AHL Module

Introduction

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

Reactions

\begin{align*} \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}}}^{\displaystyle\mathop{\downarrow}^{\text{LuxRAHL}}} \text{LuxI}\\ \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

The equations are based on mass action kinetics and basic rate laws such as Hill equation.

Assumptions

1. As LuxR is constitutively produced in this model, we considered that LuxR was constant. We used then the conservation of mass to derive the equations
2. On the other hand, the binding and unbinding of LuxR to AHL is fast compared to the synthesis of LuxI. We used the quasi steady state approximation (QSSA)

Simplified equations

\begin{align*} [LuxRAHL]&= \frac{[AHL]\cdot LuxR_\mathrm{t}}{K_{\mathrm{d-LuxRAHL}}+[AHL]}\\ \frac{d[LuxI]}{dt}&=a_{\mathrm{LuxI}}k_{\mathrm{leaky}}(LuxR_\mathrm{t}-[LuxRAHL])+\frac{a_{\mathrm{LuxI}}\left(\frac{[LuxRAHL]}{K_{\mathrm{LuxRAHL}}}\right)^2}{1+\left(\frac{[LuxRAHL]}{K_{\mathrm{LuxRAHL}}}\right)^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}}+[AHL]}\\ \frac{d[GFP]}{dt}&=a_\mathrm{GFP}k_{\mathrm{leaky}}(LuxR_\mathrm{t}-[LuxRAHL])+\frac{a_\mathrm{GFP}\left(\frac{[LuxRAHL]}{K_{\mathrm{LuxRAHL}}}\right)^2}{1+\left(\frac{[LuxRAHL]}{K_{\mathrm{LuxRAHL}}}\right)^2}-d_{\mathrm{GFP}}[GFP]\\ K_\mathrm{d-LuxRAHL} &= \frac{k_\mathrm{-LuxRAHL}}{k_\mathrm{LuxRAHL}}\\ \end{align*}