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indicate how many molecules of each substance are consumed or produced in the single reactions. Their
analysis is based on the solution of algebraic equations, and is independent of any kinetic parameter.
Of particular interest are biochemical systems persisting in a steady state (see section “Steady state
pathways, elementary modes”), i.e., in which the concentrations of their substances have reached an
equilibrium. An elementary mode (this term has been coined in Schuster and Hilgetag, 1994) can be
regarded as a minimal set of reactions (resp. of the enzymes catalyzing them) that can operate at steady
state. Knowledge about the flux rates and the elementary modes of a system allows “to define and
comprehensively describe all metabolic routes that are both stoichiometrically and thermodynamically
feasible in a given group of enzymes” [Schuster et al. , 2000a].
A metabolic system can be modeled as a Petri net in a straightforward way, as has been demonstrated
for low-level nets in Reddy et al. , 1993, and Hofestaedt, 1994, and for high-level nets in Genrich et
al ., 2001. The Petri net structure then truly reflects the biochemical topology, and the incidence matrix
of the net is identical to the stoichiometric matrix of the modeled metabolic system. Accordingly, the
mentioned elementary modes correspond almost directly to the minimal T-invariants known from the
Petri net theory. An actual account of the structural analysis of metabolic networks and the analogy to
Petri nets is given in Schuster et al. , 2000b.
The use of Petri nets for modeling quantitative (kinetic) properties of biochemical networks, especially
for genetic and cell communication processes, was discussed in Hofestaedt, 1994, and Hofestaedt and
Thelen, 1998. Other contributions followed, using various types of Petri nets like stochastic nets [Goss
and Peccoud, 1998; 1999] and hybrid nets [Matsuno et al. , 2000]. Executable high-level net models of
metabolic pathways, and their (almost automated) construction, simulation, and quantitative analysis are
described in Genrich et al. , 2001.
The application of Petri nets to this field began in the nineties with the publications of Reddy et al.
[Reddy et al. , 1993; 1996]. They present a low-level (place/transition) net to model the structure of the
combined glycolytic pathway ( GP ) and pentose phosphate pathway ( PPP ) of erythrocytes. They use
the well-known algebraic methods to compute S- and T-invariants of the net. A thorough analysis of an
extended form of this pathway was performed by Koch et al ., 2000, which forms the starting point for
this paper.
For computing conservation relations (S-invariants) and elementary modes (T-invariants) of metabolic
pathways, the software package METATOOL [Pfeiffer et al. , 1999] has been developed (by biochemists)
and successfully applied in a number of cases. However, merely the integer weighted S-invariants are
detected. Moreover, only the overall reaction equations, i. e., the net effects of a pathway execution,
can be computed, and any consideration of its dynamics, in particular of the partial order of the reaction
occurrences, is missing.
The main achievements reported in this paper rely on the use of executable high-level net models, an
executable high-level net model, and on symbolic analysis. This allows to consider the following crucial
aspects:
(a) It is well known that the detection and interpretation of invariants can substantially improve the
understanding of systems. In the context of certain problems, however, the most interesting system
properties are not invariant. In these cases, very often the divergence of these structures from an
invariant is of major importance as it indicates a defect or effect of the substructure in question.
We shall introduce and formally define these concepts for high-level Petri nets in section “Steady
state pathways, elementary modes” and use them extensively, in section “Defects, effects and
invariants”, for the symbolic structural analysis of the quite complex sample pathway in section
“Models of the glycolysis and pentose phosphate pathway”.
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