Implement the enzyme reaction
\[ \begin{align*} S + E {{k_{ {\scriptscriptstyle}1} \atop \longrightarrow} \atop {\longleftarrow \atop k_{{\scriptscriptstyle}-1}}} C \stackrel{k_{2}}{\longrightarrow} P + E. \end{align*} \]
\[ \begin{align} \dot{s} & = k_{{\scriptscriptstyle}-1} \cdot c - k_{{\scriptscriptstyle}1} s \cdot e \\ \dot{e} & = (k_{{\scriptscriptstyle}-1} + k_{{\scriptscriptstyle}2}) \cdot c - k_{{\scriptscriptstyle}1} \cdot s \cdot e \\ \dot{c} & = k_{{\scriptscriptstyle}1} \cdot s \cdot e - (k_{{\scriptscriptstyle}-1} + k_{{\scriptscriptstyle}2}) \cdot c \\ \dot{p} & = k_{{\scriptscriptstyle}2} \cdot c, \end{align} \]
\[ \begin{align} \dot{s} & = - \dot{p} = - \frac{k_{{\scriptscriptstyle}2} \mathrm{e}_{{\scriptscriptstyle}T} \cdot s}{\frac{k_{{\scriptscriptstyle}-1}}{k_{{\scriptscriptstyle}1}} + s} \\ c & = \frac{\mathrm{e}_{{\scriptscriptstyle}T} \cdot s}{\frac{k_{{\scriptscriptstyle}-1}}{k_{{\scriptscriptstyle}1}} + s}. \end{align} \]
\[ \begin{align} \dot{s} & = - \dot{p} = - \frac{k_{{\scriptscriptstyle}2} \mathrm{e}_{{\scriptscriptstyle}T} \cdot s}{\frac{k_{{\scriptscriptstyle}-1}}{k_{{\scriptscriptstyle}1}} + s} \\ c & = \frac{\mathrm{e}_{{\scriptscriptstyle}T} \cdot s}{K_{{\scriptscriptstyle}m} + s} \quad \text{mit} \quad K_{{\scriptscriptstyle}m} = \frac{k_{{\scriptscriptstyle}-1} + k_{{\scriptscriptstyle}2}}{k_{{\scriptscriptstyle}1}} \end{align} \]
Simulate the three dynamic systems with parameters \(k_1 = 0.1\), \(k_{-1} = 0.1\), and \(k_2 = 0.1\). Let the initial concentrations be \(e(0) = 10\), \(c(0) = 0\), and \(s(0) = 10, 20, ..., 100\). Hint: The return value of the ODE function is a list, the first entry being the derivatives. The list may contain more named elements, e.g., to return values of algebraic expressions of the ODE states and parameters.
Take a closer look at the initial time frame of the full system implemented by mass-action kinetics. What do you observe?
Have a look at the dynamics of all implementations in phase space \(s\) vs. \(c\).
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?