Biology Reference
In-Depth Information
FIGURE 15.8 System design space for the model in
Figure 15.7 . The regions of different color represent qualita-
tively different phenotypes. The regions are numbered arbitrarily.
The white dot represents the location of the wild-type system in the
lysogenic state; the black dot represents the location of the system
in the lytic state. Induction, caused by an increase in R, corre-
sponds to rightward movement along the horizontal black line.
Regions with a single number represent stable steady states
(regions 1, 37 and 38 correspond to lytic phenotypes; regions 11,
47, 45 and 46 to lysogenic phenotypes). The band separating these
regions represents hysteretic behavior with three steady states, one
unstable and two stable, corresponding to the two neighboring
stable regions. The color bar can be used to identify the three
phenotypes in each overlapping region (regions 7, 43 and 44
correspond to the unstable steady-state phenotypes). Wild-type
induction occurs across the critical hysteretic region (horizontal
arrow). The value of K D must lie between the dashed lines (vertical
arrow) in order to maintain the temperate life cycle operating
across the critical hysteretic region. See text for discussion.
bifurcations; nevertheless,
they do represent
transitions
Dominance Conditions
Assuming the second positive term in Eq. (12) to be
dominant over the other positive terms implies two domi-
nance conditions:
between qualitatively-distinct phenotypes.
The qualitatively distinct phenotypes represented by each
of the regions in Figure 15.8 can be readily characterized
following the procedures illustrated in the simple case of
a reversible pathway. However, it will be sufficient for our
purposes here to characterize a couple of representative cases.
g M K D < g MMax x 1
g MMax > g M K I x 1
Similarly, the assumption of dominant terms for the
remaining cases in Eqs. (12) to (14) implies the remaining
dominance conditions:
Example of a Valid Phenotype (Case 11)
If we pick the dominant terms to be the second positive
term in Eq. (12) , the first negative term in Eq. (12) , the third
positive term in Eq. (13) and the first positive term in Eq.
(14) , then the resulting S-system in steady state is
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
K D <
x p þ n
1
K n
I
K n
I
x 1
1
<
g MMax x 1 x 1
2 d M d D g 1
0
¼
x 1
2
C
x p þ n
1
¼ K n
I
0
x 2
K R >
x R
K R
0
¼
x 3
Solution
The steady-state solution for the dimer concentration D is
readily obtained as
Boundaries in System Design Space
Inserting the steady-state solution into the dominance
conditions yields the boundaries within which the solution
is valid (see Figure 15.8 ).
g MMax g C K I
2 d M d D
1
n
þ
1
D ¼ x 1 ¼
log g Mmax g C
2 d M d D
n
ð
p
1
Þ
n
þ
p
and the corresponding values for the concentration of
mRNA M and CI monomer C follow accordingly
log K D <
logK I þ
ð
þ
Þ
ð
þ
Þ
p
n
1
p
n
1
log g Mmax g C K 1
d M
2 d D K I
g C
g MMax
d M
1
1
n
n
g MMax K I
log x R <
logK R þ
n
þ
1
2a
ð
n
þ
1
Þ
M
¼
d M x I ¼
s
ð
2n
þ
1
b D þ
0 : 5 g MMax g C K I
2 d M d D
5
n þ
0
:
þ
log
½
2 d D
b D þ
d D Þ
d D
1
2a
ð
n
þ
1
Þ
C
¼
x 1
¼
g D
g D
log "
#
þ ð
n
þ
1
Þ
g D
d CMax ð
For this solution to be valid it must satisfy the domi-
nance conditions assumed above.
2a
ð
n
þ
1
Þ
b D þ
d D Þ
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