A mechanism for robust circadian timekeeping via stoichiometric balance

Enlightening the blind spot of the Michaelis-Menten rate law: The role of relaxation dynamics in molecular complex formation

The century-long Michaelis-Menten rate law and its modifications in the modeling of biochemical rate processes stand on the assumption that the concentration of the complex of interacting molecules, at each moment, rapidly approaches an equilibrium (quasi-steady state) compared to the pace of molecular concentration changes. Yet, in the case of actively time-varying molecular concentrations with transient or oscillatory dynamics, the deviation of the complex profile from the quasi-steady state becomes relevant. A recent theoretical approach, known as the effective time-delay scheme (ETS), suggests that the delay from the relaxation time of molecular complex formation contributes to the substantial breakdown of the quasi-steady state assumption. Here, we systematically expand this ETS and inquire into the comprehensive roles of relaxation dynamics in complex formation. Through the modeling of rhythmic protein-protein and protein-DNA interactions and the mammalian circadian clock, our analysis reveals the effect of the relaxation dynamics beyond the time delay, which extends to the dampening of changes in the complex concentration with a reduction in the oscillation amplitude compared to the quasi-steady state. Interestingly, the combined effect of the time delay and amplitude reduction shapes both qualitative and quantitative oscillatory patterns such as the emergence and variability of the mammalian circadian rhythms. These findings highlight the downside of the routine assumption of quasi-steady states and enhance the mechanistic understanding of rich time-varying biomolecular processes.

Development of circadian rhythms in mammalian systems

In mammals, molecular mechanisms of circadian rhythms involve a time-delayed negative feedback loop generating autonomous oscillations of ∼24 h. Most cell types in mammals possess circadian rhythms regulating temporal organization of cellular and physiological processes. Intriguingly, pluripotent stem cells do not possess circadian rhythms and oscillations arise after a defined period of differentiation. Previous studies demonstrated that post-transcriptional regulations of core clock components, CLOCK and PER2, play critical roles in inducing circadian rhythms. In this article, we review the development of circadian rhythms in mammalian systems and provide a theoretical understanding of potential mechanisms regulating the birth of circadian rhythms using mathematical modeling.

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.

Phosphorylation of DNA-binding domains of CLOCK-BMAL1 complex for PER-dependent inhibition in circadian clock of mammalian cells

In mammals, CLOCK and BMAL1 proteins form a heterodimer that binds to E-box sequences and activates transcription of target genes, including Period (Per). Translated PER proteins then bind to the CLOCK-BMAL1 complex to inhibit its transcriptional activity. However, the molecular mechanism and the impact of this PER-dependent inhibition on the circadian clock oscillation remain elusive. We previously identified Ser38 and Ser42 in a DNA-binding domain of CLOCK as phosphorylation sites at the PER-dependent inhibition phase. In this study, knockout rescue experiments showed that nonphosphorylatable (Ala) mutations at these sites shortened circadian period, whereas their constitutive-phospho-mimetic (Asp) mutations completely abolished the circadian rhythms. Similarly, we found that nonphosphorylatable (Ala) and constitutive-phospho-mimetic (Glu) mutations at Ser78 in a DNA-binding domain of BMAL1 also shortened the circadian period and abolished the rhythms, respectively. The mathematical modeling predicted that these constitutive-phospho-mimetic mutations weaken the DNA binding of the CLOCK-BMAL1 complex and that the nonphosphorylatable mutations inhibit the PER-dependent displacement (reduction of DNA-binding ability) of the CLOCK-BMAL1 complex from DNA. Biochemical experiments supported the importance of these phosphorylation sites for displacement of the complex in the PER2-dependent inhibition. Our results provide direct evidence that phosphorylation of CLOCK-Ser38/Ser42 and BMAL1-Ser78 plays a crucial role in the PER-dependent inhibition and the determination of the circadian period.

Temperature compensation through kinetic regulation in biochemical oscillators

Nearly all circadian clocks maintain a period that is insensitive to temperature changes, a phenomenon known as temperature compensation (TC). Yet, it is unclear whether there is any common feature among different systems that exhibit TC. From a general timescale invariance, we show that TC relies on the existence of certain period-lengthening reactions wherein the period of the system increases strongly with the rates in these reactions. By studying several generic oscillator models, we show that this counterintuitive dependence is nonetheless a common feature of oscillators in the nonlinear (far-from-onset) regime where the oscillation can be separated into fast and slow phases. The increase of the period with the period-lengthening reaction rates occurs when the amplitude of the slow phase in the oscillation increases with these rates while the progression speed in the slow phase is controlled by other rates of the system. The positive dependence of the period on the period-lengthening rates balances its inverse dependence on other kinetic rates in the system, which gives rise to robust TC in a wide range of parameters. We demonstrate the existence of such period-lengthening reactions and their relevance for TC in all four model systems we considered. Theoretical results for a model of the Kai system are supported by experimental data. A study of the energy dissipation also shows that better TC performance requires higher energy consumption. Our study unveils a general mechanism by which a biochemical oscillator achieves TC by operating in parameter regimes far from the onset where period-lengthening reactions exist.

Network switches and their role in circadian clocks

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

A robust ultrasensitive transcriptional switch in noisy cellular environments

Ultrasensitive transcriptional switches enable sharp transitions between transcriptional on and off states and are essential for cells to respond to environmental cues with high fidelity. However, conventional switches, which rely on direct repressor-DNA binding, are extremely noise-sensitive, leading to unintended changes in gene expression. Here, through model simulations and analysis, we discovered that an alternative design combining three indirect transcriptional repression mechanisms, sequestration, blocking, and displacement, can generate a noise-resilient ultrasensitive switch. Although sequestration alone can generate an ultrasensitive switch, it remains sensitive to noise because the unintended transcriptional state induced by noise persists for long periods. However, by jointly utilizing blocking and displacement, these noise-induced transitions can be rapidly restored to the original transcriptional state. Because this transcriptional switch is effective in noisy cellular contexts, it goes beyond previous synthetic transcriptional switches, making it particularly valuable for robust synthetic system design. Our findings also provide insights into the evolution of robust ultrasensitive switches in cells. Specifically, the concurrent use of seemingly redundant indirect repression mechanisms in diverse biological systems appears to be a strategy to achieve noise-resilience of ultrasensitive switches.

Noisy Delay Denoises Biochemical Oscillators

Genetic oscillations are generated by delayed transcriptional negative feedback loops, wherein repressor proteins inhibit their own synthesis after a temporal production delay. This delay is distributed because it arises from a sequence of noisy processes, including transcription, translocation, translation, and folding. Because the delay determines repression timing and, therefore, oscillation period, it has been commonly believed that delay noise weakens oscillatory dynamics. Here, we demonstrate that noisy delay can surprisingly denoise genetic oscillators. Specifically, moderate delay noise improves the signal-to-noise ratio and sharpens oscillation peaks, all without impacting period and amplitude. We show that this denoising phenomenon occurs in a variety of well-studied genetic oscillators, and we use queueing theory to uncover the universal mechanisms that produce it.

