Switch Model

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system: \begin{equation*} [\text{ON}]'=k_\text{H}[\text{Hbif}]^m[\text{OFF}]-k_\text{F}[\text{FimE}]^n[\text{ON}] \end{equation*}

\begin{equation*} [\text{OFF}]'=k_\text{F}[\text{FimE}]^n[\text{ON}]-k_\text{H}[\text{Hbif}]^m[\text{OFF}] \end{equation*}

\begin{equation*} [\text{FimE}]'=0 \end{equation*}

\begin{equation*} [\text{Hbif}]'=0 \end{equation*}

conservation equation: \begin{equation*} [\text{ON}]+[\text{OFF}]=1 \end{equation*}

reduced model: \begin{equation*} [\text{ON}]'=k_\text{H}[\text{Hbif}]^m-(k_\text{H}[\text{Hbif}]^m+k_\text{F}[\text{FimE}]^n)[\text{ON}] \end{equation*}

\begin{equation*} [\text{FimE}]'=0 \end{equation*}

\begin{equation*} [\text{Hbif}]'=0 \end{equation*}

steady state: \\ if $k_\text{H}[\text{Hbif}]^m+k_\text{F}[\text{FimE}]^n \neq 0$,

\begin{equation*} [\text{ON}]_\infty=\frac{k_\text{H}[\text{Hbif}]^m}{k_\text{H}[\text{Hbif}]^m+k_\text{F}[\text{FimE}]^n} \end{equation*}

\begin{equation*} [\text{OFF}]_\infty=\frac{k_\text{F}[\text{FimE}]^n}{k_\text{H}[\text{Hbif}]^m+k_\text{F}[\text{FimE}]^n} \end{equation*}

if $k_\text{H}[\text{Hbif}]^m=k_\text{F}[\text{FimE}]^n=0$, \begin{equation*} [\text{ON}]_\infty=[\text{ON}]_0 \end{equation*}

\begin{equation*} [\text{OFF}]_\infty=1-[\text{ON}]_\infty=1-[\text{ON}]_0=[\text{OFF}]_0 \end{equation*}

calibration: \begin{equation*} m=\frac{ln\left(\frac{ ln\big(\frac{[\text{ON}]_0-[\text{ON}]_{\infty_1}}{[\text{ON}]_1-[\text{ON}]_{\infty_1}}\big)\frac{[\text{ON}]_{\infty_1}}{t_1} }{ ln\big(\frac{[\text{ON}]_0-[\text{ON}]_{\infty_2}}{[\text{ON}]_2-[\text{ON}]_{\infty_2}}\big)\frac{[\text{ON}]_{\infty_2}}{t_2} }\right)}{ln\big(\frac{[\text{Hbif}]_1}{[\text{Hbif}]_2}\big)} = \frac{ln\left(\frac{ ln\big(\frac{[\text{OFF}]_0-[\text{OFF}]_{\infty_1}}{[\text{OFF}]_1-[\text{OFF}]_{\infty_1}}\big)\frac{1-[\text{OFF}]_{\infty_1}}{t_1} }{ ln\big(\frac{[\text{OFF}]_0-[\text{OFF}]_{\infty_2}}{[\text{OFF}]_2-[\text{OFF}]_{\infty_2}}\big)\frac{1-[\text{OFF}]_{\infty_2}}{t_2} }\right)}{ln\big(\frac{[\text{Hbif}]_1}{[\text{Hbif}]_2}\big)} \end{equation*}

\begin{equation*} n=\frac{ln\left(\frac{ ln\big(\frac{[\text{ON}]_0-[\text{ON}]_{\infty_1}}{[\text{ON}]_1-[\text{ON}]_{\infty_1}}\big)\frac{1-[\text{ON}]_{\infty_1}}{t_1} }{ ln\big(\frac{[\text{ON}]_0-[\text{ON}]_{\infty_2}}{[\text{ON}]_2-[\text{ON}]_{\infty_2}}\big)\frac{1-[\text{ON}]_{\infty_2}}{t_2} }\right)}{ln\big(\frac{[\text{FimE}]_1}{[\text{FimE}]_2}\big)} = \frac{ln\left(\frac{ ln\big(\frac{[\text{OFF}]_0-[\text{OFF}]_{\infty_1}}{[\text{OFF}]_1-[\text{OFF}]_{\infty_1}}\big)\frac{[\text{OFF}]_{\infty_1}}{t_1} }{ ln\big(\frac{[\text{OFF}]_0-[\text{OFF}]_{\infty_2}}{[\text{OFF}]_2-[\text{OFF}]_{\infty_2}}\big)\frac{[\text{OFF}]_{\infty_2}}{t_2} }\right)}{ln\big(\frac{[\text{FimE}]_1}{[\text{FimE}]_2}\big)} \end{equation*}

