Development and validation of computational models for mammalian circadian oscillators

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/ .

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.

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.

Mathematical modeling of circadian rhythms

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

Cancer chronotherapeutics: experimental, theoretical, and clinical aspects

The circadian timing system controls cell cycle, apoptosis, drug bioactivation, and transport and detoxification mechanisms in healthy tissues. As a consequence, the tolerability of cancer chemotherapy varies up to several folds as a function of circadian timing of drug administration in experimental models. Best antitumor efficacy of single-agent or combination chemotherapy usually corresponds to the delivery of anticancer drugs near their respective times of best tolerability. Mathematical models reveal that such coincidence between chronotolerance and chronoefficacy is best explained by differences in the circadian and cell cycle dynamics of host and cancer cells, especially with regard circadian entrainment and cell cycle variability. In the clinic, a large improvement in tolerability was shown in international randomized trials where cancer patients received the same sinusoidal chronotherapy schedule over 24h as compared to constant-rate infusion or wrongly timed chronotherapy. However, sex, genetic background, and lifestyle were found to influence optimal chronotherapy scheduling. These findings support systems biology approaches to cancer chronotherapeutics. They involve the systematic experimental mapping and modeling of chronopharmacology pathways in synchronized cell cultures and their adjustment to mouse models of both sexes and distinct genetic background, as recently shown for irinotecan. Model-based personalized circadian drug delivery aims at jointly improving tolerability and efficacy of anticancer drugs based on the circadian timing system of individual patients, using dedicated circadian biomarker and drug delivery technologies.

A robust two-gene oscillator at the core of Ostreococcus tauri circadian clock

The microscopic green alga Ostreococcus tauri is rapidly emerging as a promising model organism in the green lineage. In particular, recent results by Corellou et al. [Plant Cell 21, 3436 (2009)] and Thommen et al. [PLOS Comput. Biol. 6, e1000990 (2010)] strongly suggest that its circadian clock is a simplified version of Arabidopsis thaliana clock, and that it is architectured so as to be robust to natural daylight fluctuations. In this work, we analyze the time series data from luminescent reporters for the two central clock genes TOC1 and CCA1 and correlate them with microarray data previously analyzed. Our mathematical analysis strongly supports both the existence of a simple two-gene oscillator at the core of Ostreococcus tauri clock and the fact that its dynamics is not affected by light in normal entrainment conditions, a signature of its robustness.

Robustness of circadian clocks to daylight fluctuations: hints from the picoeucaryote Ostreococcus tauri

The development of systemic approaches in biology has put emphasis on identifying genetic modules whose behavior can be modeled accurately so as to gain insight into their structure and function. However, most gene circuits in a cell are under control of external signals and thus, quantitative agreement between experimental data and a mathematical model is difficult. Circadian biology has been one notable exception: quantitative models of the internal clock that orchestrates biological processes over the 24-hour diurnal cycle have been constructed for a few organisms, from cyanobacteria to plants and mammals. In most cases, a complex architecture with interlocked feedback loops has been evidenced. Here we present the first modeling results for the circadian clock of the green unicellular alga Ostreococcus tauri. Two plant-like clock genes have been shown to play a central role in the Ostreococcus clock. We find that their expression time profiles can be accurately reproduced by a minimal model of a two-gene transcriptional feedback loop. Remarkably, best adjustment of data recorded under light/dark alternation is obtained when assuming that the oscillator is not coupled to the diurnal cycle. This suggests that coupling to light is confined to specific time intervals and has no dynamical effect when the oscillator is entrained by the diurnal cycle. This intringuing property may reflect a strategy to minimize the impact of fluctuations in daylight intensity on the core circadian oscillator, a type of perturbation that has been rarely considered when assessing the robustness of circadian clocks.

Circadian clocks keep time of day in many living organisms, allowing them to anticipate environmental changes induced by day/night alternation. They consist of networks of genes and proteins interacting so as to generate biochemical oscillations with a period close to 24 hours. Circadian clocks synchronize to the day/night cycle through the year principally by sensing ambient light. Depending on the weather, the perceived light intensity can display large fluctuations within the day and from day to day, potentially inducing unwanted resetting of the clock. Furthermore, marine organisms such as microalgae are subjected to dramatic changes in light intensities in the water column due to streams and wind. We showed, using mathematical modelling, that the green unicellular marine alga Ostreococcus tauri has evolved a simple but effective strategy to shield the circadian clock from daylight fluctuations by localizing coupling to the light during specific time intervals. In our model, as in experiments, coupling is invisible when the clock is in phase with the day/night cycle but resets the clock when it is out of phase. Such a clock architecture is immune to strong daylight fluctuations.

Development of a two-dimension manifold to represent high dimension mathematical models of the intracellular Mammalian circadian clock

A new focus for mathematical models of the circadian pacemaker involves the encapsulation within the models of detailed biological processes responsible for generating those circadian rhythms. Representing greater biological detail requires more mathematical equations, which pose a greater challenge for the analysis of such systems. Development of a method that retains the predominant dynamics while still providing biologically detailed information is advantageous. Two high-dimension mathematical models of intracellular mammalian circadian pacemakers, Leloup-Goldbeter and Forger-Peskin, with 19 and 73 differential equations, respectively, have been published. The authors projected each of these high-dimension models onto their respective manifold using proper orthogonal functions (POFs) obtained from the empirical decomposition of the model's phase space to obtain a 2-dimension model. The resulting 2-dimension model, represented by 2 differential equations, predicts most of the salient characteristics of a biological clock including approximately 24-h oscillations, entrainment to an LD cycle, phase response curves, and the amplitude recovery dynamics that emerge following amplitude suppression. The manifold representation simplifies the mathematical analysis, since only 2 variables need to be observed and analyzed to understand the behavior of the biological clock. This reduced model derived from a model based on biological variables can be used for the development and analysis of mathematical models of the coupled mammalian oscillators to understand the dynamics of the integrated circadian pacemaker.

Fractal Stochastic Modeling of Spiking Activity in Suprachiasmatic Nucleus Neurons

Individual neurons in the suprachiasmatic nucleus (SCN), the master biological clock in mammals, autonomously produce highly complex patterns of spikes. We have shown that most (approximately 90%) SCN neurons exhibit truly stochastic interspike interval (ISI) patterns. The aim of this study was to understand the stochastic nature of the firing patterns in SCN neurons by analyzing the ISI sequences of 150 SCN neurons in hypothalamic slices. Fractal analysis, using the periodogram, Fano factor, and Allan factor, revealed the presence of a 1/f-type power-law (fractal) behavior in the ISI sequences. This fractal nature was persistent after the application of the GABAA receptor antagonist bicuculline, suggesting that the fractal stochastic activity is an intrinsic property of individual SCN neurons. Based on these physiological findings, we developed a computational model for the stochastic SCN neurons to find that their stochastic spiking activity was best described by a gamma point process whose mean firing rate was modulated by a fractal binomial noise. Taken together, we suggest that SCN neurons generate temporal spiking patterns using the fractal stochastic point process.