GAMS code of 6h-phase resetting for mammalian circadian rhythms from Reconciling periodic rhythms of large-scale biological networks by optimal control

Periodic rhythms are ubiquitous phenomena that illuminate the underlying mechanism of cyclic activities in biological systems, which can be represented by cyclic attractors of the related biological network. Disorders of periodic rhythms are detrimental to the natural behaviours of living organisms. Previous studies have shown that the state transition from one to another attractor can be accomplished by regulating external signals. However, most of these studies until now have mainly focused on point attractors while ignoring cyclic ones. The aim of this study is to investigate an approach for reconciling abnormal periodic rhythms, such as diminished circadian amplitude and phase delay, to the regular rhythms of complex biological networks. For this purpose, we formulate and solve a mixed-integer nonlinear dynamic optimization problem simultaneously to identify regulation variables and to determine optimal control strategies for state transition and adjustment of periodic rhythms. Numerical experiments are implemented in three examples including a chaotic system, a mammalian circadian rhythm system and a gastric cancer gene regulatory network. The results show that regulating a small number of biochemical molecules in the network is sufficient to successfully drive the system to the target cyclic attractor by implementing an optimal control strategy.

周期性节律是一类普遍存在的现象，可揭示生物系统循环活动的内在机制，这类节律可通过对应生物网络的循环吸引子（cyclic attractors）进行表征。周期性节律紊乱会损害生物体的自然生理行为。既往研究显示，通过调控外部信号可实现不同吸引子间的状态转换。但截至目前，多数相关研究主要聚焦于点吸引子（point attractors），却忽视了循环吸引子。本研究旨在开发一种可将异常周期性节律（如振幅减弱的昼夜节律与相位延迟）校正至复杂生物网络正常节律的方法。为此，我们构建并求解了一个混合整数非线性动态优化问题，以同时识别调控变量并确定用于状态转换与周期性节律调整的最优控制策略。我们在三个示例中开展了数值实验，涵盖混沌系统、哺乳动物昼夜节律系统以及胃癌基因调控网络（gene regulatory network）。实验结果表明，仅需调控网络中少量生物化学分子，配合最优控制策略即可成功将系统驱动至目标循环吸引子。