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In-Depth Information
γ
A
, we essentially made the
assumption that allolactose degrades much slower than mRNA and that the delays
τ
M
and
By considering the middle-of-the-range value for
τ
B
are negligible in comparison with the time for degradation and dilution
of allolactose. However, considering estimates for
γ
A
among the largest reported in
52 min
−
1
), a different Boolean model would be more
appropriate since in that case the half-life for
A
, estimated at
the literature (e.g.,
γ
A
=
0
.
h
A
260 min, is
similar to the half-life of mRNA. This situation calls for assumptions different from
those used to build themodel given by Eqs. (
2.55
): (i) If a larger time step is considered
(e.g.,
t
= 15 min), we can eliminate the variables
A
old
and
B
old
(
2
)
=
1
.
. (ii) If we consider
a much smaller time step (e.g.,
t
1 min), more additional variables will be needed
in order to account for the system delays and for the fact that multiple time steps
will be needed for dilution and degradation to bring
M
and
A
below the discretization
threshold levels.
The Boolean model defined by Eqs. (
2.56
) would be appropriate under assumption
=
(i).
f
M
=
A
f
B
old
=
M
∧
B
,
(2.56)
f
B
=
M
∨
(
B
∧
B
old
)
f
A
=
(
B
∧
L
)
∨
L
high
.
Exercise 2.6.
(a) Justify the model presented by Eqs. (
2.56
) by explaining the
logical expression defining each of the transition functions as was done above for the
Boolean model in Eqs. (
2.55
); (b) Explain why the model would be consistent with
choosing a time step
t
15 min.; (c) Use DVD to analyze the model and obtain
the state space diagram presented in Figure
2.9
; (d) Does the result from (c) imply
bistability for the system?
=
The Boolean model defined by Eqs. (
2.57
) on the other hand would be appropriate
under the assumption (ii) above
(
t
=
1min
)
. Newvariables
M
1
and
M
2
are introduced
to model the delayed effect (by
-galactosidase
and
A
1
is needed to represent the delayed action of
A
on the production of mRNA by
τ
M
=
τ
B
) of mRNA on the production of
β
1 min. As before,
M
old
and
A
old
track the loss of
M
and
A
due to dilution and
degradation. Since the loss of
B
is much slower, several “old” variables are needed. We
have used two such variables here but one would think that a much larger number of
such variables will be needed to represent the time scales accurately. In the discussion
0
.
Table 2.5
Fixed points for the Boolean model from Eqs. (
2.55
). The system
always settles in a fixed point. There are no limit cycles. Two steady states are
possible for medium lactose concentrations (the system is bistable).
Inducer Level
L
L
high
M B B
old(1)
B
old(2)
A A
old
Operon is
1
Low lactose
0
0
0
0
0
0
0
0
Off
2 High lactose
1
1
1
1
0
0
1
1
On
3 Medium lactose
1
0
0
0
0
0
0
0
Off
4 Medium lactose
1
0
1
1
0
0
1
0
On
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