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Modeling Circadian Clocks: Roles, Advantages, and Limitations Cent. Eur. J. Biol.• 6(5) • 2011 • 712-729 DOI: 10.2478/s11535-011-0062-4 Central European Journal of Biology Modeling circadian clocks: Roles, advantages, and limitations Review Article Didier Gonze1,2,* 1Unité de Chronobiologie Théorique, Service de Chimie Physique, Université Libre de Bruxelles, Campus Plaine, B-1050 Brussels, Belgium 2Laboratoire de Bioinformatique des Génomes et des Réseaux, Faculté des Sciences, Université Libre de Bruxelles Campus Plaine, B-1050 Brussels, Belgium Received 31 January 2011; Accepted 17 June 2011 Abstract: Circadian rhythms are endogenous oscillations characterized by a period of about 24h. They constitute the biological rhythms with the longest period known to be generated at the molecular level. The abundance of genetic information and the complexity of the molecular circuitry make circadian clocks a system of choice for theoretical studies. Many mathematical models have been proposed to understand the molecular regulatory mechanisms that underly these circadian oscillations and to account for their dynamic properties (temperature compensation, entrainment by light dark cycles, phase shifts by light pulses, rhythm splitting, robustness to molecular noise, intercellular synchronization). The roles and advantages of modeling are discussed and illustrated using a variety of selected examples. This survey will lead to the proposal of an integrated view of the circadian system in which various aspects (interlocked feedback loops, inter-cellular coupling, and stochasticity) should be considered together to understand the design and the dynamics of circadian clocks. Some limitations of these models are commented and challenges for the future identified. Keywords: Mathematical modeling • Circadian rhythms • Interlocked feedback loops • Temperature compensation • Molecular noise • Synchronisation © Versita Sp. z o.o. Introduction The major role of circadian rhythms is to coordinate physiological and behavioural processes with natural Did you already try to switch off your alarm clock daily variation, and in particular they allow an organism before going to bed? You will most probably wake up to anticipate daily variations of the environment [5,6]. at the usual time because your body has its own inbuilt This timekeeping mechanism has been shown to play clock. Since life exists on Earth, it is subject to the a crucial role in establishing the direction of migration periodic alternation of day and night and has developed in birds [7] and butterflies [8], to provide an adaptive mechanisms to adapt to this periodic change of the advantage to cyanobacteria [9] and to plants [10], and environment. Virtually all living organisms, ranging from to control the annual flowering time of Arabidopsis [11]. cyanobacteria to insects, plants, and mammals, exhibit In mammals, an obvious function of circadian clocks is daily changes in their behaviour. Remarkably, these to regulate the timing of our daily activity and sleeping rhythms have been demonstrated to persist even in the phases [12] through the regulation of hormone production absence of external clues (i.e. in constant darkness) [1], and metabolism [13]. Disruptions of the circadian clock indicating that they are generated endogenously by living have been related to sleep phase disorders [14], and to organisms. In constant conditions, biological rhythms are diseases such as cancer [15]. characterized by a free-running period of about 24h [2,3]. Genetic advances have demonstrated that circadian Franz Halberg qualified these rhythms as “circadian” oscillations originate at the molecular level through (from latin “circa”, around, and “dies”, day) [4]. the regulation of specific genes called “clock genes” * E-mail: [email protected] 712 D. Gonze [16,17]. Since the discovery of the first clock gene, named In mammals, the central circadian oscillator is per (for period), in Drosophila [18], many additional clock located in the suprachiasmatic nuclei (SCN) of the genes have been identified in several model organisms hypothalamus. Through the regulation of the expression including Neurospora, Drosophila, Arabidopsis and of clock-controlled genes in every SCN neuron, the mammals. Oscillations in the expression of these genes molecular clock controls the timing of physiological are caused by multiple regulatory feedback loops processes such as hormone release, blood pressure, involving a handful of key genes. The more we learn body temperature, or the sleep-wake cycle (Figure 2) [6]. about the molecular circuitry of the cellular circadian The circadian pacemaker is composed of a population clock, the more the picture resembles a complex but of cells which need to be coupled and synchronized carefully assembled system of cogs