Robustness of circadian rhythms with respect to molecular noise

Cerebrovascular autoregulation and preterm brain injury: a systematic review and meta-analysis

BACKGROUND

Preterm neonates are vulnerable to impaired cerebrovascular autoregulation (CAR), with a risk of developing cerebral injury (intraventricular hemorrhage or periventricular leukomalacia). In this meta-analysis, we evaluate the association between the degree of CAR and cerebral injury in preterm neonates, using various CAR assessment techniques mainly based on near-infrared spectroscopy.

METHODS

We systematically searched PubMed and EMBASE between 01-05-2023 and 31-01-2024. We included original articles published between 2000-2023, reporting CAR and cerebral injury in preterm neonates. The study was registered in PROSPERO (ID: 427323). We assessed the standardized mean difference in degree of CAR in groups with and without cerebral injury, with a subgroup analysis of CAR measurement techniques.

RESULTS

Seventeen articles were included, encompassing 742 patients. The overall standardized mean difference of 0.59 (95%CI 0.30, 0.88) indicates more impaired CAR in neonates with cerebral injury. Time-domain analysis produced the least heterogeneous and most significant difference of 0.61 (95%CI 0.24, 0.98). Limitations include differences in measurement techniques and a lack of randomized controlled trials.

CONCLUSION

This meta-analysis showed an association between CAR and cerebral injury, highlighting the clinical relevance of CAR assessment in preterm neonates. Future randomized studies using standardized measurement techniques should assess the feasibility of CAR-guided hemodynamic management.

IMPACT

Cerebral injury and adverse neurological outcomes are prevalent in preterm neonates. Historically, immature cerebrovascular autoregulation was thought to be one of the main causes of cerebral injury in this patient population. This is the first study to systematically review and synthetize robust evidence on the relation between cerebrovascular autoregulation and preterm brain injury. This meta-analysis further adds to the existing literature by comparing different cerebrovascular autoregulation measurement techniques. The findings can help to standardize cerebrovascular autoregulation assessment and implement it in clinical practice using large, randomized trials.

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.

Quantifying the global stability and transition dynamics of a coupled human-environment system via a landscape-flux approach

Human and environmental systems should not be viewed in isolation from each other but as a complex integrated system since humans not only influence ecosystem services and functions but also respond to changes in the ecosystem. Additionally, stochastic perturbations play a crucial role in natural systems, and stochastic factors associated with social and ecological systems can significantly affect the dynamics of coupled models, such as noise-induced tipping. In this paper, we propose a coupled human-environment model with noisy disturbances that includes the dynamics of forest conservation opinions within a population and the natural expansion and harvesting of forest ecosystems. We investigate how stochasticity triggers critical transitions between high and low forest cover states (or a stable oscillatory state) using social and ecological fitting parameters from old-growth forests in Oregon. Based on landscape-flow theory from non-equilibrium statistical mechanics, we quantify the global stability and robustness of equilibria and limit cycles using barrier height and average flux. We find that the stability of the high forest cover state weakens, and the low forest cover state becomes increasingly stable as noise intensity increases. Conversely, an increase in the intensity of injunctive social norms favors the global stability of the high forest cover state. Moreover, only a sufficiently small forest protection cost will allow forest cover to be maintained at a high level. Finally, a sensitivity analysis of the parameters of the coupled system is conducted, revealing the key factors affecting the global stability and critical transitions of high and low forest cover states.

Global niche partitioning of purine and pyrimidine cross-feeding among ocean microbes

Cross-feeding involves microbes consuming exudates of other surrounding microbes, mediating elemental cycling. Characterizing the diversity of cross-feeding pathways in ocean microbes illuminates evolutionary forces driving self-organization of ocean ecosystems. Here, we uncover a purine and pyrimidine cross-feeding network in globally abundant groups. The cyanobacterium Prochlorococcus exudes both compound classes, which metabolic reconstructions suggest follows synchronous daily genome replication. Co-occurring heterotrophs differentiate into purine- and pyrimidine-using generalists or specialists that use compounds for different purposes. The most abundant heterotroph, SAR11, is a specialist that uses purines as sources of energy, carbon, and/or nitrogen, with subgroups differentiating along ocean-scale gradients in the supply of energy and nitrogen, in turn producing putative cryptic nitrogen cycles that link many microbes. Last, in an SAR11 subgroup that dominates where Prochlorococcus is abundant, adenine additions to cultures inhibit DNA synthesis, poising cells for replication. We argue that this subgroup uses inferred daily adenine pulses from Prochlorococcus to synchronize to the daily photosynthate supply from surrounding phytoplankton.

Exploring buoyancy-driven effects in chemo-hydrodynamic oscillations sustained by bimolecular reactions

Exotic dynamics, previously associated only with reactions involving complex kinetics, have been observed even with simple bimolecular reactions A + B → C, when coupled with hydrodynamical flows. Numerical studies in two-dimensional reactors have shown that oscillatory dynamics can emerge from an antagonistic coupling between chemically-driven buoyancy and Marangoni convective flows, induced by changes in density and surface tension, respectively, as the reaction occurs. Here, we investigate reactions increasing both surface tension and density, leading to a cooperative coupling between the flows and show how, in this configuration, buoyancy-driven contribution dampens spatio-temporal oscillations of concentration. We finally identify the key parameters controlling the onset and persistence of the oscillatory instability, namely the density and surface tension gradients, and the systems height.

Analytic solutions for the circadian oscillator characterize cycle dynamics and its robustness

Circadian clocks form a fundamental mechanism that promotes the correct behavior of many cellular and molecular processes by synchronizing them on a 24 h period. However, the circadian cycles remain difficult to describe mathematically. To overcome this problem, we first propose a segmentation of the circadian cycle into eight stages based on the levels of expression of the core clock components CLOCK:BMAL1, REV-ERB and PER:CRY. This cycle segmentation is next characterized through a piecewise affine model, whose analytical study allows us to propose an Algorithm to generate biologically-consistent circadian oscillators. Our study provides a characterization of the cycle dynamics in terms of four fundamental threshold parameters and one scaling parameter, shows robustness of the circadian system and its period, and identifies critical points for correct cycle progression.

