Table 11.

Invariant expressions obtained from corrected ratios of concentration profiles of single-step substances for some linear complex mechanisms.

Linear Complex Mechanism	Single-Step Substances	Invariant Expression
${\mathrm{A}}_1\begin{array}{c}{\mathrm{k}}_1^{+}\\ \rightleftarrows \\ {\mathrm{k}}_1^{-}\end{array}{\mathrm{A}}_2\begin{array}{c}{\mathrm{k}}_2^{+}\\ \rightleftarrows \\ {\mathrm{k}}_2^{-}\end{array}{\mathrm{A}}_3$	A1, A3	$\frac{{\mathrm{A}}_{1\left(2,3\right)}-{\mathrm{A}}_{1,\mathrm{eq}}}{{\mathrm{A}}_{3\left(1,2\right)}-{\mathrm{A}}_{3,\mathrm{eq}}}=-\frac{{\mathrm{k}}_1^{-}}{{\mathrm{k}}_2^{+}}\frac{{\mathsf{\delta }}_{\mathrm{n}}}{{\mathsf{\delta }}_{\mathrm{d}}}={\mathsf{\kappa }}_{\mathrm{out}}\frac{{\mathsf{\delta }}_1}{{\mathsf{\delta }}_2}$
${\mathrm{A}}_1\begin{array}{c}{\mathrm{k}}_1^{+}\\ \rightleftarrows \\ {\mathrm{k}}_1^{-}\end{array}{\mathrm{A}}_2\begin{array}{c}{\mathrm{k}}_2^{+}\\ \rightleftarrows \\ {\mathrm{k}}_2^{-}\end{array}{\mathrm{A}}_3\begin{array}{c}{\mathrm{k}}_3^{+}\\ \rightleftarrows \\ {\mathrm{k}}_3^{-}\end{array}{\mathrm{A}}_4$	A1, A4	$\frac{{\mathrm{A}}_{1\left(3,4\right)}-{\mathrm{A}}_{1,\mathrm{eq}}}{{\mathrm{A}}_{4\left(1,2\right)}-{\mathrm{A}}_{4,\mathrm{eq}}}=-\frac{{\mathrm{k}}_1^{-}{\mathrm{k}}_2^{-}}{{\mathrm{k}}_2^{+}{\mathrm{k}}_3^{+}}\frac{{\mathsf{\delta }}_1}{{\mathsf{\delta }}_2}=-\frac{{\mathrm{k}}_1^{-}}{{\mathrm{k}}_3^{+}}\left(\frac{1}{{\mathrm{K}}_2}\right)\frac{{\mathsf{\delta }}_1}{{\mathsf{\delta }}_2}$
${\mathrm{A}}_1\begin{array}{c}{\mathrm{k}}_1^{+}\\ \rightleftarrows \\ {\mathrm{k}}_1^{-}\end{array}{\mathrm{A}}_2\begin{array}{c}{\mathrm{k}}_2^{+}\\ \rightleftarrows \\ {\mathrm{k}}_2^{-}\end{array}{\mathrm{A}}_3\begin{array}{c}{\mathrm{k}}_3^{+}\\ \rightleftarrows \\ {\mathrm{k}}_3^{-}\end{array}{\mathrm{A}}_4\begin{array}{c}{\mathrm{k}}_4^{+}\\ \rightleftarrows \\ {\mathrm{k}}_4^{-}\end{array}{\mathrm{A}}_5$	A1, A5	$\frac{{\mathrm{A}}_{1\left(4,5\right)}-{\mathrm{A}}_{1,\mathrm{eq}}}{{\mathrm{A}}_{5\left(1,2\right)}-{\mathrm{A}}_{5,\mathrm{eq}}}=-\frac{{\mathrm{k}}_1^{-}{\mathrm{k}}_2^{-}{\mathrm{k}}_3^{-}}{{\mathrm{k}}_2^{+}{\mathrm{k}}_3^{+}{\mathrm{k}}_4^{+}}\frac{{\mathsf{\delta }}_1}{{\mathsf{\delta }}_2}=-\frac{{\mathrm{k}}_1^{-}}{{\mathrm{k}}_4^{+}}\left(\frac{1}{{\mathrm{K}}_2{\mathrm{K}}_3}\right)\frac{{\mathsf{\delta }}_1}{{\mathsf{\delta }}_2}$
$\begin{array}{ccccccc}{\mathrm{A}}_1& \begin{array}{c}{\mathrm{k}}_1^{+}\\ \rightleftarrows \\ {\mathrm{k}}_1^{-}\end{array}& {\mathrm{A}}_2& \begin{array}{c}{\mathrm{k}}_2^{+}\\ \rightleftarrows \\ {\mathrm{k}}_2^{-}\end{array}& {\mathrm{A}}_3& \begin{array}{c}{\mathrm{k}}_3^{+}\\ \rightleftarrows \\ {\mathrm{k}}_3^{-}\end{array}& {\mathrm{A}}_4\\ & & & & \begin{array}{ccc}{\mathrm{k}}^{+}& ⇅& {\mathrm{k}}^{-}\end{array}& & \\ & & & & {\mathrm{A}}_5& & \end{array}$	A1, A4	$\frac{{\mathrm{A}}_{1\left(3,4\right)}-{\mathrm{A}}_{1,\mathrm{eq}}}{{\mathrm{A}}_{4\left(1,2\right)}-{\mathrm{A}}_{4,\mathrm{eq}}}=-\frac{{\mathrm{k}}_1^{-}{\mathrm{k}}_2^{-}}{{\mathrm{k}}_2^{+}{\mathrm{k}}_3^{+}}\frac{{\mathsf{\delta }}_1}{{\mathsf{\delta }}_2}=-\frac{{\mathrm{k}}_1^{-}}{{\mathrm{k}}_3^{+}}\left(\frac{1}{{\mathrm{K}}_2}\right)\frac{{\mathsf{\delta }}_1}{{\mathsf{\delta }}_2}$
