Two competiting walks with absorbing final stages

D. Michel, B. Boutin, P. Ruelle The accuracy of biochemical interactions is ensured by endothermic stepwise kinetics.
Progress in Biophysics and Molecular Biology, Volume 121, Issue 1, May 2016, Pages 35-44.


Consider a virtual beacker constitued of \(500\times 500\) loci, all initally at the stage \(0\). Each of these loci is subject to evolve through a double discrete in time random walk, with forward and backward evolutions, and with absorbing final stages \(n_1\) and \(n_2\) at the end of these two chains. These chains have respective length \(n_1\) and \(n_2\) and are figured just below.
\[ \begin{matrix} n_2 & \leftarrow[]{p_2} & \dots & 2_2 & \leftrightharpoons[1-p_2]{p_2} & 1_2 & \leftrightharpoons[1-p_2]{1/2} & 0 & \rightleftharpoons[1-p_1]{1/2} & 1_1 & \rightleftharpoons[1-p_1]{p_1} & 2_1 & \dots & \rightarrow[]{p_1} & n_1\\ \\ \end{matrix} \]
Loci at the stage \(0\) evolve

• with probability \(1/2\) to the stage \(1_1\),
• with probability \(1/2\) to the stage \(1_2\).

Let be \(j=1\) or \(j=2\). Being given a locus at the stage \(i_j\), (\(1\leq i\leq n_j-1\)),

• with probability \(p_j\), it evolves forward to the stage \(i_j+1\),
• with probability \(1-p_j\), it evolves backward to the stage \(i_j-1\).

Loci at the final absorbing stage \(n_j\) do not evolve anymore and are figured with spots (red ones for the stage \(n_1\) and blue ones for the stage \(n_2\)).


Try yourself !

1. Choose a set of parameters (\(p_1\), \(p_2\), \(n_1\) and \(n_2\)) or one in the predefined list.
2. Click 'Complex formation' to initiate the evolution of the whole beacker.
3. Click 'Competitive binding' to count the repartition of loci through the chains (normalized abscissae).