Metabolic control analysis patterns
Jannie Hofmeyr published the first catalog of control patterns in metabolic control analysis (MCA). His doctoral research.[ 1] concerned the use of graphical patterns to elucidate chains of interaction in metabolic regulation, later published in the European Journal of Biochemistry[ 2] . In his thesis, he cataloged 25 patterns for various biochemical networks. In later work, his research group, together with Carl D Christensen and Johann Rohwer , developed a Python based tool called SymCA that was part of the PySCeSToolbox toolkit [ 3] [ 4] that could generate patterns automatically and symbolically from a description of the network. This software was used to generate the patterns shown below.
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}
. This reaction rate can be influenced by changes in the concentrations of substrate
S
{\displaystyle S}
or product
P
{\displaystyle P}
. This influence is measured by an elasticity which is defined as:
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}
, then the reaction index will be
v
i
+
1
{\displaystyle v_{i+1}}
. The product elasticity will also have an index of
i
+
1
{\displaystyle i+1}
. This means that a product elasticity will have identical subscripts and superscripts making them easy to identify. The source boundary species is always labeled zero as well as the label for the first reaction.
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}}
, therefore define the following two terms:
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 .