HW
7
Solutions
1.
a) Both
a
1
and
a
2
should have the unit of 1/nM
n
b)
)
1
)(
1
(
)
(
)
1
(
)
(
)
1
/(
]
)
1
(
)
1
[(
ln
ln
2
1
2
1
2
2
2
1
2
2
2
1
1
2
A
a
A
a
a
a
A
A
a
F
a
a
A
A
a
a
A
a
a
A
a
F
A
A
F
F
A
A
F
s
+
+
−
=
+
−
=
+
+
−
+
=
∂
∂
=
∂
∂
=
c)
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12
1
1
2
1
[]
[ ]
[
]
and
,or[
]
and[
]
[
]
AO
BO
BOK
KK
A
O
O
AO
BO
K
B
==
=
=
2
2
10
1
202
01
0
2
1
02
0
20
2
00
2
S
i
n
c
e
[ ][ ][
]a
n
d
[
][ ][ ]
,
[ ]([
] [
])
([
] [
])
a
n
d
[
]
[
]
Thus, [
]
and
[
]
[
]
[
]
Let
[
]
,
OO
A
O
B
OOO
AO A
O
O OK
AO
O
KB
KO
AO
O
KA
rkA
OO k
KAKB
kkOK
r
=−
−−
++
⎛⎞
⎜⎟
⎝⎠
=
2
0
.
(
[ ])(
[ ])
kA
=
b) Then
;
so
(
[ ])(
[ ])
AB
r
c)
A
,
wherem is mRNAof A protein and
isdegradation rateconstant of the mRNA
[
]
[ ], where
is translation rate constant
Using QSSA:
0
[
],
Thus
[
]
Therefore,
A
mA
m
pA
A
p
A
m
p
m
dm
rdm
dt
d
dA
km dA
k
dt
r
m
d
kr
d
dt
d
=
(
[ ])(
[ ])
p
AA
m
kkA
Ad
A
dK AK A
d) In Figure 4, one can see that there are two steady states (one for each intersection of the blue and
red lines). The steady state at (0,0) is unstable because a slight increase from (0,0) gives rise to a greater
increase in A. In contrast, the other steady state is stable because a small decrease from the steady
state gives rise to a corresponding increase in A (synthesis > decay). Likewise, a small increase from the
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 Spring '07
 YUAN
 Steady State, Trigraph, steady states, K3, Xss. Lowering k3

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