This protocol is extracted from research article:
A mathematical model of circadian rhythms and dopamine
Theor Biol Med Model, Feb 17, 2021; DOI: 10.1186/s12976-021-00139-w

To model the production of REV and ROR as a function of BCfr, we rely on data in [12] and [31]. Experimental data [12] suggests that REV and ROR peak at the same time, with REV displaying larger fold changes. We create terms G1 and G2 for the production rates of REV and ROR.

To model the impact of REV and ROR on the clock, we create an intermediate step S, which can be thought of as Bmal1. REV-ERB and ROR compete to bind to the RORE sequence of the Bmal1 promoter, thus we have REV inhibit S and ROR activate S with competition for binding. Ikeda et al. [12] suggest that REV and ROR levels peak at the same time, with REV having a larger impact than ROR during peak levels. When they are not at their peak levels, ROR activates S. We use Eq. (2) to write f(S,REV,εs) as the percentage of free S after REV binding, with dissociation constant εs. Since 1−f(S,REV,εs) is the percentage of S bound to REV, we write the repression term Rs as

so if REV binds more to S, then Rs decreases. Following this idea, we choose to have the activation term As be dependent on the percentage of free S and amount of ROR.

We choose a simple equation for the change in S, by adding a basal production rate β, repression and activation terms Rs and As, and a degradation term proportional to the amount of S. We chose parameter values such that the behavior of S would agree with the findings in [12, 26]. The equations for REV, ROR, and S are given below.

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