Biology Reference
In-Depth Information
Going up
Going up
Going down
0
0
p
p
(a)
(b)
FIGURE 2.1
Bistability and hysteresis emerging in cellular reaction networks. Bistability can be
reversible (a) or irreversible (b). In reversible bistable systems, the steady states can
change from low to high or high to low state as the parameter
p
is tuned. However, the
steady state changes from a low state to a high state at a higher value of
p
than that for
which the change from a high steady state to a low steady state occurs. In an irreversible
bistable system, once a system shifts from a low state to a high state, it stays there even
when the value of the parameter
p
is set back to a value much lower than that where an
earlier shift took place.
In the case of the lactose (
lac
) operon in
Escherichia coli
(
E. coli
), it has been known
since the 1950s [
1
] that it exhibits a hysteresis effect: In the absence of glucose, the
operon is uninduced for low concentrations below a threshold
L
1
of extracellular
lactose and fully induced at high concentrations above a threshold of
L
2
.Inthe
interval
, both induced and uninduced cells can be observed and their status
depends on the cell history (the system response is hysteresis). The interval
(
L
1
,
L
2
)
is
defined as a region of bistability and is referred to as
maintenance concentration
. Cells
grown in an environment poor in extracellular lactose will have low levels of internal
lactose and allolactose and may remain uninduced for lactose levels in the interval
(
(
L
1
,
L
2
)
. In contrast, cells grown in a lactose-rich environment remain induced for
concentrations in the interval
L
1
,
L
2
)
. A recent discussion of the bistability feature
of the
lac
operon together with green-fluorescence and inverted phase-contrast images
of a cell population showing a bimodal distribution of
lac
expression levels can be
found in [
2
]. A survey of the quantitative approaches to the study of bistability in the
lac
operon is given in [
3
].
In this chapter we examine two types of mathematical models of the
lac
operon:
differential equation models and Boolean network models. In both cases the focus
will be on bistability and on the capability of the models to capture the bistable
behavior of the system. The chapter is organized as follows: Section
2.2
presents a
short description of the regulatory mechanism of the
lac
operon. In Section
2.3
we
(
L
1
,
L
2
)
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