Toward a detailed computational model for the mammalian circadian clock

Combining lineage correlations and a small molecule inhibitor to detect circadian control of the cell cycle

Chronotherapy offers an exciting possibility for improving cancer treatments by leveraging the influence of the circadian clock on the cell cycle. While several molecular interactions coupling the two oscillators have been identified, whether they lead to emergent control of cellular proliferation remains unclear. Using stochastic simulations, we demonstrate that the established gene networks underlying the two oscillators are sufficient to generate lineage correlations in cell cycle times, as observed in single-cell microscopy data. The interactions also create a 'therapeutic window' between cancer and normal cell proliferation peaks that can be leveraged for chronotherapy. Surprisingly, our model predicts that KL001, a clock inhibitor, minimally affects population growth but significantly alters lineage correlations. Our results suggest that clock control of the cell cycle may not be detectable by measuring changes in population dynamics, but combining measurements of lineage correlations with KL001 treatment may provide a more sensitive approach to detecting the coupling.

Dynamics of phase tumbling and the reentrainment of circadian oscillators

Circadian clocks are comprised of networks of cellular oscillators that synchronize to produce endogenous daily rhythms in gene expression and protein abundance. These clocks have evolved to align the physiology and behavior of organisms to the 24-h environmental cycles arising from Earth's rotation. Rapid travel across time zones causes misalignment between an organism's circadian rhythms and its environment, leading to sleep problems and other jet lag symptoms until the circadian system entrains to the external cycles of the new time zone. Experimental and modeling work has shown that phase tumbling, defined as desynchronizing networks of circadian oscillators prior to an abrupt phase shift of the light-dark cycle, can speed up the process of reentrainment. Here, we use a mathematical model of circadian oscillators and 2-D entrainment maps to analyze the conditions under which phase tumbling has a positive, neutral, or negative effect on reentrainment time. We find that whether or not phase tumbling is beneficial depends on the size of the external phase shift and the location of the perturbed oscillator with respect to the fixed points and invariant manifolds of the entrainment map.

A theoretical systems chronopharmacology approach for COVID-19: Modeling circadian regulation of lung infection and potential precision therapies

The COVID-19 pandemic, caused by SARS-CoV-2, has underscored the urgent need for innovative therapeutic approaches. Recent studies have revealed a complex interplay between the circadian clock and SARS-CoV-2 infection in lung cells, opening new avenues for targeted interventions. This systems pharmacology study investigates this intricate relationship, focusing on the circadian protein BMAL1. BMAL1 plays a dual role in viral dynamics, driving the expression of the viral entry receptor ACE2 while suppressing interferon-stimulated antiviral genes. Its critical position at the host-pathogen interface suggests potential as a therapeutic target, albeit requiring a nuanced approach to avoid disrupting essential circadian regulation. To enable precise tuning of potential interventions, we constructed a computational model integrating the lung cellular clock with viral infection components. We validated this model against literature data to establish a platform for drug administration simulation studies using the REV-ERB agonist SR9009. Our simulations of optimized SR9009 dosing reveal circadian-based strategies that potentially suppress viral infection while minimizing clock disruption. This quantitative framework offers insights into the viral-circadian interface, aiming to guide the development of chronotherapy-based antivirals. More broadly, it underscores the importance of understanding the connections between circadian timing, respiratory viral infections, and therapeutic responses for advancing precision medicine. Such approaches are vital for responding effectively to the rapid spread of coronaviruses like SARS-CoV-2.

Modeling of Jet Lag and Searching for an Optimal Light Treatment

The role of the hierarchical organization of the suprachiasmatic nucleus (SCN) in its functioning, jet lag, and the light treatment of jet lag remains poorly understood. Using the core-shell model, we mimic collective behavior of the core and shell populations of the SCN oscillators in transient states after rapid traveling east and west. The existence of a special region of slow dynamical states of the SCN oscillators can explain phenomena such as the east-west asymmetry of jet lag, instances when entrainment to an advance is via delay shifts, and the dynamics of jet lag recovery time. If jet lag brings the SCN state into this region, it will take a long time to leave it and restore synchronization among oscillators. We show that the population of oscillators in the core responds quickly to a rapid phase shift of the light-dark cycle, in contrast to the shell, which responds slowly. A slow recovery of the synchronization among the shell oscillators in transient states may strongly affect reentrainment in peripheral tissues and behavioral rhythms. We discuss the relationship between molecular, electrical, and behavioral rhythms. We also describe how light pulses affect the SCN and analyze the efficiency of the light treatment in facilitating the adaptation of the SCN to a new time zone. Light pulses of a moderate duration and intensity reduce the recovery time after traveling east, but not west. However, long duration and high intensity of light pulses are more detrimental than beneficial for speeding up reentrainment. The results of the core-shell model are compared with experimental data and other biologically motivated models of the SCN.

