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C
means that a number
u
of molecules of
kind
A
and the same number
u
of molecules
B
are replaced by
u
molecules of type
C
. This means that this reaction is a multiset rewriting rule. The value of
u
is the
flux of the rule application. Let us observe a system at some time steps 0
A chemical reaction such as
A
+
B
→
,
,
,...,
1
2
i
,
,
and consider a substance
x
that is produced by rules
r
1
r
3
and is consumed by rule
r
2
.If
u
1
[
i
]
,
u
2
[
i
]
,
u
3
[
i
]
are the fluxes of the rules
r
1
,
r
2
,
r
3
, respectively, in the passage
from step
i
to step
i
+
1, then the variation of substance
x
is given by:
x
[
i
+
1
]
−
x
[
i
]=
u
1
[
i
]
−
u
2
[
i
]+
u
3
[
i
]
.
Then, passing from one step to the next (after a fixed time interval), the number
of objects consumed and produced by a reaction
r
i
, for any single occurrence of
reactant/product of
r
i
,isthe
reaction flux
u
i
, which depends on the state of the
system by means of a map
ϕ
i
called the
regulator
of
r
i
. In this sense, a metabolic
system is a dynamical system, because the quantities of its substances change along
the time. In general, a
time series
is a sequence of real values intended
as “equally spaced” in time. In the following we will study metabolic systems in
a discrete mathematical perspective, by introducing
Metabolic P grammars
,or
briefly
MP grammars
. These grammars are related to
Psystems
introduced in
1998 by Gheorghe Paun [49, 50], and are a special kind of
multiset processing
grammars
[98, 92, 93, 94, 95, 97].
(
X
[
i
]
|
i
∈
N)
Definition 3.1 (MP grammar).
An MP grammar
G
is a generator of time series,
determined by the following structure (
n
,
m
∈
N
, the set of natural numbers):
G
=(
M
,
R
,
I
,
Φ
)
where:
1.
M
is a finite set of elements called
metabolites
or
substances
.
A
metabolic state
is given by a list of
n
values, each of which is associated to a
metabolite;
2.
R
=
{
x
1
,
x
2
,...
x
n
}
=
{
α
→
β
|
j
=
1
,...
m
}
is a set of
rules
,or
reactions
, with
α
j
and
β
j
mul-
j
j
m
;
3.
I
are
initial values
of metabolites, that is, a list
x
1
[
tisets over
M
for
j
=
1
,...
0
]
,
x
2
[
0
]
,...
x
n
[
0
]
providing the
metabolic state at step 0
;
4.
Φ
=
{
ϕ
1
,...,
ϕ
m
}
is a list of functions, called
regulators
, one for each rule, such
that, for 1
≤
j
≤
m
:
n
ϕ
j
:
R
→
R
.
(3.1)
G
defines, for any
x
∈
M
, a time series:
(
x
[
i
]
|
i
∈
N
,
i
>
0
)
(3.2)
in the following way. Let
s
[
i
]=(
x
[
i
]
|
x
∈
M
)
(3.3)
