Biology Reference
In-Depth Information
Evaluation of Global Performance
Characterizing local (small) changes in performance is
important for characterizing the qualitatively distinct
phenotypes in each region, but it does not address the
overall behavior when phenotypes change as a result of
large environmental stimuli or mutation in components of
the system. The partitioning of system design space into
qualitatively different phenotypic regions provides
a means, based on the boundaries, of calculating tolerances
to global (large) changes in parameter values
[49]
. These
tolerances are defined for each parameter as the ratio of its
value at the nominal steady state (normal operating point
for the system) to its value on the boundary to an adjacent
phenotypic region (or the inverse, depending on which
value is the larger). The geometry of system design space in
Figure 15.8
itself suggests additional criteria that are rele-
vant for assessing global behavior of the system.
lysogenic to lytic growth and back again across the critical
hysteretic region. This is facilitated by having a design that
places the operating point of the system between the dashed
lines in
Figure 15.8
and by having a large distance between
these limits, i.e., above the lower limit
g
D
b
D
þ d
D
g
MMax
g
C
d
M
d
CMax
2
K
lower
D
¼
and below the upper limit
g
M
g
C
2
d
M
d
D
g
MMax
g
C
d
M
d
CMax
1
=
p
K
Upper
D
¼
Furthermore, this zone of operation should be robust to
small variations in parameter values, which can be quan-
tified by the parameter sensitivities S
ð
K
lower
D
;
p
j
Þ
and
K
Upper
D
S
, and tolerant to large changes in parameter
values, which can be quantified by the global tolerances
T
D
;
ð
;
p
j
Þ
T
I
ð
and
T
D
;
T
I
ð
K
Upper
D
K
lower
D
;
p
j
Þ
;
p
j
Þ
.
Quantitative Criteria
The criterion particularly important for the natural selection of
a particular system design is large tolerances to a change in
phenotype when the change produces dysfunction. Moreover,
when these values are optimal they often are robustly so
(again, see
[49]
). Several criteria for effective global perfor-
mance that reflect this objective can be summarized as follows.
The hysteretic region (corresponding to the three over-
lapping Cases of 43, 37 and 47 or 45) in the design space of
Figure 15.8
provides a type of 'safety factor'. Once a switch
from lysogenic to lytic growth has been initiated, the phage
becomes committed in the following sense: if there is
a decrease in the SOS signal, it must be large enough to
move the operation of the system completely back across the
hysteretic region: any lesser decrease is insufficient to
prevent completion of the switch. The hysteretic region also
acts as a buffer to prevent inappropriate switching when
there is only a transient small SOS signal, one that is
insufficient to cause the system to cross the hysteretic region.
This function is augmented by the slow response time noted
above, which tends to filter out fast local transients.
These functions of the hysteretic region are enhanced
by a large horizontal distance between the inclined lines in
Figure 15.8
, which can be quantified by the value of
D
H
,
which is given by
Analysis of Global Performance
The size of the hysteretic buffer and its global tolerance to
large changes in parameter values can be calculated
analytically; the numerical results, using the values of the
parameters in
Table 15.4
, show that the breadth of this
buffer is represented by a 5.1-fold change in RecA* activity
(R). The global tolerances of the hysteretic buffer
D
H
to
large variations in parameters
½
T
D
;
T
I
ðD
H
;
p
j
Þ
for
g
Mmax
,
g
M
, p and a are [7.1,
],
respectively (
Table 15.6
). The corresponding values for
robustness to small variations in parameters of the hyster-
etic buffer S
N
], [
N
,7.1], [6.0,
N
] and [
N
,
N
4.25,
respectively. All other parameters have no influence on the
size of the hysteretic buffer, and thus their global tolerances
and local robustness are effectively infinite.
The aspects of global behavior pertaining to the main-
tenance of the temperate lifecycle requires large tolerances
to both the lower bound
ðD
H
;
P
i
Þ
are 0.83,
0.83, 0.85 and
K
lower
D
½
T
D
;
T
I
ð
;
p
j
Þ
and to the upper
K
Upper
D
bound
. These tolerances can be calcu-
lated analytically, and the numerical results, using the
values of the parameters in
Table 15.4
, are shown for Case
11 in
Table 15.6
. Most of the tolerances for change in
parameters are essentially infinite; but if these are excluded,
the resulting values range from a low of 1.6-fold to a high
of 83-fold, with an overall average of 15-fold. The local
robustness for these lower and upper bounds can be
calculated as well (data not shown).
½
T
D
;
T
I
ð
;
p
j
Þ
g
Mmax
g
M
ð
2P
1
Þ
2pa
D
H
¼
This safety factor should be robust to small parameter vari-
ations, which can be quantified by the parameter sensitivities
S
, and tolerant to large parameter changes, which
can be quantified by the global tolerances
ðD
H
;
P
j
Þ
Biological Design Principles
The phenotypes of a system involve the interaction of
the
.
For long-term survival of the temperate life cycle, the
phage must be capable of repeatedly switching from
½
T
D
;
T
I
ðD
H
;
P
j
Þ
environmental
(independent)
variables
and
the
