Combined Model

Introduction

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

Combining all of the equations from the two different modules, it yields the following system:

\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

Here, we expect to find the features from the two models: