IGEM:IMPERIAL/2007/Projects/Biofilm Detector/Modelling/Construct2/Case 2
Construct 2: Case 2
Case 2: Only initial [AHL] is controlled
[math]\displaystyle{ \ [LuxR]_0 = [LuxR] + [A] }[/math]
[math]\displaystyle{ \ \Rightarrow y = [LuxR] + [A] }[/math]
[math]\displaystyle{ \ \Rightarrow y = [LuxR] + K_{\alpha}x[LuxR] }[/math]
Set subject to [LuxR]:
[math]\displaystyle{ [LuxR] = \frac{y}{1+K_{\alpha}x}\cdots\cdots(1) }[/math]
Substitute (1) into [math]\displaystyle{ \ [A] = K_{\alpha}[AHL][LuxR] }[/math]
[math]\displaystyle{ \therefore [A] = K_{\alpha}\frac{xy}{1+K_{\alpha}x}\cdots\cdots(2) }[/math]
Substitute (2) into [math]\displaystyle{ \ [AP] = \frac{K_{\beta}[A][P]_0}{1+K_{\beta}[A]} }[/math]
[math]\displaystyle{ \therefore R_{y}(x) = \frac{K_{\alpha}K_{\beta}xy}{1+K_{\alpha}x+ K_{\alpha}K_{\beta}xy} }[/math] where [math]\displaystyle{ \ R_{y}(x) = \frac{[AP]}{[P]_0} }[/math]
[math]\displaystyle{ \Rightarrow R_{y}(x) = \frac{K_{\alpha}K_{\beta}xy}{1+x(K_{\alpha}+ K_{\alpha}K_{\beta}y)} }[/math]
As x [math]\displaystyle{ \rightarrow \infty: R_{y}(x) \rightarrow \frac{K_{\beta}y}{1+ K_{\beta}y} \lt 1 }[/math]
[math]\displaystyle{ \Rightarrow }[/math] We don't reach optimal efficiency as before, in Case 1
