# Prokaryotic autoregulatory gene network

Chemical Langevin equation for a simple system describing production of a protein that is repressing its own production. The process under consideration is a $4$-dimensional diffusion driven by an $8$-dimensional Wiener process. The stochastic differential equation takes a form:

$\dd X_t = S\left[\theta \circ h(X_t)\right]\dd t + S\odot \gamma(\theta\circ h(X_t)) \dd W_t,$

where $\circ:\RR^d\to \RR^d$ is a component-wise multiplication:

$(\mu \circ \nu)_i = \mu_i\nu_i,\quad i=1,\dots,d,$

the custom operation $\odot:\RR^{d\times d'}\to\RR^{d\times d'}$ is defined via:

$(M\odot \mu)_{i,j} = M_{i,j}\mu_j,\quad i=1,\dots,d;\, j=1,\dots,d',$

the function $\gamma:\RR^d\to \RR^d$ is a component-wise square root:

$(\gamma(\mu))_i=\sqrt{\mu_i},\quad i=1,\dots,d,$

$S$ is the stoichiometry matrix:

$S=\left[ \begin{matrix} 0 & 0 & 1 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 & -2 & 2 & 0 & -1 \\ -1 & 1 & 0 & 0 & 1 & -1 & 0 & 0 \\ -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \right]$

and the function $h$ is given by:

$h(x) = (x_3x_4, K-x_4, x_4, x_1, x_2(x_2-1)/2, x_3, x_1, x_2)^T$

The chemical Langevin equation above has been derived as an approximation to a chemical reaction network

\begin{align*} &\mathcal{R}_1:\texttt{DNA} + \texttt{P}_2\rightarrow\texttt{DNA}\cdot\texttt{P}_2, &\mathcal{R}_2:\texttt{DNA}\cdot\texttt{P}_2\rightarrow\texttt{DNA}+\texttt{P}_2\\ &\mathcal{R}_3:\texttt{DNA}\rightarrow\texttt{DNA}+\texttt{RNA}, &\mathcal{R}_4:\texttt{RNA}\rightarrow\texttt{RNA}+\texttt{P};\\ &\mathcal{R}_5:2\texttt{P}\rightarrow\texttt{P}_2, &\mathcal{R}_6:\texttt{P}_2\rightarrow 2\texttt{P},\\ &\mathcal{R}_7:\texttt{RNA}\rightarrow\emptyset, &\mathcal{R}_8\texttt{P}\rightarrow\emptyset, \end{align*}

with reactant constants given by the vector $\theta$.

Can be imported with

@load_diffusion Prokaryote

#### Example

using DiffusionDefinition
using StaticArrays, Plots

plot(X, Val(:vs_time), label=["RNA" "P" "P₂" "DNA"], size=(800,300))