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 64

of lactose , there are two fixed points and the state of the operon

depends on the initial conditions (its history): If the lactose concentration is raised to
medium for cells grown in low lactose concentrations (those cells have thus settled
in the Off state), the operon remains Off. If the concentration of lactose is decreased
to medium for cells grown in lactose-rich environment (those cells have thus settled
in the On state) the operon remains On.
For the model defined by Eqs. ( 2.57 ) the entire state space is too large to display
but we present its fixed points in Table 2.6 .
Exercise 2.7. Verify that the fixed points for the Boolean model from Eqs. ( 2.57 )
are as presented in Table 2.6 .

( L = 1 , L high = 0 )

So far, we have used designated “old” variables to separate the time scales of
the dilution and degradation processes from those of synthesis. As the next model
shows, this could also be done implicitly through the logical assumptions built into the
transition functions. If we choose toworkwith the estimate for 10 − 4 min − 1

fromWong et al. [ 17 ], the degradation time for A will be the slowest among the three
variables M

γ A = 1 . 8 ×

B , and A . The transition function for A in the next model accounts for
this by allowing for A

,

1 either because of new production or because previously
produced amounts of A are still present and have not been lost due to conversion to
glucose and galactose.

=

f M = A , (2.58)

f B = M ,

f A = ( B ∧ L ) ∨ L high ∨ ( A ∧ B ).

Exercise 2.8. Justify and analyze the model from Eqs. ( 2.58 ). Show that its long-
term behavior captures the bistability of the lac operon in medium lactose
concentrations.

2.6.2 Boolean Variants of the 5-Variable Model
We next consider Boolean variants of the model from Table 2.2 that uses five dynamic
variables, M -galactosidase, allolactose, intra-

cellular lactose, and lac permease. Recall that extracellular glucose is assumed to be
absent and extracellular lactose

, B , A , L , and P , representingmRNA, β

( L e ) present at all times. L e is a parameter for the

model.
The degradation constants for L and P are estimated to be 0min − 1

γ L = 0 .

65 min − 1 (meaning the dilution loss of L is fully due to the growth rate) and γ P = 0 . ;

τ P = τ P in the equations for P [ 9 ].

As before, we need to be able to distinguish between high, medium, and low
concentrations of external lactose, where medium concentration would roughly cor-
respond to the maintenance range estimated to be (0.027, 0.062) mM (see Figure 2.6 ).
We introduce an additional parameter L e high . Assuming that L e =

0 . 83 min is an estimate for the delay

1 stands for at least

medium external lactose, the combination L e = 1 ; L e high = 0 indicates medium