Biology Reference
In-Depth Information
whether a Petri net has one of these properties. Note that these three cases do not cover all networks. In
fact, biochemical networks are usually open systems with a throughput of mass, as described by a flux
between external metabolites. Therefore they may, depending on conditions, have a positive or negative
mass balance, so that they usually belong to none of these classes.
P
-invariants are useful in checking the property of mutual exclusion. Two transitions are called to be
in mutual exclusion if there is no reachable marking that allows the two transitions to fire simultaneously.
The first step is to identify the marking set that could characterise the simultaneous activation of the
specified transitions. This marking is reachable only if it satisfies the conservation relation given by the
P
-invariants. For example, if four places need to have at least one token each to enable two transitions
to fire simultaneously, while the conservation sum is three, mutual exclusion occurs. Such a case is
irrelevant for metabolic networks provided that the molecule numbers are large enough. Alternatively,
if the token numbers represent millimoles or the like, token numbers need not be integer, so that mutual
exclusion is no problem either.
Often, two transitions are in mutual exclusion when they compete for the same input places set. If
the tokens are indivisible and once a transition takes the existent tokens, the other transition cannot
fire. If the tokens needed to reactivate the competing transitions are simultaneously regenerated for each
conflict case, the net is called persistent. In this case, the two transitions do not deactivate each other. At
first sight, some metabolic networks seam to be non-persistent because different enzyme reactions often
compete for the same substance. However, in real metabolic nets, even if the quantity of the common
resource is very small, the concurrent reactions share it, maybe in different percentage according to the
various reactions rates. Until now, there are no techniques based on Petri nets that can model accurately
this behaviour.
Biochemical networks often reach, after some initial transient, a stationary state. More concretely, this
is the case when the kinetic properties of the networks are such that the stationary state is asymptotically
stable, as is often the case [Clarke, 1980; Heinrich and Schuster, 1996]. At steady state, the following
equation holds:
V
=0
(6)
C
where
V
stands for the vector of net fluxes. They correspond to the flow of tokens per time in Petri
nets. Special attention has to be paid to the involvement of external places in matrix
C
. If we connect
outputs with inputs by additional transitions as explained in the previous section or additional arcs are
added to create self-loops next to initial and final places,
C
can contain the coefficients both for internal
and external places. Otherwise, it should only contain the coefficients for the internal places in order for
Eq. (6) to hold true.
A
T
-invariant (transition invariant) is a vector with the property that if each transition fires as many
times as the value of the corresponding component of the vector indicates, the original marking is
restored. Algebraically, these vectors are the solutions of Eq. (6). Therefore,
T
-invariants correspond to
flux distributions in steady state.
As Petri nets usually involve irreversible transitions only, all components of a
T
-invariant must be
non-negative.
T
-invariants with this property are called true
T
-invariants. Frequently, the net direction
of all biochemical reactions in a network is known, for example, because they are irreversible or have
a defined biochemical function. In this case, the orientation of reactions can be chosen in such a way
that all (net) fluxes are non-negative. Then, only steady-state flux distributions corresponding to true
T
-invariants are relevant.
