Biology Reference
In-Depth Information
Example of an Invalid Phenotype (Case 6)
If we pick the dominant terms to be the first positive term in
Eq.
(12)
, the first negative term in Eq.
(12)
, the third
positive term in Eq.
(13)
, and the second positive term in
Eq.
(14)
. The resulting S-system in steady state is
Evaluation of Local Performance
In each region of
Figure 15.8
the steady-state equations for
the concentration of monomer CI (C), dimer CI (D) and
mRNA (M) follow analytically from the form of the orig-
inal non-linear equations, independent of the particular
numerical values for the parameters. Thus, one can in
principle characterize the various regions and compare
their behaviors analytically without knowing the values of
the parameters. This has been done for simple systems
[39,49]
, but in practice it becomes intractable for large
systems. The region of design space in which the system
operates and the numerical positioning of the boundaries
are determined when specific values for the parameters of
the model are given.
g
M
K
D
x
1
2
d
M
d
D
g
1
0
¼
x
1
2
C
x
p
þ
n
1
K
n
I
0
¼
x
2
x
R
0
¼
x
3
Solution
The steady-state solution for the dimer concentration D is
Quantitative Criteria
Two criteria often found to correlate with the natural
selection of a particular system design are the maximization
(minimization) of a particular steady-state function and the
response time following change. Moreover, when these
values are optimal they often are robustly so (e.g., see
[49]
).
Several criteria for effective local performance that reflect
these expectations can be summarized as follows.
The lysogenic state should be locally robust to pertur-
bations in the values of the parameters that define the
system, and this robustness can be quantified by the
parameter sensitivities, such as S
g
M
g
C
K
D
K
I
2
d
M
d
D
1
p
þ
n
þ
1
D
¼
x
1
¼
and the corresponding values for the concentration of
mRNA M and CI monomer C follow accordingly
2
d
D
g
C
g
M
K
D
K
I
d
M
p
þ
n
p
þ
n
þ
1
1
p
þ
n
g
M
K
D
K
I
d
M
x
p
þ
n
M
¼
¼
1
s
ð
b
D
þ
0
:
5
g
M
g
C
K
D
K
I
2
d
M
d
D
0
:
5
b
D
þ
d
D
Þ
d
D
p
þ
n
þ
1
C
¼
x
1
¼
. It should not be
influenced by fluctuations in the input signal (the level of
DNA damage), which can be quantified by the logarithmic
gain with respect to RecA* activity, such as L
ð
D
;
p
j
Þ
g
D
g
D
For this solution to be valid it must satisfy the dominance
conditions assumed above.
. The
robustness of the logarithmic gain in the face of perturba-
tions in parameter values is also expected to be small, and
this robustness can be quantified by parameter sensitivities
such as S
ð
D
;
R
Þ
Dominance Conditions
Assuming the first positive term in Eq.
(12)
to be dominant
over the other positive terms implies two dominance
conditions:
. The dimer form of CI (D)is
responsible for the primary regulatory actions, but similar
criteria hold for the concentration of the monomer form of
CI (C) and the mRNA (M) as well.
It is important to minimize the response time for
restoring the system to its nominal steady state following
small changes in the variables of the system, and this can be
quantified in terms of the dominant eigenvalue
l
dominant
.
One could of course consider other criteria, but these will
suffice for our purposes here.
½
ð
;
Þ;
p
j
L
D
R
g
M
K
D
<
g
MMax
x
1
K
D
>
K
I
x
p
þ
n
1
Similarly, the assumption of dominant terms for the
remaining cases in Eqs.
(12) to (14)
implies the remaining
dominance conditions:
2
d
D
x
1
=
2
d
C
K
R
K
1
=
2
2
d
D
x
1
=
2
d
CMax
K
1
=
2
x
1
3
x
R
x
1
>
>
1
C
1
C
3
Analysis of Local Performance
The local robustness of the system in each of the
phenotypic regions of system design space can be
calculated analytically for the model in
Figure 15.7
.
From these results, one can predict maximum local
robustness in most cases when the Hill numbers p and n
are large, although trade-offs are involved in determining
the optimum values, whereas local robustness is maximal
K
D
<
x
p
þ
n
1
K
n
I
K
n
I
x
1
1
<
x
R
1
>
K
D
>
x
p
þ
n
1
K
n
I
Note
that
the
conditions
and
K
D
<
K
I
x
p
þ
n
1
are mutually exclusive. Therefore, this
particular assumption of dominance conditions cannot be
satisfied and there is no corresponding valid phenotype.
