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.

Chemical species

Name	Description
AHL	Signaling protein, Acyl homoserine lactone (30C6-HSL)
LuxR	Regulator protein, that can bind to AHL to form a complex
LuxRAHL	Complex of LuxR and AHL, activates transcription of LuxI
LuxI	Autoinducer synthase
Aiia	AHL-lactonase, N-Acyl Homoserine Lactone Lactonase
Lact	Lactate
LacI	Lac operon repressor, DNA-binding protein, acts as a protein
IPTG	Isopropyl β-D-1-thiogalactopyranoside, prevents LacI from repressing the gene of interest
IL	Dimer formed between LacI and IPTG

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: