Activation of transcription To emphasize the contrast with a catalytic effect of a transcription factor we first have another look at this type of reaction. When we skip the intermediate step of the formation of mRNA, we can also write the reaction scheme as follows:

$$\begin{aligned}\text{A}+\text{DNA}&{\mathop{\rightleftharpoons} \limits_{k_{ - 1}}^{k_1}} \text{A:DNA}\\ \text{A:DNA} &{\mathop{\longrightarrow}\limits_{}^{k_2}} \text{DNA} + \text{P}\end{aligned}$$ and the reaction rate $r$ is then given by $$r=\frac{V_{\max}\cdot [\text{A}]}{K_m+[\text{A}]}$$ where $V_{\max}=k_2$ and $[\text{A}]$ is the concentration of the transcription factor A. You can look at this formula for the reaction rate as a scalar multiple of the fraction of bound DNA, because in the model of Michaelis and Menten we have $$\text{fraction of bound DNA}= \frac{[\text{A}]}{K_m+[\text{A}]}$$ The fraction of free DNA is equal to 1 - fraction of bound DNA, and this can be rewritten as $$\text{fraction of free DNA}= \frac{K_m}{K_m+[\text{A}]}$$

Inhibition of transcription In inhibition of protein formation by a transcription factor R can the reaction scheme be written as follows:

$$\begin{aligned}\text{R}+\text{DNA}&{\mathop{\rightleftharpoons }\limits_{k_{-1}}^{k_1}} \text{R:DNA}\\ \text{DNA} &{\mathop{\longrightarrow} \limits_{}^{k_2}} \text{P}+\text{DNA}\end{aligned}$$ Now we can just do a similar derivation of the formula for the reaction rate $r$, but it is easier to realize that now the fraction of free DNA leads to to the formation of the protein product P and that the reaction rate is proportional to the fraction of free DNA. From our earlier formulas for fractions it follows that $$r = \frac{V_{\max} \cdot K_m}{K_m+[\text{R}]}$$ where $[\text{R}]$ is the concentration of the transcription factor R that leads to inhibition.