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the discretization threshold by time
t
+
1 (that is,
B
∧
B
old
(
2
)
=
1),
B
will still be
available at time
t
1.
Transition Equation for B
old
(
1
)
+
: When no mRNA is available at time
t
,nonewly
produced
1, the amounts
of
B
produced at time
t
will be reduced once due to dilution and degradation and
B
old
(
1
)
(
β
-galactosidase will be available at time
t
+
1. By time
t
+
1.
Transition Equation for B
old
(
2
)
t
+
1
)
=
: When no mRNA is available at time
t
,nonewly
produced
1, the amounts
of
B
, already reduced once at time
t
, will be further reduced due to dilution and
degradation and
B
old
(
2
)
(
β
-galactosidase will be available at time
t
+
1. By time
t
+
1.
Transition Equation for A
: There are three possible ways for
A
to be available at
time
t
t
+
1
)
=
-galactosidase above the discretization threshold is available
together with at least medium concentration of lactose. Under those conditions,
+
1: (i) At time
t
,
β
β
-
galactosidase will convert lactose into allolactose by time
t
1. (ii) At time
t
,the
high concentration of intracellular lactose ensures that available trace amounts of
β
+
-galactosidase will convert enough lactose molecules into allolactose to bring the
concentration of allolactose at time
t
1 above the discretization threshold. (iii) At
time
t
, the concentration of
A
is high enough not to be reduced below the threshold
level at time
t
+
+
1 due to dilution and it will not be lost via conversion into glucose
and galactose (mediated by
-galactosidase).
Transition Equation for A
old
:Attime
t
, the conditions for producing
A
by time
β
t
+
1 are not met. Amounts of
A
available at time
t
will be reduced once by time
t
+
1
due to dilution and degradation and
A
old
(
1.
We analyze the Boolean model defined by Eqs. (
2.55
) using the web-based DVD
software [
18
], considering all four possible combinations for the parameter values.
When
M
t
+
1
)
=
1, the cell is producing all necessary proteins for
the metabolism of lactose. In this case we say that the operon is On. We say that the
lac
operon is Off when those proteins are not being produced (
M
=
1
,
A
=
1, and
B
=
=
0
,
A
=
0, and
0). The results are presented in Table
2.5
. Regardless of the parameter values,
the system has only fixed points and no limit cycles. As expected, low concentration
of lactose
B
=
drives the system to a single steady state in which
the operon is Off, while for high concentrations of lactose
(
L
=
0
,
L
high
=
0
)
the
system settles in a fixed point at which the operon is On. For intermediate lactose
concentrations
(
L
=
1
,
L
high
=
1
)
the model approximates the bistable nature of the
operon, qualitatively replicating the bistability results from Section
2.4
.Ifwestart
at the fixed point 1 in Table
2.5
(corresponding to low inducer concentration and
an Off state for the operon) and increase the lactose concentration to medium, the
operon remains Off (fixed point 3). If we start at fixed point 1 and increase the lactose
concentration to high, the operon turns On (fixed point 4). In a similar way, if we
start at fixed point 2 (corresponding to high inducer concentration and an On state for
the operon) and reduce the lactose concentration to medium, the operon remains On
(fixed point 4). Thus, in intermediate inducer concentrations the long-term behavior
of the system depends on its history (hysteresis).
(
L
=
1
,
L
high
=
0
)
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