(383e) Isochron-Based Phase Sensitivity Analysis of Oscillatory Biological Systems

Oscillatory behavior is common in biological systems, such as in cell cycle, neuronal activity, and circadian rhythm. The key properties of these systems, including period and phase, do not directly fit into the traditional framework of sensitivity analysis. This work presents systems theoretic tools for investigating sensitivity of biological oscillatory systems, using the circadian rhythm system as an illustrative example. The circadian rhythm controls the day and night cycle of diverse species, from unimolecular Neurospora to highly multicellular mammals, as an adaptation to the 24-hour earth rotation. The structure of the circadian gene network is remarkably preserved across different species suggesting an evolutionary convergence [1]. The rhythm manifests itself in overt behavior such as the rest-activity cycle (sleep-wake), and controls many hormonal, physiological, and psychomotor performance functions [2]. The core of this rhythm consists of an autonomous genetic oscillator with multiple feedback loops forming a limit cycle which can be entrained by environmental cues, e.g., sunlight. Disruptions to the circadian mechanism can lead to sleep disorders and possibly seasonal affective disorders [3].

Biological systems, including circadian rhythm, are known to exhibit robustness to external and internal disturbances, such as temperature fluctuations (external) [4] and inherent stochastic noise in gene regulation (internal) [5]. Here, robustness constitutes the ability of biological organisms to maintain certain phenotype under uncertainties in the environment. In circadian rhythm, the robustness of the period has been investigated using sensitivity analysis to elucidate the dependence of the states and period on the system parameters [6,7]. Another key characteristic in a circadian rhythm is the (relative) phase of the circadian clock which describes the relative position in the limit cycle. Many circadian disorders arise because of the inability to synchronize the internal phase of the circadian clock to the light-dark phase of the environment. Also, the efficacy of an environmental cue for entrainment depends on the phase at which the cue was administered. An example of such dependence appears in the phase response curve (PRC) which quantifies the phase advance/lag induced by a light pulse treatment [2]. Though arguably of higher importance, there has been little study on the phase sensitivity of circadian rhythms [8]. Note that the period sensitivity is simply the accumulated phase sensitivity over one revolution on the limit cycle.

The tools are based on the phase sensitivity analysis for oscillatory chemical systems developed by Kramer and coworkers [9]. The enabling concept for the analysis is the isochorns, the collection of points that evolve to the same position in the limit cycle, which allows quantification of phase shifts between different limit cycles. A ?master? curve for the analysis of oscillatory systems can be computed based on the isochrons, from which novel utilities are developed to analyze different characteristics of the circadian clock including the PRC and the peak-to-peak sensitivity. This master curve represents the accumulated phase shifts over time due to parameter variations, which can be efficiently computed using (one row of) the adjoint sensitivity equation. The PRC measures the phase shift induced by a light pulse treatment, which can be represented as a square-wave perturbation to the limit cycle. On the other hand, the peak-to-peak sensitivity necessitates a careful consideration of the limit cycle shape sensitivity to the parameter changes. In addition, the phase sensitivity analysis for forced oscillatory systems is developed to characterize the effects of entrainment on the system sensitivity. The developed tools provide an efficient and accurate analysis of key characteristics of circadian rhythm and other biological oscillatory systems. The analysis can be used in model identification as well as in elucidation of the design principles guiding the evolution process.

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