Mathematical modeling of circadian rhythms

Mathematical modeling of viral infection and the immune response controlled by the circadian clock

Time of day affects how well the immune system responds to viral or bacterial infections. While it is well known that the immune system is regulated by the circadian clock, the dynamic origin of time-of-day-dependent immunity remains unclear. In this paper, we studied the circadian control of immune response upon infection of influenza A virus through mathematical modeling. Dynamic simulation analyses revealed that the time-of-day-dependent immunity was rooted in the relative phase between the circadian clock and the pulse of viral infection. The relative phase, which depends on the time the infection occurs, plays a crucial role in the immune response. It can drive the immune system to one of two distinct bistable states, a high inflammatory state with a higher mortality rate or a safe state characterized by low inflammation. The mechanism we found here also explained why the same species infected by different viruses has different time-of-day-dependent immunities. Further, the time-of-day-dependent immunity was found to be abolished when the immune system was regulated by an impaired circadian clock with decreased oscillation amplitude or without oscillations.

Optimal external forces of the lock-in phenomena for flow past an inclined plate in uniform flow

We theoretically studied the optimal control, frequency lock-in, and phase lock-in phenomena due to the spatially localized periodic forcing in flow past an inclined plate. Although frequency lock-in is evident in many fluid phenomena, especially fluid-structure interactions, not many researchers have investigated it using a theoretical approach based on flow details. We obtained detailed information on the lock-in phenomena to external periodic forcing using phase reduction theory, a mathematical method for extracting the dynamics near the limit cycle. Furthermore, the optimal forces applied to the velocity field were determined under the condition of the minimum forcing energy and maximum lock-in range. The study of uniform periodic forces applied within spatially confined regions led to the conclusion that the effective lock-in position, which includes both the upstream and downstream areas of the plate, depends on the principal frequency of the force. The frequency lock-in range of these forces was analyzed and compared with theoretical predictions.

Beyond microtubules: The cellular environment at the endoplasmic reticulum attracts proteins to the nucleus, enabling nuclear transport

All proteins are translated in the cytoplasm, yet many, including transcription factors, play vital roles in the nucleus. While previous research has concentrated on molecular motors for the transport of these proteins to the nucleus, recent observations reveal perinuclear accumulation even in the absence of an energy source, hinting at alternative mechanisms. Here, we propose that structural properties of the cellular environment, specifically the endoplasmic reticulum (ER), can promote molecular transport to the perinucleus without requiring additional energy expenditure. Specifically, physical interaction between proteins and the ER impedes their diffusion and leads to their accumulation near the nucleus. This result explains why larger proteins, more frequently interacting with the ER membrane, tend to accumulate at the perinucleus. Interestingly, such diffusion in a heterogeneous environment follows Chapman's law rather than the popular Fick's law. Our findings suggest a novel protein transport mechanism arising solely from characteristics of the intracellular environment.

Environmental, social, and behavioral challenges of the human circadian clock in real-life conditions

Urban environments, in which ambient light has become a less-reliable entrainer, are challenging for the biological clock to maintain performance. As a consequence, human circadian rhythms are less robust and more variable among individuals. Assessing the individual phase of entrainment, as well as its plastic shifts in response to disturbances of the physical and social environment, is a way to measure circadian disruption. However, this is still difficult to address in real-life scenarios in which several factors modulate the circadian phase not always in a concerted manner. In this perspective, we present the contribution of two real-life situations, in which the circadian system is challenged by important alterations in entraining signals: 1) a trip to the Antarctic summer (socio-environmental challenge), and 2) dancers trained in morning/night shifts (socio-behavioral challenge). Both natural chronobiological experiments are helpful in exploring the functioning and plasticity of the circadian clock and allow for considering individual characteristics and history.

Predicting neurobehavioral performance of resident physicians in a Randomized Order Safety Trial Evaluating Resident-Physician Schedules (ROSTERS)

OBJECTIVES

Mathematical models of human neurobehavioral performance that include the effects of acute and chronic sleep restriction can be key tools in assessment and comparison of work schedules, allowing quantitative predictions of performance when empirical assessment is impractical.

