Speaker
Description
We investigate a class of chemical reaction networks models associated to a phosphorylation mechanism with three steps. This is an important mechanism in many biological cells. A chemical species, the substrat has three possible configurations: ${\mathcal S}_1$, ${\mathcal S}_2$ and ${\mathcal S}_3$. There are transformations by two types of chemical species (enzymes) ${\mathcal A}$ and ${\mathcal B}$:
$$\begin{aligned} &A{+}S_1\mathrel{\mathop{\rightleftarrows}_{\alpha_{1}^-}^{\alpha_{1}^+}} AS_1\stackrel{\lambda_1}{\rightharpoonup} A{+}S_2 \mathrel{\mathop{\rightleftarrows}_{\alpha_{2}^-}^{\alpha_{2}^+}} AS_2\stackrel{\lambda_2}{\rightharpoonup}A{+}S_3\\ & B{+}S_1\stackrel{\mu_2}{\leftharpoonup} BS_2\mathrel{\mathop{\rightleftarrows}_{\beta_{2}^+}^{\beta_{2}^-}} B{+}S_{2} \stackrel{\mu_1}{\leftharpoonup}BS_3\mathrel{\mathop{\rightleftarrows}_{\beta_{1}^+}^{\beta_{1}^-}} B{+}S_3. \end{aligned}$$ In this work, in a Markovian setting, we assume that the initial total number of copies of substrat is $N$ and that the total number of copies of each type of enzymes is proportional to $N$. We investigate the asymptotic behavior, when $N$ gets large, of the concentrations of the chemical species $(\mathcal{S}{i})$. The dependence of the possible asymptotic regimes on the reaction rates is investigated. It turns out that there are significant differences with deterministic models of the literature. The detailed stability properties of a deterministic dynamical system in ${\mathbb{R}}{+}^{4}$ plays an important role in several of our scaling results.
Joint work with Lucie Laurence (University of Bern)
