Biomedical Engineering Reference
In-Depth Information
of cilium
m
is completely described by a single parameter
ϕ
m
. The parameterization
of the filament then only implicitly depends on time,
r
(
s, t
)=
r
(
s, ϕ
(
t
))
.
(8.26)
Upon appropriately rescaling
t
and measuring time in units of the beating period
T
, one can set for the unperturbed motion
ϕ
=2
π
. We assume here that all cilia of
the array have the same intrinsic beating frequency
ω
. Arrays of cilia beating with
different frequencies will be discussed in Section 8.3.4.
Under this assumption, interacting ciliar filaments undergo the same shape de-
formations as free cilia but with a different velocity
ϕ
. Thus, the beating cilium is a
phase oscillator. As shown in Appendix B the equation of motion for two hydrody-
namically coupled cilia can again be written as
ϕ
1
+
J
(
ϕ
1
,ϕ
2
)
ϕ
2
=2
π,
(8.27)
where now
J
(
ϕ
1
,ϕ
2
)
=
∞
∞
∞
A
nmα
(sin
nϕ
1
+
a
nα
cos
nϕ
1
)(sin
mϕ
2
+
b
mα
cos
mϕ
2
)
|
r
(0)
1
−
r
(0)
2
|
α
n
=1
m
=1
α
=1
≡ Ω
(
ϕ
1
,ϕ
2
)
G
(
r
(0)
1
−
r
(0
2
)
.
(8.28)
The Fourier-Taylor coecients
A
nmα
,
a
nα
,and
b
mα
characterize the beating pat-
tern of the cilium and can (in principle) be calculated from the solution of the
Stokes equation (see Appendix B). Because hydrodynamic interactions are weak,
the coecients
A
nmα
/l
α
,
a
nα
/l
α
and
b
mα
/l
α
are of order
ε
1foratypicaldis-
tance
l
between neighboring cilia, where the dimensionless parameter
ε
measures
the strength of the hydrodynamic interactions.
This general equation of motion has been studied in detail in [32]. There, gen-
eral criteria for synchronization of ciliar beating and the dispersion relation of
metachronal waves have been derived. Here, we only illustrate these general results
in two specific models.
(1) Models with symmetric interactions
. An example is given by a cilium
described as a stokeslet (of radius
a
) moving along the line trajectory
r
t
(
ϕ
)=
(
x, y, z
)=(
x
(
ϕt
)
,
0
,h
) with the power stroke corresponding to the phase interval
(0
,π
) and the recovery stroke to (
π,
2
π
). Thus, in leading order [cf. Equation (8.28)]
one has (
x, y, z
)=(
R
sin(
ϕt
)
,
0
,h
). For two cilia at positions
r
1
and
r
2
one has
r
m
(
s, ϕ
m
)=
r
m
(
s
=0)+
r
t
(
ϕ
m
)
δ
(
s
L
)for
m
=1
,
2, where
s
=
L
is the position
of the tip. By denoting the distance between the cilia with
l
−
≡|
r
1
(
s
=0)
−
r
2
(
s
=0)
|
,
the interaction for parallel motion close to a wall is given by
R
l
,
J
(
ϕ
1
,ϕ
2
)=
−εah
α−
1
sin
κ
(
ϕ
1
)sin
κ
(
ϕ
2
)
l
α
+
O
(8.29)
where in leading order
α
= 3 if the no-slip boundary condition on the supporting
wall is taken into account. In fact, the obtained results depend only weakly on
α
.
Furthermore,
κ
=1
,
3
,
5
....
. The latter parameter characterizes the movement of the
cilium. For large
κ
, the power and recovery stroke are localized around
ϕ
i
=
π/
2
and
ϕ
i
=3
π/
2, respectively.
