IGEM:IMPERIAL/2007/Projects/Biofilm Detector/Modelling/Construct2/Case 2

=Construct 2: Case 2=

 Introduction Case 1 Case 2 Case 3 Case 4 Case 5 References  

Case 2: Only initial [AHL] is controlled $$\ [LuxR]_0 = [LuxR] + [A]$$ $$\ \Rightarrow y = [LuxR] + [A]$$ $$\ \Rightarrow y = [LuxR] + K_{\alpha}x[LuxR]$$

Set subject to [LuxR]: $$[LuxR] = \frac{y}{1+K_{\alpha}x}\cdots\cdots(1)$$

Substitute (1) into $$\ [A] = K_{\alpha}[AHL][LuxR]$$

$$\therefore [A] = K_{\alpha}\frac{xy}{1+K_{\alpha}x}\cdots\cdots(2)$$

Substitute (2) into $$\ [AP] = \frac{K_{\beta}[A][P]_0}{1+K_{\beta}[A]}$$

$$\therefore R_{y}(x) = \frac{K_{\alpha}K_{\beta}xy}{1+K_{\alpha}x+ K_{\alpha}K_{\beta}xy}$$ where $$\ R_{y}(x) = \frac{[AP]}{[P]_0}$$

$$\Rightarrow R_{y}(x) = \frac{K_{\alpha}K_{\beta}xy}{1+x(K_{\alpha}+ K_{\alpha}K_{\beta}y)}$$

As x $$\rightarrow \infty: R_{y}(x) \rightarrow \frac{K_{\beta}y}{1+ K_{\beta}y} < 1$$ $$ \Rightarrow $$ We don't reach optimal efficiency as before, in Case 1