Circadian regulation of sinoatrial nodal cell pacemaking function: Dissecting the roles of autonomic control, body temperature, and local circadian rhythmicity

Strong circadian (~24h) rhythms in heart rate (HR) are critical for flexible regulation of cardiac pacemaking function throughout the day. While this circadian flexibility in HR is sustained in diverse conditions, it declines with age, accompanied by reduced maximal HR performance. The intricate regulation of circadian HR involves the orchestration of the autonomic nervous system (ANS), circadian rhythms of body temperature (CRBT), and local circadian rhythmicity (LCR), which has not been fully understood. Here, we developed a mathematical model describing ANS, CRBT, and LCR in sinoatrial nodal cells (SANC) that accurately captures distinct circadian patterns in adult and aged mice. Our model underscores how the alliance among ANS, CRBT, and LCR achieves circadian flexibility to cover a wide range of firing rates in SANC, performance to achieve maximal firing rates, while preserving robustness to generate rhythmic firing patterns irrespective of external conditions. Specifically, while ANS dominates in promoting SANC flexibility and performance, CRBT and LCR act as primary and secondary boosters, respectively, to further enhance SANC flexibility and performance. Disruption of this alliance with age results in impaired SANC flexibility and performance, but not robustness. This unexpected outcome is primarily attributed to the age-related reduction in parasympathetic activities, which maintains SANC robustness while compromising flexibility. Our work sheds light on the critical alliance of ANS, CRBT, and LCR in regulating time-of-day cardiac pacemaking function and dysfunction, offering insights into novel therapeutic targets for the prevention and treatment of cardiac arrhythmias.

A review on gene regulatory network reconstruction algorithms based on single cell RNA sequencing

BACKGROUND

Understanding gene regulatory networks (GRNs) is essential for unraveling the molecular mechanisms governing cellular behavior. With the advent of high-throughput transcriptome measurement technology, researchers have aimed to reverse engineer the biological systems, extracting gene regulatory rules from their outputs, which represented by gene expression data. Bulk RNA sequencing, a widely used method for measuring gene expression, has been employed for GRN reconstruction. However, it falls short in capturing dynamic changes in gene expression at the level of individual cells since it averages gene expression across mixed cell populations.

OBJECTIVE

In this review, we provide an overview of 15 GRN reconstruction tools and discuss their respective strengths and limitations, particularly in the context of single cell RNA sequencing (scRNA-seq).

METHODS

Recent advancements in scRNA-seq break new ground of GRN reconstruction. They offer snapshots of the individual cell transcriptomes and capturing dynamic changes. We emphasize how these technological breakthroughs have enhanced GRN reconstruction.

CONCLUSION

GRN reconstructors can be classified based on their requirement for cellular trajectory, which represents a dynamical cellular process including differentiation, aging, or disease progression. Benchmarking studies support the superiority of GRN reconstructors that do not require trajectory analysis in identifying regulator-target relationships. However, methods equipped with trajectory analysis demonstrate better performance in identifying key regulatory factors. In conclusion, researchers should select a suitable GRN reconstructor based on their specific research objectives.

Generalized Michaelis-Menten rate law with time-varying molecular concentrations

The Michaelis-Menten (MM) rate law has been the dominant paradigm of modeling biochemical rate processes for over a century with applications in biochemistry, biophysics, cell biology, systems biology, and chemical engineering. The MM rate law and its remedied form stand on the assumption that the concentration of the complex of interacting molecules, at each moment, approaches an equilibrium (quasi-steady state) much faster than the molecular concentrations change. Yet, this assumption is not always justified. Here, we relax this quasi-steady state requirement and propose the generalized MM rate law for the interactions of molecules with active concentration changes over time. Our approach for time-varying molecular concentrations, termed the effective time-delay scheme (ETS), is based on rigorously estimated time-delay effects in molecular complex formation. With particularly marked improvements in protein-protein and protein-DNA interaction modeling, the ETS provides an analytical framework to interpret and predict rich transient or rhythmic dynamics (such as autogenously-regulated cellular adaptation and circadian protein turnover), which goes beyond the quasi-steady state assumption.

Archaic Introgression Shaped Human Circadian Traits

When the ancestors of modern Eurasians migrated out of Africa and interbred with Eurasian archaic hominins, namely, Neanderthals and Denisovans, DNA of archaic ancestry integrated into the genomes of anatomically modern humans. This process potentially accelerated adaptation to Eurasian environmental factors, including reduced ultraviolet radiation and increased variation in seasonal dynamics. However, whether these groups differed substantially in circadian biology and whether archaic introgression adaptively contributed to human chronotypes remain unknown. Here, we traced the evolution of chronotype based on genomes from archaic hominins and present-day humans. First, we inferred differences in circadian gene sequences, splicing, and regulation between archaic hominins and modern humans. We identified 28 circadian genes containing variants with potential to alter splicing in archaics (e.g., CLOCK, PER2, RORB, and RORC) and 16 circadian genes likely divergently regulated between present-day humans and archaic hominins, including RORA. These differences suggest the potential for introgression to modify circadian gene expression. Testing this hypothesis, we found that introgressed variants are enriched among expression quantitative trait loci for circadian genes. Supporting the functional relevance of these regulatory effects, we found that many introgressed alleles have associations with chronotype. Strikingly, the strongest introgressed effects on chronotype increase morningness, consistent with adaptations to high latitude in other species. Finally, we identified several circadian loci with evidence of adaptive introgression or latitudinal clines in allele frequency. These findings identify differences in circadian gene regulation between modern humans and archaic hominins and support the contribution of introgression via coordinated effects on variation in human chronotype.

Circadian Disruption in Night Shift Work and Its Association with Chronic Pulmonary Diseases

Globalization and the expansion of essential services over continuous 24 h cycles have necessitated the adaptation of the human workforce to shift-based schedules. Night shift work (NSW) causes a state of desynchrony between the internal circadian machinery and external environmental cues, which can impact inflammatory and metabolic pathways. The discovery of clock genes in the lung has shed light on potential mechanisms of circadian misalignment in chronic pulmonary disease. Here, the current knowledge of circadian clock disruption caused by NSW and its impact on lung inflammation and associated pathophysiology in chronic lung diseases, such as asthma, chronic obstructive pulmonary disease, pulmonary fibrosis, and COVID-19, is reviewed. Furthermore, the limitations of the current understanding of circadian disruption and potential future chronotherapeutic advances are discussed.

