# Construct 1: Introduction

With the chemical pathway below for construct 1 we can make form some initial equations about the system.

Our aim in modelling Infector Detector is to determine the concentration of biofilm we can detect such that we report a visible signal. Therefore we want to relate the GFP concentration to the concentration of AHL. Starting that the rate of change in GFP concentration:

$\frac{d[GFP]}{dt}=k_{6}[AP]-\delta_{GFP}[GFP]\cdots\cdots(1)$

Aussume : [AP] reaches a constant level
Reason : Treat 1st formation of AP complex as a black box - it just reaches steady state.

$\therefore\frac{d[AP]}{dt}=0=k_{4}[A][P]-k_{5}[AP]\cdots\cdots(2)$
$\Rightarrow[AP]=\frac{k_{4}}{k_{5}}[A][P]\cdots\cdots(3)$

Assume : Conservation of Plux Promoters
Reason : no damage to Promoters will occour eg. no DNA damage due to old age or cell defence mechanisms attacking the DNA

$\displaystyle[P]_{0}=[P]+[AP]\cdots\cdots(4)$
$\displaystyle[P]=[P]_{0}-[AP]\cdots\cdots(5)$

Substitute this into (3)

$[AP]=\frac{k_{4}}{k_{5}}[A]\left([P]_{0}-[AP]\right)\cdots\cdots(6)$

Solving for [AP]

$[AP]+\frac{k_{4}}{k_{5}}[A][AP]=\frac{k_{4}}{k_{5}}[A][P]_{0}\cdots\cdots(7)$
$[AP]\left(1+\frac{k_{4}}{k_{5}}[A]\right)=\frac{k_{4}}{k_{5}}[A][P]_{0}\cdots\cdots(8)$
$[AP]=\frac{k_4[A][P]_{0}}{k_{5}+k_{4}[A]}=\frac{K_{\beta}[A][P]_{0}}{1+K_{\beta}[A]}\cdots\cdots(9)$
$K_{\beta}=\frac{k_{4}}{k_{5}}$

From 9 $\because$ [AP] is a constant [A] is a constant

From Reaction pathways at top of page we can see that because 3 is in equilibrium 2 is in equilibrium

From these two statements we can say that at equilibrium:

$[A]=K_{\alpha}[AHL][LuxR]\cdots\cdots(10)$
$K_{\alpha}=\frac{k_{2}}{k_{3}}$

Substituting all of this into expression for GFP

$\frac{d[GFP]}{dt}=k_{6}\left[\frac{K_{\alpha}K_{\beta}[AHL][LuxR][P]_{0}}{1+K_{\alpha}K_{\beta}[AHL][LuxR]}\right]-\delta_{GFP}[GFP]\cdots\cdots(11)$