Biology Reference
In-Depth Information
Proof
: Directly from the definition of a non-conflicting set and from the fact that finite addition is
commutative.
Definition 8.
(Case in marking
M
)
:
For a given net
N
and its marking
M
, every maximal non-conflicting
subset of transitions will be called a
case
in
M.
Fact 2.
Every case in
M
contains only transitions active in
M.
Let
P
=
{
t
1
,
...
,
t
k
}
be a case of net
N
in marking
M
. Firing of case
P
,
Definition 9.
(Case firing):
M
, is defined as firing of all transitions from
P
, in an arbitrary order.
denoted
M
[
P
Once we formally defined the notion of case, we can use the
case semantics
of Petri nets. The
case semantics is a more coarse grained approach to the behavior of the network, which now consists
of sequential firing of cases. From now on, when referring to the Petri net semantics, unless stated
otherwise, we will mean the case semantics.
The case semantics is introduced to model the concurrency of chemical reactions. A case is a set of
reactions that are potentially independent and can take place at the same time. It should be noted that
cases of the network are not a its static property, because they depend on the current marking.
The net shown on Fig. 2 has exactly one case, that is
P
=
{
t
1
,
t
2
,
t
3
}
. As a result of case
P
being
fired, the marking of the net is left unchanged.
METHODS
One central issue that arises when studying
gene regulatory networks
is identifying their
stationary
states
.A
stationary state
can be defined in a variety of ways, however in this article, the term mean a
steady state i.e. the “attractor” of the system (similar concepts are considered in [Thomas and Kaufman,
2001a; Soule, 2003]. That is, when a system is in a stationary state, potentially many chemical reactions
may take place, but the concentrations of different substrates stay unchanged. Such a situation occurs
when products of one set of reactions are immediately consumed as substrates in another set of reactions.
The notion of a
stationary state
is of great importance in biology. Cells achieve
stationary states
as
a result of their differentiation process. It determines the type and the function of the cell. Changes in
the cell's chemical environment may lead to a change of its
stationary state
. A known example is the
behavior of
E. coli
when its environment is supplied with lactose [Stryer, 1995]. Presence of lactose
causes a rapid change in the cell's stationary state, making it possible for
E. coli
to exploit lactose as an
energy source.
Since the stationary states of a cell may change as an effect of some external factors we must assume
that the cells behavior changes over time.
A way of determining the possible stationary states of a chemical system might yield answers to
fundamental questions in medicine and biology. It would be very promising if one could check if a cell
can transform into a dangerous type (e.g. cancer) or if one could modify the cells chemical system in such
a way, that it would never reach a dangerous stationary state. Moreover when studying new organisms
a lot of useful information could be obtained if one could identify the possible stationary states of the
cells. In this paper we propose a method that might be very useful in this field.
Below we formally define the notion of a
stationary state
in terms of Petri nets. Moreover we propose
a method for finding such states based on a static analysis of the structure of the net.
