Biomedical Engineering Reference
In-Depth Information
B Derivation of the Phase-Oscillator Equation of Motion
Here, we derive the equation of motion (8.27). The velocity field
v
s
is (as a solution
of the linear Stokes equation) a linear functional of
r
(
s, t
), i.e.,
v
s
=
V
s
[
∂
t
r
(
s, t
)].
A more explicit expression for the solution can be found in [35]. To demonstrate
the influence of the interaction term
v
n
, we will now consider two neighboring ciliar
Filaments 1 and 2 at positions
r
1
=
r
1
(
s, ϕ
1
)and
r
2
=
r
2
(
s, ϕ
2
). Then,
v
n
at posi-
tion
r
1
is a linear functional of
r
2
, i.e.,
v
n
=
V
n
[
∂
t
r
2
(
s, t
)]. Equation (8.26) implies
v
s
=
ϕ
1
(
t
)
V
s
[
∂
ϕ
1
r
1
(
s, ϕ
1
)] and
v
n
=
ϕ
2
(
t
)
V
n
[
∂
ϕ
2
r
2
(
s, ϕ
2
(
t
))]. Thus, Equation
(8.25) becomes
ζ
ij
ϕ
1
∂
ϕ
1
r
1
i
(
s, ϕ
1
)
− ϕ
1
V
i
[
∂
ϕ
1
r
1
(
s, ϕ
1
)]
− ϕ
2
V
i
[
∂
ϕ
2
r
2
(
s, ϕ
2
)]
=
F
j
(
r
1
(
s, ϕ
1
)
,s
)
,
(8.46)
where
r
1
i
and
V
i
[
∂
ϕ
1
r
(
s, ϕ
1
)] denote the
i
-th component of
r
1
and
v
s
, respectively.
By projecting the forces Filament 2 exerts on Filament 1 onto the tangential vector
of the trajectory of Filament 1 and upon summing over all filament pieces (i.e.,
stokeslets) Equation (8.46) becomes
ϕ
1
F
d
(
ϕ
1
)
− ϕ
2
I
(
ϕ
1
,ϕ
2
)=
F
(
ϕ
1
)
.
(8.47)
Here,
F
d
(
ϕ
1
)=
L
0
ds ∂
ϕ
1
r
1
j
(
s, ϕ
1
)
ζ
ij
∂
ϕ
1
r
1
i
(
s, ϕ
1
)
− V
i
[
∂
ϕ
1
r
1
(
s, ϕ
1
)]
,
(8.48)
I
(
ϕ
1
,ϕ
2
)=
L
0
ds ∂
ϕ
1
r
1
j
(
s, ϕ
1
)
ζ
ij
V
i
[
∂
ϕ
2
r
(
s, ϕ
2
)]
,
(8.49)
(
ϕ
1
)=
L
0
F
ds ∂
ϕ
1
r
1
j
(
s, ϕ
1
)
F
j
(
r
1
(
s, ϕ
1
)
,s
)
,
(8.50)
where
r
1
(
s, ϕ
1
) is the solution of the Equation (8.25) without interactions [where
ϕ
=2
πt
]. Because then 2
πF
d
(
ϕ
)=
F
(
ϕ
), one finds Equation (8.27), where now
J
(
ϕ
1
,ϕ
2
)
≡−
2
πI
(
ϕ
1
,ϕ
2
)
/F
(
ϕ
1
)
.
(8.51)
Equations (8.19), and (8.48)-(8.50) are valid for arbitrary beating patterns even
for arrays of cilia with different (individual) beating patterns. This description is
a generalization of the model for interacting monocilia introduced in Section 8.2.1.
In this case,
r
(
s, ϕ
(
t
)) =
r
t
=(
R
cos
ϕ
(
t
)
,R
sin
ϕ
(
t
)
,h
)
δ
(
s − L
), with
δ
(
s
) denoting
Dirac's delta function and
s
=
L
is the position of the tip. Furthermore,
V
i
=0,
V
i
[
∂
ϕ
2
r
(
s, ϕ
2
)] =
v
12
/ ϕ
2
,where
v
12
is given by Equations (8.2) or (8.23). Then,
I
F
d
(
ϕ
)=
R
2
ζ
.
In principle, a continuous description of the bending deformation of the filament
could be used to calculate
J
(
ϕ
1
,ϕ
2
). However, here we will use a different approach.
For a system of cilia with identical beating patterns, one has
r
m
=
r
m
(
s
=0)+
r
t
(
s, ϕ
m
), where and
r
t
parameterizes the time-dependent beating pattern. Because
all cilia have the same beating pattern,
r
t
depends only on
ϕ
m
. Because 0
≤ ϕ
m
≤
2
π
the function
J
(
ϕ
1
,ϕ
2
)isperiodicinbothargumentsandisgivenbyEquation(8.28).
(
ϕ
1
,ϕ
2
)=
Rζ
t
1
·
v
12
/ ϕ
2
,
F
(
ϕ
)=
R
t
1
·
F
=
RF
in
,and
