Biomedical Engineering Reference
In-Depth Information
relative probabilities of observing different current levels (channel conformations)
e.g.,
W
j
+
k
/
k
and
j
, may show different control values as
the bilayer properties change. The corresponding free energy distribution between
different current levels can be determined by
W
j
between levels
j
+
exp
G
j
→
j
+
k
k
B
T
W
j
+
k
W
j
−
=
,
(5.25)
G
j
→
j
+
k
where
is the free energy of the channel in current level
j
+
k
relative to
level
j
.
5.3.2 Derivation of Gramicidin A Channel Lifetime (
)
in a Continuum Distribution of Local Energy Traps
τ
As discussed earlier, the relationship between the lifetime and the deformation energy
change is proposed to be:
τ
=
(
−
G
def
k
B
T
exp
)
(Eq.
5.13
) which assumes that the
G
prot
does not change considerably as the
gramicidin A subunits move apart by a distance (
λ
+
−
λ
) to dissociate from each
other [
30
]. For a particular bilayer deformation, the negative exponential energy
dependence of the channel lifetime is a valid approximation. However, in the case
where a continuum distribution of local energy traps is involved, an integration over
all trap levels is needed to find the average value of the channel lifetime
τ
av
. Here,
the appropriate formula to be used is
difference in protein transition energy
τ
exp
ρ(
−
E
k
B
T
τ
av
=
E
)
d
E
,
(5.26)
G
def
where
denotes the probability
distribution of having a trap with a particular energy level. Following the detailed
calculations provided in reference [
85
] we find that the dependence of the average
lifetime
τ
av
on deformation energy change
E
stands for
(for simplicity) and
ρ(
E
)
G
def
transforms from exponential to
a power law relation:
G
def
)
−
a
≈
((λ
+
−
λ)
F
dis
)
−
a
τ
av
≈
(
.
(5.27)
Here,
a
is a parameter which is dependent on chemical and thermodynamic condi-
tions.
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