\begin{equation*} = \frac{1}{2}ln\big(\frac{[\text{ON}]_0-[\text{ON}]_{\infty_1}}{[\text{ON}]_1-[\text{ON}]_{\infty_1}}\big)\frac{[\text{ON}]_{\infty_1}}{t_1}[\text{Hbif}]_1^{\frac{ln\left(\frac{ ln\big(\frac{[\text{ON}]_0-[\text{ON}]_{\infty_1}}{[\text{ON}]_1-[\text{ON}]_{\infty_1}}\big)\frac{[\text{ON}]_{\infty_1}}{t_1} }{ ln\big(\frac{[\text{ON}]_0-[\text{ON}]_{\infty_2}}{[\text{ON}]_2-[\text{ON}]_{\infty_2}}\big)\frac{[\text{ON}]_{\infty_2}}{t_2} }\right)}{ln(\frac{[\text{Hbif}]_1}{[\text{Hbif}]_2})}} + \frac{1}{2}ln(\frac{[\text{ON}]_0-[\text{ON}]_{\infty_2}}{[\text{ON}]_2-[\text{ON}]_{\infty_2}})\frac{[\text{ON}]_{\infty_2}}{t_2}[\text{Hbif}]_2^{\frac{ln\left(\frac{ ln\big(\frac{[\text{ON}]_0-[\text{ON}]_{\infty_1}}{[\text{ON}]_1-[\text{ON}]_{\infty_1}}\big)\frac{[\text{ON}]_{\infty_1}}{t_1} }{ ln\big(\frac{[\text{ON}]_0-[\text{ON}]_{\infty_2}}{[\text{ON}]_2-[\text{ON}]_{\infty_2}}\big)\frac{[\text{ON}]_{\infty_2}}{t_2} }\right)}{ln(\frac{[\text{Hbif}]_1}{[\text{Hbif}]_2})}} \end{equation*}

\begin{equation*} k_{H} = \frac{1}{2}ln(\frac{[\text{OFF}]_0-[\text{OFF}]_{\infty_1}}{[\text{OFF}]_1-[\text{OFF}]_{\infty_1}})\frac{1-[\text{OFF}]_{\infty_1}}{t_1}[\text{Hbif}]_1^{\frac{ln\left(\frac{ ln\big(\frac{[\text{OFF}]_0-[\text{OFF}]_{\infty_1}}{[\text{OFF}]_1-[\text{OFF}]_{\infty_1}}\big)\frac{1-[\text{OFF}]_{\infty_1}}{t_1} }{ ln\big(\frac{[\text{OFF}]_0-[\text{OFF}]_{\infty_2}}{[\text{OFF}]_2-[\text{OFF}]_{\infty_2}}\big)\frac{1-[\text{OFF}]_{\infty_2}}{t_2} }\right)}{ln(\frac{[\text{Hbif}]_1}{[\text{Hbif}]_2})}} + \frac{1}{2}ln(\frac{[\text{OFF}]_0-[\text{OFF}]_{\infty_2}}{[\text{OFF}]_2-[\text{OFF}]_{\infty_2}})\frac{1-[\text{OFF}]_{\infty_2}}{t_2}[\text{Hbif}]_2^{\frac{ln\left(\frac{ ln\big(\frac{[\text{OFF}]_0-[\text{OFF}]_{\infty_1}}{[\text{OFF}]_1-[\text{OFF}]_{\infty_1}}\big)\frac{1-[\text{OFF}]_{\infty_1}}{t_1} }{ ln\big(\frac{[\text{OFF}]_0-[\text{OFF}]_{\infty_2}}{[\text{OFF}]_2-[\text{OFF}]_{\infty_2}}\big)\frac{1-[\text{OFF}]_{\infty_2}}{t_2} }\right)}{ln(\frac{[\text{Hbif}]_1}{[\text{Hbif}]_2})}} \end{equation*}