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.

Computational modeling establishes mechanotransduction as a potent modulator of the mammalian circadian clock

Mechanotransduction, which is the integration of mechanical signals from the external environment of a cell to changes in intracellular signaling, governs many cellular functions. Recent studies have shown that the mechanical state of the cell is also coupled to the cellular circadian clock. To investigate possible interactions between circadian rhythms and cellular mechanotransduction, we have developed a computational model that integrates the two pathways. We postulated that translocation of the transcriptional regulators MRTF (herein referring to both MRTF-A and MRTF-B), YAP and TAZ (also known as YAP1 and WWTR1, respectively; collectively denoted YAP/TAZ) into the nucleus leads to altered expression of circadian proteins. Simulations from our model predict that lower levels of cytoskeletal activity are associated with longer circadian oscillation periods and higher oscillation amplitudes, which is consistent with recent experimental observations. Furthermore, accumulation of YAP/TAZ and MRTF in the nucleus causes circadian oscillations to decay in our model. These effects hold both at the single-cell level and within a population-level framework. Finally, we investigated the effects of mutations in YAP or lamin A, the latter of which result in a class of diseases known as laminopathies. In silico, oscillations in circadian proteins are substantially weaker in populations of cells with mutations in YAP or lamin A, suggesting that defects in mechanotransduction can disrupt the circadian clock in certain disease states; however, reducing substrate stiffness in the model restores normal oscillatory behavior, suggesting a possible compensatory mechanism. Thus, our study identifies that mechanotransduction could be a potent modulatory cue for cellular clocks and that this crosstalk can be leveraged to rescue the circadian clock in disease states.

Nascent RNA kinetics with complex promoter architecture: Analytic results and parameter inference

Transcription is a stochastic process that involves several downstream operations which make it difficult to model and infer transcription kinetics from mature RNA numbers in individual cell. However, recent advances in single-cell technologies have enabled a more precise measurement of the fluctuations of nascent RNA that closely reflect transcription kinetics. In this paper we introduce a general stochastic model to mimic nascent RNA kinetics with complex promoter architecture. We derive the exact distribution and moments of nascent RNA using queuing theory techniques, which provide valuable insights into the effect of the molecular memory created by the multistep activation and deactivation on the stochastic kinetics of nascent RNA. Moreover, based on the analytical results, we develop a statistical method to infer the promoter memory from stationary nascent RNA distributions. Data analysis of synthetic data and a realistic example, the HIV-1 gene, verifies the validity of this inference method.

Design patterns of biological cells

Design patterns are generalized solutions to frequently recurring problems. They were initially developed by architects and computer scientists to create a higher level of abstraction for their designs. Here, we extend these concepts to cell biology to lend a new perspective on the evolved designs of cells' underlying reaction networks. We present a catalog of 21 design patterns divided into three categories: creational patterns describe processes that build the cell, structural patterns describe the layouts of reaction networks, and behavioral patterns describe reaction network function. Applying this pattern language to the E. coli central metabolic reaction network, the yeast pheromone response signaling network, and other examples lends new insights into these systems.

A critical signal for phenotype transition driven by negative feedback loops

The biological rhythms governed by negative feedback loops have undergone extensive investigation. However, developing reliable and versatile warning signals to predict periodic fluctuations in physiological processes and behaviors associated with these rhythms remains a challenge. Here, we monitored the heart rate and tracked ovulation dates of 91 fertile women. The finding strongly links the velocity (derivative) of heart rate with ovulation in menstrual cycles, providing a predictive warning signal. Similarly, an analysis of calcium signaling in the suprachiasmatic nucleus (SCN) of mice reveals that the maximum velocity of rising calcium signal aligns with locomotor activity offsets. To demonstrate the generality of derivative-transitions link, numerical simulations using a negative feedback loop model were conducted. Statistical analysis indicated that over 90% of the oscillations exhibited a correlation between maximum velocity and transition points. Consequently, the maximum velocity derived from oscillatory curves holds significant potential as an early warning signal for critical transitions.

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.

Stochastic modeling of the mRNA life process: A generalized master equation

The mRNA life cycle is a complex biochemical process, involving transcription initiation, elongation, termination, splicing, and degradation. Each of these molecular events is multistep and can create a memory. The effect of this molecular memory on gene expression is not clear, although there are many related yet scattered experimental reports. To address this important issue, we develop a general theoretical framework formulated as a master equation in the sense of queue theory, which can reduce to multiple previously studied gene models in limiting cases. This framework allows us to interpret experimental observations, extract kinetic parameters from experimental data, and identify how the mRNA kinetics vary under regulatory influences. Notably, it allows us to evaluate the influences of elongation processes on mature RNA distribution; e.g., we find that the non-exponential elongation time can induce the bimodal mRNA expression and there is an optimal elongation noise intensity such that the mature RNA noise achieves the lowest level. In a word, our framework can not only provide insight into complex mRNA life processes but also bridge a dialogue between theoretical studies and experimental data.

