Combined Model
Introduction
Genetic design
In this section, we describe the behaviour of the combined model.
Single cell model
Overview
The single cell model is provided here to simulate the combined model.
Reactions
\begin{align*}
&\mathop{\xrightarrow{\hspace{4em}}}_{a_{LacI},K_{A,appLact}}^{\displaystyle\mathop{\downarrow}^{\text{Lact}}} \text{LacI}\\
&\mathop{\xrightarrow{\hspace{4em}}}_{a_{LuxR},K_{A,appLact}}^{\displaystyle\mathop{\downarrow}^{\text{Lact}}} \text{LuxR}\\
&\mathop{\xrightarrow{\hspace{4em}}}_{a_{LuxR},K_{R,LacI}}^{\displaystyle\mathop{\bot}^{\text{LacI}}} \text{LuxR}\\
\text{IPTG} + \text{LacI} &\mathop{\mathop{\xrightarrow{\hspace{4em}}}^{\xleftarrow{\hspace{4em}}}}_{k_{\mathrm{IL}}}^{k_{\mathrm{-IL}}} \text{IL}\\
\varnothing&\mathop{\xrightarrow{\hspace{4em}}}^{a_{\mathrm{LuxR}}} \text{LuxR}\\
\text{AHL} + \text{LuxR} &\mathop{\mathop{\xrightarrow{\hspace{4em}}}^{\xleftarrow{\hspace{4em}}}}_{k_{\mathrm{LuxRAHL}}}^{k_{\mathrm{-LuxRAHL}}} \text{LuxRAHL}\\
&\mathop{\xrightarrow{\hspace{4em}}}_{a_\mathrm{LuxI},K_{\mathrm{a,LuxRAHL}}}^{\displaystyle\mathop{\downarrow}^{\text{LuxRAHL}}} \text{LuxI}\\
&\mathop{\xrightarrow{\hspace{4em}}}_{a_\mathrm{GFP},K_{\mathrm{a,LuxRAHL}}}^{\displaystyle\mathop{\downarrow}^{\text{LuxRAHL}}} \text{GFP}\\
\text{LuxI}&\mathop{\xrightarrow{\hspace{4em}}}^{a_{\mathrm{AHL}}}\text{AHL}+\text{LuxI}\\
\text{LuxR}&\mathop{\xrightarrow{\hspace{4em}}}^{d_{\mathrm{LuxR}}}\varnothing\\
\text{AHL}&\mathop{\xrightarrow{\hspace{4em}}}^{d_{\mathrm{AHL}}}\varnothing\\
\text{LuxRAHL}&\mathop{\xrightarrow{\hspace{4em}}}^{d_{\mathrm{LuxRAHL}}}\varnothing\\
\text{LuxI}&\mathop{\xrightarrow{\hspace{4em}}}^{d_{\mathrm{LuxI}}}\varnothing\\
\text{Aiia}+\text{AHL}&\mathop{\xrightarrow{\hspace{4em}}}^{K_{\mathrm{M}},v_{\mathrm{Aiia}}}\text{Aiia}\\
\end{align*}
Equations
\begin{align*}
\frac{d[LacI]}{dt}&=\frac{a_\mathrm{LacI} \cdot (\frac{[Lact]}{K_\mathrm{A,appLact}})^{n_1}}{1+(\frac{[Lact]}{K_\mathrm{A,appLact}})^{n_1}}-d_{\mathrm{LacI}}[LacI]\\
\frac{d[LuxR]}{dt}&=\frac{a_\mathrm{LuxR} \cdot (\frac{[Lact]}{K_\mathrm{A,appLact}})^{n_1}}{1+(\frac{[Lact]}{K_\mathrm{A,appLact}})^{n_1}} \cdot \frac{1}{1+(\frac{[LacI]}{K_{\mathrm{R,LacI}}\cdot (\gamma_2+1)})^{n_\mathrm{2}}}-d_{\mathrm{LuxR}}[LuxR]\\
[LuxRAHL]&= \frac{[AHL]\cdot [LuxR]}{K_{\mathrm{d,LuxRAHL}}+[AHL]}\\
\frac{d[LuxI]}{dt}&=a_{\mathrm{LuxI}}k_{\mathrm{leaky}}([LuxR]-[LuxRAHL])+\frac{a_{\mathrm{LuxI}}(\frac{[LuxRAHL]}{K_{\mathrm{A,LuxRAHL}}})^2}{1+(\frac{[LuxRAHL]}{K_{\mathrm{A,LuxRAHL}}})^2}-d_{\mathrm{LuxI}}[LuxI]\\
\frac{d[AHL]}{dt}&=a_{\mathrm{AHL}}[LuxI]-d_{\mathrm{AHL}}[AHL]-\frac{v_\mathrm{Aiia}\cdot [AHL]}{K_{\mathrm{M,AiiA}}+[AHL]}\\
\frac{d[GFP]}{dt}&=a_\mathrm{GFP}k_{\mathrm{leaky}}([LuxR]-[LuxRAHL])+\frac{a_\mathrm{GFP}(\frac{[LuxRAHL]}{K_{\mathrm{A,LuxRAHL}}})^2}{1+(\frac{[LuxRAHL]}{K_{\mathrm{A,LuxRAHL}}})^2}-d_{\mathrm{GFP}}[GFP]\\
K_\mathrm{d,LuxRAHL} &= \frac{k_\mathrm{-LuxRAHL}}{k_\mathrm{LuxRAHL}}\\
\gamma_2 &= \frac{IPTG_{tot}}{K_{IL}}
\end{align*}
Simulation
