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the
state vector
of
G
at step
i
, which can be seen as a function from the set of
metabolites to
R
, then the flux
u
j
[
i
]
of rule
r
j
at step
i
, is given by applying the
regulator
ϕ
j
to the state
s
[
i
]
:
[
]=
ϕ
(
[
])
.
u
j
i
s
i
(3.4)
j
∈
N
[
+
]
∈
For any
i
,thevalueof
x
i
1
, for each
x
M
is given by the following equa-
tion, where
α
(
x
)
and
β
(
x
)
denote the multiplicities of
x
in the multiset
α
j
and
β
j
,
j
j
respectively:
m
j
=
1
(
β
j
(
x
)
−
α
j
(
x
))
u
j
[
i
]
.
x
[
i
+
1
]=
x
[
i
]+
(3.5)
An MP graph is a natural graphical representation of an MP grammar
G
(originated
from [91]). An MP grammar
G
becomes an
MP system
when values for
time factor
τ
are added to
G
(however, very often MP
grammar and MP system are used synonymously):
•
τ
∈
R
,
population factor
ν
,and
mass factor
μ
is the
time interval
between two consecutive steps;
•
ν
∈
R
is the number of molecules which represents a
(conventional) mole
in the
model;
•
μ
∈
R
n
is the vector of the
mole masses
of metabolites.
An MP grammar
G
is
parametric
,whenaset
P
of parameters is added to
G
,and
metabolic states include also elements of
P
(to which, the state assigns real values).
If
G
is parametric, the time series of parameters also has to be provided in order to
specify
G
.
Table 3.1 gives an example of MP grammar, where 0 denotes an empty multiset
(0-rules correspond to introduction or expulsion of matter) and substance symbols
occurring in regulators denote the corresponding substance quantities. The quan-
tities that occur as arguments of a regulator (substances or parameters) are called
the
tuners
of that regulator. In Table 3.1
p
is a parameter, because it occurs as
a tuner, but does not occur in any reaction. In the following, we are focused es-
sentially on rules and regulators, then, often, initial values and parameters (if they
are present) are given separately. Moreover,
X
will also be used for denoting the
vector of metabolite values at step
i
(in the order they are specified), while
s
[
i
]
[
i
]
is
the
state vector
extending
X
with the values of parameters at step
i
(if they are
present). For the sake of simplification, we consider equivalent expressions such as:
ϕ
(
[
i
]
.
Three aspects are essential in the notion of MP grammar:
multiset rewriting
,
time discreteness
,and
molar perspective
. In fact:
1. rules transform multisets. We remark that an implicit assumption of Eqs. (3.5) in
Definition 3.1 is that all the rules can be applied. This is a reasonable hypothesis if
fluxes have values smaller than the quantities of available substances. If this is not
the case, further strategies need to be specified which establish how to proceed
when the hypothesis of metabolite abundance does not apply. For example, if
the rules consume a metabolite quantity which is not available, then all the rules
consuming it could be blocked at that step (or some of them according to some
specified criterion);
x
4
[
i
]
,
x
3
[
i
]
,
x
1
[
i
])
or
ϕ
(
x
4
,
x
3
,
x
1
)[
i
]
,or
ϕ
(
s
[
i
])
