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give a step-by-step tutorial on how to model biochemical reactions using differential
equations. These approaches are then used in Section
2.4
to justify several widely used
differential equation models of the
lac
operon. Section
2.5
extends the material from
Chapter 1 on Boolean networks, describing Boolean models with varying elimination
times and Boolean models involving delays. In Section
2.6
we build Boolean model
approximations of the differential equation models from Section
2.4
and demonstrate
that those models capture the bistability of the system. Section
2.7
contains some
closing comments and conclusions.
2.2
THE LACTOSE OPERON OF
ESCHERICHIA COLI
The
lac
operon is a well-known example of an inducible genetic circuit. It has been
serving as a model system for understanding many aspects of gene regulations since
its discovery in the late 1950s. The
lac
operon encodes the genes for internalization
of extracellular lactose and then its conversion to glucose. A cartoon that depicts the
major regulatory components of this system is shown in Figure
2.2
.The
lac
operon
consists of a promoter/operator region
and three structural genes
lac
Z,
lac
Y, and
lac
A.
A
regulatory gene
lac
I(
I
) preceding the
lac
operon is responsible
for producing a repressor
(
P
and
O
)
protein. The molecular mechanism of this operon is
as follows: In the presence of allolactose, a complex between allolactose and the
(
R
)
FIGURE 2.2
Schematic representation of the lactose operon regulatory system.
See the text for details. Reprinted from
Methods in Enzymology
, v. 484, Yildirim, N. and
Kazanci, C., Deterministic and stochastic simulation and analysis of biochemical reaction
networks the lactose operon example, p. 371-395, (2011), with permission from Elsevier.
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