Mechanisms of Gene Regulation: Boolean Network Models of the Lactose Operon in Escherichia coli - Mathematical Concepts and Methods in Modern Biology - page 23

rationale for yourmodification and specify the biologicalmechanismormodel assump-
tions that justify the change. Use DVD to analyze the modified model. For each of
your modifications, use the number of fixed points to decide if they correspond to
biologically realistic situations. Note that there should be no limit cycles.

1.3.7 A More Refined Boolean Model of the Lac Operon
The models we have considered so far attempted to reproduce features of the lac
operonmechanisms by including a relatively small number of variables. The catabolite
repression mechanism (the CAP-cAMP positive control loop) was excluded, as was
the explicit modeling of the presence of low levels of lactose and allolactose. For
proteins and enzymes, since basal concentrations are always nonzero, we used the
Boolean value 0 to refer to basal levels and 1 for concentrations that are much higher.
This does not apply to lactose and allolactose, the concentrations of which may be
truly zero. Thus, in the case of lactose and allolactose, it is justified to look for ways
to model low but nonzero concentrations separately.
There are several possible options to do so. One of them is to consider discrete
models in which the variables take values from a set S of three or more values,
corresponding to ranges in the respective concentrations. If S = {0, 1, 2}, the value 0
may be used to represent a concentration near zero, 1 to represent a low concentration
that is not near zero, and 2 to represent a high concentration. This approach can no
longer be implemented using Boolean networks and leads to discrete models with
multiple states. When the set S has a prime number of elements, it can be shown
that the transition functions of the model have a representation as polynomials of the
model variables. The theory of polynomial dynamical systems is then applicable to
the analysis of such networks and there are many interesting mathematical questions
arising in this context (see, e.g., [ 17 - 21 ]).
Another possible approach, which we will examine here, is to stay within the
framework of Boolean networks and introduce additional Boolean variables to allow
for a separation into three concentration ranges instead of two. The model that follows
comes from Stigler and Veliz-Cuba [ 22 ] and utilizes this approach. It also includes
the CAP-cAMP positive control mechanism of the lac operon that was not considered
in the models discussed so far.
We begin by identifying the model variables and parameters. As in the minimal
model, L e and G e are model parameters, denoting the extracellular lactose and the
extracellular glucose. The variables L l and A l (the index stands for low concentration)
are introduced to facilitate the ability of the model to distinguish between “no lactose”
and “some lactose” and similarly for allolactose. This is an improvement over our
initial model because, unlike for proteins and enzymes, it would not be warranted to
assume that baseline levels of lactose or allolactose are always present. When L l and
A l have value 1, this means that at least low concentrations of lactose and allolactose,
respectively, are available in the cell. As before, L 1 stand for high

levels of lactose and allolactose. High levels of lactose or allolactose at any given

= 1 and A =