Phenotypes and Design Principles in System Design Space - Systems Biology Concepts and Insights - page 290

when a is small (data not shown). By inserting the esti-
mated values for the parameter ( Table 15.4 )intothese
analytical results one can convert them into numerical
results. Small values imply superior performance for each
of the criteria. The local robustness in each of the
phenotypic regions reveals that on average the influence
of perturbations in parameters is attenuated (magnitude
of sensitivities less than one). The results for Case 11,
shown in Table 15.5 , range from a low of 0.1 to a high of
0.6 with an overall average of 0.37 if one omits all the
cases with zero value.
The logarithmic gains of the system in response to the
environmental input signal are all equal to zero,

the dominant S-system equations. In the Case 11 region
(again assuming rapid dimerization)

" # n

ðb D þ d D Þ K I g 1 = n

dM
dt ¼

C 2n

MMax

d M M

g D

2 d D g D
ð

C 2

dC
dt ¼

g C M

b D þ d D Þ

which for local responses reduces to the following linear
system [2,12] :

d M 2nu 2

du 1
dt ¼

u 1

L ð M ; R Þ¼ L ð C ; R Þ¼ L ð D ; R Þ¼ 0

F 2 u 1 2u 2

du 2
dt ¼

i.e., fluctuations in the level of DNA damage (RecA*
levels) are subthreshold and have no influence on the Case
11 phenotype of the system. It should be emphasized that
this is robustness to local (small) changes. The system will
respond to global (large) changes, which is the subject of
global tolerance as discussed below.
The local response times in each of the phenotypic
regions of system design space are readily calculated from

n þ 1

0 : 5 g Mmax g C K I
d M

0 : 5
n þ 1

g D

n þ 0 : 5

with F 2 ¼ð 2 d D Þ

ð b D þ d D Þ

With the values of the parameters in Table 15.4 , the real
part of the dominant eigenvalue is

0.317 min -1 , which

corresponds to a half-time of 2.2 min.

TABLE 15.5 Summary of Analytical and Numerical Values for Local Robustness of the Lysogenic Phenotype

Corresponding to Region Case 11 of the System Design Space in Figure 15.8 . Numerical Results are Calculated
on the Basis of the Parameter Values in Table 15.4 , which were Determined from Experimental Data

Parameters S ð M ; p i Þ S ð C ; p i Þ S ð D ; p i Þ

K D 0 0 0 0 0 0

K I n =ð n þ 1 Þ 0.6 0 : 5n =ð n þ 1 Þ 0.3 n =ð n þ 1 Þ 0.6

K R 0 0 0 0 0 0

g MMax 1 =ð n þ 1 Þ 0.4 0 : 5 =ð n þ 1 Þ 0.2 1 =ð n þ 1 Þ 0.4

g M 0 0 0 0 0 0

g C n =ð n þ 1 Þ 0.6 0 : 5 =ð n þ 1 Þ 0.2 1 =ð n þ 1 Þ 0.4

g D 0 0 0.5 0.5 0 0

b D 0 0 0 : 5b D =ðb D þ d D Þ 0.5 0 0

d M 1 =ð n þ 1 Þ 0.4 0 : 5 =ð n þ 1 Þ 0.2 1 =ð n þ 1 Þ 0.4

d CMax 0 0 0 0 0 0

0 0 0 0 0 0

d C

d D n =ð n þ 1 Þ 0.6 0 : 5 ð nd D b D Þ 0.2 1 =ð n þ 1 Þ 0.4

ðb D þ d D Þð n þ 1 Þ

a 0 0 0 0 0 0

p 0 0 0 0 0 0

2 In 2d M d D K I
g MMax g C

2 In 2d M d D K I
g MMax g C

2 In 2d M d D K I
g MMax g C

n n 0.21 0 : 5n 0.10 n 0.21

ð n þ 1 Þ ð n þ 1 Þ ð n þ 1 Þ