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low, medium, and high levels of lactose, it does not exhibit bistability.
f
M
=
G
e
∧
(
L
∨
L
e
),
f
E
=
M
,
f
L
=
G
e
∧
((
E
∧
L
e
)
∨
(
L
∧
E
)).
(2.61)
1.
Give justification for the transition functions presented by Eqs. (
2.61
). Use DVD
to calculate the system's fixed points and present a table with the fixed points for
all possible combinations of the parameter values.
2.
Now assume that in the model from Eqs. (
2.61
),
L
e
stands for external concen-
tration of lactose that is at least medium. Then introduce a new parameter
L
e
high
to denote high levels of external lactose. The combination of parameter values
L
e
=
0 now stands for medium external lactose. Modify the tran-
sition functions from Eqs. (
2.61
) to make the modified model exhibit bistable
behavior for medium lactose concentrations.
1 and
L
e
high
=
2.7
CONCLUSIONS AND DISCUSSION
In this chapter we developed and compared a number of mathematical models of
the lactose regulatory mechanism of
E. coli
. The focus was on bistability. The con-
tinuous models were previously published in [
8
,
9
], while the Boolean models we
have considered are new. Since the discovery of the
lac
operon by Jacob and Monod
in 1960 ([
19
,
20
]), both differential equation models and Boolean models have been
introduced to describe operon system dynamics, beginning with the work of Goodwin
[
21
] and Kauffman [
22
], respectively. Bistability for the
lac
operon system has been
experimentally observed and simulated by a number of continuous models, includ-
ing those considered in this chapter. However, establishing that Boolean models are
capable of capturing the system's bistability has only been done recently [
16
,
23
].
A necessary condition for bistability when modeling with Boolean networks is to
make possible distinguishing between at least three levels of inducer concentrations:
low, medium, and high. Both deterministic models (e.g., [
16
]) and models involving
stochasticity (e.g., [
23
]) have been proposed.
In this chapter we confirmed that Boolean models can approximate delayed differ-
ential equation models, preserving critical qualitative features such as bistability. It
was noted in [
8
] that the model in [
9
] does not consider inducer exclusion or catabolite
repression (both are external-glucose-dependent mechanisms), indicating that bista-
bility is independent from the presence of glucose in the extracellular medium. The
3-variable model from Table
2.1
in [
8
] ignores the lactose permease in the operon
regulation and considers only the role of
β
-galactosidase. This shows that of all feed-
back loops in the system, the
-galactosidase regulation is the one responsible for the
bistable behavior of the operon. The fact that the Boolean models that approximate
this continuous model preserve bistability further underscores this result. In [
8
,
9
]
the authors also show that bistability of the system is preserved when the delays in
the regulatory equations from Tables
2.1
and
2.2
are ignored. The Boolean models
β
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