Biology Reference
In-Depth Information
equations can be represented locally by a second system of
equations, the S-system representation, which has a single
analytical solution that is linear in the logarithms of the
concentration variables and rate constants
[71,2]
. The
boundaries between such dominant systems according to
this method are not arbitrary, but are determined by the
parameters of the original system.
TABLE 15.4
Values of the Parameters for the CI Gene
Circuit in
Figure 15.7
Estimated from Experimental Data
for E. coli Host Cells Grown in Rich Media with
a Doubling Time of ~ 20 min
[69]
Parameters
Values
Units
K
D
130
nM
K
I
320
nM
Number of Qualitatively Distinct Phenotypes
in Design Space
Since each term of a given sign in Eqs.
(12) to (14)
is
potentially dominant, there are as many potential solutions
as there are combinations of dominant terms; hence
a bound on the total number (T) is provided by
K
R
205
*
nM
g
MMax
355
LacZ units
g
M
50
LacZ units
nM LacZ units
1
min
2
g
C
0.0173
=
ε
y
nM
1
min
1
g
D
1
Y
m
T
¼
P
i
N
i
min
1
b
D
32
:
6
=
ε
i
¼
1
min
1
d
M
0.116
where m is the number of equations and P
i
and N
i
are the
number of positive and negative terms in the ith equation.
(In the case of Eqs.
(12) to (14)
, the bound is T
¼
min
1
d
CMax
7.49
(3 * 3) *
min
1
d
C
0.0347
(3 * 1) * (2 * 1)
54.)
However, not all potential solutions are necessarily
valid. The conditions for any given term to be dominant are
provided by a set of linear inequalities in log space. A test
of each potential solution against the inequalities necessary
for its validity will determine whether or not a potential
solution is in fact a valid solution. Substituting the valid
solution into the corresponding dominance conditions
yields a set of linear inequalities that defines the boundaries
of the region in which the solution is valid. By following
this strategy for the lambda cI gene circuit, and selecting
the parameters R and K
D
for the axes of a 2D plot that
represents a slice through the design space, we obtain the
result depicted in
Figure 15.8
. Note that only 10 of the 54
possibilities are valid solutions for this particular design
space. Regions on the top of this design space
[1,37,38]
correspond to stable steady-state operating points with low
values for the rate of transcription from the promoter pRM;
we will henceforth refer to the qualitative nature of these as
lytic-like or simply lytic states. Regions on the bottom of
this design space
[11,47,45,46]
correspond to stable steady-
state operating points with high values; we will henceforth
refer to the qualitative nature of these as lysogen-like or
simply lysogenic states. The regions in the middle
[7,43,44]
correspond to steady-state operating points with interme-
diate values that exhibit saddle-point instability. The
overlapping regions thus represent hysteresis, which also is
revealed by conventional bifurcation analysis (for
a comparison of the methods see
[72]
). The system design
spaces of some models have boundaries that correspond to
other conventional bifurcations, such as the Hopf bifurca-
tion for oscillations. However, most boundaries in system
design
¼
min
1
d
D
0.0347
a
1
-
p
3
-
n
1.5
-
*The concentration of RecA* ( R) is normalized to the geometric mean of
the values for the hysteretic thresholds at the nominal value of K
D
.
y
1for a dimerization rate that is much greater than rate of loss by
dilution.
ε
<<
dD
dt
¼ g
D
C
2
b
D
D
d
D
D
(9)
K
D
þ
D
p
K
n
I
D
p
þ
n
x
2
¼
þ
(10)
K
R
þ
R
a
x
3
¼
(11)
In steady state the derivatives of Eqs.
(7) to (9)
are equal to
zero, and the equations can be combined into the following
system of non-linear algebraic equations:
g
M
K
D
x
1
g
MMax
x
1
x
1
x
p
þ
n
1
g
M
K
n
x
1
2
0
¼
þ
þ
2
2
I
x
1
d
M
d
C
g
C
K
1
=
2
K
R
x
1
=
2
2
d
M
d
D
g
1
x
1
3
C
C
1
d
M
d
CMax
g
C
K
1
=
2
x
R
x
1
=
2
X
1
3
(12)
C
1
K
D
þ
x
1
p
K
n
I
x
p
þ
n
1
0
¼
þ
x
2
(13)
K
R
þ
x
R
0
¼
x
3
(14)
g
D
;
where K
C
¼ð
R.
Each of these equations is a sum of several positive and/
or negative terms. When one term of each sign in each of
the equations is dominant (i.e., is the term with the greatest
magnitude among those of the same sign), the system of
b
D
þ
d
D
Þ
x
1
¼
D and x
R
¼
space
do
not
correspond
to
conventional
