Bistability in the Lactose Operon of Escherichia coli: A Comparison of Differential Equation and Boolean Network Models - Mathematical Concepts and Methods in Modern Biology - page 36

CHAPTER

2

Bistability in the Lactose
Operon of Escherichia coli :
A Comparison of Differential
Equation and Boolean
Network Models

Raina Robeva ∗ and Necmettin Yildirim †
∗ Department of Mathematical Sciences, Sweet Briar College, Sweet Briar, VA 24595, USA
† Division of Natural Sciences, New College of Florida, FL 34243, USA

2.1 INTRODUCTION
Cellular systems are complex and involve many interacting components, which are
ultimately responsible for cellular functions. Such complexity is capable of produc-
ing interesting dynamic behaviors. Among such behaviors, bistability is extremely
important and is a recurring pattern in biological systems. A system is called bistable
if it is capable of resting in two stable steady states separated by an unstable steady
state. A bistable system can reside in one of the two states for a long time unless
a strong external perturbation kicks in. Bistability also provides a perfect discontin-
uous switching among the stable steady states and can convert graded inputs into
switch-like outputs.
Bistability can be either reversible or irreversible. A system is irreversibly bistable
if it cannot go back to its earlier state once it settles down to a state after a perturbation.
However, a reversible bistable system can go back and forth between the stable states
as the system parameters vary. Figure 2.1 a shows a reversible bistable switch and
Figure 2.1 b is for an irreversible bistable switch. As seen in Figure 2.1 a, the value
of the parameter p for the system to switch from a lower steady state to a higher
steady state is different from the value of the parameter p for which a switch from a
higher steady state to a lower steady state occurs. This is called hysteresis. Bistability
comes with hysteresis, which can be defined as dependence of the current state of a
system not only on its current state but also on its past state. To predict future state of
a bistable system, both its current state and its history must be known.