Biology Reference
In-Depth Information
complexes. An example is shown in Fig. 2, which includes the pathway modeled in bipartite graphs
in [5]: the TGF-
β
pathway. In [5] the authors introduced in their graph representation 'virtual decom-
position' arc whose semantics is different from that of other arcs, whereas in Fig. 2 all the arcs account
for the consistent semantics: the transmission of
activity
through
molecular recognition
. The Petri net
in Fig. 2 does not describe all the states of all the components of the receptor complex. For example,
it omits TGFbR2 in a complex composed of Smad7, Smurf1, TGFb, TGFbR1, and TGFbR2. The state
of TGFbR2 can be omitted because it is regarded as not to hold
activity
and
molecular recognition
but
to serve for keeping the structure and location of the other focused components. Such simplification is
upheld by the ontological explication that activity and molecular recognition accounts for the common
essence of the
process of signaling
. Thus the Petri net units prevent the Petri net graphs from being too
complicated and unmanageable as well as from being inconsistent.
Tokens in Petri Nets for Signaling Pathways
In the Petri net representations of biological pathways reported so far,
tokens
in a
place
have accounted
for the concentration of molecules in a certain state that the place indicates. Because the Petri net
units coincide with units of transmitting
activity
, the
tokens
in the units account for the concentration
of molecules carrying activity. If activated molecules are free in a cell, then the
tokens
represent
concentration of the activated molecules themselves. If activated molecules belong to complexes, then
the
tokens
represent the concentration of the activated components in complexes.
Petri nets in Tables 1 and Fig. 2 are Condition Event nets: the most fundamental class of Petri nets,
where every
place
takes a
token
of one or zero. Therefore the
tokens
come to represent existence of
molecules of carrying activity. In other words, the
tokens
represent activation or inactivation of individual
molecules alternating progressively in a signaling pathway. Accordingly
tokens
in the initial state of a
Petri net represent a distribution of activated molecules before receiving extracellular stimuli (Fig. 2).
Tokens
in the final state of a Petri net represent a distribution of activated molecules as a consequence
of response to extracellular stimuli. It is the remarkable advantage of Petri nets that a progression of
activations is concisely visualized by execution of a
token game
in the nets.
The Petri net units can represent correct stoichiometry of every reaction at every time point, even when
they are applied to higher classes of Petri nets. When applied, one should make firing rules satisfy the
condition that places should carry the same numbers of
tokens
if the
places
are labeled external states
involving the same members of complexes. For example in Fig. 2, the number of
tokens
in the place
of TGFb(B)
[TGFb:TGFbR2]
and that of TGFbR2(B)
[TGFb:TGFbR2]
should be the same at any time
point in the
token game
.
DISCUSSION
When Petri nets are used for a conceptual formalization of causal sequences of events, the formalization
should be based on consistent semantics accounting for the common essence of the causal sequences.
CSNO provides us with such consistent semantics for signaling pathways. I would like to propose
standardization of Petri net units based on CSNO, in which every
place, transition
, and
token
is specified
on the basis of ontological definitions of the
process of signaling
. On the other hand, semantics of
existing Petri net models is left discrepant, so that some places and transitions might be unspecified and
inconsistent.