Plausible, robust biological oscillations through allelic buffering

Biological oscillators can specify time- and dose-dependent functions via dedicated control of their oscillatory dynamics. However, how biological oscillators, which recurrently activate noisy biochemical processes, achieve robust oscillations remains unclear. Here, we characterize the long-term oscillations of p53 and its negative feedback regulator Mdm2 in single cells after DNA damage. Whereas p53 oscillates regularly, Mdm2 from a single MDM2 allele exhibits random unresponsiveness to ∼9% of p53 pulses. Using allelic-specific imaging of MDM2 activity, we show that MDM2 alleles buffer each other to maintain p53 pulse amplitude. Removal of MDM2 allelic buffering cripples the robustness of p53 amplitude, thereby elevating p21 levels and cell-cycle arrest. In silico simulations support that allelic buffering enhances the robustness of biological oscillators and broadens their plausible biochemical space. Our findings show how allelic buffering ensures robust p53 oscillations, highlighting the potential importance of allelic buffering for the emergence of robust biological oscillators during evolution. A record of this paper's transparent peer review process is included in the supplemental information.

Entrainment and multi-stability of the p53 oscillator in human cells

The tumor suppressor p53 responds to cellular stress and activates transcription programs critical for regulating cell fate. DNA damage triggers oscillations in p53 levels with a robust period. Guided by the theory of synchronization and entrainment, we developed a mathematical model and experimental system to test the ability of the p53 oscillator to entrain to external drug pulses of various periods and strengths. We found that the p53 oscillator can be locked and entrained to a wide range of entrainment modes. External periods far from p53's natural oscillations increased the heterogeneity between individual cells whereas stronger inputs reduced it. Single-cell measurements allowed deriving the phase response curves (PRCs) and multiple Arnold tongues of p53. In addition, multi-stability and non-linear behaviors were mathematically predicted and experimentally detected, including mode hopping, period doubling, and chaos. Our work revealed critical dynamical properties of the p53 oscillator and provided insights into understanding and controlling it. A record of this paper's transparent peer review process is included in the supplemental information.

Holimap: an accurate and efficient method for solving stochastic gene network dynamics

Gene-gene interactions are crucial to the control of sub-cellular processes but our understanding of their stochastic dynamics is hindered by the lack of simulation methods that can accurately and efficiently predict how the distributions of gene product numbers vary across parameter space. To overcome these difficulties, here we present Holimap (high-order linear-mapping approximation), an approach that approximates the protein or mRNA number distributions of a complex gene regulatory network by the distributions of a much simpler reaction system. We demonstrate Holimap's computational advantages over conventional methods by applying it to predict the stochastic time-dependent dynamics of various gene networks, including transcriptional networks ranging from simple autoregulatory loops to complex randomly connected networks, post-transcriptional networks, and post-translational networks. Holimap is ideally suited to study how the intricate network of gene-gene interactions results in precise coordination and control of gene expression.

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.

Enhanced precision of circadian rhythm by output system

Circadian rhythms are biological rhythms with a period of approximately 24 h that persist even under constant conditions without daily environmental cues. The molecular circadian clock machinery generates physiological rhythms, which can be transmitted into the downstream output system. Owing to the stochastic nature of the biochemical reactions and the extracellular environment, the oscillation period of circadian rhythms exhibited by individual organisms or cells is not constant on a daily basis with variations as high as 10%, as reflected by the coefficient of variation. Although the fluctuations in the circadian rhythm are measured through a reporter system such as bioluminescence or fluorescence, which is an example of output systems, experimentally confirming whether the fluctuations found in the reporter system are the same as those in the circadian clock is challenging. This study investigated a coupled system of a circadian clock and its output system numerically and analytically, and then compared the fluctuations in the oscillation period of the two systems. We found that the amount of fluctuations in the output system is smaller than that in the circadian clock, assuming the degradation rate of the molecules responsible for the output system is a typical value for protein degradation. The results indicate that the output system can improve the accuracy of the circadian rhythm without the need for any denoising processes.

Systematic analysis of negative and positive feedback loops for robustness and temperature compensation in circadian rhythms

Temperature compensation and robustness to biological noise are two key characteristics of the circadian clock. These features allow the circadian pacemaker to maintain a steady oscillation in a wide range of environmental conditions. The presence of a time-delayed negative feedback loop in the regulatory network generates autonomous circadian oscillations in eukaryotic systems. In comparison, the circadian clock of cyanobacteria is controlled by a strong positive feedback loop. Positive feedback loops with substrate depletion can also generate oscillations, inspiring other circadian clock models. What makes a circadian oscillatory network robust to extrinsic noise is unclear. We investigated four basic circadian oscillators with negative, positive, and combinations of positive and negative feedback loops to explore network features necessary for circadian clock resilience. We discovered that the negative feedback loop system performs the best in compensating temperature changes. We also show that a positive feedback loop can reduce extrinsic noise in periods of circadian oscillators, while intrinsic noise is reduced by negative feedback loops.

Noise facilitates entrainment of a population of uncoupled limit cycle oscillators

Many biological oscillators share two properties: they are subject to stochastic fluctuations (noise) and they must reliably adjust their period to changing environmental conditions (entrainment). While noise seems to distort the ability of single oscillators to entrain, in populations of uncoupled oscillators noise allows population-level entrainment for a wider range of input amplitudes and periods. Here, we investigate how this effect depends on the noise intensity and the number of oscillators in the population. We have found that, if a population consists of a sufficient number of oscillators, increasing noise intensity leads to faster entrainment after a phase change of the input signal (jet lag) and increases sensitivity to low-amplitude input signals.

Design centering enables robustness screening of pattern formation models

MOTIVATION

Access to unprecedented amounts of quantitative biological data allows us to build and test biochemically accurate reaction-diffusion models of intracellular processes. However, any increase in model complexity increases the number of unknown parameters and, thus, the computational cost of model analysis. To efficiently characterize the behavior and robustness of models with many unknown parameters remains, therefore, a key challenge in systems biology.

RESULTS

We propose a novel computational framework for efficient high-dimensional parameter space characterization of reaction-diffusion models in systems biology. The method leverages the Lp-Adaptation algorithm, an adaptive-proposal statistical method for approximate design centering and robustness estimation. Our approach is based on an oracle function, which predicts for any given point in parameter space whether the model fulfills given specifications. We propose specific oracles to efficiently predict four characteristics of Turing-type reaction-diffusion models: bistability, instability, capability of spontaneous pattern formation and capability of pattern maintenance. We benchmark the method and demonstrate that it enables global exploration of a model's ability to undergo pattern-forming instabilities and to quantify robustness for model selection in polynomial time with dimensionality. We present an application of the framework to pattern formation on the endosomal membrane by the small GTPase Rab5 and its effectors, and we propose molecular mechanisms underlying this system.

