress, and couplings balance them depending on environments.

Long-term survival, toxicities, and the role of chrono-chemotherapy with different infusion rates in locally advanced nasopharyngeal carcinoma patients treated with intensity-modulated radiation therapy: a retrospective study with a 5-year follow-up

Purpose

This study aimed to evaluate 5-year outcomes and the late toxicity profile of chrono-chemotherapy with different infusion rates in patients with locally advanced nasopharyngeal carcinoma (NPC).

Methods and materials

Our retrospective analysis included 70 patients with locally advanced NPC stages III and IVB (according to the 2010 American Joint Committee on Cancer staging system). Patients were treated with two cycles of induction chemotherapy (IC) before concurrent chemoradiotherapy (CCRT) at Guizhou Cancer Hospital. The IC with docetaxel, cisplatin (DDP) and fluorouracil regimen. Patients were divided into two groups during CCRT. Using a "MELODIE" multi-channel programmed pump, DDP (100 mg/m2) was administered for 12 hours from 10:00 am to 10:00 pm and repeated every 3 weeks for 2-3 cycles. DDP was administered at the peak period of 4:00 pm in the sinusoidal chrono-modulated infusion group (Arm A, n=35). The patients in Arm B received a constant rate of infusion. Both arms received radiotherapy through the same technique and dose fraction. The long-term survival and disease progression were observed.

Results

After a median follow-up of 82.8 months, the 5-year progression-free survival rate was 81.3% in Arm A and 79.6% in Arm B (P = 0.85). The 5-year overall survival rate was not significantly different between Arm A and Arm B (79.6% vs 85.3%, P = 0.79). The 5-year distant metastasis-free survival rate was 83.6% in Arm A and 84.6% in Arm B (P = 0.75). The 5-year local recurrence-free survival rate was 88.2% in Arm A and 85.3% in Arm B (P = 0.16). There were no late toxicities of grade 3-4 in either group. Both groups had grade 1-2 late toxicities. Dry mouth was the most common late toxic side effect, followed by hearing loss and difficulty in swallowing. There was no statistically significant difference between Arm A and Arm B in terms of side effects.

Conclusion

Long-term analysis confirmed that in CCRT, cisplatin administration with sinusoidal chrono-modulated infusion was not superior to the constant infusion rate in terms of long-term toxicity and prognosis.

Contribution of membrane-associated oscillators to biological timing at different timescales

Environmental rhythms such as the daily light-dark cycle selected for endogenous clocks. These clocks predict regular environmental changes and provide the basis for well-timed adaptive homeostasis in physiology and behavior of organisms. Endogenous clocks are oscillators that are based on positive feedforward and negative feedback loops. They generate stable rhythms even under constant conditions. Since even weak interactions between oscillators allow for autonomous synchronization, coupling/synchronization of oscillators provides the basis of self-organized physiological timing. Amongst the most thoroughly researched clocks are the endogenous circadian clock neurons in mammals and insects. They comprise nuclear clockworks of transcriptional/translational feedback loops (TTFL) that generate ∼24 h rhythms in clock gene expression entrained to the environmental day-night cycle. It is generally assumed that this TTFL clockwork drives all circadian oscillations within and between clock cells, being the basis of any circadian rhythm in physiology and behavior of organisms. Instead of the current gene-based hierarchical clock model we provide here a systems view of timing. We suggest that a coupled system of autonomous TTFL and posttranslational feedback loop (PTFL) oscillators/clocks that run at multiple timescales governs adaptive, dynamic homeostasis of physiology and behavior. We focus on mammalian and insect neurons as endogenous oscillators at multiple timescales. We suggest that neuronal plasma membrane-associated signalosomes constitute specific autonomous PTFL clocks that generate localized but interlinked oscillations of membrane potential and intracellular messengers with specific endogenous frequencies. In each clock neuron multiscale interactions of TTFL and PTFL oscillators/clocks form a temporally structured oscillatory network with a common complex frequency-band comprising superimposed multiscale oscillations. Coupling between oscillator/clock neurons provides the next level of complexity of an oscillatory network. This systemic dynamic network of molecular and cellular oscillators/clocks is suggested to form the basis of any physiological homeostasis that cycles through dynamic homeostatic setpoints with a characteristic frequency-band as hallmark. We propose that mechanisms of homeostatic plasticity maintain the stability of these dynamic setpoints, whereas Hebbian plasticity enables switching between setpoints via coupling factors, like biogenic amines and/or neuropeptides. They reprogram the network to a new common frequency, a new dynamic setpoint. Our novel hypothesis is up for experimental challenge.

