Mechanisms of Gene Regulation: Boolean Network Models of the Lactose Operon in Escherichia coli - Mathematical Concepts and Methods in Modern Biology - page 2

DNA sequences controlling transcription may be found at considerable distances
from the gene to be controlled and may be brought to the vicinity of the RNA poly-
merase binding site (the promoter) by the bending of the DNA. This highly complex
structure presents significant experimental challenges in the process of understanding
and describing cellular behavior.
Mathematics provides a formal framework for organizing the overwhelming
amounts of disparate experimental data and for developing models that reflect the
dependencies between the system's components. Different types of mathematical
models have been developed in an attempt to capture gene regulatory mechanisms
and dynamics.
Various broad classifications are used in reference to such models. Deterministic
models generate the exact same outcomes under a given set of initial conditions while
in stochastic models the outcomes will differ due to inherent randomness. Dynamic
models focus on the time-evolution of a system while static models do not consider
time as part of themodeling framework. Among the dynamicmodels, time-continuous
models utilize time as a continuous variable, while in time-discrete models time can
only assume integer values. Space-continuous models refer to situations where the
model variables can assume a continuum of values while in space-discrete models
those variables can only assume values from a finite set. Space-continuous models of
gene regulation are often constructed in the form of differential equations (in the case
of continuous time) or difference equations (in the case of discrete time) and focus on
the fine kinetics of biochemical reactions. We will refer to such models as DE models .
Discrete-time models built from functions of finite-state variables are referred to as
algebraic models.
In aDEmodel, all variables assume values fromwithin biologically feasible ranges.
Modelers usually need comprehensive knowledge of the interactions between vari-
ables, which may include detailed information of recognized control mechanisms,
rates of production and degradation, minimal and maximal biologically relevant con-
centrations, and so on. In an algebraic model only values from a finite set are allowed.
The special case of a Boolean network allows only two states, e.g., 0 and 1, generally
representing the absence or presence of gene products in a model of gene regulation.
In contrast to DE models, the information necessary to construct a Boolean model
requires only a conceptual understanding of the causal links of dependency. Thus, in
general, DE models are quantitative while Boolean models are qualitative in nature.
Historically, DEmodels have been the preferred type of mathematical models used
in biology. This type of dynamical modeling has proved to be essential for problems in
ecology, epidemiology, physiology, and endocrinology, among many others. Boolean
models were first introduced to biology in 1969 to study the dynamic properties of
gene regulatory networks [ 1 ]. They are appropriate in cases where network dynamics
are determined by the logic of interactions rather than finely tuned kinetics, which
may often be unknown.
In this chapter we present some of the fundamentals of creating Boolean network
models for one of the simplest and best understood mechanisms of gene regula-
tion: the lactose ( lac ) operon that controls the transport and metabolism of lactose