Biology Reference
In-Depth Information
Table 1
Definition of the terms preset and postset
Name
Notation
Definition
Preset of t
t
{p|p ∈ P , pre ( p, t ) = 0 }
Postset of t
t
{p|p ∈ P , post ( t, p ) = 0 }
Preset of p
p
{t|t ∈ T , pre ( p, t ) = 0 }
Postset of p
p
{t|t ∈ T , post ( t, p ) = 0 }
Fig. 2. Marking and firing. M : P → N is called marking. For each place p ∈ P , M ( p ) represents the number of tokens which
exist in p . M ( p ) gives the local state, while the vector m gives the state of the system and is called vector state. A transition is
enabled/activated if M ( p ) pre( p, t ) ∀p ∈ P and K ( p ) M ( p ) pre( p, t ) + post( t, p ) ∀p ∈ P . The mapping K : P → N
represents the maximal capacity of a place, if the number of tokens is limited. After the enabled transition fires, the new state
of the system is M : P → N , so that M ( p )= M ( p ) pre( p, t ) + post( t, p ) ∀p ∈ P . In the example, the marking M = [0,
1, 1, 4, 1] is obtained from the marking M = [2, 5, 2, 1, 0] after transition t fires. The formal description is: m [ t>M .
of a substance have to react to produce how many molecules of product. The stoichiometric coefficients
are described by arc weights. Thus, the stoichiometry matrix containing these coefficients corresponds
to the incidence matrix of a Petri net (see below).
A further object - the token - was introduced to describe the dynamics of a Petri net. It is denoted by a
solid dot ( ) inside the circles representing places. In ordinary Petri nets, the tokens are indistinguishable.
They indicate the presence or absence of a condition, a signal, or a resource. In our case, the number
of tokens in a place stands for the number of molecules of that metabolite existing at a given moment.
Alternatively, tokens may correspond to any predefined unit measuring the amount of substance, such
as mole, millimole etc. However, this brings about that non-integer token numbers should be admitted.
This leads to continuous Petri nets, which are currently being developed [Alla and David, 1998; Matsuno
et al. , 2000].
The tokens that exist in the system at a given time describe the state of the system. This is called
marking, M ( P ) [Reisig, 1985; Starke, 1990]. The system state changes when a transition fires. This can
happen only if the transition is active/enabled, that means that every place from the input places set (Fig. 2
and Table 1) of the considered transition has at least as many tokens as the weight of the corresponding
arc. The set of input and output places of a transition t is denoted by
, respectively. This set
corresponds to the metabolites that act as reactants and products, respectively, in the reaction t . The new
state is obtained by subtracting from each input place of the considered transition a number of tokens
equal to the weight of the corresponding arc and adding in each output place a number of tokens equal
to the weight of the corresponding arc (Fig. 2 and Table 1). Formally, we can also speak about the input
and output transitions set of a place p
t and t
−•
p and p
, respectively, which contain all the transitions which
produce, respectively consume, the metabolite p .
It is useful to know between which places ( P ) and transitions ( T ) there exist arcs. For this purpose,
two mappings describing weights were introduced (see also Table 1): pre : PT N and post : TP N (with
N denoting the set of natural numbers). One can think about them also as matrices. The rows in pre
correspond to places and the columns to transitions, while in the matrix
post , the roles of rows and
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