Biology Reference
In-Depth Information
where (if you care to know)
1
,
3BPG is
1
,
3
−
biphosphoglycerate
,
3PG
is
3
−
phosphoglycerate
,
.
PEP
is
phosphoenolpyruvate, and
Pyr
is
pyruvate
.
Now, each (bio)chemical equation (
8.1
) can be rewritten mathematically as follows:
a
1
C
i
1
+
a
2
C
i
2
+···+
a
k
C
i
k
=
a
k
+
1
C
i
k
+
1
+···+
a
t
C
i
t
.
(8.5)
Necessarily, each
a
j
>
0, since
a
j
is the number of molecules of the metabolite
C
i
j
.
Equivalently, the equality (
8.5
) can be rewritten as
−
a
1
C
i
1
+
(
−
a
2
)
C
i
2
+···+
(
−
a
k
)
C
i
k
+
a
k
+
1
C
i
k
+
1
+···+
a
t
C
i
t
=
0
,
(8.6)
with the convention that the number
a
j
of molecules of the metabolite
C
i
j
is altered
by a minus sign when
C
i
j
is a substrate (an “input,” or a metabolite “consumed by the
reaction”), and taken to be positive when
C
i
j
is a product (an “output,” or a metabolite
“produced by the reaction”). For example, placing Eq. (
8.2
)intheformEq.(
8.6
)gives
(
−
1
)(
GLC
)
+
(
−
1
)(
ATP
)
+
(
1
)(
G6P
)
+
(
1
)(
ADP
)
=
0
.
(8.7)
It is not uncommon for the coefficients of these (mathematically written) reaction
equations to be simply
1; in fact, equations arising from biochemical networks
3
can
often be represented in this form.
Exercise 8.1.
Rewrite Eq. (
8.3
) in the form of Eq. (
8.7
) and call the result Eq. (8.8).
±
(8.8)
Now do the same for Eq. (
8.4
), and call the result Eq. (8.9).
(8.9)
The data in the three Eqs. (8.7)-(8.9) can be encoded in a matrix, a stoichiometric
matrix, which we now define.
Definition 8.1.
The
stoichiometric matrix
, for a system of
n
biochemical reactions
(
8.1
), is an
m
×
n
matrix
S
∈
M
m
,
n
(
R
)
, where
m
is the total number of metabolites
C
1
,...,
C
m
in the
n
equations. The stoichiometry matrix
S
has exactly one row
for each metabolite appearing in the system. Each column vector of the matrix
S
corresponds to a reaction, by recording the coefficients of the metabolites taken from
the reaction equation in the form (
8.6
). If a metabolite does not appear in a reaction,
the corresponding column entry is taken to be 0.
3
As opposed to “elemental equations,” reaction equations associated to the decomposition of compounds
into their basic chemical elements, like hydrogen, oxygen, etc.
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