Circadian rhythm is usually represented by the 24-hour biological cycles which are mainly driven by the daily light-dark cycle on earth. Light stimulus is widely used as the control input of the circadian system entrainment. In this paper, we study the light-based minimum-time circadian entrainment problem of mammals, Neurospora, and Drosophila based on mathematical models of clock gene transcription . These models contain 3 or even more nonlinear differential equations , and the optimal light control of the entrainment problem on the full models is hard to solve directly . Two model simplification methods are applied to multi-variable models: (1) the phase response curves (PRC) and the circadian phase dynamics of these models are generated by impulse response simulation; (2) the Principal Orthogonal Decomposition (POD) is performed to reduce these models to ones with merely two variables. The minimum-time entrainment of the circadian phase is performed in a feedback form; the minimum-time entrainment of the second-order model is treated as a two-point boundary valued problem(TPBVP), which is solved by a direct shooting algorithm in terms of the Hamiltonian and Pontryagin Minimum Principle. Eventually, the results in low-order models are used for initializing a gradient descent algorithm , which is introduced in solving the entrainment problem of the full models . In this paper, we present: (1) the application of PRC and direct shooting algorithm on multi-variable nonlinear models; (2) a general process for solving the minimum-time optimal control problem on multi-variable models; (3) the impacts of minimum-time optimal light on clock gene transcription and protein synthesis.
The light-based minimum-time circadian entrainment problem for mammals, Neurospora, and Drosophila is studied based on the mathematical models of their circadian gene regulation . These models contain high order nonlinear differential equations . Two model simplification methods are applied to these high-order models: the phase response curves (PRC) and the Principal Orthogonal Decomposition (POD) . The variational calculus and a gradient descent algorithm are applied for solving the optimal light input in the high-order models. As the results of the gradient descent algorithm rely heavily on the initial guesses, we use the optimal control of the PRC and the simplified model to initialize the gradient descent algorithm . In this paper, we present: (1) the application of PRC and direct shooting algorithm on high-order nonlinear models; (2) a general process for solving the minimum-time optimal control problem on high-order models; (3) the impacts of minimum-time optimal light on circadian gene transcription and protein synthesis.
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