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In-Depth Information
Ta b l e 3 . 4
Substance variations, in the MP grammar of 3.1, in passing from step 0 to step 1
Δ
A
[
0
]=
A
[
1
]
−
A
[
0
]=
u
1
[
0
]
−
u
2
[
0
]
−
u
4
[
0
]=
0
.
01
−
0
.
2
−
0
.
6
=
−
0
.
79
Δ
B
[
0
]=
B
[
1
]
−
B
[
0
]=
u
2
[
0
]
−
u
3
[
0
]=
0
.
2
−
0
.
1
=
0
.
1
Δ
C
[
0
]=
C
[
1
]
−
C
[
0
]=
u
4
[
0
]
−
u
5
[
0
]=
0
.
6
−
0
.
4
=
0
.
2
In fact, according to Eq. (3.6), we can compute the substance variations as reported
in Table 3.4.
In this manner we compute the substance vector at time 1:
A
[
1
]=
A
[
0
]+
Δ
A
[
0
]=
1
.
0
−
0
.
79
=
0
.
21
B
[
1
]=
B
[
0
]+
Δ
B
[
0
]=
1
.
0
+
0
.
1
=
1
.
1
C
[
1
]=
C
[
0
]+
Δ
C
[
0
]=
1
.
0
+
0
.
2
=
1
.
2
and, by applying the same method, we get the value of this vector in all the fol-
lowing steps. In other words, the metabolic grammar provides the time series of its
metabolites. If
Δ
[
]
i
is the vector of substance variation at step
i
:
Δ
[
]=(
Δ
[
]
|
∈
)
i
i
x
M
(3.7)
x
then, for any metabolite
x
:
x
[
i
+
1
]=
x
[
i
]+
Δ
x
[
i
]
(3.8)
therefore, the
dynamics generated
by a metabolic grammar with
n
substances and
m
reactions of regulators
ϕ
1
,
ϕ
2
,...,
ϕ
m
and stoichiometric matrix
r
1
r
2
...
r
m
)
,
A =(
(3.9)
is completely expressed by the following
Equational Metabolic Algorithm
:
Δ
[
i
]=A
×
U
[
i
]
(3.10)
where:
⎛
⎞
ϕ
1
(
s
[
i
])
⎝
⎠
.
])
.........
ϕ
m
(
ϕ
2
(
s
[
i
U
[
i
]=
(3.11)
s
[
i
])
In conclusion, an MP grammar
G
can be completely represented in four ways:
1. in textual format, by providing its reactions with regulators and its initial values
(see, for example, Fig. 3.1);
2. by its corresponding MP graph (see, for example, Fig. 3.1);