METHODS

Using such a model, we tested the hypothesis that resident physicians working an extended duration work roster, including 24-28 hours of continuous duty and up to 88 hours per week averaged over 4weeks, would have worse predicted performance than resident physicians working a rapidly cycling work roster intervention designed to reduce the duration of extended shifts. The performance metric used was attentional failures (ie, Psychomotor Vigilance Task lapses). Model input was 169 actual work and sleep schedules. Outcomes were predicted hours per week during work hours spent at moderate (equivalent to 16-20 hours of continuous wakefulness) or high (equivalent to ≥20 hours of continuous wakefulness) performance impairment.

RESULTS

The model predicted that resident physicians working an extended duration work roster would spend significantly more time at moderate impairment (p = .02, effect size=0.2) than those working a rapidly cycling work roster; this difference was most pronounced during the circadian night (p < .001). On both schedules, performance was predicted to decline from weeks 1 + 2 to weeks 3 + 4 (p < .001), but the rate of decline was significantly greater on extended duration work roster (p < .01). Predicted performance impairment was inversely related to prior sleep duration (p < .001).

CONCLUSIONS

These findings demonstrate the utility of a mathematical model to evaluate the predicted performance profile of schedules for resident physicians and others who experience chronic sleep restriction and circadian misalignment.

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.

Non-contact robotic manipulation of floating objects: exploiting emergent limit cycles

The study of non-contact manipulation in water, and the ability to robotically control floating objects has gained recent attention due to wide-ranging potential applications, including the analysis of plastic pollution in the oceans and the optimization of procedures in food processing plants. However, modeling floating object movements can be complex, as their trajectories are influenced by various factors such as the object's shape, size, mass, and the magnitude, frequency, and patterns of water waves. This study proposes an experimental investigation into the emergence ofrobotically controlled limit cycles in the movement of floating objects within a closed environment. The objects' movements are driven by robot fins, and the experiment plan set up involves the use of up to four fins and variable motor parameters. By combining energy quantification of the system with an open-loop pattern generation, it is possible to demonstrate all main water-object interactions within the enclosed environment. A study using dynamic time warping around floating patterns gives insights on possible further studies.

Adaptive Circadian Rhythms for Autonomous and Biologically Inspired Robot Behavior

Biological rhythms are periodic internal variations of living organisms that act as adaptive responses to environmental changes. The human pacemaker is the suprachiasmatic nucleus, a brain region involved in biological functions like homeostasis or emotion. Biological rhythms are ultradian (<24 h), circadian (∼24 h), or infradian (>24 h) depending on their period. Circadian rhythms are the most studied since they regulate daily sleep, emotion, and activity. Ambient and internal stimuli, such as light or activity, influence the timing and the period of biological rhythms, making our bodies adapt to dynamic situations. Nowadays, robots experience unceasing development, assisting us in many tasks. Due to the dynamic conditions of social environments and human-robot interaction, robots exhibiting adaptive behavior have more possibilities to engage users by emulating human social skills. This paper presents a biologically inspired model based on circadian biorhythms for autonomous and adaptive robot behavior. The model uses the Dynamic Circadian Integrated Response Characteristic method to mimic human biology and control artificial biologically inspired functions influencing the robot's decision-making. The robot's clock adapts to light, ambient noise, and user activity, synchronizing the robot's behavior to the ambient conditions. The results show the adaptive response of the model to time shifts and seasonal changes of different ambient stimuli while regulating simulated hormones that are key in sleep/activity timing, stress, and autonomic basal heartbeat control during the day.

