Biomedical Engineering Reference
In-Depth Information
From these equations, the beating motion of a single cilium can be calculated
numerically. The generalization to three dimensions is straightforward [39]. Figure
8.2 shows an example in three dimensions. The prescribed force fields are similar to
the one given in Equation (8.5) and the beating pattern consists of a power and a
recovery stroke.
Figure 8.2.
Beating of a single cilium in three dimensions consisting of a power
stroke (
t
=1
−
5ms) and a recovery stroke (
t
=6
−
21ms). The details of this beating
pattern are determined by the prescribed activity pattern of the molecular motors.
Data is for
L
=10
μ
m,
a
= 100nm,
ω
= 25Hz and
κ
=
ξ
p
k
B
T
with
ξ
p
= 6mm.
Figure is reprinted from [39]. Copyright (2001), with permission from The Royal
Society.
The beating of a cilium can also be implemented numerically in a more direct
way as shown in [40]. There, Kim and Netz have studied the pumping eciency
of a periodically beating elastic filament anchored to a solid surface in Brownian
dynamics simulations.
In these simulations, the filament is modeled as a chain of
N
connected spherical
beads of radius
a
. In Brownian dynamics simulations, the equation of motion of bead
i
at position
r
i
is given by the position Langevin equation [41]
d
r
i
d
t
=
−
j
μ
ij
∇
r
j
U
+
μ
i
2
r
12
×
τ
(
t
)
r
12
+
ξ
i
.
(8.15)
Here,
U
is the elastic potential energy, given by a worm-like chain model
r
ij
≡
|
r
i
−
r
j
|
,and
τ
(
t
) is a prescribed (time-dependent) torque consisting of a phase of