Modeling the Effects of Napping and Non-napping Patterns of Light Exposure on the Human Circadian Oscillator

In early childhood, consolidation of sleep from a biphasic to a monophasic sleep-wake pattern, that is, the transition from sleeping during an afternoon nap and at night to sleeping only during the night, represents a major developmental milestone. Reduced napping behavior is associated with an advance in the timing of the circadian system; however, it is unknown if this advance represents a standard response of the circadian clock to altered patterns of light exposure or if it additionally reflects features of the developing circadian system. Using a mathematical model of the human circadian pacemaker, we investigated the impact of napping and non-napping patterns of light exposure on entrained circadian phases. Simulated light schedules were based on published data from 20 children (34.2 ± 2.0 months) with habitual napping or non-napping sleep patterns (15 nappers). We found the model predicted different circadian phases for napping and non-napping light patterns: both the decrease in afternoon light during the nap and the increase in evening light associated with napping toddlers' later bedtimes contributed to the observed circadian phase difference produced between napping and non-napping light schedules. We systematically quantified the effects on phase shifting of nap duration, timing, and light intensity, finding larger phase delays occurred for longer and earlier naps. In addition, we simulated phase response curves to a 1-h light pulse and 1-h dark pulse to predict phase and intensity dependence of these changes in light exposure. We found the light pulse produced larger shifts compared with the dark pulse, and we analyzed the model dynamics to identify the features contributing to this asymmetry. These findings suggest that napping status affects circadian timing due to altered patterns of light exposure, with the dynamics of the circadian clock and light processing mediating the effects of the dark pulse associated with a daytime nap.

Archaic Introgression Shaped Human Circadian Traits

Introduction

When the ancestors of modern Eurasians migrated out of Africa and interbred with Eurasian archaic hominins, namely Neanderthals and Denisovans, DNA of archaic ancestry integrated into the genomes of anatomically modern humans. This process potentially accelerated adaptation to Eurasian environmental factors, including reduced ultra-violet radiation and increased variation in seasonal dynamics. However, whether these groups differed substantially in circadian biology, and whether archaic introgression adaptively contributed to human chronotypes remains unknown.

Results

Here we traced the evolution of chronotype based on genomes from archaic hominins and present-day humans. First, we inferred differences in circadian gene sequences, splicing, and regulation between archaic hominins and modern humans. We identified 28 circadian genes containing variants with potential to alter splicing in archaics (e.g., CLOCK, PER2, RORB, RORC), and 16 circadian genes likely divergently regulated between present-day humans and archaic hominins, including RORA. These differences suggest the potential for introgression to modify circadian gene expression. Testing this hypothesis, we found that introgressed variants are enriched among eQTLs for circadian genes. Supporting the functional relevance of these regulatory effects, we found that many introgressed alleles have associations with chronotype. Strikingly, the strongest introgressed effects on chronotype increase morningness, consistent with adaptations to high latitude in other species. Finally, we identified several circadian loci with evidence of adaptive introgression or latitudinal clines in allele frequency.

Conclusions

These findings identify differences in circadian gene regulation between modern humans and archaic hominins and support the contribution of introgression via coordinated effects on variation in human chronotype.

Transcription factor competition facilitates self-sustained oscillations in single gene genetic circuits

Genetic feedback loops can be used by cells to regulate internal processes or to keep track of time. It is often thought that, for a genetic circuit to display self-sustained oscillations, a degree of cooperativity is needed in the binding and unbinding of actor species. This cooperativity is usually modeled using a Hill function, regardless of the actual promoter architecture. Furthermore, genetic circuits do not operate in isolation and often transcription factors are shared between different promoters. In this work we show how mathematical modelling of genetic feedback loops can be facilitated with a mechanistic fold-change function that takes into account the titration effect caused by competing binding sites for transcription factors. The model shows how the titration effect facilitates self-sustained oscillations in a minimal genetic feedback loop: a gene that produces its own repressor directly without cooperative transcription factor binding. The use of delay-differential equations leads to a stability contour that predicts whether a genetic feedback loop will show self-sustained oscillations, even when taking the bursty nature of transcription into account.

A general model-based causal inference method overcomes the curse of synchrony and indirect effect

To identify causation, model-free inference methods, such as Granger Causality, have been widely used due to their flexibility. However, they have difficulty distinguishing synchrony and indirect effects from direct causation, leading to false predictions. To overcome this, model-based inference methods that test the reproducibility of data with a specific mechanistic model to infer causality were developed. However, they can only be applied to systems described by a specific model, greatly limiting their applicability. Here, we address this limitation by deriving an easily testable condition for a general monotonic ODE model to reproduce time-series data. We built a user-friendly computational package, General ODE-Based Inference (GOBI), which is applicable to nearly any monotonic system with positive and negative regulations described by ODE. GOBI successfully inferred positive and negative regulations in various networks at both the molecular and population levels, unlike existing model-free methods. Thus, this accurate and broadly applicable inference method is a powerful tool for understanding complex dynamical systems.

Proteostasis failure exacerbates neuronal circuit dysfunction and sleep impairments in Alzheimer's disease

Failed proteostasis is a well-documented feature of Alzheimer's disease, particularly, reduced protein degradation and clearance. However, the contribution of failed proteostasis to neuronal circuit dysfunction is an emerging concept in neurodegenerative research and will prove critical in understanding cognitive decline. Our objective is to convey Alzheimer's disease progression with the growing evidence for a bidirectional relationship of sleep disruption and proteostasis failure. Proteostasis dysfunction and tauopathy in Alzheimer's disease disrupts neurons that regulate the sleep-wake cycle, which presents behavior as impaired slow wave and rapid eye movement sleep patterns. Subsequent sleep loss further impairs protein clearance. Sleep loss is a defined feature seen early in many neurodegenerative disorders and contributes to memory impairments in Alzheimer's disease. Canonical pathological hallmarks, β-amyloid, and tau, directly disrupt sleep, and neurodegeneration of locus coeruleus, hippocampal and hypothalamic neurons from tau proteinopathy causes disruption of the neuronal circuitry of sleep. Acting in a positive-feedback-loop, sleep loss and circadian rhythm disruption then increase spread of β-amyloid and tau, through impairments of proteasome, autophagy, unfolded protein response and glymphatic clearance. This phenomenon extends beyond β-amyloid and tau, with interactions of sleep impairment with the homeostasis of TDP-43, α-synuclein, FUS, and huntingtin proteins, implicating sleep loss as an important consideration in an array of neurodegenerative diseases and in cases of mixed neuropathology. Critically, the dynamics of this interaction in the neurodegenerative environment are not fully elucidated and are deserving of further discussion and research. Finally, we propose sleep-enhancing therapeutics as potential interventions for promoting healthy proteostasis, including β-amyloid and tau clearance, mechanistically linking these processes. With further clinical and preclinical research, we propose this dynamic interaction as a diagnostic and therapeutic framework, informing precise single- and combinatorial-treatments for Alzheimer's disease and other brain disorders.