AVAILABILITY AND IMPLEMENTATION

Our code is implemented in MATLAB and is available as open source under https://git.mpi-cbg.de/mosaic/software/black-box-optimization/rd-parameter-space-screening.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Frequency dependent growth of bacteria in living materials

The fusion of living bacteria and man-made materials represents a new frontier in medical and biosynthetic technology. However, the principles of bacterial signal processing inside synthetic materials with three-dimensional and fluctuating environments remain elusive. Here, we study bacterial growth in a three-dimensional hydrogel. We find that bacteria expressing an antibiotic resistance module can take advantage of ambient kinetic disturbances to improve growth while encapsulated. We show that these changes in bacterial growth are specific to disturbance frequency and hydrogel density. This remarkable specificity demonstrates that periodic disturbance frequency is a new input that engineers may leverage to control bacterial growth in synthetic materials. This research provides a systematic framework for understanding and controlling bacterial information processing in three-dimensional living materials.

Temperature Controls Onset and Period of NF-κB Oscillations and can Lead to Chaotic Dynamics

The transcription factor NF-κB plays a vital role in the control of the immune system, and following stimulation with TNF-α its nuclear concentration shows oscillatory behaviour. How environmental factors, in particular temperature, can control the oscillations and thereby affect gene stimulation is still remains to be resolved question. In this work, we reveal that the period of the oscillations decreases with increasing temperature. We investigate this using a mathematical model, and by applying results from statistical physics, we introduce temperature dependency to all rates, resulting in a remarkable correspondence between model and experiments. Our model predicts how temperature affects downstream protein production and find a crossover, where high affinity genes upregulates at high temperatures. Finally, we show how or that oscillatory temperatures can entrain NF-κB oscillations and lead to chaotic dynamics presenting a simple path to chaotic conditions in cellular biology.

Adaptive Diversification in the Cellular Circadian Behavior of Arabidopsis Leaf- and Root-Derived Cells

The plant circadian system is based on self-sustained cellular oscillations and is utilized to adapt to daily and seasonal environmental changes. The cellular circadian clocks in the above- and belowground plant organs are subjected to diverse local environments. Individual cellular clocks are affected by other cells/tissues in plants, and the intrinsic circadian properties of individual cells remain to be elucidated. In this study, we monitored bioluminescence circadian rhythms of individual protoplast-derived cells from leaves and roots of a CCA1::LUC Arabidopsis transgenic plant. We analyzed the circadian properties of the leaf- and root-derived cells and demonstrated that the cells with no physical contact with other cells harbor a genuine circadian clock with ∼24-h periodicity, entrainability and temperature compensation of the period. The stability of rhythm was dependent on the cell density. High cell density resulted in an improved circadian rhythm of leaf-derived cells while this effect was observed irrespective of the phase relation between cellular rhythms. Quantitative and statistical analyses for individual cellular bioluminescence rhythms revealed a difference in amplitude and precision of light/dark entrainment between the leaf- and root-derived cells. Circadian systems in the leaves and roots are diversified to adapt to their local environments at the cellular level.

A spatial model of the plant circadian clock reveals design principles for coordinated timing

Individual plant cells possess a genetic network, the circadian clock, that times internal processes to the day-night cycle. Mathematical models of the clock are typically either "whole-plant" that ignore tissue or cell type-specific clock behavior, or "phase-only" that do not include molecular components. To address the complex spatial coordination observed in experiments, here we implemented a clock network model on a template of a seedling. In our model, the sensitivity to light varies across the plant, and cells communicate their timing via local or long-distance sharing of clock components, causing their rhythms to couple. We found that both varied light sensitivity and long-distance coupling could generate period differences between organs, while local coupling was required to generate the spatial waves of clock gene expression observed experimentally. We then examined our model under noisy light-dark cycles and found that local coupling minimized timing errors caused by the noise while allowing each plant region to maintain a different clock phase. Thus, local sensitivity to environmental inputs combined with local coupling enables flexible yet robust circadian timing.

Restoring circadian gene profiles in clock networks using synthetic feedback control

The circadian system-an organism's built-in biological clock-is responsible for orchestrating biological processes to adapt to diurnal and seasonal variations. Perturbations to the circadian system (e.g., pathogen attack, sudden environmental change) often result in pathophysiological responses (e.g., jetlag in humans, stunted growth in plants, etc.) In view of this, synthetic biologists are progressively adapting the idea of employing synthetic feedback control circuits to alleviate the effects of perturbations on circadian systems. To facilitate the design of such controllers, suitable models are required. Here, we extend our recently developed model for the plant circadian clock-termed the extended S-System model-to model circadian systems across different kingdoms of life. We then use this modeling strategy to develop a design framework, based on an antithetic integral feedback (AIF) controller, to restore a gene's circadian profile when it is subject to loss-of-function due to external perturbations. The use of the AIF controller is motivated by its recent successful experimental implementation. Our findings provide circadian biologists with a systematic and general modeling and design approach for implementing synthetic feedback control of circadian systems.

Normalized coefficient of variation (nCV): a method to evaluate circadian clock robustness in population scale data

Summary

Robust oscillation of clock genes is a core feature of the circadian system. Relative amplitude (rAMP) measures the robustness of clock gene oscillations but only works for longitudinal samples. We lack a method for estimating robust oscillations from human samples without labeled time. We show that the normalized coefficient of variation (nCV) of 10 clock genes is linearly correlated with their normalized rAMP, independent of time labels. We found that the mean nCV of clock genes are consistently decreased in tumors compared to nontumors, suggesting a new therapeutic target in cancer treatment by enhancing clock robustness. nCV can provide a simple measure of the clock robustness in population-level datasets.

Availability and implementation

The nCV package (https://github.com/gangwug/nCV) and web application (https://github.com/gangwug/nCVapp) are available on the GitHub repository.

Supplementary information

Supplementary data are available at Bioinformatics online.

A stochastic oscillator model simulates the entrainment of vertebrate cellular clocks by light

The circadian clock is a cellular mechanism that synchronizes various biological processes with respect to the time of the day. While much progress has been made characterizing the molecular mechanisms underlying this clock, it is less clear how external light cues influence the dynamics of the core clock mechanism and thereby entrain it with the light-dark cycle. Zebrafish-derived cell cultures possess clocks that are directly light-entrainable, thus providing an attractive laboratory model for circadian entrainment. Here, we have developed a stochastic oscillator model of the zebrafish circadian clock, which accounts for the core clock negative feedback loop, light input, and the proliferation of single-cell oscillator noise into population-level luminescence recordings. The model accurately predicts the entrainment dynamics observed in bioluminescent clock reporter assays upon exposure to a wide range of lighting conditions. Furthermore, we have applied the model to obtain refitted parameter sets for cell cultures exposed to a variety of pharmacological treatments and predict changes in single-cell oscillator parameters. Our work paves the way for model-based, large-scale screens for genetic or pharmacologically-induced modifications to the entrainment of circadian clock function.

