Jannie Hofmeyr published the first catalog of control patterns in metabolic control analysis (MCA). His doctoral research.[ 1] [ 2] Carl D Christensen and Johann Rohwer , developed a Python based tool called SymCA that was part of the PySCeSToolbox toolkit [ 3] [ 4]
The control equations, especially the numerators of the equations, can give information on the relative importance and routes by which perturbations travel through a biochemical network[ 5]
Control patterns describe how a perturbation to a given parameter affects the steady-state level of a given variable. For example, a concentration control coefficient can describe how the overexpression of a specific enzyme can influence steady-state metabolite concentrations. Flux control coefficients are similar in that they describe how a perturbation in a given enzyme affects steady-state flux through a pathway. Such coefficients can be written in terms of elasticity coefficients.
Elasticity coefficients are local properties that describe how a single reaction is influenced by changes in the substrates and products that might influence the rate. For example, given a reaction such as:
S
⟶
v
P
{\displaystyle S{\stackrel {v}{\longrightarrow }}P}
we will assume it has a rate of reaction of
v
{\displaystyle v}
substrate
S
{\displaystyle S}
product
P
{\displaystyle P}
ε
s
v
=
∂
v
∂
s
s
v
{\displaystyle \varepsilon _{s}^{v}={\frac {\partial v}{\partial s}}{\frac {s}{v}}}
To make the notation manageable, a specific numbering scheme is used in the following patterns. If a substrate has an index of
i
{\displaystyle i}
v
i
+
1
{\displaystyle v_{i+1}}
i
+
1
{\displaystyle i+1}
For example, the following fragment of a network illustrates this labeling:
X
o
⟶
v
1
S
1
⟶
v
2
S
2
⟶
v
3
{\displaystyle X_{o}{\stackrel {v_{1}}{\longrightarrow }}S_{1}{\stackrel {v_{2}}{\longrightarrow }}S_{2}{\stackrel {v_{3}}{\longrightarrow }}}
then
ε
1
2
=
∂
v
2
∂
s
1
s
1
v
2
,
ε
2
2
=
∂
v
2
∂
s
2
s
2
v
2
,
ε
2
3
=
∂
v
3
∂
s
2
s
2
v
3
{\displaystyle \varepsilon _{1}^{2}={\frac {\partial v_{2}}{\partial s_{1}}}{\frac {s_{1}}{v_{2}}},\quad \varepsilon _{2}^{2}={\frac {\partial v_{2}}{\partial s_{2}}}{\frac {s_{2}}{v_{2}}},\quad \varepsilon _{2}^{3}={\frac {\partial v_{3}}{\partial s_{2}}}{\frac {s_{2}}{v_{3}}}}
X
o
⟶
v
1
S
1
⟶
v
2
X
1
{\displaystyle X_{o}{\stackrel {v_{1}}{\longrightarrow }}S_{1}{\stackrel {v_{2}}{\longrightarrow }}X_{1}}
Assuming both steps are Irreversible
