## Pages

### Control Expressions

This page presents a list of sensitivity equations for a selection of network motifs.

## Linear Chains

#### Two step pathway:

$$X_o \stackrel{v_1}{\longrightarrow} S_1 \stackrel{v_2}{\longrightarrow} X_1$$

\begin{align*}
C^{S_1}_{E_1} &= \frac{1}{\varepsilon^2_1 - \varepsilon^1_1} \\[5pt]
C^{S_1}_{E_2} &= -\frac{1}{\varepsilon^2_1 - \varepsilon^1_1} \\[5pt]
C^J_{E_1} &= \frac{\varepsilon^2_1}{\varepsilon^2_1 - \varepsilon^1_1} \\[5pt]
C^J_{E_2} &= -\frac{\varepsilon^1_1}{\varepsilon^2_1 - \varepsilon^1_1}
\end{align*}

#### Three step pathway:

$$X_o \stackrel{v_1}{\longrightarrow}S_1 \stackrel{v_2}{\longrightarrow} S_2 \stackrel{v_3}{\longrightarrow} X_1$$

\begin{align*}
C^J_{E_1} &= \varepsilon^{2}_1 \varepsilon^{3}_2 / D \\[5pt]
C^J_{E_2} &= -\varepsilon^{1}_1 \varepsilon^{3}_2 / D \\[5pt]
C^J_{E_3} &= \varepsilon^{1}_1 \varepsilon^{2}_2 / D
\end{align*}

where $D$ the denominator is given by:

$$D = \varepsilon^{2}_1 \varepsilon^{3}_2 -\varepsilon^{1}_1 \varepsilon^{3}_2 + \varepsilon^{1}_1 \varepsilon^{2}_2$$

\begin{align*}
C^{S_1}_{E_1} &= (\varepsilon^{3}_2 - \varepsilon^{2}_2) / D \\[5pt]
C^{S_1}_{E_2} &= - \varepsilon^{3}_2 / D \\[5pt]
C^{S_1}_{E_3} &= \varepsilon^{2}_2 / D \\
\end{align*}

And for $S_2$

\begin{align*}
C^{S_2}_{E_1} &= \varepsilon^{2}_1 / D \\[5pt]
C^{S_2}_{E_2} &= -\varepsilon^{1}_1 / D \\[5pt]
C^{S_2}_{E_3} &= (\varepsilon^{1}_1 - \varepsilon^{2}_1) / D
\end{align*}