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 51

Table 2.2 The 5 variable Yildirim-Mackey Model (from [ 9 ]).

τ M n
K + K 1 e − μτ M A τ M n

K 1 e − μτ M A 1 +

dM
dt

= α M + 0 − γ M M

dB
dt

α B e − μτ B M τ B − γ B B =

dA
dt

L
K L + L − β A B

A
K A + A − γ A A

(2.52)

= α A B

dP
dt

α P e − μ(τ B + τ P ) M = − γ P P

τ + τ B P

dL
dt

L e
K L e + L e − β L 1 P

L
K L 1 + L − α A B

L
K L + L − γ L L

= α L P

2.4.2 Numerical Simulation of the Yildirim-Mackey
Models and Bistability
In this section, we will perform steady state analysis and numerical simulations for
both models. The models consist of delay differential equations with discrete time
delays due to the transcription and translation processes. Besides delay parameters,
they also involve a number of unknown parameters that need to be fixed in order
to perform steady state analysis and numerical simulations. Yildirim et al. [ 9 , 8 ]did
an extensive literature search to estimate these parameters from published articles.
The values and details on estimation of the parameters can be found in [ 9 , 8 , 13 ]. In
our simulations in this chapter, we have used two different values, as was done in

the original papers. For simulations with the 3 variable model we took

μ

μ = . ×

3 03

10 − 2 min − 1 . For the simulations with the 5 variable model, a smaller

μ

value was

10 − 2 min − 1 . These values were estimated by fitting the differential
equation models to experimental data.
Experimental results showing that the lac operon in Escherichia coli is capable
of showing bistable behavior for a range of extracellular lactose concentrations have
been available since the late 1950s [ 1 , 14 ] and more recently in [ 2 ]. To see if the 3
variable model can show bistability, steady state analysis is performed, which can be
studied by setting the left-hand side of each equation in the 3 variable model given in
Table 2.1 to zero and solving it for a range of L concentrations after keeping all the other
parameters fixed at their estimated values. The result is shown in (Figure 2.4 ). The
3 variable model predicts that there is a range for the internal lactose concentration,
which corresponds to the S -shaped curve in the figure. When the lactose concentration
is in this range, the lac operon can have three coexisting steady states.
Figure 2.5 shows how the bistable behavior arises in the time series simulation of
the 3 variable model. For this simulation, all the parameters are kept constant at their
estimated values when L

used, μ = 2 . 26 ×

10 − 3 mM. As seen from the steady state curve in
(Figure 2.4 ), there are three distinct steady states for this particular concentration of L .

= 50 ×