Biology Reference
In-Depth Information
What is the strategy of attributing colors to the tokens (molecules) along a given pathway model?
Starting with a primary source substance of the pathway, we look for
conflicts
on the way to the sink(s).
By definition,
p
is a conflict place if it has more than one output transition, and all are enabled if
p
carries
a suitable token. If one of these alternative transitions occurs, all remaining transitions are disabled. In
our context, a conflict would cause no harm as long as all but one alternative paths starting at
p
would
end up again at this
p
without any lasting marking change. In general however, this is not the case.
When looking at the metabolism in one specific organism, alternative metabolic paths most often result
in different metabolic overall reactions. Therefore, they shall be discriminated and must not be combined
deliberately. This discrimination is performed by attributing different identifiers to the molecules and by
additionally blocking certain transitions for particular molecules using guards.
This shall be demonstrated by use of the sample net
P
, treated again as a low-level net by disregarding
all arc labels and guards. Starting with the source Gluc, the first conflict is encountered at G6P which
can be the reactant of either the reaction
l
2or
m
1. A G6P-molecule with destination
l
2 gets a color,
say C, and that one with destination
m
1 gets a different color (to be decided upon later). The guard [
x
C] prevents a C-token to be consumed by
m
1. Proceeding downwards the
GP
, we examine
F6P 1
-a
fusion place. As
F6P 2
on the right-hand side has no outgoing arc, a conflict does not exist. The next
conflict on the way down is found at GAP (the fused
GAP 1
and
GAP 2
), a conflict among the three
transitions
l
6,
l
7 and
r
4. The reaction
l
6 is trivial, as the loop
GAP 1
GAP 1
returns the token to
GAP 1
without affecting any other places. The conflict between
l
7 and
r
4 is difficult
to discuss at this moment without knowledge about the situation in the
PPP
at
GAP 2
→
l
6
→
DHAP
→
l
5
→
}
. We postpone
it to the end of this paragraph. Instead, the path from G6P into the
PPP
, in the middle and right part
of the figure shall be inspected. Choosing a separate color F for the molecules of the middle part is not
mandatory because there is no conflict. The next conflict occurs at Ru5P with the choice to continue
via
r
1or
r
2. The colors of these two molecules must be different from each other and from C; so we
choose G and H. The last conflict at GAP is the postponed one. However, because the color G has been
maintained from Ru5P via R5P until
GAP 2
and the tokens on
GAP 1
have the (different) color D, their
distinction is accomplished already: the G-molecules are removed by reaction
r
4 and the D-molecules
by
l
7. The resulting model is again
P
, but now regarded as a colored net by including the arc labels and
transition guards of Fig. 2.
DEFECTS, EFFECTS AND INVARIANTS
The following calculations were made by use of an experimental software package SY, written by H.
Genrich in Standard ML for the symbolic analysis of colored Petri nets. This package supports symbolic
calculations based on the incidence matrix of an executable colored net in Design/CPN. It inspects and
adopts the internal tables produced by the Design/CPN simulator for the graphical model and its data
base. It allows, among others, to form symbolic dot products and matrix products, and to apply useful
reduction rules and different formats for presenting the results.
Defects and S-invariants
We start with looking for S-invariants in the net
P
of Fig. 2. Obviously, there are four pairs of
ubiquitous substrates which, if produced or consumed by a reaction, are transformed into each other,
namely (ADP, ATP), (NADP
+
, NADPH), (2 GSSG, GSH), and (NAD
+
, NADH). This is verified by
applying the function DEFECT of the package SY to the four S-vectors
