Biology Reference
In-Depth Information
epigenetic differences also those involved in cell differentiation [Thomas and Kaufman, 2001a & 2001b].
Another interesting approach to the stationary analysis based on structural properties of the interaction
system can be found in [Soul, 2003]. In papers [Mendoza and Alvarez-Buylla, 1998; Mendoza
et al.
,
1999], Boolean networks were used as a model, but no automatic procedure was proposed for analysis
of stationary states. Sophisticated algorithm for recovering the stationary states for boolean networks
was proposed in [Gat-Viks
et al.
, 2004]. However, the applicability of this algorithm is restricted to very
simple models. The Petri net approach was applied in [Voss
et al.
, 2003], where authors exploit the well
known notion of
S/T-invariants
.
In contrast to previous methods we propose a fully automatic effective procedure for analyzing the
stationary states of the network. Our definition of the notion of stationarity reflects the chemical
equilibrium of the system that underlies the cell differentiation process.
In the rest of this section we define the notion of Petri net and introduce some basic concepts that we use
in the sequel. Next, we formally define the notion of stationary state and introduce a concise
Presburger
arithmetic
formula that describes all such states. Finally, we show how this formal approach can be
applied to find stationary states of the gene regulatory network describing the flower morphogenesis of
A. thaliana
.
Basic definitions
A Petri net consists of
places
and
transitions
. Transitions specify the dynamics of the network, that is
how
tokens
move from one place to another. A special function, referred to as the
weight function
defines
the quantitative aspect of the networks behavior. Firing of a transition changes the state of the network
in accordance to the weight function. The process induced by the network consists of sequentially fired
transitions. Symbols
N
and
N
+
denote sets of non-negative and positive integers, respectively.
Definition 1.
(Petri net):
The network is defined as a five element tuple,
N
=
(
S
,
T
,
F
,
W
,
M
0
)
, where:
1.
S
is a set of places.
2.
T
is a set of transitions.
3.
F
⊆
(
S
×
T
)
∪
(
T
×
S
)
defines the networks topology.
4.
W
:
F
→
N
+
is the weight function.
M
0
:
S
→
N
is the initial marking
(
state
)
.
Petri nets, when applied in modeling biochemical systems can be viewed in the following way: the set
of places
S
specifies the entities involved (e.g., chemical molecules), the set of transitions
T
5.
specifies
the possible interactions (e.g., chemical reactions) between these entities and relation
F
together with
the weight function
W
specify which and how many entities are consumed and produced by these
interactions.
Let's consider a simple chemical reaction of the form
n
A
+
m
B
+
k
C
→
y
D
+
x
E. The entities
involved here, are molecules of types A, B, C, D and E. There is only one possible interaction, the above
stated chemical reaction, let's call it
R
. Figure 1 depicts an appropriate Petri net. There are three arcs
ending in
R
and two which originate in
R
. The arcs that end in
R
lead from the places representing
substrates whereas the arcs that originate in
R
lead to places representing reaction products. These arcs
make up the
F
relation. The labels on these arcs correspond to values, assigned to them by the weight
function
W
and are equal to the coefficients from the reaction.
We have already constructed a network for our reaction. We have places representing reactants and
a transition representing the reaction. These elements make up the structure of the net, they define the
