Biology Reference
In-Depth Information
Fig. 4. Gene regulatory network of
A. thaliana
flower morphogenesis. Pointed arrows reflect an activation relationship, whereas
the blunt arrows reflect a repression relationship between genes. For instance
EMF1
activates
TFL1
, but at the same time it is a
repressor of
LFY
.
to ignore the possible overflow. We adopt the latter option. To be more specific, when firing a case, we
fire all the transitions from that case, as if there was no bound, and after that we remove the overflow.
All the definitions from previous sections hold also for the newly introduced semantics of the net.
However we must slightly modify Eq. (4) and add an extra condition for variables
s
1
,
...
,
s
m
, so that the
possible markings do not exceed the bounds. In ordinary networks the state was unaffected by firing of
a case only when its balance was equal to 0. This is also true for nets with bounds, but the state can also
remain unaffected, when the balance is non zero and place bounds are reached. The modified version of
Eq. (4), referred to as balance (
s
1
,
...
,
s
m
,
t
1
,
...
,
t
n
), is as follows:
∧
m
(
t
1
(
s
i
)
∗
t
1
+
...
+
t
n
(
s
i
)
∗
t
n
=0)
∨
(
t
1
(
s
i
)
∗
t
1
+
...
+
t
n
(
s
i
)
∗
t
n
>
0
∧
s
i
=
L
(
s
i
))
(9)
1
i
RESULTS
In the previous sections we introduced the notion of a stationary state and we also proposed a way to
describe and analyze it mathematically. In this section we use an example of
A. thaliana
to demonstrate
the practical application of these methods.
Flower morphogenesis of A. thaliana
The gene regulatory mechanism, that lay behind the flower morphogenesis of
Arabidopsis thaliana
are broadly studied by biologists. The mechanisms that control the differentiation of the cells are well
understood. Extensive research allowed for construction of a gene regulatory network that depicts the
mechanism governing this process (see Fig. 4). For this reason the model of
A. thaliana
is commonly used
to test new mathematical formalisms for identifying stationary states [see Mendoza and Alvarez-Buylla,
1998; Mendoza
et al.
, 1999; Espinosa-Sotoa
et al.
, 2004].
During the flower morphogenesis the cells are subject to a differentiation. This biochemical process
ends when a chemical equilibrium is reached. The type of the equilibrium determines the type of the
cell.
The
A. thaliana
flower is build of four concentric whorls:
sepals
,
petals
,
stamens
and
carpels
. Research
has shown that cells building up different parts of the flower have different types of chemical equilibrium.
