Consider a chain of Michaelis-Menten enzyme reactions:
\[ S \stackrel{E_{1}}{\longrightarrow} S_{1} \stackrel{E_{2}}{\longrightarrow} S_{2} \stackrel{E_{3}}{\longrightarrow} S_{3} \stackrel{E_{4}}{\longrightarrow} P \]
for a constant concentration \(S=1\) and given \(V_\mathrm{max}\) and \(K_{\mathrm{M}}\) values:
\[ \begin{align*} & &V_{{\scriptscriptstyle}\mathrm{max}} &\quad K_{{\scriptscriptstyle}\mathrm{M}} \\ & E_{{\scriptscriptstyle}1} &0.1 &\quad 0.1 \\ & E_{{\scriptscriptstyle}2} &1.0 &\quad 1.0 \\ & E_{{\scriptscriptstyle}3} &1.0 &\quad 0.1 \\ & E_{{\scriptscriptstyle}4} &5.0 &\quad 5.0 \\ \end{align*} \]
Implement the system and plot the solution in config-space.
Determine the steady state concentrations \(S_{{\scriptscriptstyle}1}, S_{{\scriptscriptstyle}2}\) und \(S_{{\scriptscriptstyle}3}\) and the steady state flux \(J\) by simulating the system for long time periods.
Have a look at the function rootSolve::steady() and use it to calculate the steady state concentrations and flux.
Evaluate the steady state concentrations and flux by varying the \(V_{\mathrm{max}}\) values. Use the manipulate package for an interactive visualization and plot the original values along.
Calculate the control coefficients using the numDeriv package. Hint: have a look at outer() for calculating the outer product of two vectors.
Why is the cathedral tower at the bottom foursided while being octagonal at the top?