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The Hill funtion can be derived from statistical mechanics of binding and is often used as an aprroximation for the input function when the production rate is a function. The input function describes how the production rate of a gene depends on the transcription factor concentration.

Hill function for an activator

For an activator the Hill function is an increasing function with concentration of active activator $X^*$. The production rate rises from zero and reaches the maximal producion rate in a sigmoidal shape. \begin{align} f(X^*)&=\frac{\beta_{max} \cdot {X^*}^n}{K^n+{X^*}^n}\\ \\ \beta_{max} &\dots \text{maximal production rate}\\ n &\dots \text{Hill coefficient}\\ K &\dots \text{activation coefficient}\\ \end{align} $\beta_{max}$ is the maximal production rate of the promoter-transcirption factor complex. If the activator concentration $X^*$ equals the activation coefficient $K$, the Hill function reaches the half-maximum point. The value of $K$ is hardwired and a characteristic of the promoter, that depends on the promoter strength as well as on increased affinity through transcription factors.

Hill function for activator with parameter $\beta_{max}=1$, $K=0.5$, $X \in [0,1]$, $n=1,2,4, \infty$

The lager the Hill coefficient $n$ the more step like the Hill function becomes and in the limit $n \rightarrow \infty$ the $f(X^*)$ takes a unit step at $X^*=K$. The Hill coefficient comes from the fact the transcription factors can act as multimeres which leads to cooperative behaviour. Typical values for $n$ are $1-4$.

Hill function for a repressor

For repressors the Hill function decreases with the concentration of active repressor $X^*$. \begin{align} f(X^*)&=\frac{\beta_{max}}{1+({X^*}/K)^n}\\ \\ \beta_{max} &\dots \text{maximal production rate}\\ n &\dots \text{Hill coefficient}\\ K &\dots \text{repression coefficient}\\ \end{align}

Hill function for activator with parameter $\beta_{max}=1$, $K=0.5$, $X \in [0,1]$, $n=1,2,4, \infty$

The more repressor is available, the higher the probability that a repressor binds to the operator site, thus the expression level is more and more repressed with increasing repressor levels. Half-maximal repression occurs, when the concentration of active repressor equals the repression coefficient $X^*=K$.

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Further reading:

• Uri Alon - An Introduction to Systems Biology: Design Principles of Biological Circuits (Link)