The Leloup et al. (1999) Drosophila Model
This model consists of two genes, PERIOD (PER) and TIMELESS (TIM), which negatively regulate themselves. Both genes are modelled in the same way. The rate of transcription is described by a Hill function, with the nuclear PER-TIM complex acting as a repressor. Messenger RNA is degraded in a Michaelis Menten and a linear manner, or is translated in a linear manner.
In order to provide the necessary time delay between translation and repression of transcription, each protein must undergo two phosphorylations before combining reversibly to form the PER-TIM complex. These phosphorylations are reversible, with forward and backward reactions being modelled in a Michaelis Menten manner. The unphosphorylated, monophosphorylated and diphosphorylated forms are all degraded in a linear manner. In addition, the diphosphorylated form is degraded enzymatically in a Michaelis Menten manner.
The PER-TIM complex is also degraded in a linear manner, or is transported into the nucleus in a reversible first-order process, in order to provide negative feedback on the rate of transcription. The Hill coefficient describes the number of repressor molecules which must bind the promotor in order to be effective. Nuclear PER-TIM is also degraded in a linear manner.
Light input to this model is in the form of an increase in the maximum rate of enzymatic degradation of diphosphorylated TIM in response to illumination.
This model is described in J. Biol. Rhythms 14(6):433-48, 1999.
It should be noted that in our model the parameter kd, the rate constant of linear degradation for mRNAs and unbound proteins, has been replaced by a number of equivalent parameters, one for each entity (kdmP, kd0P, kd1P, kd2P, kdmT, kd0T, kd1T, kd2T).