Leloup & Goldbeter (2003) Mammalian Model (Version B)

This model is described in PNAS 100 (12): 7051-7056, 2003. This paper contain two similar models, one which includes the Rev-Erb-α gene, and one which does not. This is an implementation of the 19 equation model which does include this gene.

The equations model four genes, Per, Cry, Bmal1 and Rev-Erb-α, which form two interlocked feedback loops.

PER - CRY Loop

This loop consists of the Per and Cry genes. Transcription of both is modelled as a Hill function, with BMAL1 protein as an activator. Translation of both mRNAs is then modelled as a linear process producing cytoplasmic protein. mRNA degradation is modelled in a Michaelis-Menten manner.

Both proteins can then undergo a reversible phosphorylation reaction, modelled in a Michaelis-Menten manner. The phosphorylated forms are then degraded enzymatically, again modelled in a Michaelis-Menten manner. Otherwise, the unphosphorylated proteins combine reversibly to form the PER:CRY complex. This complex can also be phosphorylated and then degraded, or it can move into the nucleus in a reversible linear process. In the nucleus the complex can again be phosphorylated and degraded, or it can combine reversibly with BMAL1 to form an inactive complex. This has the effect of reducing levels of free BMAL1, and so reducing transcription of the PER and CRY genes. Thus a negative feedback loop is formed. The inactive complex is also degraded in a Michaelis-Menten manner.

BMAL1 - REV-ERB-α Loop

Transcription of the Bmal1 gene is modelled as a Hill function, with the REV-ERB-α protein as an inhibitor. Translation is modelled as a linear process producing cytoplasmic BMAL1. mRNA degradation is modelled in a Michaelis-Menten manner.

As above, BMAL1 can undergo a reversible phosphorylation reaction and then enzymatic degradation. Otherwise, unphosphorylated BMAL1  can move into the nucleus in a reversible linear process. In the nucleus it can again be phosphorylated and degraded, or it can combine with the PER:CRY complex to form the inactive complex. Free BMAL1 in the nucleus acts as an activator of the Per, Cry and Rev-Erb-α genes.

Rev-Erb-α transcription is modelled as a Hill function, with  BMAL1 as an activator. Translation is again modelled as a linear process producing cytoplasmic protein, and mRNA degradation is modelled in a Michaelis-Menten manner. REV-ERB-α protein can then be enzymatically degraded in a Michaelis-Menten manner, or it can move into the nucleus in a reversible linear process. In the nucleus, REV-ERB-α can again be degraded, or can also act as a repressor of Bmal1 transcription, so completing a negative feedback loop.

Light input into this model is in the form of an increase in the maximum rate of Per transcription .

Note that in our model we have replaced certain constants used in multiple equations with separate values for each equation. We have added extensions to these parameter names to distinguish them, separated from the original name by the use of an underscore character '_'. The constants affected are n, K_p, K_dp, K_d and  k_dn.