Biology Reference
In-Depth Information
reactions, is an
n
r
matrix where the element
N
ij
denotes the flux from the
i
-th metabolite to the
j
-th
reaction, i.e., the amount
δc
i
/
δt
of the metabolite concentration produced or consumed by that reaction.
The stoichiometric matrix of a pathway precisely corresponds to the incidence matrix of its low-level
Petri net model.
A metabolic pathway is in
steady state
if and only if the reaction rates fulfill the condition
×
r
δc
i
/δt
=
N
ij
.
j
=0
,i
=1
,...n,
or, in matrix notation
,N
·
v
=
O,
j
=1
for an integer vector
v
=
(
v
1
,
...
,
v
r
)
T
, called
flux vector
, where a component
v
j
is an integer weight
factor of the
j
-th reaction.
“Flux modes” constitute the core concept of the algebraic analysis for metabolic pathways that are
assumed to reach a steady state. “An
elementary flux mode
is a minimal set of enzymes that could operate
at steady state, with the enzymes weighted by the relative flux they carry. 'Minimal' means that if only
the enzymes belonging to this set were operating, complete inhibition of one of these enzymes would
lead to cessation of any steady-flux in the system” [Schuster
et al.
, 2000a].
Before relating biochemical analysis methods to the corresponding Petri net algorithms, two particular
questions have to be discussed. Firstly, in steady state analysis, those processes are of particular interest
that start with the source substances of the investigated pathway and finish with its sink substances. For
these external metabolites, constant concentrations have to be assumed to reach a steady state. Using
METATOOL, this is achieved by excluding external metabolites from the stoichiometry matrix, but
including the reactions affecting them: hence, their concentrations remain unchanged. In contrast, in our
Petri net models, we include the external substances and introduce an extra transition
StartEnd
that closes
the pathway to a cycle by supplying the initially needed substrates and consuming the finally produced
ones. This measure enables us to also identify those
internal
metabolites requiring initial markings and
to compute their amounts.
Secondly, we have to take into account the reversibility of reactions. An obvious solution is to admit
negative factors in the T-vector to denote the occurrences of reversible reaction transitions in the backward
direction. This would lead to T-vectors
x<
0
, not satisfying the standard definition of T-invariants for
Petri nets. However, instead of deviating from this definition, we introduce, for every reversible transition
t
, an additional complementary (reverse) transition
t
to the net (see Fig. 1). Doing that, we get - for
each reversible reaction
t
- a potentially endless loop (
t
,
t
,
t
,
t
,
...
). This slight disadvantage can be
turned into an advantage in high-level nets where we can discriminate the directions of certain reactions
according to the flux modes to which they belong. Thus we can model the situation in the cell, where
the direction of a reaction depends on the need of the cell controlled by the metabolite concentrations.
MODELS OF THE GLYCOLYSIS AND PENTOSE PHOSPHATE PATHWAY
The
glycolysis pathway
(
GP
) is a sequence of reactions that converts glucose into pyruvate with the
concomitant production of a relatively small amount of ATP. Then, pyruvate can be converted into
lactate. The version chosen for this paper is that one for erythrocytes [see Stryer, 2006]. In the Petri net
P
(Fig. 2), the
GP
consists of the reactions
l
1to
l
8.
The
pentose phosphate pathway
(
PPP
), also called
hexose monophosphate pathway
, again starts
with glucose and produces NADPH and ribose-5-phosphate (R5P) which then is transformed into
glyceraldehyde-phosphate (GAP) and fructose-phosphate (F6P) and thus flows into the
GP
. In Fig. 2,