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.

Circadian clock-based therapeutics in chronic pulmonary diseases

The circadian clock is the biochemical oscillator that orchestrates the observable circadian rhythms in physiology and behavior. Disruption of the circadian clock in the lungs during chronic pulmonary diseases is considered one of the key etiological risk factors that drive pathobiology. Preclinical studies support that pharmacological manipulation of the circadian clock is a conceivable approach for the development of novel clock-based therapeutics. Despite recent advances, no effort has been undertaken to integrate novel findings for the treatment and management of chronic lung diseases. We, therefore, recognize the need to discuss the candidate clock genes that can be potentially targeted for therapeutic intervention. Here, we aim to create the first roadmap that will advance the development of circadian- clock-based therapeutics that may provide better outcomes in treating chronic pulmonary diseases.

Rhythmic transcription of Bmal1 stabilizes the circadian timekeeping system in mammals

In mammals, the circadian clock consists of transcriptional and translational feedback loops through DNA cis-elements such as E-box and RRE. The E-box-mediated core feedback loop is interlocked with the RRE-mediated feedback loop, but biological significance of the RRE-mediated loop has been elusive. In this study, we established mutant cells and mice deficient for rhythmic transcription of Bmal1 gene by deleting its upstream RRE elements and hence disrupted the RRE-mediated feedback loop. We observed apparently normal circadian rhythms in the mutant cells and mice, but a combination of mathematical modeling and experiments revealed that the circadian period and amplitude of the mutants were more susceptible to disturbance of CRY1 protein rhythm. Our findings demonstrate that the RRE-mediated feedback regulation of Bmal1 underpins the E-box-mediated rhythm in cooperation with CRY1-dependent posttranslational regulation of BMAL1 protein, thereby conferring the perturbation-resistant oscillation and chronologically-organized output of the circadian clock.

Deciphering clock cell network morphology within the biological master clock, suprachiasmatic nucleus: From the perspective of circadian wave dynamics

The biological master clock, suprachiasmatic nucleus (of rat and mouse), is composed of ~10,000 clock cells which are heterogeneous with respect to their circadian periods. Despite this inhomogeneity, an intact SCN maintains a very good degree of circadian phase (time) coherence which is vital for sustaining various circadian rhythmic activities, and it is supposedly achieved by not just one but a few different cell-to-cell coupling mechanisms, among which action potential (AP)-mediated connectivity is known to be essential. But, due to technical difficulties and limitations in experiments, so far very little information is available about the morphology of the connectivity at a cellular scale. Building upon this limited amount of information, here we exhaustively and systematically explore a large pool (~25,000) of various network morphologies to come up with some plausible network features of SCN networks. All candidates under consideration reflect an experimentally obtained 'indegree distribution' as well as a 'physical range distribution of afferent clock cells.' Then, importantly, with a set of multitude criteria based on the properties of SCN circadian phase waves in extrinsically perturbed as well as in their natural states, we select out appropriate model networks: Some important measures are, 1) level of phase dispersal and direction of wave propagation, 2) phase-resetting ability of the model networks subject to external circadian forcing, and 3) decay rate of perturbation induced "phase-singularities." The successful, realistic networks have several common features: 1) "indegree" and "outdegree" should have a positive correlation; 2) the cells in the SCN ventrolateral region (core) have a much larger total degree than that of the dorsal medial region (shell); 3) The number of intra-core edges is about 7.5 times that of intra-shell edges; and 4) the distance probability density function for the afferent connections fits well to a beta function. We believe that these newly identified network features would be a useful guide for future explorations on the very much unknown AP-mediated clock cell connectome within the SCN.

Circadian Rhythm Regulated by Tumor Suppressor p53 and Time Delay in Unstressed Cells

Circadian function and p53 network are interconnected on the molecular level, but the dynamics induced by the interaction between the circadian factor Per2 and the tumor suppressor p53 remains poorly understood. Here, we constructed an integrative model composed of a circadian clock module and a p53-Mdm2 feedback module to study the dynamics of p53-Per2 network in unstressed cells. As expected, the model can accurately predict the circadian rhythm, which is consistent with diverse experimental observations. In addition, using a combination of theoretical analysis and numerical simulation, the results demonstrated that p53 expression enhances the phase advance of circadian rhythm and reduces the robustness of circadian rhythm. Furthermore, the time delay required for the transcription and translation of Per2 protein induces oscillations by undergoing a supercritical Hopf bifurcation, and improves the robustness of circadian rhythm. In summary, this work shows that the p53-Per2 interaction and the time delay are two essential factors for circadian functions.

A Phenomenological Mouse Circadian Pacemaker Model

Mathematical models have been used extensively in chronobiology to explore characteristics of biological clocks. In particular, for human circadian studies, the Kronauer model has been modified multiple times to describe rhythm production and responses to sensory input. This phenomenological model comprises a single set of parameters which can simulate circadian responses in humans under a variety of environmental conditions. However, corresponding models for nocturnal rodents commonly used in circadian rhythm studies are not available and may require new parameter values for different species and even strains. Moreover, due to a considerable variation in experimental data collected from mice of the same strain, within and across laboratories, a range of valid parameters is essential. This study develops a Kronauer-like model for mice by re-fitting relevant parameters to published phase response curve and period data using total least squares. Local parameter sensitivity analysis and parameter distributions determine the parameter ranges that give a near-identical model and data distribution of periods. However, the model required further parameter adjustments to match characteristics of other mouse strains, implying that the model itself detects changes in the core processes of rhythm generation and control. The model is a useful tool to understand and interpret future mouse circadian clock experiments.

Combined multiple transcriptional repression mechanisms generate ultrasensitivity and oscillations

Transcriptional repression can occur via various mechanisms, such as blocking, sequestration and displacement. For instance, the repressors can hold the activators to prevent binding with DNA or can bind to the DNA-bound activators to block their transcriptional activity. Although the transcription can be completely suppressed with a single mechanism, multiple repression mechanisms are used together to inhibit transcriptional activators in many systems, such as circadian clocks and NF-κB oscillators. This raises the question of what advantages arise if seemingly redundant repression mechanisms are combined. Here, by deriving equations describing the multiple repression mechanisms, we find that their combination can synergistically generate a sharply ultrasensitive transcription response and thus strong oscillations. This rationalizes why the multiple repression mechanisms are used together in various biological oscillators. The critical role of such combined transcriptional repression for strong oscillations is further supported by our analysis of formerly identified mutations disrupting the transcriptional repression of the mammalian circadian clock. The hitherto unrecognized source of the ultrasensitivity, the combined transcriptional repressions, can lead to robust synthetic oscillators with a previously unachievable simple design.

