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 49

Suppose that a protein P decays exponentially. If p represents the concentration of
this protein, then we have

dp
dt =− μ

p , (2.44)

where μ> 0 quantifies how fast this degradation happens. When this equation is

integrated from t − τ M to t , we obtain

e − μτ M p p ( t ) =

τ M ,

(2.45)

. This derivation explains the prefactor e − μτ M in Eq. ( 2.43 ).
Exercise 2.3. Derive the function p

where p τ M = p ( t − τ M )

( t ) in Eq. ( 2.45 )fromEq.( 2.44 ).

0 that
accounts for the basal transcription rate in the absence of the extracellular lactose.
Therefore, we use Eq. ( 2.46 ) for the dynamics of mRNA concentration in the 5 vari-
able model.

In the 5-variable model, the mRNA equation has an extra parameter

K 1 e − μτ M A

τ M n

1 +

dM
dt = α M

K 1 e − μτ M A τ M n

+ 0 − γ M M . (2.46)

K +

The basal transcription rate for mRNA is not explicitly modeled in Eq. ( 2.43 )
but is included in the parameters

α M and K , instead. The second term in Eq. ( 2.43 )
models the loss in mRNA concentration. It is a sum of two terms and given by
γ M M = γ M M + μ γ M M accounts for the loss due to mRNA degradation

M .Theterm

μ

and the term M is the effective loss due to the bacterial growth, as explained in Eq.

( 2.32 ).
Equation ( 2.47 ) models the dynamics of the β -galactosidase concentration. It can

be assumed that the rate of production of β -galactosidase is directly proportional to

the mRNA concentration at time ( t − τ B ) with a proportionality constant α B .

dB
dt = α B e − μτ B M τ B − γ B B

. (2.47)

Here τ B is the time required for the translation of mRNA, γ B = γ B + μ and

= ( − τ B )

.
Equation ( 2.48 ) governs the allolactose dynamics. The first term on the right-hand
side of this equation gives the gain in allolactose due to the conversion of lactose
mediated by

M τ B M t

-galactosidase. We assume that this conversion follows the Michaelis-
Menten equation as derived in Eq. ( 2.14 ).

β

dA
dt = α A B

L
K L +

A
K A +

L − β A B A − γ A A

.

(2.48)

The second term denotes the loss of allolactose due to its conversion to glucose and
galactose by -galactosidase, which is also assumed to follow a Michaelis-Menten

kinetics. The last term accounts for the loss in the allolactose concentration due to
its degradation and dilution. The equations for the 3-variable model are presented in
Table 2.1 .

β