Mechanisms of Gene Regulation: Boolean Network Models of the Lactose Operon in Escherichia coli - Mathematical Concepts and Methods in Modern Biology - page 21

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FIGURE 1.9
State space diagram for the model defined by the update functions in Eqs. ( 1.6 ). The
order of components for each state is ( M , E , L , L e , G e ) . See the text for further details.

1.3.6 How to Recognize a Deficient Model
In the minimal model introduced and discussed above, the testing and analysis showed
that the model can adequately represent the behavior of the operon to be On or
Off. One should remember, however, that this model is just one of many possible
models that can be created and that every model relies on a set of assumptions made
during the modeling process. For the minimal Boolean model above, we made several
assumptions both during the process of selecting the model variables and at the stage
of writing down the transition functions that determine the dynamical behavior of the
system. Different choices and assumptions would lead to different models, some of
which may work and some of which may not. In this section we present a “model”
(and the quotation marks here are meant to indicate that we will ultimately show that
it is not a good model) that may look legitimate at first. The initial testing, however,
will show that the model does not adequately capture the regulatory behavior of the
lac operon.
The model is based on the following five variables: mRNA ( M ), -galactosidase

( B ), lac permease ( P ), intracellular lactose ( L ), and allolactose ( A ). This certainly
appears to be a reasonable choice, as it includes the primary components of the lac
operon regulatory mechanism. As in the previous model, and for the same reasons as
before, the CAP-cAMP positive control mechanism is excluded from the modeling
effort. The following transition functions are proposed for describing the underlying
biology:

β

f M = A

f B = M

f A = A ∨ ( L ∧ B )

f L = P ∨ ( L ∧ B )

f P = M . (1.7)