Pharmacokinetic Model-Based Control across the Blood-Brain Barrier for Circadian Entrainment

The ability to shift circadian phase in vivo has the potential to offer substantial health benefits. However, the blood-brain barrier prevents the absorption of the majority of large and many small molecules, posing a challenge to neurological pharmaceutical development. Motivated by the presence of the circadian molecule KL001, which is capable of causing phase shifts in a circadian oscillator, we investigated the pharmacokinetics of different neurological pharmaceuticals on the dynamics of circadian phase. Specifically, we developed and validated five different transport models that describe drug concentration profiles of a circadian pharmaceutical at the brain level under oral administration and designed a nonlinear model predictive control (MPC)-based framework for phase resetting. Performance of the novel control algorithm based on the identified pharmacokinetic models was demonstrated through simulations of real-world misalignment scenarios due to jet lag. The time to achieve a complete phase reset for 11-h phase delay ranged between 48 and 72 h, while a 5-h phase advance was compensated in 30 to 60 h. This approach provides mechanistic insight into the underlying structure of the circadian oscillatory system and thus leads to a better understanding of the feasibility of therapeutic manipulations of the system.

A minimal model of peripheral clocks reveals differential circadian re-entrainment in aging

The mammalian circadian system comprises a network of endogenous oscillators, spanning from the central clock in the brain to peripheral clocks in other organs. These clocks are tightly coordinated to orchestrate rhythmic physiological and behavioral functions. Dysregulation of these rhythms is a hallmark of aging, yet it remains unclear how age-related changes lead to more easily disrupted circadian rhythms. Using a two-population model of coupled oscillators that integrates the central clock and the peripheral clocks, we derive simple mean-field equations that can capture many aspects of the rich behavior found in the mammalian circadian system. We focus on three age-associated effects that have been posited to contribute to circadian misalignment: attenuated input from the sympathetic pathway, reduced responsiveness to light, and a decline in the expression of neurotransmitters. We find that the first two factors can significantly impede re-entrainment of the clocks following perturbation, while a weaker coupling within the central clock does not affect the recovery rate. Moreover, using our minimal model, we demonstrate the potential of using the feed-fast cycle as an effective intervention to accelerate circadian re-entrainment. These results highlight the importance of peripheral clocks in regulating the circadian rhythm and provide fresh insights into the complex interplay between aging and the resilience of the circadian system.

Dynamical analysis of the effects of circadian clock on the neurotransmitter dopamine

The circadian clock is an autonomous timing system that regulates the physiological and behavioral activities of organisms. Dopamine (DA) is an important neurotransmitter that is associated with many biological activities such as mood and movement. Experimental studies have shown that the circadian clock influences the DA system and disorders in the circadian clock lead to DA-related diseases. However, the regulatory mechanism of the circadian clock on DA is far from clear. In this paper, we apply an existing circadian-dopamine mathematical model to explore the effects of the circadian clock on DA. Based on numerical simulations, we find the disturbance of the circadian clock, including clock gene mutations, jet lag and light pulses, leads to abnormal DA levels. The effects of mutations in some clock genes on the mood and behavior of mice are closely related to DA disruptions. By sensitivity analysis of DA levels to parameter perturbation, we identify key reactions that affect DA levels, which provides insights into modulating DA disorders. Sudden changes in external light influence the circadian clock, bringing about effects on the DA system. Jet lag causes transient DA rhythm desynchronization with the environment and the influence of jet lag in different directions on DA level and phase varies. Light pulses affect the amplitude and phase shift of DA, which provides a promising method for treating DA disorders through light exposure. This study helps to better understand the impact of the circadian clock on the DA system and provides theoretical support for the treatment of DA disorders.

Synchronization, clustering, and weak chimeras in a densely coupled transcription-based oscillator model for split circadian rhythms

The synchronization dynamics for the circadian gene expression in the suprachiasmatic nucleus is investigated using a transcriptional circadian clock gene oscillator model. With global coupling in constant dark (DD) conditions, the model exhibits a one-cluster phase synchronized state, in dim light (dim LL), bistability between one- and two-cluster states and in bright LL, a two-cluster state. The two-cluster phase synchronized state, where some oscillator pairs synchronize in-phase, and some anti-phase, can explain the splitting of the circadian clock, i.e., generation of two bouts of daily activities with certain species, e.g., with hamsters. The one- and two-cluster states can be reached by transferring the animal from DD or bright LL to dim LL, i.e., the circadian synchrony has a memory effect. The stability of the one- and two-cluster states was interpreted analytically by extracting phase models from the ordinary differential equation models. In a modular network with two strongly coupled oscillator populations with weak intragroup coupling, with appropriate initial conditions, one group is synchronized to the one-cluster state and the other group to the two-cluster state, resulting in a weak-chimera state. Computational modeling suggests that the daily rhythms in sleep-wake depend on light intensity acting on bilateral networks of suprachiasmatic nucleus (SCN) oscillators. Addition of a network heterogeneity (coupling between the left and right SCN) allowed the system to exhibit chimera states. The simulations can guide experiments in the circadian rhythm research to explore the effect of light intensity on the complexities of circadian desynchronization.

