This page presents a list of sensitivity equations for a selection of network motifs.
Linear Chains
Two step pathway:
![\[X_o \stackrel{v_1}{\longrightarrow} S_1 \stackrel{v_2}{\longrightarrow} X_1\]](http://blog.analogmachine.org/wp-content/ql-cache/quicklatex.com-76b47f702a7799cafad16c01d6e52ed9_l3.png "Rendered by QuickLaTeX.com")
![\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*}](http://blog.analogmachine.org/wp-content/ql-cache/quicklatex.com-6f21cb2e5b01ccadec52ea71dcda279f_l3.png "Rendered by QuickLaTeX.com")
Three step pathway:
![\[X_o \stackrel{v_1}{\longrightarrow}S_1 \stackrel{v_2}{\longrightarrow} S_2 \stackrel{v_3}{\longrightarrow} X_1\]](http://blog.analogmachine.org/wp-content/ql-cache/quicklatex.com-91a11115cb59636d30d1696a08a24e1f_l3.png "Rendered by QuickLaTeX.com")
![\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*}](http://blog.analogmachine.org/wp-content/ql-cache/quicklatex.com-9340878b9076a1f662acd3f0d7254183_l3.png "Rendered by QuickLaTeX.com")
where 
the denominator is given by:
![\[D = \varepsilon^{2}_1 \varepsilon^{3}_2 -\varepsilon^{1}_1 \varepsilon^{3}_2 + \varepsilon^{1}_1 \varepsilon^{2}_2\]](http://blog.analogmachine.org/wp-content/ql-cache/quicklatex.com-313408555d2b57b8e634bfcec0a551ad_l3.png "Rendered by QuickLaTeX.com")
![\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*}](http://blog.analogmachine.org/wp-content/ql-cache/quicklatex.com-a8f6ba418c7646f87bf90fab26bf0115_l3.png "Rendered by QuickLaTeX.com")
And for 
![\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*}](http://blog.analogmachine.org/wp-content/ql-cache/quicklatex.com-87cb2f4f114ab3a416aacd2d45e8b883_l3.png "Rendered by QuickLaTeX.com")
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