Biology Reference
In-Depth Information
INVARIANTS IN PETRI NETS
When studying a system, it is always appropriate to begin with the study of its structural invariants.
They help in analysing the system's behaviour and checking its logical properties. The same is true for
Petri nets describing biochemical networks because the structural invariants do not depend on kinetic
enzyme parameters, which vary due to external influences and internal fluctuations. Basically, there
are two types of invariants in Petri nets:
P
-invariants and
T
-invariants [Reisig, 1985; Starke, 1990].
P
-invariants (place invariants) are vectors,
Y
, with the property that multiplication of these vectors with
any place marking reachable from a given initial marking yields the same result.
If
M
0
is the initial
marking and
m
is some arbitrary marking, the relation
Y
T
M
=
Y
T
M
0
describes a
P
-invariant and
is called
relation of marking conservation
. Taking into account consecutive markings (that are obtained
by firing of only one transition), it results that
Y
T
·
·
·
col
t
(
C
)
=
0, for each transition
t
, where
C
is the
incidence matrix. That means that, algebraically, these vectors are solutions of the equation
Y
T
·
C
=0
.
(3)
Invariants in Petri nets correspond to basic concepts in traditional biochemical modelling. In particular,
P
-invariants express conservation relations for metabolites, as becomes clear in the scheme shown in
Fig. 3. This net has the
P
-invariant ATP
+
ADP
=
const. In general, Eq. (3) is known, as for metabolic
systems, as the general form of conservation relations [Horn and Jackson, 1972; Clarke, 1980, Heinrich
and Schuster, 1996]. In most cases, these relations express the conservation of atom groups [Schuster and
Hilgetag, 1995; Schuster and H ofer, 1991]. In the example in Fig. 3, the adenosine moiety is conserved.
In algebraic terms, invariants form a linear vector space. This implies that if
I
1
and
I
2
are invariants,
also
c
1
I
1
+
c
2
I
2
with
c
1
,
c
2
being real numbers, are invariants of the net [Reisig, 1985]. For example, if
a biochemical net involves the
P
-invariants ATP
+
ADP
=
const. and NAD
+
NADH
=
const. [Stryer,
1995], then also ATP
+
ADP
+
2NAD
+
2 NADH
=
const. is a
P
-invariant. Normally, one chooses
invariants with the smallest integer coefficients and tries to decompose the invariants into the minimal
terms (such as ATP
+
ADP
=
const.). This leads to the concept of minimal invariants (see below).
In order that conservation relations reflect the conservation of atom groups, the coefficients in these
relations have to be non-negative. This leads to non-negative conservation relations [Schuster and
Hilgetag, 1995; Schuster and H ofer, 1991]. They correspond to semi-positive
P
-invariants in Petri nets
[Colom and Silva, 1990]. If all substances are involved in such conservation relations, the system is
called conservative [ Erdi and T oth, 1989; Horn and Jackson, 1972]. This implies that a positive linear
combination of all concentrations (token numbers in Petri nets) is constant in time,
g
i
z
i
(
τ
)=
u, g
i
>
0
for any
i
(4)
i
where denotes time. can be, for example, the number of some sort of atoms. In a closed system, that is, a
system without external metabolites, there is always one relation of the type Eq. (4) in which represents
total mass. In addition, there may be further relations of the type Eq. (4). If there is a positive linear
combination that increases in time, the system is called superconservative [ Erdi and T oth, 1989]:
g
i
z
i
(
τ
)
>
i
g
i
z
i
(
τ
)
,τ >τ
,g
i
>
0
i
for any
(5)
i
If the sum in Eq. (5) decreases in time until it reaches zero, the system is subconservative. The terms
conservative, superconservative and subconservative have also been coined for Petri nets. The program
