An accurate mathematical model of the mammalian circadian clock provides novel insights into the mechanisms that generate 24-h rhythms. A double-negative feedback loop design is proposed for biological clocks whose period needs to be tightly regulated.

Experimental data on the circadian (Ϸ24-h) clock in mammalian cells are vast, diverse, and detailed. Mathematical models are therefore needed to piece these data together and to study overall clock behavior. Previous models have focused on Neurospora or Drosophila or can be converted to a Drosophila model simply by renaming variables. Those models used Hill-type terms for transcription regulation and Michaelis-Menten type or delay terms for posttranslation regulation. Recent mammalian experimental data call into question some of the assumptions in these approaches. Moreover, gene duplication has led to more proteins in the mammalian system than in lower organisms. Here we develop a detailed distinctly mammalian model by using mass action kinetics. Parameters for our model are found from experimental data by using a coordinate search method. The model accurately predicts the phase of entrainment, amplitude of oscillation, and shape of time profiles of clock mRNAs and proteins and is also robust to parameter changes and mutations.mathematical models ͉ eukaryotic transcription regulation ͉ PER ͉ CRY ͉ CLOCK M en and women lead scheduled lives in which the time of day dictates when to eat, sleep, work, or relax. To live this way, our bodies must have internal clocks to know when to be prepared for eating, sleeping, working, or relaxing. Amazingly, many cells in our bodies have such a clock that is coordinated by the cellular clocks within the suprachiasmatic nucleus (SCN) cells of the brain. Each cell's clock is much more complex than some might naively imagine, because it involves many interacting genes and proteins. For this reason, mathematical models are needed for any hope of understanding cellular clocks in a detailed way.At its core, the circadian clock within a cell is a series of biochemical reactions that produce Ϸ24-h oscillations. As long as the details of these reactions were largely unknown, early modeling attempts made simplifying assumptions about the reactions within the clock to keep the number of equations at a minimum. For instance, some authors represented many biochemical processes as simply a delay (1). Most models also assumed low enzyme concentrations (summarized in ref.2). Both of these simplifying assumptions made it easier for a model to oscillate.Recent advances provide a more detailed picture of mammalian cellular circadian clocks than these earlier models represent and even challenge some assumptions made in previous models. There is no pure delay in the clock. Likewise, the key enzyme in the mammalian circadian clock, casein kinase I (CKI), is expressed at a higher concentration than its substrates, the PERIOD proteins, PER1 and PER2 (3). Moreover, with the recent advances in computational speed and software designed to simulate biochemical processes (e.g., BIOSPICE, www.biospice. org), the number of equations in a model does not need to be limited by computational resources. In short, we are now able to identify many of the specific reactions involved in the clock and directly simula...

Neurons in the brain's suprachiasmatic nuclei (SCNs), which control the timing of daily rhythms, are thought to encode time of day by changing their firing frequency, with high rates during the day and lower rates at night. Some SCN neurons express a key clock gene, period 1 (per1). We found that during the day, neurons containing per1 sustain an electrically excited state and do not fire, whereas non-per1 neurons show the previously reported daily variation in firing activity. Using a combined experimental and theoretical approach, we explain how ionic currents lead to the unusual electrophysiological behaviors of per1 cells, which unlike other mammalian brain cells can survive and function at depolarized states.

Biological clocks with a period of Ϸ24 h (circadian) exist in most organisms and time a variety of functions, including sleep-wake cycles, hormone release, bioluminescence, and core body temperature fluctuations. Much of our understanding of the clock mechanism comes from the identification of specific mutations that affect circadian behavior. A widely studied mutation in casein kinase I (CKI), the CKI tau mutant, has been shown to cause a loss of kinase function in vitro, but it has been difficult to reconcile this loss of function with the current model of circadian clock function. Here we show that mathematical modeling predicts the opposite, that the kinase mutant CKI tau increases kinase activity, and we verify this prediction experimentally. CKI tau is a highly specific gain-of-function mutation that increases the in vivo phosphorylation and degradation of the circadian regulators PER1 and PER2. These findings experimentally validate a mathematical modeling approach to a complex biological function, clarify the role of CKI in the clock, and demonstrate that a specific mutation can be both a gain and a loss of function depending on the substrate.kinase ͉ systems biology ͉ phosphorylation ͉ PER ͉ degradation C ircadian rhythms govern key physiologic processes including sleep-wake cycles; glucose, lipid, and drug metabolism; heart rate; stress and growth hormones; and immunity, as well as basic cellular processes such as timing of the cell division cycle (1-6). The disruption of circadian rhythm causes significant physiologic stress, is frequently experienced in jet lag and night-shift work, and has been linked to bipolar disorder (7). Thus, circadian regulation of physiology has important consequences for health. A detailed quantitative model that makes clear, testable, and accurate predictions about the clock and how we may manipulate it can therefore have benefits for human health.Much of our understanding of clock components and their interactions began with the identification of mutations that affect circadian behavior (8, 9). In mammals, the original and most extensively studied circadian rhythm mutation is the semidominant tau, first described in 1988. Hamsters with this mutation show phase-advanced activity and have a circadian period of 20 h when homozygous mutant animals are isolated from time cues (9). This tau mutation has been identified as a missense mutation within the substrate recognition site of casein kinase I (denoted CKI tau ) (10). CKI and the closely related CKI␦ are widely expressed serine-threonine protein kinases implicated in development, circadian rhythms, and DNA metabolism (11). When tested in vitro on multiple substrates, CKI tau was shown to have a much reduced overall catalytic activity (10,12,13). This partial loss-of-function mutation and its phenotype have been difficult to reconcile with our current understanding of the molecular feedback loop that governs timing in mammalian cells (13) and recent empirical observations on clock function (14-16). For example, Dey et al. (1...