E-box binding transcription factors in cancer

E-boxes are important regulatory elements in the eukaryotic genome. Transcription factors can bind to E-boxes through their basic helix-loop-helix or zinc finger domain to regulate gene transcription. E-box-binding transcription factors (EBTFs) are important regulators of development and essential for physiological activities of the cell. The fundamental role of EBTFs in cancer has been highlighted by studies on the canonical oncogene MYC, yet many EBTFs exhibit common features, implying the existence of shared molecular principles of how they are involved in tumorigenesis. A comprehensive analysis of TFs that share the basic function of binding to E-boxes has been lacking. Here, we review the structure of EBTFs, their common features in regulating transcription, their physiological functions, and their mutual regulation. We also discuss their converging functions in cancer biology, their potential to be targeted as a regulatory network, and recent progress in drug development targeting these factors in cancer therapy.

An in silico genome-wide screen for circadian clock strength in human samples

MOTIVATION

Years of time-series gene expression studies have built a strong understanding of clock-controlled pathways across species. However, comparatively little is known about how 'non-clock' pathways influence clock function. We need a strong understanding of clock-coupled pathways in human tissues to better appreciate the links between disease and clock function.

RESULTS

We developed a new computational approach to explore candidate pathways coupled to the clock in human tissues. This method, termed LTM, is an in silico screen to infer genetic influences on circadian clock function. LTM uses natural variation in gene expression in human data and directly links gene expression variation to clock strength independent of longitudinal data. We applied LTM to three human skin and one melanoma datasets and found that the cell cycle is the top candidate clock-coupled pathway in healthy skin. In addition, we applied LTM to thousands of tumor samples from 11 cancer types in the TCGA database and found that extracellular matrix organization-related pathways are tightly associated with the clock strength in humans. Further analysis shows that clock strength in tumor samples is correlated with the proportion of cancer-associated fibroblasts and endothelial cells. Therefore, we show both the power of LTM in predicting clock-coupled pathways and classify factors associated with clock strength in human tissues.

AVAILABILITY AND IMPLEMENTATION

LTM is available on GitHub (https://github.com/gangwug/LTMR) and figshare (https://figshare.com/articles/software/LTMR/21217604) to facilitate its use.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Chronotherapy: Circadian Rhythms and Their Influence in Cancer Therapy

Living organisms present rhythmic fluctuations every 24 h in their behavior and metabolism to anticipate changes in the environment. These fluctuations are controlled by a very complex molecular mechanism, the circadian clock, that regulates the expression of multiple genes to ensure the right functioning of the body. An individual's circadian system is altered during aging, and this is related to numerous age-associated pathologies and other alterations that could contribute to the development of cancer. Nowadays, there is an increasing interest in understanding how circadian rhythms could be used in the treatment of cancer. Chronotherapy aims to understand the impact that biological rhythms have on the response to a therapy to optimize its action, maximize health benefits and minimize possible adverse effects. Clinical trials so far have confirmed that optimal timing of treatment with chemo or immunotherapies could decrease drug toxicity and increase efficacy. Instead, chronoradiotherapy seems to minimize treatment-related symptoms rather than tumor progression or patient survival. In addition, potential therapeutic targets within the molecular clock have also been identified. Therefore, results of the application of chronotherapy in cancer therapy until now are challenging, feasible, and could be applied to clinical practice to improve cancer treatment without additional costs. However, different limitations and variables such as age, sex, or chronotypes, among others, should be overcome before chronotherapy can really be put into clinical practice.

Cell size homeostasis under the circadian regulation of cell division in cyanobacteria

Bacterial cells maintain their characteristic cell size over many generations. Several rod-shaped bacteria, such as Escherichia coli and the cyanobacteria Synechococcus elongatus, divide after adding a constant length to their length at birth. Through this division control known as the adder mechanism, perturbation in cell length due to physiological fluctuation decays over generations at a rate of 2^-1 per cell division. However, previous experiments have shown that the circadian clock in cyanobacteria reduces cell division frequency at a specific time of day under constant light. This circadian gating should modulate the division control by the adder mechanism, but its significance remains unknown. Here we address how the circadian gating affects cell length, doubling time, and cell length stability in cyanobacteria by using mathematical models. We show that a cell subject to circadian gating grows for a long time, and gives birth to elongated daughter cells. These elongated daughter cells grow faster than the previous generation, as elongation speed is proportional to cell length and divide in a short time before the next gating. Hence, the distributions of doubling time and cell length become bimodal, as observed in experimental data. Interestingly, the average doubling time over the population of cells is independent of gating because the extension of doubling time by gating is compensated by its reduction in the subsequent generation. On the other hand, average cell length is increased by gating, suggesting that the circadian clock controls cell length. We then show that the decay rate of perturbation in cell length depends on the ratio of delay in division by the gating τ_G to the average doubling time τ₀ as [Formula: see text] . We estimated τ_G≈2.5, τ₀≈13.6 hours, and τ_G/τ₀≈0.18 from experimental data, indicating that a long doubling time in cyanobacteria maintains the decay rate similar to that of the adder mechanism. Thus, our analysis suggests that the acquisition of the circadian clock during evolution did not impose a constraint on cell size homeostasis in cyanobacteria.