edit
C
e
1
J
=
1
C
e
2
J
=
0
{\displaystyle C_{e_{1}}^{J}=1\qquad C_{e_{2}}^{J}=0}
C
e
1
s
1
=
1
ε
1
2
C
e
2
s
1
=
−
1
ε
1
2
{\displaystyle C_{e_{1}}^{s_{1}}={\frac {1}{\varepsilon _{1}^{2}}}\qquad C_{e_{2}}^{s_{1}}={\frac {-1}{\varepsilon _{1}^{2}}}}
Assuming both steps are Reversible
edit
C
v
1
J
=
ε
1
2
ε
1
2
−
ε
1
1
C
v
2
J
=
−
ε
1
1
ε
1
2
−
ε
1
1
{\displaystyle C_{v_{1}}^{J}={\frac {\varepsilon _{1}^{2}}{\varepsilon _{1}^{2}-\varepsilon _{1}^{1}}}\qquad C_{v_{2}}^{J}={\frac {-\varepsilon _{1}^{1}}{\varepsilon _{1}^{2}-\varepsilon _{1}^{1}}}}
C
v
1
s
1
=
1
ε
1
2
−
ε
1
1
C
v
2
s
1
=
−
1
ε
1
2
−
ε
1
1
{\displaystyle C_{v_{1}}^{s_{1}}={\frac {1}{\varepsilon _{1}^{2}-\varepsilon _{1}^{1}}}\qquad C_{v_{2}}^{s_{1}}={\frac {-1}{\varepsilon _{1}^{2}-\varepsilon _{1}^{1}}}}
X
o
⟶
v
1
S
1
⟶
v
2
S
2
⟶
v
3
X
1
{\displaystyle X_{o}{\stackrel {v_{1}}{\longrightarrow }}S_{1}{\stackrel {v_{2}}{\longrightarrow }}S_{2}{\stackrel {v_{3}}{\longrightarrow }}X_{1}}
Assuming the three steps are Irreversible
edit
Denominator:
d
=
ε
1
2
ε
2
3
{\displaystyle d=\varepsilon _{1}^{2}\varepsilon _{2}^{3}}
Assume that each of the following expressions is divided by d
C
e
1
J
=
1
C
e
2
J
=
0
C
e
3
J
=
0
{\displaystyle {\begin{array}{lll}C_{e_{1}}^{J}=1&C_{e_{2}}^{J}=0&C_{e_{3}}^{J}=0\end{array}}}
C
e
1
s
1
=
ε
2
3
C
e
1
s
2
=
ε
1
2
C
e
2
s
1
=
−
ε
2
3
C
e
2
s
2
=
0
C
e
2
s
2
=
0
C
e
3
s
2
=
−
ε
1
2
{\displaystyle {\begin{array}{ll}C_{e_{1}}^{s_{1}}=\varepsilon _{2}^{3}&C_{e_{1}}^{s_{2}}=\varepsilon _{1}^{2}\\[6pt]C_{e_{2}}^{s_{1}}=-\varepsilon _{2}^{3}&C_{e_{2}}^{s_{2}}=0\\[6pt]C_{e_{2}}^{s_{2}}=0&C_{e_{3}}^{s_{2}}=-\varepsilon _{1}^{2}\end{array}}}
Assuming the three steps are Reversible
edit
Denominator:
d
=
ε
1
2
ε
2
3
−
ε
1
1
ε
2
3
+
ε
1
1
ε
2
2
{\displaystyle d=\varepsilon _{1}^{2}\varepsilon _{2}^{3}-\varepsilon _{1}^{1}\varepsilon _{2}^{3}+\varepsilon _{1}^{1}\varepsilon _{2}^{2}}
Assume that each of the following expressions is divided by
d
{\displaystyle d}
C
e
1
J
=
ε
1
2
ε
2
3
C
e
2
J
=
−
ε
1
1
ε
2
3
C
e
3
J
=
ε
1
1
ε
2
2
{\displaystyle {\begin{array}{lll}C_{e_{1}}^{J}=\varepsilon _{1}^{2}\varepsilon _{2}^{3}&C_{e_{2}}^{J}=-\varepsilon _{1}^{1}\varepsilon _{2}^{3}&C_{e_{3}}^{J}=\varepsilon _{1}^{1}\varepsilon _{2}^{2}\\[6pt]\end{array}}}
C
e
1
s
1
=
ε
2
3
−
ε
2
2
C
e
1
s
2
=
ε
1
2
C
e
2
s
1
=
−
ε
2
3
C
e
2
s
2
=
−
ε
1
1
C
e
3
s
1
=
ε
2
2
C
e
3
s
2
=
ε
1
1
−
ε
1
2
