IGEM:IMPERIAL/2007/Projects/Biofilm Detector/Modelling/Construct2/Case 2: Difference between revisions
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<math>\therefore [A] = K_{\alpha}\frac{xy}{1+K_{\alpha}x}\cdots\cdots(2)</math><br><br> | <math>\therefore [A] = K_{\alpha}\frac{xy}{1+K_{\alpha}x}\cdots\cdots(2)</math><br><br> | ||
Substitute ( | Substitute (2) into | ||
<math>\Rightarrow R_{y}(x) = \frac{K_{\alpha}K_{\beta}xy}{1+K_{\alpha}x+ K_{\alpha}K_{\beta}xy}</math> | <math>\Rightarrow R_{y}(x) = \frac{K_{\alpha}K_{\beta}xy}{1+K_{\alpha}x+ K_{\alpha}K_{\beta}xy}</math> |
Revision as of 06:29, 17 August 2007
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{ \Rightarrow R_{y}(x) = \frac{K_{\alpha}K_{\beta}xy}{1+K_{\alpha}x+ K_{\alpha}K_{\beta}xy} }[/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