	A1, A5	$\frac{{\mathrm{A}}_{1\left(4,5\right)}-{\mathrm{A}}_{1,\mathrm{eq}}}{{\mathrm{A}}_{5\left(1,2\right)}-{\mathrm{A}}_{5,\mathrm{eq}}}=-\frac{{\mathrm{k}}_1^{-}{\mathrm{k}}_2^{-}}{{\mathrm{k}}_2^{+}{\mathrm{k}}^{+}}\frac{{\mathsf{\delta }}_1}{{\mathsf{\delta }}_2}=-\frac{{\mathrm{k}}_1^{-}}{{\mathrm{k}}^{+}}\left(\frac{1}{{\mathrm{K}}_2}\right)\frac{{\mathsf{\delta }}_1}{{\mathsf{\delta }}_2}$
	A4, A5	$\frac{{\mathrm{A}}_{5\left(3,4\right)}-{\mathrm{A}}_{5,\mathrm{eq}}}{{\mathrm{A}}_{4\left(3,5\right)}-{\mathrm{A}}_{4,\mathrm{eq}}}=-\frac{{\mathrm{k}}^{-}}{{\mathrm{k}}_3^{+}}\frac{{\mathsf{\delta }}_1}{{\mathsf{\delta }}_2}$
$\begin{array}{ccccccccc}{\mathrm{A}}_1& \begin{array}{c}{\mathrm{k}}_1^{+}\\ \rightleftarrows \\ {\mathrm{k}}_1^{-}\end{array}& {\mathrm{A}}_2& \begin{array}{c}{\mathrm{k}}_2^{+}\\ \rightleftarrows \\ {\mathrm{k}}_2^{-}\end{array}& {\mathrm{A}}_3& \begin{array}{c}{\mathrm{k}}_3^{+}\\ \rightleftarrows \\ {\mathrm{k}}_3^{-}\end{array}& {\mathrm{A}}_4& \begin{array}{c}{\mathrm{k}}_4^{+}\\ \rightleftarrows \\ {\mathrm{k}}_4^{-}\end{array}& {\mathrm{A}}_5\\ & & & & \begin{array}{ccc}{\mathrm{k}}^{-}& ⇅& {\mathrm{k}}^{+}\end{array}& & & & \\ & & & & {\mathrm{A}}_6& & & & \end{array}$	A1, A5	$\frac{{\mathrm{A}}_{1\left(4,5\right)}-{\mathrm{A}}_{1,\mathrm{eq}}}{{\mathrm{A}}_{5\left(1,2\right)}-{\mathrm{A}}_{5,\mathrm{eq}}}=-\frac{{\mathrm{k}}_1^{-}{\mathrm{k}}_2^{-}{\mathrm{k}}_3^{-}}{{\mathrm{k}}_2^{+}{\mathrm{k}}_3^{+}{\mathrm{k}}_4^{+}}\frac{{\mathsf{\delta }}_1}{{\mathsf{\delta }}_2}=-\frac{{\mathrm{k}}_1^{-}}{{\mathrm{k}}_4^{+}}\left(\frac{1}{{\mathrm{K}}_2{\mathrm{K}}_3}\right)\frac{{\mathsf{\delta }}_1}{{\mathsf{\delta }}_2}$
	A1, A6	$\frac{{\mathrm{A}}_{1\left(5,6\right)}-{\mathrm{A}}_{1,\mathrm{eq}}}{{\mathrm{A}}_{6\left(1,2\right)}-{\mathrm{A}}_{6,\mathrm{eq}}}=-\frac{{\mathrm{k}}_1^{-}{\mathrm{k}}_2^{-}}{{\mathrm{k}}_2^{+}{\mathrm{k}}^{+}}\frac{{\mathsf{\delta }}_1}{{\mathsf{\delta }}_2}=-\frac{{\mathrm{k}}_1^{-}}{{\mathrm{k}}^{+}}\left(\frac{1}{{\mathrm{K}}_2}\right)\frac{{\mathsf{\delta }}_1}{{\mathsf{\delta }}_2}$
	A5, A6	$\frac{{\mathrm{A}}_{6\left(4,5\right)}-{\mathrm{A}}_{6,\mathrm{eq}}}{{\mathrm{A}}_{5\left(3,6\right)}-{\mathrm{A}}_{5,\mathrm{eq}}}=-\frac{{\mathrm{k}}_3^{-}{\mathrm{k}}^{+}}{{\mathrm{k}}_3^{+}{\mathrm{k}}_4^{+}}\frac{{\mathsf{\delta }}_1}{{\mathsf{\delta }}_2}=-\frac{{\mathrm{k}}^{+}}{{\mathrm{k}}_4^{+}}\left(\frac{1}{{\mathrm{K}}_3}\right)\frac{{\mathsf{\delta }}_1}{{\mathsf{\delta }}_2}$