Dynamic modelling predicts lactate and IL-1β as interventional targets in metabolic-inflammation-clock regulatory loop in glioma

In an attempt to understand the role of dysregulated circadian rhythm in glioma, our recent findings highlighted the existence of a feed-forward loop between tumour metabolite lactate, pro-inflammatory cytokine IL-1β and circadian CLOCK. To further elucidate the implication of this complex interplay, we developed a mathematical model that quantitatively describes this lactate dehydrogenase A (LDHA)-IL-1β-CLOCK/BMAL1 circuit and predicts potential therapeutic targets. The model was calibrated on quantitative western blotting data in two glioma cell lines in response to either lactate inhibition or IL-1β stimulation. The calibrated model described the experimental data well and most of the parameters were identifiable, thus the model was predictive. Sensitivity analysis identified IL-1β and LDHA as potential intervention points. Mathematical models described here can be useful to understand the complex interrelationship between metabolism, inflammation and circadian rhythm, and in designing effective therapeutic strategies. Our findings underscore the importance of including the circadian clock when developing pharmacological approaches that target aberrant tumour metabolism and inflammation. Insight box  The complex interplay of metabolism-inflammation-circadian rhythm in tumours is not well understood. Our recent findings provided evidence of a feed-forward loop between tumour metabolite lactate, pro-inflammatory cytokine IL-1β and circadian CLOCK/BMAL1 in glioma. To elucidate the implication of this complex interplay, we developed a mathematical model that quantitatively describes this LDHA-IL-1β-CLOCK/BMAL1 circuit and integrates experimental data to predict potential therapeutic targets. The study employed a multi-start optimization strategy and profile likelihood estimations for parameter estimation and assessing identifiability. The simulations are in reasonable agreement with the experimental data. Sensitivity analysis found LDHA and IL-1β as potential therapeutic points. Mathematical models described here can provide insights to understand the complex interrelationship between metabolism, inflammation and circadian rhythm, and in identifying effective therapeutic targets.

Higher-order interactions shape collective dynamics differently in hypergraphs and simplicial complexes

Higher-order networks have emerged as a powerful framework to model complex systems and their collective behavior. Going beyond pairwise interactions, they encode structured relations among arbitrary numbers of units through representations such as simplicial complexes and hypergraphs. So far, the choice between simplicial complexes and hypergraphs has often been motivated by technical convenience. Here, using synchronization as an example, we demonstrate that the effects of higher-order interactions are highly representation-dependent. In particular, higher-order interactions typically enhance synchronization in hypergraphs but have the opposite effect in simplicial complexes. We provide theoretical insight by linking the synchronizability of different hypergraph structures to (generalized) degree heterogeneity and cross-order degree correlation, which in turn influence a wide range of dynamical processes from contagion to diffusion. Our findings reveal the hidden impact of higher-order representations on collective dynamics, highlighting the importance of choosing appropriate representations when studying systems with nonpairwise interactions.

Limits of entrainment of circadian neuronal networks

Circadian rhythmicity lies at the center of various important physiological and behavioral processes in mammals, such as sleep, metabolism, homeostasis, mood changes, and more. Misalignment of intrinsic neuronal oscillations with the external day-night cycle can disrupt such processes and lead to numerous disorders. In this work, we computationally determine the limits of circadian synchronization to external light signals of different frequency, duty cycle, and simulated amplitude. Instead of modeling circadian dynamics with generic oscillator models (e.g., Kuramoto-type), we use a detailed computational neuroscience model, which integrates biomolecular dynamics, neuronal electrophysiology, and network effects. This allows us to investigate the effect of small drug molecules, such as Longdaysin, and connect our results with experimental findings. To combat the high dimensionality of such a detailed model, we employ a matrix-free approach, while our entire algorithmic pipeline enables numerical continuation and construction of bifurcation diagrams using only direct simulation. We, thus, computationally explore the effect of heterogeneity in the circadian neuronal network, as well as the effect of the corrective therapeutic intervention of Longdaysin. Last, we employ unsupervised learning to construct a data-driven embedding space for representing neuronal heterogeneity.

