Biology Reference
In-Depth Information
Hydrolyzing lactose to glucose and galactose
The
lac
operon transcription and translation mechanisms are described in Fig. 7 together with the
effects of two products of the genes
lacZ
and
lacY
on hydrolyzing lactose to glucose and galactose. The
effect of the gene
lacA
is not included in this figure, since the
lacA
gene does not play a role in either
the lac operon gene regulatory mechanism or the glycolytic pathway.
The places “mRNA lacZ” (
m
19), “mRNA lacY” (
m
23), and “mRNA lacA” (
m
27) represent the
concentrations of mRNAs transcribed from the genes
lacZ
,
lacY
, and
lacA
, respectively, and the
transcription rates are given at the discrete transitions
T
66
,
T
69
, and
T
72
as the delay times.
The places “LacZ” (
m
20), “LacY” (
m
24), and “LacA” (
m
28) represent the concentrations of proteins
translated from the
lacZ
,
lacY
, and
lacA
mRNAs, and the translation rates are given at the continuous
transitions
T
67
,
T
70
, and
T
73
. As is shown in Table 3, we set the initial values of proteins LacZ, LacY,
and LacA to 5, 2.5, 1, respectively. These values are chosen according to the production ratios of LacZ,
LacY, and LacA proteins [Lewin, 1997]. Actually, the formulas
m
19,
m
32
2
m
27
5
are assigned to the
transitions
T
67
,
T
70
, and
T
73
as the speeds, according to the fact that the proteins of lacZ, lacY, and
lacA are produced in the ratio 1:
, and
1
1
5
.
Degradation rates of mRNAs (proteins) are assigned to the transitions
T
7
,
T
9
, and
T
11
(
T
8
,
T
10
, and
T
12
).
In this model, to represent the lac operon DNA, only discrete elements, discrete transitions
T
66
,
T
68
,
T
69
,
T
71
, and
T
72
, and discrete places X1 (
m
21), X2 (
m
22), X3 (
m
25), and X4 (
m
26), are used. The
discrete places X1 (X3) represents the Boolean status of the transcription of
lacZ
gene (
lacY
gene).
That is, each time transcription of
lacZ
(
lacY
)
is finished, the place X1 (X3) receives a token. At the
discrete transition
T
68
(
T
71
), the delay time 0.051 (0.065), which is required for the RNA polymerase
moving from the end of the
lacZ
gene to the beginning of the
lacY
gene (the end of the
lacY
gene
to the beginning of the
lacA
gene), is assigned. The delay times 3.075 and 1.254 are assigned to the
transitions
T
66
and
T
69
, respectively, according to the fact that the length of the
lacZ
gene (
lacY
gene)
is 3075 bp (1254 bp). The delay time 0.682 at the transition
T
72
represents the length of the
lacA
gene (the length of the
lacA
gene is 682 bp). The lengths of the genes are obtained from the website
http://genolist.pasteur.fr/Colibri/genome.cgi
.
Note that we can know the transcription status of the gene
lacY
(the gene
lacA
) by observing whether the discrete places X2 (X4) contains tokens or not.
Recall that the product of the gene
lacZ
is an enzyme which hydrolyzes lactose to glucose and
galactose. This reaction is modeled by using the places “lactose”, “galactose” (
m
30), and “glucose”,
and the transitions
T
75
,
T
16
, and
T
17
. In our model, 20 is assigned to each of the places “lactose” and
“lactose outside of a cell” as an initial value. A test arc is used from the place “lacZ” to the transition
T
75
, since the enzyme is not consumed.
We consider that the production rates of glucose and lactose depend on both the concentration of
lactose and the concentration of the product of the
lacZ
gene. The formula
2
:
m
29
m
29+
m
24
m
24
×
10
representing
×
the speed of the transition
T
74
reflects this idea.
We mentioned that the gene
lacY
encodes the permease that brings lactose into the cell. In Fig. 7, this
function is realized with the places “LacY”, “lactose outside of a cell”, and “lactose”, and the transitions
T
74
,
T
13
, and
T
14
. Since the product of
lacY
gene is an enzyme, test arc is used from the place “LacY”
to the transition
T
74
, and the speed of this transition is given by the formula
m
9
m
9+
m
20
m
20
×
10
similarly above.
The weight 2.5 (5) of the arc from the place LacY (
m
24) (LacZ (
m
20)) to the transition
T
74
(
T
75
)
corresponds to the basal concentration of LacY (LacZ) presented in Table 3.
×
