A solvable model of entrainment ranges for the circadian rhythm under light/dark cycles

A continuous model for entrainment of the circadian clock responding to light/dark cycles is analytically studied. The circadian rhythm is entrained to light/dark cycles in a certain range of periods including the 24 hour period. Entrainment ranges vary among organisms; plants show much larger ranges than mammals. To analyze entrainment manners, we exploit a simplified model in which the angular velocity of the circadian rhythm is modulated by a sinusoidal function of phase difference between the circadian rhythm and reference phases. This model contains only one parameter called entrainment strength (K). For light and dark states, we set two reference phases. Using this model, we characterized the entrainment manners of circadian rhythms under non-24 hour light/dark cycles. The conditions of the K value for the entrainment were analytically calculated and a phase diagram showing entrainment/non-entrainment boundaries was drawn. In the diagram we found a region where disordered orbitals emerge. Then, critical values for the entrainment and the disordered irregular motion of the oscillation were derived. Simulation results near a critical value were comparable with the experimental results of the entrainment manners of plant circadian rhythms, suggesting the compatibility of self-oscillation and a strong light/dark response in plants. The diagram clearly represents an overview of the relationship between entrainment strengths and entrainment manners of circadian rhythms in general, and enables us to uniformly compare strengths of periodic stimuli in the environment and degrees of responsiveness for the stimuli among various circadian rhythms.