\begin{equation*} k_{F} = \frac{1}{2}ln(\frac{[\text{ON}]_0-[\text{ON}]_{\infty_1}}{[\text{ON}]_1-[\text{ON}]_{\infty_1}})\frac{1-[\text{ON}]_{\infty_1}}{t_1}[\text{FimE}]_1^{\frac{ln\left(\frac{ ln\big(\frac{[\text{ON}]_0-[\text{ON}]_{\infty_1}}{[\text{ON}]_1-[\text{ON}]_{\infty_1}}\big)\frac{1-[\text{ON}]_{\infty_1}}{t_1} }{ ln\big(\frac{[\text{ON}]_0-[\text{ON}]_{\infty_2}}{[\text{ON}]_2-[\text{ON}]_{\infty_2}}\big)\frac{1-[\text{ON}]_{\infty_2}}{t_2} }\right)}{ln(\frac{[\text{FimE}]_1}{[\text{FimE}]_2})}} + \frac{1}{2}ln(\frac{[\text{ON}]_0-[\text{ON}]_{\infty_2}}{[\text{ON}]_2-[\text{ON}]_{\infty_2}})\frac{1-[\text{ON}]_{\infty_2}}{t_2}[\text{FimE}]_2^{\frac{ln\left(\frac{ ln\big(\frac{[\text{ON}]_0-[\text{ON}]_{\infty_1}}{[\text{ON}]_1-[\text{ON}]_{\infty_1}}\big)\frac{1-[\text{ON}]_{\infty_1}}{t_1} }{ ln\big(\frac{[\text{ON}]_0-[\text{ON}]_{\infty_2}}{[\text{ON}]_2-[\text{ON}]_{\infty_2}}\big)\frac{1-[\text{ON}]_{\infty_2}}{t_2} }\right)}{ln(\frac{[\text{FimE}]_1}{[\text{FimE}]_2})}} \end{equation*}

\begin{equation*} = \frac{1}{2}ln(\frac{[\text{OFF}]_0-[\text{OFF}]_{\infty_1}}{[\text{OFF}]_1-[\text{OFF}]_{\infty_1}})\frac{[\text{OFF}]_{\infty_1}}{t_1}[\text{FimE}]_1^{\frac{ln\left(\frac{ ln\big(\frac{[\text{OFF}]_0-[\text{OFF}]_{\infty_1}}{[\text{OFF}]_1-[\text{OFF}]_{\infty_1}}\big)\frac{[\text{OFF}]_{\infty_1}}{t_1} }{ ln\big(\frac{[\text{OFF}]_0-[\text{OFF}]_{\infty_2}}{[\text{OFF}]_2-[\text{OFF}]_{\infty_2}}\big)\frac{[\text{OFF}]_{\infty_2}}{t_2} }\right)}{ln(\frac{[\text{FimE}]_1}{[\text{FimE}]_2})}} + \frac{1}{2}ln(\frac{[\text{OFF}]_0-[\text{OFF}]_{\infty_2}}{[\text{OFF}]_2-[\text{OFF}]_{\infty_2}})\frac{[\text{OFF}]_{\infty_2}}{t_2}[\text{FimE}]_2^{\frac{ln\left(\frac{ ln\big(\frac{[\text{OFF}]_0-[\text{OFF}]_{\infty_1}}{[\text{OFF}]_1-[\text{OFF}]_{\infty_1}}\big)\frac{[\text{OFF}]_{\infty_1}}{t_1} }{ ln\big(\frac{[\text{OFF}]_0-[\text{OFF}]_{\infty_2}}{[\text{OFF}]_2-[\text{OFF}]_{\infty_2}}\big)\frac{[\text{OFF}]_{\infty_2}}{t_2} }\right)}{ln(\frac{[\text{FimE}]_1}{[\text{FimE}]_2})}} \end{equation*}