Circadian characteristics of the rest-activity rhythm, executive function, and glucose fluctuations in young adults with type 1 diabetes

Circadian alignment is an important element in individual health, and one behavioral marker, rest-activity rhythm, could influence self-management in young adults with type 1 diabetes (T1D). Little is known about the rest-activity rhythms, executive function, and glycemia among young adults with type 1 diabetes (T1D). The purpose of this study was to evaluate parametric and nonparametric circadian characteristics of the rest-activity rhythm and the associations between these variables, sleep-wake behavior, executive function, and glycemia among young adults with T1D. Young adults with T1D, recruited from diabetes clinics, wore wrist actigraphs and a continuous glucose monitor (CGM) concurrently for 6-14 days. Participants completed a 3-minute Trail Making Test on paper and electronic questionnaires - 8-item PROMIS v1.0 Emotional Distress Scale, 17-item Diabetes Distress Scale, including twice-daily Pittsburgh sleep diaries. Cosinor and nonparametric analyses were used to compute the rest-activity rhythm parameters, and linear regression modeling procedures were performed to determine the associations among the study variables. The sample included 46 young adults (mean age 22.3 ± 3.2; 32.6% male; 84.8% non-Hispanic White, HbA1c mean 7.2 ± 1.1%, BMI mean 27.0 ± 4.4 kg/m²). A number of parametric associations were observed between a stronger rhythm, better objective sleep-wake characteristics, and less daytime sleepiness. Nonparametric circadian parameters were significantly associated with several outcomes: a stronger rhythm adherence (higher inter-daily stability) with better objective sleep-wake characteristics, better executive function, lower diabetes distress, less hyperglycemia risk, and more time spent in hypoglycemia/hypoglycemia risk; and a more robust rhythm (higher relative amplitude) with better objective sleep-wake characteristics and more time spent in hypoglycemia/higher hypoglycemia risk. Future work should be directed at designs that test causality, such as interventions directed at the strength and stability of rest-activity rhythms, for the potential to improve glucoregulation and other diabetes outcomes.

Stochastic simulation of a model for circadian rhythms in plants

Circadian clocks allow living organisms to anticipate and adapt to the daily variations of the environment. The interlocked feedback loops of the transcription factors network in the plant clock generate oscillations with expression peaks at specific times of the day. In this work, we explore the effect of molecular noise on the behavior of the plant circadian clock through numerical simulations. The influence of system size, photoperiod, and mutations of clock genes on the robustness of the oscillations are discussed. Our simulations show that the oscillations remain robust when the mRNA and protein levels are in the range of a few hundreds molecules. Entrainment by light-dark cycles enhances the robustness compared to constant conditions. Multiple light inputs and inter-cellular coupling also contribute to the robustness of the oscillations. The comparison between deterministic and stochastic simulations of single and double mutants shows that stochasticity does not qualitatively affect the behaviour of mutants but that they do not have the same robustness to noise. Finally, the model shows that noise can induce transitions between two limit cycles in a birhythmic clock mutant.

Early Detection of Daylengths with a Feedforward Circuit Coregulated by Circadian and Diurnal Cycles

Light-entrained circadian clocks confer rhythmic dynamics of cellular and molecular activities to animals and plants. These intrinsic clocks allow stable anticipations to light-dark (diel) cycles. Many genes in the model plant Arabidopsis thaliana are regulated by diel cycles via pathways independent of the clock, suggesting that the integration of circadian and light signals is important for the fitness of plants. Previous studies of light-clock signal integrations have focused on moderate phase adjustment of the two signals. However, dynamical features of integrations across a broad range of phases remain elusive. Phosphorylation of ribosomal protein of the small subunit 6 (eS6), a ubiquitous post-translational modification across kingdoms, is influenced by the circadian clock and the light-dark (diel) cycle in an opposite manner. To understand this striking phenomenon and its underlying information processing capabilities, we built a mathematical model for the eS6 phosphorylation (eS6-P) control circuit. We found that the dynamics of eS6-P can be explained by a feedforward circuit with inputs from both circadian and diel cycles. Furthermore, the early day response of this circuit with dual rhythmic inputs is sensitive to the changes in daylength, including both transient and gradual changes observed in realistic light intervals across a year, because of weather and seasons. By analyzing published gene expression data, we found that the dynamics produced by the eS6-P control circuit can be observed in the expression profiles of a large number of genes. Our work provides mechanistic insights into the complex dynamics of a ribosomal protein, and it proposes a previously underappreciated function of the circadian clock, which not only prepares organisms for normal diel cycles but also helps to detect both transient and seasonal changes with a predictive power.

Alternative splicing coupled mRNA decay shapes the temperature-dependent transcriptome

Mammalian body temperature oscillates with the time of the day and is altered in diverse pathological conditions. We recently identified a body temperature‐sensitive thermometer‐like kinase, which alters SR protein phosphorylation and thereby globally controls alternative splicing (AS). AS can generate unproductive variants which are recognized and degraded by diverse mRNA decay pathways—including nonsense‐mediated decay (NMD). Here we show extensive coupling of body temperature‐controlled AS to mRNA decay, leading to global control of temperature‐dependent gene expression (GE). Temperature‐controlled, decay‐inducing splicing events are evolutionarily conserved and pervasively found within RNA‐binding proteins, including most SR proteins. AS‐coupled poison exon inclusion is essential for rhythmic GE of SR proteins and has a global role in establishing temperature‐dependent rhythmic GE profiles, both in mammals under circadian body temperature cycles and in plants in response to ambient temperature changes. Together, these data identify body temperature‐driven AS‐coupled mRNA decay as an evolutionary ancient, core clock‐independent mechanism to generate rhythmic GE.

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.

Coupling protocol of interlocked feedback oscillators in circadian clocks

Circadian rhythms (approx. 24 h) show the robustness of key oscillatory features such as phase, period and amplitude against external and internal variations. The robustness of Drosophila circadian clocks can be generated by interlocked transcriptional-translational feedback loops, where two negative feedback loops are coupled through mutual activations. The mechanisms by which such coupling protocols have survived out of many possible protocols remain to be revealed. To address this question, we investigated two distinct coupling protocols: activator-coupled oscillators (ACO) and repressor-coupled oscillators (RCO). We focused on the two coupling parameters: coupling dissociation constant and coupling time-delay. Interestingly, the ACO was able to produce anti-phase or morning-evening cycles, whereas the RCO produced in-phase ones. Deterministic and stochastic analyses demonstrated that the anti-phase ACO provided greater fluctuations in amplitude not only with respect to changes in coupling parameters but also to random parameter perturbations than the in-phase RCO. Moreover, the ACO deteriorated the entrainability to the day-night master clock, whereas the RCO produced high entrainability. Considering that the real, interlocked feedback loops have evolved as the ACO, instead of the RCO, we first proposed a hypothesis that the morning-evening or anti-phase cycle is more essential for Drosophila than achieving robustness and entrainability.

