Mechanisms of Gene Regulation: Boolean Network Models of the Lactose Operon in Escherichia coli - Mathematical Concepts and Methods in Modern Biology - page 3

in E. coli. Since the seminal work by Jacob and Monod [ 2 , 3 ], the lac operon has
become one of the most widely studied and best understood mechanisms of gene
regulation. It has also been used as a test system for virtually every mathematical
method of modeling gene regulation (see, e.g., [ 4 - 10 ]).
The rest of the chapter is organized as follows: In Section 1.2 we outline the basic
structure of the lac operon. This section is meant only as a quick introduction and
is not comprehensive in any way (see [ 11 ] for a more thorough introduction). Sec-
tion 1.3 focuses on the construction and initial testing of a mathematical model with
an emphasis on Boolean networks. A primer on Boolean algebra is included to make
the chapter self-contained. We consider several Boolean models of the lac operon,
then introduce and utilize the web-based suite of applications Discrete Visualizer of
Dynamics [ 12 ] to perform initial testing and validation of the models. In Section 1.4
we turn to the question of determining the steady states (fixed points) of Boolean
networks, casting the question in the broader context of polynomial dynamical sys-
tems and the use of Groebner bases for solving systems of polynomial equations. In
Section 1.5 we point out directions for extending and generalizing the models and
provide some concluding remarks regarding the possible use of this material in the
undergraduate mathematics and biology curricula.
1.2 E. COLI AND THE LAC OPERON
E. coli is a short rod-shaped bacterium which is a common intestinal resident of
mammals and birds. It has been the object of extensive study for decades. DNA
replication, transcription, and translation were all elucidated in E. coli before they
were studied in eukaryotic cells. Its physiology is well-understood and its entire gene
sequence is known. For an overview of its importance to the study of genetics, see,
for example, [ 13 ].
Since it lives in the intestines, any given E. coli bacterium's nutrition depends
upon the diet of the animal whose digestive tract it inhabits. Digestion of the com-
plex biomolecules in the foods consumed by the animal generally provides the bac-
terium with all of the simple biomolecules it needs. Digestion of starches provides
the monosaccharide (simple sugar) glucose, digestion of proteins provides all of the
amino acids, and, whenever milk is consumed by the host, E. coli is also exposed to
lactose (milk sugar). Lactose is a disaccharide consisting of one glucose sugar linked
to one galactose sugar. Galactose is a six carbon simple sugar which is an isomer of
glucose. It has the same chemical formula as glucose but differs in the position of one
hydroxyl group. Like glucose, galactose can be used as an energy source, although
some additional enzymatic manipulation will be required.
In order to import sugars across their plasma membranes, cells must produce
specific sugar transport proteins (e.g., glucose requires a glucose transporter; lactose
requires a lactose transporter, and so on). Once the sugar has been imported into the
cell, specific enzymes will act on the sugar, either to use it to make a required cellular
molecule (in the constructive processes collectively called anabolism ) or to break it