Biomedical Engineering Reference
In-Depth Information
This force can, in principle, be determined theoretically, although this requires a
complicated numerical calculation. If we assume that
G
prot
does not considerably
respond to bilayer deformation, changes in
F
dis
will mainly be observable as changes
in channel lifetime
τ
, which means that gramicidin A channels become molecular
force transducers embedded in the lipid bilayers [
5
]. This is so because
τ
=
1
/
k
1
,
−
where
k
1
is the dimer dissociation rate constant. The disjoining force alters
k
1
by
−
−
altering the activation energy for channel dissociation:
exp
exp
G
prot
+
G
def
G
‡
k
B
T
−
1
τ
0
·
1
τ
0
k
=
=
,
(5.9)
−
1
k
B
T
where
τ
−
1
0
G
prot
denote
the difference in bilayer deformation energy and the protein transition energy, respec-
tively, as the two subunits move apart by a distance
G
def
denotes the frequency factor for the reaction,
and
(λ
+
−
λ)(
(
d
0
−
l
))
to reach the
G
‡
is their sum. With some approxima-
transition state for dimer dissociation, and
(λ
+
−
λ)
≈
tion in the case
0 (ignoring the change in protein conformational energy
before the actual event of the real dissociation) the following equation is found:
G
def
≈
F
dis
·
(λ
+
−
λ).
(5.10)
The transition between a gramicidin A dimer (D) and monomer (M) and vice
versa, via the intermediate energy state where the dissociation/association between
two gramicidin A monomers (
D
M
) happens, is illustrated in Fig.
5.5
.
Based on the molecular dynamics simulation of gramicidin A in lipid bilayers
considering an all-atom force field [
2
], we have gained an important insight into how
a gramicidin A channel exists inside lipid bilayers. We observe here that at the binding
site of the channel bilayer interface, the lipid head group region is more effectively
regulating the lipid bilayer gramicidin A channel hydrophobic coupling. That is,
the lipid head groups, due to their physical presence, compensate for the hydropho-
bic free length (
d
0
−
↔
l
) between the bilayer thickness and the channel length in
the channel bilayer coupling interface (see the illustration of the model in Fig.
5.2
).
The bilayer, however, exerts a restoring force
F
dis
on the two longitudinal edges of the
gramicidin A channel to return it to its original thickness and, as a result, the grami-
cidin A dimer experiences destabilization. It finally dissociates from the bilayer, and
gramicidin A monomers also dissociate from each other.
Calculation of the
F
dis
acting against the bilayer gramicidin A channel coupling
has been a long-standing challenge, and the form of
F
dis
mainly depends on how
one treats a lipid bilayer membrane, such as whether it is treated as equivalent to a
perfect elastic body or as a liquid crystalline structure. Based on the theory of elastic
bilayer deformation [
38
,
40
,
66
,
67
] the bilayer deformation energy has been found
to show bi-quadratic form in terms of (
d
0
−
l
) and intrinsic monolayer curvature
c
0
parameters [
55
,
66
,
67
]
G
def
=
2
c
0
,
H
B
·
(
d
0
−
l
)
+
H
X
·
(
d
0
−
l
)
·
c
0
+
H
C
·
(5.11)
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