Detail

\[ X+Y \overset{k_{\rm unbind}}{\underset{k_{\rm bind}}{\rightleftharpoons}} Z \] \begin{cases} \dot{x}=a-bx-k_{\rm bind}xy+k_{\rm unbind}z & \\ \dot{y}=1-y-k_{\rm bind}xy+k_{\rm unbind}z & \\ \dot{z}=k_{\rm bind}xy-k_{\rm unbind}z-cz & \end{cases} \[ y=\frac{1}{2} \biggl\{ \sqrt{ \bigl( a-1+\frac{b}{\gamma} \bigl)^2 +4 \frac{b}{\gamma}} - \Bigl( a-1+\frac{b}{\gamma} \Bigl) \biggl\} 　\] \[ \gamma = \frac{k_{\rm bind}c}{k_{\rm unbind}+c} \]

Detail

\[ X+Y \overset{k_{\rm unbind}}{\underset{k_{\rm bind}}{\rightleftharpoons}} Z \] \begin{cases} \dot{x}=a-bx-k_{\rm bind}xy+k_{\rm unbind}z & \\ \dot{y}=1-y-k_{\rm bind}xy+k_{\rm unbind}z & \\ \dot{z}=k_{\rm bind}xy-k_{\rm unbind}z-cz & \end{cases} \[ y=\frac{1}{2} \biggl\{ \sqrt{ \bigl( a-1+\frac{b}{\gamma} \bigl)^2 +4 \frac{b}{\gamma}} - \Bigl( a-1+\frac{b}{\gamma} \Bigl) \biggl\} 　\] \[ \gamma = \frac{k_{\rm bind}c}{k_{\rm unbind}+c} \]