Biology Reference
In-Depth Information
DISCUSSION
Petri nets provide a special formalism to describe processes in networks. In particular, they are suitable
to model biochemical networks. Here, we have shown that several concepts from Petri net theory have
a significance for this modelling. However, there are alternative formalisms, and it is difficult to decide
which formalism is best suited. To implement the calculations on computer, one usually translates Petri
nets into matrices. So one may argue that the networks could be modelled by matrices from the very
beginning. Indeed, Petri nets have the advantage to provide a means of visualisation. On the other hand,
biochemists use a special way of visualisation for decades [Stryer, 1995; Kanehisa and Goto, 2000]
(and chemists already for centuries). Multimolecular reactions such as “A
+
B gives C
+
D
+
E” are
represented by an arrow that has two upper ends and three lower ends. This arrow can be represented, in
a formal language, as a pair of n-tuples: ((A,B), (C,D,E)). In contrast, in a normal graph as used in graph
theory, edges correspond to simple pairs of nodes. In Petri nets, the representation is “disentangled” by
introducing additional nodes and arcs. The above reaction would then be represented by five place nodes
and one transition node (T) linked by five arcs. The arcs correspond to the following pairs of nodes:
(A,T), (B,T), (T,C), (T,D) and (T,E). It is a matter of taste which representation is preferred - one pair of
n-tuples or several pairs.
Many concepts from Petri net theory have counterparts in traditional biochemical modelling, for
example,
P
-invariants (conservation relations),
T
-invariants (flux modes), and minimal
T
-invariants (el-
ementary flux modes). In metabolism, minimal
T
-invariants can be interpreted as biochemical pathways.
Detection of these in complex networks is often not straightforward. It is helpful in determining maximal
conversion yields [Rohwer and Botha, 2001; Schuster
et al.
, 2000; Van Dien and Lidstrom, 2002].
The concepts of trap, siphon, deadlock, and liveness, among others, have not been considered in
biochemical modelling so far. Here, we have shown that these are helpful to characterize special
properties of metabolic networks. For example, the test for deadlock-freeness helps to determine
whether a biochemical pathway can attain a false equilibrium, where it is blocked. From another point
of view, this situation has been referred to as the danger of a turbo design of pathways [Teusink, 1998].
The liveness of a system indicates that all transitions are able to fire infinitely often, and the processes are
not eventually restricted to a subsystem. Traps can correspond to storage metabolites that are produced
during growth of an organism and steadily increase in their concentrations.
We have here analysed the example of the energy metabolism in
t
. brucei. If the accumulations in the
trap exceed a certain amount, this can cause product inhibition of some transitions (the aldolase reaction
in the example), forcing the system to stop working. This result is of interest for elementary-modes
analysis. It has been argued that this analysis can help assert the effects of enzyme deficiencies and
knockout mutations [Klamt and Stelling, 2003; Schuster
et al.
, 2000]. The example analysed here shows
that in a deficient system, the remaining elementary modes may not be functional because of occurrence
of a trap. Therefore, pathway analysis should be refined by considering traps, siphons, deadlock-freeness
and liveness.
Siphons can correspond to storage substances when they are gradually depleted during starvation.
An analysis of traps and siphons appears to be promising in studying diseases such as obesity and
hypercholesterolemia, which are related to over-accumulation of storage substances. It will be worth
including the analysis of traps, siphons, deadlocks, and liveness in metabolic simulation packages.
So far, in Petri net theory, transitions are always considered to be unidirectional. However, many
biochemical reactions such as all isomerases are known to be reversible in that their net flow can change
sign depending on the physiological state.
If such a reaction is described by two oppositely directed