Interacting Oscillators with Fluctuating Coupling: Mode Mixing without Cross-Correlations

Coupled oscillators are one of the basic models in nonlinear dynamics. Here, we study numerically and analytically the spectra of two harmonic oscillators with stochastically fluctuating coupling and driving forces reproducing a thermostat. We show that, even at small coupling, vanishing on average, the oscillation spectra exhibit mixing, even though no cross-correlations exists between the oscillators. Our results reveal a new mechanism of mode mixing for stochastically uncorrelated systems that is crucial for analysis of spectra in various systems, from simple liquids to living systems.

Bilateral Feedback in Oscillator Model Is Required to Explain the Coupling Dynamics of Hes1 with the Cell Cycle

Biological processes are governed by the expression of proteins, and for some proteins, their level of expression can fluctuate periodically over time (i.e., they oscillate). Many oscillatory proteins (e.g., cell cycle proteins and those from the HES family of transcription factors) are connected in complex ways, often within large networks. This complexity can be elucidated by developing intuitive mathematical models that describe the underlying critical aspects of the relationships between these processes. Here, we provide a mathematical explanation of a recently discovered biological phenomenon: the phasic position of the gene Hes1’s oscillatory expression at the beginning of the cell cycle of an individual human breast cancer stem cell can have a predictive value on how long that cell will take to complete a cell cycle. We use a two-component model of coupled oscillators to represent Hes1 and the cell cycle in the same cell with minimal assumptions. Inputting only the initial phase angles, we show that this model is capable of predicting the dynamic mitosis to mitosis behaviour of Hes1 and predicting cell cycle length patterns as found in real-world experimental data. Moreover, we discover that bidirectional coupling between Hes1 and the cell cycle is critical within the system for the data to be reproduced and that nonfixed asymmetry in the interactions between the oscillators is required. The phase dynamics we present here capture the complex interplay between Hes1 and the cell cycle, helping to explain nongenetic cell cycle variability, which has critical implications in cancer treatment contexts.

The emergence of polyglot entrainment responses to periodic inputs in vicinities of Hopf bifurcations in slow-fast systems

Several distinct entrainment patterns can occur in the FitzHugh-Nagumo (FHN) model under external periodic forcing. Investigating the FHN model under different types of periodic forcing reveals the existence of multiple disconnected 1:1 entrainment segments for constant, low enough values of the input amplitude when the unforced system is in the vicinity of a Hopf bifurcation. This entrainment structure is termed polyglot to distinguish it from the single 1:1 entrainment region (monoglot) structure typically observed in Arnold tongue diagrams. The emergence of polyglot entrainment is then explained using phase-plane analysis and other dynamical system tools. Entrainment results are investigated for other slow-fast systems of neuronal, circadian, and glycolytic oscillations. Exploring these models, we found that polyglot entrainment structure (multiple 1:1 regions) is observed when the unforced system is in the vicinity of a Hopf bifurcation and the Hopf point is located near a knee of a cubic-like nullcline.

Multi-synchronization and other patterns of multi-rhythmicity in oscillatory biological systems

While experimental and theoretical studies have established the prevalence of rhythmic behaviour at all levels of biological organization, less common is the coexistence between multiple oscillatory regimes (multi-rhythmicity), which has been predicted by a variety of models for biological oscillators. The phenomenon of multi-rhythmicity involves, most commonly, the coexistence between two (birhythmicity) or three (trirhythmicity) distinct regimes of self-sustained oscillations. Birhythmicity has been observed experimentally in a few chemical reactions and in biological examples pertaining to cardiac cell physiology, neurobiology, human voice patterns and ecology. The present study consists of two parts. We first review the mechanisms underlying multi-rhythmicity in models for biochemical and cellular oscillations in which the phenomenon was investigated over the years. In the second part, we focus on the coupling of the cell cycle and the circadian clock and show how an additional source of multi-rhythmicity arises from the bidirectional coupling of these two cellular oscillators. Upon bidirectional coupling, the two oscillatory networks generally synchronize in a unique manner characterized by a single, common period. In some conditions, however, the two oscillators may synchronize in two or three different ways characterized by distinct waveforms and periods. We refer to this type of multi-rhythmicity as ‘multi-synchronization’.

