Biomedical Engineering Reference
In-Depth Information
Gueron's approach does not reveal which of the obtained results (such as the oc-
currence of synchronization and metachronal wave formation) are universal (i.e.,
would also be found with a different description of the ciliar beating) and which are
only properties of the underlying model. Generally, all approaches so far have con-
sidered only specific beating patterns and not the general aspects of hydrodynamic
interactions between cilia.
Only recently, a more general framework has been developed in [32] in which
the influence of the beating pattern on the collective effects can be systematically
investigated
2
. For this purpose, it is necessary to first focus on arrays with low
ciliar density where the hydrodynamic interactions are weak and do not alter the
(prescribed) beating pattern of the individual cilia.
We will give a short summary of this approach. Again, we directly prescribe
here the beating pattern of the cilium. This has the advantage that our approach is
independent of any microscopic details (such as the forces exerted by the molecular
motors on the filaments). However, the obtained results do not depend on the ex-
plicit form of the beating pattern, they are valid for general motion of the filaments.
We now calculate the flow field induced by a moving cilium by (formally) solving the
Stokes equation and then using this solution to systematically analyze the influence
of hydrodynamic interactions on cooperative beating. In doing so, general criteria
can be derived when these interactions lead to synchronization and metachronal
wave formation. It turns out that for low ciliar densities, the stability of the syn-
chronized state and the dispersion relation of metachronal waves are non-universal,
i.e., they depend crucially on the details of the ciliar beating pattern and the imposed
boundary conditions.
For this purpose, we parameterize the conformation of the elastic filament of
length
L
by the vector
r
(
s, t
), which is a function of arclength
s
(with 0
≤
s
≤
L
)
and time
t
. The drag forces balance all other forces acting on the cilium
ζ
ij
(
∂
t
r
i
(
s, t
)
− v
i
(
r
(
s, t
))) =
F
j
(
r
(
s, t
)
,s
)
,
(8.25)
where repeated indices have to be summed over and vector
v
has spatial components
v
i
. Here,
ζ
ij
is the friction tensor that accounts for the fact that tangential and
normal frictions of the filament are different. Furthermore,
v
(
r
(
s, t
)) is the velocity
in the fluid at position
r
. Here,
v
=
v
e
+
v
s
+
v
n
,where
v
e
is the external flow
field,
v
s
the contribution arising from the motion of other parts of the filament (at
positions
r
=
r
), and (in ciliar arrays)
v
n
is the contribution from the motion of
neighboring cilia. Finally,
F
(
r
(
s, t
)
,s
) is the sum of all forces acting on the cilium
segment at
r
(
s, t
), i.e., elastic and internal driving forces
F
=
F
el
+
F
in
. We assume
that these forces are a functional of
r
(
s, t
) and only explicitly depend on
s
but not
on
t
.
As mentioned, we assume that the beating pattern is not altered by the inter-
actions (i.e., for given
v
e
it is not influenced by
v
n
). We will discuss under which
conditions this strong assumption holds and which consequences arise from it. Thus,
the sequence of beating shapes is the same for an isolated cilium and for a cilium be-
longing to an array. However, the velocity of shape changes generally will depend on
the hydrodynamic interactions. With this assumption, the sequence of shape changes
2
It should be mentioned that there are also other effective models [45] to describe
collective effects in ciliar arrays which, however, do not explicitely take the hy-
drodynamic interactions into account.
