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 52

Table 2.3 The steady state values calculated by solving the nonlinear system
of equations by setting time derivatives to zero in the 3 variable model when
L 10 −3 mM for which there exist three steady states.

Steady States

= 50 ×

A ∗ (mM) M ∗ (mM) B ∗ (mM)

10 −3 10 −7 10 −7 I 4 . 27 × 4 . 57 × 2 . 29 ×

10 −2 10 −6 10 −7

II 1 . 16 × 1 . 38 × 6 . 94 ×

10 −2 10 −5 10 −5 III 6 . 47 × 3 . 28 × 1 . 65 ×

We calculated these steady state values numerically as in Table 2.3 . The steady state
values were calculated by solving the nonlinear system of equations obtained by set-
ting time derivatives to zero in the model equations in Table 2.1 and solving it for the
concentrations. Then six distinct initial values were chosen for the protein concen-
tration around the unstable steady state concentration (steady state II in Table 2.3 ).
Three of them were below the unstable steady state and the other three were above
it. As expected, three initials converged to the lower steady state (steady state I in
Table 2.3 ), the other three ended up settling at the higher stable steady state (steady
state III in Table 2.3 ).
Exercise 2.4. The Lac repressor protein is a tetramer of identical subunits [ 11 ]. It
has been shown experimentally that two allolactose molecules on average bind to this
protein and effectively block it so that transcription of new proteins can take place

10 −1

10 − 2

10 −3

0.01 0.02 0.03 0.04 0.05 0.06 0.07

L (mM)

FIGURE 2.4
Bistability arises in (L; A ∗ ) space in the 3 variable lac operon model ( y -axis is in loga-
rithmic scale). For a range of L concentrations there are three coexisting steady states
for the allolactose concentration. We estimated this range to be (0.039,0.055) mM of L
concentration.