Biomedical Engineering Reference
In-Depth Information
For
κ
=1,
J
given by Equation (8.29) corresponds to the simplest choice where
in-phase motion of two neighboring cilia leads to an increase in phase speed. In fact,
monocilia close to a wall interact precisely in this way, cf. Equation (8.24).
It turns out that in this model, the synchronized state and metachronal waves are
only marginally stable in linear order. Nonlinear terms destabilize the synchronized
state but stabilize the metachronal wave (under periodic boundary conditions) [32].
(2) Non-symmetric models with a power stroke
. In a somewhat more
realistic description of the ciliar beating one can take into account that during the
powerstroke0
<ϕ<π
interactions are stronger. A possible choice is
R
l
,
J
(
ϕ
1
,ϕ
2
)=
−εah
α−
1
(1 +
β
sin
ν
(
ϕ
2
))
sin
κ
(
ϕ
1
)sin
κ
(
ϕ
2
)
l
α
+
O
(8.30)
where 0
<β<
1,
ν, κ
=1
,
3
,
5
, ...
. The more complex beating pattern does not alter
the stability of the dynamical states. Again, in linear order the synchronized state
and metachronal waves are only marginally stable. Numerically, one sees that in
this case the synchronized state is also unstable, while metachronal waves are stable
[32].
Finally, we have to show that our basic assumption (namely that the hydro-
dynamic interactions do not alter the sequence of shape changes the filament un-
dergoes) is justified in arrays of low ciliar densities. Indeed, for a cilium consisting
of
N
stokeslets, one has in leading order
v
(
r
)
∼ Nr
−α
(and
α ≥
3). Thus the
beating cilium exerts a drag force
F
d
∼ Nζl
−α
on its neighbor. However, because
F
in
∼
l
α
F
d
F
in
∼ ζNa
−α
one has
1 justifying the analysis just given.
8.3.3 Discussion
This general approach shows that no synchronization of ciliar motion occurs if hydro-
dynamic interactions do not alter the (prescribed) beating pattern of the individual
cilia (which is the case in low-density arrays).
These general results, however, do not contradict the numerical findings pre-
sented in Section 8.3.1. For example, Gueron et al., who consider arrays of high ciliar
density, explicitely take into account the dependence of the ciliar beating pattern
on the flow in the surrounding liquid, see Equations (8.3), (8.8), and (8.9). In fact,
the comparison of the general results with the numerical findings of Gueron et al.
suggests that the latter effect is a requirement for the occurrence of synchronization
(and metachronal wave formation under non-periodic boundary conditions). This
is supported by the findings of [44] where it was shown that two rotating helices
synchronize only if their motion is not too much confined.
The phase-oscillator approach can be generalized to arrays with high ciliar den-
sities where hydrodynamic interaction alters the beating pattern [46]. In this case,
hydrodynamic interactions generally lead to synchronization. For example, if the
trajectories of the monocilia introduced in Section 8.3.1 have a variable radius
R
,
this additional degree of freedom already leads to synchronization of the beating of
two neighboring cilia [46]. Within our approach, it will be possible to develop gen-
eral analytical criteria for the occurrence of these phenomena and to classify their
stability for various boundary conditions.
Finally, the phase-oscillator description shows that hydrodynamic interactions
generally lead to the formation of metachronal waves. However, for weak hydrody-
