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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 transition 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
]
.
In a MP system, for any state, the flux of each rule is provided by a func-
tion, called
regulator
of the reaction. Substances, reactions, and regulators (plus
parameters which are physical variables different from substances occurring as
arguments of regulators) specify a dynamics, given at discrete steps. At every
step, the dynamics is governed by a partitioning of mass determined by the fluxes
of rules consuming it. An MP system is described entirely by an
MP grammar
where multiset rewriting reactions are given with the corresponding regulators.
An MP grammar is also specified by an MP graph where the relationships be-
tween reactions and regulators appear in a direct way (see Fig. 1).
φ
1
φ
1
r
1
r
1
φ
2
φ
2
+
=
x
y
r
2
x
y
r
2
r
3
r
3
φ
3
φ
3
Stoichiometry
Flux Regulation
MP Graph
Fig. 1.
MP graphs define the stoichiometric and regulatory components:
substances
(circles),
input
and
output
gates (green triangles) are connected by reactions arrows.
The
regulators
(rectangles) determining the flux of rules are linked with dashed arrows.
2 Arithmetical MP Systems
The Arithmetical MP Systems are a class of MP Systems describing a universal
computational model. Universality can be easily proved modelling a register
machine or, with a little more effort, the
μ
-recursive functions.
Notational Conventions.
A set of substances
S
, considered in a conventional
order, determines a vector
X
=(
x
1
,...,x
n
)
n
of substance quantities provid-
ing the metabolic state of the system. When there is no possibility of ambiguity,
for any
j,
1
∈
N
≤
j
≤
n
, we use the notation
x
j
as substance quantity rather then
substance symbol.
Let
R
be the set of reactions having the set of regulators
Φ
=
{
ϕ
r
|
r
∈
R
}
.A
reaction
r
is then represented by a left vector
r
−
∈
N
n
of reactants and a right
vector
r
+
∈
N
n
of products. The component-wise difference
r
#
=
r
+
−
r
−
is the
stoichiometry balance
of
r
.
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