Mechanisms of Gene Regulation: Boolean Network Models of the Lactose Operon in Escherichia coli - Mathematical Concepts and Methods in Modern Biology - page 9

The other quantities ( M, B, P, L , and A ) will be assumed to vary with time. Some of
these variables, however, exhibit related dynamics due to similarities in their underly-
ing biochemical structures andmechanisms of production. Thus, as described next, we
can further reduce the number of model variables based on such known dependences.
β

-galactosidase is a homo-tetramer made up of four identical LacZ polypeptides.
If we denote the LacZ polypeptide by E , the following holds for the concentrations
of 4. Further, since the translation rate of the

LacY transcript can be assumed to be the same as the rate for the LacZ transcript, the
following holds for the concentration of permease: P

β -galactosidase and LacZ: B = E /

E . Finally, we can assume
that the concentrations of internal lactose ( L ) and allolactose ( A ) are proportional,
that is A

=

pL , where p indicates the fraction of lactose converted into allolactose
and can be determined experimentally. Thus, in our first “minimal” model, we will
consider only three variables— M, E, L , and two parameters— L e and G e . Knowing
the variables M, E, and L would allow us to determine the values of P, B , and A from
the equations B

=

= E / 4 , P = E , and A = pL . The corresponding dependency graph

is depicted in Figure 1.4 .
The choice for variables and parameters discussed here is just one possibility
among many others. The model by Yildrim and Mackey for instance [ 15 ] is based
on assumptions leading to a wiring diagram including five nodes corresponding (in
our notation) to the variables M, B, P, L, and A , and a node for external lactose as
a parameter. In [ 16 ], the authors consider a reduction of this five-variable model to
a network of three nodes: M, B, and L . These models, together with their Boolean
approximations, will be introduced and discussed in Chapter 2 of this volume. Later
in this chapter we consider a Boolean model of the lac operon with nine variables
and two parameters that includes the mechanism of catabolite repression.
Once the model variables have been identified, the decision on the type of mathe-
matical model should be made. As mentioned earlier, various types of mathematical
models can be developed (including DE, algebraic, stochastic, and simulation mod-
els among others) from the same wiring diagram. In this section, we will focus on
developing a Boolean network model.

1.3.2 Boolean Network Models
Boolean variables and Boolean expressions. Boolean models allow only two states,
e.g., 0 and 1, indicating the absence or presence of the components represented by the
nodes of the wiring diagram. Since in biology trace amounts of various substances
may be present at all times, “absence” usually stands for concentrations lower than
a certain threshold value separating higher concentrations from the baseline, and
“presence” is interpreted as concentrations higher than this threshold. It may appear
that because chemical concentrations span a continuous range of values, choosing
a single threshold cutoff may not be appropriate—two concentration values may be
very close numerically with one of them falling above the threshold and the other one
falling below it. Although this may be a legitimate concern in general, it would rarely
apply to a model of gene regulation. When the gene is expressed, the concentration