Dynamical Systems in Biology

Exercise 1: Enzyme reaction chain

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.

Cathedral exercise:

Why is the cathedral tower at the bottom foursided while being octagonal at the top?