Time-keeping and decision-making in living cells: Part I

To survive and reproduce, a cell must process information from its environment and its own internal state and respond accordingly, in terms of metabolic activity, gene expression, movement, growth, division and differentiation. These signal–response decisions are made by complex networks of interacting genes and proteins, which function as biochemical switches and clocks, and other recognizable information-processing circuitry. This theme issue of Interface Focus (in two parts) brings together articles on time-keeping and decision-making in living cells—work that uses precise mathematical modelling of underlying molecular regulatory networks to understand important features of cell physiology. Part I focuses on time-keeping: mechanisms and dynamics of biological oscillators and modes of synchronization and entrainment of oscillators, with special attention to circadian clocks.

Mathematical analysis of robustness of oscillations in models of the mammalian circadian clock

Circadian rhythms in a wide range of organisms are mediated by molecular mechanisms based on transcription-translation feedback. In this paper, we use bifurcation theory to explore mathematical models of genetic oscillators, based on Kim & Forger’s interpretation of the circadian clock in mammals. At the core of their models is a negative feedback loop whereby PER proteins (PER1 and PER2) bind to and inhibit their transcriptional activator, BMAL1. For oscillations to occur, the dissociation constant of the PER:BMAL1 complex, K^d, must be ≤ 0.04 nM, which is orders of magnitude smaller than a reasonable expectation of 1–10 nM for this protein complex. We relax this constraint by two modifications to Kim & Forger’s ‘single negative feedback’ (SNF) model: first, by introducing a multistep reaction chain for posttranscriptional modifications of Per mRNA and posttranslational phosphorylations of PER, and second, by replacing the first-order rate law for degradation of PER in the nucleus by a Michaelis-Menten rate law. These modifications increase the maximum allowable K^d to ~2 nM. In a third modification, we consider an alternative rate law for gene transcription to resolve an unrealistically large rate of Per2 transcription at very low concentrations of BMAL1. Additionally, we studied extensions of the SNF model to include a second negative feedback loop (involving REV-ERB) and a supplementary positive feedback loop (involving ROR). Contrary to Kim & Forger’s observations of these extended models, we find that, with our modifications, the supplementary positive feedback loop makes the oscillations more robust than observed in the models with one or two negative feedback loops. However, all three models are similarly robust when accounting for circadian rhythms (~24 h period) with K^d ≥ 1 nM. Our results provide testable predictions for future experimental studies.

The circadian rhythm aligns bodily functions to the day/night cycle and is important for our health. The rhythm originates from an intracellular molecular clock mechanism that mediates rhythmic gene expression. It is long understood that transcriptional negative feedback with sufficient time delay is key to generating circadian oscillations. However, some of the most widely cited mathematical models for the circadian clock suffer from problems of parameter ‘fragilities’. That is, sustained oscillations are possible only for physically unrealistic parameter values. A recent model by Kim & Forger nicely incorporates the inhibitory binding of PER proteins to their transcription activator BMAL1, but oscillations in the Kim-Forger model require a binding affinity between PER and BMAL1 that is orders of magnitude larger than observed binding affinities of protein complexes. To rectify this problem, we make several physiologically credible modifications to the Kim-Forger model, which allow oscillations to occur with more realistic binding affinities. The modified model is further extended to explore the potential roles of supplementary feedback loops in the mammalian clock mechanism. Ultimately, accurate models of the circadian clock will provide better predictive tools for chronotherapy and chrono-pharmacology studies.

Quantification of protein abundance and interaction defines a mechanism for operation of the circadian clock

The mammalian circadian clock exerts control of daily gene expression through cycles of DNA binding. Here, we develop a quantitative model of how a finite pool of BMAL1 protein can regulate thousands of target sites over daily time scales. We used quantitative imaging to track dynamic changes in endogenous labelled proteins across peripheral tissues and the SCN. We determine the contribution of multiple rhythmic processes coordinating BMAL1 DNA binding, including cycling molecular abundance, binding affinities, and repression. We find nuclear BMAL1 concentration determines corresponding CLOCK through heterodimerisation and define a DNA residence time of this complex. Repression of CLOCK:BMAL1 is achieved through rhythmic changes to BMAL1:CRY1 association and high-affinity interactions between PER2:CRY1 which mediates CLOCK:BMAL1 displacement from DNA. Finally, stochastic modelling reveals a dual role for PER:CRY complexes in which increasing concentrations of PER2:CRY1 promotes removal of BMAL1:CLOCK from genes consequently enhancing ability to move to new target sites.

Promoter methylation in a mixed feedback loop circadian clock model

We investigate how epigenetic modifications to clock gene promoters affect transcriptomic activity in the circadian clock. Motivated by experimental observations that link DNA methylation with the behavior of the clock, we introduce and analyze an extension of the mixed feedback loop (MFL) model of François and Hakim. We extend the original model to include an additional methylated promoter state and allow for reversible protein sequestration, an important feature for circadian applications. First, working with the general form of the MFL model, we find that the qualitative behavior of the model is dictated by the promoter state with the highest transcription rate. We then build on the work of Kim and Forger, who analyzed the stability of the mammalian circadian clock by using a reduced form of the MFL model. We present a rigorous procedure for translating between the MFL model and the reduction of Kim and Forger. We then propose a model reduction more appropriate for the study of oscillatory promoter states, making use of a fully coupled quasi-steady-state approximation rather than the standard partially uncoupled quasi-steady-state approach. Working with the novel reduced form of the model, we find substantial differences in the transcription function and show that, although methylation contributes to period control, excessive methylation can abolish rhythmicity. Altogether our results show that even in a minimal clock model, DNA methylation has a nontrivial influence on the system's ability to oscillate.

Systematic modeling-driven experiments identify distinct molecular clockworks underlying hierarchically organized pacemaker neurons

In metazoan organisms, circadian (∼24 h) rhythms are regulated by pacemaker neurons organized in a master-slave hierarchy. Although it is widely accepted that master pacemakers and slave oscillators generate rhythms via an identical negative feedback loop of transcription factor CLOCK (CLK) and repressor PERIOD (PER), their different roles imply heterogeneity in their molecular clockworks. Indeed, in Drosophila, defective binding between CLK and PER disrupts molecular rhythms in the master pacemakers, small ventral lateral neurons (sLN_vs), but not in the slave oscillator, posterior dorsal neuron 1s (DN1_ps). Here, we develop a systematic and expandable approach that unbiasedly searches the source of the heterogeneity in molecular clockworks from time-series data. In combination with in vivo experiments, we find that sLN_vs exhibit higher synthesis and turnover of PER and lower CLK levels than DN1_ps. Importantly, light shift analysis reveals that due to such a distinct molecular clockwork, sLN_vs can obtain paradoxical characteristics as the master pacemaker, generating strong rhythms that are also flexibly adjustable to environmental changes. Our results identify the different characteristics of molecular clockworks of pacemaker neurons that underlie hierarchical multi-oscillator structure to ensure the rhythmic fitness of the organism.

