Mechanisms of Gene Regulation: Boolean Network Models of the Lactose Operon in Escherichia coli - Mathematical Concepts and Methods in Modern Biology - page 22

01000 10000 11000 10001 10010 10011 11001 01010

00000 00001 00011 01001 01011 11010 11011 01100 01110 00100

00010 00101 00110 00111 01101 01111 11100 11110 10100

10110 10101 10111 11101

11111

FIGURE 1.10
The state space transition diagram of the points ( M , B , A , L , P ) for the Boolean model
defined by Eqs. ( 1.7 ).

Here, as for the minimal model from Eqs. ( 1.4 ), it is assumed that translation and
transcription require one unit of time, protein and mRNA degradation require one
unit of time, and lactose metabolism requires one unit of time. This model does not
involve any parameters: instead, it assumes that extracellular lactose is always avail-
able, and extracellular glucose is always unavailable. The rationale and interpretation
for the functions proposed by Eqs. ( 1.12 ) comes from the biological interactions
between the main regulatory elements. For instance, the transition function f L indi-
cates that at time t + 1 lactose ( L ) will be available if permease ( P ) is available at
time t to bring the extracellular lactose into the cell or, i n case lactose ( L ) is already
available at time t , there is no

β -galactosidase at time t ( B ) to metabolize the lactose

into glucose and galactose.
Exercise 1.12. Sketch the wiring diagram (the dependency graph) for the model
described by Eqs. ( 1.7 ).

As for our earlier model, one way to initially test the model from Eqs. ( 1.7 )isto
examine its fixed points and determine if they correspond to states that are biologi-
cally feasible. The state space graph for the model defined by Eqs. ( 1.7 ) is presented
in Figure 1.10 . It has three fixed points ( M , B , A , L , P ): ( 0 , 0 , 0 , 0 , 0 ), ( 1 , 1 , 1 , 1 , 1 ) ,

and . The first two correspond to the operon being Off and On, respec-

tively. The third one, however, corresponds to a biological scenario under which the
bacterium does not metabolize the intracellular lactose, which is unrealistic (recall
that the model assumed that extracellular lactose is always available and that there is
no extracellular glucose).
The fact that the state ( M , B , A , L , P )=

( 0 , 0 , 0 , 1 , 0 )

is a fixed point for the model
indicates that the model does not represent accurately the most important qualitative
behavior of the lac operon regulation. Thus, the model fails the initial testing and is
in need of modification.
Exercise 1.13. Consider the transition functions in Eqs. ( 1.7 ) and criticize the
model. Can you find reasons to question the definitions of the transition functions?
The way to approach this is by examining each function and asking if it accurately
reflects the underlying biology and/or the model assumptions.

( 0 , 0 , 0 , 1 , 0 )

Exercise 1.14. Consider possible modifications of the transition functions in
Eqs. ( 1.7 ), aimed at eliminating the biologically infeasible fixed point. Give the