{\displaystyle {\begin{array}{ll}C_{e_{1}}^{s_{1}}=\varepsilon _{2}^{3}-\varepsilon _{2}^{2}&C_{e_{1}}^{s_{2}}=\varepsilon _{1}^{2}\\[6pt]C_{e_{2}}^{s_{1}}=-\varepsilon _{2}^{3}&C_{e_{2}}^{s_{2}}=-\varepsilon _{1}^{1}\\[6pt]C_{e_{3}}^{s_{1}}=\varepsilon _{2}^{2}&C_{e_{3}}^{s_{2}}=\varepsilon _{1}^{1}-\varepsilon _{1}^{2}\end{array}}}
X
o
⟶
v
1
S
1
⟶
v
2
S
2
⟶
v
3
S
3
⟶
v
4
X
1
{\displaystyle X_{o}{\stackrel {v_{1}}{\longrightarrow }}S_{1}{\stackrel {v_{2}}{\longrightarrow }}S_{2}{\stackrel {v_{3}}{\longrightarrow }}S_{3}{\stackrel {v_{4}}{\longrightarrow }}X_{1}}
Denominator:
d
=
ε
1
1
ε
2
2
ε
3
3
−
ε
1
1
ε
2
2
ε
3
4
+
ε
1
1
ε
2
3
ε
3
4
−
ε
1
2
ε
2
3
ε
3
4
{\displaystyle d=\varepsilon _{1}^{1}\varepsilon _{2}^{2}\varepsilon _{3}^{3}-\varepsilon _{1}^{1}\varepsilon _{2}^{2}\varepsilon _{3}^{4}+\varepsilon _{1}^{1}\varepsilon _{2}^{3}\varepsilon _{3}^{4}-\varepsilon _{1}^{2}\varepsilon _{2}^{3}\varepsilon _{3}^{4}}
Assume that each of the following expressions is divided by
d
{\displaystyle d}
C
e
1
J
=
−
ε
1
2
ε
2
3
ε
3
4
C
e
2
J
=
ε
1
1
ε
2
3
ε
3
4
C
e
3
J
=
−
ε
1
1
ε
2
2
ε
3
4
C
e
4
J
=
ε
1
1
ε
2
2
ε
3
3
C
e
1
s
1
=
−
ε
2
2
ε
3
3
+
ε
2
2
ε
3
4
−
ε
2
3
ε
3
4
C
e
2
s
1
=
−
ε
2
3
ε
3
4
C
e
3
s
1
=
−
ε
2
2
ε
3
4
C
e
4
s
1
=
ε
2
2
ε
3
3
C
e
1
s
2
=
ε
1
2
ε
3
3
−
ε
1
2
ε
3
4
C
e
2
s
2
=
−
ε
1
1
ε
3
3
+
ε
1
1
ε
3
4
C
e
3
s
2
=
−
ε
1
1
ε
3
4
+
ε
1
2
ε
3
4
C
e
4
s
2
=
ε
1
1
ε
3
3
−
ε
1
2
ε
3
3
C
e
1
s
3
=
−
ε
1
2
ε
2
3
C
e
2
s
3
=
ε
1
1
ε
2
3
C
e
3
s
2
=
−
ε
1
1
ε
2
2
C
e
4
s
2
=
ε
1
1
ε
2
2
−
ε
1
1
ε
2
3
+
ε
1
2
ε
2
3
{\displaystyle {\begin{array}{lll}C_{e_{1}}^{J}=-\varepsilon _{1}^{2}\varepsilon _{2}^{3}\varepsilon _{3}^{4}&C_{e_{2}}^{J}=\varepsilon _{1}^{1}\varepsilon _{2}^{3}\varepsilon _{3}^{4}&C_{e_{3}}^{J}=-\varepsilon _{1}^{1}\varepsilon _{2}^{2}\varepsilon _{3}^{4}&C_{e_{4}}^{J}=\varepsilon _{1}^{1}\varepsilon _{2}^{2}\varepsilon _{3}^{3}\\[4pt]C_{e_{1}}^{s_{1}}=-\varepsilon _{2}^{2}\varepsilon _{3}^{3}+\varepsilon _{2}^{2}\varepsilon _{3}^{4}-\varepsilon _{2}^{3}\varepsilon _{3}^{4}&C_{e_{2}}^{s_{1}}=-\varepsilon _{2}^{3}\varepsilon _{3}^{4}&C_{e_{3}}^{s_{1}}=-\varepsilon _{2}^{2}\varepsilon _{3}^{4}&C_{e_{4}}^{s_{1}}=\varepsilon _{2}^{2}\varepsilon _{3}^{3}\\[4pt]C_{e_{1}}^{s_{2}}=\varepsilon _{1}^{2}\varepsilon _{3}^{3}-\varepsilon _{1}^{2}\varepsilon _{3}^{4}&C_{e_{2}}^{s_{2}}=-\varepsilon _{1}^{1}\varepsilon _{3}^{3}+\varepsilon _{1}^{1}\varepsilon _{3}^{4}&C_{e_{3}}^{s_{2}}=-\varepsilon _{1}^{1}\varepsilon _{3}^{4}+\varepsilon _{1}^{2}\varepsilon _{3}^{4}&C_{e_{4}}^{s_{2}}=\varepsilon _{1}^{1}\varepsilon _{3}^{3}-\varepsilon _{1}^{2}\varepsilon _{3}^{3}\\[4pt]C_{e_{1}}^{s_{3}}=-\varepsilon _{1}^{2}\varepsilon _{2}^{3}&C_{e_{2}}^{s_{3}}=\varepsilon _{1}^{1}\varepsilon _{2}^{3}&C_{e_{3}}^{s_{2}}=-\varepsilon _{1}^{1}\varepsilon _{2}^{2}&C_{e_{4}}^{s_{2}}=\varepsilon _{1}^{1}\varepsilon _{2}^{2}-\varepsilon _{1}^{1}\varepsilon _{2}^{3}+\varepsilon _{1}^{2}\varepsilon _{2}^{3}\end{array}}}
Linear Chains with Negative Feedback
edit
Denominator:
d
=
ε
1
1
ε
2
2
−
ε
1
1
ε
2
3
+
ε
1
2
ε
2
3
−
ε
2
1
ε
1
2
{\displaystyle d=\varepsilon _{1}^{1}\varepsilon _{2}^{2}-\varepsilon _{1}^{1}\varepsilon _{2}^{3}+\varepsilon _{1}^{2}\varepsilon _{2}^{3}-\varepsilon _{2}^{1}\varepsilon _{1}^{2}}
Assume that each of the following expressions is divided by
d
{\displaystyle d}
C
e
1
J
=
ε
1
2
ε
2
3
C
e
2
J
=
−
ε
1
1
ε
2
3
C
e
3
J
=
ε
1
1
ε
2
2
−
ε
2
1
ε
1
2
C
e
1
s
1
=
ε
2
3
−
ε
2
2
C
e
2
s
1
=
−
ε
2
3
−
ε
2
1
C
e
3
s
1
=
ε
2
2
−
ε
2
1
C
e
1
s
2
=
ε
1
2
C
e
2
s
2
=
−
ε
1
1
C
e
3
s
2
=
ε
1
1
−
ε
1
2
{\displaystyle {\begin{array}{lll}C_{e_{1}}^{J}=\varepsilon _{1}^{2}\varepsilon _{2}^{3}&C_{e_{2}}^{J}=-\varepsilon _{1}^{1}\varepsilon _{2}^{3}&C_{e_{3}}^{J}=\varepsilon _{1}^{1}\varepsilon _{2}^{2}-\varepsilon _{2}^{1}\varepsilon _{1}^{2}\\[4pt]C_{e_{1}}^{s_{1}}=\varepsilon _{2}^{3}-\varepsilon _{2}^{2}&C_{e_{2}}^{s_{1}}=-\varepsilon _{2}^{3}-\varepsilon _{2}^{1}&C_{e_{3}}^{s_{1}}=\varepsilon _{2}^{2}-\varepsilon _{2}^{1}\\[4pt]C_{e_{1}}^{s_{2}}=\varepsilon _{1}^{2}&C_{e_{2}}^{s_{2}}=-\varepsilon _{1}^{1}&C_{e_{3}}^{s_{2}}=\varepsilon _{1}^{1}-\varepsilon _{1}^{2}\\[4pt]\end{array}}}
Denominator:
d
=
ε
1
1
ε
2
2
ε
3
4
−
ε
1
1
ε
2
3
ε
3
4
−
ε
3
1
ε
1
2
ε
2
3
+
ε
1
2
ε
2
3
ε
3
4
−
ε
1
1
ε
2
2
ε
3
3
{\displaystyle d=\varepsilon _{1}^{1}\varepsilon _{2}^{2}\varepsilon _{3}^{4}-\varepsilon _{1}^{1}\varepsilon _{2}^{3}\varepsilon _{3}^{4}-\varepsilon _{3}^{1}\varepsilon _{1}^{2}\varepsilon _{2}^{3}+\varepsilon _{1}^{2}\varepsilon _{2}^{3}\varepsilon _{3}^{4}-\varepsilon _{1}^{1}\varepsilon _{2}^{2}\varepsilon _{3}^{3}}
Assume that each of the following expressions is divided by
d
{\displaystyle d}
C
v
1
J
=
ε
1
2
ε
2
3
ε
3
4
C
v
2
J
=
−
ε
1
1
ε
2
3
ε
3
4
C
v
3
J
=
ε
1
1
ε
2
2
ε
3
4
C
v
4
J
=
−
ε
1
1
ε
2
2
ε
3
3
−
ε
3
1
ε
1
2
ε
2
3
C
v
1
S
1
=
ε
2
2
ε
3
3
−
ε
2
2
ε
3
4
+
ε
2
3
ε
3
4
C
v
2
S
1
=
ε
3
1
ε
2
