Chemistry Reference
In-Depth Information
The noise of the active receptor (
s
2
R
) and the active second messenger (
s
2
X
) are
given by the gain
fluctuation relationship [28] which is shown below,
2
R
R
s
1
R
¼
g
R
ð
5
:
3
Þ
2
2
X
X
2
R
R
s
1
X
þ
t
R
s
R
g
X
¼
g
X
ð
5
:
4
Þ
2
t
þ t
2
X
where
t
X
are the characteristic time constants for the ligand binding reaction
and the reaction to produce the second messenger, respectively. In Equation 5.1, the
time constants of the reactions aregivenby thereactionratesas
t
R
and
k
off
)
1
and
t
R
¼
(k
on
L
þ
k
d
)
1
. The gains, g
R
and g
X
, quantify the ampli
cation rate of the output
response to small ch
ang
es in input. These gains are given by g
R
¼
(k
p
R
þ
t
X
¼
L)
1
K
R
(K
R
þ
R
Þ
1
. The
first and second terms on the right-hand side of
Equation 5.4 are known as the intrinsic and the extrinsic noise, respectively. The
intrinsic noise represents inherently-generated noise due to the stochastic nature of
the reactions produced by the second messenger, while the extrinsic noise represents
the noise propagated from ligand
-
receptor binding reactions. Thus, Equation 5.4
describes the input
-
output relationship between noise in the active receptor concen-
tration and noise in the active second messenger concentration.
The gain-
uctuation relationship tells us that signal and noise propagation along the
signaling cascade can be characterized by the gain and the characteristic duration of
the signaling reactions (Equations 5.3 and 5.4). The reactions with higher gains
generatemore noise. Also, the propagation of the noise fromthe active receptor to the
second messenger concentration depends on the factor
and g
X
¼
K
X
ð
K
X
þ
t
= t
ð
þ t
Þ
in the second
R
X
R
term. As
X
, the
extrinsic noise decreases due to an increase in time-averaging effects. In order to
reveal how signal and noise are propagated along the signaling cascade, it is
important to determine experimentally the gains and the time constants, which is
possible by using single-molecule imaging detection and other techniques.
To evaluate the effects of the noise on gradient sensing, we studied the signal-to-
noise ratio (SNR) of the chemotactic signals. As shown in Figure 5.7B, the difference
in concentration of the ligand (
t
R
/(
t
þ t
R
) decreases with a decrease in
t
R
and/or increase in
t
X
D
L) may produce the difference in receptor occupancy
R
), whichmay in turn produce the difference in second messenger concentration
(
D
R
and
X
) between the anterior and posterior regions of chemotactic cell
s.
X
(
D
D
D
X
,
respectively. Based on the gain-
uctuation rel
atio
nship (Equ
ation
5.4), the relationship
between the relative noise intensities
2
D
2
D
R
and
should include the noise
s
R
and
s
X
around the average values
D
D
R
and
X
is given by
s
DR
=D
s
DX
=D
2
2
2
D
R
D
s
1
g
X
X
t
s
D
R
D
R
X
¼
þ
ð
5
:
5
Þ
2
R
t
þ t
R
X
D
X
R
where the
first and second terms on the right-
han
d side are derived from the intrinsic
and extrinsic noise, respectively. We de
ned
X
=s
DX
as the SNR of the chemotactic
signals, which can be calculated by using the appropriate parameter values obtained
D