Generation and Disruption of Circadian Rhythms in the Suprachiasmatic Nucleus: A Core-Shell Model

We focus our research on how the core-shell organization controls behavior of the suprachiasmatic nucleus (SCN), how the core and shell are synchronized to the environment, what impact they have on the behavior of the SCN under different lighting conditions, and what mechanisms disrupt synchronization. To this end, we use a reduced Kuramoto model, with parameters inferred from experimental observations and calibrated for mice, and perform a detailed comparison between the model and experimental data under light-dark (LD), dark-dark (DD), and light-light (LL) conditions. The operating limits of free-running and entrained SCN activity under symmetric LD cycles are analyzed, with particular focus on the phenomena of anticipation and dissociation. Results reveal that the core-shell organization of the SCN enables anticipation of future events over circadian cycles. The model predicts the emergence of a second (dissociated) rhythm for large and small LD periods. Our results are in good qualitative and quantitative agreement with experimental observations of circadian dissociation. We further describe SCN activity under LL conditions and show that our model satisfies Aschoff’s first rule, according to which the endogenous free-running circadian period observed under complete darkness will shorten in diurnal animals and lengthen in nocturnal animals under constant light. Our results strongly suggest that the Kuramoto model captures essential features of synchronization and entrainment in the SCN. Moreover, our approach is easily extendible to an arbitrary number of groups, with dynamics described by explicit equations for the group phase and synchronization index. Viewed together, the reduced Kuramoto model presents itself as a useful tool for exploring open problems in the study of circadian rhythms, one that can account for evolving views of the circadian system’s organization, including peripheral clocks and inter-hemispheric interaction, and can be translated to other nocturnal and diurnal animals, including humans.

Principles, mechanisms and functions of entrainment in biological oscillators

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

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.

Data-driven identification of dynamical models using adaptive parameter sets

This paper presents two data-driven model identification techniques for dynamical systems with fixed point attractors. Both strategies implement adaptive parameter update rules to limit truncation errors in the inferred dynamical models. The first strategy can be considered an extension of the dynamic mode decomposition with control (DMDc) algorithm. The second strategy uses a reduced order isostable coordinate basis that captures the behavior of the slowest decaying modes of the Koopman operator. The accuracy and robustness of both model identification algorithms is considered in a simple model with dynamics near a Hopf bifurcation. A more complicated model for nonlinear convective flow past an obstacle is also considered. In these examples, the proposed strategies outperform a collection of other commonly used data-driven model identification algorithms including Koopman model predictive control, Galerkin projection, and DMDc.

Synergies of Multiple Zeitgebers Tune Entrainment

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

Periodic synchronization of isolated network elements facilitates simulating and inferring gene regulatory networks including stochastic molecular kinetics

Background

The temporal progression of many fundamental processes in cells and organisms, including homeostasis, differentiation and development, are governed by gene regulatory networks (GRNs). GRNs balance fluctuations in the output of their genes, which trace back to the stochasticity of molecular interactions. Although highly desirable to understand life processes, predicting the temporal progression of gene products within a GRN is challenging when considering stochastic events such as transcription factor–DNA interactions or protein production and degradation.

Results

We report a method to simulate and infer GRNs including genes and biochemical reactions at molecular detail. In our approach, we consider each network element to be isolated from other elements during small time intervals, after which we synchronize molecule numbers across all network elements. Thereby, the temporal behaviour of network elements is decoupled and can be treated by local stochastic or deterministic solutions. We demonstrate the working principle of this modular approach with a repressive gene cascade comprising four genes. By considering a deterministic time evolution within each time interval for all elements, our method approaches the solution of the system of deterministic differential equations associated with the GRN. By allowing genes to stochastically switch between on and off states or by considering stochastic production of gene outputs, we are able to include increasing levels of stochastic detail and approximate the solution of a Gillespie simulation. Thereby, CaiNet is able to reproduce noise-induced bi-stability and oscillations in dynamically complex GRNs. Notably, our modular approach further allows for a simple consideration of deterministic delays. We further infer relevant regulatory connections and steady-state parameters of a GRN of up to ten genes from steady-state measurements by identifying each gene of the network with a single perceptron in an artificial neuronal network and using a gradient decent method originally designed to train recurrent neural networks. To facilitate setting up GRNs and using our simulation and inference method, we provide a fast computer-aided interactive network simulation environment, CaiNet.

Conclusion

We developed a method to simulate GRNs at molecular detail and to infer the topology and steady-state parameters of GRNs. Our method and associated user-friendly framework CaiNet should prove helpful to analyze or predict the temporal progression of reaction networks or GRNs in cellular and organismic biology. CaiNet is freely available at https://gitlab.com/GebhardtLab/CaiNet.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12859-021-04541-6.

