The dynamics of a protein concentration is not only determined by the protein production through gene expression, but also by the degradation of the produced protein. We assume here that the degradation can be modelled as exponential decay with characteristic time $\tau_\text{X}$. We start with protein formation via an activating transcription factor A. Then: $$\begin{aligned} \text{rate of formation of the protein X} &= \frac{V_{\max}\cdot [\text{A}]}{K_m+[\text{A}]}\\ \text{degradation rate of the protein X} &= \frac{[\text{X}]}{\tau_\text{X}} \end{aligned}$$

The differential equation is for the concentration of protein X: $$\begin{aligned}\frac{\dd[\text{X}]}{\dd t}&= \text{formation rate}- \text{degradation rate}\\ &= \frac{V_{\max}\cdot [\text{A}]}{K_m+[\text{A}]}-\frac{[\text{X}]}{\tau_\text{X}}\end{aligned}$$ This is a differential equation of limited exponential growth. We first determine the steady state solution $[\text{X}_{ss}]$ by setting $\frac{\dd[\text{X}]}{\dd t}$ equal to zero: $$\frac{\dd[\text{X}_{ss}]}{\dd t}=\frac{V_{\max}\cdot [\text{A}]}{K_m+[\text{A}]}\cdot \tau_\text{X}$$ The differential equation can be rewritten as $$\frac{\dd[\text{X}]}{\dd t}=\frac{1}{\tau_X}\left([\text{X}_{ss}]-[\text{X}]\right)$$ Then the time course of the concentration is $$[\text{X}]=[\text{X}_{ss}]\cdot\left(1-e^{-\frac{t}{\tau_\text{X}}}\right)$$