Mathematical modeling of circadian rhythms

Abstract

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 K E Y W O R D S biological oscillators, circadian clock, circadian rhythms, dynamic systems, mathematical modeling, statistical modeling BOX 1 CIRCADIAN MODELINGCircadian clocks are present within individual cells, and communication among multiple cells gives rise to emergent properties at the tissue level. In mammals, both the master circadian clock in the suprachiasmatic nucleus and tissuelevel peripheral clocks have major effects at the organism level on numerous key physiological functions, including sleep/wake cycles, metabolism, cardiovascular function, reproduction, immune function, neurobehavioral performance, and mood. Misalignment between the master clock and peripheral clocks within an organism, or misalignment between an organism's clocks and its external environment, has adverse physiological consequences. When circadian interventions are needed to improve physiological function, a multiscale understanding of circadian rhythmicity therefore is essential to accurately manipulate this complex oscillatory system. Mathematical modeling is an essential tool to study and analyze complex physiological systems. It has been used to provide insight into the circadian system at multiple levels (i.e., organism, multi-cellular, cellular, molecular, genetic), to design new experiments, and to manipulate and control the components of the system in silico with specificity that

Sleep Modelling across Physiological Levels

“…There appear to be no models at the molecular level, which is largely due to that fact that the genetic and molecular mechanisms of sleep are not yet studied sufficiently for developing a theory. This is opposed to the modelling of circadian rhythms, where majority of the models are at the molecular level and the genetic machinery of the clock is well established [44][45][46][47][48][49][50][55][56][57][58][59]. The brain parts typically addressed are the thalamocortical system and the sleep regulatory networks in the hypothalamus, brainstem and forebrain.…”

“…Modelling plays an important role in the fields of sleep and circadian research with benefits ranging from advancing our fundamental understanding of the mechanisms to guiding experiments and making predictions about sleep, circadian and alertness dynamics [36,[43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58]. However, similar to experiments, models of sleep and circadian dynamics so far mostly focused on a specific physiological level including behavioural, mean field and cellular activity.…”

“…In the past five decades more than 600 theoretical studies were published in each of these fields. A number of detailed reviews of circadian models have been published [44][45][46][47][48][49][50][55][56][57][58][59], including those focusing on modelling at different physiological levels and highlighting the need for a systems theory approach for understanding the circadian system [49,55,56]. Models of sleep regulation have been reviewed separately.…”

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

“…Models of other mammalian tissues, such as liver [ 180 ], might be particularly interesting for tumor specific modelling of circadian rhythms, given their role in the regulation of drug metabolism and detoxification pathways. Various models also exist for non-mammalian organisms, see for example [ 181 , 182 , 183 ].…”

Entrainment in forced Winfree systems

“…Some of these models are oscillator models [98,142,181,190] and some are data driven models [33]. Asgari-Targhi and Klerman [17] classify the circadian modeling at organism level [46,98,142,194], biochemical and molecular level [66] and multicellular level [49]. At molecular level, the mathematical model emphasizes the feedback loop in the circadian clock.…”