Biology Reference
In-Depth Information
Fig. 1. A sample reaction and its reverse. The transition (reaction)
r
3 consumes one R5P and one Xu5P molecule and produces
one GAP and one S7P molecule. The arc labels are variables that denoting the identities (colors) of the respective molecules.
Transition
r
3' is the reverse of
r
3. Its additional guard [
x <><>
D] demands that the color x of the GAP molecule must not
be equal to D in order to enable the transition
r
3'.
around a transition which determines the particular kind of its occurrence. A product
c
ij
x
tj
consists of
simultaneously applying the substitutions in
x
tj
to the variables in the arc expression
c
ij
, and yields an
integer linear combination of tokens from the color set of
s
i
.
Treating
x
as a one-column matrix, the state difference
Δ=
M
1
·
x
is an S-vector whose
entries are integer linear combinations of colors denoting the marking differences effected by
x
on the
different places. It is called the
effect
of
x
.
The T-vector
x
is called a
T-invariant
of
N
iff
C
−
M
2
=
C
·
·
x
=
O
.
π
M
2
performing a T-invariant leads from one state to the same state again (
M
1
=
M
2
).
It
re-generates
the state
M
1
, hence defines a
cyclic
process.
As mentioned in the introduction, the approach described in this paper concentrates on the mere
structure of the pathways, i.e., on the topology of the interconnections of metabolites via enzymatic
reactions. Hence, it is
structural
or
qualitative
as it does not deal with the kinetics of the reactions.
Constructing a Petri net of such a pathway is straightforward, representing metabolites as places, reactions
as transitions, and the stoichiometric relations by labelled directed arcs between them. Examples can be
found in Reddy
et al.
, 1993, Hofestaedt, 1994, Reddy
et al.
, 1996, Koch
et al.
, 2000, 2005, and 2008
(low-level nets), or in Genrich
et al.
, 2001, and section “Models of the glycolysis and pentose phosphate
pathway” of this paper (high-level nets). In the following, such a net is called
the
Petri net model of the
pathway. Figure 1 shows a sample high-level net model of a metabolic reaction. Jensen, 1992-1997,
gives an excellent introduction into the theory and application of colored Petri nets.
In our metabolic pathways, a distinction is made between
external
and
internal
metabolites according
to whether or not they are involved in reactions outside the system considered. External metabolites
are called
sources
resp.
sinks
of the pathway if they are produced resp. consumed by those (external)
reactions. A metabolic pathway is said to persist in a
steady state
if the concentrations of all internal
substances have reached a dynamic equilibrium: for each internal metabolite, the total rate of its
consumption is to that of its production. Assuming a constant activity of all enzymes involved in the
system, many (but not all) metabolic pathways reach such a dynamic equilibrium after some time. That
and how this happens, has been demonstrated for glycolysis, gluconeogenesis, citric acid cycle (TCA),
and combinations of them in Genrich
et al.
, 2001, by simulation runs of quantitative high-level Petri
net models. Structural analysis of metabolic systems in steady state aims at, among others, “elucidating
relevant relationships among system variables” [Heinrich and Schuster, 1998] and does not rely on
imperfectly known or doubtful kinetic data.
A formalization of steady state and related notions is given in Schuster
et al.
, 1996. For our paper,
we need the following. The
stoichiometric matrix
N
of a metabolic pathway with
n
metabolites and
r
A process
M
1
→
