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 50

Table 2.1 The 3 variable Yildirim-Mackey Model (from [ 8 ]).

τ M n
K + K 1 e − μτ M A τ M n

K 1 e − μτ M A 1 +

dM
dt = α M

− γ M M

dB

dt = α B e − μτ B M τ B − γ B B

(2.49)

dA
dt = α A B

L
K L + L − β A B

A
K A + A − γ A A

In addition to the equations in Table 2.1 , the 5-variable model includes specific
equations for the dynamics of permease and internal lactose. The equation for the
dynamics of the permease is similar to the β -galactosidase equation and it is given in

Eq. ( 2.50 ).

dP
dt = α P e − μ(τ B + τ P ) M τ B + τ P − γ P P

. (2.50)

Here M τ B + τ P . The first term models

the gain in the permease concentration due to the transcriptional induction of lac Y
gene by the external lactose, and it is assumed to be proportional to the mRNA
concentration at a time

= M ( t − (τ B + τ P )) and γ P = γ P + μ

τ B + τ P ago. Here α P is the proportionality constant for this

process, and -

galactosidase and permease, respectively. The term e − μ(τ B + τ P ) models the dilution in
mRNA concentration due to cellular growth during the translation of such enzymes.
The last term is for the loss in the permease concentration due to cellular growth and
degradation of this enzyme.
The equation governing the dynamics of internal lactose is given in Eq. ( 2.51 ).
The first three terms in this equation are in the Michaelis-Menten form. The first term
models the gain in the concentration of lactose due to the transport of external lactose
by the permease.

τ B and τ P are the times required for the translation of functional β

dL
dt = α L P

L e
K L e +

L
K L 1 +

L
K L + L e − β L 1 P L − α A B L − γ L L

. (2.51)

Here L e denotes the concentration of external lactose, which is a parameter for the
5-variable model. The permease-mediated transport of lactose through the cellular
membrane is reversible [ 12 ]. The second term in Eq. ( 2.51 ) accounts for the loss in
the internal lactose concentration due to the reversible nature of this transport process.
The third termdescribes the loss in the lactose level due to its conversion to allolactose,
which is mediated by -galactosidase. The last term models the loss in the lactose

concentration due to the cellular growth and its degradation. The equations for the
5-variable model are presented in Table 2.2 .

β