Biology Reference
In-Depth Information
of which are locally stable and the third one is unstable [
9
,
15
]. For this simulation, we
have chosen six different initial values for the protein concentrations in the neighbor-
hood of the unstable steady state (steady state II in Table
2.4
) when all the parameters
were held at their estimated values in Table
2.4
. Since we started our simulations from
a point close to the unstable steady state (but not exactly from it), three of the initials
converged to the higher steady state (steady state III in Table
2.4
) and and the other
three initials converged to the lower steady state (steady state I in Table
2.4
).
Exercise 2.5.
Consider the 5 variable model and the parameter values given in
Table I in the paper [
9
] except for the bacterial growth rate
. Bacterial growth rate
can change depending on the environmental condition. In our analysis, we estimated
this parameter to be about 30 min. Assume that you have a bacteria population that can
double in size every 100 min then compute numerically the range for the extracellular
lactose concentration and produce the bistability plot in
μ
A
∗
)
space. Furthermore,
compute and estimate the value for the bacterial growth rate from this model for which
the lactose operon is no longer capable of showing bistable behavior.
MATLAB starter code is provided for this exercise. Open the file
Code_for_Ex_2_
5_Starter.m
and add the appropriate lines of code. Note that this exercise requires the
use of the MATLAB's Global Optimization Toolbox.
(
L
e
,
2.5
BOOLEAN MODELING OF BIOCHEMICAL INTERACTIONS
Boolean models were introduced and discussed in detail in Chapter 1 of this volume
and we refer the reader to Chapter 1 for a primer on Boolean networks. We repeat
some of the basics here for easy reference. In a Boolean model, all model variables
are discretized to take values 1 or 0 (we also say that a Boolean variable can be On
or Off, respectively). A certain threshold of discretization is chosen for each variable
and a value of 0 typically represents the case when only “trace” (baseline) values of a
substance are available. A value of 1 refers to concentrations larger than the threshold
level.
As in the differential equation models, each variable in a Boolean model represents
the concentration of a molecular species (e.g., enzyme, substrate, protein, or mRNA).
A Boolean model of
n
-variables
x
1
,
x
2
,...,
x
n
consists of
n
transition Boolean
functions (also called update functions)
f
x
1
,
f
x
2
,...,
f
x
n
describing the dynamical
n
evolution of the model variables, where
f
x
j
(
x
1
,
x
2
,...,
x
n
)
:{
0
,
1
}
→{
0
,
1
}
,
j
=
1
,
2
,...,
n
. Equations (
2.53
) present a simple example.
f
x
1
=
f
1
(
x
1
,
x
2
,
x
3
,
x
4
)
=
x
3
,
f
x
2
=
f
1
(
x
1
,
x
2
,
x
3
,
x
4
)
=
x
3
∧
x
4
,
f
x
3
=
f
1
(
x
1
,
x
2
,
x
3
,
x
4
)
=
x
2
∧
x
3
,
f
x
4
=
f
1
(
x
1
,
x
2
,
x
3
,
x
4
)
=
x
1
∧
x
2
∧
x
3
.
(2.53)
Boolean models are time-discrete and the dynamical evolution of the model is
determined by iterating the transitions defined by the update functions. Starting from
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