Biology Reference
In-Depth Information
coordinated production of these two proteins when lactose alone is present. In this
chapter we show howBoolean networks can be used to model the control mechanisms
of the
lac
operon system. The chapter can be used as an introduction to mathematical
modeling without calculus. A brief primer of Boolean algebra is included, so virtu-
ally no mathematical prerequisites are required. We introduce specialized web-based
software for testing and analyzing the models, which is a convenient way to separate
theory from applications. For readers interested in the theoretical underpinnings, the
chapter provides an online appendix (Appendix 1) that outlines the use of Groebner
bases as it pertains to solving systems of polynomial equations.
The models considered in this chapter are relatively simple and capture only the
most significant qualitative behavior of the
lac
operon - its ability to turn on its lactose
utilization mechanism when glucose is absent from the external medium and lactose
is present. It is clear even from these simple models, that threshold values separating
concentrations with Boolean value 0 from those with Boolean value 1 need to be
selected carefully based on the biology. We generally assume that a value of 0 and 1
indicate, respectively, “low” and “high” concentrations, where “low” does not always
mean a concentration of zero. For concentrations of mRNA,
lac
permease, and
-
galactosidase, as well as for other proteins and enzymes involved in the
lac
operon
control, which increase by a factor of thousands when the operon is turned on in
comparison with their baseline concentrations, such threshold values can easily be
selected. The concentrations of lactose, allolactose, and glucose, however, depend
on the external environment and change more gradually. For those, “low” could truly
mean “absent” and medium-level concentrations are biologically completely feasible.
In the model from Eqs. (
1.8
), the authors take care to introduce designated Boolean
variables for low lactose and allolactose to ensure that “low” means “some but not
zero” [
22
]. Other models (see, e.g., [
26
,
27
] and Chapter 2 of this volume) consider
Boolean frameworks within which it is possible to distinguish between low, medium,
and high lactose concentrations.
Focusing on the medium range of lactose concentration is of particular interest
since the lactose operon has been shown to exhibit
bistability
for medium levels of
lactose. Bistability is the ability of a system to settle in one of two different fixed points
under the same set of external conditions. Which of these fixed points the system
will reach depends on its history. It has been known since the 1950s [
28
] that for
medium lactose concentrations, both induced and uninduced cells can be observed in
a population of
E.coli
. Cells grown in an environment poor in extracellular lactose will
likely remain uninduced for medium lactose levels while cells grown in a lactose-rich
environment will likely retain their induced state. Chapter 2 of this volume examines
Boolean network models of the
lac
operon that can capture the bistability property of
the system.
Boolean models are important from an educational perspective since they require
only a minimal mathematics background. At the introductory level, the construction
of a simple model may essentially amount to translating the system's interactions
represented by a biology “cartoon” into a directed graph (wiring diagram), followed
by a subsequent translation into logical expressions (the update rules). This makes
β
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