Synergies of Multiple Zeitgebers Tune Entrainment

Circadian rhythms are biological rhythms with a period close to 24 h. They become entrained to the Earth's solar day via different periodic cues, so-called zeitgebers. The entrainment of circadian rhythms to a single zeitgeber was investigated in many mathematical clock models of different levels of complexity, ranging from the Poincaré oscillator and the Goodwin model to biologically more detailed models of multiple transcriptional translational feedback loops. However, circadian rhythms are exposed to multiple coexisting zeitgebers in nature. Therefore, we study synergistic effects of two coexisting zeitgebers on different components of the circadian clock. We investigate the induction of period genes by light together with modulations of nuclear receptor activities by drugs and metabolism. Our results show that the entrainment of a circadian rhythm to two coexisting zeitgebers depends strongly on the phase difference between the two zeitgebers. Synergistic interactions of zeitgebers can strengthen diurnal rhythms to reduce detrimental effects of shift-work and jet lag. Medical treatment strategies which aim for stable circadian rhythms should consider interactions of multiple zeitgebers.

Harvesting random embedding for high-frequency change-point detection in temporal complex systems

Recent investigations have revealed that dynamics of complex networks and systems are crucially dependent on the temporal structures. Accurate detection of the time instant at which a system changes its internal structures has become a tremendously significant mission, beneficial to fully understanding the underlying mechanisms of evolving systems, and adequately modeling and predicting the dynamics of the systems as well. In real-world applications, due to a lack of prior knowledge on the explicit equations of evolving systems, an open challenge is how to develop a practical and model-free method to achieve the mission based merely on the time-series data recorded from real-world systems. Here, we develop such a model-free approach, named temporal change-point detection (TCD), and integrate both dynamical and statistical methods to address this important challenge in a novel way. The proposed TCD approach, basing on exploitation of spatial information of the observed time series of high dimensions, is able not only to detect the separate change points of the concerned systems without knowing, a priori, any information of the equations of the systems, but also to harvest all the change points emergent in a relatively high-frequency manner, which cannot be directly achieved by using the existing methods and techniques. Practical effectiveness is comprehensively demonstrated using the data from the representative complex dynamics and real-world systems from biology to geology and even to social science.

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.

Autonomous clocks that regulate organelle biogenesis, cytoskeletal organization, and intracellular dynamics

How do cells perceive time? Do cells use temporal information to regulate the production/degradation of their enzymes, membranes, and organelles? Does controlling biological time influence cytoskeletal organization and cellular architecture in ways that confer evolutionary and physiological advantages? Potential answers to these fundamental questions of cell biology have historically revolved around the discussion of 'master' temporal programs, such as the principal cyclin-dependent kinase/cyclin cell division oscillator and the circadian clock. In this review, we provide an overview of the recent evidence supporting an emerging concept of 'autonomous clocks,' which under normal conditions can be entrained by the cell cycle and/or the circadian clock to run at their pace, but can also run independently to serve their functions if/when these major temporal programs are halted/abrupted. We begin the discussion by introducing recent developments in the study of such clocks and their roles at different scales and complexities. We then use current advances to elucidate the logic and molecular architecture of temporal networks that comprise autonomous clocks, providing important clues as to how these clocks may have evolved to run independently and, sometimes at the cost of redundancy, have strongly coupled to run under the full command of the cell cycle and/or the circadian clock. Next, we review a list of important recent findings that have shed new light onto potential hallmarks of autonomous clocks, suggestive of prospective theoretical and experimental approaches to further accelerate their discovery. Finally, we discuss their roles in health and disease, as well as possible therapeutic opportunities that targeting the autonomous clocks may offer.

A modular approach for modeling the cell cycle based on functional response curves

Modeling biochemical reactions by means of differential equations often results in systems with a large number of variables and parameters. As this might complicate the interpretation and generalization of the obtained results, it is often desirable to reduce the complexity of the model. One way to accomplish this is by replacing the detailed reaction mechanisms of certain modules in the model by a mathematical expression that qualitatively describes the dynamical behavior of these modules. Such an approach has been widely adopted for ultrasensitive responses, for which underlying reaction mechanisms are often replaced by a single Hill function. Also time delays are usually accounted for by using an explicit delay in delay differential equations. In contrast, however, S-shaped response curves, which by definition have multiple output values for certain input values and are often encountered in bistable systems, are not easily modeled in such an explicit way. Here, we extend the classical Hill function into a mathematical expression that can be used to describe both ultrasensitive and S-shaped responses. We show how three ubiquitous modules (ultrasensitive responses, S-shaped responses and time delays) can be combined in different configurations and explore the dynamics of these systems. As an example, we apply our strategy to set up a model of the cell cycle consisting of multiple bistable switches, which can incorporate events such as DNA damage and coupling to the circadian clock in a phenomenological way.