Bayesian modeling reveals metabolite-dependent ultrasensitivity in the cyanobacterial circadian clock

Mathematical models can enable a predictive understanding of mechanism in cell biology by quantitatively describing complex networks of interactions, but such models are often poorly constrained by available data. Owing to its relative biochemical simplicity, the core circadian oscillator in Synechococcus elongatus has become a prototypical system for studying how collective dynamics emerge from molecular interactions. The oscillator consists of only three proteins, KaiA, KaiB, and KaiC, and near-24-h cycles of KaiC phosphorylation can be reconstituted in vitro. Here, we formulate a molecularly detailed but mechanistically naive model of the KaiA-KaiC subsystem and fit it directly to experimental data within a Bayesian parameter estimation framework. Analysis of the fits consistently reveals an ultrasensitive response for KaiC phosphorylation as a function of KaiA concentration, which we confirm experimentally. This ultrasensitivity primarily results from the differential affinity of KaiA for competing nucleotide-bound states of KaiC. We argue that the ultrasensitive stimulus-response relation likely plays an important role in metabolic compensation by suppressing premature phosphorylation at nighttime.

Transcriptional Structure of Petunia Clock in Leaves and Petals

The plant circadian clock coordinates environmental signals with internal processes including secondary metabolism, growth, flowering, and volatile emission. Plant tissues are specialized in different functions, and petals conceal the sexual organs while attracting pollinators. Here we analyzed the transcriptional structure of the petunia (Petunia x hybrida) circadian clock in leaves and petals. We recorded the expression of 13 clock genes in petunia under light:dark (LD) and constant darkness (DD). Under light:dark conditions, clock genes reached maximum expression during the light phase in leaves and the dark period in petals. Under free running conditions of constant darkness, maximum expression was delayed, especially in petals. Interestingly, the rhythmic expression pattern of PhLHY persisted in leaves and petals in LD and DD. Gene expression variability differed among leaves and petals, time of day and photoperiod. The transcriptional noise was higher especially in leaves under constant darkness. We found that PhPRR7, PhPRR5, and PhGI paralogs showed changes in gene structure including exon number and deletions of CCT domain of the PRR family. Our results revealed that petunia petals presented a specialized clock.

Translational thresholds in a core circadian clock model

Organisms have evolved in a daily cyclic environment, developing circadian cell-autonomous clocks that temporally organize a wide range of biological processes. Translation is a highly regulated process mainly associated with the activity of microRNAs (miRNAs) at the translation initiation step that impacts on the molecular circadian clock dynamics. Recently, a molecular titration mechanism was proposed to explain the interactions between some miRNAs and their target mRNAs; new evidence also indicates that regulation by miRNA is a nonlinear process such that there is a threshold level of target mRNA below which protein production is drastically repressed. These observations led us to use a theoretical model of the circadian molecular clock to study the effect of miRNA-mediated translational thresholds on the molecular clock dynamics. We model the translational threshold by introducing a phenomenological Hill equation for the kinetics of PER translation and show how the parameters associated with translation kinetics affect the period, amplitude, and time delays between clock mRNA and clock protein expression. We show that our results are useful for analyzing experiments related to the translational regulation of negative elements of transcriptional-translational feedback loops. We also provide new elements for thinking about the translational threshold as a mechanism that favors the emergence of circadian rhythmicity, the tuning of the period-delay relationship and the cell capacity to control the protein oscillation amplitude with almost negligible changes in the mRNA amplitudes.

Hilbert transform-based time-series analysis of the circadian gene regulatory network

In this work, the authors propose the Hilbert transform (HT)‐based numerical method to analyse the time series of the circadian rhythms. They demonstrate the application of HT by taking both deterministic and stochastic time series that they get from the simulation of the fruit fly model Drosophila melanogaster and show how to extract the period, construct phase response curves, determine period sensitivity of the parameters to perturbations and build Arnold tongues to identify the regions of entrainment. They also derive a phase model that they numerically simulate to capture whether the circadian time series entrains to the forcing period completely (phase locking) or only partially (phase slips) or neither. They validate the phase model, and numerics with the experimental time series forced under different temperature cycles. Application of HT to the circadian time series appears to be a promising tool to extract the characteristic information about circadian rhythms.

Circadian Clock Properties and Their Relationships as a Function of Free-Running Period in Drosophila melanogaster

The stability of circadian clock mechanisms under cyclic environments contributes to increased Darwinian fitness by accurately timing daily behavior and physiology. Earlier studies on biological clocks speculated that the timing of behavior and its accuracy are determined by the intrinsic period (τ) of the circadian clock under constant conditions, its stability, the period of the external cycle (T), and resetting of the clock by environmental time cues. However, most of these previous studies suffered from certain limitations, the major ones being a narrow range of examined τ values and a non-uniformity in the genetic background across the individuals tested. We present data that rigorously test the following hypotheses by employing Drosophila melanogaster fruit flies with τ ranging from 17 to 30 h in a uniform genetic background. We tested whether 1) precision (day-to-day stability of τ) is greater for clocks with τ close to 24 h; 2) accuracy (i.e., day-to-day stability of the phase relationship (ψ), where ψ is the duration between a phase of the rhythm and a phase of the external cycle) is greater for clocks with τ close to 24 h; 3) Ψ is delayed with an increase in τ; and 4) Ψ becomes more advanced with an increase in length of zeitgeber cycle (T). We show that precision is not always maximum for ~24-h clocks, but that accuracy is greatest when τ approximates T. Further, flies exhibit a delayed phase relationship with increasing τ and an advanced phase relationship under long T-cycles as compared with shorter T-cycles. We also describe relationships between activity and rest durations and how our observations fit predictions from models of circadian entrainment. Overall, we confirm that accuracy and phase of entrained rhythm are governed by both intrinsic clock period and the length of the external cycle; however, we find that the relationship between intrinsic period and precision does not fit previous predictions.

