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

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Construct 2: Case 2



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

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