3
−
ε
2
3
ε
3
4
C
v
3
S
1
=
−
ε
3
1
ε
2
2
+
ε
2
2
ε
3
4
C
v
4
S
1
=
ε
3
1
ε
2
2
−
ε
3
1
ε
2
3
−
ε
2
2
ε
3
3
C
v
1
S
2
=
−
ε
1
2
ε
3
3
+
ε
1
2
ε
3
4
C
v
2
S
2
=
ε
1
1
ε
3
3
−
ε
1
1
ε
3
4
C
v
3
S
2
=
ε
B
1
ε
3
4
+
ε
3
1
ε
1
2
−
ε
1
2
ε
3
4
C
v
4
S
2
=
−
ε
1
1
ε
3
3
−
ε
3
1
ε
1
2
+
ε
1
2
ε
3
3
C
v
1
S
3
=
ε
1
2
ε
2
3
C
v
2
S
3
=
−
ε
1
1
ε
2
3
C
v
3
S
3
=
ε
1
1
ε
2
2
C
v
4
S
3
=
−
ε
1
1
ε
2
2
+
ε
1
1
ε
2
3
−
ε
1
2
ε
2
3
{\displaystyle {\begin{array}{llll}C_{v_{1}}^{J}=\varepsilon _{1}^{2}\varepsilon _{2}^{3}\varepsilon _{3}^{4}&C_{v_{2}}^{J}=-\varepsilon _{1}^{1}\varepsilon _{2}^{3}\varepsilon _{3}^{4}&C_{v_{3}}^{J}=\varepsilon _{1}^{1}\varepsilon _{2}^{2}\varepsilon _{3}^{4}&C_{v_{4}}^{J}=-\varepsilon _{1}^{1}\varepsilon _{2}^{2}\varepsilon _{3}^{3}-\varepsilon _{3}^{1}\varepsilon _{1}^{2}\varepsilon _{2}^{3}\\C_{v_{1}}^{S_{1}}=\varepsilon _{2}^{2}\varepsilon _{3}^{3}-\varepsilon _{2}^{2}\varepsilon _{3}^{4}+\varepsilon _{2}^{3}\varepsilon _{3}^{4}&C_{v_{2}}^{S_{1}}=\varepsilon _{3}^{1}\varepsilon _{2}^{3}-\varepsilon _{2}^{3}\varepsilon _{3}^{4}&C_{v_{3}}^{S_{1}}=-\varepsilon _{3}^{1}\varepsilon _{2}^{2}+\varepsilon _{2}^{2}\varepsilon _{3}^{4}&C_{v_{4}}^{S_{1}}=\varepsilon _{3}^{1}\varepsilon _{2}^{2}-\varepsilon _{3}^{1}\varepsilon _{2}^{3}-\varepsilon _{2}^{2}\varepsilon _{3}^{3}\\C_{v_{1}}^{S_{2}}=-\varepsilon _{1}^{2}\varepsilon _{3}^{3}+\varepsilon _{1}^{2}\varepsilon _{3}^{4}&C_{v_{2}}^{S_{2}}=\varepsilon _{1}^{1}\varepsilon _{3}^{3}-\varepsilon _{1}^{1}\varepsilon _{3}^{4}&C_{v_{3}}^{S_{2}}=\varepsilon _{B}^{1}\varepsilon _{3}^{4}+\varepsilon _{3}^{1}\varepsilon _{1}^{2}-\varepsilon _{1}^{2}\varepsilon _{3}^{4}&C_{v_{4}}^{S_{2}}=-\varepsilon _{1}^{1}\varepsilon _{3}^{3}-\varepsilon _{3}^{1}\varepsilon _{1}^{2}+\varepsilon _{1}^{2}\varepsilon _{3}^{3}\\C_{v_{1}}^{S_{3}}=\varepsilon _{1}^{2}\varepsilon _{2}^{3}&C_{v_{2}}^{S_{3}}=-\varepsilon _{1}^{1}\varepsilon _{2}^{3}&C_{v_{3}}^{S_{3}}=\varepsilon _{1}^{1}\varepsilon _{2}^{2}&C_{v_{4}}^{S_{3}}=-\varepsilon _{1}^{1}\varepsilon _{2}^{2}+\varepsilon _{1}^{1}\varepsilon _{2}^{3}-\varepsilon _{1}^{2}\varepsilon _{2}^{3}\end{array}}}
At steady-state
v
1
=
v
2
+
v
3
{\displaystyle v_{1}=v_{2}+v_{3}}
α
=
v
2
v
1
1
−
α
=
v
3
v
1
{\displaystyle \alpha ={\frac {v_{2}}{v_{1}}}\quad 1-\alpha ={\frac {v_{3}}{v_{1}}}}
Denominator:
d
=
ε
s
2
α
+
ε
s
3
(
1
−
α
)
−
ε
s
1
{\displaystyle d=\varepsilon _{s}^{2}\alpha +\varepsilon _{s}^{3}(1-\alpha )-\varepsilon _{s}^{1}}