Bistability plays an important role in many biochemical processes and typically emerges from complex interaction patterns such as positive and double negative feedback loops. Here, we propose to theoretically study the effect of bistability in a larger interaction network. We explicitly incorporate a functional expression describing an S-shaped input-output curve in the model equations, without the need for considering the underlying biochemical events. This expression can be converted into a functional module for an ultrasensitive response, and a time delay is easily included as well. Exploiting the fact that several of these modules can easily be combined in larger networks, we construct a cell cycle model consisting of multiple bistable switches and show how this approach can account for a number of known properties of the cell cycle.

The circadian oscillator analysed at the single-transcript level

The circadian clock is an endogenous and self-sustained oscillator that anticipates daily environmental cycles. While rhythmic gene expression of circadian genes is well-described in populations of cells, the single-cell mRNA dynamics of multiple core clock genes remain largely unknown. Here we use single-molecule fluorescence in situ hybridisation (smFISH) at multiple time points to measure pairs of core clock transcripts, Rev-erbα (Nr1d1), Cry1 and Bmal1, in mouse fibroblasts. The mean mRNA level oscillates over 24 h for all three genes, but mRNA numbers show considerable spread between cells. We develop a probabilistic model for multivariate mRNA counts using mixtures of negative binomials, which accounts for transcriptional bursting, circadian time and cell-to-cell heterogeneity, notably in cell size. Decomposing the mRNA variability into distinct noise sources shows that clock time contributes a small fraction of the total variability in mRNA number between cells. Thus, our results highlight the intrinsic biological challenges in estimating circadian phase from single-cell mRNA counts and suggest that circadian phase in single cells is encoded post-transcriptionally.

A design principle for posttranslational chaotic oscillators

Chaos behavior has been observed in various cellular and molecular processes. Here, we modeled reversible phosphorylation dynamics to elucidate a design principle for autonomous chaos generation that may arise from generic enzymatic reactions. A comprehensive parameter search demonstrated that the reaction system composed of a set of kinases and phosphatases and two substrates with two modification sites exhibits chaos behavior. All reactions are described according to the Michaelis-Menten reaction scheme without exotic functions being applied to enzymes and substrates. Clustering analysis of parameter sets that can generate chaos behavior revealed the existence of motif structures. These chaos motifs allow the two-substrate species to interact via enzyme availability and constrain the two substrates' dynamic changes in phosphorylation status so that they occur at different timescales. This chaos motif structure is found in several enzymatic reactions, suggesting that chaos behavior may underlie cellular autonomy in a variety of biochemical systems.

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Two substrates with reversible two-site phosphorylation can exhibit chaos behavior

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The chaos does not require autocatalysis or allosteric regulation of enzymes

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The chaos is a result of the coupling of two substrates via enzyme availability

Experimental observations of synchronization between two bidirectionally coupled physically dissimilar oscillators

We experimentally study the complex dynamics of two mutually coupled physically dissimilar oscillators with two different kinds of coupling mechanisms. Specifically, an optoelectronic oscillator is coupled to a Colpitts oscillator via optical power and the Colpitts oscillator is coupled back to the optoelectronic oscillator via electric voltage. We investigate and characterize phase synchronization and generalized chaos synchronization in this coupled system. Phase synchronization is observed when both oscillators are preset to oscillate periodically prior to coupling while generalized chaos synchronization is observed when both oscillators are preset to oscillate chaotically prior to coupling. In the periodical oscillation regime, we observe a linear relationship between the strengths of the two unidirectional coupling factors at which the system transitions to a synchronized state. In the chaotic regime, we observe a transition from hyperchaos to chaos associated with the onset of generalized synchronization.

Circadian clock effects on cellular proliferation: Insights from theory and experiments

Oscillations of the cellular circadian clock have emerged as an important regulator of many physiological processes, both in health and in disease. One such process, cellular proliferation, is being increasingly recognized to be affected by the circadian clock. Here, we review how a combination of experimental and theoretical work has furthered our understanding of the way circadian clocks couple to the cell cycle and play a role in tissue homeostasis and cancer. Finally, we discuss recently introduced methods for modeling coupling of clocks based on techniques from survival analysis and machine learning and highlight their potential importance for future studies.