Biology Reference
In-Depth Information
b. For each input arc
a
i
,
T
specifies a function
f
i
(
m
1
(
t
)
,
...
,
m
p
(
t
))
>
0 which defines the speed
of consumption from
P
i
when it is firing. If
a
i
is a test input arc, then we assume
f
i
0 and no
amount is removed from
P
i
. Namely,
d
[
a
i
](
t
)
/
dt
=
f
i
(
m
1
(
t
)
,
...
,
m
p
(
t
))
, where [
a
i
](
t
)
denotes
the amount removed from
P
i
at time
t
through the continuous input arc
a
i
during the period of
firing.
c. For each output arc
b
j
,
T
specifies a function
g
j
(
m
1
(
t
)
,
...
,
m
p
(
t
))
>
0 which defines the speed
of amount added to
Q
j
at time
t
through the continuous output arc
b
j
when it is firing. Namely,
d
[
b
j
](
t
)
/
dt
=
g
j
(
m
1
(
t
)
,
...
,
m
p
(
t
))
, where [
b
j
](
t
)
denotes the amount of the contents added to
Q
j
at time
t
through the continuous output arc
b
j
during the period of firing.
2.
Discrete transition:
A
discrete transition
T
of HFPN consists of discrete/test input arcs
a
1
,
...
,
a
p
from places
P
1
,
...
,
P
p
to
T
and discrete output arcs
b
1
,
...
,
b
q
from
T
to places
Q
1
,
...
,
Q
q
.
Let
m
1
(
t
)
,
...
,
m
p
(
t
)
and
n
1
(
t
)
,
...
,
n
q
(
t
)
be the contents of
P
1
,
...
,
P
p
and
Q
1
,
...
,
Q
q
at time
t
, respectively. The discrete transition
T
specifies the following:
a. The
firing condition
is given by a predicate
c
(
m
1
(
t
)
,
...
,
m
p
(
t
))
. If this is true,
T
gets ready to
fire.
b. The
delay function
given by a nonnegative integer valued function
d
(
m
1
(
t
)
,
...
,
m
p
(
t
))
. If the
firing condition gets satisfied at time
t
,
T
fires in delay
d
(
m
1
(
t
)
,
...
,
m
p
(
t
))
. However, if the
firing condition is changed during this delay time, the transition T looses the chance of firing
and the firing condition will be reset.
c. For each input arc
a
i
,
T
specifies a nonnegative integer valued function
f
i
(
m
1
(
t
)
,
...
,
m
p
(
t
))
>
0
which defines the number of tokens (integer) removed from
P
i
through arc ai by firing. If
a
i
is
a test input arc, then we assume
f
i
0 and no token is removed.
d. For each output arc
b
j
,
T
specifies a nonnegative integer valued function
g
j
(
m
1
(
t
)
,
...
,
m
p
(
t
))
>
0 which defines the number of tokens (integer) are added to
Q
j
through arc
b
j
by firing.
In Fig. 3, examples of continuous transition and discrete transition are shown.
From the above definition, it may be obvious that in the HFPN model, the dimer-to-monomers reaction
can be intuitively represented as Fig. 2 (c). Not only this simple example but also more complex
interactions can be easily and intuitively described with HFPN. The software GON is developed and
implemented based on this HFPN architecture.
CIRCADIAN RHYTHMS IN DROSOPHILA
The control mechanism of autoregulatory feedback loops of
Drosophila
circadian rhythms has been
intensively studied [9-13,16] and some fine modelings by ODEs with detailed coefficients have also been
reported [14,15]. These ODE-based models can be easily described with HFPNs with GON. Highly
appreciating such fine modelings, we first show an HFPN realization of the model due to Ueda
et al.
[15].
Moreover, we also show that an HFPN can be designed with GON easily and intuitively by interpreting
the biological facts and observations given in [9-13,16]. GON is intended to be a naive platform where
we can create hypotheses and evaluate them by simulation. This feature is especially important when
only rough modeling is enough or enough information is not available for fine modeling.
Figure 4 shows the scheme of the regulatory mechanism of five genes contributing to the
Drosophila
circadian rhythms;
period
(
per
),
timeless
(
tim
),
Drosophila Clock
(
dClk
),
cycle
(
cyc
) and
double-time