Mathematical modeling of circadian rhythms

Circadian rhythms are endogenous ~24-hr oscillations usually entrained to daily environmental cycles of light/dark. Many biological processes and physiological functions including mammalian body temperature, the cell cycle, sleep/wake cycles, neurobehavioral performance, and a wide range of diseases including metabolic, cardiovascular, and psychiatric disorders are impacted by these rhythms. Circadian clocks are present within individual cells and at tissue and organismal levels as emergent properties from the interaction of cellular oscillators. Mathematical models of circadian rhythms have been proposed to provide a better understanding of and to predict aspects of this complex physiological system. These models can be used to: (a) manipulate the system in silico with specificity that cannot be easily achieved using in vivo and in vitro experimental methods and at lower cost, (b) resolve apparently contradictory empirical results, (c) generate hypotheses, (d) design new experiments, and (e) to design interventions for altering circadian rhythms. Mathematical models differ in structure, the underlying assumptions, the number of parameters and variables, and constraints on variables. Models representing circadian rhythms at different physiologic scales and in different species are reviewed to promote understanding of these models and facilitate their use. This article is categorized under: Physiology > Mammalian Physiology in Health and Disease Models of Systems Properties and Processes > Organ, Tissue, and Physiological Models.

Comparison of Deterministic and Stochastic Regime in a Model for Cdc42 Oscillations in Fission Yeast

Oscillations occur in a wide variety of essential cellular processes, such as cell cycle progression, circadian clocks and calcium signaling in response to stimuli. It remains unclear how intrinsic stochasticity can influence these oscillatory systems. Here, we focus on oscillations of Cdc42 GTPase in fission yeast. We extend our previous deterministic model by Xu and Jilkine to construct a stochastic model, focusing on the fast diffusion case. We use SSA (Gillespie's algorithm) to numerically explore the low copy number regime in this model, and use analytical techniques to study the long-time behavior of the stochastic model and compare it to the equilibria of its deterministic counterpart. Numerical solutions suggest noisy limit cycles exist in the parameter regime in which the deterministic system converges to a stable limit cycle, and quasi-cycles exist in the parameter regime where the deterministic model has a damped oscillation. Near an infinite period bifurcation point, the deterministic model has a sustained oscillation, while stochastic trajectories start with an oscillatory mode and tend to approach deterministic steady states. In the low copy number regime, metastable transitions from oscillatory to steady behavior occur in the stochastic model. Our work contributes to the understanding of how stochastic chemical kinetics can affect a finite-dimensional dynamical system, and destabilize a deterministic steady state leading to oscillations.

On chaotic dynamics in transcription factors and the associated effects in differential gene regulation

The control of proteins by a transcription factor with periodically varying concentration exhibits intriguing dynamical behaviour. Even though it is accepted that transcription factors vary their dynamics in response to different situations, insight into how this affects downstream genes is lacking. Here, we investigate how oscillations and chaotic dynamics in the transcription factor NF-κB can affect downstream protein production. We describe how it is possible to control the effective dynamics of the transcription factor by stimulating it with an oscillating ligand. We find that chaotic dynamics modulates gene expression and up-regulates certain families of low-affinity genes, even in the presence of extrinsic and intrinsic noise. Furthermore, this leads to an increase in the production of protein complexes and the efficiency of their assembly. Finally, we show how chaotic dynamics creates a heterogeneous population of cell states, and describe how this can be beneficial in multi-toxic environments.

It is becoming clear that the dynamics of transcription factors may be important for gene regulation. Here, the authors study the implications of oscillatory and chaotic dynamics of NF-κB and demonstrate that it allows a degree of control of gene expression and can generate phenotypic heterogeneity.

Protein sequestration versus Hill-type repression in circadian clock models

Circadian (∼24 h) clocks are self-sustained endogenous oscillators with which organisms keep track of daily and seasonal time. Circadian clocks frequently rely on interlocked transcriptional-translational feedback loops to generate rhythms that are robust against intrinsic and extrinsic perturbations. To investigate the dynamics and mechanisms of the intracellular feedback loops in circadian clocks, a number of mathematical models have been developed. The majority of the models use Hill functions to describe transcriptional repression in a way that is similar to the Goodwin model. Recently, a new class of models with protein sequestration-based repression has been introduced. Here, the author discusses how this new class of models differs dramatically from those based on Hill-type repression in several fundamental aspects: conditions for rhythm generation, robust network designs and the periods of coupled oscillators. Consistently, these fundamental properties of circadian clocks also differ among Neurospora, Drosophila, and mammals depending on their key transcriptional repression mechanisms (Hill-type repression or protein sequestration). Based on both theoretical and experimental studies, this review highlights the importance of careful modelling of transcriptional repression mechanisms in molecular circadian clocks.

Entrainment of a cellular circadian oscillator by light in the presence of molecular noise

In this paper, we consider a stochastic molecular circadian oscillator described by a sequence of biological reactions and its deterministic kinetics governed by a system of ordinary differential equations in the limit of large numbers of molecules. The oscillations in the model are generated by negative feedback regulation of a gene. The focus of this paper is the entrainment of the oscillator by a periodic light signal that affects the maximal transcription rate of the gene. We introduce two scalings of the model parameters that provide independent control over the natural frequency of the oscillator and the relative noise level. We study entrainment in two ways: by visualizing the stochastic limit cycle in various projections of the discrete phase space of the system and by evaluating the maximum of the normalized cross correlation of the light signal with the number of protein molecules in the cell. The visualization method ignores the phase of the oscillator, and we find in this way that entrainment has a subtle organizing effect on the limit cycle as a whole. The cross correlation results reveal an interval of natural frequencies of the oscillator surrounding the frequency of the light signal within which maximal entrainment occurs with rather sharp drops in entrainment at the edges of this interval. The width of the interval of maximal entrainment increases with the amplitude of the light signal. These statements are applicable both to the stochastic oscillator and to its deterministic limit, but the results are most clear-cut in the deterministic case and degrade from there as the relative noise level increases.

Modulation of dynamic modes by interplay between positive and negative feedback loops in gene regulatory networks

A positive and a negative feedback loop can induce bistability and oscillation, respectively, in biological networks. Nevertheless, they are frequently interlinked to perform more elaborate functions in many gene regulatory networks. Coupled positive and negative feedback loops may exhibit either oscillation or bistability depending on the intensity of the stimulus in some particular networks. It is less understood how the transition between the two dynamic modes is modulated by the positive and negative feedback loops. We developed an abstract model of such systems, largely based on the core p53 pathway, to explore the mechanism for the transformation of dynamic behaviors. Our results show that enhancing the positive feedback may promote or suppress oscillations depending on the strength of both feedback loops. We found that the system oscillates with low amplitudes in response to a moderate stimulus and switches to the on state upon a strong stimulus. When the positive feedback is activated much later than the negative one in response to a strong stimulus, the system exhibits long-term oscillations before switching to the on state. We explain this intriguing phenomenon using quasistatic approximation. Moreover, early switching to the on state may occur when the system starts from a steady state in the absence of stimuli. The interplay between the positive and negative feedback plays a key role in the transitions between oscillation and bistability. Of note, our conclusions should be applicable only to some specific gene regulatory networks, especially the p53 network, in which both oscillation and bistability exist in response to a certain type of stimulus. Our work also underscores the significance of transient dynamics in determining cellular outcome.

