Biology Reference
In-Depth Information
translation mechanisms are modeled by the places “mRNA repressor” (
m
14) and “repressor” (
m
15) and
the transitions
T
57
,
T
2
,
T
58
, and
T
3
.
The reaction forming a tetramer from monomers (Fig. 6(a)) can be represented by the HDN as is
shown in Fig. 6(b). (HPN can not model this type of reaction, since, any reaction speed should be
realized by assigning a function of values of the places to the continuous transition.) By comparing this
representation and the representation of Fig. 6(c), we recognize that an HFPN allows us to represent such
reaction naturally and intuitively. Tetramer formation is represented in the HFPN by places “repressor”
and “4 repressor” (
m
16) and three transitions
T
3
,
T
59
, and
T
60
. The function
5
m
1
5
is assigned
for the input (output) arcs to (from) the transition
T
59
as a flow speed. Note that the speed of the input
arc is four times faster than the speed of the output arc.
For the repressor forming tetramer, we determined that about 96% of them bind to the operator site,
about 3.99% of them bind to the other DNA sites, and about 0.01% of them do not bind to DNA. These
percentages are determined from the description in the literature [Lewin, 1997]. The places “operator
bind” (
m
7), “DNA bind” (
m
17), and “not bind” (
m
18) represent the amount of these repressors.
According to these binding rates, the firing speeds of the transitions
T
60
,
T
61
, and
T
62
were given by
96
4
×
m
15
×
m
16
100
399
m
16
10000
×
m
16
10000
respectively.
We separate the concentration of lactose to two places “lactose outside of a cell” (
m
29) and “lactose”
(
m
9) for the convenience of describing the function of the
lacY
gene in the next subsection. Concentration
of the allolactose is represented by the place “allolactose” (
m
8) whose accumulation rates are given at
the transitions
T
94
and
T
76
. It is known [Hofest adt, 1994] that allolactose is produced from the lactose
existing outside of a cell as well as the lactose inside a cell. Since it is natural to consider that a production
speed of allolactose is faster than the passing rate of allolactose through the cell membrane, the speed of
transition
T
76
(
m
9
/
5
) is set to be faster than the speed of transition
T
94
(
m
29
/
10
).
The negative regulation in Fig. 5 was realized in the following way:
The place “operator” receives tokens if
-
the concentration of the place “operator bind” exceeds the threshold 1 given at the test arc to the
transition
T
64
as a weight, and
-
the concentration of the place “allolactose” does not exceed the threshold 4 given at the inhibitory
arc to the transition
T
64
.
Four molecules of allolactose need to bind with one molecule of tetramer repressor. Accordingly the
function of input arc from the place “allolactose” to the transition
T
78
should be four times faster than
that of input arc from the place “4 repressor”. In summary, we gave formulas,
-
,
, and
4
×
m
8
×
m
16
to the arc from the place “allolactose” to T78,
-
m
8
×
m
16
to the arc from the place “4 repressor” to
T
78
, and
m
16
to given at the arc from the
T
78
to the place “complex”.
The transition
T
78
gives the complex forming rate of the allolactose and the tetramer of repressor
proteins. The place “complex” (
m
10) represents the concentration of the complex. Allolactose can
also release the tetramer of repressor proteins from the operator site by forming a complex with it. The
transition
T
77
and the arcs from/to the transition realize this mechanism. Discrete transitions are used
for the transitions
T
77
, since only a discrete transition is available for the arc from the discrete place (a
continuous amount can not be removed from a discrete place). In order to realize a smooth removal of
allolactose, a small delay time (0.5) is assigned to the transition
T
77
.
The transitions
-
m
8
×
T
4
,
T
5
,
T
6,
T
13
,
T
14
,
T
15
and
T
18
represent the natural degradation of the
corresponding substances.