Numerous studies have used the classic van der Pol oscillator, which contains a cubic nonlinearity, to model the effect of light on the human circadian pacemaker. Jewett and Kronauer demonstrated that Aschoff's rule could be incorporated into van der Pol type models and used a van der Pol type oscillator with higher order nonlinearities. Kronauer, Forger, and Jewett have proposed a model for light preprocessing, Process L, representing a biochemical process that converts a light signal into an effective drive on the circadian pacemaker. In the paper presented here, the authors use the classic van der Pol oscillator with Process L and Jewett and Kronauer's model of Aschoff's rule to model the human circadian pacemaker. This simpler cubic model predicts the results of a three-pulse human phase response curve experiment and a two-pulse amplitude reduction study with as much, or more, accuracy as the models of Jewett and Kronauer and Kronauer, Forger, and Jewett, which both employ a nonlinearity of degree 7. This suggests that this simpler cubic model should be considered as a potential alternative to other models of the human circadian system currently available.

The authors' previous models have been able to describe accurately the effects of extended (approximately 5 h) bright-light (>4000 lux) stimuli on the phase and amplitude of the human circadian pacemaker, but they are not sufficient to represent the surprising human sensitivity to both brief pulses of bright light and light of more moderate intensities. Therefore, the authors have devised a new model in which a dynamic stimulus processor (Process L) intervenes between the light stimuli and the traditional representation of the circadian pacemaker as a self-sustaining limit-cycle oscillator (Process P). The overall model incorporating Process L and Process P is intended to allow the prediction of phase shifts to photic stimuli of any temporal pattern (extended and brief light episodes) and any light intensity in the photopic range. Two time constants emerge in the Process L model: the characteristic duration for necessary bright-light pulses to achieve their full effect (5-10 min) and the characteristic stimulus-free (dark) interval that can be tolerated without incurring an excessive penalty in phase shifting (30-80 min). The effect of reducing light intensity is incorporated in Process L as an extension of the time necessary for the light pulse to be fully realized (a power-law relation between time and intensity). This new model generates a number of new testable hypotheses, including the surprising prediction that 24-h cycles consisting of 8 h of darkness and 16 h of only approximately 3.5 lux would be capable of entraining a large fraction of the adult population (approximately 45%). Experimental data on the response of the human circadian system to lower light intensities and briefer stimuli are needed to allow for further refinement and validation of the model proposed here.

Circadian (nearly 24-h) clocks are remarkably accurate at timing biological events despite the randomness of their biochemical reactions. Here we examine the causes of their immunity to molecular noise in the context of a detailed stochastic mathematical model of the mammalian circadian clock. This stochastic model is a direct generalization of the deterministic mammalian circadian clock model previously developed. A feature of that model is that it completely specifies all molecular reactions, leaving no ambiguity in the formulation of a stochastic version of the model. With parameters based on experimental data concerning clock protein concentrations within a cell, we find accurate circadian rhythms in our model only when promoter interaction occurs on the time scale of seconds. As the model is scaled up by proportionally increasing the numbers of molecules of all species and the reaction rates with the promoter, the observed variability scales as 1͞n 0.5 , where n is the number of molecules of any species. Our results show that gene duplication increases robustness by providing more promoters with which the transcription factors of the model can interact. Although PER2 mutants were not rhythmic in the deterministic version of this model, they are rhythmic in the stochastic version. molecular noise ͉ Gillespie method ͉ mathematical models ͉ eukaryotic transcription regulation ͉ phosphorylation T he unicellular organism's timing of daily (circadian) biological events like luminescence (1) and O 2 consumption (2) can be attributed to an intracellular clock consisting of oscillating protein feedback loops. Higher organisms can time sleep, hormone release, and other biological processes by means of a population of cells, each of which seems to contain a biochemical clock similar in basic design to that found in unicellular organisms (3). Isolated cellular circadian clocks can time events with an error of less than Ϯ10% of the 24-h circadian day (2, 4) and might be able to time events with an error of Ͻ1% of the circadian day (1). Because chemical reactions in cells involve finite (and often low) numbers of molecules (5), individual molecular interactions can be important. One cause of circadian mistiming is molecular noise (the fact that interactions between individual molecules are stochastic). The accuracy of circadian clocks can be essential for survival, and there is likely a strong evolutionary selection to overcome molecular noise.This article is concerned with the mechanisms by which biochemical circadian clocks maintain high accuracy with low numbers of molecules despite the stochasticity of individual molecular interactions (molecular noise). Much recent attention has focused on model predictions of the accuracy of circadian rhythms in the presence of molecular noise, and the predictions of these models have been somewhat conflicting (6, 7). Although there is a widely accepted scheme for stochastic simulation of molecular processes, the Gillespie method (8), circadian clock models often are not detailed e...

Two independent studies, one of them using a computational approach, identified CHRONO, a gene shown to modulate the activity of circadian transcription factors and alter circadian behavior in mice.