Decanalizing thinking on genetic canalization

The concept of genetic canalization has had an abiding influence on views of complex-trait evolution. A genetically canalized system has evolved to become less sensitive to the effects of mutation. When a gene product that supports canalization is compromised, the phenotypic impacts of a mutation should be more pronounced. This expected increase in mutational effects not only has important consequences for evolution, but has also motivated strategies to treat disease. However, recent studies demonstrate that, when putative agents of genetic canalization are impaired, systems do not behave as expected. Here, we review the evidence that is used to infer whether particular gene products are agents of genetic canalization. Then we explain how such inferences often succumb to a converse error. We go on to show that several candidate agents of genetic canalization increase the phenotypic impacts of some mutations while decreasing the phenotypic impacts of others. These observations suggest that whether a gene product acts as a 'buffer' (lessening mutational effects) or a 'potentiator' (increasing mutational effects) is not a fixed property of the gene product but instead differs for the different mutations with which it interacts. To investigate features of genetic interactions that might predispose them toward buffering versus potentiation, we explore simulated gene-regulatory networks. Similarly to putative agents of genetic canalization, the gene products in simulated networks also modify the phenotypic effects of mutations in other genes without a strong overall tendency towards lessening or increasing these effects. In sum, these observations call into question whether complex traits have evolved to become less sensitive (i.e., are canalized) to genetic change, and the degree to which trends exist that predict how one genetic change might alter another's impact. We conclude by discussing approaches to address these and other open questions that are brought into focus by re-thinking genetic canalization.

Biological Oscillators in Nanonetworks-Opportunities and Challenges

One of the major issues in molecular communication-based nanonetworks is the provision and maintenance of a common time knowledge. To stay true to the definition of molecular communication, biological oscillators are the potential solutions to achieve that goal as they generate oscillations through periodic fluctuations in the concentrations of molecules. Through the lens of a communication systems engineer, the scope of this survey is to explicitly classify, for the first time, existing biological oscillators based on whether they are found in nature or not, to discuss, in a tutorial fashion, the main principles that govern the oscillations in each oscillator, and to analyze oscillator parameters that are most relevant to communication engineer researchers. In addition, the survey highlights and addresses the key open research issues pertaining to several physical aspects of the oscillators and the adoption and implementation of the oscillators to nanonetworks. Moreover, key research directions are discussed.

Modeling-Based Investigation of the Effect of Noise in Cellular Systems

Noise is pervasive in cellular biology and inevitably affects the dynamics of cellular processes. Biological systems have developed regulatory mechanisms to ensure robustness with respect to noise or to take advantage of stochasticity. We review here, through a couple of selected examples, some insights on possible robustness factors and constructive roles of noise provided by computational modeling. In particular, we focus on (1) factors that likely contribute to the robustness of oscillatory processes such as the circadian clocks and the cell cycle, (2) how reliable coding/decoding of calcium-mediated signaling could be achieved in presence of noise and, in some cases, enhanced through stochastic resonance, and (3) how embryonic cell differentiation processes can exploit stochasticity to create heterogeneity in a population of identical cells.

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.

Monitoring circadian rhythms of individual cells in plants

The circadian clock is an endogenous timing system based on the self-sustained oscillation in individual cells. These cellular circadian clocks compose a multicellular circadian system working at respective levels of tissue, organ, plant body. However, how numerous cellular clocks are coordinated within a plant has been unclear. There was little information about behavior of circadian clocks at a single-cell level due to the difficulties in monitoring circadian rhythms of individual cells in an intact plant. We developed a single-cell bioluminescence imaging system using duckweed as the plant material and succeeded in observing behavior of cellular clocks in intact plants for over a week. This imaging technique quantitatively revealed heterogeneous and independent manners of cellular clock behaviors. Furthermore, these quantitative analyses uncovered the local synchronization of cellular circadian rhythms that implied phase-attractive interactions between cellular clocks. The cell-to-cell interaction looked to be too weak to coordinate cellular clocks against their heterogeneity under constant conditions. On the other hand, under light-dark conditions, the heterogeneity of cellular clocks seemed to be corrected by cell-to-cell interactions so that cellular clocks showed a clear spatial pattern of phases at a whole plant level. Thus, it was suggested that the interactions between cellular clocks was an adaptive trait working under day-night cycles to coordinate cellular clocks in a plant body. These findings provide a novel perspective for understanding spatio-temporal architectures in the plant circadian system.

Asymmetry in Signal Oscillations Contributes to Efficiency of Periodic Systems

Oscillations are an important feature of cellular signaling that result from complex combinations of positive- and negative-feedback loops. The encoding and decoding mechanisms of oscillations based on amplitude and frequency have been extensively discussed in the literature in the context of intercellular and intracellular signaling. However, the fundamental questions of whether and how oscillatory signals offer any competitive advantages-and, if so, what-have not been fully answered. We investigated established oscillatory mechanisms and designed a study to analyze the oscillatory characteristics of signaling molecules and system output in an effort to answer these questions. Two classic oscillators, Goodwin and PER, were selected as the model systems, and corresponding no-feedback models were created for each oscillator to discover the advantage of oscillating signals. Through simulating the original oscillators and the matching no-feedback models, we show that oscillating systems have the capability to achieve better resource-to-output efficiency, and we identify oscillatory characteristics that lead to improved efficiency.

Role of transcriptional bursts in cellular oscillations

Genetic oscillators are present in the cells of many organisms and control several biological processes. The common feature of such oscillators is the presence of a protein which represses the transcription of its own gene. Recently, it has been shown that for many genes transcription is not a continuous process, but that it proceeds in bursts. We study here the relationship between bursty transcription and the robustness of protein oscillations. We concentrate on the temporal profile of mRNA production by studying regimes where this profile changes but the amount of mRNA produced is kept fixed. For systems with different degrees of cooperativity we show that in general bursts are associated with more robust oscillations, but when they are too short and intense they can have the opposite effect. In other words, we show that, in terms of the regularity of the oscillations generated, there is an optimal value for the intensity of the bursts.

