Dynamical mechanism of Bmal1/Rev-erbαBmal1/Rev-erbα loop in circadian clock

• The Bmal1/Rev-erbaBmal1/Rev-erba loop in circadian clock is described by the delay differential equation model.
• The sufficient conditions for sustained oscillation of the loop are derived with Hopf bifurcation theory.
• The crucial role of delays in circadian rhythm is explained by the theoretical results.
• The period and amplitude of circadian oscillator are sensitive to the variety of delays.
• The period and amplitude have different sensitivities to the alterations of different degradation rates.

In mammals, the circadian clock is driven by multiple integrated transcriptional feedback loops involving three kinds of central clock-controlled elements (CCEs): E-boxes, D-boxes and ROR-elements. With the aid of CCEs, the concentrations of the active proteins are approximated by the delayed concentrations of mRNAs, which simplifies the circadian system drastically. The regulatory loop composed by BMAL1 and REV-ERB- αα plays important roles in circadian clock. With delay differential equations, we gave a mathematical model of this loop and investigated its dynamical mechanisms. Specially, we theoretically provided the sufficient conditions for sustained oscillation of the loop with Hopf bifurcation theory. The total of delays determines the emergence of oscillators, which explains the crucial roles of delays in circadian clock revealed by biological experiments. Numerically, we studied the amplitude and period against the variations of delays and the degradation rates. The different sensitivities of amplitude and period on these factors provide ideas to adjust the amplitude or period of circadian oscillators.