Chemistry Reference
In-Depth Information
the spontaneous curvature 1/
R
0
. Equation (3.1) has the specifi c form for each
of the given planes that is due to the specifi c physical description of the plane.
3.2.1
Luzzati Plane
The position of the Luzzati plane is defi ned by the radius of the water cylinder,
R
w
(Luzzati and Husson, 1962):
3
. )
(3.2)
(
)
R
w
=
α
1
−
φ
(
see Fig
.
3 1
l
2
π
Here
ϕ
l
is the lipid volume
fraction (Kulkarni et al., 2010). For the hexagonal phase,
α
is the lattice parameter of the H
II
phase and
is the cylinder-
center-cylinder-center distance and is related to the measured fi rst - order
Bragg spacing of the hexagonal phase,
d
hex
:
α
2
3
d
hex
(3.3)
α=
The corresponding area per molecule at the Luzzati surface (Fig. 3.1) is given
by the formula (Luzzati and Husson, 1962)
4
π
RV
wl
A
=
(3.4)
w
αφ
3
l
where
V
l
is the lipid molecular volume. The position of the Luzzati plane is
somewhere inside the hydrated structure of the polar lipid heads. It is easy to
calculate
R
w
and
A
w
using only Eqs. (3.2) and (3.4) and an appropriate experi-
mental data set (
d
hex
,
l
). Table 3.1 contains a set of calculated values of
R
w
and
A
w
for different lipids at different temperatures and water mole ratios.
Although Eq. (3.3) does not show any dependence between
d
hex
and
ϕ
l
, it
is defi nitely there. Figure 3.2 shows the experimentally determined depen-
dence
d
hex
(
ϕ
ϕ
w
), where
ϕ
w
=
1
−
ϕ
l
is the water volume fraction. The dependence
between
d
hex
and
l
exists up to the full hydration of the polar lipid heads
(Fig. 3.2a). Figure 3.2b shows only the part with clear dependence between
parameters (below full hydration). The easiest way to express the connection
between the lattice parameter and the lipid volume fraction is through the
pivotal plane parameters (Section 3.2.2). In addition, there is a model that
explains in some detail the contribution of the second lipid in the lattice
parameter
ϕ
α
(Lafl eur et al., 1996):
α
R
w
(3.5)
=−
αα
*
−
H
PH
II
2
Here,
α
H
I
*
is the sum of the contributions of the lipid chain and includes the
structural infl uence of the second component, while
α
ph
is the contribution
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