Modelling of plant circadian clock for characterizing hypocotyl growth under different light quality conditions

To meet the ever-increasing global food demand, the food production rate needs to be increased significantly in the near future. Speed breeding is considered as a promising agricultural technology solution to achieve the zero-hunger vision as specified in the United Nations Sustainable Development Goal 2. In speed breeding, the photoperiod of the artificial light has been manipulated to enhance crop productivity. In particular, regulating the photoperiod of different light qualities rather than solely white light can further improve speed breading. However, identifying the optimal light quality and the associated photoperiod simultaneously remains a challenging open problem due to complex interactions between multiple photoreceptors and proteins controlling plant growth. To tackle this, we develop a first comprehensive model describing the profound effect of multiple light qualities with different photoperiods on plant growth (i.e. hypocotyl growth). The model predicts that hypocotyls elongated more under red light compared to both red and blue light. Drawing similar findings from previous related studies, we propose that this might result from the competitive binding of red and blue light receptors, primarily Phytochrome B (phyB) and Cryptochrome 1 (cry1) for the core photomorphogenic regulator, CONSTITUTIVE PHOTOMORPHOGENIC 1 (COP1). This prediction is validated through an experimental study on Arabidopsis thaliana. Our work proposes a potential molecular mechanism underlying plant growth under different light qualities and ultimately suggests an optimal breeding protocol that takes into account light quality.

From the Belousov-Zhabotinsky reaction to biochemical clocks, traveling waves and cell cycle regulation

In the last 20 years, a growing army of systems biologists has employed quantitative experimental methods and theoretical tools of data analysis and mathematical modeling to unravel the molecular details of biological control systems with novel studies of biochemical clocks, cellular decision-making, and signaling networks in time and space. Few people know that one of the roots of this new paradigm in cell biology can be traced to a serendipitous discovery by an obscure Russian biochemist, Boris Belousov, who was studying the oxidation of citric acid. The story is told here from an historical perspective, tracing its meandering path through glycolytic oscillations, cAMP signaling, and frog egg development. The connections among these diverse themes are drawn out by simple mathematical models (nonlinear differential equations) that share common structures and properties.

Evolution of the repression mechanisms in circadian clocks

BACKGROUND

Circadian (daily) timekeeping is essential to the survival of many organisms. An integral part of all circadian timekeeping systems is negative feedback between an activator and repressor. However, the role of this feedback varies widely between lower and higher organisms.

RESULTS

Here, we study repression mechanisms in the cyanobacterial and eukaryotic clocks through mathematical modeling and systems analysis. We find a common mathematical model that describes the mechanism by which organisms generate rhythms; however, transcription's role in this has diverged. In cyanobacteria, protein sequestration and phosphorylation generate and regulate rhythms while transcription regulation keeps proteins in proper stoichiometric balance. Based on recent experimental work, we propose a repressor phospholock mechanism that models the negative feedback through transcription in clocks of higher organisms. Interestingly, this model, when coupled with activator phosphorylation, allows for oscillations over a wide range of protein stoichiometries, thereby reconciling the negative feedback mechanism in Neurospora with that in mammals and cyanobacteria.

CONCLUSIONS

Taken together, these results paint a picture of how circadian timekeeping may have evolved.

The Goodwin Oscillator and its Legacy

In the 1960's Brian Goodwin published a couple of mathematical models showing how feedback inhibition can lead to oscillations and discussed possible implications of this behaviour for the physiology of the cell. He also presented key ideas about the rich dynamics that may result from the coupling between such biochemical oscillators. Goodwin's work motivated a series of theoretical investigations aiming at identifying minimal mechanisms to generate limit cycle oscillations and deciphering design principles of biological oscillators. The three-variable Goodwin model (adapted by Griffith) can be seen as a core model for a large class of biological systems, ranging from ultradian to circadian clocks. We summarize here main ideas and results brought by Goodwin and review a couple of modeling works directly or indirectly inspired by Goodwin's findings.

A mathematical model of the circadian clock and drug pharmacology to optimize irinotecan administration timing in colorectal cancer

Scheduling anticancer drug administration over 24 h may critically impact treatment success in a patient-specific manner. Here, we address personalization of treatment timing using a novel mathematical model of irinotecan cellular pharmacokinetics and -dynamics linked to a representation of the core clock and predict treatment toxicity in a colorectal cancer (CRC) cellular model. The mathematical model is fitted to three different scenarios: mouse liver, where the drug metabolism mainly occurs, and two human colorectal cancer cell lines representing an in vitro experimental system for human colorectal cancer progression. Our model successfully recapitulates quantitative circadian datasets of mRNA and protein expression together with timing-dependent irinotecan cytotoxicity data. The model also discriminates time-dependent toxicity between the different cells, suggesting that treatment can be optimized according to their cellular clock. Our results show that the time-dependent degradation of the protein mediating irinotecan activation, as well as an oscillation in the death rate may play an important role in the circadian variations of drug toxicity. In the future, this model can be used to support personalized treatment scheduling by predicting optimal drug timing based on the patient's gene expression profile.

Universally valid reduction of multiscale stochastic biochemical systems using simple non-elementary propensities

Biochemical systems consist of numerous elementary reactions governed by the law of mass action. However, experimentally characterizing all the elementary reactions is nearly impossible. Thus, over a century, their deterministic models that typically contain rapid reversible bindings have been simplified with non-elementary reaction functions (e.g., Michaelis-Menten and Morrison equations). Although the non-elementary reaction functions are derived by applying the quasi-steady-state approximation (QSSA) to deterministic systems, they have also been widely used to derive propensities for stochastic simulations due to computational efficiency and simplicity. However, the validity condition for this heuristic approach has not been identified even for the reversible binding between molecules, such as protein-DNA, enzyme-substrate, and receptor-ligand, which is the basis for living cells. Here, we find that the non-elementary propensities based on the deterministic total QSSA can accurately capture the stochastic dynamics of the reversible binding in general. However, serious errors occur when reactant molecules with similar levels tightly bind, unlike deterministic systems. In that case, the non-elementary propensities distort the stochastic dynamics of a bistable switch in the cell cycle and an oscillator in the circadian clock. Accordingly, we derive alternative non-elementary propensities with the stochastic low-state QSSA, developed in this study. This provides a universally valid framework for simplifying multiscale stochastic biochemical systems with rapid reversible bindings, critical for efficient stochastic simulations of cell signaling and gene regulation. To facilitate the framework, we provide a user-friendly open-source computational package, ASSISTER, that automatically performs the present framework.

