Biomedical Engineering Reference
In-Depth Information
the ciliar tip. Further away from the nodal surface, the opposite is true, i.e., the
morphogen concentration is larger on the right. One way to avoid consequence from
this uneven distribution is to assign a window of activity to the morphogens [31].
However, such a mechanism is currently purely speculative. All these results are not
conclusive yet. It should be emphasized that further theoretical progress can only
be achieved if, in the calculation of the cilia-induced velocity field, all the boundary
conditions are taken into account. Only in this case can the stability of morphogen
gradients over the node be calculated reliably.
A more complex beating pattern than the simple rotational motion of the rotlets
has been considered by Buceta et al. [69]. In their model, each cilium is a string of
N
moving spheres of radius
a
connected by elastic rods and attached to a nonmoving
sphere at the nodal surface. Each sphere
i
of cilium
k
describes a trajectory
R
i,k
=
(
x
i,k
,y
i,k
,z
i,k
) and during this motion generates a velocity field corresponding to
that of a stokeslet (see Equation (8.2)). Thus, Buceta et al. also do not take into
account the no-slip boundary conditions on the wall. However, the influence of the
wall is mimicked by a time-dependent radius
a
(
t
), where
a
(
t
) is large (small) for
small (large)
z
. The spheres are also connected by elastic springs to take bending of
the ciliar filament into account.
Each cilium (i.e., each collection of spheres) is tilted towards the vertical and
undergoes a two-phase motion consisting of a power and a recovery stroke. The
velocity field created by a single cilium is simply given by the superposition of the
stokeslet fields created by the sphere
i
=1
, .., N
. However, this description is only
correct in the far field (i.e., for
r a
), because the no-slip boundary conditions on
the surfaces of the spheres are not taken into account.
By appropriately prescribing the bending of each cilium, Buceta el al. are able
to reduce the backward flow induced by the recovery stroke by reducing the drag the
cilium exerts on the fluid. However, because the recovery stroke takes place close to
the surface, it would be essential to take into account the no-slip boundary conditions
on the node. In fact, the two-phase beating might be a natural consequence of the
no-slip boundary condition on the nodal surface. Thus, by appropriately taking this
condition into account it might not even be necessary to prescribe the different
beating phases.
Within their approach Buceta et al. obtain a somewhat more realistic charac-
terization of the nodal flow. However, again, the results depend crucially on the
prescribed microscopic force fields of the beating pattern and such studies do not
reveal any universal properties of these systems. Thus, there is still quite a number
of open issues that need to be addressed in future theoretical investigations. First
of all, the no-slip boundary condition on the confining walls of the node is essential
for the analysis because it not only influences the beating pattern of the cilium but
also the backward flow due to mass conservation. Another question is how impor-
tant synchronization of ciliar beating is. Does the flow from right to left depend
on whether the cilia beat in a synchronized fashion or not? Similarly, the beating
of the individual cilia and thus the nodal flow might be influenced by membrane
fluctuations. The framework developed in [70, 71] provides the appropriate tools to
investigate such phenomena. Finally, the influence of disorder on the nodal flow has
to be analyzed. In particular, it has to be investigated whether a small number of
cilia with a negative tilting angle
ψ
or opposite rotational direction is already su-
cient to significantly alter the nodal flow as experimental studies on mutants seem
to indicate.
