A dual-feedback loop model of the mammalian circadian clock for multi-input control of circadian phase

The molecular circadian clock is driven by interlocked transcriptional-translational feedback loops, producing oscillations in the expressions of genes and proteins to coordinate the timing of biological processes throughout the body. Modeling this system gives insight into the underlying processes driving oscillations in an activator-repressor architecture and allows us to make predictions about how to manipulate these oscillations. The knockdown or upregulation of different cellular components using small molecules can disrupt these rhythms, causing a phase shift, and we aim to determine the dosing of such molecules with a model-based control strategy. Mathematical models allow us to predict the phase response of the circadian clock to these interventions and time them appropriately but only if the model has enough physiological detail to describe these responses while maintaining enough simplicity for online optimization. We build a control-relevant, physiologically-based model of the two main feedback loops of the mammalian molecular clock, which provides sufficient detail to consider multi-input control. Our model captures experimentally observed peak to trough ratios, relative abundances, and phase differences in the model species, and we independently validate this model by showing that the in silico model reproduces much of the behavior that is observed in vitro under genetic knockout conditions. Because our model produces valid phase responses, it can be used in a model predictive control algorithm to determine inputs to shift phase. Our model allows us to consider multi-input control through small molecules that act on both feedback loops, and we find that changes to the parameters of the negative feedback loop are much stronger inputs for shifting phase. The strongest inputs predicted by this model provide targets for new experimental small molecules and suggest that the function of the positive feedback loop is to stabilize the oscillations while linking the circadian system to other clock-controlled processes. Author summary The circadian clock helps to regulate many biological functions, including the sleep-wake cycle, metabolism, the cardiovascular system, and the immune response, so we can promote better health by aligning the internal body clock to the phase of the external environment. At the cellular level, the circadian clock is driven by the interactions of a core set of genes and proteins, presenting inputs to manipulate the clock. Mathematical models allow us to predict which inputs can best be used to shift the internal clock to align with the environment. In this paper, we develop a model, consisting of a system of differential equations, to describe the molecular level components which drive circadian rhythms and test the model to show it captures experimental observations. We then use this model to determine what parametric changes have the strongest resetting effect on the clock, suggesting which mechanisms the development of new experimental small molecules should target to most efficiently shift circadian phase. We demonstrate the efficacy of this approach in a model predictive control simulation. Although we build a model specific to the circadian clock, the techniques we describe for modeling and control of the circadian oscillator can be applied to many oscillating systems.