Biology Reference
In-Depth Information
repressor is formed. This complex is unable to bind to the operator region, which
allows the RNA polymerase to bind to the promoter site, starting initiation of tran-
scription of the structural genes to produce mRNA
(
M
)
. However, in the absence of
allolactose
the repressor protein
R
binds to the operator region (
O
) and blocks
the RNA polymerase from transcribing the structural genes. The translation process
takes place once the mRNA has been produced. The
lac
Z gene encodes the portion
of the mRNA for the production of the enzyme
(
A
)
and translation
of the
lac
Y gene produces the mRNA that is needed for the production of the enzyme
permease
β
-galactosidase
(
B
)
. The final portion of mRNA produced by the
lac
A gene is responsible
for the production of the enzyme thiogalactoside transacetylase, which does not play
a role in the regulation of the
lac
operon [
4
].
A bistable system often involves a positive feedback loop or a double negative
feedback. Such motifs are very common in control mechanisms in gene networks.
The lactose operon and the arabinose operon of
Escherichia coli
both have positive
feedback mechanisms [
5
,
6
]. In the lactose operon, once external lactose is converted
into allolactose, the allolactose feeds back and forms a complex with the lactose
repressor, which allows the transcription process to start. This eventually leads to
synthesis of more enzymes to transport more external lactose into the cell and then
convert it into allolactose. More information about this mechanism and about the
lac
operon in general can be found in Chapter 1.
(
P
)
2.3
MODELING BIOCHEMICAL REACTIONS
WITH DIFFERENTIAL EQUATIONS
Differential equations are widely used to study reaction kinetics. In this section we
outline the basics of reaction modeling with differential equations. This approach will
be applied in Section
2.4
to justify two well-known differential equation models of
the
lac
operon.
Two or more species can react if they come together and collide. Collision alone
is not enough for a reaction to occur since the molecules have to collide the right
way and with enough energy. Even if the molecules are orientated properly when
they collide, a reaction may not happen if the molecules do not have enough energy.
On the other hand, if the molecules collide with enough energy but their orienta-
tions are not appropriate, then the reaction still may not occur. Reactions with a
single species, such as degradation reactions, do not require orientation of collisions.
Only if they have enough energy will those reactions take place. Most of the reac-
tions involve one or two species. Reactions with three or more species are extremely
uncommon.
The rate of a reaction reflects how fast or slow it takes place. There are different
approaches and methodologies to studying reaction rates. Mass-action kinetics is a
microscopic approach to reaction kinetics and it is one of the most common methods.
Modeling based onmass-action kinetics results in a systemof differential equations as
a mathematical description of the reaction rates. This approach is fully deterministic
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