Biology Reference
In-Depth Information
Biochemical evaluation of the T-invariants
The software package METATOOL [Pfeiffer
et al.
, 1999] allows to compute the
elementary
(
flux
)
modes
(corresponding to the 'non-cyclic portions' of the minimal T-invariants) of a pathway. For each
mode, it computes (1) the T-vector, determining which reactions have to occur how often to proceed
from the sources to the sinks, and (2) the overall reaction equation.
With the colored Petri net approach and applying the package SY, we get additional information not
only about the T-invariants but also about the dynamics of the system. The symbolic treatment of
the T-vectors yields as one crucial result the marking (amount of molecules), needed at the beginning
and provided by the starting transition
s
1, to run the system without deadlock from its source to the
sink. This initial marking is 'appropriate' because it is the minimum amount of molecules necessary
for a simulation. Moreover, the stepwise construction of the symbolic parameterized T-invariants yields
knowledge not only about the frequency of transition occurrences (during a run along the invariant) but
also about the partial order in which these transitions have to occur.
An interesting question arises concerning the independence of the three T-invariants. Theoretically,
they are linearly independent because the transitions
l
2 and
l
2
are treated as not being related to each
other. If however
l
2 and -
l
2
are identified, the T-vectors get linearly dependent. This corresponds to
the observation that the overall reactions (G), (P), (R) are related to each other by the equation (P)
=
2
·
(G)
+
(R).
The problem, however, lies in the fact that a steady state process including both a reaction (
l
2) and
its
rev
erse (
l
2
) is biochemically not feasible. And on the other hand, T-vectors with negative elements
cannot be T-invariants according to the definition given in section “Steady state pathways, elementary
modes”.
The construction of the compound net
P
∗
can also be looked at from a different perspective, throwing
more light on the nature of the token colors and the conflicts. Let us discuss the three independent
modes identified in the previous subsection “Effects and T-invariants”, as separate net models. They are
depicted in a simplified version as Fig. 6, omitting all ubiquitous molecules and the 'uncritical' reactions
m
2 and
m
3.
The first mode (G), glycolysis, contains no conflict. So, only one token color, C, is needed. The
second mode (P) has two internal conflicts at Ru5P and GAP which are decided by use of the two colors
G and H. The third mode (R) contains the same two conflicts as (P), now solved by G
and H
.
These 'mode specific' conflicts describe (model) situations as happening in reality, with a great number
of molecules of every substance involved. From the definition of steady state follows that no molecule
inserted by the source may get stuck on its way to the sink. If it would, the concentration of an intermediate
substance would be increased, contradicting the definition. Looking at the right hand branch of (P) in
Fig. 6, the tokens entering that branch at Ru5P can leave it only as F6P- or GAP-molecules by means of
r
4 and
r
5. The reaction
r
4 needs one G and one H, and
r
5 one additional H. The one G or two H tokens,
resp., can only be provided by
r
1 occurring once or by
r
2 occurring twice, respectively. In organisms,
the molecules of one substance cannot be distinguished and cannot be forced to choose one out of more
alternative paths. Yet, the transitory increase of a substance concentration leads to a slowing down of
reactions producing it and an acceleration of reactions consuming it. The opposite happens in case of a
concentration decrease. So, in the long run, a relative occurrence ratio of 1:2 will be established among
r
1 and
r
2.
In contrast to the mode specific conflicts, the remaining ones are consequences of glueing the mode
nets (G), (P), and (R) into one single model
P
∗
. As these three processes are independent from each
