Mechanisms of Gene Regulation: Boolean Network Models of the Lactose Operon in Escherichia coli - Mathematical Concepts and Methods in Modern Biology - page 24

time t imply at least low levels for the next time step t + 1. The complete list of model
variables is:

M = lac mRNA L = high concentration of intracellular lactose

P = lac permease A = high concentration of allolacose (inducer)

B = β -galactosidase L l = (at least) low concentration of intracellular

lactose C = catabolite activator protein CAP

A l = (at least) low concentration of allolactose

R = repressor protein lac I

The model assumptions are:

￿ Transcription and translation require one unit of time. This means that if all neces-
sary conditions for the activation of the molecular mechanism are present at time
t , the protein production will be happening in time t +1.
￿ Degradation of all mRNA and proteins occurs in one time step.
￿ High levels of lactose or allolactose at any given time t imply at least low levels
for the next time step t +1.

The Boolean transition functions for the model reflect the dependencies between
variables according to the regulatory mechanisms of the lac operon from Section 2.
We provide justification for two of the functions; the rest are left as exercises. The
corresponding wiring diagram is depicted in Figure 1.11 .

f M =

∧

R C

f P = M f B = M

f C = G e f R = A ∧ A l (1.8)

f A = L ∧ B f A l = A ∨ L ∨ L l

f L = G e ∧

∧ =

G e ∧ (

∨

L e ).

P L e f L l L

Boolean function for R : For the concentration of the repressor protein to be high
( R = 1), there should be no allolactose present; that is A 0.

Boolean function for M: In order for production of mRNA to be high, there should
be no repressor protein ( R = 0) and the concentration of CAP should be high ( C =1).
Exercise 1.15. Consider the rest of the transition functions from Eqs. ( 1.8 ). Give
justification for the definition of each function and provide any additional assumptions
that the definition may imply.

= A l =

Exercise 1.16. There is no variable to represent cAMP in the Boolean model with
the wiring diagram depicted in Figure 1.11 and defined by Eqs. ( 1.8 ). Could you
justify this decision? How would the model change if a cAMP variable is introduced?
Do you think this change will impact the qualitative behavior of the model?

As with the earlier Boolean models we have examined, the next logical step in the
process is to find the fixed points of the model and determine if it accurately reflects
the ability of the lac operon to be On or Off. The model now has nine variables,