Biology Reference
In-Depth Information
Fig. 6. Treatment of reversible reactions. To model a reversible reaction A
+
B
←→
C
+
D, one usually introduces a transition
for each direction.
one might think of defining reversible transitions. This has, however, not been dealt with so far in the
Petri net literature.
Generally, Petri nets can be designed in different ways, called top-down and bottom-up. The first
method supposes to start with a very generalized form of the system and then, to detail it as much as
possible, until the basic units are reached. The second one starts with the “atoms”, building modules
which are then joint to model the real system. Sometimes it is advantageous to combine both approaches.
The importance of modularity is expressed by the ancient saying “
divide et impera
”.
MODELLING OF EXTERNAL METABOLITES
In metabolic networks, one needs to differentiate between internal and external metabolites. The
former are totally produced and then consumed in the given network, while the external metabolites
represent sources or sinks [Heinrich and Schuster, 1996]. Their amount is usually assumed to be
constant, due to availability in large excess or well-tuned biological regulation. If one considers the
given net as a part of a larger system, the external metabolites are a kind of boundary; or connection
points with the remaining part, in which pathways producing or consuming these metabolites exist.
An extension of the system so as to include those pathways is not useful, as the following example
illustrates. Glycolysis (the well known pathway of sugar degradation [Stryer, 1995]) contains a sequence
of reactions that transforms glucose into pyruvate, producing ATP. Glucose, pyruvate, ATP and also some
other metabolites are usually considered “external” for this pathway. We might include, in the model,
a reaction or pathway that consumes pyruvate, for example, for producing alanine. However, alanine
then would be an external metabolite. The model needs to be delimited somewhere. Algebraically, the
external metabolites can usually be identified also in the incidence matrix. Provided that each internal
metabolite is both produced and consumed within the net, the external metabolites correspond to those
rows in which all the coefficients have the same sign.
The modelling of external metabolites can be done in different ways. One of them is to fill all initial
places with an inexhaustible number of tokens (modelled by infinity). For the sink places, one could
allow them to accumulate tokens but has to take care in computing
T
-invariants (see below). If it is
preferred to use finite token numbers for the initial places, one could redefine the firing rule for the
transitions that have initial places in their preset (see Table 1) or final places in their postset, in such a