Mathematical Modeling in Circadian Rhythmicity

Circadian clocks are autonomous systems able to oscillate in a self-sustained manner in the absence of external cues, although such Zeitgebers are typically present. At the cellular level, the molecular clockwork consists of a complex network of interlocked feedback loops. This chapter discusses self-sustained circadian oscillators in the context of nonlinear dynamics theory. We suggest basic steps that can help in constructing a mathematical model and introduce how self-sustained generations can be modeled using ordinary differential equations. Moreover, we discuss how coupled oscillators synchronize among themselves or entrain to periodic signals. The development of mathematical models over the last years has helped to understand such complex network systems and to highlight the basic building blocks in which oscillating systems are built upon. We argue that, through theoretical predictions, the use of simple models can guide experimental research and is thus suitable to model biological systems qualitatively.

Uncovering the dynamics of a circadian-dopamine model influenced by the light-dark cycle

The neurotransmitter dopamine (DA) is known to be influenced by the circadian timekeeping system in the mammalian brain. We have previously created a single-cell differential equations model to understand the mechanisms behind circadian rhythms of extracellular DA. In this paper, we investigate the dynamics in our model and study different behaviors such as entrainment to the 24-hour light-dark cycle and robust periodicity versus decoupling, quasiperiodicity, and chaos. Imbalances in DA are often accompanied by disrupted circadian rhythms, such as in Parkinson's disease, hyperactivity, and mood disorders. Our model provides new insights into the links between the circadian clock and DA. We show that the daily rhythmicity of DA can be disrupted by decoupling between interlocked loops of the clock circuitry or by quasiperiodic clock behaviors caused by misalignment with the light-dark cycle. The model can be used to further study how the circadian clock affects the dopaminergic system, and to help develop therapeutic strategies for disrupted DA rhythms. .

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.

Comparison of expression patterns of six canonical clock genes of follicular phase and luteal phase in Small-tailed Han sheep

The circadian rhythm is a biological rhythm that is closely related to the rhythmic expression of a series of clock genes. Results from several studies have indicated that clock genes are associated with the estrous cycle in female animals. Until now, the relationship between estrus cycle transition and clock gene expression in reproductive-axis-related tissues has remained unknown in Small-tailed Han (STH) sheep. This study was conducted to analyze the expression patterns of six canonical clock genes (Clock, BMAL1, Per1, Per2, Cry1, and Cry2) in the follicle phase and luteal phase of STH sheep. We found that all six genes were expressed in the brain, cerebellum, hypothalamus, pituitary, ovary, uterus, and oviduct in follicle and luteal phases. The results indicated that Clock expression was significantly higher in the cerebellum, hypothalamus, and uterus of the luteal phase than that of the follicle phase, whereas BMAL1 expression was significantly higher in the hypothalamus of the luteal phase than that of the follicle phase. Per1 expression was significantly higher in the brain, cerebellum, hypothalamus, and pituitary of the luteal phase than that of the follicle phase, and Per2 expression was significantly higher in the hypothalamus, pituitary, and uterus of the luteal phase than that of the follicle phase. Cry1 expression was significantly higher in the brain, cerebellum, and hypothalamus of the luteal phase than that of the follicle phase, whereas Cry2 expression was significantly higher in the pituitary of the luteal phase than that of the follicle phase. The clock gene expression in all tissues was different between follicle and luteal phases, but all clock gene mRNA levels were found to exhibit higher expression among seven tissues in the luteal phase. Our results suggest that estrous cycles may be associated with clock gene expression in the STH sheep. This is the first study to systematically analyze the expression patterns of clock genes of different estrous cycle in ewes, which could form a basis for further studies to develop the relationship between clock genes and the estrous cycle.

Origins of oscillatory dynamics in the model of reactive oxygen species in the rhizosphere

Oscillatory processes are essential for normal functioning and survival of biological systems, and reactive oxygen species have a prominent role in many of them. A mechanism representing the dynamics of these species in the rhizosphere is analyzed using stoichiometric network analysis with the aim to determine its capabilities to simulate various dynamical states, including oscillations. A detailed analysis has shown that unstable steady states result from four destabilizing feedback cycles, among which the cycle involving hydroquinone, an electron acceptor, and its semi-reduced form is the dominant one responsible for the existence of saddle-node and Andronov-Hopf bifurcations. This requires a higher steady-state concentration for the reduced electron acceptor compared to that of the remaining species, where the level of oxygen steady-state concentration determines whether the Andronov-Hopf or saddle-node bifurcation will occur.