Inferring causality in biological oscillators

MOTIVATION

Fundamental to biological study is identifying regulatory interactions. The recent surge in time-series data collection in biology provides a unique opportunity to infer regulations computationally. However, when components oscillate, model-free inference methods, while easily implemented, struggle to distinguish periodic synchrony and causality. Alternatively, model-based methods test the reproducibility of time series given a specific model but require inefficient simulations and have limited applicability.

RESULTS

We develop an inference method based on a general model of molecular, neuronal and ecological oscillatory systems that merges the advantages of both model-based and model-free methods, namely accuracy, broad applicability and usability. Our method successfully infers the positive and negative regulations within various oscillatory networks, e.g. the repressilator and a network of cofactors at the pS2 promoter, outperforming popular inference methods.

AVAILABILITY AND IMPLEMENTATION

We provide a computational package, ION (Inferring Oscillatory Networks), that users can easily apply to noisy, oscillatory time series to uncover the mechanisms by which diverse systems generate oscillations. Accompanying MATLAB code under a BSD-style license and examples are available at https://github.com/Mathbiomed/ION. Additionally, the code is available under a CC-BY 4.0 License at https://doi.org/10.6084/m9.figshare.16431408.v1.

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.

Exploiting circadian memory to hasten recovery from circadian misalignment

Recent years have seen a sustained interest in the development of circadian reentrainment strategies to limit the deleterious effects of jet lag. Due to the dynamical complexity of many circadian models, phase-based model reduction techniques are often an imperative first step in the analysis. However, amplitude coordinates that capture lingering effects (i.e., memory) from past inputs are often neglected. In this work, we focus on these amplitude coordinates using an operational phase and an isostable coordinate framework in the context of the development of jet-lag amelioration strategies. By accounting for the influence of circadian memory, we identify a latent phase shift that can prime one's circadian cycle to reentrain more rapidly to an expected time-zone shift. A subsequent optimal control problem is proposed that balances the trade-off between control effort and the resulting latent phase shift. Data-driven model identification techniques for the inference of necessary reduced order, phase-amplitude-based models are considered in situations where the underlying model equations are unknown, and numerical results are illustrated in both a simple planar model and in a coupled population of circadian oscillators.

Targeted modification of the Per2 clock gene alters circadian function in mPer2luciferase (mPer2Luc) mice

Modification of the Per2 clock gene in mPer2Luc reporter mice significantly alters circadian function. Behavioral period in constant dark is lengthened, and dissociates into two distinct components in constant light. Rhythms exhibit increased bimodality, enhanced phase resetting to light pulses, and altered entrainment to scheduled feeding. Mechanistic mathematical modelling predicts that enhanced protein interactions with the modified mPER2 C-terminus, combined with differential clock regulation among SCN subregions, can account for effects on circadian behavior via increased Per2 transcript and protein stability. PER2::LUC produces greater suppression of CLOCK:BMAL1 E-box activity than PER2. mPer2Luc carries a 72 bp deletion in exon 23 of Per2, and retains a neomycin resistance cassette that affects rhythm amplitude but not period. The results show that mPer2Luc acts as a circadian clock mutation illustrating a need for detailed assessment of potential impacts of c-terminal tags in genetically modified animal models.

Collective Oscillations in Coupled-Cell Systems

We investigate oscillations in coupled systems. The methodology is based on the Hopf bifurcation theorem and a condition extended from the Routh–Hurwitz criterion. Such a condition leads to locating the bifurcation values of the parameters. With such an approach, we analyze a single-cell system modeling the minimal genetic negative feedback loop and the coupled-cell system composed by these single-cell systems. We study the oscillatory properties for these systems and compare these properties between the model with Hill-type repression and the one with protein-sequestration-based repression. As the parameters move from the Hopf bifurcation value for single cells to the one for coupled cells, we compute the eigenvalues of the linearized systems to obtain the magnitude of the collective frequency when the periodic solution of the coupled-cell system is generated. Extending from this information on the parameter values, we further compute and compare the collective frequency for the coupled-cell system and the average frequency of the decoupled individual cells. To compare these scenarios with other biological oscillators, we perform parallel analysis and computations on a segmentation clock model.

Na+/Ca2+ exchanger mediates cold Ca2+ signaling conserved for temperature-compensated circadian rhythms

Circadian rhythms are based on biochemical oscillations generated by clock genes/proteins, which independently evolved in animals, fungi, plants, and cyanobacteria. Temperature compensation of the oscillation speed is a common feature of the circadian clocks, but the evolutionary-conserved mechanism has been unclear. Here, we show that Na⁺/Ca²⁺ exchanger (NCX) mediates cold-responsive Ca²⁺ signaling important for the temperature-compensated oscillation in mammalian cells. In response to temperature decrease, NCX elevates intracellular Ca²⁺, which activates Ca²⁺/calmodulin-dependent protein kinase II and accelerates transcriptional oscillations of clock genes. The cold-responsive Ca²⁺ signaling is conserved among mice, Drosophila, and Arabidopsis The mammalian cellular rhythms and Drosophila behavioral rhythms were severely attenuated by NCX inhibition, indicating essential roles of NCX in both temperature compensation and autonomous oscillation. NCX also contributes to the temperature-compensated transcriptional rhythms in cyanobacterial clock. Our results suggest that NCX-mediated Ca²⁺ signaling is a common mechanism underlying temperature-compensated circadian rhythms both in eukaryotes and prokaryotes.

Mathematical modeling of mammalian circadian clocks affecting drug and disease responses

To align with daily environmental changes, most physiological processes in mammals exhibit a time-of-day rhythmicity. This circadian control of physiology is intrinsically driven by a cell-autonomous clock gene network present in almost all cells of the body that drives rhythmic expression of genes that regulate numerous molecular and cellular processes. Accordingly, many aspects of pharmacology and toxicology also oscillate in a time-of-day manner giving rise to diverse effects on pharmacokinetics and pharmacodynamics. Genome-wide studies and mathematical modeling are available tools that have significantly improved our understanding of these nonlinear aspects of physiology and therapeutics. In this manuscript current literature and our prior work on the model-based approaches that have been used to explore circadian genomic systems of mammals are reviewed. Such basic understanding and having an integrative approach may provide new strategies for chronotherapeutic drug treatments and yield new insights for the restoration of the circadian system when altered by diseases.

