Rhythmic phenomena are considered to involve a mechanism, ubiquitous among organisms populating the earth, for responding to daily and seasonal changes resulting from the planet's rotation and its orbit around the sun [1]. All these periodic responses are recorded in a circadian clock that allows the organism to anticipate rhythmic changes in the environment, thus equipping it with regulatory and adaptive machinery [2]. It is well recognized that circadian rhythms operate at all levels of biological organization, approximating a twenty-four hour period, or more accurately, the alternation between day and night [3]. Although there is a widely accepted view that the normal functions of biological processes are strongly correlated with the genes that control them, the detailed genetic mechanisms by which circadian behavior is generated and mediated are poorly understood [4].
Several studies have identified various so-called circadian clock genes and clock-controlled transcription factors through mutants in animal models [5,6]. These genes have implications for clinical trials: their identification holds great promise for determining optimal times for drug administration based on an individual patient's genetic makeup. It has been suggested that drug administration at the appropriate body time can improve the outcome of pharmacotherapy by maximizing the potency and minimizing the toxicity of the drug [7], whereas drug administration at an inappropriate body time can induce more severe side effects [8]. In practice, body-time-dependent therapy, termed chronotherapy [9], can be optimized via the genes that control expression of the patient's physiological variables during the course of a day.
With the completion of the Human Genome Project, it has been possible to draw a comprehensive picture of the genetic control of the functions of the biological clock and, ultimately, to integrate genetic information into routine clinical therapies for disease treatment and prevention. To achieve this goal, there is a pressing need to develop powerful statistical and computational algorithms for detecting genes or quantitative trait loci that determine circadian rhythms as complex dynamic traits. Unlike many other traits, rhythmic oscillations are generated by complex cellular feedback processes comprising a large number of variables. For this reason, mathematical models and numerical simulations are needed to grasp the molecular mechanisms and functions of biological rhythms fully [10]. These mathematical models have proved useful for investigating the dynamic bases of physiological disorders related to perturbations of biological behavior.
In this article, we will develop a statistical model for genetic mapping of QTL that determine patterns of rhythmic responses, using random samples from a natural population. This model is implemented by the principle of functional mapping [11], a statistical framework for mapping dynamic QTL for the pattern of developmental changes, by considering systems of differential equations for biological clocks. Simulation studies have been performed to investigate the statistical properties of the model.
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