Bistability in the Lactose Operon of Escherichia coli: A Comparison of Differential Equation and Boolean Network Models - Mathematical Concepts and Methods in Modern Biology - page 67

Transition Equation for P : When mRNA is available at time t , translation and
transcription of permease will take place to make P available at time t 1. If high

enough amounts of P produced earlier are still available by the time t (that is, by time
t they have not already been reduced enough to be below the threshold at time t

+

+ 1)

P will still be available at t 1.

When the variables M, A, L, B, and P all have values 1, the operon is On. When
they all have values 0, the operon is Off. Analyzing this model with DVD shows
no limit cycles, a single fixed point for the cases of low and high concentrations of
external lactose and two fixed points for medium concentrations. Regardless of the
initial conditions, the operon is On for high external lactose and Off for low external
lactose. When the external lactose concentration is medium, the operon can be On or
Off based on the initial conditions (Table 2.7 ).
Exercise 2.9. Assuming a time step t = 1 min, expand the Boolean model from
Eqs. ( 2.59 ) to include additional variables M 1 ,

+

M 3 , and A 1 accounting, as in the
model from Eqs. ( 2.57 ), for the delays in the 5-variable model from Table 2.2 .Use
DVD to show that introducing the delays does not affect bistability.

M 2 ,

Exercise 2.10. Reduce the Boolean model from Eqs. ( 2.59 ) by removing all “old”
variables except for L old . Explain why this would be justified. Use DVD to find the
fixed points of the system and show that the qualitative bistable behavior of this model
remains the same as for the Boolean model from Eqs. ( 2.59 ).

Exercise 2.11. In Santillán et al., 2007 [ 3 ], the authors consider a minimal ODE
model of the lac operon involving three variables: the intracellular concentration of
mRNA ( M ), lac Z polypeptide ( E ), and intracellular lactose ( L ). Since -galactosidase

is a homo-tetramer made up of four identical lac Z polypeptides and the translation
rate of the lac Y transcription is assumed to be the same as the rate for the lac Z
transcript, the following holds for the intracellular concentrations of

β

β -galactocidase

( B ) and permease ( Q ): Q 4. The model also assumes that the

concentrations of internal lactose ( L ) and allolactose ( A ) are the same. The three
ODEs for the model are given by Eqs. ( 2.60 ). External lactose

= E and B = E /

( L e ) and external

glucose ( G e ) are parameters for the model. The full model is presented and justified

in [ 3 ].

dM
dt =

Dk M P D ( G e )P R ( A ) − γ M M ,

dE
dt =

k E M − γ E E ,

dL
dt =

k L β L ( L e )β G ( G e ) Q − 2 φ M M( L ) B − γ L L . (2.60)

In Chapter 1 of this volume we examined and analyzed a Boolean model built
under the same assumptions and defined by the set of transition functions in Eqs.
( 2.61 ). Since this Boolean model does not have the ability to distinguish between