Level of constitutively expressed BMAL1 affects the robustness of circadian oscillations

The circadian rhythm is a biological oscillation of physiological activities with a period of approximately 24 h, that is driven by a cell-autonomous oscillator called the circadian clock. The current model of the mammalian circadian clock is based on a transcriptional-translational negative feedback loop in which the protein products of clock genes accumulate in a circadian manner and repress their own transcription. However, several studies have revealed that constitutively expressed clock genes can maintain circadian oscillations. To understand the underlying mechanism, we expressed Bmal1 in Bmal1-disrupted cells using a doxycycline-inducible promoter and monitored Bmal1 and Per2 promoter activity using luciferase reporters. Although the levels of BMAL1 and other clock proteins, REV-ERBα and CLOCK, showed no obvious rhythmicity, robust circadian oscillation in Bmal1 and Per2 promoter activities with the correct phase relationship was observed, which proceeded in a doxycycline-concentration-dependent manner. We applied transient response analysis to the Bmal1 promoter activity in the presence of various doxycycline concentrations. Based on the obtained transfer functions, we suggest that, at least in our experimental system, BMAL1 is not directly involved in the oscillatory process, but modulates the oscillation robustness by regulating basal clock gene promoter activity.

Circadian disruption and cancer- and treatment-related symptoms

Cancer patients experience a number of co-occurring side- and late-effects due to cancer and its treatment including fatigue, sleep difficulties, depressive symptoms, and cognitive impairment. These symptoms can impair quality of life and may persist long after treatment completion. Furthermore, they may exacerbate each other's intensity and development over time. The co-occurrence and interdependent nature of these symptoms suggests a possible shared underlying mechanism. Thus far, hypothesized mechanisms that have been purported to underlie these symptoms include disruptions to the immune and endocrine systems. Recently circadian rhythm disruption has emerged as a related pathophysiological mechanism underlying cancer- and cancer-treatment related symptoms. Circadian rhythms are endogenous biobehavioral cycles lasting approximately 24 hours in humans and generated by the circadian master clock - the hypothalamic suprachiasmatic nucleus. The suprachiasmatic nucleus orchestrates rhythmicity in a wide range of bodily functions including hormone levels, body temperature, immune response, and rest-activity behaviors. In this review, we describe four common approaches to the measurement of circadian rhythms, highlight key research findings on the presence of circadian disruption in cancer patients, and provide a review of the literature on associations between circadian rhythm disruption and cancer- and treatment-related symptoms. Implications for future research and interventions will be discussed.

Unstable eigenvectors and reduced amplitude spaces specifying limit cycles of coupled oscillators with simultaneously diagonalizable matrices: with applications from electric circuits to gene regulation

ABSTRACT

A fascinating phenomenon is the self-organization of coupled systems to a whole. This phenomenon is studied for a particular class of coupled oscillatory systems exhibiting so-called simultaneously diagonalizable matrices. For three exemplary systems, namely, an electric circuit, a coupled system of oscillatory neurons, and a system of coupled oscillatory gene regulatory pathways, eigenvectors and amplitude equations are derived. It is shown that for all three systems, only the unstable eigenvectors and their amplitudes matter for the dynamics of the systems on their respective limit cycle attractors. A general class of coupled second-order dynamical oscillators is presented in which stable limit cycles emerging via Hopf bifurcations are solely specified by appropriately defined unstable eigenvectors and their amplitudes. While the eigenvectors determine the orientation of limit cycles in state spaces, the amplitudes determine the evolution of states along those limit cycles. In doing so, it is shown that the unstable eigenvectors define reduced amplitude spaces in which the relevant long-term dynamics of the systems under consideration takes place. Several generalizations are discussed. First, if stable and unstable system parts exhibit a slow-fast dynamics, the fast variables may be eliminated and approximative descriptions of the emerging limit cycle dynamics in reduced amplitude spaces may be again obtained. Second, the principle of reduced amplitude spaces holds not only for coupled second-order oscillators, but can be applied to coupled third-order and higher order oscillators. Third, the possibility to apply the approach to multifrequency limit cycle attractors and other types of attractors is discussed.

