We start with a simple model of protein production by a gene without feedback from the produced protein.

Suppose that protein A is an activating transcription factor that binds to the promoter \(\text{p}_\text{X}\) of a gene X and forms a complex \(\text{A:p}_\text{X}\). As soon as the complex is formed it stimulates the production of mRNA which is then converted into a translation process in a protein X. The diagram below summarizes the processes together: \[\begin{aligned}\text{A}+\text{p}_\text{X}{\mathop{\rightleftharpoons}\limits_{k_{ - 1}}^{{k_1}}} \text{A:p}_\text{X} &{\mathop{\longrightarrow}\limits_{}^{k_2}} \text{p}_\text{X} + \text{mRNA}\\ \text{mRNA} &{\mathop{ \longrightarrow} \limits_{}^{k_3}} \text{X} + \text{mRNA}\end{aligned}\] The first reaction is very similar to an enzymatic reaction, where the promoter plays the role of the 'enzyme'. Let us simplify the model more and assume that the production of the protein X from mRNA takes place at a fast rater compared to the mRNA production. In other words, \(k_3\) is like the rate constants \(k_1\) and \(k_{-1}\) assumed to be large compared to \(k_2\). Under this assumption, we can summarize the process with new rate constants as \[\text{A}+\text{p}_\text{X}{\mathop{\rightleftharpoons} \limits_{k_{ - 1}}^{{k_1}}} \text{A:p}_\text{X} {\mathop{\longrightarrow} \limits_{}^{k_2}} \text{p}_\text{X} + \text{X}\] Now it looks like an enzymatic protein production with the promoter of the protein-producing gene as enzyme. We can apply Michaelis-Menten kinetics and write down the reaction rate \(r\) as \[r= \frac{V_{\max}\cdot [\text{A}]}{K_m+[\text{A}]}\] where \([\text{A}]\) is the concentration of the protein A, which acts as an activating transcription factor.