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8.3.1.
Representing dynamical schemes
The basic elements of a metabolic network are the different metabolite
concentrations and the reactions, often catalyzed by enzymes. The change of
concentration in time can be described using differential equations by
considering m metabolites and r reactions, so that we have
dS
r
∑
i
=
n v
i
=
1,...,
m
ij
j
dt
j
=
1
where
S
is the concentration of metabolite
i
,
v
is the reaction rate and
i
n
is
the stoichiometric coefficient of metabolite
i
in reaction
j
. By rearranging the
coefficients
i
n
in a matrix, we obtain the stoichiometric matrix
N
. In the case of
the network in Fig. 8.9, the stoichiometric matrix has four rows corresponding to
A, B, C, D metabolites and seven columns, corresponding to the number of
internal metabolites and reactions and is the following
1
0
0
0
−
1
−
1
0
.
0
1
0
−
1
1
0
−
1
N
=
0
0
0
0
0
1
1
0
0
−
1
1
0
0
0
Fig. 8.9. A simple network. A, B, C and D are internal metabolites;
v
4
,
v
5,
v
6,
v
7
are internal
reactions and
v
1
,
v
2,
v
3,
are external fluxes. The area delimited by the dotted line denotes the internal
metabolism.
Since all reactions may be reversible, in order to determine the signs of the
coefficients
i
n
, the directions of the arrows for reversible reactions are
considered positive if the incoming arc is directed 'from left to right' and 'from
top to bottom' by convention. For example, let us consider the second row
(metabolite B) of matrix
N
: the value in the second column is positive because
the reversible flux entering B is directed 'from top to bottom', while the
reversible reaction
v
entering B has the opposite direction (from bottom to top)
and so it has to be considered as negative.