Human Alertness Optimization with a Three-Process Dynamic Model

Circadian rhythm is an important biological process for humans as it modulates a wide range of physiological processes, including body temperature, sleep-wake cycle, and cognitive performance. As the most powerful external stimulus of circadian rhythm, light has been studied as a zeitgeber to regulate the circadian phase and sleep. This paper addresses the human alertness optimization problem, by optimizing light exposure and sleep schedules to relieve fatigue and cognitive impairment, in cases of night-shift workers and subjects with certain mission periods based on dynamics of the circadian rhythm system. A three-process hybrid dynamic model is used for simulating the circadian rhythm and predicting subjective alertness and sleepiness. Based on interindividual difference in sleep type and living habits, we propose a tunable sleep schedule in the alertness optimization problem, which allows the appropriate tuning of sleep and wake times based on sleep propensity. Variational calculus is applied to evaluate the impacts of light and sleep schedules on the alertness and a gradient descent algorithm is proposed to determine the optimal solutions to maximize the alertness level in various cases. Numerical simulation results demonstrate that the cognitive performance during certain periods can be significantly improved by optimizing the light input and tuning sleep/wake times compared to empirical data.

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.

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.

Light-based methods for predicting circadian phase in delayed sleep-wake phase disorder

Methods for predicting circadian phase have been developed for healthy individuals. It is unknown whether these methods generalize to clinical populations, such as delayed sleep-wake phase disorder (DSWPD), where circadian timing is associated with functional outcomes. This study evaluated two methods for predicting dim light melatonin onset (DLMO) in 154 DSWPD patients using ~ 7 days of sleep-wake and light data: a dynamic model and a statistical model. The dynamic model has been validated in healthy individuals under both laboratory and field conditions. The statistical model was developed for this dataset and used a multiple linear regression of light exposure during phase delay/advance portions of the phase response curve, as well as sleep timing and demographic variables. Both models performed comparably well in predicting DLMO. The dynamic model predicted DLMO with root mean square error of 68 min, with predictions accurate to within ± 1 h in 58% of participants and ± 2 h in 95%. The statistical model predicted DLMO with root mean square error of 57 min, with predictions accurate to within ± 1 h in 75% of participants and ± 2 h in 96%. We conclude that circadian phase prediction from light data is a viable technique for improving screening, diagnosis, and treatment of DSWPD.

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.

Circadian rhythms and the HPA axis: A systems view

The circadian timing system comprises a network of time-keeping clocks distributed across a living host whose responsibility is to allocate resources and distribute functions temporally to optimize fitness. The molecular structures generating these rhythms have evolved to accommodate the rotation of the earth in an attempt to primarily match the light/dark periods during the 24-hr day. To maintain synchrony of timing across and within tissues, information from the central clock, located in the suprachiasmatic nucleus, is conveyed using systemic signals. Leading among those signals are endocrine hormones, and while the hypothalamic-pituitary-adrenal axis through the release of glucocorticoids is a major pacesetter. Interestingly, the fundamental units at the molecular and physiological scales that generate local and systemic signals share critical structural properties. These properties enable time-keeping systems to generate rhythmic signals and allow them to adopt specific properties as they interact with each other and the external environment. The purpose of this review is to provide a broad overview of these structures, discuss their functional characteristics, and describe some of their fundamental properties as these related to health and disease. This article is categorized under: Immune System Diseases > Computational Models Immune System Diseases > Biomedical Engineering.

Evolutionary Constraints on Connectivity Patterns in the Mammalian Suprachiasmatic Nucleus