The circadian clock gene circuit controls protein and phosphoprotein rhythms in Arabidopsis thaliana

Twenty-four-hour, circadian rhythms control many eukaryotic mRNA levels, whereas the levels of their more stable proteins are not expected to reflect the RNA rhythms, emphasizing the need to test the circadian regulation of protein abundance and modification. Here we present circadian proteomic and phosphoproteomic time series from Arabidopsis thaliana plants under constant light conditions, estimating that just 0.4% of quantified proteins but a much larger proportion of quantified phospho-sites were rhythmic. Approximately half of the rhythmic phospho-sites were most phosphorylated at subjective dawn, a pattern we term the “phospho-dawn.” Members of the SnRK/CDPK family of protein kinases are candidate regulators. A CCA1-overexpressing line that disables the clock gene circuit lacked most circadian protein phosphorylation. However, the few phospho-sites that fluctuated despite CCA1-overexpression still tended to peak in abundance close to subjective dawn, suggesting that the canonical clock mechanism is necessary for most but perhaps not all protein phosphorylation rhythms. To test the potential functional relevance of our datasets, we conducted phosphomimetic experiments using the bifunctional enzyme fructose-6-phosphate-2-kinase/phosphatase (F2KP), as an example. The rhythmic phosphorylation of diverse protein targets is controlled by the clock gene circuit, implicating posttranslational mechanisms in the transmission of circadian timing information in plants.

•

Circadian (phospho)proteomics time courses of plants with or without functional clock.

•

Most protein abundance/phosphorylation rhythms require a transcriptional oscillator.

•

The majority of rhythmic phosphosites peak around subjective dawn (“phospho-dawn”).

•

A phosphorylated serine of the metabolic enzyme F2KP has functional relevance.

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.

Modelling the functional roles of synaptic and extra-synaptic γ-aminobutyric acid receptor dynamics in circadian timekeeping

In the suprachiasmatic nucleus (SCN), γ-aminobutyric acid (GABA) is a primary neurotransmitter. GABA can signal through two types of GABAA receptor subunits, often referred to as synaptic GABAA (gamma subunit) and extra-synaptic GABAA (delta subunit). To test the functional roles of these distinct GABAA in regulating circadian rhythms, we developed a multicellular SCN model where we could separately compare the effects of manipulating GABA neurotransmitter or receptor dynamics. Our model predicted that blocking GABA signalling modestly increased synchrony among circadian cells, consistent with published SCN pharmacology. Conversely, the model predicted that lowering GABAA receptor density reduced firing rate, circadian cell fraction, amplitude and synchrony among individual neurons. When we tested these predictions, we found that the knockdown of delta GABAA reduced the amplitude and synchrony of clock gene expression among cells in SCN explants. The model further predicted that increasing gamma GABAA densities could enhance synchrony, as opposed to increasing delta GABAA densities. Overall, our model reveals how blocking GABAA receptors can modestly increase synchrony, while increasing the relative density of gamma over delta subunits can dramatically increase synchrony. We hypothesize that increased gamma GABAA density in the winter could underlie the tighter phase relationships among SCN cells.

An effective model of endogenous clocks and external stimuli determining circadian rhythms

Circadian endogenous clocks of eukaryotic organisms are an established and rapidly developing research field. To investigate and simulate in an effective model the effect of external stimuli on such clocks and their components we developed a software framework for download and simulation. The application is useful to understand the different involved effects in a mathematical simple and effective model. This concerns the effects of Zeitgebers, feedback loops and further modifying components. We start from a known mathematical oscillator model, which is based on experimental molecular findings. This is extended with an effective framework that includes the impact of external stimuli on the circadian oscillations including high dose pharmacological treatment. In particular, the external stimuli framework defines a systematic procedure by input-output-interfaces to couple different oscillators. The framework is validated by providing phase response curves and ranges of entrainment. Furthermore, Aschoffs rule is computationally investigated. It is shown how the external stimuli framework can be used to study biological effects like points of singularity or oscillators integrating different signals at once. The mathematical framework and formalism is generic and allows to study in general the effect of external stimuli on oscillators and other biological processes. For an easy replication of each numerical experiment presented in this work and an easy implementation of the framework the corresponding Mathematica files are fully made available. They can be downloaded at the following link: https://www.biozentrum.uni-wuerzburg.de/bioinfo/computing/circadian/ .

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.

Random heterogeneity outperforms design in network synchronization

A widely held assumption on network dynamics is that similar components are more likely to exhibit similar behavior than dissimilar ones and that generic differences among them are necessarily detrimental to synchronization. Here, we show that this assumption does not generally hold in oscillator networks when communication delays are present. We demonstrate, in particular, that random parameter heterogeneity among oscillators can consistently rescue the system from losing synchrony. This finding is supported by electrochemical-oscillator experiments performed on a multielectrode array network. Remarkably, at intermediate levels of heterogeneity, random mismatches are more effective in promoting synchronization than parameter assignments specifically designed to facilitate identical synchronization. Our results suggest that, rather than being eliminated or ignored, intrinsic disorder in technological and biological systems can be harnessed to help maintain coherence required for function.

