Biomedical Engineering Reference
In-Depth Information
where time
t
is measured now in units of the period
T
of the unperturbed motion,
i.e.,
ϕ
(
T
)=2
π
. It is also assumed that the unperturbed motion of each cilium has
the same intrinsic frequency
ω
. The function
J
(
ϕ
1
,ϕ
2
)isgivenby
3
4
a
t
2
·
t
1
+(
t
1
·
n
12
)(
t
2
·
n
12
)
J
(
ϕ
1
,ϕ
2
)
≡−
,
(8.20)
r
12
where
n
12
=
r
12
/|
r
12
|
,and
r
12
is the vector pointing from Cilium 2 to 1. For arrays
with low ciliar densities,
R/l
1where
l
is the distance between the centers of the
trajectories. One has then
R
l
.
3
a
4
l
J
(
ϕ
1
,ϕ
2
)=
−
[2 cos(
ϕ
1
) cos(
ϕ
2
) + sin(
ϕ
1
)sin(
ϕ
2
)] +
O
(8.21)
From Equation (8.20) one immediately sees that two hydrodynamically interact-
ing monocilia do not synchronize. By subtracting the equation of motion for Cilium
1 from that for Cilium 2, one finds
(
ϕ
1
− ϕ
2
)[1
− J
(
ϕ
1
,ϕ
2
)] = 0
,
(8.22)
where it has been used that the drag Cilium 1 exerts on Cilium 2 is given by
ζR ϕ
1
J
(
ϕ
2
,ϕ
1
)and
J
(
ϕ
2
,ϕ
1
)=
J
(
ϕ
1
,ϕ
2
). Thus,
ϕ
1
− ϕ
2
=
const.
and any ini-
tial phase difference persists in time and no synchronization occurs. A similar phe-
nomenon has been found for two rotating helices [44].
The no-slip boundary condition on the wall modifies the velocity field created
by the cilia leading to
v
12
=12
h
2
n
12
(
s
·
n
12
)
r
12
+
O
(
r
−
5
12
)
,
(8.23)
where higher order terms and the contributions perpendicular to the trajectory have
been neglected. In this case, the interaction becomes
R
l
.
J
(
ϕ
1
,ϕ
2
)=
−
9
a
h
2
l
3
[cos(
ϕ
1
) cos(
ϕ
2
)] +
O
(8.24)
Thus, even if the boundary conditions on the wall are taken into account, the inter-
action
J
(
ϕ
1
,ϕ
2
) is symmetric in
ϕ
1
and
ϕ
2
and again, two cilia do not synchronize.
Similar results can be obtained for systems with more than two cilia.
In fact, as will be discussed later in more detail the absence of synchronization is
a direct consequence of the basic assumption that hydrodynamic interactions only
alter the velocity of the shape changes but not the beating pattern itself. Under this
assumption, even more complicated beating patterns (which could be modeled by
taking several stokeslets into account, see Figure 8.1) do not lead to synchronization.
As we will see in the next section such additional contributions do not alter the
general form of the equation of motion (8.19), they only modify the interaction
function
J
(
ϕ
1
,ϕ
2
).
On the other hand, in the specific models of Gueron et al. and Kim and Netz,
synchronization of ciliar beating occurs. For example, in [37] the synchronization of
the beating of two neighboring cilia was observed numerically (in two dimensions).
Synchronization occurred within two beat cycles. Also metachronal waves form spon-
taneously. Figure 8.4 shows an example for a three-dimensional array with nine cilia
in the model of Gueron. In the simulations, such a phase shift between neighboring
