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approaches because of potential advantages of Petri nets providing an intuitive graphical representation
and capabilities for mathematical analysis.
Signaling pathways regulate elaborated cell communication mechanisms by controlling various alter-
ation procedures of cell behavior such as cell growth, survival, proliferation, and apoptosis. By such
cellular communication mechanisms, cell activities could be subtly governed and maintained in a good
condition along with other biochemical interactions and processes.
Recently, Li
et al.
[8] proposed a qualitative modeling method of signaling pathways by paying
attention to the molecular interactions and mechanisms using discrete Petri nets. Further, they proposed
a timed Petri net based method of determining the
firing delay time
(or
delay time
for short) of transitions,
i.e., the time each transition takes to fire [9]. Their method to determine delay times in a timed Petri net
is based on the assumption that, for any reaction, a total amount of consumed substrates is equal to a total
amount of products of the reaction. Although this assumption ensures the concentration equilibrium of
the reaction, their method can only produce “exact delay time” for any transition in the Petri net model,
i.e., no time range is allowed for any transition in determining delay time. This means that their method
eliminates the variety of possible reaction speeds, which commonly exists in an ordinary signaling
pathway. Furthermore, their method is designed based on the strategy that in the conflict situation at
a certain place, “the same firing frequencies” should be assigned to the transitions going out from that
place. This strategy should be improved, since it does not reflect real reactions in a cell.
This paper proposes a new method to determine delay times in a signaling pathway, which resolves
above two problems, “exact delay time” and “the same firing frequency,” while keeping smooth signal
flows of a signaling pathway. A new concept “retention-free” is introduced to the Petri net. In a
retention-free Petri net, the total token amount flowed out of a place per time unit should not be less
than the one that flowed into the place. This new concept allows us to resolve the former problem,
providing a delay time in some range to any of the transitions. On the other hand, the latter problem is
resolved by introducing stochastic factors into transitions coming from a certain place where a conflict
happens. These stochastic factors reflect the reaction rates of corresponding biological reactions, making
it possible to realize more realistic signal flows of a signaling pathway.
The organization of this paper is as follows. After giving necessary definitions of timed Petri nets for
the discussion of this paper, a Petri net modeling method of a signaling pathway is introduced with the
example of the IL-1 signaling pathway according to the method proposed by Li
et al.
[8]. We then present
formulas in two cases, conflict-free transitions and conflict transitions, which express the conditions for
smooth token flows in a timed Petri net, i.e., smooth signal transductions in a signaling pathway. Based
on these formulas, we propose an algorithm to determine the delay time of transitions in a timed Petri
net model of a signaling pathway. The availability of this algorithm was confirmed by the results of
applying this algorithm to the timed Petri model of the IL-1 signaling pathway.
DEFINITIONS OF PETRI NETS
Petri net technology provides a powerful approach to modeling and simulating various concurrent
systems [1], which has been widely employed as a description method for biologists and computational
biologists owing to following advantages [10]:
1. “firm mathematical foundation” enabling formal and clear description of biological pathways as
well as their structural analysis, and
2. “visual representation of networks” which provides intuitive understanding of biological pathways
without any mathematical descriptions that are basically difficult for ordinary biologists.