The mammalian suprachiasmatic nucleus (SCN) comprises about 20,000 interconnected oscillatory neurons that create and maintain a robust circadian signal which matches to external light cues. Here, we use an evolutionary game theoretic framework to explore how evolutionary constraints can influence the synchronization of the system under various assumptions on the connection topology, contributing to the understanding of the structure of interneuron connectivity. Our basic model represents the SCN as a network of agents each with two properties-a phase and a flag that determines if it communicates with its neighbors or not. Communication comes at a cost to the agent, but synchronization of phases with its neighbors bears a benefit. Earlier work shows that when we have "all-to-all" connectivity, where every agent potentially communicates with every other agent, there is often a simple trade-off that leads to complete communication and synchronization of the system: the benefit must be greater than twice the cost. This trade-off for all-to-all connectivity gives us a baseline to compare to when looking at other topologies. Using simulations, we compare three plausible topologies to the all-to-all case, finding that convergence to synchronous dynamics occurs in all considered topologies under similar benefit and cost trade-offs. Consequently, sparser, less biologically costly topologies are reasonable evolutionary outcomes for organisms that develop a synchronizable oscillatory network. Our simulations also shed light on constraints imposed by the time scale on which we observe the SCN to arise in mammals. We find two conditions that allow for a synchronizable system to arise in relatively few generations. First, the benefits of connectivity must outweigh the cost of facilitating the connectivity in the network. Second, the game at the core of the model needs to be more cooperative than antagonistic games such as the Prisoner's Dilemma. These results again imply that evolutionary pressure may have driven the system towards sparser topologies, as they are less costly to create and maintain. Last, our simulations indicate that models based on the mutualism game fare the best in uptake of communication and synchronization compared to more antagonistic games such as the Prisoner's Dilemma.

An Optimal Time for Treatment-Predicting Circadian Time by Machine Learning and Mathematical Modelling

Tailoring medical interventions to a particular patient and pathology has been termed personalized medicine. The outcome of cancer treatments is improved when the intervention is timed in accordance with the patient's internal time. Yet, one challenge of personalized medicine is how to consider the biological time of the patient. Prerequisite for this so-called chronotherapy is an accurate characterization of the internal circadian time of the patient. As an alternative to time-consuming measurements in a sleep-laboratory, recent studies in chronobiology predict circadian time by applying machine learning approaches and mathematical modelling to easier accessible observables such as gene expression. Embedding these results into the mathematical dynamics between clock and cancer in mammals, we review the precision of predictions and the potential usage with respect to cancer treatment and discuss whether the patient's internal time and circadian observables, may provide an additional indication for individualized treatment timing. Besides the health improvement, timing treatment may imply financial advantages, by ameliorating side effects of treatments, thus reducing costs. Summarizing the advances of recent years, this review brings together the current clinical standard for measuring biological time, the general assessment of circadian rhythmicity, the usage of rhythmic variables to predict biological time and models of circadian rhythmicity.

The Role of Light Sensitivity and Intrinsic Circadian Period in Predicting Individual Circadian Timing

There is large interindividual variability in circadian timing, which is underestimated by mathematical models of the circadian clock. Interindividual differences in timing have traditionally been modeled by changing the intrinsic circadian period, but recent findings reveal an additional potential source of variability: large interindividual differences in light sensitivity. Using an established model of the human circadian clock with real-world light recordings, we investigated whether changes in light sensitivity parameters or intrinsic circadian period could capture variability in circadian timing between and within individuals. Healthy participants (n = 12, aged 18-26 years) underwent continuous light monitoring for 3 weeks (Actiwatch Spectrum). Salivary dim-light melatonin onset (DLMO) was measured each week. Using the recorded light patterns, a sensitivity analysis for predicted DLMO times was performed, varying 3 model parameters within physiological ranges: (1) a parameter determining the steepness of the dose-response curve to light (p), (2) a parameter determining the shape of the phase-response curve to light (K), and (3) the intrinsic circadian period (tau). These parameters were then fitted to obtain optimal predictions of the three DLMO times for each individual. The sensitivity analysis showed that the range of variation in the average predicted DLMO times across participants was 0.65 h for p, 4.28 h for K, and 3.26 h for tau. The default model predicted the DLMO times with a mean absolute error of 1.02 h, whereas fitting all 3 parameters reduced the mean absolute error to 0.28 h. Fitting the parameters independently, we found mean absolute errors of 0.83 h for p, 0.53 h for K, and 0.42 h for tau. Fitting p and K together reduced the mean absolute error to 0.44 h. Light sensitivity parameters captured similar variability in phase compared with intrinsic circadian period, indicating they are viable targets for individualizing circadian phase predictions. Future prospective work is needed that uses measures of light sensitivity to validate this approach.

