Dynamical Systems in Biology

Tutorial 9

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?