A Slow-Regulation Model for Transcription

Regarding RNA, its dynamics are quite similar to those of the constitutive transcription. RNA molecules are synthesized via a Poisson process with rate 

The synthesis and degradation of RNA molecules is a process which, even in the stationary state, is out of equilibrium. The reason is the unbalance of chemical potentials between the substrates out of which RNA is synthesized and the pool of RNA molecules, and between the RNA pool and the products of RNA degradation. These unbalances further implies the existence of a heat dissipation rate given by 

Finally, it is not hard to prove that the differential equation governing the dynamics of the average count of RNA molecules, N(t) is  

A comparison of the above results with those of the previous section reveals that the sole effect of adding repressors is decreasing the effective rate for RNA synthesis. Other than that, the models for constitutive transcription and for regulated transcription with rapid repressor-promoter interaction behave in the same way. 

In the above section we studied a model that could nicely explain how the expression of a gene can be controlled by modifying the number of regulatory molecules repressors. The problem with this model is that it relies upon an assumption that 

---