Biomedical Engineering Reference
In-Depth Information
origin of its rotation is probably of chemical nature (namely the periodic motion in
a “reaction chamber”) it nevertheless induces hydrodynamic interactions in arrays
giving rise to collective effects that will be discussed in Section 8.5.
In this review we summarize recent theoretical efforts to understand collective
effects in biological hydrodynamics. We will focus on cilia and rotating motors.
Summaries of flagellar hydrodynamics can be found in [21, 22, 23].
From a theoretical point of view it is one of the key questions what causes the
cooperative behavior in ciliar arrays. Is the cooperative motion triggered biochem-
ically [24] or are hydrodynamic interactions strong enough to couple the beating
pattern of neighboring cilia [25]? In the latter case, the beating of a cilium is in-
fluenced by the fluid flow created by the other cilia. This question is not only of
biological relevance. From the statistical physics point of view, motile bacteria and
cells represent beautiful realizations of non-equilibrium systems exhibiting ordering
transitions and pattern formation.
Since the seminal work of Blake [26] there have been considerable efforts in mod-
eling ciliar motion and calculation of the induced velocity fields from first principles
(for an overview see [27]). Generally, one has to distinguish between “discrete cilia
models” [26, 28] and “envelope models” [29, 30]. In the former, each cilium is treated
separately, and the total velocity field in the system is given by the superposition
of the velocity fields induced by each cilium. In contrast, in the envelope models it
is assumed the cilia are closely packed so that they primarily interact with the sur-
rounding water. However, as shown by Blake and Sleigh [10], the envelope approach
is only valid for symplectic metachronal waves.
For this reason, we present here only approaches that are in the spirit of the
discrete ciliar models. Therefore, we start with descriptions of the beating pattern
of a
single
cilium. Then, the collective effects in ciliar arrays can be studied by
analyzing how the beating of a single cilium is influenced by the beating of its
neighbors. This review is outlined correspondingly. We start with a summary of
the efforts to theoretically describe the beating of a single cilium in Section 8.2.
Next, it is shown how these results can be used to study collective effects in ciliar
arrays. In Section 8.4, the role of nodal flow in vertebrate development is discussed.
Finally, in the last section, we analyze the importance of hydrodynamic interactions
in collections of rotatory motors. We conclude with some (possible) technological
applications of these hydrodynamic effects.
8.2 Beating of a Single Cilium
To study collective effects in ciliar arrays, we first have to analyze the forces a
beating cilium exerts on the surrounding fluid. In doing so, we employ an approach
based on the “discrete cilia model” where one is interested in the velocity field
created by a single cilium. Because the length
L
5
−
10
μ
m of a cilium is much
larger than its thickness (radius
R
100nm [27]), one can describe the motion
of the cilium as being created by a set of forces localized at the centerline of the
cilium. Furthermore, frequencies are of the order
ω
10s
−
1
[10, 31] and the ciliar
10
−
3
(see Appendix A
for a short introduction into the relevant hydrodynamical concepts). Therefore, the
inertial terms in the Navier-Stokes equation (8.39) are much smaller than the viscous
term (and the pressure gradient). For suciently small systems (typical extension
beating is thus characterized by the Reynolds number, Re
