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be used to quantify the global tolerance to large changes in
genotype and environment as discussed below.
Global Tolerance
Our generic definition of qualitatively distinct phenotypes
and their boundaries in system design space provides
a natural approach to the characterization of the system's
response to large changes in its parameters
[49]
. We define
global tolerance to such changes as the value of a parameter
at the boundary between adjacent phenotypic regions
relative to the normal operating value within a region (or
the inverse if the normal value is greater than the value at
the boundary). We will use the expression '
Logarithmic Gain
Signal transfer functions are used in various disciplines to
characterize the response of dependent variables (such as
concentrations and fluxes) to changes in the value of an
independent variable. These amplification or gain func-
tions actually represent logarithmic gain when they are
defined in terms of a relative derivative
[12,43]
.For
example, using the intermediate concentration X,
pathway flux V and independent substrate concentration
S, representative logarithmic gains for the pathway in
Figure 15.3
are:
'
to describe the global tolerances of a given phenotype to
changes in the parameter P
j
, where T
D
¼
½
T
D
;
T
I
ð
l
;
P
j
Þ
tolerance to
a fold decrease and T
I
¼
tolerance to a fold increase (since
boundaries can be crossed either by decreasing or
increasing the value of a parameter).
Thus, systems located well within a physiological
('good') phenotypic region will be tolerant to large (global)
changes in the values of their genotypically influenced
parameters and environmentally influenced variables,
because only large changes will be sufficient to move the
system's location in system design space across a boundary
into an adjacent region representing a qualitatively distinct
(potentially pathological) phenotype.
v
Log X
v
Log S
¼
v
X
v
S
S
X
L
ð
X
;
S
Þ¼
v
Log V
v
Log S
¼
v
V
v
S
S
V
L
ð
V
;
S
Þ¼
Values
>
1 signify amplification of
the input signal,
whereas values
1 signify attenuation of the signal;
a positive sign indicates changes in the same direction,
whereas a negative sign indicates changes in the opposite
direction.
<
Comparison of Phenotypes
Each phenotype has a characteristic set of values for its
logarithmic gains, parameter sensitivities, response times
and global tolerances. By making use of these values, the
performance of the various phenotypes can be readily
compared on the basis of relevant quantitative criteria.
Parameter Sensitivity
The response of dependent concentrations and fluxes to
a change in the value of the parameters that define the
structure of the system (e.g., the rate constants here) are
defined by the relative derivative of the explicit steady-state
solution
[12,50]
. For example,
Criteria for Functional Effectiveness
For the purposes of this example, assume the criteria for
functional effectiveness of the pathway operating in the
forward direction are those listed in
Table 15.1
.
v
Log X
v
Log k
S
¼
v
X
v
k
S
k
S
S
ð
X
;
k
Þ¼
S
X
v
Log V
v
Log k
P
¼
v
V
v
k
P
k
P
S
ð
V
;
k
Þ¼
p
V
Local Performance
A comparison of the local behavior for the relevant pheno-
types is shown in
Table 15.2
. Based on this hypothetical
scenario, wewould conclude that the best overall phenotypic
region is on the boundary betweenRegion 2 and the right side
of Region 4. (In this situation Regions 2 and 4 are essentially
identical, and Region 1 has a greater intermediate concen-
tration and slower response time than the other two regions;
they trade advantages with respect to criteria 4 and 5, and all
are the samewith respect to criteria 1 and 2.) The equilibrium
state for the mutarotation of glucose in solution is located
on the positive y-axis (log
10
ð
where the interpretation of magnitudes and signs is the
same as for logarithmic gains. Small magnitudes (param-
eter 'insensitivity') imply that the system is robust to small
(local) perturbations in the genotypically influenced
parameters. Because parameter sensitivity and logarithmic
gain factors have the same mathematical form, small
magnitudes for logarithmic gains imply that the system is
robust to small (local) perturbations in the environmentally
influenced variables.
Response Time
The time it takes the system to respond to a local pertur-
bation is determined by the inverse of the real part of the
dominant eigenvalue
[2,12]
.
25
[40]
)in
Figure 15.4
. If the reaction were to be driven in the forward
direction (
b
-
D
-glucose
k
=
k
Þ
z
0
:
S
P
/ a
-
D
-glucose), then its operation
would improve until it reaches the boundary with Region 2,