Recent advances in modeling sleep: from the clinic to society and disease

In the past few decades, advances in understanding sleep-wake neurophysiology have occurred hand-in-hand with advances in mathematical modeling of sleep and wake. In this review, we summarize recent updates in modeling the timing and durations of sleep and wake, the underlying neurophysiology of sleep and wake, and the application of these models in understanding cognition and disease. Throughout, we highlight the role modeling has played in developing our understanding of sleep and its underlying mechanisms. We present open questions and controversies in the field and propose the utility of individualized models of sleep for precision sleep medicine.

Reconciling periodic rhythms of large-scale biological networks by optimal control

Periodic rhythms are ubiquitous phenomena that illuminate the underlying mechanism of cyclic activities in biological systems, which can be represented by cyclic attractors of the related biological network. Disorders of periodic rhythms are detrimental to the natural behaviours of living organisms. Previous studies have shown that the state transition from one to another attractor can be accomplished by regulating external signals. However, most of these studies until now have mainly focused on point attractors while ignoring cyclic ones. The aim of this study is to investigate an approach for reconciling abnormal periodic rhythms, such as diminished circadian amplitude and phase delay, to the regular rhythms of complex biological networks. For this purpose, we formulate and solve a mixed-integer nonlinear dynamic optimization problem simultaneously to identify regulation variables and to determine optimal control strategies for state transition and adjustment of periodic rhythms. Numerical experiments are implemented in three examples including a chaotic system, a mammalian circadian rhythm system and a gastric cancer gene regulatory network. The results show that regulating a small number of biochemical molecules in the network is sufficient to successfully drive the system to the target cyclic attractor by implementing an optimal control strategy.

Off the Clock: From Circadian Disruption to Metabolic Disease

Circadian timekeeping allows appropriate temporal regulation of an organism’s internal metabolism to anticipate and respond to recurrent daily changes in the environment. Evidence from animal genetic models and from humans under circadian misalignment (such as shift work or jet lag) shows that disruption of circadian rhythms contributes to the development of obesity and metabolic disease. Inappropriate timing of food intake and high-fat feeding also lead to disruptions of the temporal coordination of metabolism and physiology and subsequently promote its pathogenesis. This review illustrates the impact of genetically or environmentally induced molecular clock disruption (at the level of the brain and peripheral tissues) and the interplay between the circadian system and metabolic processes. Here, we discuss some mechanisms responsible for diet-induced circadian desynchrony and consider the impact of nutritional cues in inter-organ communication, with a particular focus on the communication between peripheral organs and brain. Finally, we discuss the relay of environmental information by signal-dependent transcription factors to adjust the timing of gene oscillations. Collectively, a better knowledge of the mechanisms by which the circadian clock function can be compromised will lead to novel preventive and therapeutic strategies for obesity and other metabolic disorders arising from circadian desynchrony.

Sleep Modelling across Physiological Levels

Sleep and circadian rhythms are regulated across multiple functional, spatial and temporal levels: from genes to networks of coupled neurons and glial cells, to large scale brain dynamics and behaviour. The dynamics at each of these levels are complex and the interaction between the levels is even more so, so research have mostly focused on interactions within the levels to understand the underlying mechanisms—the so-called reductionist approach. Mathematical models were developed to test theories of sleep regulation and guide new experiments at each of these levels and have become an integral part of the field. The advantage of modelling, however, is that it allows us to simulate and test the dynamics of complex biological systems and thus provides a tool to investigate the connections between the different levels and study the system as a whole. In this paper I review key models of sleep developed at different physiological levels and discuss the potential for an integrated systems biology approach for sleep regulation across these levels. I also highlight the necessity of building mechanistic connections between models of sleep and circadian rhythms across these levels.