# 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