and gears of a to be able to generate a robust overt rhythm [19,20]. mechanical watch (Figure 1). As for the watch, biologists Feedback from the clock to the input pathways believe that the complexity of the molecular mechanism (sometimes called gating) and from the outputs to the should provide the biological clocks with useful clock make the understanding of the circadian system properties, like ensuring a high precision or a fine tuning even more challenging (Figure 2) [21,22]. of the clock-controlled processes. Although working Because of their involvement at different levels of autonomously, this molecular clock is also under the the organization, circadian rhythms are extensively influence of the environment. In particular it responds to studied by geneticists, molecular biologists, structural light, food, and temperature. biologists, physiologists and... mathematicians. Long Figure 1. Cogs and gears of (A) a mechanical clock and (B) a biological clock. Circadian clocks are composed of mulitple genes and feedback loops which, as the cogs and gears of mechanical clocks, may be required for a high precision in the timing, or a reliable response to external signals. Figure 2. Scheme of the circadian system. Three parts are commonly distinguished in this system: the inputs (environmental signals such as the light, the temperature, or the food), the core oscillator (or pacemaker) and the output (clock-controlled processes such as the blood pressure, hormones release and the sleep-wake cycle). The circadian pacemaker is composed of a population of cells (grey circles), each of them being able to produce circadian oscillations through a molecular mechanism involving a few clock genes and multiple feedback loops. Cells are synchronized through inter-cellular coupling. The complexity of the circadian system results from the integration of multiple inputs by the pacemaker, the tight regulation of various clock-controlled processes, the complexity of the molecular mechanism generating oscillations at the single cell level, the complexity of the cellular network, and the feedbacks from the pacemaker to the input pathways and from the outputs to the pacemaker. 713 Modeling circadian clocks: Roles, advantages, and limitations before the discovery of the first clock genes, modeling In the accompanying paper, I have described approaches had been pursued. Before 1960, there were mathematical models developed in the circadian field already more than 600 modeling papers on circadian [25] and explained how these models are derived clocks [23]. Theoretical models are now widespread and and how they are analyzed. I have shown how to constitute a valuable tool to understand the complexity compute bifurcation diagrams, entrainment, and phase of the circadian organisation. response curves. In the present paper, I will illustrate Modeling and computer simulations can be used how mathematical models have shed light on key to test hypotheses and to make predictions but also questions raised in the circadian field. These examples to explain unintuitive dynamic behaviour. Some roles will also show that understanding the cogs and gears and advantages of modeling are listed in Table 1. As of circadian clocks requires an integrated view of the illustrated by this non-exhaustive list, the roles of circadian system in which various aspects (molecular modeling go much beyond the simple reproduction of circuitry, intercellular coupling, and stochasticity) may experimental data. Building a model implies gathering play complementary and inter-dependent roles. I will and synthesizing multiple and disparate information on also comment on several limitations of the modeling a system of interest, identification of key interactions approach, and identify challenges for the future. and connections between the different components of the system, and discrimination between essential and superfluous components. This information-gathering Key circadian problems and insights work thus provides a complete view of the system and from modeling highlights missing pieces in the puzzle. The scheme is then set up in a mathematical Origin of the 24-hour period framework and analyzed, most often, by numerical With a period of 24 hours, circadian oscillations represent simulations. Computer analyses have the advantage of the biological rhythm with the longest period known to being carried out quickly and without any cost and can lead be generated at the molecular level. The origin of this to predictable hypotheses, to the explanation of unintuitive particularly long period puzzled geneticists. Before the behavior, and can provide answers to questions that actual genetic components of the circadian clock were cannot be addressed easily by experiment. To understand identified, circadian biologists
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