Exercise 1: Enzyme dynamics

Implement the enzyme reaction

\[ \begin{align*} S + E {{k_{ {\scriptscriptstyle}1} \atop \longrightarrow} \atop {\longleftarrow \atop k_{{\scriptscriptstyle}-1}}} C \stackrel{k_{2}}{\longrightarrow} P + E. \end{align*} \]

\[ \begin{align} \dot{s} & = k_{{\scriptscriptstyle}-1} \cdot c - k_{{\scriptscriptstyle}1} s \cdot e \\ \dot{e} & = (k_{{\scriptscriptstyle}-1} + k_{{\scriptscriptstyle}2}) \cdot c - k_{{\scriptscriptstyle}1} \cdot s \cdot e \\ \dot{c} & = k_{{\scriptscriptstyle}1} \cdot s \cdot e - (k_{{\scriptscriptstyle}-1} + k_{{\scriptscriptstyle}2}) \cdot c \\ \dot{p} & = k_{{\scriptscriptstyle}2} \cdot c, \end{align} \]

\[ \begin{align} \dot{s} & = - \dot{p} = - \frac{k_{{\scriptscriptstyle}2} \mathrm{e}_{{\scriptscriptstyle}T} \cdot s}{\frac{k_{{\scriptscriptstyle}-1}}{k_{{\scriptscriptstyle}1}} + s} \\ c & = \frac{\mathrm{e}_{{\scriptscriptstyle}T} \cdot s}{\frac{k_{{\scriptscriptstyle}-1}}{k_{{\scriptscriptstyle}1}} + s}. \end{align} \]

\[ \begin{align} \dot{s} & = - \dot{p} = - \frac{k_{{\scriptscriptstyle}2} \mathrm{e}_{{\scriptscriptstyle}T} \cdot s}{\frac{k_{{\scriptscriptstyle}-1}}{k_{{\scriptscriptstyle}1}} + s} \\ c & = \frac{\mathrm{e}_{{\scriptscriptstyle}T} \cdot s}{K_{{\scriptscriptstyle}m} + s} \quad \text{mit} \quad K_{{\scriptscriptstyle}m} = \frac{k_{{\scriptscriptstyle}-1} + k_{{\scriptscriptstyle}2}}{k_{{\scriptscriptstyle}1}} \end{align} \]

Cathedral exercise:

On the historic depiction of the cathedral on a control box in front of the Hof-Apotheke, corner KaJo / Münsterstrasse, one can see that four steps lead to the main portal of the minster. Why is it only one today?