Assume that each of the following expressions is divided by
d
{\displaystyle d}
C
e
1
J
1
=
ε
s
3
(
1
−
α
)
+
ε
s
2
α
C
e
1
J
1
=
−
ε
s
1
α
C
e
1
J
1
=
−
ε
s
1
(
1
−
α
)
+
ε
s
2
α
{\displaystyle {\begin{array}{lll}C_{e_{1}}^{J_{1}}=\varepsilon _{s}^{3}(1-\alpha )+\varepsilon _{s}^{2}\alpha \\C_{e_{1}}^{J_{1}}=-\varepsilon _{s}^{1}\alpha \\C_{e_{1}}^{J_{1}}=-\varepsilon _{s}^{1}(1-\alpha )+\varepsilon _{s}^{2}\alpha \end{array}}}
^ Hofmeyr, Jan-Hendrik (1986). Studies in steady-state modelling and control analysis of metabolic systems . University of Stellenbosch. {{cite book }}: CS1 maint: location missing publisher (link )^ Hofmeyr, J.-H. S. (1989). "Control-pattern analysis of metabolic pathways: Flux and concentration control in linear pathways" . Eur. J. Biochem . 186 (1–2): 343–354. doi :10.1111/j.1432-1033.1989.tb15215.x PMID 2598934 . ^ Christensen, Carl D; Hofmeyr, Jan-Hendrik S; Rohwer, Johann M (1 January 2018). "PySCeSToolbox: a collection of metabolic pathway analysis tools". Bioinformatics . 34 (1): 124–125. doi :10.1093/bioinformatics/btx567 . PMID 28968872 . ^ Rohwer, Johann; Akhurst, Timothy; Hofmeyr, Jannie (2008). "Symbolic Control Analysis of Cellular Systems". Beilstein-Institut . S2CID 9216034 . ^ Christensen, Carl D.; Hofmeyr, Jan-Hendrik S.; Rohwer, Johann M. (28 November 2018). "Delving deeper: Relating the behaviour of a metabolic system to the properties of its components using symbolic metabolic control analysis" . PLOS ONE . 13 (11): e0207983. Bibcode :2018PLoSO..1307983C . doi :10.1371/journal.pone.0207983 PMC 6261606 PMID 30485345 .
![{\displaystyle {\begin{array}{ll}C_{e_{1}}^{s_{1}}=\varepsilon _{2}^{3}&C_{e_{1}}^{s_{2}}=\varepsilon _{1}^{2}\\[6pt]C_{e_{2}}^{s_{1}}=-\varepsilon _{2}^{3}&C_{e_{2}}^{s_{2}}=0\\[6pt]C_{e_{2}}^{s_{2}}=0&C_{e_{3}}^{s_{2}}=-\varepsilon _{1}^{2}\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa0f88a3d48a7cbe73c947c84ddea80e5d7e766c)
![{\displaystyle {\begin{array}{lll}C_{e_{1}}^{J}=\varepsilon _{1}^{2}\varepsilon _{2}^{3}&C_{e_{2}}^{J}=-\varepsilon _{1}^{1}\varepsilon _{2}^{3}&C_{e_{3}}^{J}=\varepsilon _{1}^{1}\varepsilon _{2}^{2}\\[6pt]\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c17dbdd00ed8a7fcb0011d2a0c3829448b1ea7a)
