Biomedical Engineering Reference
In-Depth Information
Fig. 10.2.
MS-RBC membrane model with
N
v
=
100, 500, and 3000 from left to right, respectively
The in-plane elastic energy mimics the elastic spectrin network, and is given by
k
B
Tl
m
(
3
x
j
−
2
x
j
)
k
p
∑
V
in
−
plane
=
+
,
(10.16)
4
p
(
1
−
x
j
)
l
n
−
1
j
(
n
−
1
)
j
∈
1
...
N
s
where
l
j
is the length of the spring
j
,
l
m
is the maximum spring extension,
x
j
=
l
m
,
p
is the persistence length,
k
B
T
is the energy unit,
k
p
is the spring constant, and
n
is
a power. Note that the spring forces in membrane are a combination of conservative
elastic forces, that may be expressed in terms of the energy potential above, and
dissipative forces to be defined below. The first term in (10.16) corresponds to the
attractive wormlike chain (WLC) potential, and the second term defines a repulsive
force for
n
l
j
/
0 to be called the power force (POW), so that we abbreviate this spring
model as WLC-POW. Note that if
n
>
=
1 the power force energy should be defined as
−
k
p
log
(
l
j
)
. A non-zero equilibrium spring length is defined by the balance of these
two forces.
The bending energy represents the bending resistance of the lipid bilayer and is
defined as
∑
j
∈
1
...
N
s
V
bending
=
k
b
[
1
−
cos
(
θ
j
−
θ
0
)]
,
(10.17)
where
k
b
is the bending constant,
θ
j
is the instantaneous angle between two adjacent
triangles having the common edge
j
,and
θ
0
is the spontaneous angle. The above
bending energy is a discretization [88] of the macroscopic Helfrich model [52].
The area and volume conservation constraints which account for area-incompres-
sibility of the lipid bilayer and incompressibility of the inner cytosol, respectively,
are expressed as
A
tot
0
2
2
k
a
(
A
−
)
k
d
(
A
j
−
A
0
)
+
∑
j
V
area
=
,
(18a)
2
A
tot
0
2
A
0
∈
1
...
N
t
V
tot
0
2
(
−
)
k
v
V
V
volume
=
,
(18b)
2
V
tot
0
where
k
a
,
k
d
and
k
v
are the global area, local area and volume constraint coefficients,
respectively. The terms
A
and
V
are the total area and volume of RBC, while
A
tot
0
and
V
tot
0
are the specified total area and volume, respectively. Note that the above ex-
