with.
2 An obvious alternative is to actively bend membranes and directly measure
their curvature elastic response. There are clearly many ways to deform a membrane; here we will describe two possibilities that have been proposed in the past as
convenient methods for obtaining the bending modulus.
Harmandaris and Deserno [113] proposed a method that relies on simulating
cylindrical membranes. Imagine a membrane of area A that is curved into a cylinder
of curvature radius R. Its length L satisfies 2πRL ¼ A, and the curvature energy per
area of this membrane is:
e ¼
1
2
κ
1
R
2
¼
1
2
κ
2πL
A
2
:
ð4Þ
Because changing the length of the cylinder at constant area will also change the
curvature radius, and thus the bending energy, there must be an axial force
F associated with this geometry. Its value is given by:
F ¼
∂eA
∂L
A
¼ A κ
2πL
A
2π
A
¼
2πκ
R
:
ð5Þ
Hence, measuring both the axial force and the cylinder radius yields the bending
modulus as κ ¼ FR/2π. Notice that within quadratic curvature elasticity, the radius
of the cylinder does not matter: Both small and large radii will lead to the same
modulus. In other words, FR is predicted to be a constant. Of course, it is conceivable that higher order corrections to the Helfrich Hamiltonian Eq. (1) matter once
curvatures become really strong. For the present geometry there is only one term,
which enters at quartic order, and one would write a modified energy density
e ¼
1
2 κK
2
þ
1
4 κ 4 K
4 . This modified functional leads to FR/2π ¼ κ + κ 4 /R
2
κ eff (R),
which can be interpreted as an effective curvature-dependent bending modulus.
Simulations using different models with different levels of resolution have indeed
both seen a small dependence of κ eff on R [111, 113]. They find softening at high
curvature, which would indicate that κ 4 is negative. In contrast, Li et al. [38]
recently studied the elastic properties of self-assembled copolymeric bilayers by
self-consistent field theory in cylindrical and spherical geometry, and found κ 4 to be
positive. The details of nonlinear elastic corrections thus depend on the specifics of
the model under study, but the present studies suggest that as long as the radius of
curvature is bigger than a few times the membrane thickness, these corrections are
negligible. For example, Li et al. [38] found the deviations from linear to be less
than 2% both in the cylinder and sphere geometry, as long as the reduced curvature
was less than K 0 d ¼ 0.6 (where d is the bilayer thickness).
2 It is easy to see that δh hh r
ð Þ
2 i
1=2 ¼ L
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k B T=16π 3 κ
p
% L=100 (assuming κ ’ 20 k B T ), which is
a few Ångstro ¨m for typical simulation sizes.
246
M. Deserno et al.
2 An obvious alternative is to actively bend membranes and directly measure
their curvature elastic response. There are clearly many ways to deform a membrane; here we will describe two possibilities that have been proposed in the past as
convenient methods for obtaining the bending modulus.
Harmandaris and Deserno [113] proposed a method that relies on simulating
cylindrical membranes. Imagine a membrane of area A that is curved into a cylinder
of curvature radius R. Its length L satisfies 2πRL ¼ A, and the curvature energy per
area of this membrane is:
e ¼
1
2
κ
1
R
2
¼
1
2
κ
2πL
A
2
:
ð4Þ
Because changing the length of the cylinder at constant area will also change the
curvature radius, and thus the bending energy, there must be an axial force
F associated with this geometry. Its value is given by:
F ¼
∂eA
∂L
A
¼ A κ
2πL
A
2π
A
¼
2πκ
R
:
ð5Þ
Hence, measuring both the axial force and the cylinder radius yields the bending
modulus as κ ¼ FR/2π. Notice that within quadratic curvature elasticity, the radius
of the cylinder does not matter: Both small and large radii will lead to the same
modulus. In other words, FR is predicted to be a constant. Of course, it is conceivable that higher order corrections to the Helfrich Hamiltonian Eq. (1) matter once
curvatures become really strong. For the present geometry there is only one term,
which enters at quartic order, and one would write a modified energy density
e ¼
1
2 κK
2
þ
1
4 κ 4 K
4 . This modified functional leads to FR/2π ¼ κ + κ 4 /R
2
κ eff (R),
which can be interpreted as an effective curvature-dependent bending modulus.
Simulations using different models with different levels of resolution have indeed
both seen a small dependence of κ eff on R [111, 113]. They find softening at high
curvature, which would indicate that κ 4 is negative. In contrast, Li et al. [38]
recently studied the elastic properties of self-assembled copolymeric bilayers by
self-consistent field theory in cylindrical and spherical geometry, and found κ 4 to be
positive. The details of nonlinear elastic corrections thus depend on the specifics of
the model under study, but the present studies suggest that as long as the radius of
curvature is bigger than a few times the membrane thickness, these corrections are
negligible. For example, Li et al. [38] found the deviations from linear to be less
than 2% both in the cylinder and sphere geometry, as long as the reduced curvature
was less than K 0 d ¼ 0.6 (where d is the bilayer thickness).
2 It is easy to see that δh hh r
ð Þ
2 i
1=2 ¼ L
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k B T=16π 3 κ
p
% L=100 (assuming κ ’ 20 k B T ), which is
a few Ångstro ¨m for typical simulation sizes.
246
M. Deserno et al.