Chemistry Reference
In-Depth Information
where
V
V
9
8
πχ
σ
(
)
=
p
ϑσχ
V
,
,
arccos
φ
(3.13)
i
p
0
l
,
i
3
l
0
Differentiating Eq. (3.12) by 1/
A
p
, an internal solution for
A
p
can be derived
for a given set of values (
V
p
,
σ
0
,
χ
):
(
)
2
πϑ
+
V
,
σχ
,
1
i
p
0
∑
N
14
−
cos
2
i
=
1
()
2
2
σφ
a
3
(
)
=
i
l
,
i
AV
,
σχ
,
2
σ
V
(3.14)
pp
0
0
l
(
)
πϑ
+
V
,
σχ
,
a
a
i
i
p
0
∑
i
N
14
−
cos
2
=
1
()
σφ
2
3
i
l
,
i
The correctness of the result could be proved by setting
N
1 and comparing
the derived expression to Eq. (3.11). By means of Eq. (3.14), the merit function
set of parameters can be reduced by one:
=
2
(
)
N
1
2
σ
σχφ
V
πϑ
+
V
,
σ
,
χ
∑
(
)
=
i
p
0
0
l
Φ
V
,
σχ
,
a
−
14
−
cos
2
p
0
i
(
)
()
2
σ
a
AV
,
,
3
i
pp
0
l
,
i
i
=
1
(3.15)
The process of deriving
V
p
,
is a subject of the chosen routine for
nonlinear optimization, most often one of the following: downhill simplex
algorithm, Powell's method, conjugate gradient (CG) algorithm, Broyden-
Fletcher - Goldfarb - Shanno (BFGS), and Newton - conjugate - gradient (Newton -
CG) (Press et al., 2007).
σ
0
, and
χ
3.2.2.2 Hexagonal Inverse Phase
In hexagonal inverse phase (H
II
), the
position of the pivotal plane is given by (Marsh, 2011)
3
V
V
R
p
(3.16)
=
α
1
−
φ
p
l
2
π
l
Here
V
p
is the pivotal volume (Fig. 3.1). The position of the pivotal plane,
R
p
,
is offset from the position of the lipid-water interface,
R
w
:
1
V
V
p
RR
=
1
−
φ
(3.17)
p
w
l
1
−
φ
l
l
It is easy to predict that if
V
p
=
V
l
then
R
p
=
R
w
. The corresponding area per
lipid molecule at the pivotal surface is
4
π
RV
pl
A
=
(3.18)
p
3
αφ
2
l
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