![{\displaystyle {\begin{array}{ll}C_{e_{1}}^{s_{1}}=\varepsilon _{2}^{3}-\varepsilon _{2}^{2}&C_{e_{1}}^{s_{2}}=\varepsilon _{1}^{2}\\[6pt]C_{e_{2}}^{s_{1}}=-\varepsilon _{2}^{3}&C_{e_{2}}^{s_{2}}=-\varepsilon _{1}^{1}\\[6pt]C_{e_{3}}^{s_{1}}=\varepsilon _{2}^{2}&C_{e_{3}}^{s_{2}}=\varepsilon _{1}^{1}-\varepsilon _{1}^{2}\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7941d60e2288d3dac30c8f56b35fa1cacef7dda)
![{\displaystyle {\begin{array}{lll}C_{e_{1}}^{J}=-\varepsilon _{1}^{2}\varepsilon _{2}^{3}\varepsilon _{3}^{4}&C_{e_{2}}^{J}=\varepsilon _{1}^{1}\varepsilon _{2}^{3}\varepsilon _{3}^{4}&C_{e_{3}}^{J}=-\varepsilon _{1}^{1}\varepsilon _{2}^{2}\varepsilon _{3}^{4}&C_{e_{4}}^{J}=\varepsilon _{1}^{1}\varepsilon _{2}^{2}\varepsilon _{3}^{3}\\[4pt]C_{e_{1}}^{s_{1}}=-\varepsilon _{2}^{2}\varepsilon _{3}^{3}+\varepsilon _{2}^{2}\varepsilon _{3}^{4}-\varepsilon _{2}^{3}\varepsilon _{3}^{4}&C_{e_{2}}^{s_{1}}=-\varepsilon _{2}^{3}\varepsilon _{3}^{4}&C_{e_{3}}^{s_{1}}=-\varepsilon _{2}^{2}\varepsilon _{3}^{4}&C_{e_{4}}^{s_{1}}=\varepsilon _{2}^{2}\varepsilon _{3}^{3}\\[4pt]C_{e_{1}}^{s_{2}}=\varepsilon _{1}^{2}\varepsilon _{3}^{3}-\varepsilon _{1}^{2}\varepsilon _{3}^{4}&C_{e_{2}}^{s_{2}}=-\varepsilon _{1}^{1}\varepsilon _{3}^{3}+\varepsilon _{1}^{1}\varepsilon _{3}^{4}&C_{e_{3}}^{s_{2}}=-\varepsilon _{1}^{1}\varepsilon _{3}^{4}+\varepsilon _{1}^{2}\varepsilon _{3}^{4}&C_{e_{4}}^{s_{2}}=\varepsilon _{1}^{1}\varepsilon _{3}^{3}-\varepsilon _{1}^{2}\varepsilon _{3}^{3}\\[4pt]C_{e_{1}}^{s_{3}}=-\varepsilon _{1}^{2}\varepsilon _{2}^{3}&C_{e_{2}}^{s_{3}}=\varepsilon _{1}^{1}\varepsilon _{2}^{3}&C_{e_{3}}^{s_{2}}=-\varepsilon _{1}^{1}\varepsilon _{2}^{2}&C_{e_{4}}^{s_{2}}=\varepsilon _{1}^{1}\varepsilon _{2}^{2}-\varepsilon _{1}^{1}\varepsilon _{2}^{3}+\varepsilon _{1}^{2}\varepsilon _{2}^{3}\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c64aea5495bb235c7ca26be5aa2e24107e0497b8)
![{\displaystyle {\begin{array}{lll}C_{e_{1}}^{J}=\varepsilon _{1}^{2}\varepsilon _{2}^{3}&C_{e_{2}}^{J}=-\varepsilon _{1}^{1}\varepsilon _{2}^{3}&C_{e_{3}}^{J}=\varepsilon _{1}^{1}\varepsilon _{2}^{2}-\varepsilon _{2}^{1}\varepsilon _{1}^{2}\\[4pt]C_{e_{1}}^{s_{1}}=\varepsilon _{2}^{3}-\varepsilon _{2}^{2}&C_{e_{2}}^{s_{1}}=-\varepsilon _{2}^{3}-\varepsilon _{2}^{1}&C_{e_{3}}^{s_{1}}=\varepsilon _{2}^{2}-\varepsilon _{2}^{1}\\[4pt]C_{e_{1}}^{s_{2}}=\varepsilon _{1}^{2}&C_{e_{2}}^{s_{2}}=-\varepsilon _{1}^{1}&C_{e_{3}}^{s_{2}}=\varepsilon _{1}^{1}-\varepsilon _{1}^{2}\\[4pt]\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ad18d784421974ed5973dcea9c1de4ee3a1589e)
