Dynamical Systems in Biology

Exercise 1: Positive cooperative feedback

Consider a protein $A$ being phosphorylated by a stimulus $S$ with rate $k_1$ and dephosphorylated with rate $k_{-1}$. The phosphorylation is cooperatively enhanced by a positiv feedback with hill coefficient $k=4$ and rate $k_2$:

$$\begin{align} \frac{\mathrm{d}}{\mathrm{d}t} pA= k_{1} \cdot S \cdot A + k_{2} \cdot A \cdot \frac{pA^4}{K_{m}^4 + pA^4} - k_{-1}\cdot pA. \end{align}$$

Further assume that the total amount of protein equals 1 ($pA + A = A_{\mathrm{total}} = 1$):

$$\begin{align*} \frac{\mathrm{d}}{\mathrm{d}t} pA = \left( k_{1} \cdot S + k_{2} \cdot \frac{pA^4}{K_m^4 + pA^4}\right) \cdot (1-pA) - k_{ -1}\cdot pA. \end{align*}$$

The parameters are given by $k_{1} = 0.1$, $k_{-1} = 1$, $k_{2} = 2$ and $K_{m} = 0.3$.

• Implement the system and plot the solution in config-space for $S=1$. Compare the solution with the solution without a positive feedback.

• Stimulate the system with $S\in[0,2]$ starting with $S = 0$, increasing $S$ in small steps until $S = 2$ and then reducing $S$ in small steps until $S = 0$ again. In each step, simulate or compute the steady state of $pA$ for each value of $S$ either by simulating the system for long times or by rootSolve::steady() (see last exercise) and take its value as initial value for the next $S$ value. Start with $pA=0$ for $S=0$.
  • How is the observed phenomenon called in physics?
  • What is its meaning in biological systems?
• Generate a rate balance plot to illustrate the mode of action: plot the build and decay fluxes of $pA$ as a function of $pA$.
  • How can you extract information on the qualitative dynamics, especially the stability of the fixed points, from this plot?

• Evaluate the system for smaller values of $k_2$ and discover another, qualitatively different behavior of the system. How can this behavior be explained?

Cathedral exercise:

What is the story of the most famous waterspout?