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 Misalignment and Metabolic Disorders: A Story of Twisted Clocks

Simple Summary

In mammals, many physiological processes follow a 24 h rhythmic pattern. These rhythms are governed by a complex network of circadian clocks, which perceives external time cues (notably light and nutrients) and adjusts the timing of metabolic and physiological functions to allow a proper adaptation of the organism to the daily changes in the environmental conditions. Circadian rhythms originate at the cellular level through a transcriptional–translational regulatory network involving a handful of clock genes. In this review, we show how adverse effects caused by ill-timed feeding or jet lag can lead to a dysregulation of this genetic clockwork, which in turn results in altered metabolic regulation and possibly in diseases. We also show how computational modeling can complement experimental observations to understand the design of the clockwork and the onset of metabolic disorders.

Abstract

Biological clocks are cell-autonomous oscillators that can be entrained by periodic environmental cues. This allows organisms to anticipate predictable daily environmental changes and, thereby, to partition physiological processes into appropriate phases with respect to these changing external conditions. Nowadays our 24/7 society challenges this delicate equilibrium. Indeed, many studies suggest that perturbations such as chronic jet lag, ill-timed eating patterns, or shift work increase the susceptibility to cardiometabolic disorders, diabetes, and cancers. However the underlying mechanisms are still poorly understood. A deeper understanding of this complex, dynamic system requires a global holistic approach for which mathematical modeling can be highly beneficial. In this review, we summarize several experimental works pertaining to the effect of adverse conditions on clock gene expression and on physiology, and we show how computational models can bring interesting insights into the links between circadian misalignment and metabolic 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.

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.

A Computational Model for the Cold Response Pathway in Plants

Understanding the mechanism by which plants respond to cold stress and strengthen their tolerance to low temperatures is an important and challenging task in plant sciences. Experiments have established that the first step in the perception and transduction of the cold stress signal consists of a transient influx of Ca²⁺. This Ca²⁺ influx triggers the activation of a cascade of phosphorylation-dephosphorylation reactions that eventually affects the expression of C-repeat-binding factors (CBFs, notably CBF3), which were shown in many plants to control resistance to cold stress by regulating the expression of cold-regulated (COR) genes. Based on experimental observations mostly made on Arabidopsis thaliana, we build a computational model for the cold response pathway in plants, from the transduction of the cold signal via the transient influx of Ca²⁺ to the activation of the phosphorylation cascade leading to CBF3 expression. We explore the dynamics of this regulatory network by means of numerical simulations and compare the results with experimental observations on the dynamics of the cold response, both for the wild type and for mutants. The simulations show how, in response to cold stress, a brief Ca²⁺ influx, which is over in minutes, is transduced along the successive steps of the network to trigger the expression of cold response genes such as CBF3 within hours. Sometimes, instead of a single Ca²⁺ spike the decrease in temperature brings about a train of high-frequency Ca²⁺ oscillations. The model is applied to both types of Ca²⁺ signaling. We determine the dynamics of the network in response to a series of identical cold stresses, to account for the observation of desensitization and resensitization. The analysis of the model predicts the possibility of an oscillatory expression of CBF3 originating from the negative feedback exerted by ZAT12, a factor itself controlled by CBF3. Finally, we extend the model to incorporate the circadian control of CBF3 expression, to account for the gating of the response to cold stress by the plant circadian clock.

A dual-feedback loop model of the mammalian circadian clock for multi-input control of circadian phase

The molecular circadian clock is driven by interlocked transcriptional-translational feedback loops, producing oscillations in the expressions of genes and proteins to coordinate the timing of biological processes throughout the body. Modeling this system gives insight into the underlying processes driving oscillations in an activator-repressor architecture and allows us to make predictions about how to manipulate these oscillations. The knockdown or upregulation of different cellular components using small molecules can disrupt these rhythms, causing a phase shift, and we aim to determine the dosing of such molecules with a model-based control strategy. Mathematical models allow us to predict the phase response of the circadian clock to these interventions and time them appropriately but only if the model has enough physiological detail to describe these responses while maintaining enough simplicity for online optimization. We build a control-relevant, physiologically-based model of the two main feedback loops of the mammalian molecular clock, which provides sufficient detail to consider multi-input control. Our model captures experimentally observed peak to trough ratios, relative abundances, and phase differences in the model species, and we independently validate this model by showing that the in silico model reproduces much of the behavior that is observed in vitro under genetic knockout conditions. Because our model produces valid phase responses, it can be used in a model predictive control algorithm to determine inputs to shift phase. Our model allows us to consider multi-input control through small molecules that act on both feedback loops, and we find that changes to the parameters of the negative feedback loop are much stronger inputs for shifting phase. The strongest inputs predicted by this model provide targets for new experimental small molecules and suggest that the function of the positive feedback loop is to stabilize the oscillations while linking the circadian system to other clock-controlled processes.

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.

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

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