Feedback, Mass Conservation and Reaction Kinetics Impact the Robustness of Cellular Oscillations

Oscillations occur in a wide variety of cellular processes, for example in calcium and p53 signaling responses, in metabolic pathways or within gene-regulatory networks, e.g. the circadian system. Since it is of central importance to understand the influence of perturbations on the dynamics of these systems a number of experimental and theoretical studies have examined their robustness. The period of circadian oscillations has been found to be very robust and to provide reliable timing. For intracellular calcium oscillations the period has been shown to be very sensitive and to allow for frequency-encoded signaling. We here apply a comprehensive computational approach to study the robustness of period and amplitude of oscillatory systems. We employ different prototype oscillator models and a large number of parameter sets obtained by random sampling. This framework is used to examine the effect of three design principles on the sensitivities towards perturbations of the kinetic parameters. We find that a prototype oscillator with negative feedback has lower period sensitivities than a prototype oscillator relying on positive feedback, but on average higher amplitude sensitivities. For both oscillator types, the use of Michaelis-Menten instead of mass action kinetics in all degradation and conversion reactions leads to an increase in period as well as amplitude sensitivities. We observe moderate changes in sensitivities if replacing mass conversion reactions by purely regulatory reactions. These insights are validated for a set of established models of various cellular rhythms. Overall, our work highlights the importance of reaction kinetics and feedback type for the variability of period and amplitude and therefore for the establishment of predictive models.

Rhythmic behavior is omnipresent in biology and has many crucial functions. In cells the activation levels and abundances of signaling molecules such as NF-κB, p53, EGFR or calcium repeatedly increase and decrease in response to stimuli. Such a dynamic behavior can also be observed monitoring the concentrations of mRNAs and proteins in the circadian clock and the cell cycle. Period and amplitude which are the time span between peaks and the peak height, respectively, as well as their variabilities are important features of oscillations. The circadian period is very stable allowing for proper time keeping, whereas in calcium signaling the period is very variable encoding different stimulation strengths. Our goal is to examine the origin of differences in sensitivities of periods and amplitudes using a computational approach. We use prototype oscillators and demonstrate that they can be used to derive general principles that explain the degree of robustness in period and amplitude for a set of commonly used models of cellular oscillators. Our findings imply that the robustness of oscillating systems can be influenced by feedback type and kinetic properties to which special attention should be paid when designing mathematical models of cellular rhythms.

Evidence for Weakened Intercellular Coupling in the Mammalian Circadian Clock under Long Photoperiod

For animals living in temperate latitudes, seasonal changes in day length are an important cue for adaptations of their physiology and behavior to the altered environmental conditions. The suprachiasmatic nucleus (SCN) is known as the central circadian clock in mammals, but may also play an important role in adaptations to different photoperiods. The SCN receives direct light input from the retina and is able to encode day-length by approximating the waveform of the electrical activity rhythm to the duration of daylight. Changing the overall waveform requires a reorganization of the neuronal network within the SCN with a change in the degree of synchrony between the neurons; however, the underlying mechanisms are yet unknown. In the present study we used PER2::LUC bioluminescence imaging in cultured SCN slices to characterize network dynamics on the single-cell level and we aimed to provide evidence for a role of modulations in coupling strength in the photoperiodic-induced phase dispersal. Exposure to long photoperiod (LP) induced a larger distribution of peak times of the single-cell PER2::LUC rhythms in the anterior SCN, compared to short photoperiod. Interestingly, the cycle-to-cycle variability in single-cell period of PER2::LUC rhythms is also higher in the anterior SCN in LP, and is positively correlated with peak time dispersal. Applying a new, impartial community detection method on the time series data of the PER2::LUC rhythm revealed two clusters of cells with a specific spatial distribution, which we define as dorsolateral and ventromedial SCN. Post hoc analysis of rhythm characteristics of these clusters showed larger cycle-to-cycle single-cell period variability in the dorsolateral compared to the ventromedial cluster in the anterior SCN. We conclude that a change in coupling strength within the SCN network is a plausible explanation to the observed changes in single-cell period variability, which can contribute to the photoperiod-induced phase distribution.

Design Principles of Biological Oscillators through Optimization: Forward and Reverse Analysis

From cyanobacteria to human, sustained oscillations coordinate important biological functions. Although much has been learned concerning the sophisticated molecular mechanisms underlying biological oscillators, design principles linking structure and functional behavior are not yet fully understood. Here we explore design principles of biological oscillators from a multiobjective optimization perspective, taking into account the trade-offs between conflicting performance goals or demands. We develop a comprehensive tool for automated design of oscillators, based on multicriteria global optimization that allows two modes: (i) the automatic design (forward problem) and (ii) the inference of design principles (reverse analysis problem). From the perspective of synthetic biology, the forward mode allows the solution of design problems that mimic some of the desirable properties appearing in natural oscillators. The reverse analysis mode facilitates a systematic exploration of the design space based on Pareto optimality concepts. The method is illustrated with two case studies: the automatic design of synthetic oscillators from a library of biological parts, and the exploration of design principles in 3-gene oscillatory systems.

Noise Induces Hopping between NF-κB Entrainment Modes

Oscillations and noise drive many processes in biology, but how both affect the activity of the transcription factor nuclear factor κB (NF-κB) is not understood. Here, we observe that when NF-κB oscillations are entrained by periodic tumor necrosis factor (TNF) inputs in experiments, NF-κB exhibits jumps between frequency modes, a phenomenon we call “cellular mode-hopping.” By comparing stochastic simulations of NF-κB oscillations to deterministic simulations conducted inside and outside the chaotic regime of parameter space, we show that noise facilitates mode-hopping in all regimes. However, when the deterministic system is driven by chaotic dynamics, hops between modes are erratic and short-lived, whereas in experiments, the system spends several periods in one entrainment mode before hopping and rarely visits more than two modes. The experimental behavior matches our simulations of noise-induced mode-hopping outside the chaotic regime. We suggest that mode-hopping is a mechanism by which different NF-κB-dependent genes under frequency control can be expressed at different times.

Synchronizing stochastic circadian oscillators in single cells of Neurospora crassa

The synchronization of stochastic coupled oscillators is a central problem in physics and an emerging problem in biology, particularly in the context of circadian rhythms. Most measurements on the biological clock are made at the macroscopic level of millions of cells. Here measurements are made on the oscillators in single cells of the model fungal system, Neurospora crassa, with droplet microfluidics and the use of a fluorescent recorder hooked up to a promoter on a clock controlled gene-2 (ccg-2). The oscillators of individual cells are stochastic with a period near 21 hours (h), and using a stochastic clock network ensemble fitted by Markov Chain Monte Carlo implemented on general-purpose graphical processing units (or GPGPUs) we estimated that >94% of the variation in ccg-2 expression was stochastic (as opposed to experimental error). To overcome this stochasticity at the macroscopic level, cells must synchronize their oscillators. Using a classic measure of similarity in cell trajectories within droplets, the intraclass correlation (ICC), the synchronization surface ICC is measured on >25,000 cells as a function of the number of neighboring cells within a droplet and of time. The synchronization surface provides evidence that cells communicate, and synchronization varies with genotype.