Complementary phase responses via functional differentiation of dual negative feedback loops

Multiple feedback loops are often found in gene regulations for various cellular functions. In mammalian circadian clocks, oscillations of Period1 (Per1) and Period2 (Per2) expression are caused by interacting negative feedback loops (NFLs) whose protein products with similar molecular functions repress each other. However, Per1 expression peaks earlier than Per2 in the pacemaker tissue, raising the question of whether the peak time difference reflects their different dynamical functions. Here, we address this question by analyzing phase responses of the circadian clock caused by light-induced transcription of both Per1 and Per2 mRNAs. Through mathematical analyses of dual NFLs, we show that phase advance is mainly driven by light inputs to the repressor with an earlier expression peak as Per1, whereas phase delay is driven by the other repressor with a later peak as Per2. Due to the complementary contributions to phase responses, the ratio of light-induced transcription rates between Per1 and Per2 determines the magnitude and direction of phase shifts at each time of day. Specifically, stronger Per1 light induction than Per2 results in a phase response curve (PRC) with a larger phase advance zone than delay zone as observed in rats and hamsters, whereas stronger Per2 induction causes a larger delay zone as observed in mice. Furthermore, the ratio of light-induced transcription rates required for entrainment is determined by the relation between the circadian and light-dark periods. Namely, if the autonomous period of a circadian clock is longer than the light-dark period, a larger light-induced transcription rate of Per1 than Per2 is required for entrainment, and vice versa. In short, the time difference between Per1 and Per2 expression peaks can differentiate their dynamical functions. The resultant complementary contributions to phase responses can determine entrainability of the circadian clock to the light-dark cycle.

A mathematical model of circadian rhythms and dopamine

Background

The superchiasmatic nucleus (SCN) serves as the primary circadian (24hr) clock in mammals and is known to control important physiological functions such as the sleep-wake cycle, hormonal rhythms, and neurotransmitter regulation. Experimental results suggest that some of these functions reciprocally influence circadian rhythms, creating a highly complex network. Among the clock’s downstream products, orphan nuclear receptors REV-ERB and ROR are particularly interesting because they coordinately modulate the core clock circuitry. Recent experimental evidence shows that REV-ERB and ROR are not only crucial for lipid metabolism but are also involved in dopamine (DA) synthesis and degradation, which could have meaningful clinical implications for conditions such as Parkinson’s disease and mood disorders.

Methods

We create a mathematical model consisting of differential equations that express how the circadian variables are influenced by light, how REV-ERB and ROR feedback to the clock, and how REV-ERB, ROR, and BMAL1-CLOCK affect the dopaminergic system. The structure of the model is based on the findings of experimentalists.

Results

We compare our model predictions to experimental data on clock components in different light-dark conditions and in the presence of genetic perturbations. Our model results are consistent with experimental results on REV-ERB and ROR and allow us to predict the circadian variations in tyrosine hydroxylase and monoamine oxidase seen in experiments. By connecting our model to an extant model of dopamine synthesis, release, and reuptake, we are able to predict circadian oscillations in extracellular DA and homovanillic acid that correspond well with experimental observations.

Conclusions

The predictions of the mathematical model are consistent with a wide variety of experimental observations. Our calculations show that the mechanisms proposed by experimentalists by which REV-ERB, ROR, and BMAL1-CLOCK influence the DA system are sufficient to explain the circadian oscillations observed in dopaminergic variables. Our mathematical model can be used for further investigations of the effects of the mammalian circadian clock on the dopaminergic system. The model can also be used to predict how perturbations in the circadian clock disrupt the dopaminergic system and could potentially be used to find drug targets that ameliorate these disruptions.

A Sextuple Knockout Cell Line System to Study the Differential Roles of CRY, PER, and NR1D in the Transcription-Translation Feedback Loop of the Circadian Clock

The transcription-translation feedback loop (TTFL) is the core mechanism of the circadian rhythm. In mammalian cells, CLOCK-BMAL1 proteins activate the downstream genes by binding on the E-box sequence of the clock-controlled genes. Among these gene products, CRY1, CRY2, PER1, PER2, NR1D1, and NR1D2 can regulate the CLOCK-BMAL1-mediated transcription to form the feedback loop. However, the detailed mechanism of the TTFL is unclear because of the complicated inter-regulation of these proteins. Here, we generated a cell line lacking CRY1, CRY2, PER1, PER2, NR1D1, and NR1D2 (Cry/Per/Nr1d_KO) to study TTFL. We compared the Dbp transcription after serum-shock and dexamethasone-shock between Cry/Per/Nr1d_KO cells and cells expressing endogenous CRY (Per/Nr1d_KO) or NR1D (Cry/Per_KO). Furthermore, we found that CRY1-mediated repression of Dbp could persist more than 24 h in the absence of other proteins in the negative limb of the TTFL. Our Cry/Per/Nr1d_KO cells is a suitable system for the studying of differential roles of CRY, PER, and NR1D in the TTFL.

Optimal adjustment of the human circadian clock in the real world

Which suggestions for behavioral modifications, based on mathematical models, are most likely to be followed in the real world? We address this question in the context of human circadian rhythms. Jet lag is a consequence of the misalignment of the body’s internal circadian (~24-hour) clock during an adjustment to a new schedule. Light is the clock’s primary synchronizer. Previous research has used mathematical models to compute light schedules that shift the circadian clock to a new time zone as quickly as possible. How users adjust their behavior when provided with these optimal schedules remains an open question. Here, we report data collected by wearables from more than 100 travelers as they cross time zones using a smartphone app, Entrain. We find that people rarely follow the optimal schedules generated through mathematical modeling entirely, but travelers who better followed the optimal schedules reported more positive moods after their trips. Using the data collected, we improve the optimal schedule predictions to accommodate real-world constraints. We also develop a scheduling algorithm that allows for the computation of approximately optimal schedules "on-the-fly" in response to disruptions. User burnout may not be critically important as long as the first parts of a schedule are followed. These results represent a crucial improvement in making the theoretical results of past work viable for practical use and show how theoretical predictions based on known human physiology can be efficiently used in real-world settings.

Jet lag, a significant problem for travelers and shift workers, occurs when our body’s internal circadian (~24-hour) clock is misaligned with the time of day in the environment. Such circadian misalignment can lead to decreased performance, impaired sleep, and increased risk for severe health conditions, ranging from cancer to cardiovascular disease. Previous work has proposed mathematically optimal schedules, based on mathematical models of the human circadian pacemaker, to overcome jet lag in minimal time. Here, we use data collected from over 100 travelers by a mobile app to track when users followed or deviated from optimal schedules. Better adherence to the schedules yielded better outcomes. We also propose more practical schedules, which can be adjusted to the real-world challenges in overcoming jet lag. Our work sets the stage for changing human behaviors in other domains by computing personalized recommendations from mathematical models.